1.
Chapter 1 - INTRODUCTION TO THE THEORIES OF LIQUID STATE
1.1. BRIEF REVIEW OF THE THEORIES OF LIQUIDS .
The development of liquid state theories was quite retarded compared to the theories of
gases state and solid state (or crystalline state). However during the last forty years there has
been a considerable progress in the liquid state theories. The recent theories are able to
predict quantitative results compared to the 'order of magnitude' results obtained earlier
[1,2],
A theory of liquid state being essential for further developments of many branches
of physics and chemistry, should be in a position to explain certain physical properties of
liquids. For example the liquid state theory should be in a position to explain the thermo­
dynamic behaviour of simple liquids, equilibrium and transport properties of liquids, and
results of X-ray and neutron diffraction techniques.
The literature on liquid state theories is too extensive. Attempts were made by
various workers including those of our school [1, 3-8] to explain the developments of the
liquid state theories. Salient aspects of the theories of liquid state are presented in the
following paragraphs. The main approaches to explain the liquid state can be classified
broadly [4] into five categories:
(1) Vender Wall's (corrected)-equation of state model [9]
(2) Distribution function methods [10-12]
(3) Total correlation function models [13,14]
(4) Perturbation function based distribution function [15] and
(5) Model approach through partition function [16],
1.1.1. VANDER WALL'S CORRECTED - EQUATION OF STATE MODEL:
Liquids representing a peculiar compromise between order and disorder, the
theoretical analysis was started first with crude approximations to make them either too
rigid or too compressible. Initial studies treated them as continua and a few macroscopic
properties
were investigated for order of magnitude'. When kinetic theory of gases
attained sufficient sophistication, attempts were made to apply vander Wall's correctedequation of state [9] to describe liquid phase properties but with partial success [2]. The
finite density of matter implies that molecules must repel one another at short distances.
2.
A pair ofmolecules can not approach much closer than a certain distance, the effective
diameter, while at somewhat greater distances they are attracted by cohesive forces [2], It
was realised that the bulk properties
of matter should be related to the functions
describing the interactions of the individual molecules with one another.
From X-ray
scattering of liquids, a few traces of the crystalline lattice structure were found to survive.
1.1.2. DISTRIBUTION FUNCTION METHODS :
The probability
of finding
sets
of neighboring molecules in a particular
configuration, known is fee 'distribution function* was formulated to explain fee behaviour of
liquids. The best of fee distribution function theories follow from the laws of classical
mechanics and fundamental theories of statistical mechanics using a set of well-defined
approximations. Usefulness of such theories is determined by fee precision wife which these
approximations can be stated i.e., gross physical assumptions were unsuitable to explain
fee behaviour of liquids and liquid mixtures.
The important approaches of fee radial
distribution functions of well known models [10-12,17,18] are described by Temperley and
others [2].
1.1.3. TOTAL CORRELATION FUNCTION MODEL:
A slightly different function called fee total correlation function has also been
defined in certain other models [13,14], which are more popularly known as 'molecular
correlation function approaches*.
1.1.4. PERTURBATION FUNCTION BASED DISTRIBUTION FUNCTION :
The distribution function theories were successful in explainiqg properties of
simple fluids which can be assumed as hard spheres. To explain more complex system of
liquids, fee perturbation theories were developed. The potential pertaining to such theories
is a sum of simple partition function and a weak perturbation potential. The effect of this
latter fraction was investigated by many [15,19-21] and fee perturbation functions were
evaluated.
1.1.5. MODEL APPROACH THROUGH PARTITION FUNCTION :
Another approach is known as fee model approach. A particular physical model
of fee liquid is assumed and the so-called partition function* is formulated to evaluate fee
properties of liquid.
Various
models
were tried and tested by
comparison
wife
experimental results. Even though these models were quite insufficient to describe fee overall
3.
behaviour of the liquid, this approach
distribution
function theories.
seems
to have a definite
advantage over
the
One virtue is that the mathematical representation is
(relatively) simple compared with
the
distribution function approach.
The model
approaches have been very widely used and are reasonably successful even in the case of
liquid mixtures containing molecules differing in size. There will be a continued useful role of
these theories until die theories based on distribution function are perfected [2].
1.1.6. MODEL APPROACH - BASIC ASPECTS :
The model approach was quite successful Several theories were proposed using
different approaches.
The theories based on model approach evaluate a partition fhnction Z of the system
and relate the thermodynamic properties using the expression : F = Heimhotz free energy = kT In Z The entropy (S) and pressure (P) can be related to the appropriate derivatives of
F. This partition function Z mainly depends on position coordinates (pn) and die momenta
coordinates (q#) of the molecule [16]
z==Z(pN> + Z(qH)
...
(1.1)
...
(1.2)
...
(1.3)
on evaluation
Z (pn) = - 3/2 NkT In (2nmkT/h2)
and
Z (qM) = -kT In [1/Nlexp (-U/kl^J
where U is the total potential energy depending on the coordinate, N is the number of
molecules and other constants have their usual meaning [16]. Z (p«) depends on temperature
only and can be readily evaluated. Z (qn), a function of both temperature and volume. Its
evaluation is a chief source of difficulty. The model is assumed to evaluate this latter
partition functioa A few of the popular model approaches are briefly described here.
1.1.7. MODEL APPROACH - CELL THEORY:
Cell or free volume model is probably the simplest of all the models [22]. A
typical molecule, supposed to be confined to a small region, spends much of its time confined
to its neighbors. Hie central molecule and its neighbors form the molecular environment
which is similar to that the solid Tlie long-range order however is not prevalent. Leonard-
4.
Jones and Devonshire [10] modified Ihe model by taking the attractive potential between
the molecules into account These developments [22-24] are not sufficient to extend them to
the case of simple polyatomic molecules like CH« aid CCI4 The eccentric distribirtion of
polarisable electrons necessities
modification
of
die potential function Kihara [23].
Hainan and Lambert [26], and Pitzer [27,28] suggested modified intennolecular potentials.
The modified potential due to Pitzer [27,28], for die attractive and repulsive terms were more
steeply dependent on the intennolecular distances than in the other models. Flory [29-31]
extended these ideas using the Hirschfeider-Eyring ceil potential partition function [32-34]
as basis for treating intennolecular energies in liquids. In the case of large particles, the
range of interactions being small compared to the molecular diameter, die intennolecular
energy is treated as arising effectively from interactions between the surfaces of die adjoining
molecules. This treatment has been used to explain the thermodynamic behaviour of nonspherical molecules differing in size and was found to predict correct remits.
1.1.8. MODEL APPROACH - LATTICE MODEL:
Another model approach is more popularly known as lattice model approach.
Melting or subsequent heating of a liquid leads to a decrease, in the apparent number of
neighbors and density. The entropy increase and change in volume suggest that holes or vacant
lattice cites are formed. If a lattice model is used to explain the behaviour of a liquid the
free volume and the associated energy difference of the hole have to be accounted. Eyring's
original model [24,32-34] along with certain modifications [3S-37] were foimd to predict
the properties of liquids. The 'tunnel theory proposed by Barker [13,23,38] assumes that
the subsystems of die molecules move almost in one-dimensional tunnels whose walls are
formed by neighboring molecules. It represents a kind of irregular lattice and the partition
function was found to predict better results than the Eyring's original model. A detailed
account of the well-known lattice models was summarised in a recent book by Barker [39].
A more sophisticated approach has been extensively studied by Eyring and his
collaborators [16,39], According to this theory, a typical liquid molecule spends part of its
time in a solid like environment, reminiscent of the cell model and die other part of its time in
a gas-1 ike environment
significant structures :
Eyring significant structure approach [16,40] consists of three
5.
(1) molecules with solid - like degrees of freedom,
(2) positional degeneracy in the solid - like structures.
and
(3) molecules with gas - like degrees of freedom. It attempts to describe the real
situation by interpolating die limiting cases of solid and vapour so as to explain die
behaviour of liquids. A partition function is formulated with this view which in turn gives
the Helmholtz free energy. The thermodynamic properties can be calculated by taking die
corresponding derivatives of die Helmholtz free energy.
1.1.9. MODEL APPROACH - EQUILIBRIUM MIXTURES :
A different approach known as 'equilibrium mixture approach', which considers a
liquid as an equilibrium mixture of single molecules, pairs, triplets etc. consisting of 2,3 or
more molecules each interacting appreciably with one another, was proposed by Temperley
[2]
U.10. MOLECULAR FIELD THEORY:
Molecular field theories similar to Wiens theory of ferromagnetism were formulated
to explain the behaviour of liquids. These are based on the assumption that interactions
between the molecules are almost independent of
the variations in the intermolecular
resistances. Even though these theories are a rough approximation of the real molecules, they
have the following advantages :
(a) the mathematics is easy to work oitt
(b) they give an insight of the problem
and (c) they can be readily adopted to deal with different physical situations.
The similarities of magnetic theories and molecular theories have been discussed
by Tempereley [1J.
1.1.11. TESTING OF LIQUID STATE THEORIES:
The theories of liquid state are tested primarily by comparing the
theoretically
evaluated thermodynamic properties of liquids (or liquid mixtures) with the experimental
values. A brief review of die theories of liquid mixtures is presented in the next section
which describes the extension of liquid state theories to binary any liquid mixtures.
6.
1.2. BRIEF REVIEW OF THE THEORIES OF LIQUID MIXTURES :
The theories of liquid state are usually applied to binary or higher order li<piid
mixtures soon after their application to simple liquids. The binary liquid mixtures can be
classified as ideal or non-ideal systems depending upon whether the experimental thermo­
dynamic parameters are close to those obtained through necessary additive laws. When the
property under study differ from the so called ideal value, the difference is known as excess
thermodynamic function. These aspects were systematically dealt by Scatchard [41] initially
even though the departure was known right from Vander Wall [9].
Hie departure from the ideal behaviour has been attributed to the interaction of the
molecular species of different kind. The salient aspects of various liquid state theories as
applied to liquid mixtures are presented in the following paragraphs.
1.2.1. VANDER WALLS EQUATION OF STATE :
The initial theories of liquid mixtures basing on van der walls equation of state [9]
treated die characteristic quantities 'a' and
dependent averages.
'b' of die equation of state as composition
The deviations from the ideal solution behaviour were taken as a
consequence of differences in covolume'. This theory was successful only in explaining
certain excess properties in the critical region, of die liquid mixtures.
1.2.2. CHEMICAL EQUILIBRIA:
hi 1908 DolezaJek [40] attributed die departure from ideal behaviour to the effects of
chemical equalibria, association and solvation involving the molecular species in the mixture.
In 1916 Hildebrand [43] pointed out that die assumptions of Dolezalek were inaccurate since
the components of many non-ideal mixtures were quite normal molecules (not associated or
solvated) and Van Laaris theory [44] erred in not recognising the role of polarity of the
components in solution. He attributed the non-ideal behaviour of liquid mixture to differences
in cohesive energy densities where targe polarity differences were absent.
1.2.3. MODEL APPROACH CELL THEORY :
In 1931 Scatchard [41] developed die quantitative aspect of the treatment of
Hildebrand on the following assumptions :
7.
(1) the mutual energy of two molecules depends only upon the distance between
them and their relative orientation and not on the nature of the other molecules between or
around them or on the temperature,
(2) the distribution of molecules in position and in orientation is random and (3) the
change of volume on mixing at constant pressure is zero. He developed an equation for
energy mixing in terms of cohesive energy densities. If one wants to apply this
relation,
the thermodynamic functions determined at constant pressures should be converted into die
values at constant volumes as the latter can not be determined easily.
1.2.4. MODEL APPROACH - LATTICE THEORY :
Many rigorous theories of solutions have been developed with the aid of statistical
thermodynamics. These involve die evaluation of the partition function to be related to the
thermodynamic functions. Two broad approaches have been made : (1) those based on an a
priori' model of solutions, the partition function of the mixture being simplified to
manageable proportions by introducing adequate physical models, and (2) those derived
from radial distribution functions.
X-ray studies of the liquid state have shown that a liquid is much more like a solid
than like a gas and it can be treated as a quasi-crystalline one in which short - range order
and long-range disorder exist. This has prompted a number of workers to base the theories of
liquid mixtures on a lattice model. A detailed formulation of the classical lattice model was
presented by Guggenheim [45] in his book 'Mixtures'. In its simple form, the theory is based
on the following assumptions :
(a) the motion of the molecules
may be treated
as oscillations about some
equilibrium positions,
(b) the factorisation of die partition function is valid and
the
internal
partition
function is volume independent and the free energy of mixing depends only on configurational
partition function,
(c) the lattice is treated as rigid
and (d) each molecule occupies a single lattice point
A partition function was
developed for the binary mixuhsres containing molecules of the two components of die same
size. According to these assumptions the volume change on mixing has to be zero which, as we
8.
known is fir from the behaviour of real solutions. And also the theory could not predict
quantitatively the thermodynamic properties tike free energy of mixing.
1.2.5. MODEL APPROACH PRIGOGINE CELL MODEL :
Prigogine and co-worfters [46-49] employed the cell model of liquid state forward
by Eyring et al [22,30] in correlatiqg the thermodynamic properties of liquid mixtures with
intennoleculer energy parameters. The intermolecular energy for a pairof unchanged molecules
is represented in the cell model as
e = s’[-2(r*/r)4
+ (r7r)12
...
(1.4)
where s and r represent intermolecular energy and separation respectively. The starred
quantities represent the characteristic quantities, hi a mixture of
two components the
characteristic values can be obtained by the combination rule
S12* =(sn*S22*),/2 and
n2* = 1/2 (ru' + ra*)
(1.5)
where su* etc., with same subscript represent die quantities of die like molecules while those
with different subscripts £12’ represent unlike molecules. The Leonard-Jones and Devonshire
potential [8] was assumed to be valid for 1-2 interactions and the excess functions were
evaluated for ru* = r^*. With these potential functions configurational integrals are obtained.
Prigogine and Mathot [47] used the smoothed potential model in which a square well potential
was substituted in the dill Lennard-Jones and Devonshire model. This theory predicted a
positive excess free energy coupled with a negative volume change and was confinned by
experiments with CC1 +C(CH) [50] and CCHCH [49] mixtures. This theory was extended to
the mixtures, where r aid r are unequal, by Prigogine aid Be 11 emails [48]. They assume that
ceils of two sizes exist in liquid mixtures, cells containing molecules of species i being
different in size from those containing molecules of species j. The ratio of the diameters of the
cells was then chosen so as to minimise the free energy of mixing. Since the existence of
cells of different sizes is really incompatible with cells having die simple closed structure
of pure liquids, the Prigogine and Bellemans treatment can be a valid approximation only if
die sizes of the two species of different components do not differ very much One important
point established by this theory was that a relatively lage positive excess volume would
result from quite a small difference in the molecular size. In the cell theories described so
fir, discrepancies between the experimental and theoretical values of excess functions could
9.
arise from : (a) the failure of the model to predict the correct values of the thermodynamic
functions for any liquid aid / or (b) from the errors introduced in foe assumptions made
about foe molecular movement in foe pure component and in foe mixture. As a result of this,
a number of theories have been developed based on foe theorem of corresponding states
[50] which are liable to foe second type of error, to these theories, foe properties of mixtures
and of foe pure components are expressed in terms of those of a suitable reference liquid
1.2.6. CONFORMAL SOLUTIONS :
The theory of conformal solutions presented by Looguet-Higgins [53] is based on
foe theorem of corresponding states in its modern form due to Pitzer [54]. Using a simple
perturbation approach, Loqguet-Higgins [53] showed that as long as one retains foe
first order perturbation terms foe properties of foe mixture can be directly deduced from
foe assumptions about intermolecular forces (i.e., foe excess properties are proportional to
foe differences between foe force parameters of foe two components) with out restoring to
any specific model. This theory predicts that all excess functions are proportional to each
other and are of foe same sign. In this form foe theory could not explain foe results of CCI4
+C(CHj)4 and CO+CH4 systems for which <3® and V6 have opposite signs. One of foe
reasons for foe discrepancies might be foe neglect of foe higher order terms in Taylor's
series. Prigogineand co-workers [55] and Scott [56], therefore, considered higher order
terms in foe series for foe excess functions.
The higher order terms which are proportional to (S22* - su*)2 and (rjj* - r^*)2 could
be evaluated only on foe basis of the cell model theory. This theory combines foe basis ideas
underlying the theory of conformal solutions with those of foe cell model. Deviations from foe
laws of ideal solutions are explained as due to differences in foe depths of interactions, foe
sizes of foe molecules and their masses. Evaluation of these interactions is fundamentally a
many-body problem. However, Prigogine and co-woricers [47] have reduced it to essentially a
one - body problem. They envisaged a chosen molecule as moving in the average field of
neighbors and evaluated foe interaction energy. This was then combined with foe theorem
of corresponding states to write foe detailed expressions for foe excess functions. In foe crude
approximation of foe theory it is assumed that all cells are of the same size whether
occupied by a molecule of component one or by component two. In the refined version,
foe cell occupied by foe two different types of molecules are no longer assumed to be of foe
10.
same size. This form of the theory takes into account the local inhomogenities of structure
introduced in the liquid mixture due to differences in size of the molecules. Since the theory
has many limitations, the agreement between theoretical and experimental values may be
excepted only in the order of magnitude. In a good number of mixtures of simple substances
the theoretical values of the excess functions were found to be of correct sign and order of
magnitude. Theories on similar lines capable of dealing
with dipolar forces have been
developed by Pople [55], Rowlinson and co-workers [56,57] and Balescu [58].
1.2.7. DIPOLE INTERACTION:
In the mixture containing a dipolar liquid and a non-polar component, it is assumed that
if the dipolar forces are sufficiently weak when compared to the central forces, the rotation
of the molecule is almost free and die degree of disorder is sufficient to result in random
mixing n. In this case one can evaluate the macroscopic effects with an average of dipolar
interactions over all die relative orientations of the molecules. The computation of physical
properties then becomes a simple
perturbation calculation. The central forces represent
the main effect and the dipolar interactions prochice small deviations from the properties of
non-polar systems. This approximation forms the central idea of all the existing theories of
small orientational forces.
1.2.8. REFINED VERSION OF PRIGOGINE'S MODEL :
Balescu [58] made a synthesis of Rowlinson's theory and avenge potential theory and
gave a treatment of non-central perturbations of this type in the mixture. He developed die
potential interaction energy equation for dipolar molecules by superposing die LeonardJones potential on die electrostatic interaction energy of their dipole moment To calculate
die average value of the dipolar interactions over all orientations, he used the canonical
average by taking only the leading terms. Excess functions thus obtained were composed of
three terms, one being the contribution of the central forces mid having the same form as
that of Prigogine and co-workers while the other two were due to orientational forces of
dipolar origin. Rowlinson [59] has pointed out that the canonical average used by Balescu
is incorrect for an elliptical molecule and the average of free energy as shown by
Onsager [60] and Riisbbrooke [61] would be better. This method gives the electrostatic
II.
interaction energy which is half of that given by Balescu for the fist order nonvanishing
terms of the exponential expansion.
The solubility parameter theory introduced by Hildebrand and co-woricers [62,63]
was used for interpretation of non-electrolyte solutions. The free energy of mixing can be
represented by
F=RTf(Xj In xj + X2 In Xa)+( xj V| + xa V2)](6j - 62 )y i y 2
(1.6)
where V is the molar volume, x the mole fraction, y die volume fraction and 8 is the solubility
parameter respectively. The solubility parameter 5 is file square root of the cohesive energy
density of the respective pure components. This theory is involved with certain assumptions
regarding the randomness of position of molecules and does not depend on any specific
model. The first term of file energy expression represents file configurational part of the free
energy and is identified with the enthalpy of mixing. A solution with ideal entropy of mixing
and non-zero heat of mixing is called a regular solution’. The second terms of the equation is
a measure of the change in the molecular interaction energy upon mixing.
The theory
predict free energy data satisfactorily. However, in the case of the heat mixing it is not quite
successful.
1.2.9. CONFIGURATIONAL INTEGRAL:
The statistical thermodynamical approach for mixtures is based on the evaluation of a
configurational integral Q of file system. One approach is to evaluate Q of file pure components
and to relate them to that of the mixture. It was found that it suffices to relate the Q of the
mixture with the thermodynamic properties of the pure liquids and evaluation of individual
configurational integrals is unnecessary. This approach assumes random mixing of the species.
Prigogine and his colleagues [53], Scott [54], Byers Brown [64], and Kirkwood and his
colleagues [65] formulated the theory of random mixtures. The methods of improvement and
other details were discussed by Rowlinson [66]. The main drawback of this approach is the
assumption of random mixing while the real situation required some local order in mixing.
Later Bemi-random mixing was proposed and the theory was called the second revised
version of the average potential model' by Bellemans, Mathot and Simon [67]. A comparative
study of this model against average potential model and van der wall's model based on
Lorentz-Berthelot rule was presented by Rowlinson [66]. Prigogine and co-workers [47]
12
were the first to extend the principle of corresponding states to chain molecules on the basis
of the cell model. A principle of corresponding states was formulated by regarding a chain
molecule as a series of quasi spherical segments whose interactions with the neighboring
chains is characterised by the individual molecular energy and separation. Hie chain is then
specified by a number of segments, the external contacts that these make with other segments,
and the external degrees of freedom. The reduced parameters (volume, energy and entropy) are
related to the above specifications of the chain [68,69]. The law of corresponding states for
chain molecules has met with a noteworthy success.
Barker [70,71] proposed a theory for mixtures in which the volume for each molecule
remains constant It is necessary to assume certain geometric factors and contact relations for
the molecules in order to adjust the energy parameters characterising the interactions between
die molecules. One of the experimental excess parameters‘like excess energy has to be
assumed to evaluate die characteristic energy parameter. An attractive feature of the theory
is die possibility of deducing a standard set of contact energies which can be used in predicting
the thermodynamic properties of other similar systems. The estimation of these energies is
likely to introduce additional uncertainties but provide an independent estimate of the excess
functions. For better accuracy it is necessary to adjust the energy parameters of the liquid
mixture by assuming one of the thermodynamic excess properties. This procedure is similar to
that used in Balescu's treatment ofdie average potential model and the Floras theory [72,74].
1.2.10. MODEL APPROACH FLORY THEORY :
Flory and co-workers [27-29] adapted the same procedure wed for pure liquids
in treating the liquid mixtures of non-polar molecules differing in size. The model
configurational integral for mixture can be formulated in terms of liquid properties. Hie
characteristic parameters of the mixture can be evaluated from the parameters of the pure
liquids and a knowledge of interaction energy of the two species. Interaction energy can
be evaluated wing the procedure followed by Flory [72]. The theory was initially
developed for
chain
molecules
like
normal
paraffin hydrocarbons and their binary
solutions. Later the treatment was extended to the
study of thermodynamic properties of
mixtures of small non-polar molecules [28]. The partition function treatment was
13
generalised and extended to explain the behaviour of binary mixtures of non-polar molecules
differing in size [72-74],
1.2.11. MODEL APPROACH EYRINO'S THEORY :
Eyring et al [16,77,78] allied the significant structure theory, originally meant for
liquids, to the binary liquid mixtures with the following assumptions :
(a) non-random mixing is negligible,
(b) the characteristic temperatures of vibration q, are assumed to be die same for pure
liquids and their mixtures,
(c) molecules of both components continue to possess their gas-like translational
degrees of freedom,
(d) the degeneracy term has the same form as that for a pure liquid and
(e) the parameters E, V, n & a may be taken as suitable averages of the parameters
for the pure components. The evaluation ofthe factors involved in these expressions require
more
exactly determined experimental solid state data and involve a lot of numerical
calculation.
1.2.12. CELL THEORY-POLYMERS:
Patterson [70,71,79] studied certain mixtures of polymers in the light of recent
developments of r-mer theories due to Prigogine [47] and Hijmans [80,81]. They showed
that, using Hijman's treatment ofthe corresponding states for chain molecules (on the basis of
Prigogine's original treatment), die relations as derived by Flory for pure components can
also be evolved as a special case. Patterson and his colleagues [79] have reviewed the
Holy's treatment of chain molecules in die light of their own treatment
1.2.13. PRESENT STUDY:
In die present investigation excess volume have been estimated from the density
measurements. By measuring ultrasonic velocity and density iseniropic compressibilities have
been derived. Using literature values of heat capacity mid other required parameters of die
present study, isothermal compressibilities and evaluated The studies are confined to benzene,
chlorobenzene, toluene, acetic acid, propionic acid, butyric acid and tri-ethylamine. Five
ternary systems of benzene + chlorobenzene + third component (as one of liquids: toluene,
acetic acid, propionic acid, butyric acid or tri-etylamine). Using the excess volume (derived
14
from density) other excess parameters of the binary and ternary systems have been estimated
through the theory.
1.3. PREVIOUS EXPERIMENTAL WORK ON THERMODYNAMIC BEHAVIOUR
OF BINARY AND TERNARY LIQUID MIXTURES :
A brief review of the experimental studies in binary mixtures and temaiy liquid
mixtures of selected confounds is required. The field is quite vast and works that attempt to
correlate liquid state theories using ultrasonic measurements have been already tabulated [8].
A review article by Rubin Battino [83] on volume changes on mixing covers the work done in
the field till 1969. Methods of measurements, accurate theories of liquid mixtures and such
other details were discussed in sufficient detail. A quick and organised literature reference
soiree for the data published on mixing properties and excess properties (Vs and HE )
between 1900 and early 197? has been published by Jaime Wftnick and Abraham Tamir [84]
in their book. Mixing
and Excess Thermodynamic Properties-A literature Source Book', hi
view of these comprehensive reviews, no attempt has been made in the present work to
review the previous work on volume of mixing and heat of mixing (V6 and HE).
McGlashan [85] has considered many aspects of the process of heats of mixing and
reviewed die earlier work on heats of mixing of binary mixtures (till 1960). Wisniak and
Tamir [84] made an equally comprehensive review on heats of mixing between 1900 and
early 1977. hi view of these article, no attempt has been made here to review the work on
heats of mixing or volume of mixing.
15
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Rowlinson and G.S.Rushbrooke), North-Holland publishing conpany, Amsterdam (1968).
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[11] J.KPercus and GJ.Yevick, Phys. Rev., 110,1 (1958).
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[14] H.LFrish and J.LLebo-Witz (ed), "The Equilibrium Theory of Classical Fluids",
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