Table Of ContentMolecular Theory of Solutions
Dedicated to Ruby and Kaye
Molecular Theory
of Solutions
Arieh Ben-Naim
Department of Physical Chemistry
The Hebrew University, Jerusalem
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Preface
The aim of a molecular theory of solutions is to explain and to predict the
behavior of solutions, based on the input information of the molecular
properties of the individual molecules constituting the solution. Since Prigo-
gine’sbook(publishedin1957)withthesametitle,aimingtowardsthattarget,
therehasbeenconsiderablesuccessinachievingthatgoalformixturesofgases
andsolids,butnotmuchprogresshasbeenmadeinthecaseofliquidmixtures.
This is unfortunate since liquid mixtures are everywhere. In almost all indus-
tries and all biological sciences, we encounter liquid mixtures. There exists an
urgent need to understand these systems and to be able to predict their
behavior from the molecular point of view.
The main difficulty in developing a molecular theory of liquid mixtures, as
compared to gas or solid mixtures, is the same as the difficulty which exists in
the theory of pure liquids, compared with theories of pure gases and solids.
Curiouslyenough,thoughvariouslatticetheoriesoftheliquidstatehavefailed
to provide a fair description of the liquid state, they did succeed in char-
acterizing liquid mixtures. The reason is that in studying mixtures, we are
interested in the excess or the mixing properties – whence the problematic
characteristicsoftheliquidstateofthepurecomponentspartiallycancelout.In
other words, the characteristics of the mixing functions, i.e., the difference
between the thermodynamics of the mixture, and the pure components are
nearlythesameforsolidsandliquidmixtures.Muchofwhathasbeendoneon
thelatticetheoriesofmixturewaspioneeredbyGuggenheim(1932,1952).This
work was well documented by both Guggenheim (1952) and by Prigogine
(1957), as well as by many others.
Anotherdifficultyindevelopingamoleculartheoryofliquidmixturesisthe
relativelypoorknowledgeoftheintermolecularinteractionsbetweenmolecules
of different species. While the intermolecular forces between simple spherical
particles are well-understood, the intermolecular forces between molecules of
different kinds are usually constructed by the so-called combination rules, the
most well-known being the Lorentz and the Berthelot rules.
In view of the aforementioned urgency, it was necessary to settle on an
intermediatelevelofatheoryy.Insteadoftheclassicalaimofamoleculartheory
y By intermediate level of theory, I do not mean empirical theories which are used mainly by
chemicalengineers.
vi PREFACE
of solutions, which we can write symbolically as
I: Molecular Information ! Thermodynamic Information
An indirect route has been developed mainly by Kirkwood, which involves
molecular distribution functions (MDF) as an intermediate step. The mole-
culardistributionfunctionapproachtoliquidsandliquidmixtures,foundedin
theearly1930s,graduallyreplacedthevariouslatticetheoriesofliquids.Today,
lattice theories have almost disappeared from the scene of the study of liquids
and liquid mixturesy. This new route can be symbolically written as
II: Molecular Information + MDF ! Thermodynamic Information
Clearly,routeIIdoesnotremovethedifficulty.Calculationofthemolecular
distribution functions from molecular properties is not less demanding than
calculation of the thermodynamic quantities themselves.
Nevertheless, assuming that the molecular distribution functions are given,
then we have a well-established theory that provides thermodynamic infor-
mationfromacombinationofmolecularinformationandMDFs.Thelatterare
presumed to be derived either from experiments, from simulations, or from
some approximate theories. The main protagonists in this route are the pair
correlation functions; once these are known, a host of thermodynamic quan-
tities can be calculated. Thus, the less ambitious goal of a molecular theory of
solutions has been for a long time route II, rather route I.
Between thetimes of Prigogine’s book upto the present, several books have
been published, most notably Rowlinson’s, which have summarized both the
experimental and the theoretical developments.
During the 1950s and the 1960s, two important theories of the liquid state
weredeveloped,initiallyforsimpleliquidsandlaterappliedtomixtures.These
are the scaled-particle theory, and integral equation methods for the pair
correlationfunction.Thesetheoriesweredescribedinmanyreviewsandbooks.
In this book, we shall only briefly discuss these theories in a few appendices.
Except for these two theoretical approaches there has been no new molecular
theorythatwasspecificallydesignedanddevelopedformixturesandsolutions.
This leads to the natural question ‘‘why a need for a new book with the same
title as Prigogine’s?’’
To understand the reason for writing a new book with the same title, I will
first modify route II. The modification is admittedly, semantic. Nevertheless,
it provides a better view of the arguments I am planning to present below.
y Perhapsliquidwaterisanexception.Thereasonisthatwater,intheliquidstate,retainsmuch
ofthestructureoftheice.Therefore,manytheoriesofwaterandaqueoussolutionhaveusedsome
kindoflatticemodelstodescribethepropertiesoftheseliquids.
PREFACE vii
We first rewrite route II as
III: Microscopic Properties ! Thermodynamic Properties
RoutesIIandIIIareidenticalinthesensethattheyusethesametheoretical
tools to achieve our goals. There is however one important conceptual dif-
ference.Clearly,molecularpropertiesaremicroscopicproperties.Additionally,
all that has been learned about MDF has shown that in the liquid phase, and
not too close to the critical point, molecular distribution functions have a local
characterinthesensethattheydependuponandprovideinformationonlocal
behavior around a given molecule in the mixture. By local, we mean a few
molecular diameters, many orders of magnitude smaller than the macroscopic,
or global, dimensions of the thermodynamic system under consideration. We
thereforerewrite,onceagain,routeIIindifferentwords,butmeaningthesame
as III, namely
IV: Local Properties ! Global Properties
Even with this modification, the question we posed above is still left unan-
swered:Whyanewbookonmoleculartheoryofsolutions?Afterall,evenalong
route IV, there has been no theoretical progress.
Here is my answer to this question.
TwoimportantandprofounddevelopmentshaveoccurredsincePrigogine’s
book, not along route I, neither along II or III, but on the reverse of route IV.
The one-sided arrows as indicated in I, II, and III use the tools of statistical
thermodynamics to bridge between the molecular or local properties and
thermodynamicproperties.Thisbridgehasbeenerectedandhasbeenperfected
for many decades. It has almost always been used to cross in a one-way
direction from the local to the global.
Thenewdevelopmentusesthesametool–thesamebridge–butinreversed
direction; to go backwards from the global to the local properties. Due to its
fundamental importance, we rewrite IV again, but with the reversed directed
arrow:
(cid:1)IV: Global Properties ! Local Properties
It is along this route that important developments have been achieved
specifically for solutions, providing the proper justification for a new book
with the same title. Perhaps a more precise title would be the Local Theory of
Solutions.However,sincethetoolsusedinthistheoryareidenticaltothetools
used in Prigogine’s book, wefind it fitting to use the sametitle for the present
book. Thus, the tools are basically unchanged; only the manner in which they
are applied were changed.
viii PREFACE
There are basically two main developments in the molecular theory of
solutions in the sense of route (cid:1)IV: one based on the inversion of the
Kirkwood–Buff (KB) theory; the second is the introduction of a new measure
to study solvation properties. Both of these use measurable macroscopic, or
global quantities to probe into the microscopic, or the local properties of the
system. The types of properties probed by these tools are local densities, local
composition, local change of order, or structure (of water and aqueous solu-
tions) and many more. These form the core of properties discussed in this
book.Bothuseexactandrigoroustoolsofstatisticalmechanicstodefineandto
calculate local properties that are not directly accessible to measurements,
from measurable macroscopic quantities.
ThefirstdevelopmentconsistsoftheinversionoftheKirkwood–Bufftheory.
The Kirkwood–Buff theory has been with us since 1951. It was dormant for
more than 20 years. Though it is exact, elegant and very general, it could only
be applied when all the pair correlation functions are available. Since, for
mixtures, the latter are not easily available, the theory stayed idle for a long
time. It is interesting to note that both Prigogine (1957) and Hill (1956)
mentionedtheKBtheorybutnotanyofitsapplications.Infact,Hill(1956),in
discussingtheKirkwood–Bufftheory,writesthatitis‘‘necessarilyequivalentto
the McMillan–Mayer (1945) theory, since both are formally exact.’’ I disagree
with the implication of that statement.Ofcourse, anytwo exact theories must
be, in principle, formally equivalent. But they are not necessarily equivalent in
theirrangeandscopeofapplicabilityandintheirinterpretativepower.Ibelieve
that in all aspects, the Kirkwood–Buff theory is immensely superior to the
McMillan–Mayer theory, as I hope to convince the reader of this book. It is
somewhat puzzling to note that many authors, including Rowlinson, com-
pletely ignored the Kirkwood–Buff theory.
One of the first applications of the Kirkwood–Buff theory, even before
its inversion, was to provide a convincing explanation of one of the most
mysteriousandintellectuallychallengingphenomenonofaqueoussolutionsof
inert gases – the molecular origin of the large and negative entropy and
enthalpy of solvation of inert gases in water. This was discussed by Ben-Naim
(1974, 1992). But the most important and useful application of the KB theory
beganonlyafterthepublicationofitsinversion.Asearchintheliteratureshows
thatthe‘‘KBtheory’’wasusedaspartofthetitleofarticlesontheaverage,only
onceayearuntil1980.Thishasescalatedtoabout20–25ayearsince1980,and
it is still increasing.
Ever since the publication of the inversion of the KB theory, there had
been an upsurge of papers using this new tool. It was widely accepted and
PREFACE ix
appreciated and used by many researchers as an efficient tool to study local
properties of mixtures and solutions.
The traditional characterization and study of the properties of liquid
mixtures by means of the global excess thermodynamic functions has been
gradually and steadily replaced by the study of the local properties. The latter
providesricherandmoredetailedinformationontheimmediateenvironment
of each molecule in the mixture.
Theseconddevelopment,notlessimportantanddramatic,wasinthetheory
of solvation. Solvation has been defined and studied for many years. In fact,
there was not only one but at least three different quantities that were used to
study solvation. The problem with the traditional quantities of solvation was
that it was not clear what these quantities really measure. All of the three
involve a process of transferring a solute from one hypothetical state in one
phase,toanotherhypotheticalstateinasecondphase.Sincethesehypothetical
states have no clear-cut interpretation on a molecular level, it was not clear
what the free energy change associated with such transfer processes really
means. Thus, within the framework of thermodynamics, there was a state of
stagnation,wherethreequantitieswereusedastoolsforthestudyofsolvation.
Noonewasabletodecidewhichthepreferredoneis,orwhichisreallytheright
tool to measure solvation thermodynamics.
Asit turned out, there wasno right one. In fact, thermodynamics couldnot
provide the means to decide on this question. Astonishingly, in spite of their
vagueness,andinspiteoftheinabilitytodeterminetheirrelativemerits,some
authors vigorously and aggressively promoted the usage of one or the other
toolswithouthavinganysolidtheoreticalsupport.Someoftheseauthorshave
also vehemently resisted the introduction of the new tool.
The traditional quantities of solvation were applicable only in the realm of
verydilutesolutions,whereHenry’slawisobeyed.Ithadbeenfoundlaterthat
someoftheseareactuallyinadequatemeasuresofsolvationy.Thenewmeasure
that was introduced in the early 1970s replaced vague and hazy measures by
a new tool, sharply focusing into the local realm of molecular dimensions.
Thenewquantity,definedinstatisticalmechanicalterms,isasharp,powerful,
and very general tool to probe local properties of not only solutes in dilute
solutions, but of any molecule in any environment.
The new measure has not only sharpened the tools for probing the
surroundings aroundasinglemolecule, butitcould alsobe appliedtoavastly
larger range of systems: not only a single A in pure B, or a single B in pure A,
y InfactusingdifferentmeasuresledtoverydifferentvaluesofthesolvationGibbsenergy.Inone
famousexamplethedifferenceintheGibbsenergyofsolvationofasmallsoluteinH OandD Oeven
2 2
haddifferentsigns,inthedifferentmeasures.
