Table Of ContentWATER AND
AQUEOUS SOLUTIONS
Introduction to a Molecular Theory
WATER AND
AQUEOUS SOLUTIONS
Introduction to a Molecular Theory
Arieh Ben-Nairn
Institute of Chemistry
The Hebrew University of Jerusalem
Jerusalem, Israel
PLENUM PRESS • NEW YORK AND LONDON
Library of Congress Cataloging in Publication Data
Ben-Na'im, Aryeh.
Water and aqueous solutions.
Bibliography: p.
1. Solution (Chemistry) 2. Water. 3. Molecular theory. I. Title.
QD541.B46 546'.22 74-7325
ISBN-13: 978-1-4615-8704-0 e-ISBN-13: 978-1-4615-8702-6
001: 10.1007/978-1-4615-8702-6
Acknowledgments
Thanks are due to the following for permission to reproduce figures from their
publications: D. Eisenberg and W. Kauzmann (Figs. 6.4, 6.5, 6.6); A. H. Narten and
H. A. Levy (Figs. 6.7,6.10); A. Rahman and F. H. Stillinger (Figs. 6.32, 6.33); J. A.
Barker and R. O. Watts (Fig. 6.30); North-Holland Publishing Company (Figs. 6.19,
6.30,8.23); The Journal of Chemical Physics (Figs. 1.3, 5.2, 5.4, 5.5,5.6, 5.7, 5.8, 5.9,
6.2,6.7,6.9,6.14,6.18,6.19,6.20,6.21,6.22,6.24, 6.25, 6.26, 6.27, 6.28, 6.29,
6.31,6.32,6.33,8.1,8.4,8.7,8.8,8.12,8.13,8.15, 8.17, 8.18); The Clarendon Press,
Oxford (Figs. 6.4,6.5,6.6); The Journal of Solution Chemistry (Figs. 7.4, 7.5,7.6,
8.11,8.20,8.21,8.22); Molecular Physics (Taylor and Francis, Ltd.) (Figs. 6.21,
6.23); Chemical Physics Letters (6.19,6.30,8.23); J. Wiley and Sons, Inc. (Figs. 2.6,
2.7,2.8,4.3,6.13,6.14, 7.1,7.2>'
@ 1974 Plenum Press, New York
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Dedicated to Talma
Preface
The molecular theory of water and aqueous solutions has only recently
emerged as a new entity of research, although its roots may be found in
age-old works. The purpose of this book is to present the molecular theory
of aqueous fluids based on the framework of the general theory of liquids.
The style of the book is introductory in character, but the reader is presumed
to be familiar with the basic properties of water [for instance, the topics
reviewed by Eisenberg and Kauzmann (1969)] and the elements of classical
thermodynamics and statistical mechanics [e.g., Denbigh (1966), Hill
(1960)] and to have some elementary knowledge of probability [e.g., Feller
(1960), Papoulis (1965)]. No other familiarity with the molecular theory
of liquids is presumed.
For the convenience of the reader, we present in Chapter 1 the rudi
ments of statistical mechanics that are required as prerequisites to an under
standing of subsequent chapters. This chapter contains a brief and concise
survey of topics which may be adopted by the reader as the fundamental
"rules of the game," and from here on, the development is very slow and
detailed.
Excluding the introductory chapter, the book is organized into three
parts. The first, Chapters 2-4, presents the general molecular theory of
fluids and mixtures. Here the notions of molecular distribution functions
are developed with special attention to fluids consisting of nonspherical
particles. We have included only those theories judged to be potentially
useful in the study of aqueous fluids, so this part may not be considered as
an introduction to the theory of fluids per se.
With this objective in mind we did not survey the recent developments
in the theory of simple fluids. Instead we present ample illustrative examples
vii
viii Preface
to stress the contrast between simple fluids, on one hand, and the more
complex, aqueous fluids on the other. Of course, the particular choice of
topics is a matter of personal taste and has no absolute significance. For
instance, the theory of solutions may be developed either along the Mc
Millan-Mayer (1945) theory or along the Kirkwood-Buff (1951) theory.
Both are exact and equivalent from the formal point of view. However,
the latter is judged to be the more suitable for problems arising in the
theory of aqueous fluids.
The second part consists of Chapter 5 alone, which comprises a bridge
connecting the formal theory of fluids, on the one hand, and its application
to water and aqueous solutions on the other. The construction of this
bridge is rendered possible through the generalization of the ideas of mo
lecular distribution functions, which lays the foundation for the so-called
mixture-model approach to the theory of fluids. The latter may be viewed
as the formal basis for various ad-hoc mixture models for water and aqueous
solutions that have been suggested by many authors.
The third part, Chapters 6-8, presents the treatment of essentially
three systems, namely pure water with zero, one, and two simple solutes,
respectively. Chapter 6 includes a brief survey of the properties of water.
We have avoided excessive duplication of material which has been fully
discussed by Eisenberg and Kauzmann (1969). The emphasis is mainly
on the various theoretical approaches, both old and of recent origin, to
explain the anomalous properties of this unique fluid. Chapter 7 is con
cerned with very dilute solutions of simple nonelectrolytes which, from the
formal point of view, reflect the properties of pure water with a single
solute particle. Both experimental facts and theoretical attempts at inter
pretation are surveyed. Special attention is devoted to elaboration on the
exact meaning and significance of "structural changes" induced by a solute
on the solvent.
The last chapter deals with small deviations from very dilute solutions.
The problem of hydrophobic interaction, considered to be of crucial
importance in biochemical processes, is formulated, and methods of es
timating the strength of solute-solute interaction in various solvents are
discussed. Preliminary attempts at interpretation, based on concepts de
veloped in the preceding chapters, are also surveyed.
Although the framework of this book could have easily accommodated
a chapter on ionic solutions we chose not to include this topic, as several
works already exist dealing with it exclusively.
The entire subject of aqueous solutions is still subject to vigorous
debate, and many approaches, theories, and interpretations are highly
Preface ix
controversial. We have expended a mild effort to represent a reasonable
spectrum of opinions advanced by various authors. However, a book on
such a subject must inevitably reflect the author's own bias. The common
thread linking the subject matter included in Chapters 5-8 is the application
of the mixture-model approach to the theory of fluids. It is the author's
opinion that this theoretical tool is particularly useful in treating aqueous
fluids and, hopefully, will help us to understand these systems on both the
molecular and the macroscopic levels.
I am very much indebted to many friends and colleagues who encour
aged me in undertaking the writing of this book. Thanks are due to Drs.
R. Battino, D. Henderson, H. S. Frank, A. Nitzan, D. Shalitin, F. H.
Stillinger, and R. Tenne for reading parts of the manuscript and kindly
offering helpful comments and suggestions.
Arieh Ben-Nairn
Jerusalem, Israel
Contents
Chapter 1. Introduction and Prerequisites
1.1. Introduction
1.2. Notation. . 2
1.3. Classical Statistical Mechanics 6
1.4. Connections between Statistical Mechanics and Thermo-
dynamics 9
1.4.1. T, V, N Ensemble 9
1.4.2. T, P, N Ensemble 10
1.4.3. T, V, p, Ensemble 10
1.5. Basic Distribution Functions in Classical Statistical Mechanics 13
1.6. Ideal Gas . . . . . . . . . . . . . 15
1. 7. Pair Potential and Pairwise Additivity 17
1.8. Virial Expansion and van der Waals Equation 25
Chapter 2. Molecular Distribution Functions 29
2.1. Introduction . . . . . . . . . . 29
2.2. The Singlet Distribution Function 30
2.3. Pair Distribution Function 36
2.4. Pair Correlation Function . 39
xi
xii Contents
2.5. Features of the Radial Distribution Function 43
2.5.1. Ideal Gas 44
2.5.2. Very Dilute Gas . . . . . . . . . . . . 45
2.5.3. Slightly Dense Gas . . . . . . . . . . . 45
2.5.4. Lennard-Jones Particles at Moderately High Densities. 49
2.6. Further Properties of the Radial Distribution Function 53
2.7. Survey of the Methods of Evaluating g(R) 65
2.7.1. Experimental Methods. 65
2.7.2. Theoretical Methods. . . . . . . 68
2.7.3. Simulation Methods. . . . . . . 69
2.8. Higher-Order Molecular Distribution Functions 75
2.9. Molecular Distribution Functions (MDF) in the Grand Ca-
nonical Ensemble. . . . . . . . . . . . . . . 78
Chapter 3. Molecular Distribution Functions and Ther
r.nodynar.nics . . . . . . . . . . . . . . . .. 81
3.1. Introduction . . . . . . . . . . . . 81
3.2. Average Values of Pairwise Quantities 82
3.3. Internal Energy. . . . 85
3.4. The Pressure Equation . . 88
3.5. The Chemical Potential . . 91
3.6. Pseudo-Chemical Potential . 99
3.7. Entropy . . . . . . . . . 101
3.8. Heat Capacity . . . . . . 102
3.9. The Compressibility Equation 104
3.10. Local Density Fluctuations 109
3.11. The Work Required to Form a Cavity in a Fluid. 114
3.12. Perturbation Theories of Liquids. . . . . . . . 120
Chapter 4. Theory of Solutions 123
4.1. Introduction . . . . . . . . . 123
4.2. Molecular Distribution Functions in Mixtures; Definitions. 124
4.3. Molecular Distribution Functions in Mixtures; Properties. 127
Contents xiii
4.4. Mixtures of Very Similar Components. . . . . . . . . .. 135
4.5. The Kirkwood-Buff Theory of Solutions. . . . . . . . .. 137
4.6. Symmetric Ideal Solutions; Necessary and Sufficient Conditions 145
4.7. Small Deviations from Symmetric Ideal (SI) Solutions 153
4.8. Dilute Ideal (DI) Solutions. . . . . . . . . 155
4.9. Small Deviations from Dilute Ideal Solutions 159
4.10. A Completely Solvable Example . . . . . . 164
4.10.1. Ideal Gas Mixture as a Reference System. . 167
4.10.2. Symmetric Ideal Solution as a Reference System 167
4.10.3. Dilute Ideal Solution as a Reference System. . 168
4.11. Standard Thermodynamic Quantities of Transfer 170
4.11.1. Entropy. 174
4.11.2. Enthalpy. 176
4.11.3. Volume . 176
Chapter 5. Generalized Molecular Distribution Functions
and the Mixture-Model Approach to Liquids 177
5.1. Introduction 177
5.2. The Singlet Generalized Molecular Distribution Function 179
5.2.1. Coordination Number (CN). 180
5.2.2. Binding Energy (BE) 183
5.2.3. Volume of the Voronoi Polyhedron (VP). 184
5.2.4. Combination of Properties 186
5.3. Illustrative Examples of GMDF's 187
5.4. Pair and Higher-Order GMDF's . 194
5.5. Relations between Thermodynamic Quantities and GMDF's 195
5.5.1. Heat Capacity at Constant Volume . 197
5.5.2. Heat Capacity at Constant Pressure. 198
5.5.3. Coefficient of Thermal Expansion. 200
5.5.4. Isothermal Compressibility 200
5.6. The Mixture-Model (MM) Approach; General Considerations 201
5.7. The Mixture-Model Approach to Liquids; Classifications Based
on Local Properties of the Molecules . 208
5.8. General Relations between Thermodynamics and Quasicom-
ponent Distribution Functions (QCDF) . 211
