Solution Thermodynamics and its Application to Aqueous Solutions: A Differential Approach

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Solution Thermodynamics

and its Application

to Aqueous Solutions

A Differential Approach

This page intentionally left blank

Solution Thermodynamics

and its Application

to Aqueous Solutions

A Differential Approach

Yoshikata Koga Suiteki Juku (Water Drop Institute) and

Department of Chemistry, The University of British Columbia

Vancouver, B.C. Canada

Amsterdam • Boston • Heidelberg • London • New York • Oxford

Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands Linacre House, Jordan Hill, Oxford OX2 8DP, UK First edition 2007 Copyright © 2007 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-444-53073-8 For information on all Elsevier publications visit our website at books.elsevier.com Printed and bound in The Netherlands 07

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Working together to grow libraries in developing countries www.elsevier.com | www.bookaid.org | www.sabre.org

To my family

of

the past, the present and the future

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TABLE OF CONTENTS

Preface

xi

Acknowledgements

xiii

Chapter 0 Introduction [0-1] Introduction [0-2] Qualitative judgments and sciences [0-3] Outline of this book

1

1

2

3

Part A

7

A Differential Approach to Solution Thermodynamics

Chapter I Basics of thermodynamics – Derivatives of Gibbs energy, G [I-1] System, state, first and second laws of thermodynamics [I-2] Giles’ derivation of entropy [I-2-1] State, process, and irreversibility of process [I-2-2] Entropy, and potentials – Defining T, and p [I-3] Logical (mathematical) deduction [I-4] Stability criteria [I-5] Multi-component system – Partial molar quantities [I-6] Excess quantities [I-7] Response functions [I-8] Thermodynamic quantities – Order of derivative [I-9] Interaction functions – Third derivatives

9

9

11

11

12

14

16

17

18

19

20

21

Chapter II Solution thermodynamics – Use of the second and third

derivatives of G [II-1] Mixture [II-2] Gibbs-Duhem relation [II-3] Vapor pressures [II-4] Raoult’s Law and Henry’s Law [II-5] Process of mixing – Mixing entropy [II-6] Conversion from nB  nW  to xB  N variable systems [II-7] Interaction functions due to the ideal mixing entropy

23

23

25

30

33

35

37

40

viii

[II-8] [II-9] Chapter [III-1] [III-2] [III-3] [III-4] [III-5] [III-6] [III-7]

Phase separation – Critical point (UCST or LCST) Azeotrope III Determination of the partial molar quantities Introduction Calculation of HiE from HmE Experimental determination of excess partial molar enthalpy Experimental determination of excess partial molar volume Excess partial molar entropy – Excess chemical potential Boissonnas analysis – Excess chemical potential Partial pressures of 1-propanol (1P)−H2 O

41

48

51

51

51

54

58

58

60

66

Chapter IV Fluctuation and partial molar fluctuation –

Understanding H2 O [IV-1] Introduction [IV-2] Fluctuation functions – Coarse grain [IV-3] H2 O vs. n-hexane [IV-4] Site-correlated percolation model of H2 O [IV-5] Concentration fluctuations and Kirkwood-Buff integrals

69 69

69

73

78 84

Part B Studies of Aqueous Solutions using the Second and the Third

Derivatives of G

87

Chapter [V-1] [V-2] [V-3] [V-4] [V-5]

V Mixing schemes in aqueous mono-ols Mixing schemes in 2-butoxyethanol (BE)–H2 O Mixing schemes in other mono-ols (AL)–H2 O Fluctuation functions – More about Mixing Scheme I Concentration fluctuations – Mixing Scheme II Mixing Scheme III – Second and third derivative quantities

in the alcohol-rich region [V-6] Mixing schemes of aqueous alcohols (AL) studied by other

techniques

Chapter VI Mixing schemes in aqueous solutions of non-electrolytes [VI-1] Introduction [VI-2] Type (a) – Aqueous solutions of iso-butoxyethanol (iBE) at 20  C

and acetonitrile (ACN) at 6 to 45  C [VI-3] Type (d) – Aqueous solutions of glycerol (Gly), acetone (AC),

1,3-propanediol (13P), and tetramethyl urea (TMU) [VI-4] Type (b) – Aqueous solutions of iso-butyric acid (IBA)

and 2-butanone (BUT)

89

89 106

117

130

134

147

151

151

155

160

166

ix

[VI-5] [VI-6]

Type (c) – Aqueous solutions of dimethylsulfoxide (DMSO) and 1,2-propanediol (12P) Mixing schemes of aqueous non-electrolytes studied by other techniques

Chapter VII Effects of non-electrolytes on the molecular organization of H2 O: 1-propanol (1P) probing methodology [VII-1] Introduction – 1-propanol (1P) probing methodology [VII-2] Effects of methanol (ME), 2-propanol (2P) and tert-butanol (TBA) E on H2 O as probed by the H1P−1P pattern change [VII-3] Effects of urea (U), tetramethyl urea (TMU) and acetone (AC) E on H2 O as probed by the H1P−1P pattern change [VII-4] Effects of ethylene glycol (EG), 1,2- and 1,3-propanediol (12P and 13P), glycerol (Gly) and poly (ethylene glycol) (PEG) E on H2 O as probed by the H1P−1P pattern change [VII-5] Concluding remarks – Summary

170

175 175 179 187

193 200

Chapter VIII [VIII-1] [VIII-2] [VIII-3] [VIII-4] [VIII-5] [VIII-6] [VIII-7]

The effects of salts on the molecular organization of H2 O: 1-propanol (1P)-probing methodology Introduction – Hofmeister series Effects of NaF and NaCl on H2 O as probed by the E H1P−1P pattern change Effects of NaBr and NaI on H2 O as probed by the E H1P−1P pattern change Effects of Na2 SO4  NaOOCCH3 (NaOAc), NaClO4 , E and NaSCN on H2 O as probed by the H1P−1P pattern change Effects of CaCl2  NH4 Cl and tetramethyl ammonium chloride E (TMACl) on H2 O as probed by the H1P−1P pattern change Hydration number of glycine and its salts as probed E by the H1P−1P pattern change Concluding remarks in relation to other studies on aqueous electrolytes in the literature [VIII-7-1] Ion pairing [VIII-7-2] Hydration number, nH [VIII-7-3] Hofmeister series

168

Chapter IX Interactions in ternary aqueous solutions – General treatment [IX-1] Introduction [IX-2] Solute–solute interactions in tert-butanol (TBA) – Dimethylsulfoxide (DMSO)–H2 O [IX-3] Solute–solute interactions in 2-butoxyethanol (BE)–dimethyl sulfox­ ide (DMSO)–H2 O

205 205 207 211 218 225 229 234 234 235 237

241 241 242 256

x

Chapter X In closing – Executive summary on the effect

of solute on H2 O

265

Appendix A

Graphical differentiation by means of B-spline

267

Appendix B

Gibbs-Konovalov Correction

269

Appendix C

Heat capacity anomalies associated with phase

transitions – Two level approximation

271

Appendix D

Freezing point depression

281

Appendix E

Titration calorimetry with dilute titrant

285

References

287

Index

295

PREFACE

At the risk of being overly simplistic, the last century was the century of quantum mechanics. A handful of geniuses, driven just by curiosity, have learned how atoms and electrons behave, how an atom is made up, how a nucleus is held together, and so on, and constructed what we know as quantum mechanics. While it was developed purely for the fulfillment of human curiosity, it has led to its application to various technologies to the extent unimaginable to early mankind. Examples include the usage of nuclear power, the invention of transistor and various electronic devices, and drug design, to name but a few. How many of these geniuses who served an essential role in bringing in modern “conveniences” (and hence huge businesses) would have dreamed of the results of their endeavor? A philosophical spin-off of quantum mechanics, on the other hand, is that human beings have learned to look into details at the nanoscopic level of constituent elements. After all, we live in a macroscopic world which consists of components, sub-components, sub-sub-components and so on down to the elementary particles. And we are now good at, or at least comfortable in, studying the nature of what we believe to be the smallest components in detail. However, we all know that the nature of an assembly is not just the sum of the properties of all the constituent elements. Rather, each assembly has the specific extra characteristics due to the interactions among elements – the so-called “Many-Body Problem”. Recognition of such characteristics of assemblies is more common in social sciences, a theory in ecology based on Holism (Merchant, 1992), for example. In science, there is an effort to extract hitherto unknown laws purely by observing the peculiar characteristics of various many-body problems including biological systems and “life” itself. (Makishima, 2001). Thermodynamics, developed in the 19th century, on the other hand, deals with the macroscopic world. A number of geniuses, driven equally by curiosity, extracted the laws of thermodynamics. Since it deals exclusively with macroscopic phenomena, recog­ nition of the many-body nature is inherent, as typified by the common use of the excess thermodynamic functions in solution thermodynamics, excess over the sum of the ther­ modynamic functions of all constituents. At this point in time, therefore, I believe it is worthwhile to revisit thermodynamics in order to tackle the many-body problems, since quantum mechanics is not so well designed to do so. However, conventional usage of thermodynamics is basically phenomenological in nature and generally lacks the power to elucidate underlying mechanisms. For the last 20 years or so, we have been making an attempt at improving this point with some success. Our approach is purely experimental.

xii

In contrast to conventional thermodynamic studies, we make an effort to obtain ther­ modynamic quantities that are proportional to at least the second and possibly the third and higher derivatives of Gibbs function. We introduce the interaction functions and the partial molar fluctuations, both being the third derivatives. Without resorting to any model system, we learn the detailed thermodynamic behavior of the system in question, and hence the underlying mechanisms, directly from the composition or the temperature dependence of these higher order derivative quantities. With this methodology, we learn the mixing schemes or the “solution structures” in aqueous solutions in more detail than hitherto possible. In some cases, the findings by the present novel approach to solution thermodynamics more than complement those by other techniques including modern spectroscopic studies. In particular, the power of the present methodology in elucidating the molecular processes in dilute aqueous solutions is remarkable, as will be demonstrated throughout this book. This book has emerged from notes given to new students in our research group. All of them have had some exposure to the first formal introduction to thermodynamics in their undergraduate courses. Thus, it starts off with a brief review of the basics, followed by the development suitable for our approach. Then, what we learned about aqueous solutions using our thermodynamic methodology is reviewed. As appropriate, the findings about the mixing scheme or the “solution structures” by other methods are compared. Thus, the main part of this book is still growing and may necessarily be revised later. I wish to dedicate this book to my family of the past, the present and the future. For me to reach the present point in my career, I owe a great deal to my mentors, collaborators, postdoctoral fellows and students. I list their names on the following pages in an approximately chronological order. I wish to thank each and all sincerely. The work has been mainly supported by the Natural Science and Engineering Research Council (NSERC) of Canada. Although there has been a continuous bureaucratic reluctance, I thank NSERC all the same. Vancouver January, 2007 Yoshikata Koga

ACKNOWLEDGEMENTS

I thank:Shoji Makishima Charles A. McDowell Hiroshi Suga Akira Inaba Loren G. Hepler Chris G. Vonk Robert M. Miura Thierry Michael Rita Kasza Adam Chong Margaret Slater L. Le Joan S. C. Kam Frankie Lau Catherine Hui Christa Trandum Kathy Puhacz Mark Nitz Mark Klippenstein Yvonne Yau Betty Lam Anu Saini Ichiro Fujihara Candy Cheung Ami Hui Shigeomi Takai Daniel Chen Jesper Hansen Hiroshi Matsuo Denise Wong Pui Ming Chu

Eiji Ootsuka Hideaki Chihara Tooru Atake G. C. Benson Peter Westh Ichiro Arakawa Sachio Murakami William W. Y. Siu Lisa C. F. Chao Kirby Wong-Moon Mahiko Nagao Raymond Huie Yingnian Xu Stephen Cheng Heidi Lam Hikari I. Yoshihara Alice Ho Steven Tanaka Ken Suh Jerri Lai Sandy Hundall Paul Yenson Ole G. Mouritsen Yet Chow H. Kawaji K. Kawasaki Eric To Yoshihiro Taniguchi Jianhua Hu Wesley Chiang Yasutoshi Kasahara

Lionel G. Harrison James A. Morrison Keiko Nishikawa Aase Hvidt Jesper Kristiansen R. Troy Lassau Katsutoshi Tamura Terrance Y. H. Wong John T. W. Lai David Fraser Anselm Wong Laurence Beach Jenny Chuang Johny Shim Damon Robb Susan Chan James V. Davies Virginia Loo Arezoo Nasiry Paul Cheung Rena Kare Kazuko Mizuno Kenneth Chong George Makris T. Tojo Y. Ootsuka Christian Svane Seiji Sawamura Charles A. Haynes Jason Sze Matthew T. Parsons

xiv

Kim B. Anderson Jonathan Lau Mark Chen Kitty Chan Hideki Katayanagi Mette N. Kristensen Naoki Toshima Aurelien Perera Ken-ichi Tozaki David Siu Pang Hiew

Eric Yee Kevin Ralloff Amy Wu Harris Huang Hideki Shimozaki Hitoshi Kato Yukihide Shiraishi Franjo Sokolic Yasutaka Kido Lion Kit

Allen Liu Karin Liltorp Andrew Racaza Takehiko Ichioka Kumiko Miki Fumie Sebe Masao Kanemaru Laszlo Almasy Lori Leong Kenneth Wong

Chapter 0

INTRODUCTION

[0-1] INTRODUCTION As the title implies, this book describes an application of thermodynamics to solutions – multi-component systems. The original formulation of thermodynamics for multi-component systems was made by Gibbs (1993) and we basically follow his deriva­ tion here. We then emphasize the importance and usefulness of higher order derivatives of Gibbs energy, G, than have hitherto been utilized. In particular, we show that the second and third derivatives of G, if determined within reasonable accuracy, are pow­ erful in providing information towards molecular level understanding of the “mixing schemes” or the “solution structures” of aqueous solutions. At this point in time, the response functions (heat capacity, compressibility, thermal expansivity), the second derivative quantities, can be experimentally determined with a high accuracy. At the risk of over-generalization, however, it is common practice to integrate these measured second derivative quantities. The resulting very accurate ther­ modynamic quantities that are first derivatives of G are then compared with theoretical predictions. We argue that this practice is a waste of the qualitative information contained in the second derivative quantities themselves. For example, a heat capacity, Cp , provides information regarding the entropy fluctuation of the system, while its integrated quantity, the entropy, gives the global average. We suggest that the information contained in the second derivative quantities be taken advantage of by itself. Our originality is that we further take one more derivative of the measured second derivative quantities graphi­ cally without resorting to any fitting function. We have thus model-free, experimentally accessible third derivative quantities. Of course, with this treatment the resulting third derivatives are not without increased uncertainties. As shown below, they could amount to several per cent with the measured second derivative data determined within 0.1%. However, this quantitative disadvantage is well compensated for by the gain in higher quality information which is powerful in elucidating the molecular processes in aqueous solutions. We will demonstrate this throughout the book. The importance of H2 O and aqueous solutions on this planet earth requires no empha­ sis. A vast amount of effort has been invested in understanding the nature of aqueous solutions, yet we are still far from a complete understanding, if such is at all possible. We believe this new approach to solution thermodynamics adds, and will continue to add, some new insights into this perpetual human question.

2

Fig. 0-1

[0-2] QUALITATIVE JUDGMENTS AND SCIENCES Natural sciences are believed to be quantitative. Or are they? Let us consider the process of determining the mass of a sample, the most fundamental quantity in chemistry. We are given a balance and a set of standard weights. We place a sample on the lefthand side of the balance, and we discover that the left side is heavier, as shown in Fig. 0-1(a). A 10 g standard weight is placed on the right-hand pan, and we judge that a 10 g standard is too heavy for the sample, Fig. 0-1(b). We replace the 10 g standard with two 1 g weights, and realize the sample is heavier than 2 g, Fig. 0-1(c). We place another 1 g standard on the right pan, and learn that 3 g is too heavy, Fig. 0-1(d). Thus we conclude that the mass of the sample is between 2 and 3 g. These processes are repeated m times using 0.1 g standards until we can conclude that the mass of the sample is 2 + m − 1 × 01 g < mass < 2 + m × 01 g. We continue until a required uncertainty within which we wish to know the mass of the sample. As this example hints, the mass is determined within the required accuracy by a series of qualitative judgments as to

(b)

(a)

10 g

(c)

2g

(d)

3g

Fig. 0-1. Weighing a sample using a balance and standard weights, 10 g, 1 g, 0.1 g, 0.01 g    .. Weighing continues while a qualitative judgment is made as to which way the balance tips until we know the range of the mass within the required accuracy. See the text, Section [0-2].

3

which way the balance tips. Here we assume that the balance is sensitive enough for the required accuracy and that a set of standard weights are available down to the required sensitivity. As we see below, qualitative judgments are prevalent throughout this book.

[0-3] OUTLINE OF THIS BOOK In Part A, we discuss a differential approach to solution thermodynamics. It is new only because the second and third order derivatives of G are used as the basis of qualitative judgment as to what the thermodynamic quantities imply. At first, we briefly review the foundation of thermodynamics. To this end, we basically take the same attitude as Gibbs (1993), Hashizume (1981) and Callen (1985), whereby the existence of entropy and its nature are taken for granted, and build thermodynamics relations analytically. However, since a much smoother and more straightforward introduction of entropy is available by the “order-theoretical” approach (Giles, 1964; Lieb & Yngvason, 1999), we very briefly introduce Giles’ approach for entropy, sacrificing the rigor of the original. We then follow in Chapter II the derivation of the second derivatives of G – partial molar quantities, and the third derivatives – intermolecular interaction functions. Their physical meanings are briefly discussed in Part A, but their actual power in elucidating the nature of mixing schemes or “solution structures” is demonstrated in Part B, where these quantities are evaluated for actual aqueous solutions. For some unknown reasons, the experimental determination of the partial molar quantities, the other second derivative quantities than the response functions mentioned above, has been largely ignored, though the technical capability exists. Hence we devote Chapter III to the experimental methods of determining the partial molar quantities. Also included in the same chapter is a graphical differentiation to obtain the interaction functions – the third derivatives. A computer graphic technique, the “cubic B-spline”, could be used for this purpose. The present state of the cubic B-spline technique is discussed in Appendix A. Chapter IV deals with other second derivatives – response functions. We introduce the fluctuation functions that could provide qualitative information about the intensity (amplitude) as well as the extensity (wavelength) of fluctuations. We take one more derivative and introduce the partial molar fluctuation functions, the effect of additional solute on the fluctuation characteristics of the system. Part B describes our studies up to the present time on aqueous solutions using the second and third derivative quantities, and thus demonstrates their power in elucidating the mixing schemes. Chapter V deals with the aqueous mono-ols to which we have applied our methodology so far, while Chapter VI is about the aqueous solutions of some other non-electrolytes. In these two chapters the thermodynamic signatures and hence the mixing schemes of hydrophobic and hydrophilic solutes are discussed. These understandings, however, become deeper with our “three component 1-propanol probing methodology” described in Chapter VII and Chapter VIII. The concept of partial molar quantities is most useful in multi-component systems in that they refer only to a target component even in the presence of many other components. Namely, it is only a

4

target component that is required to be perturbed. Furthermore, the special nature of H2 O dictates that the solution in the H2 O–rich composition range is governed by the interactions of each solute via bulk H2 O, rather than by direct contacts among solutes themselves. This led to our “three component 1P-probing methodology” in which we use thermodynamic properties (in terms of third derivative quantities) of 1-propanol (1P) in a mixed aqueous solution including a third component in the 1P – a third component – H2 O system. We use the induced changes in the third derivative quantity of 1P by adding a third component as a probe to learn the effect of the third component on the molecular organization of H2 O. We apply this methodology to elucidate the effects of amphiphiles on H2 O. We learn that the hydrophobic and the hydrophilic moieties of an amphiphile compete or cooperate towards interaction with H2 O. The net result is that the effects of each moiety manifest themselves competitively or cooperatively in the behavior of the third derivative quantities. We thus establish a hydrophobicity/hydrophilicity scale within the methodology. In Chapter VIII, we apply the same 1P probing methodology to some electrolyte salts, and learn the effects of cations and anions on the molecular organization of H2 O. The effects of ions on the structure and the function of biopolymers in aqueous solutions have been studied extensively. As a result, ions have been ranked as the Hofmeister series (Hofmeister, 1887) according to the propensity and the strength of the effect of each ion. The generality of this ion ranking for a wide range of processes has sparked a massive interest in the possible underlying mechanism. However, the molecular level understanding of the Hofmeister effect is still fragmentary. We make an attempt at sorting out the ranking in terms of the hyrophobicity/hydrophilicity scale within our methodology. In addition, we show that our method could identify some ions as simple hydration centers without altering the nature of bulk H2 O. The hydration number of these ions could be evaluated by this method. We show for such ions that the Hofmeister ranking is consistent with the Hofmeister’s original idea, “water withdrawing power” (Hofmeister, 1887). In Chapter IX, we discuss the treatment by the present methodology of the three component systems in general. We introduce the heterogeneous interaction functions between two different species in the second and third derivative levels. We show that the heterogeneous interactions also occur via bulk H2 O competitively or cooperatively but not linearly additively. We then provide the executive summary of our findings in the last chapter, Chapter X. We had to omit from this book our recent unfinished works using the 1-propanol probing methodology on the nature of aqueous solutions of “ionic liquids” and that of aqueous “osmolytes”. Both works will be dealt with in a future revised edition. As appropriate, we review and compare recent findings about the mixing schemes, or the solution structures, in aqueous solutions by other techniques; spectroscopic, scat­ tering, molecular dynamics and modern ultra fast techniques. We demonstrate that the present thermodynamic methodology is just as powerful as, if not more than, those modern techniques. Throughout this book, all the figures, tables and equations are numbered in Arabic numerals preceded by a Roman number denoting the chapter. Thus, eq. (III-21) is the 21st equation in Chapter III. However, when cited within the same chapter, the first Roman

5

numerals are omitted. We make frequent references to equation and figure numbers in the text. For the convenience of a reader, we make entries of equation and figure numbers on the top of each page. When the table is referred, its page number is given. As for any other thermodynamic discussions, we use partial differentiations frequently. When obvious, the lists of variables that remain constant are often omitted. Readers are asked to interpret that all the variables other than those of differentiation are kept constant. Indeed, we exclusively dwell in the Gibbs variable system (p, T , ni ), and it should be obvious which are kept constant.

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Part A A DIFFERENTIAL APPROACH TO SOLUTION THERMODYNAMICS

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Eq. (I-1) to (I-3)

Chapter I

BASICS OF THERMODYNAMICS – DERIVATIVES OF GIBBS ENERGY, G

[I-1] SYSTEM, STATE, FIRST AND SECOND LAWS OF THERMODYNAMICS Consider a “system” surrounded by a “surrounding”. A system can be “open” or “closed” depending on whether it can exchange matter with the surroundings or not. An “isolated” system cannot exchange “energy” with the surroundings. Here we accept “energy” as one of the fundamental quantities in describing the “state” of a system. Historically, however, careful observation and precise logical deduction by a number of geniuses on macroscopic phenomena regarding heat and work acting on a closed but not isolated system have led to the discovery of the energy or the internal energy, E, described by equation, dE = q − w = qrev − wrev �

(I-1)

This is the well known first law of thermodynamics. Eq. (1) is the result of the realization that the heat received by the system, q, and the work done on the surroundings by the system, w, are equivalent and that the combination of the two determines the change in the internal energy, E, which is a state function. Among an infinite number of possible combinations of the values of q and w resulting in the same value of dE, there is the unique value, qrev and wrev respectively, such that �qrev � ≥ �q� and

�wrev � ≥ �w��

(I-2)

Namely, the absolute value of qrev takes the maximum, and so does that of wrev under the special situation called the reversible, or the quasi-static process. This unique value of qrev defines another thermodynamic state function, entropy, S, by eq. (I-3) – the second law of thermodynamics. dS =

qrev ≥ 0� T

(I-3)

In the advent of statistical mechanics, we learned that S represents the randomness of the system. While this description based on statistical mechanics is easier to understand

10

Fig. I-1

Surrounding

System closed, not isolated

w work out

dE q heat in

dE = q - W = q i - Wi etc. = qrev - Wrev

Fig. I-1. A closed but not isolated system receives heat q from, and gives work w to the surround­ ing. As a result, the internal energy of the system, E, changes by dE = q − w. For a given value of dE, there are countless combinations of the values of q and w. Out of these combinations, there is the unique combination, qrev and wrev , such that the absolute value of qrev or wrev is the largest, pertaining to the reversible process, i.e. the interaction between the system and the surrounding takes place “reversibly”.

than the original derivation by Clausius leading to eq. (3) (Kestin, 1976), thermody­ namics must be self-contained within a macroscopic description. Below, we very briefly introduce Giles’ method (1964) of introducing entropy, S, which we find is much more straightforward than Clausius’ original derivation (Kestin, 1976), without extensive use of gedanken experiments on heat engines. Indeed, in textbooks and lectures on ther­ modynamics, chemistry students are exposed for the first time to careful consideration about heat engines only at the beginning. Once the entropy is introduced, heat engines are completely forgotten and the difference in thermodynamic functions between two states then takes center stage throughout. For example, the Gibbs free energy difference is emphasized between 2 mole of H2 and 1 mole of O2 gas put together without mixing them and their gaseous mixture, or between the latter gaseous mixture and 2 mole of H2 O. Using Giles’ approach we avoid discussion on heat engines, as shown below. The second inequality of eq. (3) means that the system completely isolated can only become more disorganized spontaneously; otherwise, dS = 0, the system is in equilibrium. This will lead to the stability criteria of various thermodynamic functions discussed in Section [I-4]. If we limit our attention to the case where work is done only by volume expansion, wrev = p dV�

(I-4)

Eq. (I-4) to (I-6d)

11

where p is the pressure of the surroundings and V is the volume of the system. Equa­ tions (1), the first half of (3) and (4) yield the first law of thermodynamics, dE = T dS − p dV�

(I-5)

This is the most fundamental relation in thermodynamics and is the starting point for further discussion.

[I-2] GILES’ DERIVATION OF ENTROPY [I-2-1] State, process, and irreversibility of process We follow primarily the route of the “order–theoretical” approach, introduced by Giles (1964), but later more rigorously treated by Lieb and Yngvason (1999). Consider a closed and isolated system, A, that could take a state labeled as a, b, c � � � . Consider, for example, a system consisting of 2 mole of H and 1 mole of O atoms. The system could be in a state labeled as a, which consists of 1 mole of H2 and 1/2 mole of O2 . The same system could be filled with a mole of H2 O vapor, which is called state b. Since the system is isolated and closed, the total amount of H and O remains the same as well as the internal energy E. To ensure that there is no work done on the system in addition to no heat exchange, the volume of the system should be the same, V . We now consider a process a→b. Suppose we have some means of judging whether such a process indeed occurred or not. We do not care what happened during the course of the process, as long as E and V and the total amount of H and O are all kept the same at the beginning and the end of the process. We could thus add a pinch of catalysts into the system and remove them at the end without disturbing the above conditions. We can summarize the results of the observation by the following statements: �a� reversible�

if a→b

and

b→a�

�b� natural irreversible�

if a→b

and

b −→ a�

NOT

�c� anti-natural irreversible� if a −→ b �d� impossible�

NOT

if a −→ b

NOT

and

b→a�

and

b −→ a�

NOT

NOT

(I-6a) (I-6b) (I-6c) (I-6d)

Here → denotes the process that actually occurred, while −→ means never materialized. We now consider the association of two systems, A and X, with the resulting com­ pound system A + X, and they take states, a, x, and a + x. The two systems are sitting side by side, but not in direct contact with each other, within the same surroundings. They do not exchange E, V , or matter between them. We can imagine x being the state

12

in which 1 mole of H2 and 1 mole of S are enclosed in system X, and in state y 1 mole of H2 S is present. Suppose we know that a→b, and x→y are both natural irreversible, NOT NOT namely, b −→ a, and y −→ x separately. For convenience of further discussion, we define process � ≡ (a, b), and � ≡ (x, y). “≡” signifies the definition. Assume that both � and � are natural irreversible independently. Let us now consider a new process � − � ≡ (a + y, b + x) for the combined system A + X. If such a process � − �. is observed to be natural irreversible, we then conclude that � is more irreversible than �. If we assign a value to each process representing its “irreversibility” as, I���, I��� etc, we can write I��� > I���. If, on the other hand, � − � is found to be anti-natural irreversible, then � is less irreversible than � and I��� < I���. We note in passing by comparing with eq. (6) above that I��� = 0 for � reversible, I��� > 0 for natural irreversible, and I���< 0 for anti-natural irreversible. With positive integers k and m, we may consider a process k� − m�, namely, we associate k systems of A and m systems of X, and connect states ka + my and kb + mx. If it is found that k� −m�. is natural irreversible, then kI���> mI���, or I��� > (m/k)I���. If we add one more �. reversed, i.e. we try the process k� − (m + 1)�, and the result turns out to be anti-natural irreversible, we con­ clude that (m/k)I��� < I��� < {(m + 1)/k}I���. With sufficiently large integers of k and m, we could sandwich the value of I��� with a sufficiently small interval in reference to the irreversibility of a “standard” process �.

[I-2-2] Entropy, and potentials – defining T, and p Since � connects two states a and b, and since I��� is determined within a required accuracy relative to a standard I���, we define a state function S for each state such that, I��� ≡ S�b� − S�a��

(I-7)

Hence, following (6), we make the following statements as, S�b� = S�a�� if � is reversible�

(I-8a)

S�b� > S�a�� if � is �natural� irreversible�

(I-8b)

For an infinitesimally small process, i.e. a ≈ b, we may rewrite eq. (8) as, �S = 0� if reversible� �equilibrium��

(I-9a)

�S > 0

(I-9b)

if irreversible� �non-equilibrium��

For the above two statements, anti-natural and impossible cases are omitted for obvious reasons. The definition of an equilibrium state is given by eq. (9). Namely, if an

Eq. (I-7) to (I-13)

13

infinitesimal deviation from a state could naturally bring back to the original state, such a state is said to be in equilibrium. We now have a state function S, in addition to E, and V considered above. There is obviously another variable describing the amount of each component in the system, ni . At present, however, we fix ni and ignore its variation. Hence we could consider S to be a function, a differentiable function, of E and V , and write as, S = S�E� V ��

(I-10)

Then its perfect differential is written as, � dS =

�S �E

�

�

�S dE + �V V

� dV�

(I-11)

E

We define “potentials” following Giles (1964) as, � �

�S �E �S �V

� ≡ 1/T�

(I-12)

≡ p/T�

(I-13)

V

� E

We note in passing that if ni is kept as a variable, we could have defined the chemical potential of the i-th component as ��S/�ni � ≡ �i /T . Here we are keeping ni constant. Later we will introduce chemical potential, by introducing the variation in ni . With these definitions (12) and (13), eq. (11) is rewritten as, dE = T dS − p dV�

(I-5)

Thus we recover eq. (5). The original derivation by Giles (1964) is much more rigorous than the present brief description. However, this brief account will provide the essence of the “order­ theoretical” method of introducing entropy without resorting to the use of heat engines including an unreal Carnot cycle. These heat engines are not familiar to chemistry students, and in most textbooks of thermodynamics, such heat engines are used only once at the beginning for introducing the entropy; in all the remaining applications to chemistry no heat engines are mentioned. Giles’ treatment is not without a problem. The connection of two states via → and the differentiability of S are taken for granted. Lieb & Yngvason (1999) claim, however, that both premises are provable.

14

[I-3] LOGICAL (MATHEMATICAL) DEDUCTION Since eq. (5) can be rewritten as, � dE =

�E �S

�

� dS + V

�E �V

� dV�

(I-5’)

S

it follows that � T=

�E �S

� �

(I-14)

V

which recovers eq. (12) above and � p= −

�E �V

� �

(I-15)

S

If we invent enthalpy, H, following Gibbs as, H = E + PV �

(I-16)

dH = dE + p dV + V dp = T dS + V dp�

(I-17)

it follows using eq. (5) that

The motivation for adding the pV term in eq. (16) is to change the independent variable from V to p, as is evident in eq. (17). Namely, in eq. (5), we change V (together with S� and seek the induced change in terms of E, while in (17) we change p infinitesimally. The choice of the independent variable, p or V , is a matter of convenience. For theoretical works, V is more convenient conceptually, while p is the choice of experimentalists. Eq. (17) then implies that � T=

�H �S

� �

(I-18)

�

(I-19)

p

and � V=

�H �p

� S

We have Helmholtz energy, A, and Gibbs energy, G, thanks to these geniuses. A = E − TS�

(I-20)

Eq. (I-14) to (I-25)

15

and G = H − TS�

(I-21)

Here again, the TS term is introduced to convert the independent variable from S to T , the convenience of which is immediately obvious. For Gibbs function, G, eqs. (5), (16) and (21) yield, dG = −S dT + V dp�

(I-22)

Eq. (22) demonstrates the experimental convenience of G, since its variation is deter­ mined by changing T and p, both of which are controllable experimentally. The discus­ sion of the stability criteria in Section [I-4] below shows its further convenience in the (p, T ) variable system. It follows from eq. (22) that, � S=−

�G �T

� �

(I-23)

�

(I-24)

p

and � V=

�G �p

� T

We see that � H = G + TS = G − T

�G �T

�

� = −T 2 p

��G/T� �T

� �

(I-25)

p

We note in eqs. (22)–(24) that G is a convenient function in the system of the independent variables (p� T ), and that S and V are the first derivative of G with respect to T and p. Its convenience becomes more convincing below when we discuss the stability criteria in Section [I-4]. We point out from eq. (25) that H is not strictly the first derivative of G. However, it contains the first derivative of G also. Equivalent relations using A, eq. (20), instead of G can be obtained for the variable system, (V� T ). For the convenience of experimentalists, however, we limit our attention to the (p� T ) variable system. p and T are experimentally controllable and are also both intensive quantities. In Section [I-5], we expand our discussion to the (p� T� ni ) variable system, where ni is the amount of the i-th component of the mixture – the main subject of this book, the solution thermodynamics. We note that ni is also experimentally controllable.

16

[I-4] STABILITY CRITERIA The inequality of the second law, eq. (3) or (9), above dictates that for an isolated and closed system any spontaneous process produces entropy, dS ≥ 0�

(I-26)

This is unconditionally true as long as the system under consideration is isolated and closed. Thus, if a process were to be associated with dS < 0, such a process will never be materialized. When dS = 0 the system is then in equilibrium and nothing apparently changes. If dS > 0, the system then undergoes a spontaneous change. Thus, eq. (26) shows the spontaneity of the process, or the stability of its final state, and is called a stability criterion. It would also be useful if we have some indication of spontaneity in terms of other state functions, E, H, V, and G� For this purpose, we combine eqs. (1), (3) and (4) and obtain, dE ≤ T dS − p dV�

(I-27)

Therefore, dE ≤ 0� if

dS = dV = 0�

(I-28)

dS ≥ 0� if

dE = dV = 0�

(I-29)

Since H = E + pV, eq. (27) is rewritten as, dH ≤ T dS + V dp�

(I-30)

Therefore, dH ≤ 0� if

dS = dp = 0�

(I-31)

dS ≥ 0�

dH = dp = 0�

(I-32)

if

Since G = H – TS, eq. (30) is rewritten as, dG ≤ −SdT + V dp�

(I-33)

Therefore, dG ≤ 0� if

dT = dp = 0�

(I-34)

Eq. (I-26) to (I-38)

17

Eq. (34) shows the usefulness of G in the (p� T ) variable system, as pointed out above. Namely, we can control p and T to be constant and seek a change in state such that dG ≤ 0. We know then that such a state is stable to any disturbances. As discussed for dS above, the equal sign indicates the equilibrium situation. We point out in passing that the stability criterion for dV cannot be drawn in a meaningful manner. Readers might like to check this out.

[I-5] MULTI-COMPONENT SYSTEM – PARTIAL MOLAR QUANTITIES We now consider a multi-component system (Koga, 2004b). Up to this point the amount of material in the system was left implicit and fixed. Now the independent variables are (p� T� ni �, where ni is the amount of the i-th component, and i = 1, 2, 3, � � � k, for a k-component system. Thus, we have a convenient Gibbs function, G, as a function of all these variables. G = G�p� T� ni ��

(I-35)

When the independent variables are perturbed, the resulting total differential of G is written as, � dG =

�G �T

�

� dT +

p�ni

�G �p

� dp + T�ni

� � � �G dn � �ni p�T�nj�=i i i

(I-36)

The first two partial derivatives on the right of eq. (36) are given in (23) and (24). The third and onward derivatives are called by Gibbs “the chemical potential of the i-th component”. � �i =

�G �ni

� �

(I-37)

p�T�nj�=i

� i) where the partial derivative is taken with respect to ni keeping p� T� and nj � j = constant. The operational meaning of �i is the response of the entire system in terms of G when the system is perturbed in ni . Thus, �i implies the actual contribution of the i-th component towards the entire system in terms of G or the actual situation of the i-th component in terms of G in the mixture. As is obvious from eq. (37), �i is the first derivative of G. By differentiating both sides of eqs. (23)–(25) with respect to ni and using eq. (37), we have � Si ≡

�S �ni

�

�

nj�=i

��i =− �T

� � p�nj=i �

(I-38)

18

� Vi ≡

�V �ni

�

� = nj�=i

��i �p

� �

(I-39)

T�nj=i �

� � � � �H ���i /T� 2 Hi ≡ = −T � �ni nj�= i �T p�nj=i �

(I-40)

They are called, respectively, the partial molar entropy, volume and enthalpy of the i-th component. Similar to �i , they show the actual situation of the i-th component in the mixture in terms of entropy, volume and enthalpy, respectively.

[I-6] EXCESS QUANTITIES The property of a mixture is not just the sum of those of constituent components. Aside from the mixing entropy discussed below, it is only an ideal, and hence a non-real situation, when the sum of the properties of the constituent components fully describes the property of the resulting mixture. For real cases, an assembly of many components often shows some characteristics unimaginable from the properties of each component. Such a special characteristic manifests itself through strong interactions among the constituents – the so-called “many-body problem”. In terms of enthalpy, for example, the total enthalpy of a mixture, H, is written as, H=

�

ni Hi∗ + H E �

(I-41)

ni

where Hi∗ is the enthalpy per unit amount of the i-th component in its pure state. For an ideal, and hence a non-real mixture, the total enthalpy is just the sum of those of each component, i.e. H E = 0. In reality, however, H E �= 0, as a manifestation of intermolecular interactions among constituents. Equivalently, for other extensive quantities, V=

�

ni Vi∗ + V E �

(I-42)

i

� S = ni �Si∗ − R ln�ni /N �� + S E �

(I-43)

i

� G = ni �G∗i + RT ln�ni /N �� + GE �

(I-44)

i

The second term on the right of the last two equations� R ln(ni /N ), is the mixing entropy arising from the fact that each component is distinguishable from the others and hence their local accessibility increases by mixing even in the ideal solution. We introduce the mixing entropy in consideration of the process of mixing below in [II-5]. Here, N = �ni .

Eq. (I-39) to (I-51)

19

The key concept in understanding the nature of a mixture is the intermolecular interactions. Thus, the solution thermodynamics deals mostly with the excess functions. Therefore, instead of eqs. (37)–(40), we use the following excess partial molar quantities of the i-th component more often. � �Ei

≡ �

SiE ≡

ViE

�GE �ni �S E �ni

� �

(I-45)

p�T�nj�=i

�

� =−

p�T�nj�=i

��Ei �T

� �

(I-46)

p�nj�=i

� E� � E� �V ��i ≡ = � �ni p�T�nj �p T�nj=�i �

HiE =

�

�H E �ni

� = −T 2

p�T�nj�=i

(I-47)

���Ei /T� �T

� �

(I-48)

p�nj�=i

Equations (45)–(48) indicate �Ei is the first derivative and SiE , ViE , HiE are the second derivative of GE .

[I-7] RESPONSE FUNCTIONS There are three more second derivatives of G, in the (p� T� ni � variable system. They are called collectively the response functions: heat capacity, Cp , isothermal compressibility, �T , and isobaric thermal expansivity, �p . � Cp =

�T = − 1 �p = V

�H �T 1 V �

�

�

� =T

p

�

�V �p

�V �T

=− T

� T

�

�S �T

1 = V

� = −T

p

1 V

�

�

� �T

�2 G �p2

� � p

�2 G �T 2

� �

(I-49)

p

� �

(I-50)

T

�G �p

� �

(I-51)

T

In all the above derivatives, ni are kept constant. From our experience, we know that Cp > 0, and �T > 0. We have not encountered a system in which the temperature decreases on addition of heat or the volume increases on applying pressure. The nonnegativity of Cp and �T is called the mechanical stability criterion. As will be discussed

20

in Chapter IV, Cp is proportional to the mean square (hence non-negative) fluctuation of entropy and �T to that of volume. Hence they cannot be negative. [I-8] THERMODYNAMIC QUANTITIES – ORDER OF DERIVATIVE So far we have seen thermodynamic quantities that contain the first and second deriva­ tives of G with respect to independent variables in the Gibbs ensemble, the (p� T� ni � variable system. We summarized them in Table I-1. They are classified by the highest order of derivative that each thermodynamic quantity contains. The first column shows the order of derivative of the leading term and the number of the quantity in each class is shown in parentheses. The second column is the notation. Those in the upper portion of the table are commonly accepted notations, while towards the bottom of the table new notations are indicated. The list in { } indicates the variables of differentiation. Thus, Cp , Table I-1. Thermodynamic quantities in �p� T� ni � variable system. 0th

G

1st (4)

H � �T �a� S � �T � V � �p� �i � �ni �

H = G − T��G/�T� S = −��G/�T� V = ��G/�p� �i = ��G/�ni �

H E = GE − T��GE /�T� S E = −��GE /�T� V E = ��GE /�p� �Ei = ��GE /�ni �

2nd (7)

Hi � �T� ni � Si � �T� ni � Vi � �p� ni � Cp � �T� T � �T � �p� p� �p � �T� p� �i−j � �ni � nj �

Hi = ��H/�ni � Si = ��S/�ni � Vi = ��V/�ni � Cp = ��H/�T� = T��S/�T� �T = −��V/�p�/V �p = ��V/�T�/V �i−j ≡ N���i /�nj �

HiE = ��H E /�ni � SiE = ��S E /�ni � ViE = ��V E /�ni � S � ≡ Cp /V , S � = S�R/V V � ≡ T�T , V � = V�R/V SV � ≡ T�p , SV � = SV�R/V �i−j E ≡ N���Ei /�nj �

3rd (11)

Hi−j � �T� ni � nj � Si−j � �T� ni � nj � Vi−j � �p� ni � nj � S �i , S �i V �i , V �i SV �i , SV �i

Hi−j ≡ N��Hi /�nj � Si−j ≡ N��Si /�nj � Vi−j ≡ N��Vi /�nj � S �i ≡ N��S �/�ni �, S �i ≡ N��S �/�ni � V �i ≡ N��V �/�ni �, V �i ≡ N��V �/�ni � SV �i ≡ N��SV �/�ni �, SV �i ≡ N��SV �/�ni � ��Cp /�T� ��Cp /�p� = ���p /�T� ���p /�p� = −���T /�T� ���T /�p� ���i−j /�nk �

Hi−j E ≡ N��HiE /�nj � Si−j E ≡ N��SiE /�nj � Vi−j E ≡ N��ViE /�nj �

4th (16) a�

� � � � � � � � � � � � � � � ..

The variable in { } is the variable of differentiation.

Eq. (I-52)

21

for example, contains two T , indicating G is differentiated with respect to T twice to obtain Cp . The third column contains the definition equality. In this book, we more often use the quantities in the fourth column, the excess functions, for the reasons discussed above. Furthermore, instead of using the response functions, Cp , �T , and �p , directly we introduce (Koga, 1999; Koga & Tamura, 2000) and use q � and q �(q = S, V , or SV). q � signifies the amplitude or the intensity of the mean square fluctuation in quantity q, while q � contains qualitative information about the wavelength or the extensity of fluctuation in q. For q = SV, both fluctuation functions, SV � and SV �, show the cross fluctuation between S and V� Unlike normal liquids, H2 O has a negative contribution in the SV cross fluctuation, due to a putative formation/destruction of ice-like patches. Hence SV � and SV � are important quantities in studying aqueous solutions. The process of modification of the molecular organization of H2 O by solutes could be monitored by these quantities. We discuss fluctuation functions in more detail in Chapter IV. Having tabulated the thermodynamic quantities up to the second derivatives, it is natural to expand the table to include higher order derivatives, as shown in Table 1 (p. 20). In the subsequent discussion of the study of aqueous solutions, we fully utilize the quantities listed in the second column of the third derivative class. We have not yet exploited the bottom five third derivative quantities and thus no new notations are given. It is our challenge to experimentally determine these and even higher order derivatives in the future. G, the zeroth order derivative, dictates the fate of an equilibrium state, by the stability criterion discussed above, eq. (34). The first derivatives, H and S� indicate the ingredient of G. A second derivative, Cp for example, would indicate the degree of fluctuation in H or S� Thus, the higher the order of the derivative, the more detailed information it provides. Our approach is, therefore, to obtain quantities of the second derivative by direct measurements. As much as the situation permits, we take one more derivative graphically without resorting to any fitting function. We then learn what the nature tells us directly from the data of the third derivative quantities that are model-free and experimentally accessible.

[I-9] INTERACTION FUNCTIONS – THIRD DERIVATIVES For an extensive excess function, F E (=GE � H E � S E or V E �, the excess partial molar quantity of the i-th component, FiE , is written following eqs. (45)–(48) as, � FiE =

�F E �ni

� �

(I-52)

p�T�nj�=i

For historical reasons (since Gibbs used the notation), FiE = �Ei for F E = GE ; otherwise FiE = HiE , SiE , or ViE . The physical meaning of the excess partial molar quantities of the i-th component is the actual situation that the i-th component experiences in terms

22

of GE , H E , S E , and V E respectively within the mixture. We now define the following derivative. � E� �Fi E Fi−j ≡ N � (I-53) �nj p�T�n k�=j

This quantity, which we call the interaction function, is the change in FiE due to a unit perturbation in nj , keeping all the other independent variables constant. Therefore, FiE−j signifies the effect of the j-th component on the actual situation of the i-th component in terms of F E . Thus, �Ei−j is the i-j interaction in terms of chemical potential. It is clear from the definition that �Ei−j is the second derivative of G (see Table 1 (p. 20)). HiE−j , SiE−j , and ViE−j are, on the other hand, the third derivatives of G, and they are indicative of the i-j interactions in terms of H E � S E , and V E respectively. According to the stability criteria, if �Ei−j or HiE−j is negative, the j-th component makes the chemical potential or the enthalpic situation of the i-th component more favorable. Therefore we may say that the i-j interaction is favorable or “attractive” in terms of chemical potential or enthalpy. If it is positive, then the i-j interaction is said to be unfavorable or “repulsive”. Of course, i and j are interchangeable and j could be equal to i also. Since the stability criterion for entropy is the opposite of enthalpy or Gibbs function, a positive value of SiE−j is indicative of a favorable (or attractive) i-j interaction in terms of excess entropy. For volume, there is no stability criterion. Thus, E Vi−j provides simply the degree of the effect of the j-th component on the volumetric situation of the i-th component in the mixture. We note in passing, � � ��i �i−i ≡ N � (I-54) �ni is inversely related to the concentration fluctuation accessible by scattering experiments. (Note this is not the excess function, �Ei−i , eq. (53) with F = G with i = j). As will be discussed in Chapter IV, this is inversely proportional to the mean square (hence non-negative) concentration fluctuation. The non-negativity of �i−i , or the right-hand side of eq. (54), is known as the diffusional stability criterion. It is also related to the Kirkwood-Buff (1951) integrals. If the excess partial molar quantity, FiE , is determined accurately and in small incre­ ments in nj , then it is possible to evaluate FiE−j graphically within an acceptable accuracy, several per cent. (See [III-4], for graphical differentiation). Thus, we have FiE−j , giving important information about the i-j interaction, purely from experiment without resorting to any model. This information is crucial in understanding the nature of a non-ideal, and hence a real mixture. We will make extensive use of this model-free, experimentally accessible information about the i-j interactions in order to study the molecular processes in aqueous solutions.

Eq. (I-53) to (II-6)

Chapter II

SOLUTION THERMODYNAMICS – USE OF THE SECOND AND THIRD DERIVATIVES OF G

[II-1] MIXTURE Consider � a gas mixture consisting of ni of the i-th component, i = 1� 2� 3� � � � k, with N = ni . The mixture occupies the volume, V , under the total pressure, p. We define i

the mole fraction of the i-th component, xi , as, ni � N

xi =

(II-1)

The partial pressure of the i-th component, pi , is defined as, pi = xi p�

(II-2)

and hence, p=

�

pi �

(II-3)

i

A perfect gas by definition obeys the Boyle-Charles law, a special form of the state function, V = f�p� T� ni �� V=

NRT � p

(II-4)

V=

ni RT � pi

(II-5)

Using eqs. (1), (2) and (3),

Differentiating eq. (4) with respect to ni , keeping p, T, and nj =i constant and using  eq. (I-39), we obtain, � � �V RT V = = � (II-6) Vi = �ni p�T�nj=i p N

24

Recalling the second equality of eq. (I-39), and using eq. (6), � � ��i RT � = Vi = �p T�nj=i p

(II-7)

Integration of eq. (7) yields, �i =

�0i + RT

�pi 1 p0

p

dp�

(II-8)

Thus, � � p �i = �0i + RT ln i � p0

(II-9)

where p0 is the pressure of a standard state, which is normally taken to be 1 atm, and �0i is the value of �i at p = p0 . We note in passing that for non-perfect gases, the state function, eq. (4), will have virial correction terms. In such a case, eq. (9) should be written as, � � f (II-10) �i = �0i + RT ln i � f0 where fi and f0 are fugacities. By virtue of eq. (2), eq. (9) is rewritten as, �i = �∗i + RT ln xi �

(II-11)

where �∗i is the chemical potential of the i-th component in its pure state, i. e. xi = 1, with � ∗� p �∗i = �0i + RT ln i p0 where pi∗ is the vapor pressure of pure liquid i at the same temperature. Eq. (11) is valid strictly for a perfect gas mixture or an ideal mixture. A perfect gas mixture is characterized by the total lack of intermolecular interaction among constituent molecules. This situation is not at all real. For a real non-perfect gas mixture, an artificial introduction of fugacity, fi , eq. (10), instead of measurable pi is one way to cope with the situation. Another is to introduce an extra term, the excess chemical potential, �Ei , of i to the right of eq. (11), as shown in eq. (12) below. We take the latter route. We then experimentally determine �Ei . �i = �∗i + RT ln xi + �Ei �

(II-12)

Eq. (II-7) to (II-17)

25

�Ei is called the excess chemical potential of the i-th component, and takes a non-zero value reflecting the real situation in a mixture in which intermolecular interactions are prevalent. We note that by the present choice of �∗i , �i must be equal to �∗i and hence �Ei = 0 at xi = 1. This is equivalent to choosing the reference state, where �Ei = 0, to be the pure state for all the components. Hence this is often called the symmetric reference system, since all the components are treated equally. Any other choices are possible, but the dilute ideal solution is often taken as the reference. Namely, �Ei is taken as zero at the infinite dilution, i. e. all the mole fractions of the solutes are zero. This is a convenient choice for studying exclusively on dilute solutions. Here we choose the symmetric reference system, since we are interested in the mixing schemes in the entire composition range. Instead of eq. (12), a more conventional way is to write �i = �∗i + RT ln ai �

(II-13)

where ai is the activity of the i-th component in the mixture and a i = x i �i �

(II-14)

�i is called the activity coefficient. Comparing eq. (12) with (13) and (14), �Ei = RT ln �i �

(II-15)

We use eq. (12) throughout this book, instead of the more common eq. (13), and determine �Ei experimentally. The usage of ai seems to us to have some apologetic connotation; apologetic in the fact that the reality does not follow eq. (11).

[II-2] GIBBS-DUHEM RELATION From eqs. (I-23), (I-24), (I-36) and (I-37), dG = −S dT + V dp +

�

�i dni �

(II-16)

i

With T and p being kept constant, dG =

� i

�i dni �

(II-17)

26

Since G is an extensive quantity, the value of G is proportional to the amount of the system. Namely, if we increase ni to rni �i = 1� 2� 3� � � � k�, where r is a positive constant, then the value of the Gibbs function of the r-fold system, G , will be equal to rG. Or, G �rni � = rG�ni ��

(II-18)

The chemical potential, on the other hand, is an intensive variable and independent of the size of the system, as long as xi ’s are constant. To show this, we calculate the chemical potential of the i-th component in the r-fold system, �i , as, � �i =

�G ��rni �

�

� =

nj

��rG� ��rni �

�

� = nj

�G �ni

� = �i �

Thus, the chemical potential of the i-th component is invariant to the change from ni to rni , i.e. �i is an intensive quantity. The same argument leads also to the fact that other partial molar quantities, eq. (I-38) to eq. (I-40), are all intensive quantities. We now integrate eq. (17) from r = 0 to r = 1. The result is written with the zero integration constant as, G=

�

n i �i �

(II-19)

i

A similar deduction leads to the relation for the excess Gibbs function as, GE =

�

ni �Ei �

(II-20)

i

The same arguments as those above also lead to similar expressions, HE =

�

ni HiE �

(II-21)

ni SiE �

(II-22)

ni ViE �

(II-23)

i

SE =

� i

VE =

� i

We now take the differential of eq. (19) mathematically. Namely, dG =

� i

�i dni +

� i

ni d�i �

(II-24)

Eq. (II-18) to (II-30)

27

For this equation to be physically consistent with the original eq. (17), the second term on the right of eq. (24) must be identically equal to zero. �

ni d�i = 0�

(II-25)

i

This is the Gibbs-Duhem relation. We make use of this relation later in analyzing vapor pressure data to calculate the excess chemical potentials. Similar deduction for the other excess functions yields, �

ni d�Ei = 0�

(II-26)

ni dHiE = 0�

(II-27)

ni dSiE = 0�

(II-28)

ni dViE = 0�

(II-29)

i

� i

� i

� i

Thus, the changes in (excess) partial molar quantities are not completely independent. These relations are useful in checking the self-consistency in experimental data. For example, the excess partial molar enthalpies of each component in a binary mixture E and HWE , must be related thanks to of ethanol (abbreviated as ET) and H2 O (W), HET eq. (27) as, E xET �HET + xW �HWE = 0�

(II-30)

where xET and xW are the mole fraction of ET and W respectively. Fig. (II-1) shows the E , and that of W, HWE , actually measured by the excess partial molar enthalpy of ET, HET method described in Chapter III (Tanaka et al. 1996; Liltorp et al., 2005). Thus the slopes E and �HWE , are related by eq. (30). The at a given mole fraction shown in the figure, �HET thermodynamic consistency of these data is therefore checked by eq. (30). The results of the left-hand side of eq. (30) are listed in Table II-1. As is evident, the left-hand sides of eq. (30) are indeed zero within ±0�02 kJ mol−1 , which is comparable with the E and HWE , ±0�03 kJ mol−1 . Thus, this set of data is thermodynamically uncertainty of HET E available, eq. (30) consistent. If, on the other hand, there are only the data for HET E E provides the means to estimate HW . Since HW is identically zero at xW = 1, or xET = 0, the increase in HWE � �HWE for a small increment in xET � �xET is given by eq. (30) knowing E E � �HET . Thus, HWE can be calculated stepwise from xET = 0 to the the increment of HET entire composition range.

28

Fig. II-1

2

H ETE

H ETE or H WE / kJ mol-1

0

-2

H WE

-4

-6

-8

-10

-12 0.0

0.2

0.4

0.6

0.8

1.0

xET

Fig. II-1. Excess partial molar enthalpies of ethanol (ET) and H2 O(W) in ET-H2 O at 25  C. The slopes of the tangents (shown by straight lines) at a fixed mole fraction of ET, xET , (indicated by E the dotted line) are related by the Gibbs-Duhem relation, xET �HET + xW �HWE = 0. The data from Tanaka et al. (1996). Table II-1. Gibbs–Duhem for the excess partial molar enthalpy of ethanol (ET) and water (W) in aqueous ethanol at 25 C. xET

E HET kJ/mol

HWE kJ/mol

Error 0 0�01 0�02 0�03 0�04 0�05 0�06 0�07 0�08 0�09 0�1 0�11 0�12 0�13

0�03 −10�11 −9�72 −9�26 −8�71 −8�07 −7�34 −6�52 −5�68 −4�86 −4�1 −3�37 −2�73 −2�2 −1�74

0�03 0 0 −0�01 −0�03 −0�07 −0�1 −0�13 −0�19 −0�24 −0�31 −0�39 −0�47 −0�54 −0�62

xET

Gibbs-Duhem Sum kJ/mol

0�005 0�015 0�025 0�035 0�045 0�055 0�065 0�075 0�085 0�095 0�105 0�115 0�125 0�135

0�002 −0�003 −0�006 −0�016 0�004 0�017 −0�002 0�015 0�001 −0�003 −0�004 −0�001 −0�013 −0�011

29

Table II-1. (Continued) xET

E HET kJ/mol

HWE kJ/mol

xET

Gibbs-Duhem Sum kJ/mol

0�14 0�15 0�16 0�17 0�18 0�19 0�2 0�22 0�24 0�26 0�28 0�3 0�32 0�34 0�36 0�38 0�4 0�42 0�44 0�46 0�48 0�5 0�52 0�54 0�56 0�58 0�6 0�62 0�64 0�66 0�68 0�7 0�72 0�74 0�76 0�78 0�8 0�82 0�84 0�86

−1�37 −1�04 −0�78 −0�54 −0�35 −0�2 −0�08 0�06 0�15 0�19 0�23 0�23 0�23 0�23 0�22 0�21 0�2 0�18 0�16 0�13 0�105 0�08 0�06 0�04 0�02 0 −0�02 −0�04 −0�06 −0�08 −0�09 −0�1 −0�1 −0�1 −0�1 −0�1 −0�1 −0�1 −0�09 −0�08

−0�69 −0�74 −0�8 −0�83 −0�875 −0�9 −0�93 −0�97 −1�01 −1�01 −1�01 −1�01 −1�01 −1�01 −1 −0�99 −0�98 −0�96 −0�94 −0�92 −0�9 −0�89 −0�87 −0�85 −0�82 −0�8 −0�78 −0�76 −0�73 −0�7 −0�68 −0�66 −0�63 −0�6 −0�59 −0�59 −0�59 −0�6 −0�62 −0�65

0�145 0�155 0�165 0�175 0�185 0�195 0�21 0�23 0�25 0�27 0�29 0�31 0�33 0�35 0�37 0�39 0�41 0�43 0�45 0�47 0�49 0�51 0�53 0�55 0�57 0�59 0�61 0�63 0�65 0�67 0�69 0�71 0�73 0�75 0�77 0�79 0�81 0�83 0�85 0�87

0�005 −0�010 0�015 −0�004 0�007 −0�001 −0�002 −0�010 0�010 0�011 0�000 0�000 0�000 0�003 0�003 0�002 0�004 0�003 −0�003 −0�001 −0�007 0�000 −0�001 0�003 −0�003 −0�004 −0�004 −0�001 −0�002 0�000 −0�001 0�009 0�008 0�003 0�000 0�000 −0�002 0�005 0�004 0�002 (Continued)

30

Fig. II-2

Table II-1. (Continued) xET 0�88 0�9 0�92 0�94 0�96 0�98 1

E HET kJ/mol

−0�07 −0�055 −0�04 −0�03 −0�02 −0�01 0

HWE kJ/mol

xET

−0�7 −0�77 −0�85 −0�95 −1�1 −1�26 −1�48

0�89 0�91 0�93 0�95 0�97 0�99

Gibbs-Duhem Sum kJ/mol 0�006 0�006 0�002 0�002 0�005 0�008

[II-3] VAPOR PRESSURES Consider the equilibrium between a liquid and a gas mixture with p and T kept constant. At the equilibrium, �i �gas� = �i �liquid��

(II-31)

This is shown by the following argument. Transfer an infinitesimal amount �ni from the gas phase to the liquid phase, as shown in Fig. II-2. If the entire system is in equilibrium, the change in the total Gibbs function is zero, �G�total� = 0 = �G�gas� + �G�liquid��

(II-32)

�G�gas� = −�G�liquid��

(II-33)

Therefore,

Since, �G�gas� = �i �gas��ni � and, �G�liquid� = �i �liquid��−�ni �� we immediately recover eq. (31). We point out in passing that eq. (31) leads to the Gibbs phase rule. In the system consisting of C-components in equilibrium with P-phases with no chemical reactions among components in all phases, the variance, or the number of degrees of freedom of variables, f , is written as, f = C − P + 2�

(II-34)

Eq. (II-31) to (II-37)

31

Gas phase

p i (i = 1, 2, 3,...,k) δn i

ni (i = 1, 2, 3,...,k)

Liquid phase

Fig. II-2. A k-component mixture is in a closed and isolated system consisting of two phases: gas and liquid. Each phase is open and non-isolated to the other. They can exchange material and energy. pi is the partial pressure of the i-th component in gas phase and ni the amount of the i-th component in liquid mixture.

The deduction is straightforward and most elementary textbooks contain a paragraph devoted to this discussion. �i �gas� and �i �liquid� are rewritten by virtue of eqs. (9) and (12) as, � � p (II-35) �i �gas� = �0i + RT ln i � p0 and, �i �liquid� = �∗i + RT ln xi + �Ei �

(II-36)

with the condition for the equilibrium between the gas and liquid phases, �i �gas� = �i �liquid��

(II-37)

Here, we assume that the gas phase mixture can be treated as an ideal mixture in comparison with the non-ideality of liquid mixture. Since the density of gas is generally 1000-fold smaller than that of liquid, the intermolecular interactions in gas could be

32

Fig. II-3

negligibly small in comparison with those in liquid. This assumption is shown below to be acceptable. Consider now separately the equilibrium between the pure liquid and the pure gas of the i-th component at the same temperature. Instead of eqs. (35) and (36), we have for gas, �

�i �pure� gas� =

�0i + RT

� pi∗ ln � p0

(II-38)

where pi∗ is the vapor pressure of the pure liquid at the same temperature. For liquid, �i �pure� liquid� = �∗i �

(II-39)

with the condition for equilibrium, �i �pure� gas� = �i �pure� liquid��

(II-40)

Combination of eqs. (35) to (40) yields, � �Ei = RT ln

� pi � xi pi∗

(II-41)

This provides the method of evaluating �Ei in the liquid mixture by measuring the equilibrium partial pressure of the i-th component, pi . When non-ideality of the gas phase mixture must be taken into account, eq. (41) needs another term with the virial coefficients of all the components, Bii , together with all the cross terms, Bij in the gas phase. Thus, instead of eq. (41), � �Ei = RT ln

� pi + Ci � xi pi∗

(II-42)

with Ci = �Bii − Vi∗ ��p − pi � + p�1 − pi /p�z�

(II-43)

where z = 2B12 − B11 − B22 for a binary mixture consisting of components 1 and 2. However, for the aqueous solutions of our main interest, the first term on the right-hand side of eq. (42) gives a large non-zero value and the additional virial correction term is negligible. We therefore use eq. (41) for most cases. For aqueous tert-butanol, for example, the excess chemical potential of tert-butanol, �ETBA calculated by eq. (41) was about 6000 J/mol in the H2 O-rich region, while the virial correction term in eq. (42) amounted only to several J/mol. (Koga et al., 1990a; Koga et al., 1990c).

Eq. (II-38) to (II-45)

33

[II-4] RAOULT’S LAW AND HENRY’S LAW For the very special case of an ideal mixture which does not exist in reality, �Ei = 0. Then from eq. (41), pi = xi pi∗ �

(II-44)

This is Raoult’s law, which says that the other components work only as diluents on the vapor pressure of the i-th component as a result of no intermolecular interaction. For such a binary mixture of component 1 and 2, the total pressure is written as, p = p1 + p2 = x1 p1∗ + x2 p2∗ �

(II-45)

Fig. II-3 shows the vapor pressure of an ideal, and hence a non-real, mixture, following eq. (44) and eq. (45). In reality, however, the vapor pressures depend on the mole fraction in a much more complicated manner. See Fig. II-4, for the mole fraction dependence of the vapor pressures for tert-butanol (TBA) – H2 O at 25  C. pTBA and pW are the partial pressures of tert-butanol (TBA) and H2 O(W), respectively. (Koga et al., 1990a) This complex mole fraction dependence is the manifestation of the intermolecular interactions

p 2*

p, p1, or p 2

Ideal mixture 1 - 2

p = p1 + p 2

p2

p1*

p1

0

1

x2

Fig. II-3. Total pressure, p, and the partial pressures over the ideal mixture 1–2. p1 and p2 are the partial pressures of 1 and 2. p1∗ and p2∗ are the vapor pressures of pure 1 and pure 2, respectively, at the same temperature.

34

Fig. II-4

60

50

p, p TBA, or p w / Torr

p = p TBA + p w 40

p TBA

30

20

pW

10

0 0.0

0.2

0.4

0.6

0.8

1.0

x TBA

Fig. II-4. Vapor pressures of a real mixture, for example, tert-butanol (TBA)−H2 O(W) at 25  C. pTBA , and pW are the partial pressures of TBA and W. p is the total pressure. The broken lines would be the partial pressures of TBA and W if the mixture were to be ideal. Note that pTBA is close to the partial pressure of the ideal solution near xTBA = 1, and pW also near xW = 1, or xTBA = 0. The data from Koga et al. (1990a).

occurring in the mixture and is the very key to understanding the molecular mechanisms, as will be discussed throughout this book. We note in Fig. II-4, however, that the partial pressure of H2 O, pW , approaches the broken line in the figure, the hypothetical Raoult’s law vapor pressure, as xTBA → 0. Namely, the other component, TBA, is so dilute that the interactions from TBA do not affect H2 O much, and thus TBA molecules are working just as diluents for H2 O. Thus, the partial pressure of H2 O, pW , in this region is approximated as, ∗ � pW ≈ �1 − xTBA �pW

(II-46)

This approximation will be used to calculate the partial pressures from the total vapor pressure data for a binary system. Whether eq. (46) becomes strictly the equality, ∗ � pW = �1 − xTBA �pW

(II-47)

in a very dilute but a finite concentration range is an unresolved issue. (Koga, 1995a; Westh et al, 1998). If eq. (47) holds at all in a finite range or even in an asymptotic

Eq. (II-46) to (II-54)

35

condition of xTBA → 0 and xW → 1, then due to the Gibbs-Duhem relation, the partial pressure of TBA must also be proportional to xTBA in the same range as, pTBA = kH xTBA �

(II-48)

kH is called Henry’s constant. Observe in Fig. II-4 that for xTBA → 0, pTBA is almost on a straight line against xTBA . The same argument applies for pW in the region of xW → 0, or xTBA → 1.

[II-5] PROCESS OF MIXING – MIXING ENTROPY Let us consider mixing nB mol of B and nW mol of W resulting in the total N�= nB + nW � mol of the mixture with T and p kept constant. Namely, �nB mol of pure B� + �nW mol of pure W� → �Mixture� N = nB + nW ��

(II-49)

The change in the Gibbs energy associated with this process, �mix G, is the difference in the total Gibbs energy between the right and the left of the process (49). Hence, �mix G = G�nB � nW � − �G∗ �nB � + G∗ �nW ���

(II-50)

where G�nB � nW � is the Gibbs energy of the mixture, while G∗ �nB � and G∗ �nW � those of nB of pure B and nW of pure W, respectively. According to eq. (19), G�nB � nW � is written as, G�nB � nW � = nB �B + nW �W �

(II-51)

G∗ �nB � = nB �B ∗ �

(II-52a)

G∗ �nW � = nW �W ∗ �

(II-52b)

and

The chemical potentials are written due to eq. (12) as, �B = �B ∗ + RT ln�xB � + �EB �

(II-53a)

�W = �W ∗ + RT ln�xW � + �EW �

(II-53b)

�mix G = �nB �EB + nW �EW � + RT �nB ln�xB � + nW ln�xW ���

(II-54)

Thus,

36

The above process, eq. (49), is nothing but the defining process of the Gibbs energy of the mixture at p, T, nB and nW , if we choose the pure states of each constituent as the reference, i.e. G∗ �nB � = G∗ �nW � = 0. Hence, G�p� T� nB � nW � = �mix G = GE + GIdeal�

(II-55)

with GE = nB �EB + nW �EW �

(II-56a)

GIdeal = RT �nB ln�xB � + nW ln�xW ���

(II-56b)

The superscript “Ideal” on the left of eq. (56b) is due to the fact that this term remains non-zero, even if the mixture is ideal with �EB = �EW = 0. By differentiating with respect to T , and recalling eq. (I-23), we obtain, � � �G = S E + S Ideal � (II-57) S�p� T� nB � nW � = − �T with, S E = nB SBE + nW SWE �

(II-58a)

S Ideal = −R�nB ln�xB � + nW ln�xW ���

(II-58b)

where �

SBE

��EB =− �T

�

�

� and

SWE

��EW =− �T

� �

(II-59)

from eq. (I-46). By virtue of the definition, G = H–TS, H�p� T� nB � nW � = H E + H Ideal �

(II-60)

H E = nB HBE + nW HWE �

(II-61a)

H Ideal = 0�

(II-61b)

with

where � HBE = −T 2

� � � ����EB /T� ����EW /T� � and HWE = −T 2 � �T �T

(II-62)

Eq. (II-55) to (II-65)

37

from eq. (I-48). From the relation, V = ��G/�p�, eq. (I-24), V = V E + V Ideal �

(II-63)

V E = nB VBE + nW VWE �

(II-64a)

V Ideal = nB VB∗ + nW VW∗ �

(II-64b)

with

where � VBE =

��EB �p

�

� � and VWE =

��EW �p

� �

(II-65)

from eq. (I-47). We note that S Ideal , eq. (58b), is always non-zero for any mixture. Even if the solution is free from interactions and S E is zero, this term is positive since xB and xW are both less than unity. This is called the mixing entropy, arising from the fact that species B and W are distinguishable. Namely when B and W are added together, they spontaneously tend to mix randomly, even if there is no interaction between them. This brings about the finite negative value of GIdeal , eq. (56b); the spontaneity of the process of mixing B and W even if there is no interaction between them. In Statistical Mechanics, the mixing entropy, eq. (58b) is nothing but the increase in accessibility of distinguishable B and W species in a completely random mixture. Due to the choice of the reference states, H Ideal , eq. (61b) is apparently zero. Namely, for an ideal solution lacking in interaction there is no enthalpy of mixing. V Ideal , eq. (64b), on the other hand, is non-zero and is the sum of the volumes of nB of B and nW of W in their pure states. This apparent difference simply comes from the fact that volumes have an absolute scale while energies and enthalpies have not. In reality, the excess functions, GE , eq. (56a), S E , eq. (58a), H E , eq. (61a), and V E , eq. (64a) are non­ zero due to interactions among the constituents of the mixture. Thus, the experimental determination of these excess functions and their higher order derivatives is crucial in obtaining information about the interactions in order to elucidate the mixing schemes or the solution structures.

[II-6] CONVERSION FROM �nB � nW � TO �xB � N� VARIABLE SYSTEMS The excess partial molar quantities, eq. (I-45) to eq. (I-48), are defined in the (p, T , nB , nW ) variable system for a binary mixture of B and W. For experimentalists, however, it is more convenient to use the (p, T , xB , N ) variable system, in order to present the results in a graphical form. Otherwise, the entire composition range from the pure B to

38

the pure W end cannot be realistically covered in the (nB , nW ) scale. Consider now any excess thermodynamic function, F E , in both variable systems. Thus, keeping p, and T constant, F E �nB � nW � = F E �xB � N��

(II-66)

Recalling eq. (1), xB =

nB n = B� n B + nW N

(II-67)

The total differential of F E is identical in each variable system, as long as the condition eq. (67) holds. Hence, dF E �nB � nW � = dF E �xB � N��

(II-68)

and � dF E �nB � nW � = � dF �xB � N� = E

�F E �nB

�F E �xB

�

� dnB +

nW

�

�F E �nW

�

�F E dxB + �N N

� dnW �

(II-69)

nB

� dN�

(II-70)

xB

Inserting eq. (69) and eq. (70) into eq. (68) and dividing both side of eq. (68) by �nB keeping nW constant, i.e. �nW = 0, we obtain, �

�F E �nB

�

� Since from eq. (67), �

� =

nW

�xB �nB

�F E �xB

�

�F E �nB

nW

nW

N

�xB �nB

�

�

nW

�1 − xB � , and N

= �

� �

�1 − xB � = N

�

�F E + �N �

�F E �xB

�N �nB

� � xB

�N �nB

� �

(II-71)

nW

� = 1, eq. (71) is rewritten as, nW

�

�

�F E + �N N

� �

(II-72)

xB

If F E is an extensive quantity, such as GE � H E � S E , and V E , the second term on the right of eq. (72) is identically equal to F E /N which is the molar excess quantity, FmE . However, if F E is an intensive quantity as �Ei , HiE , SiE , and ViE , then the second term of eq. (72) is identically zero. Hence, for an extensive quantity, �

�F E �nB

� nW

�

�FmE = �1 − xB � �xB

� + FmE �

(II-73)

Eq. (II-66) to (II-76d)

39

and for an intensive quantity, �

�F E �nB

�

�1 − xB � N

= nW

�

�F E �xB

� �

(II-74)

These equations, eq. (73) and eq. (74) provide the means to raise the order of derivative of a thermodynamic function by one step in a convenient �xB � N� variable system. For F ≡ H, for example, eq. (73) is rewritten as, �

HBE

�HmE = �1 − xB � �xB

� + HmE �

(II-75)

Hence if HmE = H E /N is experimentally determined as a function of xB , HBE should be calculated by eq. (75). As will be discussed in Chapter III, this route of obtaining HBE , a second derivative of G, is problematic and it is much more advantageous to determine HBE experimentally. Recall eq. (I-53), in which we introduced the thermodynamic interaction functions. Eq. (74) above provides the basis for calculating them in the �xB � N� variable system. For F E = �EB , HBE , SBE , or VBE , we redefine the interaction functions in terms of G, H, S, or V as, � E� � E� ��B ��B � (II-76a) = �1 − xB � �EB−B ≡ N �nB �xB � HBE−B ≡ N � SBE−B ≡ N � VBE−B ≡ N

�HBE �nB

�SBE �nB

�

� = �1 − xB �

�

�VBE �nB

� = �1 − xB �

�

�SBE �xB

� = �1 − xB �

�HBE �xB

� �

(II-76b)

�

�VBE �xB

�

(II-76c)

� �

(II-76d)

As the definitions imply, these interaction functions signify the effect of a unit increase of B on �EB , HBE , SBE , or VBE , the actual situation of existing B in the mixture. Hence they represent the B-B interaction. Due to the stability criteria discussed in [I-4], the sign of the interaction function, eq. (76), indicates that an addition of B results in a favorable or a non-favorable situation of existing B in terms of each quantity. We call the B-B interaction “attractive” or “repulsive” for short. Thus, the B-B interaction is attractive (or repulsive) if �EB−B and HBE−B are negative (or positive) in terms of chemical potential and enthalpy, respectively. The statements hold for SBE−B except that the sign is reversed. Since there is no stability criterion for volume, VBE−B simply shows the effect of an additional B on the volumetric situation of B.

40

We use the word “interaction” in accordance with the sense conveyed by eq. (76) above. Hence, it is not necessarily limited to such conventional meanings as the potential or the force field. The latter and many other effects are included in a holistic manner in our usage. So far, we have dealt only with a binary mixture (B, W). Generalization of the above discussion to a multi-component system �1� 2� 3� � � � � k� is straightforward following the route starting at eq. (66) and keeping track of which variables are kept constant. Thus an equivalent expression for eq. (72) is written as, �

�F E �n1

� = n2 �n3 � � � � �nk �nk+1

�1 − x1 � N �

�F E + �N

�

�F E �x1

� − x2 �x3 � � � � �xk

� � k � � xm � �F E �xm xj=m m=2 N

� �

(II-77)

xi

[II-7] INTERACTION FUNCTIONS DUE TO THE IDEAL MIXING ENTROPY As we have discussed so far, the excess functions, GE � H E � S E , and V E contain the information about the solute-solute interaction in real mixtures crucial for understanding the underlying molecular processes – mixing schemes. However, there is the ideal mixing entropy that dictates the complete random mixing even in the absence of the excess entropy term, S E . The ideal mixing entropy, S Ideal , is written due to eq. (58b) as, S Ideal = −R�nB ln�xB � + nW ln�xW ���

(II-78)

and its molar quantity, Smid , is Smid = −R�xB ln�xB � + xW ln�xW ���

(II-79)

Then the partial molar ideal mixing entropy of B, SBid , can be written following eq. (72) as, � SBid = �1 − xB �

� �Smid + Smid �xB

= −Rln�xB ��

(II-80)

Thus, the B-B interaction in terms of the ideal mixing entropy, SBid−B could be calculated by eq. (74) as, SBid−B = −R

�1 − xB � � xB

(II-81)

Eq. (II-77) to (II-83)

41

This SBid−B remains negative in the entire composition range. Namely, SBid−B drives B molecules apart though its degree is small except for the region sufficiently near xB = 0. Similar arguments lead to the B-B interaction in terms of the ideal chemical potential, �id B−B , written as, �id B−B = RT

�1 − xB � � xB

(II-82)

which is always positive, indicating repulsion in terms of ideal chemical potential. This effect is dominant only at a very dilute region over that by the excess functions. The E E contributions from SBid−B and/or �id B−B over corresponding SB−B and/or �B−B will be discussed when we determine the latter interaction functions for actual aqueous solutions of non-electrolytes in Part B below.

[II-8] PHASE SEPARATION – CRITICAL POINT (UCST OR LCST) In this and the next section we discuss the topics that will be useful for Part B. At constant pressure and temperature, the thermodynamic stability criterion for G is dG ≤ 0, eq. (I-34) As will be discussed in Chapter IV, the heat capacity, Cp , and the isothermal compressibility, �T , can never be negative. This is completely consistent with human experience and is often called the mechanical stability criteria. Another inequality, called the stability criterion, is applicable in a binary mixture (B, W) such that � diffusional � � ��B �xB must never be negative. This together with the mechanical stability criteria are supported by the fact that these quantities are related to the mean square fluctuations, as discussed in Chapter IV, i.e. they are never negative, being squared quantities. Namely, �

��B �xB

� ≥ 0�

(II-83)

Note, however, that the xB derivative of the excess chemical potential of B, �EB , could be either positive or negative. Indeed, this is related to the B-B interaction in terms of excess chemical potential, �EB−B , as shown in eq. (76a). The sign of �EB−B is an important piece of information for understanding that the B-B interaction is favorable or not in terms of excess chemical potential, as will be used repeatedly throughout this book. Eq. (83) is concerned with the sign of the derivative to the total chemical potential, not the excess chemical potential. The difference is, of course, the term due to the mixing entropy, as discussed above in Section [II-7]. When the interaction among the constituents becomes sufficiently strong in a binary (B, W) mixture, we may observe a phase separation consisting of two liquid phases; one rich in B and the other in W. Under a constant pressure condition, the region of phase separation is depicted in the temperature-mole fraction field as shown in Fig. II-5. Two typical cases are given in Fig. 5(a) and 5(b). The former, Fig. 5(a), is

42

Fig. II-5(a) to II-5(c)

(a) Single phase mixture

UCST

T

Two Liquids

XB

Fig. II-5(a). Phase separation with the upper critical solution temperature (UCST). The necessary but not sufficient conditions for UCST to occur are HBE−B < 0, and SBE−B < 0. (See text; see also Figs. II – 7(a) and 7(b).)

(b)

Two Liquids

T LCST

Single Phase Mixture

XB

Fig. II-5(b). Phase separation with the lower critical solution temperature (LCST). The necessary but not sufficient conditions for LCST to occur are HBE−B > 0 and SBE−B > 0. (See also Figs. II – 8(a) and 8(b).)

the case in which phase separation occurs at low temperatures with a critical point. The temperature above which the mixture forms a single phase is called the upper critical solution temperature (UCST). This case is easily understandable in that a strong attraction between like species in terms of enthalpy would bring a phase separation at

Eq. (II-84)

43

70

(c)

60

temperature / °C

50

2-butanone - H2O

40 30

Two liquids 20 10 0

0.0

0.2

0.4

0.6

0.8

1.0

x BUT

Fig. II-5(c). Liquid phase diagram for 2-butanone (BUT) – H2 O. Phase boundaries are due to Felino et al. (1983). Freezing prevents observation of LCST, which is estimated from −22 to −6  C (Francis, 1961). (See also Wong et al. (1992).)

low temperatures, while at higher temperatures the total entropy effect drives the system to a random mixture. Fig. 5(b) shows the case in which phase separation occurs at high temperatures with the lower critical solution temperature (LCST). This case is not so easily explained as the UCST case above. However, the following argument using the diffusional stability criterion, eq. (83), provides a general understanding for both the UCST and the LCST cases. Since �B = �B ∗ + RT ln�xB � + �EB , eq. (12), eq. (83) is rewritten as, � � � E� RT ��B ��B = + ≥ 0� (II-84) �xB xB �xB The first term on the right of eq. (84) is always positive. If the second term is positive, i.e. ���EB /�xB � > 0, then the inequality eq. (83) or eq. (84) is always satisfied and the system is in a single phase. However, if the second term is sufficiently negative, i.e. ���EB /�xB � T , the point Q is dxB UCST, and if TC < T , point Q is LCST. The scale of ordinate is arbitrary.

45

a van der Waals fluid (Chapter XVI in Prigogine & Defay, 1954), the area encircled by AMBA and that by DMCD are the same. We note that points B and C, the edge of meta-stability, are called the spinodal points. We also note that at a certain condition (temperature), eq. (83) is always satisfied, but at point Q in Fig. 6(b), ���B /�xB � = 0. This is the critical point, the end point of phase separation. Thus, a necessary condition for phase separation to occur is that ���EB /�xB � be suffi­ ciently negative. For this to occur, there are two simple possibilities depicted in Figs. II-7 and II-8. For Case I, Fig. 7(a), HBE > TSBE with both ��HBE /�xB � and ��SBE /�xB � are negative, hence ���B /�xB � < 0 with the possibility of phase separation. However, if temperature increases sufficiently, then a situation may arise such that HBE < TSBE as shown in Fig. 7(b). Hence ���EB /�xB � > 0 and there is no phase separation. Namely, phase separation could occur at low temperatures with a UCST. In this case, the B-B interactions introduced by eq. (76) are attractive in terms of enthalpy, but repulsive in entropy. This is consistent with the intuitive discussion above in that the attractive enthalpic B-B interaction dominates at low temperatures and provides the possibility for phase separation. At high temperatures the repulsive entropic B-B interaction coming from the excess entropy, together with that from the ideal mixing entropy, dictates that the system mixes well. Case II, Fig. 8, shows the opposite situation, whereby the B-B interaction is repulsive in terms of enthalpy but attractive entropy-wise. At high temperatures, HBE > TSBE and hence ���EB /�xB � < 0, as shown in Fig. 8(a), providing the possibility of phase separation. At sufficiently low temperatures, however, HBE < TSBE resulting in ���EB /�xB � > 0 with no chance for phase separation. Thus, if phase separation is to occur, it must be with an LCST, as shown in Fig.5(b). In summary, the system with a UCST, Fig. 5(a), is consistent with the case HBE−B < 0 and SBE−B < 0. We note that the phase boundary in the low xB side slants with the slope dT/dxB > 0, with dHBE /dxB < 0, and dTSBE /dxB < 0. The system with an LCST, Fig. 5(b), on the other hand, has the situation HBE−B > 0 and SBE−B > 0 with the phase boundary slanting with a negative slope, dT/dxB < 0, with dHBE /dxB > 0, and dTSBE /dxB > 0. There are many binary systems with various types of phase diagrams and complete generalization is not possible. There are cases including aqueous 2-butanone. As shown in Fig. II-5(c), there appears to be an LCST, but it is hidden by the formation of solid phases. In this and more general cases, it is possible to draw the relationships between the sign of HBE−B and SBE−B in a single phase of small xB side with the sign of the slope of the phase boundary, �dT/dxB �, with the two phases on the larger xB side. The results are consistent with the discussion above; namely, if HBE−B > 0 and SBE−B > 0 the phase boundary slants with a negative slope, dT/dxB < 0, but if HBE−B < 0 and SBE−B < 0, then the phase boundary has a positive slope, dT/dxB > 0. Note, however, the inverse is not necessarily true. We will not repeat the discussion here; readers may refer to Wong et al. (1992). The UCST and the LCST are the critical points and have specific characteristics of the so-called critical phenomena (Stanley, 1971). As with any other critical points at the super critical region in a single phase domain, the solution shows a large fluctuation

46

Fig. II-7(a) to II-8(b)

H BE and TS BE

H BE

(a) Low temperature Possible phase separation

TS BE

µ BE 0 0

1

xB

Fig. II-7(a). The case in which HBE > TSBE at low temperature, resulting in the required behavior of �EB for possible phase separation. Note that HBE−B < 0 and TSBE−B < 0.

H BE and TS BE

TS BE

(b) High temperature No phase seperation

H BE

0

µ BE 0

1

xB

Fig. II-7(b). The case in which HBE < TSBE at high temperature resulting in the behavior of �EB which gives no chance for phase separation. Note HBE−B < 0, and SBE−B < 0.

47

µ BE, H BE, and TS BE

µ BE 0

H BE

TS BE (a) High temperature

Possible phase separation

0

1

xB

Fig. II-8(a). The case in which HBE > TSBE at high temperature resulting in the required behavior of �EB for phase separation. Note that HBE−B > 0 and SBE−B > 0. 0

µ B E, H BE, and TS BE

µ BE

TS BE

H BE (b)

Low temperature No phase separation

0

1

xB

Fig. II-8(b). The case in which TSBE > HBE resulting in the behavior of �EB which gives no chance for phase separation. Note HBE−B > 0, and SBE−B > 0.

48

Fig. II-9

in xB which diverges to infinity at the critical point. Namely the solution in the super critical region consists of two kinds of clusters; one rich in B and the other in W. These clusters are nanoscopic in size but their size grows as the condition approaches that of the critical point and eventually diverges to infinity at the critical point. Within the two-phase region then, these clusters acquire macroscopic size and the system becomes a two-phase liquid mixture. Indeed, scattering experiments reveal dramatically enhanced concentration fluctuations in the super critical tert-butanol−H2 O (with a possible LCST) (Koga, 1984; Nishikawa et al., 1987 and 1989) and acetonitrile−H2 O with the UCST (Nishikawa et al., 2002).

[II-9] AZEOTROPE The azeotrope occurs when the liquid phase composition and the gas phase composition are identical at the equilibrium, or the total vapor pressure as a function of xB shows an extremum, as depicted in Fig.II-9, i. e. dp/dxB = 0 with p = pB + pW . It follows then that, �

dp dxB

�

� =

� � � dpB dpW + = 0� dxB dxB

(II-85)

p

Azeotrope

xB

Fig. II-9. The azeotrope, where the total pressure has the extremum, dp/dxB = 0. It can be shown that the composition of the vapor phase and that of the liquid phase are identical, i. e. xB /xW = pB /pW .

Eq. (II-85) to (II-87)

49

Taking into account the Gibbs-Duhem relation. eq. (26) and the expression for the chemical potential using the partial pressure, eq. (41), eq. (85) is rewritten as, �

dp dxB

�

� =

dpB dxB

��

xB xW − pB pW

� = 0�

(II-86)

Recall that at a critical point �d�B /dxB � = 0 and hence �dpB /dxB � = 0. Thus, unless the system is at a critical point, the following relation holds at an azeotrope, xB p = B� xW pW

(II-87)

This occurs quite commonly in aqueous alcohols. As discussed in Chapter III, this situation is problematic in analyzing the vapor pressure data to obtain chemical potentials by the Boissonnas method (1939). We will circumvent this difficulty by using the concentration fluctuation data accessible from scattering experiments.

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Chapter III

DETERMINATION OF THE PARTIAL MOLAR QUANTITIES

[III-1] INTRODUCTION While a vast amount of the excess (integral) molar enthalpy data of mixtures exists in the literature, the excess partial molar enthalpy of a component has been rather rarely used in thermodynamic studies, despite its obvious power of providing more detailed information of intermolecular interactions. Arnett et al. (1965a, 1965b, 1966 and 1969) determined the differential molar heats of solution of various alcohols and amines, which are the same as the excess partial molar enthalpies of the solute at the infinite dilution. From their temperature dependence, the heat capacities of solution were calculated at the infinite dilution. This quantity is a third derivative of G and has already been shown to be powerful in providing evidence for the so-called “iceberg” formation. Unfortunately, however, the determination was limited to the infinite dilution. Bertrand et al. (1966), on the other hand, determined the excess partial molar enthalpies of methanol and ethanol in the respective binary aqueous solution for the entire composition range. They used the data for calculating the heats of neutralization of H+ + OH− → H2 O in the mixed solvent of aqueous alcohol. The potential usefulness of these partial molar enthalpy data for understanding the molecular processes in aqueous solutions was not explored until the late 1980s (Koga, 1986; Anderson and Olofsson, 1988). Thus, it may be of some use to review in this chapter the method of determining partial molar enthalpy, together with partial molar entropy and partial molar volume. First, however, we discuss the relationship between the excess (integral) molar enthalpy, HmE , that has been commonly measured, and the excess partial molar enthalpy of the i-th component, HiE , in the way of conversion of the variable system.

[III-2] CALCULATION OF HiE FROM HmE Fig. III-1 shows the plots of the excess (integral) molar enthalpy, HmE , data for ethanol(ET)−H2 O by Costigan et al.(1980). Recall eq. (II-75), which provides the E relationship between the excess partial molar enthalpy of ethanol (or H2 O), HET (or HWE ), and HmE .

52

Fig. III-1 to III-2

Ethanol(ET) - H2 O at 25 °C

0

H mE

R

P

-200

HET E

H wE

-400

-600

Q

-800

-1000 0.0

0.2

0.4

0.6

0.8

1.0

xET

Fig. III-1. The excess molar enthalpy, HmE , of ethanol (ET)−H2 O(W) at 25  C, according to E Costigan et al., (1980). The excess partial molar enthalpy of ET, HET , at point P is the value of the intercept at xET = 1 of the tangent at point P. The excess partial molar enthalpy of W, HWE , is the value of the intercept at xET = 0 of the tangent at point P. Reproduced with permission from the Australian Journal of Chemistry 33(10):2103–2119 (Costigan, MJ et al.). Copyright CSIRO 1980. Published by CSIRO PUBLISHING, Melbourne, Australia – http://publish.csiro.au/ nid/52/issue/3652.htm.

�

E HET

=

HmE

�HmE + �1 − xET � �xET �

HWE = HmE + �1 − xW �

�HmE �xW

� �

�

(III-1a) �

= HmE − xET

�HmE �xET

� (III-1b)

E What eq. (1) indicates graphically is that the value of HET at point P in the figure is the E value at the intercept at the xET = 1 axis, R, while HW is that at the xET = 0 axis, Q. The line QPR in the figure is the tangent at point P. If we know the function form of HmE , it is E analytically. Unfortunately, however, it is generally theoretically possible to obtain HET difficult to find the exact and true function form covering the entire composition range. The goodness of curve-fitting with a trial function form is judged by the sum of the square deviations between the experimental values and the fitted values. When interest lies in a dilute, H2 O-rich region, for example, the values of HmE are close to zero. For these data points the absolute deviation could be small in comparison with the middle composition region, but the relative deviations could be systematic and substantial. This difficulty in the dilute region could give a devastating error when differentiated

Eq. (III-1a) to (III-2)

53

analytically. Ott et al. (1986) devised the following fitting function to the HmE data for ET−H2 O determined by Costigan et al. (1980), shown in Fig. III-1. HmE = x�1 − x��exp�−ax��bj �1 − x�j + �1 − exp�−ax���ck �1 − x�k ��

(III-2)

This function turned out to be quite successful in reproducing the excess partial molar E E . By virtue of eq. (1) and eq. (2), the calculated values of HET , enthalpy of ethanol, HET with j = 0 to 2, and k = 0 to 4, are shown in Fig. III-2 together with our data (Tanaka et al. 1996) measured directly by the method described in the next section. As is evident from Fig. 2, the match is satisfactory. Such success in finding the correct function for HmE in the entire composition range is quite rare. We note that eq. (2) contains the exponential term, which seems to work as a switching function from the first term to the second term dominance. We have applied a similar equation to eq. (2) using the exponential term to the HmE data of the 2-butoxyethanol �BE�−H2 O carefully determined by Davis et al. (1988) without success. It appears that a much stronger switching function is required for BE−H2 O. Indeed, Davis (1983, 1984, 1985) devised the four segment model in which four different fitting functions are used in each of four mole fraction regions. Douheret et al. (1989) thus fitted a total of 13 parameters including the segment boundaries with reasonable success, 2

0

HET E / kJ mol-1

-2

-4

-6

-8

Measured -10

-12 0.0

0.2

0.4

0.6

0.8

1.0

xET

Fig. III-2. The excess partial molar enthalpy of ethanol (ET) in ET−H2 O at 25  C. The hollow circles are the measured by Tanaka et al. (1996). The line is calculated by curve-fitting by Ott et al. (1986) of the HmE data by Costigan et al. (1980). A rare case of successful curve-fitting to E obtain reasonable values of HET in the entire composition range (see text). The measured data from Tanaka et al. (1996). Reproduced with permission from the Canadian Journal of Chemistry 74, 713, by Tanaka et al., Copyright (1996), with permission from NRC Research Press.

54

Fig. III-3 to III-4

100

BE - H2 O at 25 °C 0

Measured Line - 4 segment fit Broken line - fit by eq(III-3)

HmE / J mol-1

-100

-200

-300

-400

-500 0.0

0.2

0.4

0.6

0.8

1.0

xBE

Fig. III-3. The excess molar enthalpy, HmE , of 2-butoxyethanol �BE�−H2 O against mole fraction of BE, xBE , at 25  C. The open circles indicate the data by Davis et al. (1988). The line is the four-segment fit by Douheret et al. (1989). Four different fitting functions were used bounded at about the arrows to cover the entire composition range. The broken line is fitted by eq. (III-3). (See Koga(2004a) for details.) Reproduced with permission from 3.1.3 Determination of Partial Molar Quantities, (Koga, Y) in Comprehensive Handbook of Calorimetry and Thermal Analysis, (Sorai, M. Editor-in-chief), Copyright (2005), John Wiley & Sons.

as shown in Fig. III-3. There are, however, small but definite systematic errors in the seg­ E ment boundaries. These systematic errors will lead to incorrect values of HBE . This effect is shown in Fig. III-6 in the next section. We have also tried another fitting function of the all-purpose Abbott-Van Ness type (Van Ness & Abbott, 1982) for the BE−H2 O data, � i

E Hm

ax (III-3) = � i j � bj x x�1 − x� E The resulting HBE data were not satisfactory in the H2 O-rich region, as shown in Fig. III-6 below (see Koga (2004a) for details). Thus, it would be much simpler to determine the excess partial molar enthalpy experimentally.

[III-3] EXPERIMENTAL DETERMINATION OF EXCESS PARTIAL MOLAR ENTHALPY The excess partial molar enthalpy of the i-th component is, by definition, eq. (I-48), the slope of the tangent drawn on the curve H E vs. ni at point P in Fig. III–4. Experimentally,

Eq. (III-3)

55

n j (j other than i) are fixed

R

HE

δH E/δn i H iE P Q

δn i /2

δni /2 ni

� E� Fig. III-4. Approximation of the slope of tangent at point P, HiE = �H , by the slope of �ni n j=i � E� (see text). Reproduced with permission from 3.1.3 Determination of Partial the cord QR, �H �ni Molar Quantities, (Koga, Y) in Comprehensive Handbook of Calorimetry and Thermal Analysis, (Sorai, M. Editor-in-chief), Copyright (2005), John Wiley & Sons.

however, the amount of the i-th component, ni , must be perturbed by a small but finite amount �ni and the resulting change in the excess enthalpy, �H E is determined at the con­ stant pressure. This corresponds to approximating the slope of the tangent at P with the slope of the cord QR in Fig. 4. The goodness of this approximation could be checked by reducing the size of �ni and seeing if the resulting quotient �H E /�ni converges to a constant value. This converged value could be safely assumed to be the desired slope of the tangent. Fig. III-5 is a sketch of the calorimeter for this purpose according to Wadso (1966). It is commonly called an isoperibolic titration calorimeter. It is operated under a semi-adiabatic condition. The air space separating the cell and the water bath, the temperature of which is controlled within ±0�001 C, serves as a semi-adiabatic shield. A small amount of heat leak, however, is monitored by a thermal sensor and corrected for. The target component is titrated into the cell containing the mixture. Generally, for non-electrolyte−H2 O systems, a titre of less than 0.02 mol is sufficiently small for the above approximation if the mix­ ture in the cell is about 5 mol. There are a number of commercial titration calorimeters that use a similar or smaller titrant to titrand ratio. The thermal response of the cell on titra­ tion is picked up by a thermal sensor which is separately calibrated by a known amount of electric energy. For the calorimeter shown in Fig. 5 a thermistor of 2 k� is satisfactory. The process of measurement to titrate �nB of B into a solution made of nB of B and nW of W can be described as, �nB of B + nW of W� + ��nB of B� → ��nB + �nB � of B + nW of W�

56

Fig. III-5 to III-6

Dry nitrogen

Buret

Heat exchanger

Bath Heater

Thermistor

Bath Air space

Fig. III-5. An isoperibolic titration calorimeter. Cell contains about 100 mL of titrand into which about 0.5 mL of a titrant is delivered. A thermistor of about 2 k� and a standard heater calibrate the thermal effect of titrant.

The thermal effect �qp associated with the process is written as, �qp = H�nB + �nB � nW �−H�nB � nW � − �nB HB ∗ �

(III-4)

where H is the total enthalpy and HB ∗ is the molar enthalpy of pure B. It follows then, �qp H�nB + �nB � nW � − H�nB � nW � = − HB ∗ � �nB �nB

(III-5)

and for a sufficiently small �nB , �qp = �nB

�

�H �nB

� − HB ∗ = HBE � nW

(III-6)

Eq. (III-4) to (III-6)

57

0 -2

HBE E / kJ mol-1

-4 -6

(a)

-8 -10

Measured

-12

Line - calc. by 4 segment fit -14

Broken line - Abbott/vanNess fit

-16 -18 0.00

0.02

0.04

0.06

0.08

0.10

0.8

1.0

xBE 1

HBE E / kJ mol-1

0

-1

(b)

-2

Measured Line - calc by 4-segment fit

Broken line - calc. by eq.(III-3) -3

-4 0.0

0.2

0.4

0.6

xBE

Fig. III-6. The excess partial molar enthalpy of 2-butoxyethanol (BE) in BE−H2 O at 25  C. The open circles are the measured data by Siu & Koga (1989). The line is calculated by the foursegment fit by Douheret et al. (1989). Broken line is calculated using the Abbott-van Ness fitting, E eq. (III-3). Clearly both fittings are not satisfactory for calculating HBE , although the fit for HmE looks almost acceptable (see Fig. III – 3).

If we titrate W instead of B, we obtain HWE . Fig. III-6 shows the results of such E , the excess partial molar measurements for 2-butoxyethanol�BE�−H2 O at 25  C, HBE E enthalpy of BE. For comparison, the calculated values of HBE (Douheret et al. 1989) E E using the four segment fit for Hm (Davis et al. 1988) and the values of HBE calculated using eq. (3) (Koga, 2004a) are also shown.

58

Fig. III-7

[III-4] EXPERIMENTAL DETERMINATION OF EXCESS PARTIAL MOLAR VOLUME The method described for determining the excess partial molar enthalpy is completely applicable to volume. Thus, the volume change �V is determined on perturbing the amount of the i-th component. Namely, for the process, �nB of B + nW of W� + ��nB of B� → ��nB + �nB � of B + nW of W�� the volume change �V is written equivalently to eq. (6) above as, �V = �nB

�

�V �nB

� − VB ∗ = VBE �

(III-7)

nW

where VB ∗ is the molar volume of pure B. �V could be determined directly by dilatometer (Davies et al., 1994). Alternatively, if the density data of the solution are determined accurately with five to six significant figures, using for example an automatic vibrating tube densimeter (Dethlefsen, 1984; Dethlefsen et al., 1984), then the excess partial molar volume can be determined by using the relation, � VBE = VmE + �1 − xB �

�VmE �xB

� �

(III-8)

The derivative of the second term on the right is taken by graphical differentiation (Koga et al., 1993). The pros and cons of graphical and manual differentiation as opposed to the computer garphics technique using B-spline is discussed in Appendix A.

[III-5] EXCESS PARTIAL MOLAR ENTROPY – EXCESS CHEMICAL POTENTIAL In theory, the excess partial molar entropy of the i-th component can be determined by measuring the resulting entropy change of the system on perturbing the amount of the target component i. In practise, however, the direct experimental determination of entropy has been difficult, if not impossible. Instead, we determine the excess chemical potential of the i-th component, �Ei , and take advantage of the relation, �Ei = HiE − TSiE �

(III-9)

to calculate SiE . Eq. (II-41) shows the way to obtain the chemical potential of a component from its equilibrium partial pressure, pi , i.e.

Eq. (III-7) to (III-10)

59

� �Ei = RT ln

� pi � xi pi∗

(III-10)

where pi∗ is the vapor pressure of pure i at the same temperature. The first step that we take is to determine the total vapor pressure of the head space gas mixture over a given liquid mixture directly. For this purpose capacitance manometers can be used to advantage. A typical sensitivity is 1 mTorr with a full scale 100 Torr. The temperature variation of the sample can be corrected for by the Gibbs-Konovalov correction, shown in Appendix B. Thus, the total pressures are obtained at a fixed temperature, 298.15 K, for example. A simple vacuum system capable of pumping down to 10−6 Torr made of Pyrex glass or stainless steal is sufficient. A typical vapor pressure apparatus is shown in Fig. III-7. The region encircled by a broken line is an air bath, the temperature of which is kept higher than the temperature of the cells, in order to avoid gas condensation. For the same purpose, tape heaters are wound around the tubes connecting cells to the measuring system, in order to keep the temperature of the connecting tubes higher than that of the cells. An amount of each component can be measured in the gas handling manifold with the known volume, temperature and pressure, then transferred Vaccum line Capacitance Manometer Air Bath 5 L bulb 0.5 L

Sample 3

Sample 1

Sample 2 Band Heater

Bellow Valves Cell Water bath Up/Down

Fig. III-7. Schematic diagram of vapour pressure apparatus (see text).

60

to the cell by cooling the latter with liquid nitrogen. During this operation the water bath is lowered. The bath is then raised; the liquid mixture is thawed and mixed with a magnetic spin bar. After thermal equilibration, the pressure is measured by a capacitance manometer. After determining the equilibrium pressure, the gas phase sample may be transferred back to the cell by freezing the cell with liquid nitrogen, or may be sent to gas chromatographic analysis to determine the gas phase composition. For the latter operation, the amount of gas phase lost must be recorded and corrected for the subsequent determination of the mole fraction. For binary systems, numerical analysis based on the Gibbs-Duhem relation is possible due to Boissonnas (1939) without gas analysis. For some ternary systems a similar numerical analysis is also possible. The total pressures of the ternary system must behave monotonously as a function of all the composition variables without showing any extremum. We will show in Chapter IX such an analysis for a tert-butanol−dimethyl sulfoxide−H2 O system (Trandum et al., 1998).

[III-6] BOISSONNAS ANALYSIS – EXCESS CHEMICAL POTENTIAL Table III-1 lists the vapor pressure data at 293.15 K for acetone(AC)−H2 O, determined by Perera et al. (2005). The total pressure, p, increases monotonously as a function of the mole fraction of AC, xAC . This is an ideal situation for the following analysis, as will become evident shortly. Our task is to evaluate the partial pressures of AC and H2 O, pAC and pW , from the data of p against xAC . The Gibbs-Duhem relation, eq. (II-26), is described for the present binary system as, xAC ��EAC + xW ��EW = 0�

(III-11)

Using partial pressures, eq. (11) is rewritten as, � xAC

�pAC pAC

�

� + xW

�pW pW

� = 0�

(III-12)

With p = pAC + pW , and xAC + xW = 1, eq. (12) is rewritten by eliminating �pW as, �pAC = �p�AC �

(III-13)

where �AC is what we call the Boissonnas coefficient and is defined as, �AC = 1/�1 − �pW /pAC ��xAC /xW ���

(III-14)

If we know the values of pAC , pW , xAC and xW at one data point, then the increment in pAC for the next point is calculated by eq. (13) by measuring the change in the total pressure, �p. We repeat the process stepwise until all the data points are solved. To start this iteration, however, we have to assume that at the most dilute data point, the

Eq. (III-11) to (III-14)

61

Table III-1. Vapor pressure analysis by the Boissonnas method Acetone (AC) - H2O (W) at 20.00 degC Perera et al. (2005). Starting at xAC = 0 xAC units

p(meas) Torr

pW Torr

pAC Torr

0 0�00791 0�01491 0�02173 0�02883 0�03604 0�0433 0�05147 0�06048 0�07033 0�08089 0�09192 0�10308 0�1134 0�12346 0�13177 0�15353 0�18451 0�22605 0�27538 0�32203 0�35752 0�41168 0�50181 0�59569 0�6981 0�75982 0�81295 0�86031 0�90702 0�9421 0�97658 0�99336 0�99778 1

17�527 25�497 32�293 38�675 45�012 51�256 57�251 63�592 70�191 76�932 83�671 89�915 95�725 100�631 102�053 105�112 112�145 120�051 128�446 135�746 140�955 143�956 148�237 153�642 158�513 163�72 167�13 170�43 173�883 177�659 180�658 183�588 185 185�34 185�474

17�527 17�388 17�270 17�157 17�043 16�928 16�815 16�694 16�564 16�427 16�286 16�150 16�018 15�902 15�867 15�786 15�592 15�354 15�065 14�771 14�517 14�344 14�063 13�635 13�106 12�308 11�519 10�510 9�129 7�173 5�085 2�474 0�605 0�277 0

0�000 8�109 15�023 21�518 27�969 34�328 40�436 46�898 53�627 60�505 67�385 73�765 79�707 84�729 86�186 89�326 96�553 104�697 113�381 120�975 126�438 129�612 134�174 140�007 145�407 151�412 155�611 159�920 164�754 170�486 175�573 181�114 184�395 185�063 185�474

Starting at xW = 0 fai(AC)

1�01739 1�0177 1�01803 1�01842 1�01878 1�01918 1�0197 1�02029 1�02097 1�02173 1�02267 1�02364 1�0246 1�02662 1�02756 1�03017 1�03432 1�04038 1�04866 1�05768 1�06563 1�07915 1�10877 1�15313 1�23149 1�30579 1�39981 1�51803 1�69613 1�89133 2�32334 1�96534

xW 1 0�99209 0�98509 0�97827 0�97117 0�96396 0�9567 0�94853 0�93952 0�92967 0�91911 0�90808 0�89692 0�8866 0�87654 0�86823 0�84647 0�81549 0�77395 0�72462 0�67797 0�64248 0�58832 0�49819 0�40431 0�3019 0�24018 0�18705 0�13969 0�09298 0�0579 0�02342 0�00664 0�00222 0

pW Torr −3�850 −3�839 −3�826 −3�811 −3�795 −3�779 −3�760 −3�738 −3�715 −3�690 −3�665 −3�641 −3�619 −3�612 −3�597 −3�558 −3�507 −3�444 −3�377 −3�321 −3�284 −3�221 −3�115 −2�986 −2�792 −2�632 −2�441 −2�195 −1�850 −1�497 6�634 0�984 0�278 0�000

pAC Torr

fai(W)

29�347 0�00161 36�132 0�002 42�501 0�00231 48�823 0�00257 55�051 0�00279 61�030 0�00302 67�352 0�00324 73�929 0�00347 80�647 0�0037 87�361 0�00395 93�580 0�00419 99�366 0�00442 104�250 0�00479 105�665 0�005 108�709 0�00555 115�703 0�00638 123�558 0�00757 131�890 0�00914 139�123 0�01081 144�276 0�01226 147�240 0�01466 151�458 0�01962 156�757 0�02652 161�499 0�03732 166�512 0�04676 169�762 0�05783 172�871 0�0713 176�078 0�09136 179�509 0�11792 182�155 2�77502 176�954 −4�0016 184�016 −2�07583 185�062 185�474

62

Fig. III-8 to III-9

second point (xAC = 0�007907 for the AC−H2 O vapor pressure data listed in Table 1), the system is in the Henry’s law region. Hence, the major component W must obey ∗ ∗ �1 − xAC �, where pW is Raoult’s law, as discussed in Section [II-4]. Namely, pW = pW ∗  the vapor pressure of pure H2 O at 20 C, i, e. pW = 17�527 Torr. Then, pAC = p − pW . Thus the second data point is now solved. Using eq. (13) the third and the subsequent data points are solved point by point. The results are shown in the third and fourth columns of Table 1, and plotted in Fig. III-8. For the last point at xAC = 0�9978, the table shows pW = 0�277 Torr and pAC = 185�063 Torr, which are favorably compared with pW = 0�278 Torr and 185.062 Torr calculated by assuming Henry’s law in this most AC-rich point, since the sensitivity of the pressure determination of the gauge used is ±0�001 Torr. Hence, we conclude the analysis was satisfactory. We note that the values of �AC in column 5 are about 1. Hence the measured increment in p is carried almost directly to the increment of pAC . This means that the uncertainties in �pAC are

200

150

p / Torr

Total p pW

100

p AC

50

0 0.0

0.2

0.4

0.6

0.8

1.0

xAC

Fig. III-8. Vapor pressures of acetone (AC)−H2 O at 20  C. The total pressure (measured) increases monotonously as the mole fraction of AC, xAC , increases. Thus, the Boissonnas analysis is possible for the entire composition range. The data are according to Perera et al. (2005). (see text.) Reproduced with permission from the Journal of Chemical Physics 123, 024503 (Perera, A. et al.), Copyright (2005), American Institute of Physics.

63

about the same as those in �p. The excess chemical potentials for AC and H2 O, �EAC and �EW are calculated by eq. (10) and plotted in Fig. III-9. This method is due to Boissonnas (1939). There are many other methods of solving the total vapor pressure data; see Prausnitz et al (1967), for example. However, most of these methods assume a fitting function for the excess Gibbs function, GE , a priori, and hence there is a danger of artificially forcing the thermodynamic property of the system in question to behave in the form of the chosen fitting function. The Boissonnas method, on the other hand, being purely a numerical analysis based on the first principle, the Gibbs-Duhem relation, is free from such a danger. The Boissonnas method, however, starts with the assumption that the most dilute point is in the Henry’s law region. Is this assumption correct? This issue was dealt with by us and it was concluded that the Henry’s law region does not exist even at the mole fraction of a solute as low as 10−5 . (Koga, 1995a; Westh et al., 1998). This still leaves the possibility that Henry’s law is obeyed below 10−5 mole fraction. It is, however, very impractical for determining the vapor pressure at this low mole fraction. About 10−3 is a convenient mole fraction at which to measure the most dilute solution, as is the case for the above AC-H2 O, as well as the alcohol−H2 O (Hu et al., 2003). Thus, the partial

2.0

μ ACE/RT or μ WE/RT

1.5

μ WE/RT 1.0

μ ACE/RT 0.5

0.0

0.0

0.2

0.4

0.6

0.8

1.0

xAC

Fig. III-9. The excess chemical potential of acetone (AC), �EAC , and H2 O (W), �EW , in AC-H2 O at 20  C due to Perera et al. (2005). They are calculated by eq. (III-10), using the partial pressure data obtained by the Boissonnas method (see text). Reproduced with permission from the Journal of Chemical Physics 123, 024503 (Perera, A. et al.), Copyright (2005), American Institute of Physics.

64

Fig. III-10

pressures at the point determined on the assumption of Henry’s law are in fact in error. This error may be at first sight carried systematically through the entire process of the above analysis. It turned out, however, that eq. (13) or eq. (14) contain a self-correcting feature. (Koga, 1995a). Namely, if the value of pAC is erroneously large, the value of �AC , eq. (14), will be unduly small. It follows then that the increment for pAC , �pAC , calculated by eq. (13) becomes smaller than the proper value. Thus the value of pAC for the next point will be closer to the real value. If this correction is not enough and the value of pAC is still too high, then the next one could be more correct, and so on. The same corrective feature applies for the case of too low a value of pAC . Depending on how dilute the first point is and also on the size of increments, several points appear to be sufficient to achieve stable (and hence most likely correct) results (Koga, 1995a). For the present AC−H2 O, the first five points of �EAC below xAC 2 in the range, 0�3 < x1P < 0�6, are not reliable. Hollow symbols indicate non-reliable analyses (see text). The data from Hu et al. (2003). Reproduced with permission from the Canadian Journal of Chemsitry 81, 141, (Hu, J. et al.), Copyright (2003), NRC Research Press.

Eq. (III-20) to (III-21)

67

1.0

0.8

y1P

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

x1P

Fig. III-12. The mole fractions of 1-propanol (1P) in 1P−H2 O at 25  C. x1P is that in liquid phase and y1P in the equilibrium gas phase. The data of hollow symbols in the figure are not reliable since the Boissonnas coefficients are larger than 2 (see Fig. III–11 and text). To correct for this, a smooth curve is drawn through all the good points and the value of y1P is read off the smooth curve drawn. From the intersect of the curve and the line of y1P = x1P , we determine the azeotropy to be at x1P = 0�390 ± 0�005 (See Hu et al. (2003) for details). Reproduced with permission from the Canadian Journal of Chemsitry 81, 141, (Hu, J. et al.), Copyright (2003), NRC Research Press.

any model. Here again, we prefer this manual graphing method to the computer graphic technique, B-spline, as discussed in Appendix A. The resulting p1P data were used to calculate the excess chemical potential of 1P following eq. (10) as, � �E1P = RT ln

p1P ∗ p1P x1P

� �

(III-20)

The total chemical potential of 1P is then, �1P = �∗1P + RT ln�x1P � + �E1P �

(III-21)

The resulting chemical potential is now compared with the result obtained by a small angle x-ray scattering (SAXS) study (Hayashi et al., 1990). The concentration fluctuation

68

Fig. III-13

5

SAXS, Hayashi et al.(1990) From vapour pressure

RT(1-x1P)/(dμ 1P /dx1P)

4

3

2

1

0

0.0

0.2

0.4

0.6

0.8

1.0

x1P 1P�� � Fig. III-13. The ordinate quantity, RT �1−x ��1P , is the concentration fluctuation that can be deter­ �x1P

mined from the forward scattering intensity of small angle x-ray scattering (SAXS). The hollow square symbols in the figure are the result by Hayashi et al. (1990). The filled circles are calculated from �E1P data. See text [III-7] for details. Reproduced with permission from the Canadian Journal of Chemsitry 81, 141, (Hu, J. et al.), Copyright (2003), NRC Research Press.

mentioned in Section [I-9], is equal to the so-called partial structure factor, Sxx(0) (Bhatia & Thornton, 1970), which is directly obtained from the forward scattering intensity. Thus, �1 − x � � � Sxx�0� = RT � 1P ��1P �x1P

(III-22)

Fig. III–13 shows the values of the left of eq. (23) from SAXS measurements (Hayashi et al., 1990) and the right of eq. (23) obtained from the above vapor pressure analysis (Hu et al., 2003). The match between the two sets of data is satisfactory including the mid region of the above y1P correction, 0�3 < x1P < 0�6, assuring the soundness of both sets of determination.

Eq. (III-22) to (IV-3)

Chapter IV

FLUCTUATION AND PARTIAL MOLAR FLUCTUATION – UNDERSTANDING H2 O

[IV-1] INTRODUCTION So far we have dealt with the excess partial molar quantities, �Ei � HiE � SiE , and ViE . While �Ei is the first derivative of G, the rest are the second derivatives (see Table I-1 (p. 20)). We then took one more compositional derivative and obtained the interaction functions, �Ei−j � HiE−j � SiE−j , and ViE−j . Of course, �Ei−j is the second and HiE−j � SiE−j and E are the third derivatives. As shown in Table I-1 (p. 20), there are three more second Vi−j derivative quantities, i.e. the heat capacity, Cp , the isothermal compressibility, �T , and the isobaric expansivity, �p . They are collectively called the response functions, and defined as, � � � 2 � �S � G Cp ≡ T � (IV-1) = −T �T �T 2 �T ≡ − �p ≡

1 V

1 V �

�

�V �p

�V �T

� =−

� =

1 V

1 V

�

�

� �2 G � �p2

�2 G �p�T

(IV-2)

� �

(IV-3)

The partial derivatives are taken keeping all the other variables constant in the �p� T� ni � variable system. There is another second derivative, �i−j , in Table I-1 (p. 20). This quantity is inversely proportional to the concentration fluctuations and is related to the forward scattering intensity in light, x-ray and neutron scattering experiments. This is dealt with in Section [IV-5] below.

[IV-2] FLUCTUATION FUNCTIONS – COARSE GRAIN The response functions, eq. (1) to eq. (3), are related to the mean-square fluctuation func­ tions in V and S through standard statistical thermodynamics. (Landau & Lifshits, 1980; Debenedetti, 1986; Panagiotopoulos & Reid, 1986; Yan & Chen, 1992). For convenience

70

in dealing with fluctuations, we impose the conditions such that the independent variables �p� T� ni � are fixed and their conjugates, S = −��G/�T � and V = ��G/�p� are allowed to fluctuate. The mean-square fluctuation, for example, is evaluated by taking the aver­ age of the square of the variation of a local/instantaneous value from the global average. Since S and V are extensive quantities, a local/instantaneous value must be evaluated in a coarse grain consisting of an equal amount of each component nzi �i = 1� 2� � � � � m, � for m-component system). The total size of the coarse grain, N z = nzi , must be large enough that the thermodynamic quantities S and V can be defined, but small enough that the fluctuation can be detected, as pointed out by Debenedetti (1986) and Panagiotopoulos and Reid (1986). The fundamental problem at hand is that there is no a priori knowledge about N z . It may not even be universal. For now, however, we use an unspecified N z for the size of coarse grain and rewrite the thermodynamic relations between fluctuations and response functions, taking account of N z explicitly (Koga et al., 1996; Koga, 1999; Koga & Tamura, 2000; Siu & Koga 2005). Namely, � = kT � = kT � = kT

�V �p �S �T �V �T

� = kT VN Z �T �

(IV-4)

= kCpN Z �

(IV-5)

= kT VN Z �p �

(IV-6)

� �

� signifies the variation of the quantity in a coarse grain from its global average; k is the Boltzmann constant; VN Z is the average volume of the coarse grain; and CpN Z is the average heat capacity of the coarse grain containing N z . We stress that we have no way of knowing N z and that N z may not be universal and may depend on the conditions. For example, as a demixing critical point is approached from a super-critical region, fluctuation becomes progressively vigorous not only in its intensity (amplitude) but also its extent (wavelength). Eventually at the critical point, the extensity (wavelength) reaches a macroscopic size, and hence N z must diverge at the critical point. N z could also be different from material to material. Indeed, it has been pointed out that in liquid H2 O a large number of H2 O molecules fluctuate collectively due to hydrogen bonding (Tanaka & Ohmine, 1987; Ohmine, 1995; Iijima & Nishikawa, 1994). Hence, it is expected that the size of the coarse grain in H2 O is necessarily larger than that in a normal liquid without a hydrogen bond network. Thus, when the values of the mean square fluctuations, eq. (4) to eq. (6) are quoted, the value of N z , hence those of VN Z and CpN Z , must be specified. To circumvent this difficulty, it is common practice (though not explicitly explained) to divide both sides of eq. (4) to eq. (6) by the average volume of a coarse grain, = VN Z , and we obtain

Eq. (IV-4) to (IV-13)

71

V

� ≡ /k = T �T �

(IV-7)

S

� ≡ /k = CpN Z /VN Z = Cp m /Vm �

(IV-8)

� ≡ /k = T�p �

(IV-9)

SV

� (for q = V� S, or SV) is the notation for convenience. Cp m and Vm are the molar heat capacity and the molar volume. We call these q �’s, eq. (7) to eq. (9), the volume, the entropy and the entropy-volume cross “fluctuation density”, respectively. It is immedi­ ately evident that these quantities, q �’s, are devoid of information about N z , and hence they may be indicative of the intensity or the amplitude of fluctuation. q �’s are second derivatives of G. We then define third derivatives, the partial molar fluctuation density for a binary mixture as, � q � � q � � � � � q = �1 − xi � � (IV-10) �i ≡ N �xi �ni q

which signifies the effect of the i-th component on the respective fluctuation functions. When the fluctuation propensities are to be compared between aqueous and non­ aqueous solutions, the extensity of fluctuation could be important. As mentioned above, molecular dynamic studies (Tanaka & Ohmine, 1987; Ohmine, 1995) and an x-ray scattering investigation (Iijima & Nishikawa, 1994) indicated that in H2 O collective fluctuations involving a large number of molecules are prevalent. This suggests that H2 O has a characteristically larger extensity of fluctuation than normal liquids. It is therefore expected, though not rigorously proven, that the size of coarse grain is necessarily larger for H2 O than for normal liquids. Is there any way of identifying this effect of H2 O? It turns out that the mean square normalized fluctuations, q �, defined below by eq.(14) to eq.(16) serve this purpose though qualitatively (Koga, 1995b; Koga et al., 1996; Koga and Westh, 1996; Tamura et al., 1999; Koga, 1999; Koga & Tamura; 2000; Siu & Koga, 2005). First, we normalize the deviation of a local/instantaneous value in a coarse grain from the global average by the mean volume of coarse grain. It could very well be more meaningful to normalize by the actual volume of coarse grain. However, for mathematical simplicity, here we use the average volume to normalize with. Thus, � � �V 2 > = kT �T / = kT �T /�N z Vm �� (IV-11)
= kT �p / = kT �p /�N z Vm ��

(IV-12)

(IV-13)

72

The second equalities use the relation, = VN Z = N z Vm , and = CpN Z = N z Cp m . On the other hand, we define the normalized fluctuation functions as, V

� ≡ RT �T /Vm �

(IV-14)

S

� ≡ RCp m /Vm 2 �

(IV-15)

SV

� ≡ RT �p /Vm �

(IV-16)

and the partial molar normalized fluctuation functions as, �

q

� �q � �i ≡ �1 − xi � � �xi

(IV-17)

We now wish to compare two systems, �A� and �B�. For this we consider the ratio of the left-hand sides of eqs. (11) to (13). Using eqs. (14) to (16), the ratios between �A� and �B� are written as, �
/
�B� = �2 �B�

>

� � �� � � SV ��A� � SV ��B� �V �S �V � � �A� �B� > /< > = / �

Nz Nz (IV-20)

Here superscripts [A] and [B] designate each system. If we assume that the left-hand sides of eq. (11) to eq. (13) are universal and independent of the system under consideration, then the left-hand sides of eq. (18) to eq. (20) are identically equal to unity. If so, N z�A� > N z�B� if q ��A� > q ��B� �

(IV-21)

and the converse is also true. Namely, if we see q ��A� > q ��B� when calculated by eq. (14) to eq. (16), then we conclude N z�A� > N z�B� , and hence the extensity or the wavelength of fluctuation is larger for sample [A] than for [B]. If we relax the above assumption and allow the ratios of the left-hand sides of eq. (18) to eq. (20) to take a positive value less than unity (for the case where the ratio is larger than 1, the labels �A� and �B� can be reversed), it follows that the above statement is partly true. Namely, if we find q �A� � > q ��B� , then it is necessary but not sufficient that N z�A� > N z�B� . Thus, the relative size of q � between the two systems in comparison provides a qualitative indication as

Eq. (IV-14) to (IV-21)

73

to the relative size of coarse grain, which in turn could be parallel to the wavelength of fluctuation of each system. We used both q �’s and q �’s and their temperature and pressure dependences to contrast the nature of H2 O from that of n-hexane (Koga et al.; 1996; Koga & Tamura, 2000). Within the limited applicability of q � discussed above, we learned about the collective nature of fluctuation in H2 O in contrast to a normal liquid, n-hexane, as described below in Section [IV-3].

[IV-3] H2 O vs. n-HEXANE It is well known that H2 O is unique in comparison with normal liquids, its uniqueness being due to its hydrogen bonding capability. (Eisenberg & Kauzmann, 1969; Franks, 1972; Suzuki, 1980). A number of models have been devised for understanding this peculiar nature of H2 O in the past. They may be crudely classified into two categories represented by the mixture model (Roentgen, 1892) and the bent hydrogen bond model (Bernal & Fowler, 1933). The former postulates that liquid H2 O is a mixture of two basic components: One is a bulky ice-like and the other a dense normal liquid. This model has been developed further by many others, recently by Robinson et al. (Bassey et al., 1987; Vedamuthu et al., 1994 & 1995). In the bent hydrogen bond model, liquid H2 O is regarded as forming a more or less completely hydrogen bonded network with a wide distribution in the hydrogen bond strength due to bending – fluctuating widely. This concept has also been extended, an example being a series of work by Sceates et al. aided by spectroscopic data (Sceates et al., 1980). Here we compare H2 O and n-hexane, and see the differences in terms of fluctuation functions developed above. Figs. IV-1, IV-2 and IV-3 are plotted using the data in our earlier paper (Koga & Tamura, 2000). The raw data were taken from the literature (Pruzan, 1991; TerMinassian et al., 1981; Grindley & Lind, 1971; Kell, 1975). If n-hexane, a typical van der Waals liquid, is regarded as an assembly of harmonic oscillators, the mean square volume fluctuation is proportional to the absolute temperature and inversely proportional to the force constant of the harmonic oscillator (Landau & Lifshits, 1980; Yan & Chen, 1992). Hence, the entropy and the entropy-volume cross fluctuation are zero for strictly harmonic oscillators. The volume fluctuations for n-hexane, V � and V �, in Fig. 1 seem to approach the temperature axis at −273  C, though curling up towards high temperature. The latter curling up effect is no doubt due to non-harmonicity. Hence entropy, Fig. 2, and entropy-volume cross fluctuations, Fig. 3, take small non-zero values. As pressure increases, the force constant increases, and hence V � and V � decreases as shown in Fig. 1. Fig. 1(a) suggests that the amplitude of volume fluctuation density, V �, for H2 O is in fact smaller than that of n-hexane. At high pressures, 500 MPa for example, the values of V � are almost the same and show almost the same temperature-dependence between H2 O and n-hexane, consistent with earlier suggestions that H2 O becomes the same as normal liquids at high pressures (Stillinger & Rahman, 1974; Jonas et al., 1976; Wilbur et al., 1976). V � in Fig. 1(b), on the other hand, shows that the value of V �

74

Fig. IV-1(a) to IV-2(b)

1.2

hollow: n-hexane filled: H2O

1.0

0.1 MPa 50 MPa 200 MPa 500 MPa

0.8

0.6

V

δ / K MPa-1

(a)

0.4

0.2

0.0 -40

-20

0

20

40

60

80

100

120

140

60

70

temp / °C hollow: n-hexane, filled: H2O 0.1 MPa 50 MPa 100 MPa 200 MPa 300 MPa 500 MPa

0.12

0.10

V

Δ

0.08

(b)

0.06

0.04

0.02

0.00

-20

-10

0

10

20

30

40

50

temp / °C

Fig. IV-1. (a) The volume fluctuation density, V �, (the amplitude of volume fluctuation) and (b) the normalized volume fluctuation, V �, (the amplitude and the qualitative wavelength of volume fluctuation), for H2 O (filled symbols) and n-hexane (hollow symbols). The dashed line is a presumed contribution of the normal liquid-like part of H2 O with no ice-like patches. Reproduced with permission from the Netsusokutei (Journal of the Japan Society of Calorimetry and Thermal Analysis) 27, 195, (Koga, Y. and Tamura K.), Copyright (2000), Japan Society of Calorimetry and Thermal Analysis.

75

5

filled: H2O, hollow: n-hexane

0.1 MPa 200 MPa 500 MPa

3

S

δ / JK-1 cm-3

4

(a)

2

1

-40

-20

0

20

40

60

80

100

120

140

120

140

temp / °C

3

2

0.1 MPa

S

Δ / J2K-2 cm-6

filled: H2O, hollow: n-hexane

200 MPa

1

500 MPa

(b)

0 -40

-20

0

20

40

60

80

100

temp / °C

Fig. IV-2. (a) The entropy fluctuation density, S �, (the amplitude of entropy fluctuation), and (b)

the normalized entropy fluctuation, S �, (the amplitude and the qualitative wavelength of entropy

fluctuation), for H2 O (filled symbols) and n-hexane (hollow symbols). Reproduced with permission

from the Netsusokutei (Journal of the Japan Society of Calorimetry and Thermal Analysis) 27,

195, (Koga, Y. and Tamura K.), Copyright (2000), Japan Society of Calorimetry and Thermal

Analysis.

76

Fig. IV-3

filled: H2O, hollow: n-hexane 0.8

0.1 MPa 50 MPa

(a)

100 MPa

0.6

SV

δ

500 MPa 0.4

0.2

0.0

-40

-20

0

20

40

60

80

100

120

140

120

140

temp / °C

filled: H2O, hollow: n-hexane 0.1 MPa 50 MPa 100 MPa 200 MPa 300 MPa 500 MPa

SV

Δ / J K-1 cm-3

0.15

0.10

(b) 0.05

0.00

-40

-20

0

20

40

60

80

100

temp / °C

Fig. IV-3. (a) The entropy-volume cross fluctuation density, SV �, (the amplitude of the entropyvolume cross fluctuation), and (b) the normalized entropy-volume cross fluctuation, SV �, (the amplitude and the qualitative wavelength of the entropy-volume fluctuation) for H2 O (filled symbols) and n-hexane (hollow symbols) The dashed line indicates a presumed contribution of the normal liquid-like part of H2 O with no icebergs. Reproduced with permission from the Netsusokutei (Journal of the Japan Society of Calorimetry and Thermal Analysis) 27, 195, (Koga, Y. and Tamura K.), Copyright (2000), Japan Society of Calorimetry and Thermal Analysis.

77

for H2 O becomes generally larger than that of n-hexane, indicating that the wavelength of volume fluctuation, or the size of coarse grain, is qualitatively larger for H2 O, as discussed above in Section [IV-2]. At low pressures and temperatures (less than 200 MPa and 50  C), both V � and V � for H2 O show an anomalous initial decrease as temperature increases. Dashed lines in Fig. 1 are drawn as a guide for the presumed normal behavior for 0.1 MPa. Thus, the anomalous positive contribution from the dashed line that decreases on increasing temperature is consistent with the putative formation/destruction of ice-like patches of the mixture model of liquid H2 O. At higher temperature the probability of hydrogen bond decreases and hence the formation of ice-like patches is retarded. At pressures higher than 300 MPa, this anomalous positive contribution to V � and V � is no longer present. As will be discussed below regarding entropy fluctuation, this is due rather to retardation of forming ice-like patches according to the Le Chatelier principle than a direct reduction of hydrogen bond probability by pressure. Fig. 2 indicates that the entropy fluctuations for H2 O are larger by 3-fold for S � and by about 10-fold for S �. The entropy fluctuation is associated with the enthalpy fluctuation, i.e. = T 2 , which in turn is indicative of the bond strength fluctuation. Accordingly, this observation suggests that for H2 O the bond strength fluctuates more widely not only in the amplitude but also in the wavelength than that for n-hexane, consistent with the “bent hydrogen bond model”. A high resolution inelastic neutron scattering study suggested that there are two kinds of hydrogen bond strength with the force constant ratio of 2:1 even in ice Ih (Li & Ross, 1993; Li et al., 1994a and 1994b). While criticisms were raised of this “exotic” model by some (Kuo et al., 2005; Tse & Klug, 1995), it seems to be agreed by all that the proton position in ice Ih is highly disordered. For liquid, therefore, an even wider distribution of the hydrogen bond strength is expected, i.e. the bent hydrogen bond model. Both S � and S � for H2 O decrease on increasing temperature, in contrast for n-hexane. This is due to a decrease in the probability of intact hydrogen bond for H2 O on temperature increase. Namely, as the temperature increases, the bent hydrogen bonds are bent excessively to be broken. The molecules without hydrogen bonds contribute very little to S � and S � as n-hexane. On the other hand, the pressure generally increases S � and S � for both H2 O and n-hexane, except for the low pressure and temperature region, below 200 MPa and 50  C for H2 O. The latter region is where the ice-like patches are prevalent as discussed above. Thus, the pressure alters the force constant in a similar manner for both H2 O and n-hexane, and may not reduce the hydrogen bond probability directly. Otherwise, S � and S � should decrease for H2 O on pressure increase, as in the case for the temperature increase discussed above. For such normal liquids as n-hexane, when fluctuation causes an increase in volume of a coarse grain, then inevitably its entropy increases. Hence the entropy-volume cross fluctuation is always positive. Since both volume and entropy fluctuations, Figs. 1 and 2, increase on temperature increase, the cross fluctuation should also increase on temperature increase, which is indeed the case for n-hexane, Fig. 3. The effect of pressure on volume, Fig. 1, and the entropy, Fig. 2, fluctuations are the opposite of each other.

78

Fig. IV-4 to IV-5

The resulting cross fluctuations decrease on pressure increase, Fig. 3. Apparently the pressure effect on volume fluctuation is stronger. The peculiar behavior for H2 O in Fig. 3 is naturally interpreted as being due to the putative formation/destruction of ice-like patches at the low pressure and temperature region. Namely, unlike n-hexane, a positive volume fluctuation may be partly due to the formation of ice-like patches which is associated with an entropy decrease, and vice versa. Hence the cross term will have a negative contribution. This negative contribution is prevalent in the low temperature and pressure region where ice-like patches are putatively formed and destroyed. As the temperature and the pressure increase above 50  C and 200 MPa, this negative contribution diminishes therefore rapidly. At 500 MPa, SV � and SV � behave in a similar manner as those for n-hexane. Comparison between Fig 3(a) and Fig 3(b) suggests that the fluctuation wavelength is qualitatively larger for H2 O than n-hexane. We have thus demonstrated the usefulness of the fluctuation functions introduced above in recovering the general understanding of liquid H2 O (Eisenberg & Kauzmann, 1969; Franks, 1974; Suzuki, 1980), and that the mixture model and the bent hydrogen bond model are only two sides of a coin (Stanley & Teixeira, 1980; Blumberg et al., 1984). Furthermore, we have pointed out the importance of distinguishing the amplitude and the wavelength of fluctuation.

[IV-4] SITE-CORRELATED PERCOLATION MODEL OF H2 O As we saw in the entropy fluctuation functions, S � and S �, in [IV-3], H2 O is char­ acterized by a large fluctuation in hydrogen bond strength in terms of the intensity (amplitude) as well as the extensity (wavelength) in comparison with a normal liquid, n-hexane. This could be understood to be due to the molecular structure of H2 O, together with its hydrogen bonding capability. As shown in Fig. IV-4, the H−O−H angle of

H 104.5°

O

H

Fig. IV-4. H2 O molecular structure. The H-O-H angle is known to be 104�5 in the gaseous state, indicating that the sp3 configuration is not perfect. In solid and liquid states, this angle does not expand much (see text and Fig. IV-5). Reproduced with permission from Water and Aqueous Solutions (Suzuki, K.), Copy right (1980), Kyouritsu Shuppan, Tokyo.

79

a gaseous molecule is 104�5 , which does not increase much in the condensed phase. (Suzuki, 1980, p. 53). In ice Ih, the bend angle is cited as 107 ± 1 (Kuhs & Lehman, 1986), 106�6 ± 1�5 (Petrenko & Whiteworth, 1999), 106�3 ± 0�4 (Kuo et al., 2005) and 108�1 ± 0�8 (Fanourgakis & Xantheas, 2006). When H2 O freezes to ice Ih, O atoms are organized in a tridymite type structure with four hydrogen bonds to the adjacent O atoms as sketched in Fig. IV-5. The lattice arrangement dictates that the O−O−O angle must be 109�5 . The discrepancy by a few degrees between the O−O−O angle and the molecular H−O−H angle is the key for a large fluctuation in S � and S �. Namely, if one of the two H’s belonging (momentarily) to the central O is on line of the adjacent O−O direction to form a proper hydrogen bond, the other becomes inevitably off line by a few degrees, resulting in a bent and weak hydrogen bond. Indeed, high resolution inelastic neutron scattering experiments showed that there are two kinds of hydrogen bonds in ice; one is weaker than the other. (Li and Ross, 1993; Li et al., 1994a and 1994b). Of course, this distribution occurs momentarily, and it switches back and forth, i.e. the hydrogen bond strength is fluctuating even in ice. While this “exotic” model has been criticized (Kuo et al., 2005; Tse & Klug, 1995), there is no doubt that the location of proton in ice Ih is highly disordered. In liquid H2 O, the lattice structure of O atoms is disorganized. In addition, the bend angle in liquid is shown to be 106�1 ± 1�8 by neutron scattering (Ichikawa et al., 1991) and 106�3 ± 4�9 by simulation (Fanougakis & Xantheas, 2006). The pair correlation function obtained by x-ray scattering experiments

H 109°

106° H

O

Fig. IV-5. The local molecular arrangement of ice Ih. The O atom arrangement is that of tridymite and the O−O−O angle is 109 . The location of the H atom is highly disordered and the average H-O-H angle remains less than 109 . Another way to describe the highly disordered H arrangement is that when one H aligns with one of the four nearest O then the second H will be off-aligned, and hence there are two kinds of hydrogen bonding, one strong and the other weak (see text).

80

Fig. IV-6(a) to IV-6(b)

shows the first nearest population is at about 2�8 A and the second at 4�5 A with an interstitial population at 3�5 A . The total number of O atoms in these configurations is now 4.4, as opposed to 4 in ice. (Eisenberg & Kauzmann, 1980; Suzuki, 1980). This suggests that there is a substantial hydrogen bond network still intact in liquid, with a wide distribution in the hydrogen bond strength, consistent with the bent hydrogen bond model of H2 O (Bernal & Fowler, 1933; Sceates & Rice, 1980). This results in large S � and S � for H2 O as we saw in Section [IV-3]. The behaviors of the volume fluctuation functions, V � and V �, reflect the mixture model of H2 O (Roentgen, 1892; Bassey et al., 1987; Vedamuthu et al., 1994 and 1995), as we saw above in Section [IV-3]. Thus, both models represent the two possible and simplified views of the reality, which of course is the very purpose of devising a model. Stanley et al., on the other hand, have paid attention to the connectivity nature of the hydrogen bond network of H2 O (Stanley & Texeira, 1980; Blumberg et al., 1984). They suggested that for any reasonable definition of the “intact” hydrogen bond, the hydrogen bond network at normal temperatures is bond percolated. Namely, a snapshot of a given moment would show that the hydrogen bond network is connected throughout the entire bulk of liquid H2 O. As the global probability of the “intact” hydrogen bond decreases as temperature increases (or certain solutes are added as is the main theme of this book), a threshold, the bond percolation threshold, could be reached whereby the network connectivity is lost. This connectivity idea turned out to be useful in understanding the effect of a non-electrolyte solute on H2 O in the present studies using the second and the third derivatives of G, as will become evident in Part B. We now follow Stanley and Texeira (1980) and study the bond percolation nature using a simplified two-dimensional square lattice. First they assume a certain (reasonable) cut­ off in the energies of hydrogen bonds, so that all possible hydrogen bonds are classified as “intact” or “broken”. Then the global average of the “intact” hydrogen bond can be calculated. For a given hydrogen bond probability, the network connectivity is shown in Fig. IV-6 as a snapshot of a given instance. At every corner of the square lattice an O atom is placed. With a given probability, 25% say, a line is drawn at random for each of four possible directions from each O atom. The results are shown in Fig. IV-6(a). In addition, the O atom which has all four hydrogen bonds intact is marked with an asterisk. As is evident in Fig. IV-6(a), the hydrogen bonds are not connected throughout the entire square, and we cannot walk on bonds through from the top to the bottom of the square. This says that the (hydrogen) bond percolation is not yet established. Figs. IV-6 (b), (c) and (d) are the results of the same exercise for the hydrogen bond probability of 45, 55 and 75%, respectively. We see that the bond percolation is not yet established for (b), but in (c) we can walk through the square lattice from one end to the other, i e. the bond percolation is on. Therefore, the threshold value of the probability above which the bond percolation is established must be between 45 and 55%. It is known that the bond percolation threshold is 50% for the two-dimensional square lattice with four bonds. (Parsonage & Staveley, 1978) Furthermore we note in Fig. IV-6(c) and (d) that the asterisks, i.e. the O atom with four hydrogen bonds, show a clustering tendency, while areas without asterisks are also present. These clusters of O atoms with four bonds can

81

(a) Bond probability 0.25 | o

o | o -- o -- o

o

o | o -- o

o

o -- o -- o o

o

o | o

o -- o o

o

o

o

o

o o o | o -- o

o | o

o -- o | o -- o

o | o | o -| o

o | o

o

o

o

o

o -- o -- o | o o -- o

o

o

o -- o

o o

o

| o -- o -- o | o -- o o | o

| o

o

o

o

-- o o o o | o o -- o | o o o

o -- o o

o

o

o -- o | o o |

o | o

o

o o

| o -- o |

o

o

o |

o -- o | o o | o o

o -- o -- o o

o -- o o

o

o | o

o | o

| o

| o -- o

| o

|

o -- o

o

o

o

o

o |

o

o o

o o | o

o -- o o |

o

o -- o

o o

o o

o o -| |

-- o -- o -- o -- * -- o | o o o o o -| -- o o o -- o o | |

o -- o -- o -- o o -| o -- o o o -- o | | o o o o o | -- o o o -- o -- o | | -- o -- o o o o o --

| o -- o

o

o

o |

o

o

o

o

o

o

o |

o o |

o -- o -- o |

o -- o

o

o

|

o

o

o

o

o -­

o -- o | o o |

o o --

o |

o

o

o

o

o -­

o

o

o -­

o

o o -|

o -- o |

o |

o o -|

-- o | o -- o

o

o -- o o o -- o o | o o | o

o | o

o -­

o

o

o

o

|

o

Fig. IV-6(a). The bond percolation of 256 molecules of the two dimensional square “H2 O” with 25% hydrogen bond probability. o is the oxygen atom of H2 O. -- shows the intact lateral hydrogen bond.  shows the vertical intact hydrogen bond. ∗ is the oxygen of H2 O which has all four hydrogen bonds intact. Note there is only one ∗ . (b) Bond probability 0.45 o -- o -- o o | | | | o o -- o o | -- o o o -- o | | o o -- o o | o -- o -- o o | | | -- o o -- o -- o | o o o o | o o o -- o | | -- * -- o o o | | -- o -- o o -- o | | o o o -- o | | | -- o o o o | | -- * -- o -- o -- o | o -- o -- o o | o o o -- o | | o o -- * -- o | | |

o -- o o -| | o o -- o | | | o o -- o | | -- o o o | | o o -- o -| | o -- o -- o --

| o | o -- o | o -- o o

| | o -- o o | | o -- o o | o o -- o | | o -- o o -- o | o o o -- o | o -- o -- o -- o | o -- o o -- o | | | o o o -- o | | * -- o o -- o | | | * -- o -- o o | | | | o o o o

o o | | o -- o | o o --

o -- o o | o o o | | o -- o o | | -- o -- o -- o o o o | | | | -- o -- o o -- o o -- o | | | o o o -- o -- o o | | o -- o o o o -- o | o o o -- o o o | o o o -- o o -- o | -- o o -- o -- o o o | | | o -- o -- o -- * -- o -- o | | -- o o o o o o | | | -- o -- o o -- o -- o -- o |

---

---

o o | o -- o

o | o | o -- o

| | -- o -- o -­ o

o

-- o -- o | o o -­ | o -- o | | o -- o o o -­ | -- o o | -- o -- o -- o o | | o -- o

o -- o -- o | | o -- o | -- o o -- o o -­ | -- o -- o -- o o -- o o -­ | | | -- o o -- o -- o o -- o | | o

Fig. IV-6(b). The bond percolation of 256 molecules of the two dimensional square “H2 O” with 45% hydrogen bond probability. o is the oxygen atom of H2 O. -- shows the intact lateral hydrogen bond.  shows the vertical intact hydrogen bond. ∗ is the oxygen of H2 O which has all 4 hydrogen bonds intact. Note there are 6 ∗ out of which two of them are next to each other. The bond percolation is not yet established hence one cannot walk through the lattice from the top to the bottom on the bonded path.

82

Fig. IV-6(c) to IV-7

(c) Bond probability 0.55

Fig. IV-6(c). The bond percolation of 256 molecules of the two dimensional square “H2 O” with 55% hydrogen bond probability. o is the oxygen atom of H2 O. -- shows the intact lateral hydrogen bond.  shows the vertical intact hydrogen bond. ∗ is the oxygen of H2 O which has all 4 hydrogen bonds intact. Note the bond percolation is now established; one can walk through the lattice from the top to the bottom. Furthermore there is a clear tendency that oxygen molecules with four bonds cluster together. The bond percolation threshold for the two dimensional square lattice with 4 possible bonds is 0.50. (d) Bond probability 0.75

Fig. IV-6(d). The bond percolation of 256 molecules of the two dimensional square “H2 O” with 75% hydrogen bond probability. o is the oxygen atom of H2 O. -- shows the intact lateral hydrogen bond.  shows the vertical intact hydrogen bond. ∗ is the oxygen of H2 O which has all 4 hydrogen bonds intact. Not only bond percolation is intact but also the clusters of ∗ -oxygens are almost connected, i.e. the situation is close to forming ice. These four pictures, (a), (b) (c) and (d), are generated by a Macro in MSExcel created by Matthew T. Parsons following the idea by Stanley and Teixeira (1980). I thank Matthew T. Parsons for kindly allowing me to use the Macro. Reproduced with permission from the Journal of Chemical Physics 73, 3404, (Stanley, H. E. and Teixeira, J.), Copyright (1980), American Institute of Physics.

83

naturally be regarded as putative ice-like patches that are continually forming/breaking here and there. This is consistent with the mixture model. This clustering tendency of four-bonded O atoms occurs from the topological requirement, in spite of the fact that intact hydrogen bonds are placed randomly with a given probability. In reality, once the hydrogen bond is formed from an O atom to one direction donating H to the adjacent O, then a lone pair of the same O atom has a preference for accepting H from the other direction, i.e. the cooperativity of hydrogen bonding in H2 O (Frank and Wen, 1957; Frank, 1958). Thus, the clustering tendency is stronger in reality than is depicted in Fig. IV-6. The topological connectivity of real H2 O is, of course, not a two-dimensional square lattice, but is of the three-dimensional tridymite type. For this, the percolation threshold is known to be 39% bond probability. (Parsonage & Stavely, 1978). Fig. IV-7 shows a number of estimates of the temperature dependence of the hydrogen bond probability. These estimates are based on a variety of principles under a variety of assumptions. While there is no priority as to which estimate is closer to reality, we note that the estimate by Stanley using the density data (Stanley & Texeira, 1980) reaches 39% at about 80  C. We show below in Section [V-1] that the hydrogen bond probability of bulk 1.0 1

Hydrogen bond probability

0.8

2 3 4

0.6

0.4

65 7 8 9

0.2

0.0 0

20

40

60

80

100

120

Temperature / °C

Fig. IV-7. The temperature dependence of the hydrogen bond probability in H2 O, except for the broken lines which are for D2 O. 1. Haggis et al. (1952). 2, and 3. Stanley & Teixeira (1980). 4. Walrafen (1966). 5. Nemethy & Scheraga (1964). 6. Nemethy & Scheraga (1962). 7 and 8. Marcus & Ben-Naim (1985). 9. Grjotheim & Krogh-Moe (1954). Reproduced with permission from The structure and properties of water, (Eisenberg, D. and Kauzmann, W.), Copy right (1969), Oxford University Press, Oxford.

84

H2 O away from solutes is reduced on addition of 2-butoxyethanol and at a threshold concentration, the behavior of the third derivative thermodynamic quantities suggests the breakdown of the hydrogen bond percolation. This concentration threshold decreases on increasing temperature and it extrapolates to the zero solute concentration (pure H2 O) at about 83  C. This coincidence of about 80  C between the two sets of observations is remarkable and is fully discussed in Chapter V.

[IV-5] CONCENTRATION FLUCTUATIONS AND KIRKWOOD-BUFF INTEGRALS In considering the concentration fluctuations, we choose a coarse grain such that nzj �j = i� are fixed and the amount of the target component ni only is allowed to fluctuate. As � discussed above, the size of the coarse grain, N z = nzj , must be large enough for thermodynamic quantity such as �i to be defined and small enough so that the fluctuation can be detected. However, its value remains unspecified. Thus, the mean square variation of ni � �ni , can be written as (Debenedetti, 1986), � = RT/

� ��i � �ni

(IV-22)

The partial derivative is taken keeping p� T , and nj =i constant. When nj for j =  i are  kept constant, �xi = �ni �1 − xi �/N� and �

� � � ��i ��i = ��1 − xi �/N � � �ni �xi

Hence eq. (22) is rewritten as, � N = RT�1 − xi �/

� ��i � �xi

(IV-23)

The left-hand side of eq. (23) is called the concentration fluctuation of the i-th com­ ponent and is equal to the concentration structure factor at the long wavelength limit, Sxx(0), which is accessible by light, x-ray, or neutron scattering experiments. Namely, it is calculated by using the forward scattering intensity, the scattering amplitude, the partial molar volume of each component, the total volume and the compressibility of the mixture (Bhatia & Thornton, 1970; Koga, 1984; Nishikawa et al., 1987; Perera et al., 2005). The right-hand side of eq. (23) can be calculated using the thermody­ namic data of �Ei as discussed in Section [III-7] for 1-propanol �1P�−H2 O. Thus, in

Eq. (IV-22) to (IV-25c)

85

determining the concentration fluctuation, the route via scattering experiments and that from the thermodynamic determination of the chemical potential are complementary. In particular, consideration of relative errors reveals immediately that when the value of the concentration fluctuation is large, scattering experiment is advantageous in that the relative uncertainty is small; about the same as the uncertainty of the scattering inten­ sity measurements. From the thermodynamic route, on the other hand, the denominator, ���i /�xi �, is small near the maximum of the concentration fluctuation and its relative uncertainty inevitably becomes large. The concentration fluctuation is closely related to the Kirkwood-Buff integral, Gij , defined as, Gij � ≡

�

4�r 2 �gij �r� − 1�dr�

(IV-24)

0

where gij �r� is the pair correlation function between i-j pair of molecules with a distance r apart. On the other hand, Gij for a two component system consisting 1 and 2 can be calculated by using thermodynamic quantities as (Kirkwood & Buff, 1951; Bhatia & Thornton, 1970; Koga, 1984; Nishikawa et al., 1987; Perera et al., 2005), G12 = RT�T −

V1 V2 � Vm D

� Vj Gii = G12 + − Vm /xi � D

(IV-25a)

�

(IV-25b)

where D is defined as, D≡

x 1 x2 � N

(IV-25c)

and Vi is the partial molar volume of the i-th component (i = 1, or 2, and j = 2, or 1), Vm the total molar volume and �T the isothermal compressibility. The denominator on the right of eq. (25c) is the concentration fluctuation given by eq. (23). We note that all these thermodynamic quantities contain the second derivatives of the Gibbs energy, G, as seen in Table I-1 (p. 20). Eq. (25) is a remarkable relation. The right-hand side is calculated from the macro­ scopic thermodynamic quantities while the left contains the microscopic pair correlation function, albeit in the integral form. While there is some skepticism about the power of Gij because of the integration, (Donkersloot, 1979; Zaitsev et al., 1989), eq. (25) in turn suggests that if we could manage to take a derivative of both sides of eq. (25), then we could have third derivative quantities accessible from thermodynamic measurements on the right-hand side. The “derivative” of Gij on the left might provide some microscopic information of the level of the pair correlation function. This is the main theme of this

86

study as briefly mentioned in Section [I-9] (see eq. (I-53) and the discussion imme­ diately following it). Throughout the remainder of this book, we obtain various third derivatives and demonstrate their usefulness in elucidating the molecular processes in aqueous solutions.

Part B STUDIES OF AQUEOUS SOLUTIONS USING THE SECOND AND THE THIRD DERIVATIVES OF G

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Chapter V

MIXING SCHEMES IN AQUEOUS MONO-OLS

[V-1] MIXING SCHEMES IN 2-BUTOXYETHANOL (BE)–H2 O To describe molecular processes in aqueous (or any other) solutions, the phrase “solution structures” is often used. The word “structure”, however, implies an ordered solidlike structure, so instead we use the term “mixing scheme” throughout this book. 2-Butoxyethanol (abbreviated as BE), C-C-C-C-O-C-C-OH, is the largest mono-ol that mixes with H2 O in the entire composition range at room temperature, although at tem­ peratures higher than 50 C a phase separation occurs with the lower critical solution point (LCST) (Ellis, 1967; Ito et al., 1983; Quirion et al., 1990). At low temperatures, the mixture freezes into ice and the addition compound, BE�H2 O�38 , or a solid solution consisting of the addition compound and perhaps the solid BE, depending on the compo­ sition range (Koga et al., 1994) The phase diagram is shown in Fig. V-1. Our conclusion at the end of this section [V-1] is that there are three composition regions labeled as I, II, and III within a single liquid phase domain in the phase diagram, in each of which the mixing scheme is qualitatively different from those in the other two regions. The hint for this finding came immediately from the composition dependence of the excess E E E HBE � HWE � TSBE , partial molar enthalpies and entropies (times T ) of BE and H2 O� TSBE E and TSW . These quantities are determined experimentally as discussed in Chapter III, and are the second derivatives of G. Prior to these second derivative quantities, what were available as thermodynamic quantities are the excess molar (integral) enthalpy and E + xW HWE , for example (see entropy, HmE and SmE , the first derivative of G � HmE = xBE HBE  Table I-1 (p. 20) in Section [I-8]). These values at 25 C are shown in Fig. V-2, together with GEm = HmE − TSmE . Evidently, on mixing BE and H2 O there is a modest enthalpy gain, but the loss of entropy surpasses the enthalpy gain resulting in a positive GEm . Note that in the figure T is multiplied to SmE and its units are the same as HmE and GEm . This practice of using the product of T and entropy is adopted throughout this book. Thus, we could conclude that the BE−H2 O system is rather hard to mix. The degree of difficulty of mixing, i.e. the positive value of GEm , is not large enough, so that the ideal mixing entropy term, RT �xBE ln xBE + xW ln xW �, is larger in its absolute value than GEm , and the resulting Gm = GEm + RT �xBE ln xBE + xW ln xW � is negative. Thus, the system remains mixed in the entire composition. Is there anything more that Fig. 2 suggests? Theoretically, all the thermodynamical information, at least its mole fraction dependence, should be contained in Fig. 2. Being human, however, we have difficulty in extracting all the information from

90

Fig. V-1 to V-3

100 80

Two Liquids

temp. / °C

60 40

III

I

20

II

0

Glass ?

Two Solids -20

Solid Solution 0.0 S

0.1

0.2

0.3

0.4

0.5

0.6

xBE

Fig. V-1. Phase diagram for 2-butoxyethanol (BE)−H2 O. S is an addition compound, possibly a clathrate, BE�H2 O�38 . The dotted lines indicate the boundary between Mixing Schemes I and II, and that between II and III. The details of Mixing Schemes I, II, and III are discussed throughout this section [V-1]. The data for phase boundaries are from Koga et al. (1994), Ellis (1967), Ito et al. (1983) and Quirion et al. (1990). Reproduced with permission from the Journal of Physical Chemistry, 100, 5172, (Koga Y.), Copyright (1996), American Chemical Society.

BE - H2O, 25 °C

GmE, HmE, or TSmE / kJ mol -1

1.0

G mE 0.5 0.0

H mE

-0.5

TS mE

-1.0 -1.5 -2.0

RT{x BEln(x BE) + x Wln(x W)}

-2.5 0.0

0.2

0.4

0.6

0.8

1.0

x BE

Fig. V-2. The excess molar functions, GEm � HmE , and TSmE for 2-butoxyethanol (BE)−H2 O at 25  C, together with the ideal mixing free energy, RT �xBE ln�xBE � + xW ln�xW ��.

91

Fig. 2 alone. Often, a model system is constructed and GEm � HmE or TSmE are calculated accordingly. The resulting GEm etc. is then compared with those determined experimen­ tally. We do not follow this route: rather, we determine the second and third derivative quantities directly, without resorting to any model system, and learn from the data. E E We thus show HBE and TSBE in Fig. V-3, and HWE and TSWE , in Fig. V-4, experimentally  determined at 25 C (Koga, 1995c;, 1996;, 2003a). It becomes immediately obvious here that the thermodynamic behaviors and hence the mixing schemes are qualitatively different in three composition regions approximately bounded by arrows in Fig. 3. We label these three regions I, II, and III from the H2 O-rich side, and call mixing schemes operating in each region Mixing Scheme I, II, and III, respectively. In the most BE-rich region, the BE mole fraction, xBE , larger than about 0.6, the E E values of HBE and TSBE are almost zero in comparison with the H2 O-rich region. This

0

H BEE or TS BEE (kJ mol

-1

)

-10

H BEE

-20

TS BEE

(a)

-30 0.0

0.2

0.4

0.6

0.8

1.0

0

-10

-20

(b) -30 0.00

0.05

0.10

0.15

x BE E Fig. V-3. The measured excess partial molar enthalpy and entropy (times T ) of BE, HBE and E TSBE for 2-butoxyethanol (BE)−H2 O at 25  C; (a) for the entire composition range, and (b) for H2 O-rich region. See text for arrows. Data from Koga (1996). Reproduced with permission from the Journal of Physical Chemistry, 100, 5172, (Koga Y.), Copyright (1996), American Chemical Society.

Fig. V-4

H WE

1

TS WE

H WE or TS WE (kJ mol

-1

)

92

0

-1

-2

-3 0.0

0.2

0.4

0.6

0.8

1.0

x BE E Fig. V-4. The measured excess partial molar enthalpy and entropy (times T ) of H2 O� HWE and TSW for 2-butoxyethanol (BE)−H2 O at 25  C. See text for arrows. Data from Koga (1996). Reproduced with permission from the Journal of Physical Chemistry, 100, 5172, (Koga Y.), Copyright (1996), American Chemical Society.

suggests that BE molecules in the mixture are in almost the same environment locally as in the pure state, and hence BE molecules must exist as clusters of themselves, perhaps in a micellar form. Fig. 4 shows in the same range that the value of HWE is small and constant, while TSWE shows modest composition dependence but the values themselves are also small noting the scale of the ordinate in Fig. 4. These observations suggest that H2 O molecules break away from liquid H2 O and interact with the existing BE clusters with very small enthalpy and entropy changes. If the BE clusters discussed above are of micelle type with −OH group pointing outward, then it could be probable that H2 O molecules interact with −OH’s of the BE clusters one by one, without much influence among H2 O molecules themselves. We call this mode of mixing Mixing Scheme III for BE-H2 O. In the intermediate region, BE-H2 O has a phase separation with the LCST at 50  C and at xBE = 0�06. At 25  C, therefore, this composition region is super-critical. Indeed, the intensities of light scattering (Ito et al., 1981; 1983) and small angle x-ray scattering (Hayashi and Udagawa, 1992b) show a sharp maximum towards the composition of the LCST at room temperature. On increasing temperature to that of the LCST, the maximum intensity grows rapidly. There is thus no doubt that clusters are present in this composition-temperature range, which grow in size and eventually reach the macroscopic E E and TSBE have positive slopes as xBE increases size at the LCST. Furthermore, that HBE in this region (though much less sharp in comparison with those in the H2 O-rich region) is consistent with the system having a potential of phase separation with an LCST, as discussed in detail in Section [II-8]. Thus, we conclude that BE-H2 O in this region is a mixture consisting of two kinds of clusters: One rich in H2 O and the other in BE. Below

93

50  C these clusters are still sub-macroscopic in size, but grow to the macroscopic scale on reaching the LCST. Above this, two phases exist with compositions at each end of the phase boundary on the isothermal tie line. We call this Mixing Scheme II. The BE-rich clusters are no doubt reminiscent of the clusters in Mixing Scheme III discussed above. E E We now turn to the H2 O-rich region, where both HBE and TSBE change sharply. We E recall that measurements of HBE are made by titrating a minute amount of BE into a large quantity of a given mixture of BE and H2 O. Thus, at the infinite dilution, the value E of HBE is the enthalpic effect of the process in which one mole of BE breaks away from its pure state and mixes into the infinite amount of H2 O. Its value is evidently E −17 kJ mol−1 , but that of TSBE is −26 kJ mol−1 . Thus the above process is associated with a large enthalpy gain, but a larger entropy (time T ) loss, hinting that some ordering occurs in the resulting mixture. This magnitude is only discernable if we consider a strong organization caused in H2 O by BE. Thus, the enhancement of the hydrogen bond network of H2 O in the immediate vicinity of BE, i.e. the “iceberg” formation, is a reasonable interpretation. The “iceberg” concept was formulated by Frank and Evans (1945) using the partial molar entropy (a second derivative of G!) data at the infinite dilution available at the time. It is therefore applicable only in dilute aqueous solutions. Nonetheless, this concept has been taken overly seriously and each time an experimental observation has been made contrary to this concept, it has been criticized or challenged, irrespective of the concentration of solute (for example, Dixit et al., 2002a). Indeed, we suggest throughout this section that it is only in this H2 O-rich region that the “iceberg” concept is partially true. Another confusion in the literature arises regarding the size of the solute in question. The original suggestion was based on the data of non-electrolyte solutes with the molecular weight 100 at most (Frank and Evans, 1945). Thus, the blind application of this concept for functions and structures of large bio-molecules in H2 O is an over-interpretation. Indeed, there is a theoretical work suggesting that the “iceberg” concept is applicable for small hydrophobic moieties. For solutes with large surface areas and curvatures, the hydrogen bonds are broken at all temperatures (Southall and Dill, 2000). There are mixed conclusions regarding the structure of H2 O in the vicinity of flat hydrophobic surfaces (infinite curvatures) (Ball, 2004). See Koga et al. (2004a) for a more detailed discussion. Fig. V-5 shows the isotope effect of the excess partial molar enthalpy of BE (Siu et al., 1992), which supports indirectly the “iceberg” formation. We titrated either deuterated C-C-C-C-O-C-C-OD (BD) or non-deuterated BE into either H2 O or D2 O. The four sets of measurements were carried out at 30  C. The results indicate clearly that it is the solvent (H2 O or D2 O) that dictates the value of the excess partial molar enthalpy of BE or BD, while it makes no difference whether the titrant is BE or BD up to the boundary to Mixing Scheme II. It is well known that the average hydrogen bond strength is higher for D2 O than for H2 O (Eisenberg and Kauzmann, 1969; Suzuki, 1980). The fact that the E E and HBD are larger in D2 O than in H2 O indicates that the main absolute values of HBE thermodynamic effect in Mixing Scheme I is occurring in solvent H2 O or D2 O. This indirectly supports the “iceberg” formation. In Mixing II, on the other hand, whether

94

Fig. V-5 to V-6

H BDE or H BHE / kJ mol -1

0 -2

BH into D2O

-4

BD into D2O

-6 -8 -10 -12 -14

BH into H2O

-16

BD into H2O

-18 0.00

0.01

0.02

0.03

x BD or x BH

Fig. V-5. The measured isotope effect of the excess partial molar enthalpy of 2-butoxyethanol in aqueous solution at 30  C. The deuterated 2-butoxyethanol, C-C-C-C-O-C-C-OD, (BD), or normal 2-butoxyethanol, C-C-C-C-O-C-C-OH, (BH), was titrated into D2 O or H2 O. See text for implication of the results. Data from Siu et al. (1992). Reproduced from the Journal of Chemical Thermodynamics, 24, 159, (Siu, W. W. Y. et al.), Copyright (1992), with permission of Elsevier.

the titrant is BD or BE, and not whether the titrand is D2 O or H2 O, seems to be more E E or HBD . This suggests the onset of BE or BD clustering, important in the values of HBE pertaining to the BE-rich clusters putatively present in this intermediate region. On increasing xBE further, the latter clusters dominate, or rather the other H2 O-rich clusters disappear, and Mixing Scheme III sets in. E at the infinite dilution decreases, On increasing temperature, the absolute value of HBE and the mixing scheme boundary to II seems to occur at a progressively smaller value of xBE , as shown in Fig. V-6 (Koga et al., 1990b). Since the hydrogen bond probability decreases on temperature increase, this observation also suggests that it is H2 O that dictates the thermodynamics in Mixing Scheme I. Returning to the very H2 O-rich region, it is striking that the thermodynamic situation is rapidly changing as xBE increases, as is evident in Figs. 3 and 5; namely the absolute E is rapidly decreasing as xBE increases. If the first bunch of BE molecules value of HBE has gained a large enthalpy, then the second bunch must settle in the system far away from the first so as to gain enthalpy equally, i.e. enthalpy repulsion. Entropy-wise, however, since the first bunch has lost a large entropy (times T ), the second tends to settle in the vicinity so as not to lose as much entropy as the first, i.e. entropy attraction. If these two effects compensate completely, then the ideal mixing entropy dictates a random distribution of BE. However, the entropy attraction is larger than enthalpy repulsion, hence BE molecules tend to attract each other at this composition region, except for the extremely dilute region where the ideal mixing entropy dictates repulsion, as discussed in Section [II-7].

Eq. (V-1a) to (V-1b)

95

0 -2

HBEE / kJ mol-1

-4 -6 -8

29.90 °C 40.00 °C 50.09 °C 56.15 °C 60.13 °C

-10 -12 -14 -16 -18 0.000 0.005

0.010

0.015

0.020

0.025

0.030

xBE

Fig. V-6. The temperature dependence of the excess partial molar enthalpy of 2-butoxyethanol (BE) in BE−H2 O. Koga et al. (1990b). Reproduced with permission from the Journal of Physical Chemistry, 94, 3879, (Koga, Y. et al.), Copyright 1990, American Chemical Society.

Fig. V-7 and Fig. V-8 show the BE-BE interaction functions, the third derivatives, E E HBE−BE and TSBE−BE defined as (see eq. (I-53), eq. (II-76b), and eq. (II-76c)), � E � � E � �HBE �HBE E HBE−BE = �1 − xBE � � (V-1a) ≡N �nBE �xBE � E � � E � �SBE �SBE E TSBE−BE = T�1 − xBE � � ≡ NT �nBE �xBE

(V-1b)

E E Both HBE−BE and TSBE−BE are positive, and hence the BE-BE interaction is repulsive (unfavorable) in enthalpy but attractive (favorable) entropy-wise. The value of the latter is larger, and hence the entropy attraction surpasses the enthalpy repulsion. We note, however, at the very dilute solution, the BE-BE entropic repulsion coming from the ideal mixing entropy is large, as discussed in Section [II-7]. The entropic attraction by the excess entropy, as calculated by eq. (1b), compensates the repulsion from the ideal mixing entropy at about xBE = 0�0065, as indicated in Fig. 8(b). The most striking feature in Figs. 7 and 8 is the sharp initial increase to point X and the E E and TSBE−BE , third derivatives of G. This equally sharp drop to point Y in both HBE−BE resembles the heat capacity anomaly associated with the typical second order phase tran­ sition. See Appendix C for various types of anomalies in heat capacity associated with phase transitions and the commentary about the “order” of phase transition. Desnoyers et al. (Roux et al., 1978a & b) observed that the partial molar heat capacity of BE (a third derivative of G) show a sharp peak for BE-H2 O. They called this phenomenon in the

96

Fig. V-7(a) to V-8(b)

1200

25.00 °C 29.90 °C 40.00 °C 50.09 °C 60.13 °C

X 1000

HBE-BEE / kJ mol-1

Error 800 600

(a)

400 200

II I

Y

0

0.00

0.01

0.02

0.03

0.04

0.05

0.06

xBE E Fig. V-7(a). The enthalpic interaction between solutes in 2-butoxyethanol (BE)−H2 O� HBE−BE , at various temperatures. The lines drawn are a guide for the eye; they are used to locate point X, the presumed maximum, which is equivalent of the inflection point in the H2 O-rich region for the E HBE vs. xBE curves shown in Figs. 3, 5, and 6. Point X marks the onset of transition from Mixing Scheme I to II, and point Y is the end point of transition. Mixing Scheme II is operative to the right of point Y. Mixing Scheme III is apparent in Fig. 7(b). Koga et al. (1990b). Reproduced with permission from the Journal of Physical Chemistry, 94, 3879, (Koga, Y. et al.), Copyright 1990, American Chemical Society.

50

(b) 25 °C

HBE-BEE / kJ mol-1

40

30

II

20

III

10

0

-10 0.0

0.1

0.2

0.3

0.4

0.5

0.6

xBE E Fig. V-7(b). The same as Fig. 7(a) but an expanded view for a small HBE−BE and a large xBE region for 25  C only. The boundary between Mixing Schemes II and III is now apparent. Tanaka et al. (1996). Reproduced with permission from the Canadian Journal of Chemistry, 74, 713, (Tanaka, S. H. et al.), Copyright (1996), NRC Research Press, National Research Council of Canada, Ottawa.

97

1400

X

1200

(a)

TSBE-BEE / kJ mol-1

1000 800 600 400

II

I

200

Y 0 0.00

0.01

0.02

0.03

0.04

0.05

xBE

Fig. V-8(a). The entropic interaction (times T ) between solutes in 2-butoxyethanol E E E (BE)−H2 O� TSBE−BE , at 25  C. Note that the values of TSBE−BE is larger than those of HBE−BE − shown in Fig. 7. Namely, the entropic BE BE attraction is stronger than the enthalpic repulsion, E in terms of excess quantity. Calculated using TSBE data (Koga, 1991).

TSBE-BEid or -TS BE-BEE / kJ mol-1

0 -200 -400

(b) -600 -800 -1000

TS BE-BEid

-1200

-TSBE-BEE

-1400 0.00

0.01

0.02

0.03

0.04

0.05

xBE

Fig. V-8(b). Comparison between the excess entropic attraction between solutes in 2­ E id butoxyethanol (BE)−H2 O� TSBE−BE , and the ideal mixing entropic repulsion, TSBE−BE . See text for detail.

98

Fig. V-8(c) to V-9

100

(c)

TSBE-BEE / kJ mol-1

80

60

II

40

III

20

0 0.0

0.2

0.4

0.6

0.8

1.0

xBE E Fig. V-8(c). An expanded version of Fig. 8(a) in small TSBE−BE and large xBE range. The boundary E between Mixing Schemes II and III is apparent. Calculated using TSBE data (Koga, 1991).

BE-H2 O the “pseudo-phase” transition, by noting its resemblance to the anomaly of heat capacity (a second derivative quantity) for a phase transition. Here we suggest that the anomalies of the third derivatives of G shown in Figs. 7 and 8 are associated with the transition of mixing scheme, the change-over from Mixing Scheme I to II. The “phase” is a macroscopic entity and its transition is understood as the change in some long-range order parameters, which is accompanied by anomalies of the second derivatives of G. Here, the “mixing scheme” is a molecular arrangement of the mixture within a single liquid phase, and the anomalies are observed in the third derivatives. Hence it is more subtle, perhaps associated with some short- or medium-range order parameters. Thus, the boundary from Mixing Scheme I to II in the BE−H2 O is associated with a peak type anomaly in third derivatives of G. E E and TSBE−BE extrapolate Another point to note in Figs. 7 and 8 is that both HBE−BE E E to non-zero values at xBE = 0; or equivalently, HBE and TSBE do not show zero slope towards xBE = 0 in Fig. 3. If there were a finite range where their slopes were zero, it would be the Henry’s law region as discussed in Section [II-4]. Since we did not observe this at least down to the first measurement at xBE ≈ 0�001, we expanded the determination E of HBE to xBE ≈ 1 × 10−5 without any sign of Henry’s law (Westh et al., 1998). This means that the BE-BE interaction is of extremely long range, or a BE molecule interacts with another BE in the presence of 2 × 105 molecules of H2 O. Therefore, if we attempt to understand Mixing Scheme I by means of computer simulation, the size of a coarse grain ought to be at least an order of magnitude larger than this number, i.e. at least 106 molecules of H2 O are required. As discussed in Section [IV-4] above, liquid H2 O has its characteristic bond-percolated hydrogen bond network, and such a long-range BE-BE

99

Hydrogen Bond Probability

interaction might be occurring via the hydrogen bond network. This is indeed the case as will be discussed below in Section [V-3]. We now make an attempt to understand the details of molecular processes in Mixing E E Scheme I. Returning to Fig. 3, the absolute values of HBE and TSBE decrease sharply as xBE increases. The first bunch of BE alters the molecular organization of H2 O by enhancing the hydrogen bond network of H2 O in the immediate vicinity, but there must be some additional effects. One possibility for such a long-range interaction could be that the hydrogen bond probability profile may be established such as sketched in Fig. V-9. Namely, the “first BE” enhances the probability to unity in its immediate vicinity. At the same time, it affects the probability profile to a long-range as depicted in Fig. 9 over the average probability for pure H2 O. Then the second needs to exert a lesser effect on the hydrogen bond probability profile. If this process continues as xBE increases, when it completes its course, the icebergs may completely fill the entire system. At this point there is no room to accommodate BE any longer, marking the end of Mixing Scheme I. More BE introduced in the system now must cluster together pertaining to the BE-rich cluster in Mixing Scheme II. If this picture is correct and the ice-like structure fills the entire space, then the ionic conductivity of H+ and OH− should increase in the BE-H2 O as xBE increases up to the boundary between Mixing Schemes I and II. Such a structure would facilitate H+ hopping along the completed hydrogen bond network. Contrary to this expectation, the ionic conductivity in fact decreased (Koga et al., 1995a), as shown in Fig. V-10. The ionic conductivity of H+ OH− was calculated from those of KCl, KOH, and HCl at the infinite dilution. This result was the first hint that the iceberg does

1.0

0.5

Average for pure H2O

1st BE

2nd BE

0.0

Space

Fig. V-9. A presumed snapshot (Davies et al., 1994) of the hydrogen bond probability profile for Mixing Scheme I in 2-butoxyethanol (BE)−H2 O. As will become evident, this picture is wrong (see text [V-1]). See Fig. 14 for a more correct picture at the present stage. Reproduced from the Journal of Solution Chemistry, 23, 339, (Davies, J. V. et al.), Copyright (1994), with kind permission from Springer Science and Business Media.

100

Fig. V-10 to V-11(b)

35 30

0Λ' j

/ Ω-1 cm-1

25

j=

20

H+OH­

15

H+Cl10

K+OH­ K+Cl-

5

I 0 0.00

II

0.01

0.02

0.03

0.04

0.05

xBE

Fig. V-10. The mole fraction conductivity of the j-th species, 0 �j , in 2-butoxyethanol (BE)−H2 O at 25  C. j = HOH, HCl, KOH, or KCl (Koga et al., 1995a). See text. Reproduced with permission from the Canadian Journal of Chemistry, 73, 1294, (Koga Y. et al.), Copyright (1995), NRC Research Press, National Research Council of Canada, Ottwa.

not fill the entire system, but rather the hydrogen bond probability of bulk H2 O away from iceberg-clad BE may in fact be decreasing as xBE increases. The remainder of this section is devoted to a discussion of this issue. Fig. V-11 shows another second derivative of G, the excess partial molar volumes of E (Koga et al., 1993). As is evident in Fig. 11(a), VWE increases H2 O and BE, VWE and VBE at first, supporting the iceberg formation, but then it decreases before the I-II boundary. Furthermore, if the iceberg would fill the entire system at the boundary as argued above, −1 the value of VWE would be close to that of ice, about 1�5 cm3 mol ! Fig. 11(a) shows an −1 increase of the order of 0�001 cm3 mol , some 1500 times smaller. There must therefore be a different kind of additional effect from what we argued above operating to reduce VWE progressively and to reduce the ionic conductivity of H+ OH− . Before settling this issue, we now turn to the temperature dependence of the boundary between Mixing Schemes I and II. As we argued above, the anomaly of the third deriva­ E E tive quantity, in particular, point X of HBE−BE and TSBE−BE , represents the change-over point. As introduced in Section [I-8], we label these third derivatives by listing the E independent variables of differentiation as �T� nBE � nBE �. Fig. V-12 shows VBE−BE ; �p� nBE � nBE � (Koga, 1992) defined as, � E VBE−BE ≡N

E �VBE �nBE

�

� = �1 − xBE �

E �VBE �xBE

� �

(V-2)

Eq. (V-2)

101

0.02 0.00

VWE / cm3 mol-1

-0.02 -0.04 -0.06 -0.08 -0.10

(a)

-0.12 -0.14 0.00

0.01

0.02

0.03

0.04

xBE

Fig. V-11(a). Excess partial molar volume of H2 O in 2-butoxyethanol (BE)−H2 O� VWE , at 25  C (Koga et al., 1993). The arrow indicates the transition of mixing scheme from I to II. Reproduced from the Journal of Chemical Thermodynamics, 25, 51, (Koga et al.), Copyright (1993), with permission from Elsevier.

-3 -4

VBEE / cm3 mol-1

-5 -6 -7 -8

(b)

-9 -10 -11 0.00

0.01

0.02

0.03

0.04

xBE E Fig. V-11(b). Excess partial molar volume of 2-butoxyethanol (BE) in BE−H2 O� VBE , at 25  C (Koga et al., 1993). The arrow indicates the transition of mixing scheme from I to II. Reproduced from the Journal of Chemical Thermodynamics, 25, 51, (Koga et al.), Copyright (1993), with permission from Elsevier.

102

Fig. V-12 to V-13

800

15 °C 25 °C

VBE-BEE / cm3 mol-1

600

400

200

0

-200

-400 0.00

0.01

0.02

0.03

0.04

xBE E Fig. V-12. The volumetric interaction between solutes in 2-butoxyethanol (BE)−H2 O� VBE−BE , at 15 and 25  C (Koga, 1992). Reproduced with permission from the Journal of Physical Chemistry, 96, 10466, (Koga, Y.), Copyright (1993), American Chemical Society.

E An anomaly similar to that of HBE−BE is apparent with point X being at about the same locus of xBE . We calculated from the second derivative quantities a variety of third derivatives at various temperatures. They include the nW -derivative of the excess partial molar volume of W, �p� nW � nW � (Koga, 1992); the nBE -derivative of the excess partial molar heat capacity of BE, �T� T� nBE � (Roux et al., 1978a & b); the nBE -derivative of the compressibility, �p� p� nBE � (Koga et al., 1995b); the nBE -derivative of the thermal expansivity, �p� T� nBE � (Davies et al., 1992); the T -derivative of the heat capacity at E at a given xBE � �T� T� T � (Westh et al., 1994; Atake and Koga, 1994), and HBE−BE various temperatures, �T� nBE � nBE � (Koga et al., 1990b). They all show similar peak anomalies except for �T� T� T � which showed a step type of anomaly. The loci of anomalies (point X) of all these third derivatives of G are plotted in Fig. V-13 (Koga et al, 2004a) together with the phase diagram for BE-H2 O (Koga et al., 1994; Ellis, 1967). The plots seem to form a single line that separates the xBE -T field into the regions of Mixing Schemes I and II. This line is sometimes called the Koga-line (Hvidt, 1993; Perera, 2002; Nishikawa et al., 2003). In the region above and to the right of the Koga-line (beyond point Y), Mixing Scheme II is operative in which two kinds of clusters are mixed with each other, one rich in H2 O and the other in BE. In terms of the percolation nature of the hydrogen bond network of H2 O, Mixing Scheme II is devoid of the hydrogen bond percolation. This hints that the Koga-line marks the end of the hydrogen bond percolation nature that exists in Mixing Scheme I all the way down to xBE = 0, the pure H2 O. We recall the site-correlated percolation model of pure H2 O by Stanley & Teixeira (1980) discussed in Section [IV-4]. We also recall that their estimate for the average hydrogen bond

103

100

L+L

80 60 40

Temperature (°C)

20

I

II

I

II

(a)

0 -20 20

10

(b)

0

tP -10

-20

Solid Sol.

Ice + S

-30 0.00

0.02

0.04

S

0.06

0.08

0.10

xBE

Fig. V-13. Mixing scheme and phase diagram for 2-butoxyethanol (BE)−H2 O (Koga et al., 2004a). S is an addition compound of BE�H2 O�38 . tp is its incongruent melting point. Reproduced with permission from the Journal of Physical Chemistry A, 108, 3873, (Koga, Y. et al.), Copyright (2004), American Chemical Society.

probability decreases on increasing temperature, and reaches 39%, the bond percolation threshold for ice Ih (or tridymite) type bond connectivity at about 80  C. Fig. 13 indicates that the Koga-line seems to cut the xBE = 0 axis at about 83  C. This general coincidence is a circumstantial support for the above conjecture that the Koga-line is the hydrogen bond percolation threshold in BE-H2 O. Thus, Mixing Scheme I is such that BE enhances the hydrogen bond network of H2 O in the immediate vicinity of BE, i.e. the iceberg formation. At the same time the hydrogen bond probability of bulk H2 O away from the iceberg-clad BE decreases concomitantly as if the global temperature increases. As xBE increases, and more icebergs form, the hydrogen bond probability of bulk H2 O away from BE decreases progressively to 39%, whereupon the bulk H2 O loses its integrity as the liquid H2 O and Mixing Scheme II sets in. This will explain the decrease in the H+ OH− conductivity (Fig. 10) and the minute initial increase in VWE in Fig. 11(a). E On the other hand, the initial decrease in VBE (Fig. 11(b)) is the consequence of the E initial increase in VW (Fig. 11(a)), according to the Gibbs-Duhem relation, but could be

104

Fig. V-14 to V-15(a)

Hydrogen Bond Probability

explained at the molecular level as follows. The molecular size of BE is much larger by about 8-fold than that of H2 O. Hence “sands filling the gaps of pebbles” analogy would result in a negative excess partial molar volume of BE. However, H2 O is not just a small molecule, but is bulkier due to the hydrogen bond network with putative ice-like E and the net result is the actual patches. This bulkiness contributes positively to VBE E value of VBE at xBE = 0. As xBE increases, the hydrogen bond probability of bulk H2 O E deceases decreases where incoming BE settles. Hence the positive contribution to VBE E E progressively, resulting in the initial decrease in VBE . The sharp increase in HBE−BE and E must then be coming from the competition between the short-range effect of TSBE−BE iceberg formation and the long-range effect from the reduction of the hydrogen bond probability of bulk H2 O. Thus, the hydrogen bond probability profile (Fig. V-14) may be more appropriate in place of the scenario of the iceberg filling the entire system (Fig. 9). Another important point to note in Fig. 13 is that the Koga-line ends at the incongruent melting point, −3�7  C, of an addition compound labeled as “S” in the phase diagram. The composition of S translates to BE�H2 O�38 , and 38 H2 O molecules are just enough to cover the surface of BE, using the respective molar volumes and assuming both are spherical. This suggests that at the ambient condition within Mixing Scheme I the solution is preparing to form the addition compound on freezing, providing more circumstantial evidence for the existence of the icebergs with BE in the middle in solution. On cooling BE-H2 O in Mixing Scheme I, H2 O in the bulk away from the icebergs precipitates as ice at the liquidus curve, and eventually the addition compound BE�H2 O�38 forms

1.0

Average for pure H2O

0.5

Percolation Threshold

1st BE

2nd BE

0.0

Space

Fig. V-14. A snapshot of the hydrogen bond probability profile for Mixing Scheme I of 2­ butoxyethanol (BE)−H2 O. In the immediate vicinity of BE the probability is one, but in the bulk H2 O away from solutes the probability is reduced from the average for pure H2 O. When the bulk average reaches the hydrogen bond percolation threshold, the hydrogen bond network loses its infinite connectivity and the mixture consists of two kinds of clusters, one rich in H2 O and the other in BE, i.e. Mixing Scheme II sets in.

105

at −3�7  C. Since the hydrogen bond probability of bulk H2 O is reduced by the presence of the iceberg-clad BE, it requires a lower temperature than 0  C to form ice. This is a molecular level understanding of the freezing point depression normally deduced by thermodynamic reasoning alone; see Appendix D for thermodynamic arguments for freezing point depression. In summary, Mixing Scheme I is such that BE molecules are surrounded by a number of H2 O molecules whose hydrogen bond probability is enhanced, i.e. iceberg formation. Concomitantly, however, the hydrogen bond probability of bulk H2 O away from the iceberg-clad BE is reduced, as shown in Fig. 14 as a snapshot of the hydrogen bond probability profile. The probability is still high enough initially until it reaches the bond percolation threshold. Hence, the hydrogen bond percolation is intact up to the threshold composition. In other words, H2 O protects its integrity as the liquid H2 O though sacrificing the hydrogen bond probability of bulk H2 O against invading BE. When xBE reaches the threshold value, the hydrogen bond percolation is broken and Mixing Scheme II sets in. As will be shown in Setion [V-3] below, the H2 O molecules in the iceberg do not participate in the characteristic hydrogen bond strength fluctuation; namely, icebergs are solid units that would eventually form the addition compound. Fig. V-15 shows a sketch of the Mixing Schemes I, II and III, by representing H2 O as a two-dimensional square lattice shown in Fig. 6. One remaining point to discuss about the boundary between Mixing Schemes I and E E (and TSBE−BE ) starts at point X and seems to end II is that the anomaly in HBE−BE at point Y as seen in Figs. 7(a) and 8(a). Why does the boundary seem to have this width? Our conjecture is that the width reflects the fluctuating hydrogen bond strength distribution. Point X in Figs. 7(a) and 8(a) is where the strongest hydrogen bond in distribution reaches the bond percolation threshold, while at point Y, the weakest and hence all the hydrogen bonds lose the network connectivity. At present there is no experimental evidence to support this, and until such evidence becomes available, it remains just a conjecture. The boundary from Mixing Scheme II to III, on the other hand, is not as conspicuous as that between I and II. As shown in Figs. 7(b) and 8(c), Mixing Scheme I:

Fig. V-15(a). A snapshot of Mixing Scheme I (Parsons, 2001). H2 O is represented by the twodimensional square lattice shown in Fig. IV-6. Large filled circles represent BE molecules. In their immediate vicinity the hydrogen bond network is enhanced (iceberg formation). The bulk H2 O away from BE has a lower hydrogen bond probability than in the pure state, but the hydrogen bond percolation is still intact. The BE−BE interaction is occurring via the percolated hydrogen bond network. See text.

106

Fig. V-15(b) to V-17

Mixing Scheme II

H2O rich cluster

BE rich cluster

Fig. V-15(b). Mixing Scheme II is such that the system is a mixture of two kinds of clusters, one rich in H2 O, and the other in BE. The H2 O-rich clusters are reminiscent of the icebergs in Mixing Scheme I, and the BE-rich clusters are predominant in Mixing Scheme III. Parsons (2001).

Mixing Scheme III:

Fig. V-15(c). Mixing Scheme III for 2-butoxyethanol (BE)−H2 O (Parsons, 2001). BE molecules form clusters of its own kind and H2 O interacts with the BE clusters as a single molecule. For BE, the cluster could be of micelle type with OH groups pointing outward. H2 O molecules interact predominantly with OH group one by one, without lateral interaction with other H2 O. The large circle represents OH group, small circle H2 O molecule and wiggled line the alkyl chain in BE.

the boundary is located as the intersection of two straight lines crossing with a shallow E angle. Hence locating the II-III boundary using the xBE dependence of HBE−BE and/or E E E TSBE−BE is problematic. This is due to the fact that the values of HBE and TSBE are small in Mixing Scheme III. (This fact was instrumental in arriving at the interpretation of Mixing Scheme III.) This difficulty is circumvented below in Section [V-5].

[V-2] MIXING SCHEMES IN OTHER MONO-OLS (AL)–H2 O E Figs. V-16, 17 and 18 show the excess partial molar enthalpy, HAL , entropy (times T ), E E − TSAL , and volume, VAL , of alcohol (AL) in the binary AL H2 O at 25  C, respectively. Alcohols studied are methanol (ME), ethanol (ET), 1-propanol (1P), 2-propanol (2P) and tert-butanol (TBA). Together with 2-butoxyethanol (BE) discussed above in Section [V–1], these alcohols mix with H2 O in the entire composition range at about room

107

5

HALE / kJ mol-1

0

-5

ME ET

-10

1P 2P

-15

TBA

-20 0.0

0.2

0.4

0.6

0.8

1.0

xAL (AL = ME, ET, 1P, 2P, or TBA) E . Alcohols Fig. V-16. The excess partial molar enthalpy of alcohol (AL) in AL−H2 O at 25  C� HAL are methanol (ME), ethanol (ET), 1-propanol (1P), 2-propanol (2P), and tert-butanol (TBA). Data E are from Tanaka et al. (1996), except for TBA. The values of HTBA are extrapolated to 25  C using E  the data at 26�90 C (Koga, 1986; 1988a) and the T -dependence of HTBA . See Koga et al. (1990a).

5

T SALE / kJ mol-1

0

-5

ME -10

ET 1P

-15

2P -20

TBA

-25

0.0

0.2

0.4

0.6

0.8

1.0

xAL (AL = ME, ET, 1P, 2P, or TBA) E Fig. V-17. The excess partial molar entropy (times T ) of alcohol (AL) in AL−H2 O at 25  C� TSAL . The data are taken from Koga et al. (2003) except for TBA (Koga et al., 1990a). Reproduced with permission from the Canadian Journal of Chemistry, 81, 150, (Koga, Y. et al.), Copyright (2003), NRC Research Press, National Research Council of Canada, Ottawa.

108

Fig. V-18 to V-19

2

VALE / cm3 mol-1

0 -2 -4

ME, 25 °C ET, 25 °C 1P, 25 °C 2P, 25 °C TBA, 26 °C

-6 -8 -10 -12 0.0

0.2

0.4

0.6

0.8

1.0

xAL (AL = ME, ET, 1P, 2P, or TBA) E Fig. V-18. The excess partial molar volume of alcohol (AL) in AL−H2 O� VAL . The density data for methanol (ME), ethanol (ET), 1-propanol (1P) are taken from Benson and Kiyohara (1980). The density data for 2-propanol (2P) are from Sakurai (1988), and those for tert-butanol (TBA) are from Sakurai (1987).

temperature. There are no other mono-ols that mix with H2 O in the entire composition. As is immediately evident from the figures, their thermodynamic behaviors are qualita­ E E and TSAL (Figs. 16 and 17) show a tively the same as that for BE−H2 O: Namely, HAL sharp increase in the H2 O-rich region, while both stay almost constant and close to zero E (Fig. 18) in the AL-rich region, xAL >0�6. The excess partial molar volume of AL, VAL shows the characteristic initial decrease as for BE−H2 O (Fig. 11(b)). It is thus likely that Mixing Schemes I and III are equally operative in all AL−H2 O systems. As for the E E shows a slow increase for all AL as for BE-H2 O� HAL intermediate region, while TSAL decreases for all AL but ME. However, as Fig. V-19 indicates, the excess chemical potential of AL is a decreasing function of xAL for all AL including BE, which is a requirement for a phase separation to occur. Furthermore, as will be discussed below in Section [V-4], there are a number of light or x-ray scattering works showing a large scattering intensity in this intermediate region for all AL including BE, suggesting the presence of clusters. Thus, we tentatively suggest that Mixing Schemes I, II, and III are equally operative in all AL that mix with H2 O at room temperature, the same as in BE−H2 O. We discuss Mixing Scheme II in Section [V-4] and III in Section [V-5] in some more detail, and concentrate on Mixing Scheme I in this section. E E E Table V-1 lists the values of �EAL � HAL � TSAL and VAL at the infinite dilution together with limited existing data for comparison. (The other existing data are mostly limited at E is always larger than the infinite dilution.) As for BE-H2 O, the absolute value of TSAL E E that of HAL , which is the consequence of �AL to be positive. The latter is a requirement for the mixture to have the possibility of phase separation as discussed above and in

109

10

ME ET 1P 2P TBA

μALE / kJ mol-1

8

6

BE

4

2

0

-2

0.0

0.2

0.4

0.6

0.8

1.0

xAL (AL = ME, ET, 1P, 2P, TBA or BE )

Fig. V-19. The excess chemical potential of solute in alcohol−H2 O� �EAL , at 25  C. The data for methanol (ME), ethanol (ET), 1-propanol (1P), and 2-propanol (2P) are taken from Hu et al. (2003), those for tert-butanol (TBA) from Koga et al. (1990a), and those for 2-butoxyethanol (BE) from Koga (1991). Reproduced with permission from the Canadian Journal of Chemistry, 81, 141, (Hu, J. et al.), Copyright (2003), NRC Research Press, National Research Council of Canada, Ottawa.

Section [II-8]. Also shown in the list are the data for 1-butanol (1B), 1-pentanol (1Pe), 1-hexanol (1Hx), 1-Octanol (1Oc), and 1-Decanol (1De). All these latter alcohols have de-mixing gaps with H2 O, which are listed in Table V-2. For 1B−H2 O, it was shown that within the narrow solubility limit in the H2 O-rich region, x1B 0�45. See text. Reproduced with permission from the Canadian Journal of Chemistry, 74, 713, (Tanaka S. H. et al.), Copyright (1996), NRC Research Press, National Research Council of Canada, Ottawa.

400 350

TS AL-ALE / kJ mol-1

300

ME

250

ET 1P

200

2P TBA

150 100 50 0 -50

0.0

0.1

0.2

0.3

0.4

0.5

xAL (AL = ME, ET, 1P, 2P, or TBA) E Fig. V-22. The entropic interaction (times T ) between solute alcohols (AL), TSAL−AL , at 25  C. The data are taken from Koga et al. (2003) for all but TBA. The latter data are calculated using E TSTBA data at 25  C from Koga et al.(1990a). Reproduced with permission from the Canadian Journal of Chemistry, 81, 150, (Koga, Y. et al.), Copyright (2003), NRC Research Press, National Research Council of Canada, Ottawa.

Eq. (V-3)

113

50

40

TS ME-MEE / kJ mol-1

X 30

20

III

II

I 10

Y 0 0.0

0.2

0.4

0.6

0.8

1.0

xME E Fig. V-23. The entropic interaction between solutes in methanol (ME)−H2 O, TSME−ME , at 25  C (Koga et al., 2003). The boundary between Mixing Schemes II and III is less conspicuous. Reproduced with permission from the Canadian Journal of Chemistry, 81, 150, (Koga, Y. et al.), Copyright (2003), NRC Research Press, National Research Council of Canada, Ottawa.

a smaller value of xAL to bring the system to point X as well as to point Y. We may rank the hydrophobicity of each alcohol by these sets of data, which give the consistent ranking as, ME 2P > ME, holds for hydrophobicity. Figs. VII-10 and VII-11 are the results of an attempt to find the origin of hydrophobicity within the present 1P-probing methodology. We focus on the x1P -locus of point X, since the shift of point X to the left seems to be purely due to hydrophobicity of S. A simplistic view is that the degree of hydrophobicity depends on the number of carbon atoms in the alkyl chain. Thus, in Fig. 10 the value of x1P at point X is plotted against 0 the carbon atom fraction, xcarbon (= xS0 × the number of carbon atoms). It is apparent in the figure that these plots seem to converge into a single curve within the estimated uncertainty. In turn, the degree of the left shift of point X will serve as a hydrophobicity scale within the present methodology. A recent examination indicated that the standard

0.08

ME Error

2P

x 1P at point X

0.06

TBA

0.04

ME 0.02

2P I TBA

0.00 0.00

0.02

0.04

0.06

0.08

x S0 (S = ME, 2P or TBA)

Fig. VII-9. The x1P - locus of point X of the H1EP−1P pattern in 1P−S−H2 O at 25  C. The raw data are from Koga (2003b) for ME, Hu et al. (2001) for 2P, and Miki et al. (2005) for TBA. The vertical dotted line shows the upper limit of xS0 below which Mixing Scheme I is operative in binary S−H2 O for S = ME, 2P or TBA, i.e. below which the present 1P-probing methodology is valid.

186

Fig. VII-10 to VII-12

0.06

ME

x 1P at point X

2P TBA 0.04

0.02

Error

0.00 0.00

0.05

0.10

0.15

0.20

x carbon0

Fig. VII-10. The x1P locus of point X in the H1EP−1P pattern in 1P−S−H2 O (S = ME, 2P or TBA) at 25  C. The abscissa is the mole fraction of carbon atom. See text.

ME

0.06

2P

x 1P at point X

TBA

0.04

0.02

Error

0.00 0.0

0.1

0.2

0.3

0.4

0.5

x (C-H bond)0

Fig. VII-11. The x1P locus of point X in the H1EP−1P pattern in 1P−S−H2 O (S = ME, 2P, or TBA) at 25  C. The abscissa is the mole fraction of C−H bonds. See text

187

free energy of transfer from H2 O to hexadecane of alkanes, alkenes, alkadienes and arenes scale nicely with the number of C–H bonds (Kyte, 2003). This argument uses the zeroth derivative of G, while ours uses the x1P -dependence of the third derivative quantity, H1EP−1P . Nonetheless, Fig. 11 shows the same plots vs. xC0 −Hbonds . These plots also seem to converge to a single curve. We discuss this issue further below.

[VII-3] EFFECTS OF UREA (U), TETRAMETHYL UREA (TMU) AND ACETONE (AC) ON H2 O AS PROBED BY THE H1EP−1P PATTERN CHANGE =O and the −NH2 work as hydrophilic Urea (U) lacks in alkyl group, while the >C= =O as a proton accep­ centers. U could readily form hydrogen bonds to H2 O with >C= tor and −NH2 a donor. It is generally known that the O · · · H−O∗ type hydrogen bond is stronger than the N–H · · · O∗ , where O∗ is the oxygen of H2 O. The H· · · O distance is shorter for the former case than that of the latter (Huheey, 1978). Thus, =O� is lopsided in the strength of the proton donor �−NH2 � and the acceptor �>C= U. Nonetheless, urea could be regarded as a typical hydrophilic solute. Indeed the excess partial molar volume of U in the binary U−H2 O (Fig. VII-12) shows no sign 2

0

ViE / cm3 mol-1

-2

-4

-6

U TMU

-8

AC -10 0.0

0.1

0.2

0.3

0.4

xi (i = U, TMU, or AC)

Fig. VII-12. The excess partial molar volume of i, ViE (i = U, TMU, or AC) in the binary i−H2 O at 25  C. Calculated using the density data by Boeje & Hvidt (1971) for U, Phillips et al. (1974) and Jakli & Van Hook (1996) for TMU, and Boeje & Hvidt (1971) for AC.

188

Fig. VII-13 to VII-14

of the initial decrease typical of a hydrophobic solute. Also shown in Fig. 12 are the excess partial molar volume of tetramethyl urea (TMU) and that of acetone(AC). We will discuss below how the effect of S changes, when the −NH2 of urea is con­ verted to −N�CH3 �2 and further to −CH3 on the molecular organization of H2 O. Since U is a solid at room temperature, the excess partial molar enthalpy of U, HUE , has not been determined directly, although it is possible to titrate aqueous urea instead and convert the data to HUE , as discussed in Appendix E. The heats of solution of urea into H2 O have been determined in the literature (Egan & Luff, 1966) but their data are not so closely spaced in composition as to determine HUE with confidence, let alone HUE−U . Hence, we have no information as to the location of point X, the onset of transition from Mixing Scheme I to II, if any, in U−H2 O. The present 1P­ probing methodology, on the other hand, is convenient in that 1P is readily titrated in the binary U−H2 O mixed solvent to learn about the effect of U on H2 O in the H2 O-rich region. Fig. VII-13 shows H1EP in the ternary 1P−U−H2 O (To et al., 1998). It is immediately clear that the xS0 (S = U)-dependence is qualitatively differ­ ent from those for S = ME, 2P, and TBA shown in Figs. 2 to 4. The H1EP−1P data were re-evaluated using the original H1EP data; the results are shown in Fig. VII-14. As is evident in the figure, the H1EP−1P pattern reserves type (a) dependence up to xS0 �S = U� = 0�2, confirming that Mixing Scheme I is still operative. The figure indicates

0

H 1PE / kJ mol-1

-2

xU0 = 0

-4

0.03040 0.04985

-6

0.1020 0.1501

-8

0.2035 -10

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

x1P

Fig. VII-13. The excess partial molar enthalpy of 1P, H1EP , in 1P−U−H2 O at 25  C. The data are from To et al. (1998). Reproduced with permission from the Journal of Physical Chemistry B, 102, 10958, (To, E. C. H. et al.), Copyright (1998), American Chemical Society.

189

250

xU0 = 0 Error

X

0.03040 0.04985 0.1020 0.1501 0.2035

H 1P-1PE / kJ mol-1

200

150

100

50

0

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

x1P

Fig. VII-14. The 1P–1P enthalpic interaction, H1EP−1P , in 1P−U−H2 O at 25  C. Revaluated using the original H1EP data by To et al. (1998).

that the x1P -locus of point X remains the same, while the value of H1EP−1P is suppressed as xS0 (S=U) increases up to 0.2. The solubility in the binary U−H2 O at 25  C is 0.266 in the mole fraction of U, beyond which is the macroscopic manifestation of Mixing Scheme III. Unfortunately, we have no information at this stage regarding the transition from Mixing Scheme I to II, if any. From the observation from Fig. 14, however, it is likely that the point X, the onset of the transition in binary U−H2 O, could be in the range 0�2 < xU < 0�27. Nevertheless, it is clear that U does not change the x1P locus of point X. Namely, it requires the same amount of 1P to drive the system to point X, the onset of the loss of the percolation nature of the hydrogen bond network of H2 O. We suggest, therefore, that U does not alter the connectivity of the hydrogen bond network of H2 O. This is probably due to urea’s capability of forming hydrogen bonds with the existing network of H2 O. The suppression of the value of H1EP−1P , or equivalently that of SV �1P , by the presence of U shown in Fig. 14, could be attributed to the lesser decrease of the negative contribution to SV �1P , or to the direct loss of the positive part of SV �1P . The former case is, as discussed above, due to the loss of ice-like patches in the bulk H2 O, and was due to the hydrophobic effects of the solute in question. If the reduction in H1EP−1P is due to a weaker hydrophobe as ME discussed above, then the x1P -locus must also be shifted to the left. Since it remains the same, as is evident in Fig. 14, we suggest that the

190

Fig. VII-15 to VII-16

effect of U on the value of H1EP−1P , or that of SV �1P is due purely to its hydrophilicity and to reducing the positive part of SV �1P . Namely, the degree of fluctuation inherent in liquid H2 O is reduced by the presence of U. This may occur by U participating in hydrogen bonding with the existing network of H2 O, and thus keeping the hydrogen bond connectivity intact. The intervening U in the network, however, breaks the existing proton donor/acceptor symmetry of H2 O. As a result, the degree of fluctuation, the positive part of SV-cross fluctuation inherent in H2 O is reduced, while the hydrogen bond percolation is retained. Indeed, Idrissi et al. (2000) concluded that the addition of U leads to an overall isotropy and stiffening of the short-time dynamics of both species, as mentioned above. This finding is consistent with our interpretation based on Fig. 14. The capability of forming hydrogen bonds of U is no doubt due to the −NH2 =O group for H-acceptor. Therefore, the group working as the H donor and the >C= H donor/acceptor symmetry which is the very source of the unique nature of H2 O is severely retarded by the presence of U. Not only may the geometry of the hydrogen bond network, but also the strength of the H-donor/acceptor, be affected. To see how these effects manifest themselves within the present methodology, we use tetramethyl urea (TMU) and acetone (AC) for S in the present 1P−S−H2 O methodology. TMU replaces all H’s in −NH2 with −CH3 ’s and AC changes the two −NH2 ’s to −CH3 ’s. Figs. VII-15

250

xTMU0 = 0 0.01245 0.02502

H 1P1PE / kJ mol-1

200

0.05006

150

0.07507 0.1003

100

50

0

0.00

0.02

0.04

0.06

0.08

0.10

0.12

x1P

Fig. VII-15. The 1P–1P enthalpic interaction, H1EP−1P , for 1P−TMU−H2 O at 25  C. The data are from Chen et al. (2001). Reproduced from the Fluid Phase Equilibria, 189, 31, (Chen, D. H. C. et al.), Copyright (2001), with permission from Elsevier.

191

250

xAC0 = 0

X

0.03205

200

0.06103 H 1P1PE / kJ mol-1

0.09006 150

0.1604 0.2005

100

50

0

-50 0.00

0.02

0.04

0.06

0.08

0.10

0.12

x1P

Fig. VII-16. The 1P–1P enthalpic interaction, H1EP−1P , in 1P−AC−H2 O at 25  C. The data are from Chen et al. (2001). Reproduced from the Fluid Phase Equilibria, 189, 31, (Chen, D. H. C. et al.), Copyright (2001), with permission from Elsevier.

and VII-16 show the H1EP−1P data for 1P−TMU−H2 O and for 1P−AC−H2 O, respec­ tively (Chen et al., 2001). As expected, the effect of the hydrophobic moieties is now apparent in that point X is shifted to the left for both TMU and AC. We show the x1P and the H1EP−1P loci of point X in Fig. VII-17, together with those for U. Fig. 17(a) shows the effect of hydrophobic effects on the x1P -locus of point X by TMU and AC. Clearly, the effect is stronger for TMU with four −CH3 groups than that for AC with two. The effect of the number of −CH3 is not twice, however. This may be due to the difference between N and C, to which −CH3 is bonded. Fig. 12 shows the excess partial molar volume signature of hydrophobicity for TMU and AC. The initial decrease in ViE indicates that the former is a substantially stronger hydrophobe than the latter. U, on the other hand, does not show the initial decrease in VUE (Fig. 12) and does not alter the x1P -locus (Fig. 17(a)). Fig. 17(b) indicates that the effects of U, TMU, and AC on the value of H1EP−1P and hence those of SV �1P are surprisingly similar, considering the uncertainty. Perhaps, the hydrophilic effect of TMU is stronger than U and AC. This observation suggests =O, and the contribution from that the hydrophilic effect is predominantly due to >C= −NH2 in U towards hydrophilicity is marginal. As discussed above, it is known that the O · · · H−O∗ type hydrogen bond is stronger than the N−H · · · O∗ type, where O∗ is the oxygen of H2 O (Huheey, 1978).

192

Fig. VII-17(a) to VII-18

0.08

Error

(a)

x 1P at point X

0.06

0.04

Urea TMU

0.02

AC

0.00 0.00

0.05

0.10

0.15

0.20

0.25

0.30

xs0 (S = U, TMU, or AC)

Fig. VII-17(a). The x1P locus of point X in the H1EP−1P pattern for 1P−S−H2 O (S = U, TMU, or AC) at 25  C. The data arew from To et al. (1998) for U, and from Chen et al. (2001) for TMU and AC. 250

Error H 1P1PE / kJ mol-1 at point X

U 200

TMU AC

(b) 150

100

50

0.00

0.05

0.10

0.15

0.20

0.25

xs0 (S = U, TMU or AC)

Fig. VII-17(b). The value of H1EP−1P at point X in the H1EP−1P pattern of 1P−S−H2 O (S = U, TMU, AC) at 25  C. The data are from To et al. (1998) for U, and Chen et al. (2001) for TMU and AC.

193

[VII-4] EFFECTS OF ETHYLENE GLYCOL (EG), 1,2- AND 1,3-PROPANEDIOL (12P AND 13P), GLYCEROL (GLY) AND POLY (ETHYLENE GLYCOL) (PEG) ON H2 O AS PROBED BY THE H1EP−1P PATTERN CHANGE In this section, we study the effect of poly-ols on H2 O by the same 1P-probing method­ ology. Poly-ols are clearly more hydrophilic than the probing 1P. We will learn the effect of −OH on the H1EP−1P pattern. We will also learn, as shown below, how the alkyl backbone manifests its hydrophobicity in the H1EP−1P pattern. Poly(ethylene glycols) (PEG) are known to have a strong affinity to H2 O, i.e. hydrophilic. This effect is often attributed to a good match of the inter-oxygen distance in liquid H2 O and that between the neighboring ether oxygen atoms in the PEG helicoidal chain (Giordano et al., 1995). We wish to see the difference, if any, between the end −OH and the ether −O− on the respective hydrophilic effect. Figs. VII-18, VII-19, VII-20 and VII-21 are the H1EP−1P plots for S = EG(2C/2OH), 13P (3C/2OH), 12P(3C/2OH), and Gly(3C/3OH) in the ternary 1P−S−H2 O. The number of the C atoms in the alkyl chain and that of the hydroxyl −OH groups are shown in brackets for convenience. As is evident from the figures, the induced changes of the H1EP−1P pattern by the presence of S are in both ways in terms of point X, one shifting point X to the left or reducing the x1P -locus, and the other lowering the value of H1EP−1P . 200

xEG0 = 0

H 1P-1PE / kJ mol-1

150

0.02033 0.03984

X

0.08089

100

0.1687 0.5238

50

0

-50 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

x1P

Fig. VII-18. The 1P–1P enthalpic interaction, H1EP−1P , for 1P−EG−H2 O at 25  C. The data are from Koga (2003b). Reproduced from the Journal of Solution Chemistry, 32, 803, (Koga, Y.), Copyright (2003), with kind permission from Springer Science and Business Media.

194

Fig. VII-19 to VII-21

H 1P-1PE / kJ mol-1

200

X

x13P0 = 0 0.05008 0.1007

150

100

50

0

0.00

0.05

0.10

0.15

0.20

x1P

Fig. VII-19. The 1P–1P enthalpic interaction, H1EP−1P , in 1P–1,3-propanediol (13P)−H2 O at 25  C. The data are from Parsons & Koga (2002). Reproduced with permission from the Journal of Physical Chemistry B, 106, 7090, (Parsons, M. T. and Koga, Y.), Copyright (2002), American Chemical Society.

200

X

x12P0 = 0

H 1P-1PE / kJ mol-1

0.02513 150

0.05030 0.07529 0.09998

100

50

0

0.00

0.05

0.10

0.15

0.20

x1P

Fig. VII-20. The 1P–1P enthalpic interaction, H1EP−1P , in 1P–1,2-propanediol (12P)−H2 O at 25  C. The data are from Parsons and Koga (2002). Reproduced with permission from the Journal of Physical Chemistry B, 106, 7090, (Parsons, M. T. and Koga, Y.), Copyright (2002), American Chemical Society.

195

H 1P-1PE / kJ mol-1

200

X

xGly0 = 0

150

0.02493 0.05249 100

0.07541 0.10125

50

0 0.00

0.05

0.10

0.15

0.20

x1P

Fig. VII-21. The 1P–1P enthalpic interaction function, H1EP−1P , in 1P–glycerol (Gly)−H2 O at 25  C. The data are from Parsons et al. (2001). Reproduced from the Journal of Solution Chemistry, 30, 1007, (Parsons, M. T. et al.), Copyright (2001), with kind permission from Springer Science and Business Media.

The increase/decrease of the value of H1EP−1P at the start, x1P = 0, is complex due to the similar competition discussed for ME above. Namely, the increase due to the parallel shift to the left and the decrease due to hydrophilicity compete, and the net results seem complex as shown in Figs. 18 to 21. We focus for the remainder of this section on the behavior of point X only. From the discussion in the above two sections, the left shift is the signature of hydrophobicity and the decrease in the height of the peak is that of hydrophilicity of S, relative to the probing 1P. Indeed, the latter effect seems stronger for Gly (3C/3OH) (Fig.21) than EG(2C/2OH), 13P(3C-2OH), or 12P(3C/2OH) (Figs. 18–20). This point will be clearer when we show the locus of point X as a function of xS0 below. Figs. VII-22 and VII-23 are the H1EP−1P plots for S = PEG’s. Fig. 22 is for PEG with the average molecular weight of 203, abbreviated as PEG-2, which contains 8.4 carbon and 5.2 oxygen atoms, including the ether −O−. The PEG used for Fig. 23 is of the average molecular weight of 600 (PEG-6) and the number of carbon atoms is 26.4 and that for oxygen is 14.2. Of course, the two end oxygen atoms in both PEG are in the hydroxyl form −OH. The figures show that xS0 < 0�03 for S = PEG-2 (Fig. 22) and for xS0 < 0�01 for PEG-6, the H1EP−1P pattern remains as that of type (a). This suggests that within these H2 O-rich regions the integrity of H2 O is retained and Mixing Scheme I is operative in the respective binary PEG−H2 O system. Similar to poly-ols (Figs. 18–21), PEG’s also cause point X to shift to the left as well as downward.

196

Fig. VII-22 to VII-24

250

error

xPEG-20 = 0 0.008088 0.01588 0.02299 0.04179 0.07099 0.1205

X

H 1PE / kJ mol-1

200

150

X

100

50

0

-50 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

x1P

Fig. VII-22. The 1P–1P enthalpic interaction, H1EP−1P , in 1P–poly(ethylene glycol) 200 (PEG­ 2)−H2 O at 25  C. The data are from Miki et al. (2005). Reproduced with permission from the Journal of Physical Chemistry B, 109, 3873, (Miki, K. et al.), Copyright (2005), American Chemical Society. 250

H 1P-1PE / kJ mol-1

200

X

xPEG-60 = 0

error

0.004595 0.009511 0.01649 0.04247

150

X 100

50

0

-50 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

x1P

Fig. VII-23. The 1P–1P enthalpic interaction, H1EP−1P , in 1P–poly(ethylene glycol) 600 (PEG­ 6)−H2 O at 25  C. The data are from Miki et al. (2005). Reproduced with permission from the Journal of Physical Chemistry B, 109, 3873, (Miki, K. et al.), Copyright (2005), American Chemical Society.

197

240

PEG-2

Error

220

H 1P1PE at point X / kJ mol-1

PEG-6 200

EG Gly

180

13P ME

160

2P TBA

140

12P 120 100 80 0.00

xs

0.02

0.04

0.06

0.08

0.10

0 (S=PEG-2,PEG-6,EG,Gly,12P,

0.12

0.14

0.16

13P, ME, 2P, or TBA)

Fig. VII-24. The value of H1EP−1P at point X of the H1EP−1P pattern for 1P−S−H2 O at 25  C. S = PEG-2, PEG-6, EG, Gly, 12P, 13P, ME, 2P, or TBA. The abscissa is the initial mole fraction of S, xS0 . See text. Reproduced with permission from the Journal of Physical Chemistry B, 109, 3873, (Miki, K. et al.), Copyright (2005), American Chemical Society.

Fig. VII-24 summarizes the relative hydrophilicity, in terms of the value of H1EP−1P at point X, of various mono-ols, poly-ols including PEG. The PEG-6 requires xS0 (S = PEG-6) ≈ 0.015 to reduce the value of H1EP−1P to about one half of the original, while Gly needs xS0 (S = Gly) ≈ 0.07, about 5-fold, to do the same. The ratio of the oxygen in each molecule is 14�2/3 ≈ 5. This rather suggests that it is the number of oxygen that is responsible for suppression of H1EP−1P regardless of whether O is the end OH or the ether −O−. Indeed, the most effective H1EP−1P reducer apparent in Fig. 24 is PEG-6, having the largest number of O in the solutes studied. We note that ME, 2P, 12P and TBA contain −CH3 group. For those, hollow symbols are used for convenience in Fig. 24 and the subsequent three figures. Fig. VII-25 shows the plots of the x1P -locus against xS0 . As we showed in Fig. 9, the more hydrophobic the sample solute S, the more negative the slope. Similarly, from Fig. 25 a general trend may be detected. Again. PEG-6 having the largest number of C seems to be most effective for the left shift, i.e. the most hydrophobic. Thus, the number of oxygen and carbon atoms could be an important factor characterizing hydrophobicity and hydrophilicity of the present set of amphiphiles. Fig. VII-26 shows the plots of the value of H1EP−1P against the initial mole fraction 0 of oxygen atom, xoxygen (= xS0 × the number of O in S). Except for 2P and TBA, which together with ME and possibly 12P may be operating by the hydrophobic mechanism,

198

Fig. VII-25 to VII-27

0.06

PEG-2 PEG-6 EG Gly 13P

x1P at point X

Error

0.04

0.02

ME 2P TBA 12P

0.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

xs0 (S=PEG-2,PEG-6,EG,Gly, 12P, 13P, ME, 2P, or TBA)

Fig. VII-25. The x1P locus of point X in the H1EP−1P pattern for 1P−S−H2 O at 25  C. S = PEG-2, PEG-6, EG, Gly, 12P, 13P, ME, 2P, or TBA. The abscissa is the initial mole fraction of S, xS0 . See text. Reproduced with permission from the Journal of Physical Chemistry B, 109, 3873, (Miki, K. et al.), Copyright (2005), American Chemical Society.

PEG-2 PEG-6 EG Gly 13P

H 1P1PE at point X / kJ mol-1

Error 200

ME 2P TBA 12P

150

100

0.0

0.1

0.2

0.3

0.4

xOxygen0

Fig. VII-26. The value of H1EP−1P at point X in 1P−S−H2 O at 25  C. S = PEG-2, PEG-6, EG, Gly, 13P, ME, 2P, TBA, or 12 P. The abscissa is the initial mole fraction of oxygen atom in S. See text. Reproduced with permission from the Journal of Physical Chemistry B, 109, 3873, (Miki, K. et al.), Copyright (2005), American Chemical Society.

199

all seem to converge into a single curve within the estimated uncertainty. Namely, the hydrophilicity, which is caused by the hydroxyl −OH and the ether −O−, scales with the number of oxygen and the difference in the effect between the end −OH and the ether −O− is not significant within the present methodology. For ME (1C(methyl)/1OH) and 12 P (3C with 1methyl/2OH), the balance of the hydrophobic and the hydrophilic 0 moieties is such that the H1EP−1P vs. xoxygen falls on almost the same curve. Fig. VII-27 is the hydrophobic propensity in terms of the x1P -locus of point X against 0 xcarbon (= xS0 × number of carbon in S). The plots seem to converge into two curves, one group without methyl group and the other with methyl groups. A recent analysis of the standard free energies of transfer from water to hexadecane of alkanes, alkenes, alkadienes, and arenes scales well with the number of C−H bonds (Kyte, 2003). On converting the abscissa of Fig. 27 to the mole fraction of the number of C−H bonds, however, the two curves did not converge. Tanford (1973) used the logarithm of the partition coefficient of a solute in H2 O and in an organic solvent and showed a linear relationship with the number of carbon atoms in the solute molecule. However, for a relative ranking covering from hydrophobicity to hydrophilicity of amphiphiles and ions (which will be dealt with in Chapter VIII), the choice of the organic solvent makes a devastating effect on the results (Marcus et al., 1988; Taylor et al. 1991). Furthermore, the logarithm of the partition function is equal to the excess chemical potential difference 0.06

PEG-2 Error

PEG-6 EG Gly 13P

x1P at point X

0.04

0.02

ME 2P TBA 12P 0.00 0.0

0.1

0.2

0.3

0.4

xCarbon0

Fig. VII-27. The x1P locus of point X in 1P−S−H2 O at 25  C. S = PEG-2, PEG-6, EG. Gly, 13P, ME, 2P, TBA, or 12P. The abscissa is the initial mole fraction of carbon atom in S. See text. Reproduced with permission from the Journal of Physical Chemistry B, 109, 3873, (Miki, K. et al.), Copyright (2005), American Chemical Society.

200

Fig. VII-28 to VII-29

of the solute in H2 O and in the chosen organic solvent. We use here the behavior of the third derivative of G to monitor hydrophobicity and hydrophilicity, as opposed to the first derivative, chemical potential. Hence, the more subtle difference could be detected by the present method. Indeed, the present method could distinguish the methyl group containing solutes from those without.

[VII-5] CONCLUDING REMARKS – SUMMARY The present 1P-probing methodology is a powerful tool for sorting out hydrophobicity and/or hydrophilicity of a non-electrolyte. For a hydrophobic solute, its effect on the H1EP−1P pattern could be summarized in Fig. VII-28. The original pattern for the binary 1P−H2 O [O] in the figure shifts parallel to the left, in the presence of an equally hydrophobic S. Thus, the value of H1EP−1P at point X remains the same, the x1P locus decreases, and the value of H1EP−1P at the start, x1P = 0, increases steadily as xS0 increases. For a weaker (stronger) hydrophobe, in addition to the shift to the left, the value of

[O] Binary [A] Equal hydrophobe

X

[B] Weaker Hydrophobe

1P)

[C] Stronger Hydrphobe

H1P-1PE (or

SVΔ

[O]

[A]

[C]

[B]

0 0

x1P

Fig. VII-28. The induced changes in the H1EP−1P pattern by the presence of a hydrophobic S, in 1P−S−H2 O. [O] is the H1EP−1P pattern for the binary 1P−H2 O; [A] shows that with an equally hydrophobic S as the probing 1P; [B] is that with a weaker, and [C] that with a stronger hydrophobe. Reproduced with permission from Bulletin of the Chemical Society of Japan, 79, 1347, (Koga et al.), Copyright (2006), the Chemical Society of Japan.

201

E H1P−1P decreases (increases) in the entire range of x1P within Mixing Scheme I. As a result, the value of H1EP−1P at point X decreases (increases), and the rate of increase of H1EP−1P at x1P = 0 decelerates (accelerates). The mechanism for these effects on the H1EP−1P pattern comes from the way a hydrophobic S affects the molecular organization of H2 O within Mixing Scheme I. Namely, a hydrophobic S enhances the hydrogen bond network of H2 O in its immediate vicinity (“iceberg” formation) and concomitantly reduces the hydrogen bond probability of bulk H2 O away from S. This in turn reduces the chance of putative formation of ice-like patches. As a result, the negative contribution to the SV cross fluctuation decreases. The effect of 1P on the rate of this decrease increases as x1P increases, resulting in the initial increase in SV �1P and hence in H1EP−1P . At the same time, the probing 1P−1P interaction occurs via the bulk H2 O away from S, which is modified by S to its own strength and is influenced by the mixed ways in which S and 1P modify the bulk H2 O. Hence, if S is a weaker(stronger) hydrophobe than 1P, the value of H1EP−1P is suppressed (enhanced). The effect of a hydrophobe is thus mainly on the retardation of “ice-like patches” in the bulk H2 O away from “icebergs”, reflecting the “mixture model” of H2 O. Fig. VII-29 is a sketch of the effect of a hydrophile on the H1EP−1P pattern. The x1P locus remains unchanged but the value of H1EP−1P decreases by the presence of a

[O] Binary [D] Hydrophile

1P)

X

[E] Stronger Hydrphile

H1P-1PE (or

SVΔ

[O]

[E]

[D]

0 0

x1P

Fig. VII-29. The induced changes in the H1EP−1P pattern by the presence of a hydrophilic S, in 1P−S−H2 O. Reproduced with permission from Bulletin of the Chemical Society of Japan, 79, 1347, (Koga et al.), Copyright (2006), the Chemical Society of Japan.

202

Fig. VII-30

hydrophile. A hydrophile forms hydrogen bonds to the hydrogen bond network of H2 O, and thus the connectivity is retained, but the H donor/acceptor symmetry of H2 O is broken, not only in the geometry but also in the hydrogen bond strength. As a result, the hydrogen bond network becomes more rigid and loses the degree of fluctuation inherent in liquid H2 O. Namely, the regular positive part of SV cross fluctuation decreases in the presence of a hydrophilic S. Thus, its effect is more readily understood in terms of the “bent hydrogen bond” model of H2 O. For amphiphiles, their effects on the H1EP−1P pattern are some combination of the above two cases, depicted in Fig. 28 and Fig. 29. Fig. VII-30 sketches the resulting change in H1EP−1P by an amphiphile. A typical case is seen in Fig. 20 for S = 12P. For poly-ols including poly(ethylene glycols), the hydrophobic effect on the x1P locus of point X seems to scale with the number of carbon in the alkyl chain. For −CH3 containing solutes, however, their hydrophobic effect is found to be stronger. The hydrophilic effect on the value of H1EP−1P seems to scale with the number of oxygen, regardless of the hydroxyl −OH, or the ether −O−. The effects of other hydrophilic

[O] Binary [F] Amphiphile

SVΔ ) 1P

X

H1P-1PE (or

[O]

[F] 0 0

x1P

Fig. VII-30. The induced changes in the H1EP−1P pattern by the presence of an amphiphile S, in 1P−S−H2 O. The hydrophobic part shifts point X to the left and raises the value of H1EP−1P at the start, x1P = 0. The hydrophilic part, on the other hand, lowers the value of H1EP−1P in the entire range including at point X. The net result is a combination of these two effects. Reproduced with permission from the Netsusokutei (Journal of Japan Society of Calorimetry and Thermal Analysis), 34, 3 (Koga, Y.), Copyright (2007), the Japan Society of Calorimetry and Thermal Analysis.

203

moieties than −OH or −O− are yet to be studied. Preliminary indication shows that the =O appears stronger than the proton donor −NH2 within the present proton acceptor >C= methodology. As demonstrated, this 1P-probing methodology is powerful and convenient. It would therefore find wide application in aqueous solutions of biological importance. Biopoly­ mers are large molecules and almost colloidal. Their surfaces are covered with hydropho­ bic and hydrophilic moieties and each part would interact with H2 O differently. Hence, we suggest that the same methodology using another probing species, one more hydrophilic than the present 1P, is mandatory for further studies.

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Chapter VIII

THE EFFECTS OF SALTS ON THE MOLECULAR ORGANIZATION OF H2 O: 1-PROPANOL (1P)-PROBING METHODOLOGY

[VIII-1] INTRODUCTION – HOFMEISTER SERIES In this chapter, we examine the effects of various salts (and the constituent ions) on the molecular organization of H2 O using the same 1P-probing methodology success­ fully used in Chapter VII. We showed in Chapter VII that the relative hydrophobic­ ity/hydrophilicity of a sample amphiphile (S) could be sorted out and ranked in terms of the induced changes in the H1EP−1P pattern in the ternary 1P−S−H2 O system. By this method, we could distinguish the effect of the hydrophobic part and that of the hydrophilic part within the same molecule of an amphiphile. Here we apply the same methodology for S = salts. What electrolytes do to H2 O is an important and interesting subject in its own right. However, the relative influences of anions and cations on the properties and functions of biopolymers have been compared and ranked as the so-called Hofmeister series initiated in 1887 by the Hofmeister group (Kunz et al., 2004). The original ranking (Hofmeister, 1887) showed the effectiveness of the ion in precipitating the hen egg white from aqueous solution. Since then almost the same ranking applies, particularly for anions, to ion effects on a wide range of processes in aqueous solutions; not only biochemical processes but also colloidal sciences. The generality of this ion ranking has suggested that the effects of ions on H2 O are important. So modified mixed solvents of aqueous electrolyte in turn influence the behaviors of the examined processes. This belief triggered a massive interest in basic studies of binary aqueous solutions of various salts. This approach has provided important advances through analysis of various factors including surface tension increments, lyotropic number, viscosity B-coefficients, zetapotentials, salt activities, etc. However, not all of these physical properties carry the hallmarks of the Hofmeister effects. Thus, the molecular level understanding is only fragmentary (Kunz et al., 2004; Collins & Washabaugh, 1985; Cacace et al., 1997). There is another issue regarding the generality of the effects of ions on H2 O over the specificity of the examined processes. While the effects of anions are generally stronger and the specificity may not obscure the ranking, cations tend to exert weaker effects on the molecular organization of H2 O. Hence, the cation ranking in the literature is mixed. Table VIII-1 lists a number of the Hofmeister ranking in randomly selected reviews and

206

Fig. VIII-1

Table VIII-1. Hofmeister series. #

← Kosmotropic, salting out, stabilizing

Destabilizing, salting in, chaotropic →

Anions a

SO4 2− >HPO4 2− >citrate>tartrate>CrO4 − ≈ Cl− >NO3 − >ClO3 −

b

citrate>tartrate>SO4 2− >acetate>Cl− >NO3 − >Br − >I− >SCN−

c

SO4 2− ≈ HPO4 2− >F− >Cl− >Br − >I− ≈ ClO4 − >SCN−

d

SO4 2− >HPO4 2− >acetate>citrate>tartrate>Cl− >NO3 − >ClO3 − >I− >ClO4 − >SCN−

e

F− >PO4 3− >SO4 2− >acetate>Cl− >Br − >I− >SCN−

f

citrate>SO4 2− >HPO4 2− >F− > acetate>Cl− >Br − >I− >NO3 − >ClO4 − >SCN− Cations

a�

Na+ >K+

b�

Th4 >Al3+ >H+ >Ba2+ >Sr 2+ >Ca2+ >K+ >Na+ >NH4 +

d�

NH4 + >K+ >Na+ >Li+ >Mg2+ >Ca2+ >guadinium

e�

+ + 2+ + 2+ 2+ + �CH3 �4 N+ >�CH3 �2 NH2 + > NH+ 4 >K >Na >Cs >Li >Mg >Ca >Ba

f�

+ Mg2+ >Ca2+ >H+ >Na+ >K+ >Rb+ >Cs+ >NH+ 4 >�CH3 �4 N

a, a� :(Hofmeister, 1887). b, b� :(Marshall, 1978). c:(Collins & Washabaugh, 1985). d, d� :(Creighton, 1993). e, e� :(Cacace et al., 1997). f, f � :(Lopez-Leon et al.,2003).

textbooks. As is evident from the table, the anion ranking is more consistent than that for cations. Even among anions, the relative position between SO4 2− and F− varies. In scheme # c and # f, SO4 2− >F− , while in #e the opposite ranking F− >SO4 2− is shown. For cations, the rankings of # b� and of # d� are almost completely opposite, so are those of # e� and # f� . This discrepancy is no doubt due to the fact that the specificity of the examined process is more important when the difference in the ion effect becomes subtle. Indeed, Collins and Washabaugh (1985) did not list the cation ranking although (or because) the reference covered in the review is massive. In spite of these discrepancies, the generality of the Hofmeister ranking suggests that an important factor is how each salt modifies the molecular organization of H2 O. As we pointed out above, and will be doing so throughout the book, the mixing scheme in aqueous solutions is crucially dependent of the composition. It is in the H2 O-rich region only where the integrity of liquid H2 O is maintained. In this chapter we study what the sample salt (S) does to H2 O by the 1P-probing methodology. Namely, we determine H1EP−1P in the ternary 1P−S−H2 O, and monitor first that the H1EP−1P pattern remains of type (a) (Fig. VI-1(a)). This ensures that our attention is limited within Mixing Scheme I, in which the integrity of H2 O is maintained even in the presence of a given salt. We then examine the induced changes in the H1EP−1P pattern (still in type (a)) to learn about the effect of the salt on H2 O.

207

[VIII-2] EFFECTS OF NaF AND NaCl ON H2 O AS PROBED BY THE H1EP−1P PATTERN CHANGE Figs. VIII-1 and VIII-2 show the H1EP data for 1P−S−H2 O with S = NaF and NaCl, respectively (Westh et al., 2006). The arrows in the figure indicate approximate points where the thermal response after titration slows down considerably due to liquid–liquid phase separation. We do not include for further analysis the series with the phase separation prematurely before the inflection point in the H1EP plots against x1P . The corresponding H1EP−1P plots are shown in Figs. VIII-3 and VIII-4. As is evident in the figures, for xS0 �S = NaF�OAc− >Cl− >Br − >I− >ClO4 − >SCN− �

(VIII-2)

except that the relative order between SO4 − and F− is mixed (see Table 1, # c and # f vs. # e). Figs. VIII-14, VIII-15, VIII-16 and VIII-17 show the H1EP−1P pattern in 1P−S−H2 O at 25 � C with S = Na2 SO4 , NaOAc, NaClO4 , and NaSCN. Na2 SO4 (Fig. 14) shows the hallmark of a hydration center in two respects: The H1EP−1P value at the start, x1P = 0 remains constant and the x1P -locus of point X shifts to the left, to smaller values of x1P , as xS0 �S = Na2 SO4 � increases. Unlike NaF or NaCl, however, the value of H1EP−1P at point X increases. If the effects of cations and anions are additive, as we are taking as a premise, then that of Na+ would not increase the value of H1EP−1P at point X. Thus, the observed increase in H1EP−1P at point X is due to SO4 2− . We recall that tert-butanol 300

xNa2SO40 = 0

X 250

0.004840

200

H 1P1PE / kJ mol-1

0.01032

X

0.01488 150

100

50

0

0.00

I Error 0.02

II 0.04

0.06

0.08

0.10

0.12

0.14

x1P

Fig. VIII-14. The 1-propanol(1P)−1P enthalpic interaction, H1EP−1P , in 1P−Na2 SO4 −H2 O at 25 � C. The data from Koga et al. (2004b). Reproduced with permission from the Journal of Physical Chemistry A, 108, 8533, (Koga, Y. et al.), Copyright (2004), American Chemical Society.

Eq. (VIII-2)

219

250

X

H 1P-1PE / kJ mol-1

xNaOAc0 = 0

X

200

0.01302 0.02715 0.04176

150

100

50

I 0

0.00

Error

II

0.02

0.04

0.06

0.08

0.10

0.12

0.14

x1P

Fig. VIII-15. The 1-propanol (1P)–1P enthalpic interaction, H1EP−1P , in 1P−NaOAc−H2 O at 25 � C. The data from Koga et al. (2004b). Reproduced with permission from the Journal of Physical Chemistry A, 108, 8533, (Koga, Y. et al.), Copyright (2004), American Chemical Society.

250

xNaClO40 = 0 200

Error

0.01077

X

H 1P-1PE / kJ mol-1

0.01989 0.03138

150

0.04771 0.09014

100

X 50

I 0

II 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

x1P

Fig. VIII-16. The 1-propanol (1P)–1P enthalpic interaction, H1EP−1P , in 1P−NaClO4 −H2 O at 25 � C. The data from Koga et al. (2004b). Reproduced with permission from the Journal of Physical Chemistry A, 108, 8533, (Koga, Y. et al.), Copyright (2004), American Chemical Society.

220

Fig. VIII-17 to VIII-18

200

X

xNaSCN0 = 0

Error

0.01094

150

H 1P-1PE / kJ mol-1

0.03094 0.04890 100

0.1002 X

50

0

I II

-50 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

x1P

Fig. VIII-17. The 1-propanol (1P)–1P enthalpic interaction, H1EP−1P , in 1P−NaSCN−H2 O at 25 � C. The data from Koga et al. (2004b). Reproduced with permission from the Journal of Physical Chemistry A, 108, 8533, (Koga, Y. et al.), Copyright (2004), American Chemical Society.

(TBA), a stronger hydrophobe than the probing 1P, makes the value of H1EP−1P increase both at the start and at point X (Fig. VII-7). Since the value of H1EP−1P remains the same at the start, SO4 2− is not a typical hydrophobe. Thus, the detail of the effect of SO4 2− is not clear at present. It could be that SO4 2− works as a hydration center in the absence of 1P in the system, resulting in the unchanged value of H1EP−1P at x1P = 0. As x1P increases, however, a direct interaction between SO4 2− and 1P starts to operate, in addition to the H2 O-mediated 1P−SO4 2− interaction. If so, then the Hofmeister ranking depends crucially on the test biopolymer, and thus the ranking could become unstable for SO4 2− . In this context also, a series of similar studies using a probe other than 1P is awaited. We plan to do this, using glycerol (Gly) as a probe and follow the induced E changes in the HGly−Gly pattern. Further studies on Na2 SO4 such as a dielectric relaxation spectroscopy on aqueous MgSO4 (Buchner et al, 2004) are awaited. Nonetheless, from the fact that the H1EP−1P values at x1P = 0 remains constant, we may classify SO4 2− as a hydration center. Thus, from the mixing scheme boundary (Fig. VIII-18), the hydration number, nH , is estimated as 27 ± 3 for Na2 SO4 . It follows using the reference nH = 5�2 for Na+ � nH for SO4 2− is 17 ± 3. They are listed in Tables 2 (p. 211) and 3 (p. 216). Fig. 15 indicates that NaOAc works basically as a hydrophobic species with a strength equal to the probing 1P, as shown in Fig. VII-6 for 2-propanol (2P). However, there is a tendency of slight suppression of the increase in the value of H1EP−1P at x1P = 0 than that shown for the effect of a typical hydrophobe as type [A] in Fig. VII-28.

221

0.05

S+L S+L+L

xNa2SO40

0.04

0.03

L+L 0.02

0.01

I II 0.00 0.00

0.02

0.04

0.06

0.08

0.10

0.12

x1P

Fig. VIII-18. The mixing scheme and phase diagram for 1-propanol (1P)−Na2 SO4 −H2 O at 25 � C. The solid line is the phase boundary. The broken line is a presumed phase boundary. The dotted line is the mixing scheme boundary, the onset of the transition from Mixing Scheme I to II. The data from Koga et al. (2004b). Reproduced with permission from the Journal of Physical Chemistry A, 108, 8533, (Koga, Y. et al.), Copyright (2004), American Chemical Society.

The latter suppression must be due to the hydration of Na+ , which tends to keep the H1EP−1P value constant at x1P = 0. Namely, the hydrophobic behavior apparent in Fig. 15 is due to OAc− . Thus, the observed left shift of point X is due to a combination of hydration by Na+ and hydrophobicity by OAc− . Clearly the −CH3 group is the main cause of this. Why then an expected hydrophilicity from −COO− is not apparent is yet to be elucidated. We speculate that there could be some ion pairing operative such that the hydrophilic moiety is consumed by this pairing leaving the hydrophobic moieties pointing outward. Fig. VIII-19 shows the loci of point X in 1P−NaOAc−H2 O, the onset of transition from Mixing Scheme I to II as probed by H1EP−1P , together with the phase boundary. The thick dotted line in the figure indicates the loci of point X if the shift to the left of point X is due only to hydration by Na+ with nH �Na+ � = 5�2. The actual plots with the thin dotted line lie below this due to the hydrophobicity of OAc− . Fig. 16 �S = NaClO4 � and Fig. 17 (S = NaSCN) show that both effects are similar to that by glycerol (Gly) shown in Fig. VII-21. Namely, in addition to some tendency of the left shift, there is a clear decrease in the value of H1EP−1P at point X. Thus, both salts give a hydrophilic propensity. The left shift, on the other hand, could be due to the hydration by Na+ or some hydrophobicity inherent in each anion. This point is

222

Fig. VIII-19 to VIII-21

0.12

S+L 0.10

xNaOAc0

0.08

L+L

0.06

0.04

0.02

0.00 0.00

II

I

0.02

0.04

0.06

0.08

0.10

0.12

0.14

x1P

Fig. VIII-19. The mixing scheme and phase diagram for 1-propanol (1P)−NaOAc−H2 O at 25 � C. The solid line is the phase boundary. The broken line is a presumed phase boundary. The thick dotted line is the expected loci of point X if the shift to the left of point X is due only to hydration of Na+ . The thin dotted line is the observed onset of the transition from Mixing Scheme I to II. The data from Koga et al. (2004b). Reproduced with permission from the Journal of Physical Chemistry A, 108, 8533, (Koga, Y. et al.), Copyright (2004), American Chemical Society.

discussed below. Judging from the figure, however, the hydrophilic effects are more − conspicuous for both ClO− 4 and SCN . Fig. VIII-20 shows the mixing scheme boundary for 1P−S−H2 O (S = NaClO4 or NaSCN). The thick dotted line corresponds to the case where the left-shift is due only to hydration by Na+ . The actual loci of point X seem to lie below this, suggesting that both ClO4 − and SCN− contain some hydrophobic propensity, or at least some mechanisms to reduce the hydrogen bond probability of the bulk H2 O away from the solute. However, this point requires further clarification including repeats of H1EP measurements. Fig. VIII-21 is a summary of the effects of salts on the value of H1EP−1P at point X including halide salts. Those showing the signature for a hydration center (includ­ ing Na2 SO4 ) and a hydrophobe are indicated by hollow symbols in the figure. The hydrophiles are shown by filled symbols. Within these hydrophiles, the relative effect of the anion is represented by the slope in the figure, since the counter cation is fixed at Na+ . Accordingly, the slope becomes more negative in the order of, Br − NH4 + �1��

(VIII-4)

where the number given in brackets is the respective hydration number. From the original idea by Hofmeister (1887) of the “H2 O withdrawing power”, then the above ranking

228

Fig. VIII-26 to VIII-28

0.14

Solid + ? 0.12

xcacl20

0.10 0.08

Liquid - Liquid 0.06 0.04 0.02

I

II

0.00 0.00

0.05

0.10

0.15

0.20

x1P

Fig. VIII-26. The mixing scheme and phase diagram in 1-propanol (1P)−CaCl2 −H2 O at 25 � C. The solid line is the phase boundary. The broken line is a presumed phase boundary. The dotted line is the onset of the transition from Mixing Scheme I to II. The data from Koga et al. (2006). Reproduced with permission from Bulletin of the Chemical Society of Japan, 79, 1347, (Koga et al.), Copyright (2006), the Chemical Society of Japan. 0.25

0.20

xNH4Cl0

Solid + ? 0.15

0.10

Liquid - Liquid

0.05

II

I 0.00 0.00

0.05

0.10

0.15

0.20

x1P

Fig. VIII-27. The mixing scheme and phase diagram in 1-propanol (1P)−NH4 Cl−H2 O at 25 � C. The solid line is the phase boundary. The broken line is a presumed phase boundary. The dotted line is the onset of the transition from Mixing Scheme I to II. The data from Koga et al. (2006). Reproduced with permission from Bulletin of the Chemical Society of Japan, 79, 1347, (Koga et al.), Copyright (2006), the Chemical Society of Japan.

Eq. (VIII-5)

229

0.35

0.30

xTMACl0

0.25

0.20

0.15

Liquid - Liquid

0.10

0.05

I 0.00 0.00

II 0.05

0.10

0.15

0.20

x1P

Fig. VIII-28. The mixing scheme and phase diagram in 1-propanol (1P)−TMACl−H2 O at 25 � C. The solid line is the phase boundary. The dotted line is the onset of the transition from Mixing Scheme I to II. The data from Koga et al. (2006). Reproduced with permission from Bulletin of the Chemical Society of Japan, 79, 1347, (Koga et al.), Copyright (2006), the Chemical Society of Japan.

coincides in decreasing order of the precipitating power, or the order from the kosmotrope to the chaotrope. Since TMA+ is found to be a hydrophile, analogous to the effect of SCN− or ClO4 − discussed above, we suggest TMA+ is a chaotrope. Thus the overall ranking could be written as, Ca2+ >Na+ >NH4 + >TMA+ �

(VIII-5)

from the kosmotropic to the chaotropic end. This ranking is consistent with some (# b’ and # f’) but not with others (# d’ and # e’) in Table 1 (p. 206). The cations’ effects are weaker than anions’, and the ranking is not unequivocal, influenced by the specificity of the test system: Proteins, temperature, pH, concentrations etc.

[VIII-6] HYDRATION NUMBER OF GLYCINE AND ITS SALTS AS PROBED BY THE H1EP−1P PATTERN CHANGE We applied the same 1P-probing methodology to experimentally estimate the hydration number of glycine and its salts. Glycine in aqueous solution is known to take pre­ dominantly the zwitter ion form, − OOC-CH2 -NH3 + . We abbreviate glycine as GL and

230

Fig. VIII-29 to VIII-30

its zwitter ion GL+− . There have been a number of theoretical works in the literature regarding the proton transfer to give rise to the GL+− form from the neutral GL that is stable in gas phase. Some of the studies support a direct intra-molecular transfer from the carboxylic to the amino groups (Rzepa & Yi, 1991; Tortonda et al., 1998; Nagaoka et al., 1998; Tunon et al., 1998). More recent theoretical works, however, favor H2 O-mediated proton transfer mechanisms (Fernandez-Ramos et al., 2000; Chaban & Garber, 2001; Balta & Aviente, 2003; Balta & Aviente, 2004; Ahm et al., 2005; Leung & Rempe, 2005). The latter studies consider a super-molecule of the type GL�H2 O�m and show that the zwitter ion GL+− �H2 O�m is the stable form. However, the value of m varies from 1 (Chaban & Garber, 2001) to 6 (Balta & Aviente, 2004) depending on the theoretical method and the chosen potential field. The recent ab initio calculation concludes that the stable m changes from 5 for the neutral GL to 8 for GL+− (Leung & Rempe, 2005). We apply here the 1P-probing methodology with S = GL+− � NaOOC−CH2 −NH2 (NaGL), and HOOC−CH2 −NH3 Cl (GLCl) in the ternary 1P−S−H2 O at 25 � C. Figs. VIII-29, VIII-30 and VIII-31 show the H1EP data. The arrows in Fig. 30 indicate the onset of liquid–liquid phase separation. The corresponding H1EP−1P results are shown in Figs. VIII-32, VIII-33 and VIII-34. Figs. 32 and 33 carry the hallmarks of the proper hydration center, as sketched for [A] in Fig. 22. This suggests that both GL+− and Na+ GL− are the proper hydration centers. Thus, the hydration numbers, nH , are esti­ mated from the x1P -loci of point X vs. xS0 (S = GL+− and Na+ GL− ) plots, shown in Fig. VIII-35. The results are nH = 7 ± 0�6 for GL+− and 4�8 ± 1 for GL− , taking into account that nH = 5�2 for Na+ . They are listed in Table 3 (p. 216). Fig. 34 indicates that GL+ Cl− has a hydrophilic propensity, while there is a weak shift to the left on point X. If the latter part is due to hydration of both GL+ and Cl− , then the slope shown in Fig. 35 for GL+ Cl− provides the total hydration number being 5.0 ± 0.5, out of which 2.3 ± 0.6 is due to Cl− . It follows then that nH for GL+ 2.7 ± 1, if the shift to the left of point X is due to hydration to GL+ . Alternatively, this left shift could be due to a hydrophobicity of GL+ , if the latter is an amphiphile. There is no way of unequivocally distinguishing the two possibilities within the present methodology. We point out, however, that such a typical hydrophilic amphiphile as tetramethyl urea or acetone, the behavior of the value of H1EP−1P at the start, x1P = 0, is complex because of the competition between an increase due to hydrophobicity and a decrease due to hydrophilicity (see Figs. VII-15, VII-16 and VII-30). In contrast, the value of H1EP−1P at x1P = 0 in Fig. 34 decreases monotonously as xS0 �S = GL+ Cl− � increases. We are thus inclined to favor the hydration scenario for the left shift, and hence to conclude that GL+ works as a hydration center as well as a hydrophile, and its hydration number is 2.7 ± 1. These results are summarized in Table 3. We note in Table 3 (p. 216) that the sum of nH for GL− and nH of GL+ amounts to 7.5 ± 2, the same as that for GL+− , 7.0 ± 0.6. It is therefore tempting to speculate that the −COO− part hydrates 4.8 and the −NH3 + part hydrates 2.7 molecules of H2 O. If so, the un-dissociated −COOH in GL+ works as a hydrophile resulting in the reduction of H1EP−1P as seen in Fig. 34. On the other hand, −NH2 in GL− would be a very weak hydrophile with an almost

231

2 0

H 1PE / kJ mol-1

-2 -4

xGL0 = 0 0.01253

-6

0.02490 -8 -10 -12 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

x1P

Fig. VIII-29. The excess partial molar enthalpy of 1-propanol (1P), H1EP , in 1P–glycine(GL)−H2 O at 25 � C (Parsons & Koga, 2005). Reproduced with permission from the Journal of Chemical Physics, 123, 234504, (Parsons, M. T. and Koga, Y.), Copyright (2005), American Institute of Physics. 2 0

H 1PE / kJ mol-1

-2 -4

xNaGL0 = 0 -6

0.01998 0.04020

-8

0.05918 -10 -12 0.00 0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

x1P

Fig. VIII-30. The excess partial molar enthalpy of 1-propanol (1P), H1EP , in 1P–sodium glycinate (NaGL)−H2 O at 25 � C (Parsons & Koga, 2005). The arrows indicate the sign of phase separation. Reproduced with permission from the Journal of Chemical Physics, 123, 234504, (Parsons, M. T. and Koga, Y.), Copyright (2005), American Institute of Physics.

232

Fig. VIII-31 to VIII-34

2 0

H 1PE / kJ mol-1

-2 -4

xGLCl0 = 0

-6

0.02007 0.04003 0.05986

-8 -10 -12 0.00 0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

x1P

Fig. VIII-31. The excess partial molar enthalpy of 1-propanol (1P), H1EP , in 1P–glycine hydrogen chloride (GLCl)−H2 O at 25 � C (Parsons & Koga, 2005). Reproduced with permission from the Journal of Chemical Physics, 123, 234504, (Parsons, M. T. and Koga, Y.), Copyright (2005), American Institute of Physics. 250

X

200

H 1P1PE / kJ mol-1

xGL0 = 0 0.01253

150

0.02490 100

50

Error

0

I 0.00

0.02

II 0.04

0.06

0.08

0.10

0.12

0.14

x1P

Fig. VIII-32. The 1-propanol (1P)–1P enthalpic interaction, H1EP−1P , in 1P–glycine (GL)−H2 O at 25 � C (Parsons & Koga, 2005). Reproduced with permission from the Journal of Chemical Physics, 123, 234504, (Parsons, M. T. and Koga, Y.), Copyright (2005), American Institute of Physics.

233

300

250

H 1P1PE / kJ mol-1

xNaGL0 = 0

X

200

0.01998 0.04020

150

0.05918 100

Error

50

0

I 0.00

0.02

II 0.04

0.06

0.08

0.10

0.12

0.14

x1P

Fig. VIII-33. The 1-propanol (1P)–1P enthalpic interaction, H1EP−1P , in 1P–sodium glycinate (NaGL)−H2 O at 25 � C (Parsons & Koga, 2005). Reproduced with permission from the Journal of Chemical Physics, 123, 234504, (Parsons, M. T. and Koga, Y.), Copyright (2005), American Institute of Physics. 250

X

200

H 1P1PE / kJ mol-1

Error

xGLCl0 = 0 0.02007

150

0.04003 0.05986 100

50

II I 0

0.00

0.02

0.04

0.06 0.08

0.10

0.12

0.14

0.16

x1P

Fig. VIII-34. The 1-propanol(1P)–1P enthalpic interaction, H1EP−1P , in 1P–glycine hydrogen chlo­ ride (GLCl)−H2 O at 25 � C (Parsons & Koga, 2005). Reproduced with permission from the Journal of Chemical Physics, 123, 234504, (Parsons, M. T. and Koga, Y.), Copyright (2005), American Institute of Physics.

234

Fig. VIII-35

0.07

xS0 (S = GL, NaGL, or GLCI)

0.06

GL NaGL

0.05

GLCl 0.04

0.03

II

0.02

0.01

0.00 0.030

I

0.035

0.040

0.045

0.050

x1P

Fig. VIII-35. The mixing scheme boundary, the loci of the onset of the transition from Mixing Scheme I to II, in 1P−S−H2 O at 25 � C, where S = glycine (GL), sodium glycinate (NaGL), or glycine hydrogen chloride (GLCl) (Parsons & Koga, 2005). Reproduced with permission from the Journal of Chemical Physics, 123, 234504, (Parsons, M. T. and Koga, Y.), Copyright (2005), American Institute of Physics.

negligible hydrophilic effect on the H1EP−1P pattern seen for NaGL in Fig. 33, in which there is no sign of a decrease in the value of H1EP−1P at point X. We recall a similar =O and of −NH2 . conclusion in Section [VII-3], comparing the hydrophilic effects of >C= Thus, all our findings discussed above seem to be self-consistent within the 1P-probing methodology.

[VIII-7] CONCLUDING REMARKS IN RELATION TO OTHER STUDIES ON AQUEOUS ELECTROLYTES IN THE LITERATURE [VIII-7-1] Ion pairing Salts (S) in aqueous solutions have been regarded as completely dissociated at least in dilute aqueous solutions. However, there is an increasing number of studies suggesting some form of ion-pairing even at a concentration low enough that the system is still in the Mixing Scheme I region as probed by the type (a) pattern of H1EP−1P . For example, dielectric relaxation spectra were shown to be deconvoluted into a number of Debye relaxation processes. The slower dispersion processes were assigned to the H2 O-shared

235

ion pairs. This is true in the range of xS0 0�33, and xB = 0�12 and xD >0�23, etc. due to the trouble of oscillation mentioned above. The excess chemical potentials of i (i = B, D, or W) are calculated as a function of �xB � xD � using the data in Table 2 (p. 246) by, � � pi � (IX-11) �Ei = RT ln xi pi∗ where i = B, D, or W, and pi∗ is the vapor pressures of pure i. Figs. IX-1, IX-2 and IX-3 show the results. In Fig. 1, the xB dependence of �EB shows a subtle change from xD = 0�15 (up-triangle) to 0.23 (down triangle). This boundary corresponds to the onset of transition from Mixing Scheme I to II in binary DMSO �D�−H2 O� xD = 0�21 (see Table VI-2 (p. 159)). Similarly in Fig. 2, the xD dependence of �EB may show a subtle change at about xB = 0�05 (down triangle), which corresponds to the boundary region of the I–II transition for binary TBA �B�−H2 O (see Table V-3 (p. 114)). This is no doubt due to changes in the manner by which each solute modifies H2 O within Mixing Scheme I, and the effect of transition from Mixing Scheme I to II. These subtle differences in the behavior of �Ei will be more conspicuous in the second derivative quantities, �Ei−j , HiE and SiE , and more so in third derivatives, HiE−j and SiE−j . The HBE data determined experimentally for a given initial mole fraction of D, xD0 , of the mixed solvent D−H2 O are shown in Fig. IX-4. The data are obtained as a small amount of B was successively titrated into the mixed solvent D−H2 O. Fig. IX-5 is the values of TSBE , calculated from �EB and HBE . Comparing the ordinate scale with that in Fig. 1, the enthalpy–entropy compensation is apparent (Lumry, 2003; Lumry & Rejender, 1970). Similarly, Figs. IX-6 and IX-7 show HDE and TSDE .

246

Table IX-2. Partial pressures, pi / Torr (i = B, D, or W) in tert-butanol (B)–dimethyl sulfoxide (D)–H2 O. xD \xB

0.0

0.005 23.62 0.0 2.448

0.01 23.50 0.0 4.775

0.015 23.39 0.0 6.981

0.02 23.28 0.0 9.060

0.025 23.16 0.0 11.27

0.03 23.05 0.0 13.38

0.035 22.95 0.0 15.32

0.04 22.85 0.0 17.06

0.045

0.0

pW 23.68 pD 0.0 pB 0.0

22.77 0.0 18.56

0.01

pW 23.52 23.46 23.34 23.22 23.11 22.99 22.88 22.78 22.68 22.60 pD .000649 .000649 .000666 .000669 .000672 .000674 .000676 .000679 .000681 .000683 pB 0.0 2.421 4.774 7.030 9.154 11.39 13.50 15.44 17.17 18.65

0.03

pW 23.11 23.05 22.90 22.78 22.67 22.55 22.43 22.33 22.24 22.15 pD .001608 .001608 .001667 .001665 .001670 .001675 .001683 .001684 .001689 .001694 pB 0.0 2.400 4.839 7.181 9.380 11.64 13.74 15.68 17.38 18.82

0.05

pW 22.59 22.53 22.41 22.28 22.16 22.03 21.92 21.81 21.72 21.63 pD .002777 .002777 .002778 .002799 .002820 .002840 .002851 .002872 .002884 .002896 pB 0.0 2.411 4.885 7.309 9.570 11.84 13.92 15.84 17.50 18.88

0.07

pW 21.99 21.93 21.78 21.65 21.53 21.41 21.29 21.19 21.10 21.02 pD .004243 .004243 .004304 .004331 .004360 .004384 .004414 .004423 .004442 .004458 pB 0.0 2.441 4.998 7.475 9.773 12.02 14.07 15.96 17.55 18.87

0.09

pW 21.30 21.24 21.09 20.96 20.83 20.71 20.61 20.50 20.42 20.34 pD .006107 .006107 .006192 .006234 .006274 .006310 .006330 .006368 .006391 .006414 pB 0.0 2.482 5.093 7.607 9.924 12.14 14.14 15.98 17.52 18.77

0.11

pW 20.55 20.49 20.33 20.20 20.08 19.97 19.86 19.76 19.68 19.61 pD .008440 .008440 .008558 .008610 .008659 .008700 .008752 .008767 .008797 .008822 pB 0.0 2.522 5.175 7.704 10.02 12.19 14.14 15.92 17.39 18.57

0.13

pW 19.74 pD .01133 pB 0.0

19.68 .01133 2.556

19.52 .01148 5.229

19.39 .01154 7.753

19.28 .01160 10.05

19.17 .01165 12.17

19.07 .01167 14.05

18.97 .01172 15.77

18.89 .01176 17.17

18.82 .01179 18.29

0.15

pW 18.88 pD .01487 pB 0.0

18.83 .01487 2.577

18.67 .01504 5.248

18.55 .01510 7.746

18.44 .01516 10.00

18.33 .01520 12.06

18.23 .01528 13.89

18.15 .01529 15.53

18.07 .01533 16.86

18.01 .01536 17.92

0.17

pW 17.99 pD .01915 pB 0.0

17.94 .01915 2.580

17.79 .01933 5.226

17.67 .01940 7.680

17.56 .01946 9.891

17.46 .01952 11.87

17.38 .01952 13.63

17.29 .01958 15.19

17.22 .01962 16.46

17.16 .01967 17.48

0.19

pW 17.07 pD .02427 pB 0.0

17.03 .02427 2.562

16.88 .02447 5.161

16.78 .02452 7.551

16.68 .02456 9.696

16.59 .02458 11.59

16.50 .02466 13.30

16.42 .02467 14.78

16.36 .02470 15.97

16.30 .02471 16.95

0.21

pW 16.15 pD .03034 pB 0.0

16.10 .03034 2.523

15.97 .03051 5.050

15.87 .03054 7.360

15.78 .03057 9.429

15.69 .03062 11.24

15.62 .03057 12.88

15.55 .03061 14.28

15.48 .03063 15.41

15.42 .03068 16.37

0.23

pW 15.22 pD .03743 pB 0.0

15.17 .03743 2.462

15.05 .03757 4.896

14.96 .03757 7.115

14.88 .03756 9.095

14.81 .03749 10.81

14.73 .03754 12.40

14.67 .03752 13.72

14.61 .03753 14.80

14.56 .03752 15.72

0.25

pW 14.29 pD .04562 pB 0.0

14.25 .04562 2.380

14.15 .04570 4.704

14.07 .04563 6.816

13.99 .04556 8.700

13.91 .04558 10.35

13.86 .04547 11.85

13.79 .04545 13.10

13.74 .04542 14.12

13.69 .04547 15.05

247

0.05

0.055

0.06

0.07

0.08

0.09

0.10

0.11

0.12

xB \xD

22.69 0.0 19.80

22.63 0.0 20.75

22.59 0.0 21.48

22.52 0.0 22.41

22.47 0.0 22.97

22.44 0.0 23.35

22.42 0.0 23.54

22.41 0.0 23.66

22.40 0.0 23.74

pW pD pB

0.0

22.53 .000683 19.86 22.08 .001701 19.95

22.47 .000686 20.79 22.03 .001700 20.84

22.42 .000687 21.49 21.98 .001708 21.58

22.36 .000690 22.41 21.91 .001719 22.42

22.31 .000693 22.97 21.86 .001731 22.98

22.28 .000698 23.36 21.82 .001738 23.40

22.26 .000700 23.54 21.79 .001763 23.57

22.25 .000703 23.67 21.77 .001784 23.75

22.24 .000709 23.74 21.75 .001799 23.81

pW pD pB pW pD pB

0.01

21.57 .002901 19.95

21.51 .002921 20.82

21.47 .002927 21.45

21.39 .002951 22.35

21.34 .002978 22.92

21.28 .003022 23.38

21.25 .003045 23.56

21.21 .003077 23.76

21.18 .003130 23.85

pW pD pB

0.05

20.95 .004482 19.90

20.90 .004485 20.72

20.85 .004510 21.33

20.77 .004550 22.24

20.71 .004596 22.83

20.66 .004623 23.29

20.61 .004697 23.51

20.55 .004767 23.74

20.52 .004821 23.85

pW pD pB

0.07

20.28 .006424 19.75

20.22 .006465 20.55

20.17 .006478 21.15

20.09 .006530 22.05

20.02 .006587 22.65

19.95 .006688 23.14

19.90 .006747 23.39

19.85 .006816 23.63

19.80 .006926 23.79

pW pD pB

0.09

19.54 .008867 19.52

19.49 .008871 20.29

19.45 .008917 20.88

19.35 .008995 21.80

19.27 .009093 22.43

19.21 .009144 22.90

19.15 .009273 23.20

19.07 .009420 23.47

19.02 .009536 23.66

pW pD pB

0.11

18.76 .01181 19.20

18.70 .01187 19.96

18.66 .01190 20.54

18.57 .01199 21.46

18.50 .01207 22.10

18.40 .01226 22.61

18.34 .01238 22.95

18.28 .01250 23.21

18.21 .01267 23.43

pW pD pB

0.13

17.94 .01543 18.82

17.89 .01543 19.54

17.84 .01549 20.13

17.75 .01562 21.06

17.65 .01580 21.74

17.59 .01588 22.22

17.52 .01605 22.59

17.43 .01630 22.89

17.36 .01651 23.15

pW pD pB

0.15

17.11 .01967 18.34

17.04 .01977 19.07

16.99 .01982 19.65

16.90 .01995 20.58

16.83 .02004 21.26

16.73 .02031 21.78

16.65 .02054 22.19

16.59 .02071 22.48

16.52 .02092 22.74

pW pD pB

0.17

16.23 .02481 17.81

16.19 .02480 18.50

16.14 .02487 19.09

16.05 .02501 20.03

15.94 .02531 20.75

15.88 .02545 21.26

15.80 .02565 21.67

15.71 .02598 22.00

15.62 .02635 22.30

pW pD pB

0.19

15.38 .03065 17.19

15.32 .03078 17.89

15.27 .03085 18.48

15.17 .03106 19.43

15.10 .03112 20.13

15.01 .03142 20.67

14.92 .03178 21.11

14.85 .03203 21.43

14.80 .03217 21.70

pW pD pB

0.21

14.49 .03765 16.55

14.45 .03763 17.22

14.41 .03768 17.81

14.33 .03777 18.75

14.22 .03817 19.50

14.14 .03846 20.05

14.06 .03870 20.47

13.97 .03914 20.83

-------

pW pD pB

0.23

13.65 .04536 15.83

13.60 .04547 16.51

13.55 .04555 17.10

13.44 .04592 18.07

13.37 .04604 18.79

13.29 .04632 19.35

13.19 .04678 19.79

13.12 .04708 20.14

-------

pW pD pB

0.25

0.03

(Continued)

248

Fig. IX-1 to IX-2

Table IX-2. (Continued) xD \xB

0.0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.27

pW pD pB

13.39 .05495 0.0

13.35 .05495 2.282

13.26 .05496 4.481

13.18 .05486 6.485

13.11 .05474 8.271

13.06 .05457 9.820

12.99 .05453 11.27

12.94 .05445 12.44

12.89 .05442 13.43

12.84 .05434 14.32

0.29

pW pD pB

12.51 .06543 0.0

12.47 .06543 2.171

12.39 .06538 4.241

12.32 .06517 6.126

12.27 .06493 7.802

12.20 .06488 9.288

12.15 .06469 10.66

12.10 .06460 11.77

12.05 .06452 12.71

12.00 .06457 13.61

0.31

pW pD pB

11.67 .07698 0.0

11.63 .07698 2.054

11.56 .07683 3.988

11.50 .07658 5.761

11.44 .07643 7.346

11.39 .07613 8.740

11.34 .07599 10.04

11.29 .07585 11.09

11.24 .07578 12.01

11.21 .07559 12.87

0.33

pW pD pB

10.87 .08946 0.0

10.84 .08946 1.938

10.77 .08934 3.749

10.71 .08908 5.415

10.66 .08872 6.890

10.60 .08863 8.231

10.55 .08844 9.447

10.50 .08836 10.46

10.45 .08834 11.35

10.38 .08859 12.21

0.35

pW pD pB

10.13 0.1026 0.0

10.10 0.1026 1.832

10.03 0.1025 3.531

9.972 0.1022 5.095

9.909 0.1022 6.497

9.860 0.1019 7.763

9.809 0.1018 8.888

9.766 0.1016 9.855

9.726 0.1014 10.71

9.712 0.1008 11.48

7

xD 0.03 0.07 0.15 0.23 0.27 0.31 0.35

µ BE / kJ mol-1

6

5

4

3 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

xB

Fig. IX-1. The excess chemical potential of TBA, �EB , against mole fraction of TBA, xB , in TBA�B�−DMSO �D�−H2 O (W) at 24�95  C. The data from Trandum et al. (1998). Reproduced with permission from the Journal of Physical Chemistry, 102, 5182, (Trandum, Ch. et al.), Copyright (1998), American Chemical Society.

249

0.05

0.055

0.06

12.78 .05451 15.12

12.74 .05452 15.78

11.97 .06438 14.36

xD \xB

0.07

0.08

0.09

0.10

0.11

0.12

12.69 .05461 16.37

12.63 .05455 17.29

12.54 .05486 18.03

12.45 .05517 18.61

12.40 .05531 19.02

-------

-------

pW pD pB

0.27

11.92 .06439 15.01

11.89 .06437 15.58

11.77 .06487 16.55

11.68 .06531 17.30

11.56 .06607 17.93

11.37 .06771 18.47

-------

-------

pW pD pB

0.29

11.14 .07580 13.63

11.09 .07595 14.29

11.02 .07630 14.88

10.93 .07657 15.82

10.84 .07701 16.56

10.79 .07702 17.13

-------

-------

-------

pW pD pB

0.31

10.34 .08851 12.93

10.30 .08851 13.57

10.28 .08825 14.10

10.22 .08806 15.00

10.21 .08747 15.66

-------

-------

-------

-------

pW pD pB

0.33

9.685 0.1005 22.54

9.668 0.1001 12.77

9.633 0.1000 13.29

9.574 .09982 14.18

9.445 0.1009 14.94

-------

-------

-------

-------

pW pD pB

0.35

This table is reproduced with permission from the Journal of Physical Chemistry, 102, 5182, (Trandum et al.), Copyright (1998), Americal Chemical Society.

7

µ BE / kJ mol-1

6

5

4

xB 2 0.00 0.05

0.10

0.06 0.08 0.10 0.12

0.01 0.02 0.035 0.05

3

0.15

0.20

0.25

0.30

0.35

0.40

xD

Fig. IX-2. The excess chemical potential of TBA, �EB , against mole fraction of DMSO, xD , in TBA �B�−DMSO �D�−H2 O (W) at 24�95  C. The data from Trandum et al., (1998). Reproduced with permission from the Journal of Physical Chemistry, 102, 5182, (Trandum, Ch. et al.), Copyright (1998), American Chemical Society.

250

Fig. IX-3 to IX-6

0

µ DE / kJ mol-1

-2

-4

-6

-8

-10 0.00

0.21 0.27 0.31 0.35

xD 0.03 0.09 0.15 0.02

0.04

0.06

0.08

0.10

0.12

0.14

xB

Fig. IX-3. The excess chemical potential of DMSO, �ED , against mole fraction of TBA in TBA �B�−DMSO �D�−H2 O (W) at 24�95  C. The data from Trandum et al. (1998). Reproduced with permission from the Journal of Physical Chemistry, 102, 5182, (Trandum, Ch. et al.), Copy­ right (1998), American Chemical Society. 10

H BE / kJ mol-1

5

0

x D0 = 0 0.0478 0.1031 0.1999 0.2341 0.2735 0.3540

-5

-10

-15

-20 0.00

0.02

0.04

0.06

0.08

0.10

0.12

xB

Fig. IX-4. The excess partial molar enthalpy of TBA, HBE , against mole fraction of TBA, xB , in TBA �B�−DMSO �D�−H2 O (W) at 25  C. xD0 is the initial mole fraction of DMSO in the mixed solvent of DMSO−H2 O into which TBA was titrated to obtain HBE . The data from Trandum et al. (1998). Reproduced with permission from the Journal of Physical Chemistry, 102, 5182, (Trandum, Ch. et al.), Copyright (1998), American Chemical Society.

251

5

TS BE / kJ mol-1

0

-5

xD0 = 0

-10

0.0478 0.1031 0.1999 0.2341 0.2735 0.3540

-15

-20

-25 0.00

0.02

0.04

0.06

0.08

0.10

0.12

xB

Fig. IX-5. The excess partial molar entropy of TBA (times T ), TSBE , against mole fraction of TBA, xB , in TBA �B�−DMSO �D�−H2 O (W) at 25  C. The data from Trandum et al. (1998). xD0 is the initial mole fraction of the mixed solvent, D–W. Reproduced with permission from the Journal of Physical Chemistry, 102, 5182, (Trandum, Ch. et al.), Copyright (1998), American Chemical Society. -2 -4

H DE / kJ mol-1

-6 -8

xB0 = 0

-10

0.00954 -12

0.0302

-14

0.0507 0.0724

-16 -18 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

xD

Fig. IX-6. The excess partial molar enthalpy of DMSO, HDE , against mole fraction of DMSO, xD , in TBA �B�−DMSO �D�−H2 O (W) at 25  C. The data from Trandum et al. (1998). xB0 is the initial mole fraction of TBA in the mixed solvent of TBA–W, into which DMSO was titrated to obtain HDE . Reproduced with permission from the Journal of Physical Chemistry, 102, 5182, (Trandum, Ch. et al.), Copyright (1998), American Chemical Society.

252

Fig. IX-7 to IX-9

0

TS DE / kJ mol-1

-2

-4

xB0 0.00954

-6

0.0302 0.0507

-8

0.0724 -10 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

xD

Fig. IX-7. The excess partial molar entropy of DMSO (times T ), TSDE , against mole fraction of DMSO, xD , in TBA �B�−DMSO �D�−H2 O (W) at 25  C. The data from Trandum et al. (1998). xB0 is the initial mole fraction of the mixed solvent B–W before D is mixed. Reproduced with permission from the Journal of Physical Chemistry, 102, 5182, (Trandum, Ch. et al.), Copyright (1998), American Chemical Society.

Next, we determine the i−j interaction functions in terms of chemical potential by eq. (1) assuming that the xB and xD increments are small enough. The plots of �EB−B against xB are shown in Fig. IX-8 and those of �EB−D against xB are shown in Fig. IX-9. The self-consistency of the partial pressure data was checked by noting that the values of �EB−D were equal to those of �ED−B within ±0�01 kJ mol−1 (see Trandum et al. (1998) for detail). It is now clearer in Fig. 8 that the xD dependence of �EB−B changes from xB = 0�0425 to 0.065, corresponding to point X (0.045) and point Y (0.065) in binary TBA �B�−H2 O (see Table V-3 (p. 114)). Similarly, the xB depen­ dence of �EB−D shows a change from xD = 0�24 to 0.34, the Mixing Scheme transition region from I to II, for binary D–W (see Table VI-2 (p. 159)). The B—B interac­ tions in terms of enthalpy and entropy, HBE−B and TSBE−B are shown in Fig. IX-10. In calculating them by eqs. (2) and (3), the HBE data at a given xD0 were interpolated at fixed values of xD , noting that xD = xD0 (1−xB � (see Westh & Koga (1996) for detail). The HBE−B pattern changes on addition of D are similar to what we saw in the 1P-probing methodology discussed in Chapter VII. The H1EP−1P pattern is replaced here with HBE−B . Accordingly, Fig. 10 indicates that DMSO (D) is an amphiphile in comparison with the probing TBA (B). From the degree of the left shift of point X of the HBE−B pattern, DMSO seems to contain a rather strong hydrophobic effect, in addition to the reduction of the peak height which indicates a hydrophilic con­ tribution. We discuss the relative effects of the hydrophobic and the hydrophilic

253

5

0.0125 0.0225 0.0325 0.0425 0.053 0.065 0.085 0.105

xB

0

µ BBE / kJ mol-1

-5 -10 -15 -20 -25

-30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

xD

Fig. IX-8. The TBA–TBA interaction in terms of chemical potential, �EB−B , in TBA �B�−DMSO �D�−H2 O (W) against mole fraction of DMSO, xD , at 25  C. The data from Trandum et al. (1998). Reproduced with permission from the Journal of Physical Chemistry, 102, 5182, (Trandum, Ch. et al.), Copyright (1998), American Chemical Society.

2

µ BDE / kJ mol-1

0

-2

-4

xD -6

-8 0.00

0.02

0.04

0.06

0.08

0.04 0.08 0.12 0.16 0.24 0.34 0.10

0.12

0.14

xB

Fig. IX-9. The TBA–DMSO interaction in terms of chemical potential, �EB−D , in TBA �B�−DMSO �D�−H2 O (W) against mole fraction of TBA, xB , at 25  C. The data from Trandum et al. (1998). Reproduced with permission from the Journal of Physical Chemistry, 102, 5182, (Trandum, Ch. et al.), Copyright (1998), American Chemical Society.

254

Fig. IX-10 to IX-12

xD = 0 0.0478 0.1031 0.1999 0.3540

HBBE or TSBBE / kJ mol-1

500

400

X

300

200

100

0

-100 0.00

0.02

0.04

0.06

0.08

0.10

xB

Fig. IX-10. The TBA–TBA interaction in terms of enthalpy, HBE−B , (filled symbol) and entropy, TSBE−B , (hollow symbols) in TBA �B�−DMSO �D�−H2 O (W) at 25  C. Point X is determined using HBE−B data. The original data from Trandum et al. (1998). Reproduced with permission from the Journal of Physical Chemistry, 102, 5182, (Trandum, Ch. et al.), Copyright (1998), American Chemical Society.

contribution within DMSO in the next section, where we use 2-butoxyethanol (BE) instead of TBA for B in B–D–W. We note in Fig. 10 that the HBE−B pattern pre­ serves type (a) anomaly (Fig. VI-1(a)) within Mixing Scheme I of D–W, xD < 0�2, suggesting that the B–B interaction is bulk H2 O-mediated which is modified by the presence of both B and D. We have discussed this repeatedly in the previous chap­ ters. Another point to note in the figure is the strong enthalpy–entropy compensation effect. The heterogeneous B–D interactions in terms of enthalpy and entropy, HBE−D and TSBE−D are also calculated by eqs. (2) and (3) in the same manner as the B–B interactions. The results are shown in Fig. IX-11. The enthalpy–entropy compensation effect is also apparent in the figure. The xB dependence pattern of HBE−D shows type (b) (Fig. VI-1(b)) (there is a hint of type(a) anomaly) at a low xD composition. As xD increases, however, the pattern changes to type (c), the same as HDE−D for binary DMSO−H2 O, as shown in Fig. VI-14. This supports the fact that the B–D interaction is also bulk H2 O-mediated. In Fig. IX-12, we plot the values of HBE−D extrapolated to xB = 0 against xD . While the data points are scarce, its xD dependence seems to take type (c) as HDE−D for the binary DMSO �D�−H2 O, as shown in Fig. VI-14. This indicates that HBE−D at xB = 0 reflects how DMSO modifies H2 O, and thus the B–D interaction is operative via the bulk H2 O-modified by the presence of both solutes, as long as the composition of solutes is low enough for the system to be within Mixing Scheme I. This point has been stressed throughout Chapters V to VIII.

255

140

xD 0.0239 0.0478 0.1031 0.1999 0.2735

X

HBDE or TSBDE / kJ mol-1

120 100 80 60 40 20 0

0.00

0.02

0.04

0.06

0.08

0.10

xB

Fig. IX-11. The TBA–DMSO interaction in terms of enthalpy, HBE−D , (filled symbols) and entropy, TSBE−D , (hollow symbols) against mole fraction of TBA, xB , in TBA �B�−DMSO �D�−H2 O (W) at 25  C. The data from Trandum et al. (1998). Reproduced with permission from the Journal of Physical Chemistry, 102, 5182, (Trandum, Ch. et al.), Copyright (1998), American Chemical Society. 140

HB-DE (xB=0) / kJ mol-1

120 100 80 60 40 20 0

0.0

0.1

0.2

0.3

0.4

0.5

xD

Fig. IX-12. The TBA–DMSO enthalpic interaction, HBE−D , at the infinite dilution of TBA, xB = 0, in TBA �B�−DMSO �D�−H2 O (W) against mole fraction of DMSO, xD , at 25  C. The original data from Trandum et al. (1998). The xD -dependence is type (c) pattern discussed in Section [VI-1] (see Fig. VI-1(c)). The DMSO–DMSO enthalpic interaction HDE−D in binary DMSO−H2 O takes the type (c) pattern also. See text.

256

Fig. IX-13 to IX-14

[IX-3] SOLUTE–SOLUTE INTERACTIONS IN 2-BUTOXYETHANOL (BE)–DIMETHYL SULFOXIDE (DMSO)–H2 O The vapor pressure analysis successful in the previous section is not applicable to this BE�B�−DMSO �D�−H2 O�W� system, since the system has an azeotrope at about xB ≈ 0�005. For binary BE−H2 O, the Boissonnas method was applied from the solute-rich end down to xBE = 0�01 (see the difficulty associated with the presence of an azeotrope in applying the Boissonnas method discussed in Sections [III-6] and [III-7]). For ternary systems, the following methods could be possible solutions: (1) At a given �xB � xD �, the total pressure, p, is determined by the vapor pressure apparatus described in Section [III-5]. Then the gas phase composition of B, D, and W is determined by gas chromatography. The sampling for gas chromatography could be problematic, but if successful, this is the most reliable method. (2) If component D is much less volatile so that the pD is a few orders of magnitude smaller than pB , the absolute value of pB alone can be determined by head-space gas chromatography (Halpern, 1997). The binary B−H2 O, in which the values of pB are already known, could be used as the standard samples for gas chromatography. Thus, we obtain �EB as a function of xB and xD . This was used for the present BE−DMSO−H2 O. (3) If we limit ourselves to the very dilute aqueous solution, we can assume that the partial pressure of B, pB , obeys Henry’s law and hence that of the mixed solvent D–W follows Raoult’s law, being the major component. Thus the partial pressure of D–W is calculated by, �pW + pD � = �pW + pD �0 �1 − xB �, where �pW + pD �0 is the total pressure of the binary D–W(i.e. xB = 0) at the identical xD . This is exactly what we did in the previous section to evaluate the partial pressures of column 2 in Table 1, in order to start the iteration. If in addition we limit ourselves to the case in which component D is non-volatile, such as a salt, then it follows that pW = �pW �0 �1 − xB � and pB = p − pW with pD = 0. This route was used recently to calculate �E1P−S in 1-propanol (1P)–salt (S)–H2 O (Miki et al., 2006). We used method (2) above to determine the partial pressure of BE, pB , in BE (B)– DMSO (D)–H2 O (W). The resulting excess chemical potential of BE, �EB , is shown in Fig. IX-13, as a function of the mole fraction of BE, xB , and the initial mole fraction of D in the initial mixed solvent D–W, xD0 . Fig. IX-14 shows the plots of HBE . From these, HBE−B and �EB−B are calculated by eqs. (2) and (1) at the grid of �xB � xD � and are plotted in Fig. IX-15. Similarly, HBE−D and �EB−D are calculated and plotted in Fig. IX-16. The value of �Ei−j is an order of magnitude smaller than HiE−j . This is the consequence of the enthalpy–entropy compensation (Lumry, 2003; Lumry & Rajender, 1970). The general behaviors of HiE−j are all similar to the equivalent plots for TBA−DMSO−H2 O discussed in the previous section. Some quantitative differences are due to the difference in the hydrophobicity between TBA and BE. The values of HBE−D at xB = 0 are obtained by extrapolating the data in Fig. 16, as we did in the previous section. Alternatively, we

257

9

xD0 = 0 0.0159

8

µ BE / kJ mol-1

0.0517 7

0.0899

6

5

4

3 0.00

0.01

0.02

0.03

0.04

0.05

0.06

xB

Fig. IX-13. The excess chemical potential of BE, �EB , against mole fraction of BE, xB , in BE�B�−DMSO�D�−H2 O (W) at 25  C. xD0 is the initial mole fraction of DMSO in D–W, into which a small amount of B was added for gas chromatographic analysis. The data from Westh & Koga (1996). Reproduced with permission from the Journal of Physical Chemistry, 100, 433, (Westh, P. and Koga, Y.), Copyright (1996), American Chemical Society.

5

HBE / kJ mol-1

0

x D0 = 0 0.0126 0.0250 0.0545 0.0899 0.2006 0.3284 0.6607

-5

-10

-15

x D0 = 1 0.00

0.02

0.04

0.06

0.08

xB

Fig. IX-14. The excess partial molar enthalpy of BE, HBE , against mole fraction of BE, xB , in BE�B�−DMSO�D�−H2 O at 25  C. xD0 is the initial mole fraction of DMSO−H2 O into which BE is titrated to measure HBE . The data from Westh & Koga (1996). Reproduced with Permission from the Journal of Physical Chemistry, 100, 433, (Westh, P. and Koga, Y.), Copyright (1996), American Chemical Society.

258

Fig. IX-15 to IX-17

1200

xD = 0 0.02 0.04 0.06 0.08

µ B-BE or H B-BE / kJ mol-1

1000 800

HB-BE

600 400 200 0

-200 -400 0.00

µ B-BE 0.01

0.02

0.03

0.04

0.05

xB

Fig. IX-15. The BE–BE interaction in terms of chemical potential, �EB−B , and enthalpy, HBE−B , against mole fraction of BE in BE�B�−DMSO�D�−H2 O (W). The data from Westh & Koga (1996). Reproduced with permission from the Journal of Physical Chemistry, 100, 433, (Westh, P. and Koga, Y.), Copyright (1996), American Chemical Society. 160

X

xD = 0 0.02 0.04 0.06 0.08

µ B-DE or H B-DE / kJ mol-1

140 120 100 80

HB-DE

60 40 20 0 -20 0.00

E

µ B-D 0.01

0.02

0.03

0.04

0.05

xB

Fig. IX-16. The BE–DMSO interaction in terms of chemical potential, �EB−D , and enthalpy, HBE−D , against mole fraction of BE, xB , in BE−DMSO−H2 O at 25  C. The data from Westh & Koga (1996). Reproduced with permission from the Journal of Physical Chemistry, 100, 433, (Westh, P. and Koga, Y.), Copyright (1996), American Chemical Society.

259

10

HBE at xB = 0/kJ mol-1

5

0

Point Y for DMSO - H2O

-5

-10

Point X for DMSO - H2O

-15

-20 0.0

0.2

0.4

0.6

0.8

1.0

xD

Fig. IX-17. The extrapolated value of the excess partial molar enthalpy of BE, HBE , to the infinite dilution of BE, xB = 0, against mole fraction of DMSO, xD , in BE−DMSO−H2 O at 25  C. It is striking that the xD -dependence resembles that for binary DMSO−H2 O. The data from Westh & Koga (1996). Reproduced with permission from the Journal of Physical Chemistry, 100, 433, (Westh, P. and Koga, Y.), Copyright (1996), American Chemical Society.

first extrapolate the HBE data (Fig. 14) to xB = 0 to obtain HBE data at xB = 0. We then plot them against xD as shown in Fig. IX-17. It is striking that the values of HBE at the infinite dilution of B in the presence of D makes a clear change in the xD dependence slope at about point X and Y in HDE−D of purely binary D–W, as is evident in Fig. 17. From the smooth curve drawn in Fig. 17, we evaluate graphically HBE−D at xB = 0 by eq. (2). The results from both routes, from Fig. 16 and from Fig. 17, are plotted in Fig. IX-18. As for B = TBA shown in Fig. 12, Fig. 18 shows a type (c) pattern, Fig. VI-1(c), similar to HDE−D for binary DMSO−H2 O (Fig. VI-14). Namely, the hetero­ geneous B–D interaction at xB = 0� HBE−D �xB = 0�, shows the same xD dependence of the homogeneous D–D interaction, HDE−D , for binary D–W. This suggests that upon addition of the infinitely small amount of B in the mixed solvent D–W, the HBE−D �xB = 0� merely reflects the mixing situation of D–W, or the HBE−D �xB = 0� probes the mixing scheme in D–W. This suggests again that the solute–solute interaction, whether homogeneous or heterogeneous, is bulk H2 O-mediated within Mixing Scheme I. For the heterogeneous interaction, the effects of two species act competitively or cooperatively. Furthermore, the HDE−D for D–W could also be said to probe the state of mixing scheme modified by D at the infinite dilution. Thus, quantitative differences in the HBE−D �xB = 0� pattern and that of HDE−D for binary D–W indicate the relative differences in the way in which TBA, BE, and DMSO modify H2 O at the infinite dilution. Fig. IX-19 compares these. Thus the values of

260

Fig. IX-18 to IX-19

160

From data in Fig. IX-17 From data in Fig. IX-16

HB-DE at xB = 0/kJ mol-1

140 120 100

X 80 60 40

Y 20 0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

xD

Fig. IX-18. The BE–DMSO enthalpic interaction, HBE−D , at the infinite dilution of BE, xB = 0, against mole fraction of DMSO, xD , in BE−DMSO−H2 O at 25  C. This shows type (c) pattern, [VI-1]. The data from Westh & Koga (1996). 160

HD-DE for D-W

140

H TBA-DE at x TBA = 0

HB-DE at xB = 0 or HD-DE

HBE-DE at xBE = 0

XBE

120 100 80

XTBA 60 40

X

20

Y

0 -20 0.0

0.1

0.2

0.3

0.4

0.5

xD

Fig. IX-19. Comparison between HBE−D at xB = 0 for B−D−H2 O for B = BE and TBA and HDE−D for binary D – W at 25  C. The data from Westh & Koga (1996) for B = BE, and Trandum et al. (1998) for B = TBA. Those for the binary D–W from Lai et al. (1995).

Eq. (IX-12)

261

HBE−D �xB = 0� are 150, and 122 kJ mol−1 for B = BE and TBA at xD = 0, while that of E HD−D is 51 kJ mol−1 at the infinite dilution in D–W. (See Table VI-2 (p. 159)) The xD = BE, 0.17 for B = = TBA and 0.21 for binary loci of point X in Fig. 19 are 0.09 for B = D–W. In the previous chapters, we showed that the xi locus of the onset of transition to Mixing Scheme II, i.e. point X, occurs at a smaller value for a stronger hydrophobe. Furthermore the value of HiE−i at point X is larger for a stronger hydrophobe, and smaller if the hydrophilicity is stronger. Accordingly, the above observations indicate that the net hydrophobicity of the solute in the H2 O-rich region is ranked as, BE>TBA>DMSO�

(IX-12)

Thus, the heterogeneous interaction may be useful in ranking the species belonging to the different classes; BE and TBA are mono-ols and show type (a) HiE−i pattern, while DMSO has type (c) HiE−i . This inter-class comparison was not possible in the previous chapters, in which only the homogeneous interaction functions were used. While the heterogeneous interaction is the net result of the competition or cooperation of the two different species, it may not be linearly additive. If so, the value of HBE−D at the infinite dilution, xB = xD = 0, would be equal to the average of HDE−D at xD = 0 and HBE−B at xB = 0 in the respective binary systems. As is evident in Table IX-3, it is not the case, while both values are parallel. The number of examples, however, is limited at this point in time. Fig. 10 for B = TBA and Fig. 15 for B = BE are equivalent to the H1EP−1P pattern changes shown repeatedly in Chapter VII in the presence of an amphiphile. Here we use TBA or BE as a probe to elucidate the effect of DMSO on H2 O instead of 1-propanol (1P) in Chapter VII. As discussed in Chapter VII, the net effect of DMSO can be separated into the hydrophobic and the hydrophilic contributions relative to the probing B. Namely, the degree of the left shift of point X in HBE−B pattern reflects hydrophobicity and that of the decrease in HBE−B at point X shows the strength of hydrophilicity. Fig. IX-20 shows the xB locus of point X of HBE−B for B–D–W, normalized to that for binary B–W �xD = 0�. The figure indicates that DMSO has a hydrophobic moiety stronger than BE and even more so than TBA. This surprising indication may support the view discussed in Section [VI-5] that in Mixing Scheme I, DMSO molecules exist as = O pointing CH3 -groups outward a small tight cluster bound together by dipoles of >S = of these clusters. Fig. IX-21 shows the change in the value of HBE−B at point X. The Table IX-3. The values of HBE−D at xB = xD = 0 in B−DMSO−H2 O. B TBA BE a b

HBE−D �xB = xD = 0� kJ mol−1 122 150

See Table V-3 (p. 114) See Table VI-2 (p. 159)

�HBE−B �xB = 0�a + HDE−D �xD = 0�b �/2 kJ mol−1 70

220

262

Fig. IX-20 to IX-21

1.1

BE TBA

xB / xB * at point X

1.0

0.9

0.8

0.7

0.6

0.5 0.00

0.02

0.04

0.06

0.08

0.10

0.12

xD

Fig. IX-20. The xB locus of point X in the HBE−B pattern in B–D–W against xD . xB ∗ is that for binary B–W, xD = 0. The data from Westh & Koga (1996) for B = BE, and from Trandum et al. (1998) for B = TBA. 1.1

BE TBA

HB-BE / HB-BE* at point X

1.0

0.9

0.8

0.7

0.6

0.5

0.4 0.00

0.02

0.04

0.06

0.08

0.10

0.12

xD ∗

Fig. IX-21. The value of HBE−B at point X in the HBE−B pattern in B–D–W at 25  C. HBE−B is that for binary B–W. The data from Westh and Koga (1996) for B = BE and Trandum et al. (1998) for B = TBA.

263

figure suggests that DMSO has a hydrophilic moiety stronger than TBA and more so than BE, which is understandable, in view of the hydrophobicity ranking, BE>TBA, as discussed in Chapter V. Thus, it is clear that the heterogeneous interaction functions in many other ternary or multi-component aqueous solutions will be useful in understanding the molecular processes in aqueous solutions, and particularly in those of biological importance. More generally, model-free thermodynamic data of the second, third and higher order deriva­ tives of G are powerful in advancing the molecular level understanding of aqueous or any other solutions. The accumulation of such data without resorting to any fitting function is awaited.

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Chapter X

IN CLOSING – EXECUTIVE SUMMARY ON THE EFFECT OF SOLUTE ON H2 O

The nature of an aqueous solution is crucially dependent on its composition; there are three distinct composition regions, in each of which the mixing scheme is qualitatively different. The unique characteristics of liquid H2 O are retained in the H2 O-rich region even in the presence of solute. In the solute-rich composition range, solute molecules tend to cluster together, and thus the individuality of the solute dictates the nature of the mixture. In the intermediate region, the aqueous solution consists of two kinds of clusters, one rich in H2 O and the other in the solute molecules. These clusters are reminiscent of the mixing schemes operative in the H2 O-rich and the solute-rich regions. For convenience, we call these three mixing schemes operative in each region Mixing Scheme I, II and III from the H2 O-rich end. Mixing Scheme I is such that the integrity of liquid H2 O is retained. In particular, the hydrogen bond network is connected throughout the entire bulk (bond-percolated) and the hydrogen bond angle and its strength fluctuates widely. If the solute in question is a hydrophobe, classically called a “structure-maker”, the hydrogen bond network of H2 O is enhanced in the immediate vicinity of the solute corresponding to the classical “iceberg formation”. In addition, however, the hydrogen bond probability of bulk H2 O away from “icebergs” is reduced progressively. If a hydrophile, a “structure-breaker”, is added, it forms hydrogen bonds directly to the existing hydrogen bond network of H2 O and the solute acts as an impurity center within the network. Consequently, the degree of fluctuation inherent in liquid H2 O is suppressed and the hydrogen bonds are partly broken. These effects are due to the loss of H donor/acceptor symmetry enjoyed in pure liquid H2 O. If the solute is an amphiphile, the competition or cooperation of the effects of the hydrophobic and the hydrophilic moieties results in the net effect on H2 O. Either way, as long as the composition is low enough, the global hydrogen bond probability is still high enough that the bond percolation of H2 O is intact. When the solute composition increases to a threshold value, depending on the nature of solute, the global average of the hydrogen bond probability reaches the bond-percolation threshold for the bond connectivity of the ice Ih type. Thus, the integrity of liquid H2 O is lost and Mixing Scheme II sets in. Within Mixing Scheme I, the solute–solute interaction is bulk H2 O-mediated and of an extremely long range. The transition from Mixing Scheme I to II is associated with anomalies in the third derivatives of Gibbs energy, G. This contrasts with “phase” transitions for which the second derivative quantities show anomalies. Thus, the transition in the

266

Fig. A-1

mixing scheme is more subtle than “phase” transitions. The mixing scheme transition occurs at a lower solute composition as temperature increases. Its loci in the mole fraction–temperature field extrapolate to about 83 C at the infinite dilution, close to the hydrogen bond percolation threshold in terms of the average probability of hydrogen bond in pure H2 O estimated by Stanley et al. (Stanley & Teixeira, 1980). For tert-butanol or 2-butoxyethanol, the boundary ends towards low temperatures at the incongruent melting point of an addition compound of BH2 Om type, where B is tert-butanol or 2-butoxyethanol and m is about 7 or 38 respectively. This mixing scheme boundary is sometimes called the “Koga line”. Similar conclusions are drawn for electrolytes. Some ions (e.g. acetate) act as a hydrophobe, while others as a hydrophile (e.g. Br −  I−  ClO4 − , and SCN− ). In addition, some other ions (e.g. Na+  NH4 +  F− , and Cl− ) are the hydration centers and they hydrate a number of H2 O molecules. The bulk H2 O away from the hydration shell, however, is not affected by the presence of this type of ions. We thus learned that there are basically three kinds of effects of solute on the molecular organization of H2 O, as long as the solution is within Mixing Scheme I. In relation to the Hofmeister ranking of anions, kosmotropes (salting out, stabilizing agents) are either hydration centers or hydrophobes. Within hydration centers, a stronger kosmotrope hydrates a larger number of H2 O. Chaotropic ions, on the other hand, are hydrophiles; the stronger its hydrophilicity, the more chaotropic the ion. In ternary (and multi-component) solutions within Mixing Scheme I, the solute– solute interactions are also bulk H2 O-mediated, and reflect how each solute modifies H2 O. Just as for an amphiphile described above, the net interaction, whether it is homogeneous (inter-same-species) or heterogeneous (inter-different-species), is the result of competitive or cooperative effects of the combination of the above three kinds, depending on the solutes in question. The competition or cooperation between two different species does not seem linearly additive, however. All of these findings by the present thermodynamic studies are completely model-free. They are our natural interpretation of the behavior of the second and third derivatives of G that are determined experimentally and/or numerical differentiation without resorting to any fitting function. These findings based on higher order derivatives than hitherto utilized provide the molecular understanding equal, if not superior in some cases, to those elucidated by modern spectroscopic techniques.

Appendix A

GRAPHICAL DIFFERENTIATION BY MEANS OF B-SPLINE

In computer graphics, it is common to draw a smooth curve through data points using the B-spline method. We have been doing manual graphing using a flexible ruler followed by reading the data off the smooth curve drawn at a given interval. These data are used to approximate the slope of the mid point. The reliability of such an approximation has been discussed elsewhere at some length (Parsons et al., 2001;Koga, 2004a). The advantage of manual graphing is that human judgment in the quality of each data point is readily applicable. Fig. A-1 shows the enthalpic interaction between 1-propanol (1P) molecules, H1EP−1P , in binary 1P–H2 O (Koga et al., 2006). This is the best data set to our knowledge. Also shown in Fig. A-1 are the results by B-spline on the same data set. As is evident, the standard cubic B-spline and our manual graphing provide the same

350 300

Manual B-spline

H 1P-1PE / kJ mol-1

250 200 150 100 50 0 -50 0.00 0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

x1P

Fig. A-1. Cubic B-spline vs. manual graphing. A satisfactory case. Note, however, the two very bad points by B-spline at x1P ≈ 012. Reproduced with permission from the Bulletin of the Chemical Society of Japan, 79, 1347, (Koga et al.), Copyright (2006), the Chemical Society of Japan.

268

Fig. A-2

160 140

H 1P-1PE / kJ mol-1

120 100 80 60 40

Manual

20

B-spline

0 -20 0.00 0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

x1P

Fig. A-2. Cubic B-spline vs. manual graphing. An unsatisfactory case. See text. Reproduced with permission from the Bulletin of the Chemical Society of Japan, 79, 1347, (Koga et al.), Copyright (2006), the Chemical Society of Japan.

H1EP−1P plots within an acceptable uncertainty. It is apparent from the figure, however, that in the region where the slopes are close to zero, i.e. x1P >008, the resulting slopes by B-spline are unstable. Note the two bad points at about x1P ≈ 012. Another data set (Koga et al., 2006), in which a number of data points were known to be taken at a temperature higher than the set point by an unknown degree, were subjected to the cubic B-spline procedure. The results are shown in Fig. A-2 together with the manual graphing method applying human judgment as to how much to correct for these data points. There are more recent advancements in B-spline which allow uncertainties in both abscissa and ordinate and also flexible boundaries (Nunhez et al., 2004). However, for this newest method to work, the curve to be interpolated must not contain an inflection point. This is not suitable for the present purpose, since we seek an inflection point in our data. Further developments in a more robust computer interpolation technique are awaited, in view of the tediousness of manual graphing and subsequent reading of the data off the smooth curve drawn.

Eq. (B-1) to (B-5)

Appendix B

GIBBS-KONOVALOV CORRECTION

Consider the equilibrium in a binary system (B, W) between the gas and the liquid phases (see Fig. II-2). The thermodynamic quantities for the gas phase are with prime  , and those for liquid are without. Since the system is in equilibrium, B  = B

and

W  =  W 

(B-1)

The independent variables in the liquid phase are p T xB . xB  for the gas phase with common p and T . Consider now a small variation in chemical potential due to small changes in the independent variables along the equilibrium line. Recalling eq. (I-36), we � � i write the resulting variation for the gas phase, d , instead of dG in eq.(I-36) as, T � d

�

i T

=−

V Hi 1 dT + i dp + 2 T T T

�

� i  dxB   xB 

(B-2)

recalling also eq. (I-39) and eq. (I-40). Here i = B or W. Similarly for the liquid phase, d

� � i

T

=−

Hi V 1 dT + i dp + 2 T T T

�

� i dxB  xB

(B-3)

Since we consider the above variation along the equilibrium line, �

i  d T

� =d

� � i

T



(B-4)

for i = B and W. Now we wish to keep the composition of the liquid phase constant, i.e. dxB = 0. Since the partial molar volume of gas phase is much larger than that of the liquid mixture, we can ignore the latter, i.e. Vi  Vi ≈ 0. Furthermore, we assume that the gas phase mixture is ideal in comparison with the degree of non-ideality of the liquid mixture. Therefore, following eq. (II-6), Vi =

RT  p

(B-5)

270

and using eq. (II-11), �

B  xB 

� =

RT  xB 

� and

W  xB 

� =−

RT  1 − xB  

(B-6)

By virtue of eqs. (2), (3), (4), (5) and (6), we write two equations for B and W as, �

� hB R R dT = dp +  dxB   2 T p xB � � hW R R dT = dp − dx   2 T p 1 − xB   B hi = Hi − Hi = vap Hi∗ − HiE 

for i = B and W

(B-7)

(B-8) (B-9)

where vap Hi∗ is the heat of vaporization of pure i, and HiE is the excess partial molar enthalpy of i. By multiplying eq. (7) by xB  and eq. (8) by 1 − xB   and by adding them in order to eliminate dxB  , we obtain, �

dp dT

� = xB

pB hB + pW hW   RT 2

(B-10)

This equation provides the means to correct for temperature variation in determining the total pressure by the method described in Section [III-5]. While the temperature of the bath is controlled to within ±0001 K during the measurement, the day-to-day variation of ±002 K appears inevitable. To correct for the latter temperature variation, we first calculate pB and pW by the Boissonnas method (Section [III-6]), ignoring the temperature variation. We use these values to correct for the total pressure, p, by eq. (10) at the fixed mole fraction in the liquid phase, xB , using HiE separately determined. Using these corrected values of p, we recalculate pB and pW . We could reiterate the process until the values of pB and pW converge, but our experience suggests that one iteration appears sufficient. We note that a general treatment on the displacements along an equilibrium line, eq. (10) above being an example, is given in Prigogine & Defay (1954), though their approach is unique and non-standard.

Eq. (B-6) to (B-10)

Appendix C

HEAT CAPACITY ANOMALIES ASSOCIATED WITH PHASE TRANSITIONS – TWO LEVEL APPROXIMATION

In the literature, there is a voluminous accumulation of anomalies in heat capacity, Cp , the second derivative of G, associated with phase transitions. In most of these studies, some concurrent information is given such that the transition in question shows the changes in certain structural, spectroscopic, or other macroscopic properties. Some examples are shown in Fig. C-1. As shown in Fig. C-1(a), the antiferromagnetic transition of FeF2 is associated with a -type anomaly in Cp and its value seems to diverge at the transition point. Hence this is clearly the second order transition by Ehrenfest’s classification. While CoF2 may also show the same tendency in the figure, for MnF2 the value of Cp at the transition point appears finite at about 2.5R (R is the gas constant), and the pattern of anomaly seems smooth and round at the top (Stout & Catalano, 1955). Hence, the next derivative with respect to T would not show discontinuity and this transition is not readily classified by the Ehrenfest scheme. It is, however, the antiferromagnetic ordering, i.e. a change in a macroscopic property occurs across the transition point. Fig. C-1(b) is that of Cp associated with the ferromagnetic transition of EuO, which also does not show divergence at the Curie point, although the pattern is skewed resembling the -type (Kornblit & Ahlers, 1975), just as MnF2 above. Fig. C-1(c) shows the Cp anomaly associated with the changes in the twist angle of the benzene ring in p-quaterphenyl, which also has the finite value of about the same size of that for EuO, but the Cp pattern is almost symmetric (Saito et al., 1985). A similar change in the molecular conformation in biphenyl (Saito et al., 1988) results in the Cp anomaly with a small round symmetric hump of about 0.1R at 40 K (Fig. C-1(d)), while at 17 K the lock-in phase transition gives an extremely small symmetric Cp anomaly as shown in Fig. C-1(e). Fig. C-1(f) is a more recent example associated with the superconducting transition of -BEDT-TTF2 CuNCS2 (Yamashita et al., 2005). Thus, it seems reasonable to state that phase transitions are associated with anomalies in Cp , the second derivative of G, no matter how small or smooth they might be. Hence the anomalies in the third derivative quantities observed throughout this book are associated with more subtle changes than “phase” transitions.

272

Fig. C-1(a) to C-1(d)

60

Cp (elect.) / J mol-1 K-1

50

(a)

40

30

FeF2 MnF2

20

CoF2 10

0

0

20

40

60

80

100

120

140

160

T/K

Fig. C-1(a). The electronic part of heat capacity anomaly associated with the antiferromagnetic phase transitions. The data listed in Stout & Catalano (1955) were used to draw the graph. Reproduced with permission from the Journal of Chemical Physics, 23, 2013, (Stout, J. and Catalano, E.), Copyright (1955), American Institute of Physics.

50

Cp (J mole-1 K-1)

SAMPLE I SAMPLE II

40

30

65

67

69

71

T(K)

Fig. C-1(b). Heat capacity anomaly near the Curie point of EuO. Reproduced with permission from the Physical Reviews B, 11, 2678, (Kohnblit, A. & Ahlers, G.), Copyright (1975), American Physical Society.

273

15

ΔCp / J K-1 mol-1

(c)

10

5

0

160

180

200

220

240

260

280

300

T/K

Fig. C-1(c). Anomalous part of heat capacity associated with the molecular conformational change in p-quaterphenyl. The data listed in Saito et al. (1985) were used to draw the graph. Reproduced from the Journal of Chemical Thermodynamics, 17, 539, (Saito et al.), Copyright (1985), with permission from Elsevier.

ΔCp / J K-1 mol-1

0.6

0.4

1% of Cp

0.2

0

30

40

50

T/K

Fig. C-1(d). The excess heat capacity anomaly due to the twist transition of biphenyl. Reproduced with permission from Bulletin of the Chemical Society of Japan, 61, 679, (Saito et al.), Copyright (1988), the Chemical Society of Japan.

ΔCp

Fig. C-1(e) to C-1(f)

J K-1 mol-1

274

+0.1 0

1% of Cp

-0.1

θD / K

134

132

130

14

15

16

17

18

19

T/K

Fig. C-1(e). The excess heat capacity anomaly due to the lock-in transition of biphenyl. The lower part of the graph shows the Debye characteristic temperature, the deviation from the smooth curve of which was converted to the excess heat capacity. Reproduced with permission from Bulletin of the Chemical Society of Japan, 61, 679, (Saito et al.), Copyright (1988), the Chemical Society of Japan.

0.8

Cooling rate slow

ΔCp / J K-1 mol-1

fast 0.6

0.4

0.2

0.0 8

9

10

T/K

Fig. C-1(f). The heat capacity anomaly associated with the superconductive phase transition of a single crystal of -BEDT-TTF2 CuNCS2 determined by Yamashita et al. (2005). I thank Pro­ fessor Nakazawa for providing the raw data from which the above graph was drawn. Reproduced from the Thermochimica Acta, 431, 123, (Yamashita, et al.), Copyright (2005), with permission of Elsevier.

Eq. (C-1) to (C-3)

275

Here we show that a variety of Cp anomalies associated with phase transitions can be understood even within the mean-field approximation by considering the degeneracy difference between the ground and the excited states in the two-level thermodynamic treatment originated by Strassler & Kittel (1965). They consider the system consisting of a large number, N , of subsystems that could be in the ground state A or the excited state B. As T increases, the population density, , of B increases cooperatively. The cooperativity is taken into account only in the energy of excitation such that the energy gap decreases in proportion to the square of , i.e. in the mean-field approximation. However, they stressed the importance of the difference in the degeneracy between the two states in relation to the order of phase transition. We applied this idea and calculated the Cp anomalies. The results showed surprisingly rich features depending crucially on the degeneracy difference, or the standard entropy difference of the two states (Koga, 1975 & 1978; Takai et al, 1994). Here we reiterate our earlier treatment (Koga, 1978). Consider a system of N sub­ systems which are either in the ground state A or the excited one B. If the subsystems interact with each other (which of course is necessary for the system to exhibit any phase transition), the partition function, Q, is written as, Q=

N! N N q Aq BQ  NA !NB ! A B c

(C-1)

Here NA and NB are the number of subsystems in state A and B respectively, and N = NA +NB . qA and qB denote the subsystem partition functions of A and B. The system free energy for a mole of subsystems, Fm , is calculated using eq. (1) as, Fm = −RT ln Q, which yields, Fm = 1 −  fA∗ + RT ln1 −  +  fB∗ + RT ln  + f EX

(C-2)

where  = NB /N fA∗ = −RT ln qA fB∗ = −RT ln qB , and f EX = −RT ln Qc . fA∗ and fB∗ are the free energy of pure A and pure B respectively, and f EX is the excess free energy due to inter-subsystem interactions. The configurational partition function is written as in the mean-field approximation, � � Qc = exp− ij /RT 

� � where ij is the average nearest neighbor interaction energy. If for simplicity, the nearest neighbor subsystems have the� attractive interaction, BB < 0, only when both � of them are in the excited state, then ij can be expressed as, � � Nm k ij = f EX = BB 2 = − 2

2 2

(C-3)

276

Fig. C-2 to C-3

where m is the number of the nearest neighbors and k = −Nm BB > 0. We define the energy gap ∗ and the degeneracy of the ground and the excited states gA and gB , hence fB∗ − fA∗ = ∗ − Ts∗

(C-4)

where s∗ = R lngB /gA . Now by minimizing eq. (2) with respect to , using eq. (3) and eq. (4), � ∗

0 =  − k − RT

� s∗  − ln  R 1−

(C-5)

Eq. (5) provides the value of  given ∗ , s∗ and k at a required temperature T , and the thermodynamics of the system is fully described. In particular, the total molar energy for the system, Em , and the molar heat capacity, Cm , can be written as, Em = ∗  − k/22

(C-6)

Cm = ∗ − k / T 

(C-7)

The solution of  in eq. (5) is the intersection of the following two curves in the

 − z plane. 

1 −  � � k ∗ s∗ z= − +  RT k R z = ln

(C-8)

(C-9)

Fig. C-2 shows the curve of eq. (8) against 0 <  < 1. Eq. (9) is a straight line with the slope k/RT with the fixed point W ∗ /k s∗ /R. Here ∗ /k > 05 is assumed, otherwise the roles of state A and B are reversed. If the values of ∗ s∗ and k are such that the fixed point W is above the tangent PQR to the curve of eq. (8) at point Q(0.5, 0), the system displays the characteristic first order phase transition. Namely, at T = 0, the stable solution is  = 0, and all the subsystems are at the ground state. As temperature increases, the value of  increases and that of heat capacity, Cm , changes according to eq. (7). At a certain temperature, three solutions for  exist as shown by a broken line in Fig. 2. However, the lowest value gives the smallest free energy, until T = T ∗ at which the straight line of eq. (9) cuts point Q(0.5, 0). At this temperature, the values of Fm become the same at the two end solutions 1 and 2 , and the first order transition associated with the jump from 1 to 2 occurs. Thus T ∗ is equal to the first order phase transition temperature. When point W is on the line segment QR, the second order phase transition is apparent in the Ehrenfest classification. Fig. C-3 shows the three cases with fixed point

Eq. (C-4) to (C-9)

277

10

W

T = infinity

ρ2

5

z

Q

T

=

R

T*

0

T=0

P

ρ1

-5

-10 0.0

0.5

ρ

1.0

1.5

Fig. C-2. Graphical solution of eq. (5). Line PQR is the tangent to eq. (8) at point Q(0.5, 0). See text. Reproduced with permission from the Collective Phenomena, 3, 1, (Koga, Y.), Copyright (1978), Taylor & Francis.

14 12

2(a)

10

2(b)

Cm /R

2(c) 8 6 4 2 0 0.0

0.5

1.0

1.5

T / T*

Fig. C-3. The molar heat capacity anomaly associated with the second order phase transition within the mean-field two state approximation for cases 2(a), 2(b), and 2(c). See Table C-1 for the chosen parameters. Reproduced with permission from the Collective Phenomena, 3, 1, (Koga, Y.), Copyright (1978), Taylor & Francis.

278

Fig. C-4

Table C-1. The chosen values for k/RT∗ , ∗ /k, and s∗ /R for the examples of the second and the higher(?) order phase transitions. Reproduced with permission from the Collective Phenomena, 3, 1, (Koga, Y.), Copyright (1978), Taylor & Francis. k/RT ∗

Case

Second order transitions 2(a) 4 2(b) 4 2(c) 4 Higher order(?) transitions 3(d) 25 3(e) 38 3(f) 2

6

∗ /k

s∗ /R

05 06 14

0 04 36

15 06 06

25 038 02

3(d) 3(e)

Cm / R

3(f) 4

2

0 0.0

0.5

1.0

1.5

T / T*

Fig. C-4. The heat capacity anomaly associated with the higher (?) order phase transition within the mean-field two state approximation for cases 3(d), 3(e) and 3(f). See Table C-1 for the chosen parameters. Reproduced with permission from the Collective Phenomena, 3, 1, (Koga, Y.), Copyright (1978), Taylor & Francis.

279

W at (0.5, 0), (0.6, 0.4) and (1.4, 3.6). All these are on line QR. Case 2(a) with W (0.5, 0), i.e. s∗ = 0 or gA = gB , corresponds to the Bragg–Williams approximation for the order–disorder transition, the Weiss–Heisenberg theory for spin 1/2 ferromagnets, and other equivalent theories with the classical mean-field approximation without taking into account of the degeneracy difference between the ground and the excited states. As is evident from the figure, Cp shows discontinuity but does not diverge. If we relax the condition of the iso-degeneracy, the -type Cp anomaly is recovered (see case 2(b)). If we make s∗ to be large, then the Cp pattern appears almost symmetric (case 2(c)). In these cases also, T ∗ is identified as the transition temperature. Thus, in addition to the above requirement that point W lies on line QR, k/RT∗ is equal to the slope of the tangent PQR, i.e. 4. Table C-1 lists the values for the examples of the second order phase transitions shown in Fig. 3. We note in passing that a required condition for the system to exhibit the first order phase transition is k/RT ∗ > 4 (Strassler & Kittel, 1965). When the fixed point W lies below line QR but above the abscissa ∗ /k > 0, the Cp anomaly does not show discontinuity. The Cp pattern appears as a smooth and round hump and its size and shape depends crucially on the values of k ∗ /k and s∗ as shown in Fig. C-4. See Table 1 for the chosen values for the three cases shown in Fig. 4. In this case T ∗ does not coincide with the peak position and the Ehrenfest classification is not at all applicable. Thus a variety of patterns and sizes in Cp anomalies associated with phase transitions, only a small number of examples of which are shown in Fig. 1, could be understood even within the mean-field approximation by additionally taking into consideration the differences in degeneracy of the ground and the excited states.

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Eq. (D-1) to (D-8)

Appendix D

FREEZING POINT DEPRESSION

Consider a dilute solution of B in solvent W at equilibrium with the pure solid W. Imagine the situation in which a minute amount of pure W precipitates on cooling the solution. This occurs at T . The pure solid W is in equilibrium with the solution. Hence, S

L

 W = W  L

(D-1)

S

where W and W are the chemical potentials of W in the liquid and the solid phases L respectively (see eq. (II-31)). Following eq. (II-12), W is written as, ∗L

L

W = W + RT ln xW + EW 

(D-2) ∗L

where xW is the mole fraction of W in the liquid phase, and W is the chemical potential of pure liquid W. EW is the excess chemical potential of W due to the non-ideality of the system, and depends on the nature of inter-molecular interactions in solution B–W. Since we take the symmetric reference system (see Section [II-1]), EW = 0 at xW = 1. For the solid phase, the pure solid W precipitates separately from the mother liquor. Hence ∗S

S

W =  W  ∗S

where W

(D-3)

is the chemical potential of the pure solid W. Eqs. (1), (2) and (3) yield, ∗L

∗S

− ln xW − EW /RT = W − W /RT = fus HW∗ − Tfus SW∗ /RT

(D-4)

with ∗L

W

∗S

∗L

= HW

∗L

− TSW 

∗S

∗S

∗L

− HW 

W = HW − TSW  fus HW∗ = HW

∗L

∗S

∗S

fus SW∗ = SW − SW

(D-5) (D-6) (D-7) (D-8)

282

Generally, both heat and entropy of fusion of pure W are positive, i.e. fus HW∗ > 0 and fus SW∗ > 0. Consider now the equilibrium between the pure liquid W and the pure solid W. This occurs at the freezing (or melting) point of pure W, T ∗ . Since EW = 0 at xW = 1, eq. (4) yields, 0 = fus HW∗ − T ∗ fus SW∗ /RT ∗ 

(D-9)

We assumed here that the values of fus HW∗ and fus SW∗ do not change from T to T ∗ , which is an acceptable assumption since T ≈ T ∗ . Due to eq. (9), eq. (4) is rewritten as, − ln xW − EW /RT = fus HW∗ − Tfus HW∗ /T ∗ /RT or, T ∗ − T = −RT ∗ 2 /fus HW∗ ln xW + EW /RT 

(D-10)

where we used a numerical approximation TT ∗ ≈ T ∗ 2 . We now take an arithmetic approximation that ln xW = ln1−xB  ≈ −xB . For example, if xW = 0 99 and xB = 0 01, then ln0 99 = −0 01005. If xW = 0 999 ln0 999 = −0 0010005. We make another more serious assumption that the solution in question is dilute enough that the solvent obeys Raoult’s law (Section [II-4]), i.e. EW = 0. Thus, eq. (10) is rewritten as, T ∗ − T  = RT ∗ 2 /fus HW∗ xB

(D-11)

Since fus HW∗ > 0 T ∗ − T  > 0, i.e. the freezing point depression by addition of a minute amount of B. Since, the proportionality factor, RT ∗ 2 /fus HW∗ , concerns only solvent W, eq.(11) indicates that it is only the quantity of solute, xB , but not the quality of B, that determines the freezing point depression as long as the solvent is fixed. Thus, the freezing point depression is called one of the “colligative properties”, together with the boiling point elevation, the vapor pressure suppression, the osmotic pressure etc. This is true only when the Raoult’s law assumption is correct, while this assumption is approximately acceptable in dilute enough solutions. As discussed in Sections [III-6] and [V-1], this does not occur even at xB ≈ 10−5 . Thus, strictly speaking, the “colligative property” is not realistic. If the solute dependent EW is non-zero in eq. (10), it should be rewritten with the activity coefficient of W, W  EW /RT = ln W , T ∗ − T  = −RT ∗ 2 /fus HW∗  ln xW W

(D-12)

Eq. (D-9) to (D-14)

283

With the osmotic coefficient, , defined as,  = 1 + ln W / ln xW 

(D-13)

eq. (12) is written as, together with the arithmetic approximation for lnxW , T ∗ − T  = −RT ∗ 2 /fus HW∗  xB  and W are now dependent, though weakly, on what the solute B is.

(D-14)

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Eq. (E-1) to (E-10)

Appendix E

TITRATION CALORIMETRY WITH DILUTE TITRANT

Isothermal titration calorimetry is designed for a liquid sample of low viscosity. Thus, the method is not applicable for a solid sample. If the titrating sample is a viscous liquid, the titration becomes problematic in quantitative delivery. In such cases, a dilute solution of the sample, instead of pure liquid can be titrated. In this manner, a solid sample could also be used. The conversion of the raw data to the excess partial molar enthalpy is straightforward (Westh et al., 1998; Koga, 2003b). t Consider a small amount of an aqueous solution of B of the fixed mole fraction xB t t with r = xB /1 − xB  titrated into the titrand mixture consisting of nB of B and nW of W. Thus, the process is described as, nB of B nW of W + nB of B nW of Wt → nB + nB  of B nW + nW  of W

(E-1) t

where super-script (t) stands for the titrant whose composition is fixed at xB . The total enthalpy difference for the above process (eq. (1)), is determined as q. Hence, q = HnB + nB  nW + nW  − HnB  nW  − H t nB  nW 

(E-2)

with t

t

H t nB  nW  = nB HB + nW HW 

(E-3)

It follows then that � � q HnB + nB  nW + nW  − HnB  nW + nW  h≡ = nB nB t

+

HnB  nW + nW  − HnB  nW  H t − HB − W  rnW r

(E-4)

286

We used the relationship r = nB /nW in deriving eq. (4). If nB and nW are sufficiently small, the right-hand side of eq. (4) becomes, t

h = HB +

HW H t − HB − W  r r

(E-5)

where HB and HW are the partial molar enthalpy of B and W respectively. The last two terms on the right of eq. (5) are associated with the titrant and constant. A small increment of eq. (5), h, can be written, using the Gibbs–Duhem relation, as, � � HW x = HB 1 − B  (E-6) h = HB + rxW r Since we take the symmetric reference states, eq. (6) is rewritten by using the excess partial molar enthalpies, HBE = HB − HB∗ etc. as, h =

HBE

� � xB HWE E  + = HB 1 − rxW r

(E-7)

where HB∗ is the enthalpy of pure B, which is taken as zero. The same is true for W. Therefore, HBE = �

h � xB 1− rxW

(E-8)

It follows then, HBE =

�

HBE + constant

(E-9)

The integration constant on the right of eq. (9) can be determined if a value of HBE is available at one point. For ethylene glycol (EG)−H2 O, the values of HmE are available E in the literature and they show a minimum. Its minimum value was used as HEG (Koga, 2003b). Recall eq. (II-75), � E� Hm E + HmE  HB = 1 − xB  xB The B−B enthalpic interaction, HBE−B , is then readily calculated using HBE data obtained by eq. (9) as, � E� HB  (E-10) HBE−B = 1 − xB  xB

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INDEX

1-propanol probing methodology, 175–179, 180–202, 205–239 Activity, 25 coefficient, 25, 236, 282 Amplitude (intensity) of fluctuation, 70, 71 Anomalies in second derivatives of G, 271–9 in third derivatives of G, 102–103, 116, 143, 155–70, 265 Azeotropy, 48–9, 65, 66 Boissonnas method, 49, 60–5, 66, 241, 244–5, 256, 270 Chemical potential of the i-th component, 13, 17, 24, 25, 26, 58 Coarse grain, 69–78, 84, 98, 118, 149 Colligative property, 282 Concentration fluctuation, 22, 48–9, 67–9, 84–5, 130, 134 Dielectric relaxation spectroscopy, 147, 220, 238 Diffusional stability criterion, 22, 41, 43 Ehrenfest’s classification of phase transition, 271 Entropy-Giles’ derivation, 11–13 Excess chemical potential of the i-th component, 24, 25, 58–60 Excess partial molar enthalpy, 27, 51, 53, 54–7, 93, 106, 135, 168, 176, 188, 236, 285

Excess partial molar entropy, 58–60 Excess partial molar quantities, 19, 21, 27, 37, 69, 110 Excess partial molar volume, 58, 100, 102, 104, 108, 155, 156, 160, 167–8, 175–6, 187–8, 191 Excess quantities, 18–19 Extensity (wavelength) of fluctuation, 21, 70, 71, 72, 73 Extensive quantity, 26, 36, 38 Femto second pump-probe spectroscopy, 238 First law of thermodynamics, 9, 11 Fluctuation density, 71, 117, 119 Fluctuation functions, 69–78, 117–30 Freezing point depression – molecular level interpretation, 104, 105 Freezing point depression, 281–3 Giles’ derivation of entropy, 11–13 Gibbs-Duhem relation, 25–30, 35, 49, 60, 63, 103, 241–3, 286 Gibbs energy, 9, 14, 35, 36, 85, 150, 239, 265 Gibbs-Konovalov correction, 59, 269–70 Gibbs phase rule, 30 Heat capacity, 19, 41, 69, 70–1, 95, 98, 102, 156, 271–8 anomaly, 95, 271–8 Helmholtz energy, 14 Henry’s law, 33–5, 62 Higher order derivatives, 1, 21, 37, 151, 170, 263

296

Hofmeister series (or effects), 4, 205–206, 217, 218, 224, 237–8 Hydration center (or number), 207–11, 224–5, 227, 229–37 Hydrophilicity, 211–7, 222, 261 Hydrophobicity, 109, 113, 145–6, 151, 162, 166, 179, 183, 185, 191, 193, 195, 197, 199, 205, 214, 221, 230, 256, 261 ranking, 179, 183, 263

Mixing Scheme II, 93, 99, 102, 105, 109, 118, 130, 145, 149, 160, 162, 168, 173, 265 Mixing Scheme III, 92, 94, 134, 143, 147, 151, 171, 189

I-j interaction chemical potential, 22, 242 enthalpy, entropy, and volume, 22 Iceberg formation, 51, 93, 100, 103–5, 111, 116, 118, 151, 165, 175, 201, 265 Intensity (amplitude) of fluctuation, 70, 71 Intensive quantity, 26, 38–9, 178 Interaction function, 3, 4, 21–2, 39–41, 69, 95, 110–1, 252, 261–3 Ion pairing, 221, 234–5 Irreversibility of process, 11, 12, 13 Isobaric thermal expansivity, 19 Isothermal compressibility, 19, 41, 69, 85

Partial molar enthalpy, 18, 27, 51–4, 57–8, 93, 103, 135, 168, 176, 188, 236, 285–6 Partial molar entropy, 18, 51, 58–60 Partial molar fluctuation density, 71 Partial molar normalized fluctuation, 72 Partial molar quantity, 17–18, 21, 22 Partial molar volume, 51, 58, 84–5, 100, 102, 104, 108, 135, 155–6, 160, 167, 168, 175–6, 187, 188, 191, 269 Partial structure factor, Sxx(0), 68, 130

Kirkwood-Buff integral, 22, 84, 85–6 Koga-line, 102, 103, 104, 116, 148, 156, 158, 266 Lower critical solution temperature (LCST), 41–8 Many-body problem, 18 Mean-square fluctuation, 69–70 Mechanical stability criterion, 19, 41 Mixing entropy, 18, 35–7, 40–5 Mixing Scheme, 1, 3, 25, 37, 40, 89–179, 220–3, 261, 266 Mixing Scheme I, 91, 93, 94, 98, 102, 103, 104, 105, 109, 116, 117, 118, 130, 147–150, 179, 183, 188, 207, 221, 261, 266

Neutron diffraction, 148–9, 209, 214, 217, 238 Normalized fluctuation, 71, 72, 117, 120, 177 Order-theoretical approach, 3, 11, 13

Raoult’s law, 33–5, 62, 236, 244, 256, 282 Response functions, 1, 3, 19, 21, 69–70 Second law of thermodynamics, 9 Small angle X–ray scattering, SAXS, 67, 68, 92, 130, 156 Spinodal point, 45 Stability criteria, 10, 15, 16–17, 22, 39, 41 Symmetric reference, 25, 281, 286 Third derivatives, 1, 3, 21–2, 23, 69, 71, 80, 86, 95, 98, 100, 102, 135, 143, 165, 242, 245, 265–6 Titration calorimetry with dilute titrant, 285–6 Transition of mixing scheme from I to II, 95–8, 101, 114 from II to III, 101, 114, 146 Upper critical solution temperature (UCST), 41–8, 156, 166–8 Wavelength (extensity) of fluctuation, 3, 72–3, 78, 122, 165