Ionic Interactions. From Dilute Solution to Fused Salts

411 Pages·1971·35.973 MB·English

by  S. Petrucci (Eds.)

Ionic Interactions
From Dilute Solutions to Fused Salts
Edited by
S. PETRUCCI
Department of Chemistry
Polytechnic Institute of Brooklyn
Brooklyn, New York
VOLUME I
Equilibrium and Mass Transport
1971
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List of Contributors
Numbers in parentheses indicate the pages on which the authors' contributions begin.
JERRY BRAUNSTEIN (179), Reactor Chemistry Division, Oak Ridge Na-
tional Laboratory, Oak Ridge, Tennessee
W. EBELING (1, 61), Sektion Physik der Universität Rostock, Rostock,
German Democratic Republic
H. FALKENHAGEN (1, 61), Sektion Physik der Universität Rostock,
Rostock, German Democratic Republic
W. D. KRAEFT (61), Sektion Physik der Universität Rostock, Rostock,
German Democratic Republic
CORNELIUS T. MOYNIHAN (261), Vitreous State Laboratory and Chemical
Engineering Department, Catholic University of America, Washing-
ton, D. C.
S. PETRUCCI (117), Department of Chemistry, Polytechnic Institute of
Brooklyn, Brooklyn, New York
Preface
About one hundred years ago the study of electrolytes was synonymous
with physical chemistry. Since that time, physical chemistry has expe-
rienced a multiplication of fields and interests, and electrolyte research
now encompasses such diverse topics as the spectra of ions in solution,
thermodynamic properties, chemical kinetics, and reaction mechanisms.
More important, on the concentration scale, several subfields such as
the study of mass transport of dilute ionic solutions and of fused salts
have been created, and often seem to exist quite independently of one
another.
This book is an effort to present the reader with a broad spectrum of
approaches to the study of ionic systems. It is hoped, for instance, to
invite the thermodynamicist, interested in solvation studies of ions in
solution, to look into the results generated from the study of ion-solvent
interactions by crystal field and molecular orbital theories. Further,
there is an attempt, where the development of the discipline has made
it possible, to show how an increase in the concentration of an electrolyte
often requires an altered approach to the problems under study. This
point is especially evident when one compares the chapters on equilibrium
and mass transport properties written in the two concentration extremes
(dilute solutions and fused salts) by research people in their respective
fields. It is hoped that this will encourage a symbiosis of ideas among
its readers, a synthesis that in the ultimate analysis will lead to a deeper
understanding of solution phenomena.
This present work attempts specifically to study ionic interactions
(both ion-ion and ion-solvent). Formally it is divided into two volumes,
the first composed of five chapters and subtitled ''equilibrium and mass
properties/' the second volume composed of four chapters and subtitled
"kinetics and structure.,,
In the first volume, the first two chapters deal with equilibrium and
mass transport of dilute ionic electrolytes. They have been written by
H. Falkenhagen and his associates, W. Ebeling and W. D. Kraeft. Some
xi
xii Preface
interesting new developments of association theory have been included.
Chapter 3 by S. Petrucci deals with the statistical thermodynamic treat-
ment of ionic association and complexation (with a schematic presentation
of crystal field theory). J. Braunstein deals with equilibrium properties
of concentrated solutions and fused salts in Chapter 4. This is basically
a statistical mechanical treatment of ionic interactions in hydrated salts
and fused salts. The fifth chapter by C. Moynihan develops the topic
of mass transport properties of fused salts and their theoretical treat-
ment. A section on ultrasonic and dielectric relaxation methods is par-
ticularly important in showing the latest developments in the study
of the relaxation of viscosity and conductance of melts.
Volume II opens with a chapter by C. Langford on the chemical kinetics
and mechanisms of the substitution of ligands in the first coordination
sphere of metal cations in dilute solution. Chapter 7 by S. Petrucci
deals with the kinetic approach to ionic association and complex-
ation in dilute solutions studied by relaxation kinetics. Chapter 8 is
a presentation of ultraviolet and visible electronic spectra (and their
theoretical interpretation by crystal field theory and molecular orbital
theory) of electrolyte solutions and fused salts by S. Holt. Finally,
vibrational spectra and their interpretation are presented in the last
chapter by D. E. Irish.
Although this presentation is not exhaustive (other experimental ap-
proaches to the study of ionic interactions, such as NMR and X-ray results
are not presented), it is believed that within the framework of the mate-
rial presented, a unified study of equilibrium, transport properties, kinet-
ics, and the spectra of ionic liquids emerges. It was the editor's feeling
that such a text, especially with regard to the comparison of dilute solu-
tions and fused salts, had not been previously attempted despite an ob-
vious need.
The editor wishes to thank the contributors, who with their knowledge-
able cooperation made it possible to realize this project. Thanks are also
expressed to Dr. E. M. Loebl of the Polytechnic Institute of Brooklyn
for his encouragement in launching the idea of this book and to the staff
of the Academic Press for their invaluable help.
Contents of Volume II
6. Complex Formation and Solvent-Exchange Kinetics: A Dynamic
Approach to Electrolyte Solutions
Cooper H. Langford
7. Kinetic Approach to the Study of Ionic Association and Complex-
ation: Relaxation Kinetics
S. Petrucci
8. Ultraviolet and Visible Spectra of Electrolyte Solutions and Fused
Salts
S. L. Holt
9. Vibrational Spectral Studies of Electrolyte Solutions and Fused Salts
D. E. Irish
Author Index—Subject Index
xiii
Chapter 1
Equilibrium Properties
of Ionized Dilute Electrolytes
H. FALKENHAGEN AND W. EBELING
Sektion Physik der Universität Rostock
Rostock, German Democratic Republic
I. Preliminaries 2
A. Introduction 2
B. Phenomenological Description of Strong Ionized Electrolytes . . .. 3
II. Debye-Hückel Theory of Ionic Solutions 5
A. The Ionic Atmosphere 5
B. Limiting Laws 10
C. Solutions of Charged Spheres 13
III. Foundations of Modern Statistical Theories 16
A. Remarks on the Development of Modern Electrolyte Theory . . .. 16
B. Molecular Distribution Functions 17
C. Thermodynamic Excess Functions 19
IV. Bogoliubov Theory of Molecular Distribution Functions 21
A. Molecular Distribution Functions and Thermodynamic Potentials for
Coulomb Systems 21
B. Foimal Density Expansions of Molecular Distribution Functions . .. 25
V. Theory of Ionic Systems Including Long-Range and Short-Range Inter-
actions 26
A. Short-Range Forces between Ions 26
B. Convergent Expansions of the Binary Distribution Function 27
C. Cluster Expansion of the Free Excess Energy 29
VI. Thermodynamic Functions for the Model of Rigid Charged Spheres . .. 31
A. Calculation of Cluster Integrals of Second Order 31
B. Discussion of First Deviations from Debye's Limiting Law 36
C. Comparison with Experimental Data 40
1
2 H. Falkenhagen and W. Ebeling
VII. Statistical Thermodynamics of Symmetrical Electrolytes with Large
Bjerrum Parameters 43
A. The Chemical Model of Associating Electrolytes 43
B. Evaluation of the Association Constant 46
C. Thermodynamic Functions and Degree of Dissociation 52
VIII. Conclusions 56
References 57
I. Preliminaries
A. INTRODUCTION
Michael Faraday was the first to recognize the important part played
by ions in the behavior of an electrolytic solution. Fifty years later, Ar-
rhenius proposed and demonstrated that the molecules of an electrolyte
are dissociated to a certain extent into free ions even when no external
field is acting on them. He was led to this conclusion by van't HofPs
theory of solutions and his own extensive researches on the conductivity
of electrolytic solutions. The theory of Arrhenius is, however, only valid
for weak electrolytes in which only a small proportion of the molecules
are dissociated into ions. The behavior of strong electrolytes shows con-
siderable departures from Arrhenius's theory. The ideas of Milner,
Debye, Hückel, Onsager, and Falkenhagen made it possible to explain
the properties of strong electrolytes, at least in sufficiently dilute solutions.
Even today it is only with difficulty that the theory can be extended from
the field of dilute solutions to embrace more concentrated electrolytes.
Such an extended theory must take into account the short-range forces
between the ions. The theoretical treatment of these complicated factors
has, so far, been only partially successful. Various attempts to treat
theoretically the properties of more concentrated solutions were given
since the pioneering work of Debye and his co-workers (1923). Although
these theories have led to valuable insight into the nature of ionic so-
lutions, there remain many unsolved questions, e.g., the nature of the
short-range forces between the ions.
The plan of this contribution is to give a survey of various statistical
derivations of the thermodynamic excess functions of electrolytic so-
lution. The calculation of the thermodynamic functions of the ideal
solution lies outside the scope of this article. Therefore, all questions
that are connected with the ionic solvation and related problems are not
1. Equilibrium Properties of Dilute Electrolytes 3
considered. The thermodynamic excess functions of electrolytes can be
derived in several ways. In each case the mathematical difficulties en-
countered in proceeding beyond the Debye-limiting-law stage are very
formidable. In our opinion only Mayer's (1950) method of diagramatic
expansions and Bogoliubov's (1946, 1962) method of solving the equa-
tions for the distribution functions have been brought to sufficient com-
pletion to provide a basis for rigorous calculations from various models
of the ionic solution. Mayer's method is described extensively in the
excellent monograph by Friedman (1962). For this reason we prefer, in
this chapter, an extension of the method of Bogoliubov, that yields the
same results as Mayer's method (Schmitz, 1966, 1967, 1968; Ebeling,
1966, 1967). Finally, we want to underline that this work is restricted
to the statistical theory of the thermodynamic excess functions of dilute
ionic solution. Several other topics of electrolyte research may be studied
in the other chapters of this book and in several monographs (Falkenhagen
et aL, 1934, 1953; Falkenhagen et aL, 1970; Harned and Owen, 1958;
Robinson and Stokes, 1959; Mikulin, 1968; Miscenko and Poltorazki,
1968; etc.).
B. PHENOMENOLOGICAL DESCRIPTION OF STRONG IONIZED ELECTROLYTES
In this section we review the formulation of thermodynamic excess
functions customarily employed in work with electrolyte solutions. An
electrolyte solution is an example of a mixture with an essentially un-
symmetrical relation among its components; in such a case the desig-
nation of one of the species as solvent and others as solutes reflects the
physical realities. We consider a system consisting of N neutral solvent
s
molecules and Ν ions of species a (a = 1,2, . . ., s — 1). The condition
Ά
of electrical neutrality requires
2^ = 0. (1)
The concentration of the several species we describe by the number
densities
n = NJV; n = NJV,
s a
where V is the volume of the system. The neutral solute molecules do
not appear in this description. They are considered to be composite
subsystems, consisting of a group of ions. Therefore, the neutral solute
molecules appear in our model, which will be called the physical model
4 H. Falkenhagen and W. Ebeling
of the electrolyte, as quasi-particles with a finite lifetime. This model
will give reasonable results if the neutral solute molecules are rather
sparse, i.e., the concentration of ion complexes is very low. In other
words, we want to limit our consideration to strong electrolytes, e.g.,
solutions of alkali halides in water. Only in the last section of this chapter
shall we introduce ion pairs in an explicit way in order to construct a
bridge to the contribution of Petrucci on ionic association and complexa-
tion (Chapter 3).
We designate the excess free energy of the ionic system as
F* = F*(T, ν,Ν ). (2)
Λ
The excess free energy corresponds to the part of the free-energy change
in the real system that arises from interactions among the solute species.
Therefore, we have
F* = F- Fid, (3)
where F is the total free energy of the solution and Fid is the free energy
of a corresponding ideal system with the same concentration of the species
a. The osmotic pressure of the system will be
P=P*-(dF*ldV) , , (4)
TNli
where Pid is the ideal osmotic pressure.
Now we note the definition of the chemical potential
^ = (dFldN) y . (5)
k Ti iNl
In the case of ideal systems we have
μ$ = μην,Τ) + ΗΤ\ηχκ, (6)
where
/ s
*k = «k / Σ »i
1 i=l
is the mole fraction of the kth species. For real systems we define the
activities a by the relation [see, e.g., Prigogine and Defay (1954)]
k
μ^ = μην.Τ) + Ι*Τ]ηα . (7)
1Ιί
From Eqs. (5)-(7) follows
Λ = «k/*k = cx [(llkT)(dF*ldN)], (8)
P k