45

CHAPTER: 2
THEORETICAL MODELS
2. 0: INTRODUCTION:
In the present study, the different experimental techniques have been used in
conjunction with several theoretical equations in the evaluation of several parameters.
Hence it is necessary to present the elementary treatment of the models that generate
these equations. This is essential to understand the interpretations as well as the lapses in
the respective models.The Debye- Huckel model is the basic model of the electrolytic
solutions, proposed in the Year 1923. The lapses in this model were attempted to be
corrected in several ways in the last one (nearly) Century. Still the status did not improve
much, because of several new concepts, that came to lime light. One such concept is the
ETIPM (Eigen Tomm Ion Pair Mechanism). The present chapter starts with an
elementary treatment of the D- H Model and critically examines the lapses in it. The
Pitzer s model that was proposed with the usage of virial coefficients was applied earlier
by several authors, to several features of the electrolytic solutions, and it has successfully
explained the happenings in the higher ranges of concentration in many such cases, as per
the findings of the literature. So the author used this model and applied the dielectric data
to evaluate the activity coefficient, osmotic coefficient and excess Gibbs energy data.
This necessitated the presentation of the Pitzers model along with several drawbacks
involved therein to present the interpretations later. Following this logic all the basic
details of all the fundamental parameters to which the data is applied, are given in this
chapter in detail.
2. 1: Thermodynamic study:
Ionic solutions are so important that they are often ignored. Chemical reactions
have been studied for more than a century in ionic solutions [1,2,3]. Life occurs in
electrolyte solutions made of mixtures of bio-ions (sodium Na+, potassium K+, calcium
Ca
2+
, and chloride Cl), along with many other charged components. Ionic solutions are
neutral systems formed by a solute of positive and negative ions immersed in a neutral
46

polar solvent. This category includes systems of very different degree of complexity,
ranging from electrolyte solutions with cations and anions of comparable size and charge,
to highly asymmetric macromolecular ionic liquids in which macro ions like polymers,
micelles, proteins coexist with microscopic counter ions. Classical thermodynamics and
statistical mechanics describe non- interacting systems.
Electrolytes and salt solutions are ubiquitous [4] in chemical industry, biology,
and nature. However, their thermodynamic properties and applications have not been
adequately covered in a single book. Wolynes observes that the ideal theory of ionic
mobility would explain the mechanism of ion motion in terms of, the ion solvent
interaction, the structure of the solvent around the ion and the nature of the motion of the
solvent molecules. Reasonably bulky data about the Physiochemical properties of ionic
liquids are collected by several authors, but data pertaining to the Dielectric Constant as
the point of interest is of limited availability. Due to the complicated anisotropic nature
of the ion-solvent interactions and the complex nature of the local solvent
rearrangements, molecular theories are still in their infancy, in contrast to the majority of
continuum models.
The author pursued the theoretical and experimental studies, about long and short
range interactions, of the ionic liquids, in the Dielectric ,Thermodynamic ,Ultrasonic,
Electromagnetic and Densimetric aspects and this thesis is the culmination of such
attempts.
2. 1. 1: Debye - Huckel Model:
An earlier effort in describing dissolved salts was undertaken by Debye and
Huckel [5] historically, which is the first comprehensive theory developed to explain the
findings about the ionic liquids, using the Columbic interactions as the prime building
block, supported by Maxwell-Boltzmann distribution, used for the distribution of the
charged ions in the system [6]. Poisson Boltzmann equation [7] later invoked, to estimate
the potential at a central ion. This was necessitated because Electromagnetism is the
force of chemistry. Combined with the consequences of quantum and statistical
mechanics, electromagnetic forces maintain the structure and drive the processes of the
47

chemistry around and inside us. Due to the long-range nature of Columbic interactions,
electrostatics plays a particularly vital role in intra- and intermolecular interactions of
chemistry and biochemistry [8, 9, and 10]. A new concept is conceived about the
formation of ion atmosphere around the central ion, which is chosen as the point of
interest, for all the happenings. However, it was not until 1970s, liquid state theories
were well developed that a common approach to electrolyte and non- electrolyte solutions
became possible and the common basis for both happen to be the molecular distribution
functions.
Arrhenius [11, 12, and 13] suggested that electrolytes, when dissolved in water,
are dissociated to varying degrees into electrically opposite positive and negative ions.
The degree to which this dissociation occurred depended above all on the nature of the
substance and its concentration in the solution. In 1923, Debye- Huckel starting from
Arrhenius theory of dissociation and using the concept of chemical potential and excess
Gibbs energy developed the first complete model of a solution containing the electrolytic
ions in the dissolved state. A primitive model is applied, in which the ions are regarded as
charged hard spheres and the solvent is replaced by a dielectric continuum with dielectric
constant, through the whole medium. They conceived the ion atmosphere, and presented
a mathematical analysis for their model by considering inter-ionic attraction and
repulsion in general and the electro- phoretic and relaxation fields in particular in a
hydrodynamic and electrostatic continuum. They concluded that conductance varies
directly with the square root of concentration. Three factors cause ions to move in a
medium, they are 1. Thermal motion of random nature, 2.Flow of the medium as a whole
and 3. The forces acting on the ions, which maybe external in case of a field applied or
internal when charged ingredients are located in the solvent medium having dielectric
properties, gradients of viscosity, concentration or temperature.


48

2. 1. 2: The Ionic Atmosphere Concept:
Metal ions cannot exist by themselves in aqueous solutions. The principle of
electro- neutrality requires the presence of anions. Consider a positively charged ion in
solution. Due to the Columbic force of attraction between oppositely charged ions, the
positive ion will be surrounded by an ionic atmosphere consisting of an excess of
negatively charge ions as illustrated in Figure 2. 1. Similarly, a negatively charged ion
will be surrounded by its own ionic atmosphere of excess positively charged ions. Thus,
surrounding an ion of a certain charge Z, it will always find a higher than average density
of ions of the opposite charge. While enhancing the local density of the opposite charges,
this screening layer will greatly reduce the Columbic force of this ion Z on charges far
away. The thickness of the screening layer, which is given the symbol
-1
in Debye-
Huckel theory, effectively cuts off the long- ranged nature of the Coulomb interactions.
Beyond the screening length
-1
, ions are effectively non- interacting while within
-1
,
Debye- Huckel theory provides an estimate for the charge density, and the interaction
between charge density and the charge Z can be used to determine the chemical potential
of each ion. The inter- space between the spherical ionic layer, and the central ion is
called Co- sphere. This co-sphere is treated as a dielectric continuum by Debye and
Huckel who assumed the dielectric constant of this co-sphere continuum to be a constant,
and this constant appears at all the prominent places of the D- H model, especially in
which represents the reciprocal of the dimension of the ionic atmosphere. This was
modified by Glueckauff [14] who pointed out that the electrostatic field in this region of
co- sphere enhances with the increment in the charge of the ions, as the concentration of
the solute in the solvent increases. This leads to the lowering of the dielectric constant of
the continuum. The detailed mathematical model is presented in the following
paragraphs.

49


Fig: 2.1.1 Approximate picture of the ionic atmosphere surrounding a central ion without
any external field applied (not drawn to scale).

A B C
Fig: 2.1.2. A: The very nearly true situation of the water molecules of the solvent of an
aqueous ionic liquid. B: The neighborhood of a solute molecule in the non-ionized state,
C: Situation around the solute molecule, after dissociation into ions [in the absence of any
external field].
50

2. 1. 3: Concepts proposed by Debye and Huckel:
The Debye- Huckel model treated ions as point charges, allows one to describe
the main features of structural and dynamical properties of very dilute electrolyte
solutions [15]. A quantitative treatment of ionic interactions requires (a) the spatial
distribution of the ions, (b) The forces which act on the ions as a result of their ionic
atmospheres. These two aspects of the problem are inter- dependent. The inter-ionic
forces depend on the distribution of the ions.
In order to develop the necessary expressions for ionic distribution and inter-ionic
forces, Debye- Huckel made the following assumptions in their model:
(a) Strong electrolytes are completely dissociated into ions.
(b) Inter- ionic interactions obey Coulombs Law.
(c) Ionic distribution is controlled by Coulombs forces and thermal motion.
(d) There are no external forces.
(e) Solvent is a continuous dielectric medium with dielectric constant .
When an electrolyte is free of all external forces, the ionic atmosphere around
each ion must be the same in all directions, i.e., it is spherically symmetrical. Thus the
ionic distribution functions as well as the electric field are functions of distance r, but not
of direction. All the equations presented use standard notation of symbols.
The magnitude of the Columbic force between the two charged ions can be representd as,

2 2
1 2 0
( )
r
F z z e r c c =
(2.1.1)

where z
1
and z
2
: valencies of the two ions under consideration, located at a distance r,
e : electronic charge ,

0
: permittivity of free space,

r
: relative permittivity (dielectric constant ) of the solvent medium.
51

The force F acts inwards into the central ion under consideration, if it is attractive or
outwards away from the central ion if it is repulsive.
The charge density at the central ion and the divergence of the gradient of potential are
given by Poisson Boltzmann equation (in Cartesian coordinate system) as:
[Div (grad
0
)] =j
4np
c
s
[ (2.1.2)
where
o
: the potential ,

c
:

the average charge density at the central ion .
The chemical potential or partial molar free energy of an ionic species A
j
is given by

j
=
o
j
+RT ln {A
j
} (2.1.3)
Where
o
j

: the standard chemical potential and is constant under specified conditions of
temperature and pressure,
{A
j
}: the activity of species A.
The activity may be considered as an idealized or effective concentration.
Another way of expressing the chemical potential is

j
=
o
j
+RT ln
j
m
j (2.1.4)

where m
j :
molal concentration of species A
j
(i.e., moles of A per kg H
2
O),

j
: corresponding activity coefficient.
The activity coefficient is a measure for the deviation from ideal solubility.
Max Margules [16] introduced a simple thermodynamic model for the excess Gibbs free
energy of a liquid mixture. After Lewis had introduced the concept of the activity
coefficient, this model could be used to derive an expression for the activity coefficients
y

of a compound i in a liquid.
52

The ionic strength is calculated using:
2 2
1
[ ]
2
c
I z c z c
+ +
= +

(2.1.5)
where z+and z are the charge of the positive and negative ions, respectively and c
+
and
c
-
are their concentrations. It is important to note that z+and z ions under consideration
that make up the solution are not necessarily the same charges as those involved in the
chemical reaction. In other words, excess ions in solution contribute to the ionic
atmosphere.

2.1.4: Debye-Huckel calculation of potential and charge density for the electrolytic
system:
Consider an electrolyte solution consisting of different ionic species represented
by the subscripts 1, 2...s., the bulk concentrations n
1
, n
2
,. . . n
s
ions/cm
3
, and the values
of the charges z
1
, z
2
,... z
s
respectively. Further choose a coordinate system with one of
the j ions at the origin. The electrical potential
j
I at a point r from the origin is due to the
interaction between the central ion j and its ionic atmosphere. Let n
1

n
2
. . . n
s
be the
local concentrations of ions 1 to s in the ionic atmosphere.
The local ionic concentrations can be related to the bulk concentrations through
Boltzmanns distribution law as:
n
i
(r) =n
i
[ ]
i
U
kT
e

(2.1.6)
Further it can write U
i
as the potential energy of the i
th
ion at point r. If we assume
that this potential energy is due solely to the electrical field then,
U
i
=z
i
e
j
I (2.1.7)
Leading to
[ ]
'
( )
i j
z e
kT
i i
r n e n

=

(2.1.8)

53

The charge density (charge per cm
3
) at a distance r from the j ion is given by
'
1
( ) ( )
s
j i i
i
r n r z e
=
=

(2.1.9)
For z
i
e
j
<< kT, by expanding the exponential terms in the summation,
' 2
1 1
1
( ) ( ) {1 [ ] ......}
2
s s
j i i i i
i i
i i j i i j
r n r z e z n e
z e z e
kT kT

= =
= = +


(2.1.10)
the first term in above equation vanishes because of the condition of electro neutrality,
i.e.,
1
s
i i
i
z ne
=

=0 (2.1.11)
Taking only the linear term:
2 2
2
1
( )
4
s
i i j
j j
i
z e n
r
kT

ck

t
=
= =

(2.1.12)
Where has dimensions of reciprocal length and is given by (to be proved later in the
derivation) :
kT
z n e 4
i
2
1 1
2
2
c
t
= k


Another relation between electrical potential and charge density as given by Poissons
equation:
2
4t

c
V =

(2.1.13)
converting for a spherically symmetrical field and combining the linearized distribution
function, with the substitution [U =
j
r ] , gives after using the boundary conditions for
54

the potential, which must be finite at great values of r ( if a is the closest distance of
approach to the central ion),
2
4
j j
a
r dr z e t

=
}

(2.1.14)
This implies that due to electro neutrality, the net charge in the ionic atmosphere
of the central j- ion must be equal and opposite to the charge on the central j- ion
resulting to

j
=
z
j
e
e

e
ka
1+ka

e
-kr
r
(2.1.15)
Equation (2. 1. 15) represents the primary Debye- Huckel relation. It describes
the manner in which the potential changes with distance from a central ion of charge z
j
e
in the absence of external forces.
The ionic atmosphere cannot approach the central j-ion any further than r =a.
Thus a represents the effective diameter of the central j-ion. It can be seen from
equation (2.1.15) that
j
decreases with distance r and the highest value of
j
is obtained
when r equals the closest distance of approach a,

j
(a) =
j
z e
a c

1
1 a k +
(2.1.16)
Rearranging the above equation,
1 1
( ) [ ]
(1/ )
j
j
z e
a
a a

c k
=
+

(2.1.17)
Thus
j
(a) consists of two terms, as

j
(a) =
j"
(a) +
j'
(a) (2.1.18)
where
j"
(a) =
1
j
z e
a c

55

and
'
( ) [ ]
(1/ )
a
j
j
z e
e
a
a
k

c k
=
+

The electrical potential due to an isolated charge q in a medium of dielectric
constant is given by
I =q/ r (2.1.19)
Recalling that the charge of the central j- ion is z
j
e and that of the ionic
atmosphere is z
j
e, it follows from above three equations that
j"
(a) represents the
electrical potential at r =a due to the central j-ion while
j'
(a) is the potential due to the
ionic atmosphere at r =(a +1/). Thus the effect of the ionic atmosphere is equivalent to
the electric potential due to a charge z
j
e and distributed over a sphere of distance (a +
1/) from the central j-ion. The quantity 1/ is called the thickness of the ionic
atmosphere. The parameter can be expressed in terms of the ionic strength (I) of an
electrolyte solution as

2
=8e
2
N I/1000 kt (2.1.20)
where I is defined as
2
1
2
i i
i
I mz =


The Gibbs free energy change of an aqueous ion, can be evaluated, from these
equations and can be given as

j
e
=-
ka 1
k
e 2
N e z
2 2
j
+

(2.1.21)
This parameter is used to analyze the results in conjunction with the Bromleys
model, and Lu and Maurer model discussed in the coming subtitles of the same chapter.

2. 1. 5: Inconsistencies in the D-H equation for the potential function and related
parameters that form the basis for this studies:
56

The approximations made to maneuver the mathematical hurdles, and to make the
approaches of evaluation naive leave several inconsistencies in the potential function
itself, and surprisingly, attempts are yet to be seen in the literature, to overcome this
lapse. To point out the main lapses, the following eight points are to be observed.
(1) The assumption that z
i
e
j
<<kT is not too serious for points far from the central j-
ion, since in these regions,
j
should be small. However it involves errors in the
regions nearest to the j-ion at the origin.
(2) The linearization means that
n
i
'
=n
i
(1
z
i
ey
j
kT
) (2.1.22)
which is referred as approximate distribution [17].
The quantity [
z
i
ey
j
kT
] measures the ratio of the electrical energy of an ion to its
thermal energy, starting with this expression Bjerrum arrived at a very important
parameter called Bjerrums critical distance, q, an important parameter in connection
with ion-pair formation. It is given by Fuoss and Kraus using Bjerrums hypothesis [18]
as

2
1 2
2
z z e
q
kT c
=
(2.1.23)
(3) Equation (2.1.15) fails to realize that the presence of the charged bodies in the
ionic atmosphere further entails the electric field and the dielectric constant of the
medium in the ion atmosphere or dielectric continuum. It becomes concentration
dependent, leading to a lowering with increment in the number of charged ions ,
added to the solvent . This aspect was theoretically studied by Glueckauff .
(4) Further observed that the total potential decreases monotonically with the distance
from the central ion. Since the assumptions of D- H premise that the ion- cloud
must be fine grained, expecting many ions within a distance (1/) from the central
57

ion. This makes the potential oscillate with increasing distance from the central
ion, instead of decreasing steadily throughout the cloud.
(5) The presence of the Potential function shown in equation (2.1.15) that it is not
uniquely defined with respect to position of the central ion because the effect due
to all the ions is incorporated . Hence for a given central ion (say J), if two more
different ions considered as central ions [say (J-1) and (J -2)], for a given fixed
volume element (dV) around them, the equation (2.1.15) may not define the
potential uniquely for them always.
(6) In the case of symmetrical single electrolyte systems, the approximations have
less limitations, under such conditions, z
1
=-z
2
, n
1
=n
2
so that,
1 1 3
1 1 1 1
1
0 2 [ ] 0 [ ] .........
3
j j
j
z e z e
n z e n z e
kT kT

= + +
(2.1.24)

i.e., all terms in even powers vanish and therefore no errors of (z
i
e
j
/kT)
2
etc. are
involved.
(7) Even though the model uses a distribution function for the ions on the basis of
Maxwell Boltzmann statistics, it was realized that, it was necessary to review the
distribution of the charge carriers, with an alternative distribution function.
(8) Several futile attempts for this model are available in literature by many authors,
most prominent of these are: Muller [19], Gronwall [20], Eigen and Wicke [21],
all of whom attempted to preserve the higher order terms to a greater extent, only
to lead to several complications.
Sengupta et.al., [22] used an intuitively deduced formula of Dutta and Bagchi
et.al, [23] which uses the formalism developed by Dutta [24]. This could increase the
range of calculation, but did not improve to any extent.
And also there are several parameters which influence the measurable
characteristics of electrolytic solutions which were not accounted in the electrolytic
theory of Debye-Huckle. They are:
58

1. The electrostatic short range interactions between ions of electrolyte, solvent Co-
sphere , and the cluster of ions and the single central ion.
2. The electro static field modifications are caused around any central ion which is
modified as the density of the increased charged ions and these results in a major
modification to the dielectric constant of the solvent continuum. Consequently all
the electrolytic state equations of different parameters are to be subjected to a
thorough modification, considering the lowering of dielectric constant of the
solution around the central ion.
2. 1. 6: Deviations from ideality:
The D-H theory deals with departures from ideality in electrolyte solutions [25,
26, and 27]. The main experimental evidence for this non- ideality is that:
1. Concentration equilibrium constants are variables.
2. Rate constants depend on concentration.
3. Molar conductivities for strong electrolytes vary with concentration.
4. Freezing points of electrolytes are different from what would be expected for
ideal behaviour.
These departures from ideality reduce as the concentration decreases and they
were attributed to the Columbic law of electrostatic interaction.
The activities of ions in solution are relatively large compared to neutral
compounds. Ions interact through a Columbic potential which varies as reciprocal of r,
where r is the distance between ions. Neutral solutes interact through London dispersion
forces that vary as r
-6
. The greater the charge on the ions, the larger the deviation from
ideality.
Pytkowicz et. al., proposed a partial long-range order model for aqueous
electrolyte solutions to avoid contradictions present in the Debye- Huckel theory. The
partial long-range order increases with increasing salt concentration because, as the ions
59

are closer together, the Columbic energy of interaction which generates, a quasi-lattice
increases. Furthermore, the order decreases with increasing temperature because the
thermal energy increases relative to the columbic attraction of the ions.
The net charge density
j
around a central ion j, according to the D-H Model can
be written as
( ) 2
( )
4 1
j
D
r
a
j
j
D j
z
e e
a r
k
k
c
k

t k

=
+
(2.1.25)
Where Z
j
is the valence of the ion at j, c is the electronic charge, kis the reciprocal
of ionic atmosphere (which happens to be a function of the dielectric constant of the ionic
atmosphere),
D
a is the Debye radius and
j
r is the distance of the point under
consideration. Hence
( ) j
decreases monotonically from a maximum value at
j
r =
D
a
i.e., the surface of the ion j, to zero.
The potential due to the effect of all the ions located in a volume element dV
in the solution situated at a distance
j
r from the central j ion is not a unique function of
position, since the relative permittivity of the medium
r,
is responsible to control the
columbic interaction, between the charged entities. Consequently the depression in the
permittivity, with the concentration of the ionic system requires to be counted upon,
through the Glueckauffs and Von Hippel s models. This observation was reported and
provided the necessary data, evaluated and experimentally established in this study.
2. 1. 7: Chemical Potential, Activity Coefficient, Gibbs free energy and Osmotic
Coefficient of an ion in solution:
(i) Chemical potential:
While studying the formation of ionic solutions, the most useful quantity [28] to
describe is chemical potential
i
, defined as the partial molar Gibbs energy of the i th
component in a substance:
60

(2.1.26)

where x
i
can be any unit of concentration of the component: mole fraction, molality,
while for gases,
i
is chemical potential which is a function of temperature T, pressure P
and mole fraction x
i
of the system, given as
ln
i i i
kT a
o
= +
(2.1.27)

where
i
a is the activity by Lewis, designated by the symbol a. The activity gives an
indication of how active a substance is relative to its standard state as it provides a
measure of the difference between the substances chemical potential at the state of
interest and at its standard state. It depends on the composition. The standard state is
indicated by a superscript .
(ii) Activity coefficient:
In electrolyte solutions the activity coefficient is a measure of the deviation of the
idealized concentration {A
j
} from the stoichiometric or analytical concentration, m
j
. An
activity coefficient is a factor used in thermodynamics to account for deviations from
ideal behavior in a mixture of chemical substances. This deviation is a result of the
electrostatic forces of attraction and repulsion which are present in electrolyte solutions.
A discussion of the concept of activity coefficient requires an analysis of the electrostatic
interactions in electrolyte solutions.
The activity coefficient
i
is defined as:
i i i
a x = ,
Reducing 2. 1. 27 as
ln
i i i i
kT x
o
= +
(2.1.28)

61

The mean molal activity coefficient, [29] of an electrolyte solution is the
geometrical mean of the activity coefficients of the ions in the solution. For a solution
containing n
i
mol of ion i:
1 1
ln ( ) ln ln ( ) ln
i i w i i
ions ions
n x n f
n n
o

= = +


(2.1.29)
Mean molal activity coefficients can be measured directly by potentiometric methods.
The contribution of Debye and Huckel is remarkable in their ability to explain the
(I) behaviour of the activity coefficients, where I is the ionic strength [30, 31], which
arises due to the long-range Coulomb interaction and cannot be accounted for by classical
thermodynamic theories. It is the Debye- Huckel limiting law. Brahmajirao [32], used it
in the study of several parameters in several models, where in vivid evidences about ion
pair formation mechanism, were established. Most of the equations used by him were
incomplete with respect to the concepts involved. However the author of this thesis used
more refined models, for the enhanced exactness. Several deficiencies were observed in
the process. Debye-Huckel universal limiting law has been shown to be exact on
statistical mechanical grounds by Kirkwood and Poirier [33]. The limit of validity of the
Debye-Huckel model has been set by Frank and Thompson [34, 35] at 0.001 M (mole l-
1) for aqueous solutions of 1:1 electrolytes at 25 C, with the concentration limit at which
the thickness of the ionic cloud equals the average inter-ionic distance. It limits the
range of applicability of the D-H model, to study Eigen-Tamm mechanism.
Formulation of the modified equations for the activity and osmotic coefficients
due to Pitzer et.al, [36- 48] in his application, was considered in this thesis. Pitzer has
correlated salt solution data for many common salts with a virial type equation. This
method will not change the basic nature of the terms in the D- H equation. Several
authors attempted to adopt these equations and work out modifications later.
Consequently the Pitzer equations preserve an element of hope for research.
In this study, Bromley model is used [49, 50] for the calculation of both activity
and osmotic coefficients. The model by Lu and Maurer [51], is also used to understand
the results with Gibbs free energy calculations.
62

(iii) Gibbs free energy:
The Gibbs potential measures the amount of work a system can do at constant
temperature (T) and pressure (P), given by:
G = U - TS + p V (2.1.30)
Hence Gibbs potential measures the amount of non-mechanical work a system
can do at constant pressure and temperature, or the amount of chemical work a system is
able to do [52].
The enthalpy measures the amount of work a system can do in an adiabatic,
isobaric process. This is the amount of non- mechanical work, since the pressure has to
be kept constant. Using Gibbs free enthalpy G, with standard convention, using the
standard definition of chemical potential [53]
i i i i
i i
dG dN N d = +

and
i i
i
VdP SdT N d =

(2.1.31)

where N
i
indicates the number of moles of the i
th
species.
Equations (2.1.31) are Gibbs- Duhem relations used to check the validity of the models.
The Gibbs free energy change which arises when an ion enters an electrolyte solution is
given by:
G =z
j
e (
j' -j"
) (2.1.32)
and can be positive or negative depending on the nature of the interactions between ions
and the solvent. This parameter is dependent upon the potential experienced by the
central ion itself, for a given concentration of the solute in the electrolyte system and that
governs all the mechanisms that come into existence, and is directly related to the
physiochemical consequences. The activity coefficient is related to the molar excess
Gibbs energy G
E
, by:
63

, ,
ln [ ]
j
E
i T P n
i
nG
RT
n

c
=
c
(2.1.33)

In this equation, all the symbols have their conventional meaning. The excess
Gibbs energy of a solution is the difference between the actual Gibbs energy and the
Gibbs energy of the ideal solution at the same temperature T, pressure P and composition.
Anderko et.al [54], reviewed a large variety of models, in connection with Gibbs free
energy. Number of models with the D-H expression, like the UNIQUAC model [55] and
the NRTL models [56], using two different frame works, i.e., McMillan-Mayer (M-M)
framework, and Lewis- Randal (L-R) framework exist for the description of the excess
Gibbs energy of mixtures of molecular components. Some electrolyte models are
extended as models for the excess Gibbs energy calculation. Irrespective of the specific
model used to describe the individual thermodynamic properties as a function of factors
like concentration, temperature; it should be necessary that, the relationships between the
various functions must be thermodynamically consistent. Hence thermodynamic
parameter chosen for interpretation of the final result is to be evaluated with the
application of the fundamental definition of the same. In this thesis, Lu and Maure model
was used to calculate excess Gibbs energy values of chosen electrolytes from calculated
data of activity and osmotic coefficients from Pitzer model.
(iv) Osmotic coefficient:
In dilute aqueous solutions, the water activity and the water activity coefficient
are very close to unity. In order to report water activities without a large number of
significant digits, the osmotic coefficient is commonly used and is represented by the
symbol , defined by Hammer and De Wane [57] as an enhancement in the velocity of
the central ion, in the direction of the applied external field, as a result of more collisions
on the central ion from ions behind the central ion than from ions in front of it. Jean
Vidal [58] gave a very detailed exposition about the thermodynamical aspects of osmotic
coefficient and its relation with several parameters.

64

2. 1. 8: Extended Debye-Huckel Equations and Debye-Huckel limiting law, for the
activity coefficient, and its limitations:
The basic concepts of D- H equation for the activity coefficient are given below.
The Debye- Huckel equation and Debye- Huckel limiting law were derived by Peter
Debye and Erich Huckel who developed a theory to calculate activity coefficients of
electrolyte solutions. Taking into account, the ion size parameter expressed in angstroms,
the original Debye- Huckel equation for the mean activity coefficient (), for a
completely dissociated binary electrolyte, consisting of ions (mole of solute)
-1
, with
charges Z
+
and Z
-
, Robinson and stokes [59] can be written as:
1 1
2 2
-
] log [ ]/[1 Z Z A I aB I

+
= + (2.1.34)
where I is the ionic strength in molality (m) units of concentration.
( / 2) I Z Z m v
+
= , and A

and B

are parameters dependent on density and permittivity

of the solvent givenby,

2
3 3
( )
2 2
2 1
( )
1000
2.303 { }
N e
A
k T

t
c
=

in mole
-1/2
. I

. (deg K)
3/2
(2.1.35)

2
1
2
1
2
8 1
{ } [ ]
1000
( )
Ne
B
k
T

t
c
=

in cm
-1
.mole
-1/2
I
1/2
(deg K)
1/2
(2.1.36)
For low values of I , the term [
1
2
aB I

], ultimately becomes negligibly smaller than

unity then

1
2
-
log [ ] Z Z A I

+
=
(2.1.37)

which is known as D-H Limiting law. The equation for the activity coefficient was
modified by K. S. Pitzer et. al.,. This model was most widely applied and reliable.
Several modifications were tried starting with the Pitzers model, to complement some of
the shortcomings of the assumptions of this model, and were tested by several authors. So
65

these tested equations of Pitzer model were used, for the calculation of activity
coefficient and osmotic coefficient to ensure the preliminary development over the
Debye- Huckel model, into which the dielectric constant variation was impregnated
suitably along with Bromleys model.
2. 1. 9: Recent improvements over the Debye- Huckel equation: Pitzer model:
The activity is proportional to the concentration by a factor known as the activity
coefficient y, and takes into account the interaction energy of ions in the solution. It is
important to note that because the ions in the solution act together, the activity coefficient
obtained from the equation is actually a mean activity coefficient. The other approaches
based on this model are given by John Edwards [60], Maya B. Mane [61], Lloyd [62],
Bronsted [63], Guggenheim [64], Scatchard [65], McMillan Mayer [66], Chen et.al., [67-
72] and Krebs[73].
Detailed presentation of several models starting from lapses of Debye- Huckel
model are given by Horaico R Corti et.al., [74]. The electrolyte Non-Random Two-
Liquid model [75] (NRTL) is an activity coefficient model that correlates the activity
coefficients y

of a compound i with its mole fractions x

in the liquid phase concerned.

The concept of NRTL is based on the hypothesis of Hunenberger [76] that the local
concentration around a molecule is different from the bulk concentration. The NRTL
model of Chen et.al., combines a Debye- Huckel term with the NRTL local composition
model. The local composition concept is modified for ions, and the model parameters are
salt specific.The difference is due to a the interaction energy of the central molecule with
the molecules of its own kind U
ii
and that with the molecules of the other kind U
ij
. The
energy difference also introduces non-randomness at the local molecular level. The
NRTL model belongs to the so- called local- composition models. Other models of this
type are the Wilson model, the UNIQUAC model and the group contribution model
UNIFAC. These local composition models are not thermodynamically consistent due to
the assumption that the local composition around molecule i is independent of the local
composition around molecule j. McDermott and Flemmer [77] established that this
assumption is not true.
66

McMillan and Mayer developed a theory starting from the virial equation of state.
Solutions of uncharged molecules can be treated by a modification of the McMillan-
Mayer theory. When a solution contains electrolytes, electrostatic interactions must also
be taken into account. The central ion surrounded by a spherical cloud that is made up of
ions of the opposite charge is used and expressions were derived for the variation of
single-ion activity coefficients as a function of ionic strength. Debye- Huckel theory
takes no account of the specific properties of ions such as size or shape.
Later, other types of models have been developed to extend the Debye- Huckel
law to higher concentrations [78]. The first extension was to impose a restriction for a
closest approach distance to ions as the ions in the cloud could not approach the central
ion by more than some distance.
Earlier to the Debye- Huckel theory, Brnsted had independently proposed
empirical equations for activity and osmotic coefficients where the activity coefficient
depended not only on ionic strength, but also on the concentration, m, of the specific ion
through the parameter .
Activity Coefficient: ln =m
1/2
2m
Osmotic Coefficient: 1 =(/3) m
1/2
+m
This is the basis of SIT theory, used to estimate single- ion activity coefficients in
electrolyte solutions at relatively high concentrations, by taking into consideration
interaction coefficients between the various ions present in solution. Interaction
coefficients are determined from equilibrium constant values obtained with solutions at
various ionic strengths. The determination of SIT interaction coefficients also yields the
value of the equilibrium constant at infinite dilution.
Guggenheim and Scatchard extended the theory to allow the interaction
coefficients to vary with ionic strength. Bromley model, the Pitzer model, were
developments from this models. Pitzers model took the expressions of the osmotic
coefficient obtained from the extended Debye-Huckel law and applied a virial expansion
in molality, as suggested by other theories.
67

The success of the Pitzer model lies in the fact that it opens the way to the
description of highly concentrated solutions, up to 6 or 10 mol/kg, with a few parameters.
However it was observed by many authors (mainly Guggenheim and Turgeon) that the
Pitzers model fails to recognise substantial major discrepancies that encounter even at
concentrations greater than 0.1 mol/kg. It fails to reveal intricate aspects of happenings at
lower concentrations than the above successful range of concentrations because of the
apparent crudeness in assumptions. At lower concentrations, the results reported even
with D- H model of electrolytic solutions would probably get masked because of the
nature of virial terms of expansion and the approximations during their application. It
curtails to reveal ions of finer traits involved in some of the interactions attempted for
study, and hence the outcome of the application of this model cannot be comprehensive.
It is to be realised that due to these conspicuous drawbacks, even the similar models like
the Bromleys model, cannot deliver the expected improvements, leading to non-
comprehensive outcomes. The apparent inherent short comings traced by the author, are
detailed in the article 2.1.11, of this chapter. Nevertheless, the Pitzers parameters have
limited physical meaning, since the virial expansion was empirically introduced in the
model. It retains the basic physical concepts involved in the D- H model, to explain the
mechanisms involved.
Another method of investigation has also been explored by using the statistical
mechanics to obtain thermodynamic quantities for ionic solutions and it has been carried
out with the help of the Ornstein- Zernicke (OZ) equation and by treating the solution in
the McMillan Mayer formalism. The OZ equation treats statistically the interactions
between particles by taking Pitzers model. As it is the best of the earlier models, it was
chosen for application of the earlier published dielectric data in the present study.
Thomsen [79] developed equations for multi-phase flash calculations required for
aqueous electrolyte process simulations in his doctoral work and multiple steady states
were detected and analysed. Their equations are essentially similar to those of Nicolaisen
et.al., [80]. They pointed out that in order to apply the Debye- Huckel theory, to non-ideal
systems; it has to be primarily combined with a term for short-range interactions.
68

Thomsen et. al., developed the extended UNIQUAC model in which the Debye- Huckel
contribution to the excess Gibbs energy was deduced.
Pessa Filho et.al., [81] observed that mixtures containing compounds that
undergo hydrogen bonding show large deviations from ideal behaviour. These deviations
can be accounted for chemical theory, proposed by Dolezalek [82] according to which the
formation of a hydrogen bond can be treated as a chemical reaction. This chemical
equilibrium needs to be taken into account when applying stability criteria and carrying
out phase equilibrium calculations. They illustrated the application of the stability criteria
to establish the conditions under which a liquid- phase split may occur and the
subsequent calculation of liquid- liquid equilibrium using Flory- Huggins equation
modified by a chemical- theory to describe the non- ideality of aqueous two-phase
systems composed of poly ethylene glycol and dextran. The model was found to be able
to correlate ternary liquid-liquid diagrams reasonably well by simple adjustment of the
polymer-polymer binary interaction parameter. Chemical theory to modify excess Gibbs
energy models began with Kretschmer and Richard Wiebe [83], who used this theory to
modify the Flory- Huggins equation [84, 85]. During l970s, Waisman and Lebowitz
[86] solved the MSA for the restricted primitive model. A rapid succession of new results
on more sophisticated models of ionic solutions was developed. The MSA equations have
been solved for asymmetrical ions, mixtures of charged hard spheres and dipolar spheres,
and charged particles near a wall. The MSA itself has been improved in many ways by
the techniques of cluster resumptions [87]. This work was later supplemented and
arranged in a more complete form by Renon and Prausnitz [88], Nagata and Kawamura
[89], Nagata [90], Nath and Bender [91, 92], Brandani [93] and Evangelista [94]. They
have also used chemical theory to modify the UNIQUAC equation of Abrams and
Prausnitz et.al., [95], the parameters and various thermodynamic properties necessary for
performing calculations with the extended UNIQUAC model and published in a number
of papers [96- 102]. The main difference between these models lies in the way the
chemical equilibrium between clusters formed through hydrogen bonding is calculated.


69

2. 1. 10: Omissions in Pitzer model:
The Pitzer model was developed as an improvement upon earlier model
proposed by Guggenheim, involving some approximations about the concentration limit
in connection with the osmotic coefficient of a reference salt which worked well at low
electrolyte concentrations [103], but contained discrepancies at higher concentrations
(>0.1M). Scatchard suggested some modifications to these concepts, to provide a
compact summary of the experimental data but failed to cater to the desired requirements.
On the basis of the insight provided by the improvements of Friedman and
collaborators [104], Card and Valleau [105], the Pitzer model came to light and attempted
to resolve these discrepancies, without resorting to excessive arrays of higher- order
terms. Pitzer used A

, called the Debye- Huckel term, for simplification of the equations

applicable to 2- 2 electrolytes. The equations of Pitzer used in the present study are given
hereunder.
In the Pitzer model, the activity coefficient of each ion is described by two terms.
The first one is due to long-range interactions (electrostatic or Debye- Huckel behaviour)
given by f

, the second one is due to short-range interactions which are expressed in
terms of second and third coefficients of a virial equation [106].
Pitzer stated that, some marked omissions are inherent in his model. They are
1: The model starts with the system proposed by Guggenheim which follows the
principle of Specific interaction of Bronsted, excluded terms related to short range
forces between the like sign ions which is fairly inaccurate when we want to
consider the electrostatic contributions for the evaluation of depression in the
dielectric constant, as conceived in the Glueckauffs model. A similar line of
thought is continued in Pitzer s model which throws its impact on the final results
of findings reported in this thesis because mechanisms like the ion-pairing, polar
interactions etc. need cognizance.
2: The virial coefficients used in the equations are from the experimental origin;
their statistical method of evaluation would not consider the ionic level
70

happenings and their dependence on intricate properties like the relative
permittivity on the concentration of the electrolyte in the solvent. Consequently
the mechanisms that take place within ionic atmosphere during solvation and also
the solute- solvent interactions are not accounted in the statistical methods
involved for the virial coefficients of the Pitzers model.
3: Two assumptions about phenomena and concepts are erroneous. They are:
(i) The direct interactions of solute species at short distances and changes in
solvation with concentration which influence the effective inter ionic potential of
average force are not distinguished,
(ii) The distinction between molality and concentration is ignored, because mainly
the statistical calculations are used, based upon the molecular models, to develop
empirical form of qualitative understanding. Especially molality is preferred by
Pitzer since he presumes that the temperature variations influence the activity
and osmotic coefficient to a lesser extent with molality.
However happenings around the proximity of the ion in the co- sphere are not
compatible with these approximations, as the ion-pair formation around the central ion is
sensitive to even the least potential contribution either in the proximity or at a reasonable
distance from the site of mechanism.
4: Guggenheim and Turgeon pointed out that substantial discrepancies arise even at
concentrations greater than 0.1mol/kg. Several authors made variety of attempts
to accommodate this, but to no avail [107]. Scatchard and his co-workers [108]
extended and elaborated Guggenheim equations in several ways. First they
subdivided the Debye- Huckel term in the expression for excess Gibbs Energy
into a series of terms with different coefficients of I
1/2
in r corresponding to
different distances of closest approach for the solute components .
5: Appropriate derivatives later yield complex formulae for the coefficients and .
Bronsted s principle of Specific interaction is abandoned and terms are
introduced for the short range interaction of ions of like sign. Arrays of third and
71

fourth virial coefficients are added by Lietzke and Stoughton [109]. The osmotic
coefficient data for 20 pure electrolytes was represented by them to a fair
accuracy. Several systems of mixed electrolytes were treated also, resulting in a
system of equations is unwieldy and raises other complications, because the
physical interpretation of the involved terms and parameters become complex.
These omissions during the development of the theoretical equations of the model
were attempted in a pragmatic way, by making the equations simpler with more
meaningful parameters introduced in the Pitzers work.
The fundamental deficiency in the basic Debye- Huckel theory is the recognition
of the parameter the distance of closest approach, a in the calculation of the electrostatic
energy in the distribution of ions. However the kinetic effect of hard core on and is
ignored. Kirkwood points out that the hard-core effects cannot be treated rigorously in the
traditional way, by calculating the free energy. Terms were added by Rysselberghe and
Eisenbergto[110] improve hard core effects, only for the feature of the particular interest
to be lost. Sengupta, Bagchi and Dutta attempted the application of new statistics and
developed a new set of distribution equations. Dutta and Bagchi deduced theoretically
( )
,
1
1
r
z kT
r
e
b
n
v c

+

(
(
+
(


=
(2.1.38)

where
,r
n

: the number density of positive or negative ions,

b

: the volume of the covering spheres of positive or negative ions ,

v

: parameters of distribution, z

valence of the positive or negative ions and

, k and T have their usual significance.
The formalism used in this equation has been developed by Dutta and Bagchi, and
rests on the assumption that ions having characteristic volume of their own are under the
influence of their mutual electrostatic field in solution. Eigen and Wicke used a similar
72

distribution formula in their theory of strong electrolytes in a different form with slightly
different explanation for the parameters involved.
Sengupta et.al, evaluated the available electric energy, through the evaluation of
where
kT
c

(
=
(

. The Poisson-Boltzmann equation was then solved. An expression for
the Activity coefficient was then deduced using separate ionic radii for the ions.
Hill [111] using Statistical mechanics proposed a convenient equation for hard
core potential and was used in the Pitzers model for the osmotic coefficient. However
the equations of the statistical origin involve errors in the qualitative as well as
quantitative observations. This is to be accommodated at some juncture to make the
Pitzers model better. Attempts of other workers in literature are not concentrating on
these aspects. Consequently there will be marked deviations between the results of the
application of the present theories and experimental findings. The explanation of the
mechanisms like Eigen and Tamm Ion pair formation leave dissonance between theory
and experiment as is found in some results in the present study, discussed in the chapter
results and discussion. The application of Pitzer model, to the dielectric data obtained
using a very accurate operational amplifier technique is presented below.
2. 1. 11: Pitzers retention of the " " A
|
or slope of Debye-Huckel limiting law:
The Pitzers model uses primarily the concepts of the Debye- Huckel model. An
important parameter that appears in the Pitzer model isA
|
called the Debye-Huckel term,
given by
3
2 2
0
2 1
3 1000
dN e
A
kT
|
t
c
(
=
(

(2.1.39)
(Terms are explained in article:4.2.2) This is the Debye- Huckel limiting law
slopes from the basic equation of the osmotic coefficient and is a function of temperature
given by a polynomial correlation in T, used by Morteza Baghalha [112], taken from
Archer and Wang [113). This parameter was obtained from a careful evaluation of
73

density and dielectric constant data by Pitzer, Royf Sylvester , Holmes and Mesmer
[114]. Debye and Huckel originally coined this parameter, and were later modified by a
host of workers, considering the very accurate experimental data from more than one
source which gives a comprehensive picture about the ion pair formation mechanism that
will be discussed later.
2. 1. 12: The Pitzer equations for activity coefficient and osmotic coefficient:
Pitzer started with the fundamental quantity of excess Gibbs free energy and
begins with a virial expansion. Further, the free energy is expressed the sum of chemical
potentials, or partial molal free energy, and an expression for the activity coefficient is
obtained by differentiating the virial expansion with respect to molality.
For a simple electrolyte M
p
X
q
, at a concentration m, made up of ions M
z+
and X
z
,
the parameters f
|
,
MX
B
|
and
MX
C
|
are defined by
( ' )
2
f
f
I
f
|

=

(2.1.40)
' ( ){ ' } ( ){ ' }
2 2
MX MX MX MM MM XX XX
p q
B I I I
q p
|
= + + + + + (2.1.41)
3
{ }( )
MX MMX MXX
C p q
pq
|
= +

(2.1.42)
The B parameter was found empirically to show an ionic strength, I dependence
(in the absence of ion-pairing) which could be expressed as
(0) (1) I
MX MX MX
B e
| o
| |

= +

(2.1.43)
For 2-2 electrolytes, activity coefficient given by Pitzer and Mayorga [41] as,
2
ln 4
MX MX
f mB m C

= + +
(2.1.44)

74

Where
1/2 1/2 1/2
0
[ /( ) (2/ )ln(1 )] f A I I bI b I bI

= + + + +

and I =1/2 mz
2
1/ 2
1
1/ 2
2
2
2 2 (0) (1) 1/ 2
1 1 1
2 2 (1) 1/2
2 2
1
2 (2 )[1 ( ) ]
2
1
(2 )[1 ( ) ]
2
I
MX MX MX
I
MX
B I I I I e
I I I I e
o
o
| | o o o
| o o o

= + + +
+
And
(3/2)
MX MX
C C
|
= ,
3
2 2
0
2 1
3 1000
dN e
A
kT
|
t
c
(
=
(


Osmotic coefficient is given by
-1 =4f

+mB

MX
+m
2
C

MX
(2.1.45)
Where
1 1
2 2
[ /(1 ) ] f A I bI
|
|
= +
And
1 1
2 2
1 2
0 (1) (2) I I
MX MX MX MX
B e e
o o |
| | |

= + +
Similarly for (2-1), (3-1) Electrolytes,
ln =|z
M
z
X
| f

+m
M X
2v v
v
MX B

+m
2
( )
3/2
M X
2 v v
v
MX C

(2.1.46)

in which
M
and
X
are the numbers of the M and X ions in the formula and also
=
M
+
X,
and z
M
and z
X
give their respective charges in electronic units and
-1=|z
M
z
X
| f

+m
M X
2v v
v
MX

B
+m
2
( )
3/2
M X
2 v v
v
MX

C
(2.1.47)
2. 1. 13: B, and relations used in the model:
Pitzer used a set of virial coefficients
( ) ( ) 0 1
, , B
|
| | and
( ) 2
| which can be expressed
in the form of an equation

( ) ( ) ( )
1 1
2 2
1 2
0 1 2 I I
B e e
o o |
| | |

= + +
75

where
(0)
| and
(1)
|

represent the effect of short range forces.
(2)
| and
2
o
correspond to the anomalous behavior of the 2-2 electrolytes. The parameters
1
o and
2
o
are assigned with a series of values by Pitzer et.al. The values of these coefficients are
obtained by Pitzer, from experimental data statistically, and are available in literature.
Besides the set of parameters obtained by Pitzer et. al., in 1970s, Kim [115] and
Frederick [116] published the Pitzer parameters for 304 single salts in aqueous solutions
at 298.15 K, extended the model to the concentration range up to the saturation point.
Those parameters are widely used for many complex electrolytes including organic
anions or cations, which are very significant in some related fields, were not summarized
in their paper.
In the original notation used by Pitzer et.al., all the parameters that are adjusted
for each electrolyte are given in the subscript MX. The symbols , (1), and (0) etc are
labels but not exponents, and b =1.20, assumed by Pitzer (1973), in his original paper.
The data for the four parameters
(0)
,
(1)
,
(2)
, and
C

were fitted by least squares
adjustment,
1
o
=
1.4 and
2
o = 12.0 was used in all the calculations of this Thesis. The
values of these parameters used in the calculations are given in Table: 4.2.1
These equations were applied by Pitzer et.al, with minor suitable modifications to
an extensive range of experimental data at 25 C with excellent agreement to about 6 mol
kg
1
for various types of electrolytes. The treatment can be extended to mixed
electrolytes and to include association equilibria. Values for the parameters
(0)
,
(1)
and
C for inorganic and organic acids, bases and salts have been tabulated. Temperature and
pressure variation is also discussed at appropriate places by them.
In these Equations, the B
MX
term corrects for ion-ion interactions, whereas the
C
MX
term corrects for ion- ion- ion interactions. Pitzer et. al., have obtained values of

(o)
,
(1)
and C

for several electrolytes and obtained good agreement with

experimental data (up to 6M) when the assumption is made that for all electrolytes, =
2.0 and b =1.2. The C

values are generally small, which means that in most cases, C

MX
can be ignored.
76

2. 1. 14:The Bromley Model:
Friedman [117] deduced a set of equations for conversion between two frame
works of Lewis- Randall and McMillan-Mayer for the thermodynamic excess functions
of solutions. Several inconsistencies came to lime light. Martin Breil [118] finds that the
Friedman's equations only are valid for dilute solutions makes his results limited and not
generally applicable.
A generalized analytic correlation is presented for activity coefficient, osmotic
coefficient, enthalpy, and heat capacity of single and multi- component strong aqueous
solutions, by Bromley who showed that it is possible to assign individual values of to
ions i.e;

MX
=
M
+
X
(2.1.48)

Bromleys one parameter model is much easier to use than Pitzer s three-
parameter equation even though the latter is established in this thesis to be more accurate
in many ways, by virtue of the fact that the Pitzer treatment is more comprehensive.
Bromley pointed out that his method does not perform satisfactorily for divalent metal
sulphates, sulfuric acid and zinc and cadmium halides. These electrolytes are all partially
associated in solution. It is this association which makes it difficult to estimate
12
values
by a model, since this equation does not take the degree of dissociation into account. The
true ionic strength will depend on the association equilibrium and it would nevertheless
be more convenient, if a way could be found to estimate activity coefficients using the
stoichiometric or apparent ionic strengths. Pitzer and Mayorga have presented such a
model for 2- 2 electrolytes. Basically, they used their earlier equations. Bromleys
equations of the one parameter model: originate from the activity coefficient equation
1 2
1
2
1
1 2
1
2
2
(0.06 0.6 )
log
1.5
(1 )
1
z z
A Z Z I B I BI
Z Z
I
I
Z Z

+
+
+
= + +
+
+
(2.1.49)

Bromley, [49, 50] presented values of B for 1- 1, 1- 2, 1- 3, 2- 1, 2- 2, 3- 1, 3-2
and 4-1 electrolytes which predict

upto I =6 with a maximum error not exceeding
77

5%. The B values are in the range -0.50 <B <0.25. And for an electrolyte they can be
estimated with the equation,
B =B
+
+B
-
+
+

-
(2.1.50)
The values of B for chosen electrolytes were given in table 4.2.1. Starting with above
equation, it is possible to write equations for a number of other thermodynamic properties
using rigorous thermodynamics. The resulting equation for osmotic coefficient is given
by
1
1
2
2
1 2.303 ( ) 2.303(0.06 0.6 ) ( )
3 2
I I
A z z I B Z Z aI

| o
+ +
= + 2.303
2
I
B
(2.1.51)

where, A

is Debye- Huckle coefficient, given by 3A

and
1.0 = ,
1.5
a
Z Z
+
= ,
1 1 1
2 2 2
1 1
3
2 2
3 1
( ) [1 2ln(1 )
( ) [1 ]
I I I
I I
o

= + +
+

2
2 1 2 ln(1 )
( ) [ ]
(1 )
aI aI
aI
aI aI aI

+ +
=
+


2. 1. 15: Electro negativity model of Keyan Li et. al., :
Keyan Li et. al., [119], proposed a model taking electro negativity scale into
consideration. Incorporating the solvent effect into the Born effective radius, they
proposed an electro negativity scale of metal ions in aqueous solution with the most
common oxidation states and hydration coordination numbers in terms of the effective
ionic electrostatic potential. They concluded that the metal ions in aqueous solution are
poorer electron acceptors compared to those in the gas phase. It is interesting to note that
the frame work of D-H Model is retained even in this model. They claim that this
solution-phase electro negativity scale showed its efficiency in predicting some important
properties of metal ions in aqueous solution such as the aqueous acidities of the metal
ions, the stability constants of metal complexes, and the solubility product constants of
the metal hydroxides. They elaborated that the standard reduction potential and the
78

solution-phase electro negativity are two different quantities for describing the processes
of metal ions in aqueous solution to soak up electrons with different final states. This
work provides a new insight into the chemical behaviours of the metal ions in aqueous
solution, indicating a potential application of this electro negativity scale to the design of
solution reactions. It intends to study and apply this model, in the future extension work
with experimental studies using suitable technique.

Fig: 2.1.3 Electro negativity model developed by Keyan Li et, al, (2012).
2.1.16: Lu and Maurer model:
During 1993, Xiaohua Lu and Maurer [120] published a model for the excess
Gibbs energy G
E
of aqueous solutions containing mixed electrolytes. This was extended
[121] from temperatures 298 K to temperatures up to 573 K in 1996. The extension is
achieved by introducing a universal relation for the influence of temperature on some
model parameters. Although parameters are still ion specific, the influence of temperature
on those parameters is universal. No additional parameters are required for describing
aqueous solutions of mixed electrolytes. The model accurately predicts activity
coefficients at high temperatures and at concentrations up to the solubility limit in
electrolyte aqueous solutions containing, K+, Na+, NH
4
+
,(SO
4
)
2-
,Cl- and NO
3
-
.
79

The model is to describe the excess Gibbs energy and related properties like
osmotic coefficient and mean activity coefficients of dissolved electrolytes in aqueous
solution.
It is assumed that dissolving strong electrolytes in water results in a mixture of
water molecules, unsolvated and solvated ions. Solvation equilibria are used to calculate
the true concentrations of solvated and unsolvated ions from the overall concentrations of
the dissolved electrolytes. Physical interactions between all species are taken into account
by combining the Debye- Huckel law with the UNIQUAC model of Abrams and
Prausnitz [122]. The emperical temperature dependence of this model was carefully
tested by the prediction of excess enthalpy, with the Gibbs-Helmholtz relation.This
model describes excess Gibbs energy and related properties. However the deppression in
the dielectric constant of the dielectric continuum of the solvent on the dissolution of the
electrolyte does not figure in the calculations, of Lu and Maurer. Application of
experimental data in this study, for this lowering of dielectric constant to (1) the Bromley
model (2) Lu and Maurer model for evaluation of activity coefficient and osmotic
coefficient, and later evaluation of Gibbs free energy, gave inconsistant results, that
would be discussed in the chapter Results and Discussion. However concrete evidence
of the ion pair formation expected by Eigen and Tamm mechanism, was traced with the
Pitzers model to a greater extent.
The Excess Gibbs energy of the solution was evaluated in this thesis using the
notation of Lu et.al. , in terms of osmotic coefficient and activity coefficient y

.The
relationship can be expressed as:
(1 ln )
E
w
G n mRT |

= +
(2.1.52)
where n
w
is mole number of water and the remaing terms have their usual notation.
The relative enthalpy is related to
E
G

by
2
,
/
[ ]
E
P m
G T
L T
T
c
=
c
(2.1.53)
80

The Apparent Relative Molar Enthalpy that requires experimental data is given by

2
,
ln
[( ) ( ) ]
P m
w
L
L RT
n m T T
|
|
u

c c
= =
c c
(2.1.54)
In all the above equations, the notation of Prigogine and Defay and Xiaohua Lu and
Maurer was used.
2.2. Dielectric Study:
Attention was focussed on Glueckauff s model in this thesis, and evaluated the
dielectric data using this model. An apparatus, using latest operational amplifier circuitry,
was designed and fabricated for the accurate dielectric measurements to obtain the data of
the lowering of dielectric constant of chosen aqueous electrolytic solutions. This
experimental data showed an excellent agreement with the experimental, theoretical
computation of the variation of the dielectric constant using Glueckauff s model. The
author of this thesis used the equations and applied his experimental dielectric data for
the relevant parameters from the several electrolytic models developed by Pitzer,
Bromley, Lu and Maurer, to evaluate chain of thermodynamic parameters like activity
coefficient, osmotic coefficient and excess Gibbs energy. The details of the present set up
are given in the chapter Experimental Techniques. However the frequency
determination was also not reasonably accurate in the dielectric constant determination.
The researchers reported the usage of an electronic network made up of vacuum tube
circuits then and even the frequency determination is better and improved now. The
experimental dielectric , acoustic and thermodynamic data for the short range as well as
long range interactions not only revealed valuable information about ion pair formation,
which is the Noble prize winning hypothesis of Manfred Eigen and Tamm for the year
1967 [123] but also in a position to provide valuable leads, for the dissemination of the
information.


81

2. 2. 1: Glueckauffs Dielectric model, and Lowering of Dielectric Constant of the
Solvent on addition of Solute:
A careful observation of the equations for the , and obtained above, reveals
that a big lapse has occurred in the development of the Debye- Huckel model. It is the
electrostatic influence of the ions of the solute (carrying charge), on the electric field
caused at the central ion, with respect to the electrostatic permittivity of the dielectric
continuum of the solvent surrounding the central ion and incorporating the ionic
atmosphere. Various other attempts for the modification of the D- H model would result
in a half- hearted approach unless we take regard of this main lapse. This was done by
Glueckauff in his model for the depression in the permittivity of the solvent to which
solute ions are added.
Robinson Wright [124], Robinson and Stokes, Falkenhagen, and Harnedand
observed that the internal field around the central ion of the ionic atmosphere created by
the charged solute ions in the solvent of the electrolytic solution tend to reduce the value
of the dielectric constant of the electrolytic solution. This aspect is overlooked and the
departures exhibited in the properties of electrolytic solutions, as the concentration
increases, cannot be satisfactorily explained, by the Debye- Huckel model (adopting the
idea of self-consistent field) even with its several recent modifications proposed by a host
of workers with this objective. It was pointed out by Hasted, Ritson and Collie [125, 126]
that a variation of dielectric constant would have a significant, if not dominating, effect
on the properties of concentrated salt solutions. Further, Sack made a theoretical estimate
of the lowering of dielectric constant that would be expected due to the saturation of the
dielectric in the neighbourhood of an ion. Both these studies are of great theoretical
interest. They made dielectric measurements with more than one method at microwave
frequencies. Wolynes observed that the earlier accurate experimental data on
measurements of electrical conductivity and electric potential, and other related data on
several interesting phenomena became a point of interest and many a time a discrete
controversy. The long range ion- ion interactions could be used to arrive at the limiting
theoretical results that showed an agreement with experiment.

82

Wang and Anderko [127], developed a general model for calculating the dielectric
constant of mixed solvent electrolyte solutions, by the empirical modification of
Kirkwood theory for multi- component systems. A simple relationship between dielectric
constant of mixed solvents with solvent composition and temperature was proposed by
Abolghasem Jouyban et.al., [128].
Wear [129] reported for pure liquids a technique for relative permittivity
determination. The Sargent oscillometer technique is good enough but not compatible for
temperature controlled measurements. Even though this method is a development for the
oscillometer method, its accuracy of measurement of temperature is restricted, mainly
because a temperature bath is used to regulate the temperature by a liquid circulation
technique. Till now the dielectric properties of the ion conducting systems have not been
examined, despite their fundamental importance. Chemists are keen to obtain information
on the structure of solutions which can be provided by permittivity measurements.
Theoreticians hope to get more insight into the dynamic and transport properties of
electrolyte solutions from knowledge of relaxation times and dielectric decrements.
Theoretically [130] the equation for the temperature dependence of relative permittivity
is given as,
e =(e
0
) c
-L1
(2.2.1)
where is the dielectric constant , T is the absolute temperature , and L and
0
are
constants for a given solvent .The theoretical value of has been evaluated using this
equation.
A similar method was [131] used for the water- ethanol system also. The water-
methanol data of Wear recorded discrete departure between theory and experiment .This
is expected, as per the observations of other prominent findings of different [132-134]
techniques, like U-V visible spectro- photometry, ion selective electrode potentiometry,
dielectric relaxation spectroscopy and titration calorimetry.
Von Hippel [135] and others [136, 137] showed that the dielectric constant in the
vicinity of the central ion is modified by the volume fraction of the ions in layer of the
solvent around the central ion. Glueckauff deduced an expression for the lowering of the
83

dielectric constant of the solvent when a solute is added to it, on the basis of
thermodynamic and electrostatic expressions given by Booth [138].
The dependence of the dielectric constant on the field strength in the vicinity of an
ion as per Glueckauff can be given as (following his own notation),
2 2
0
3 1 1
( ) ( )
tanh
D n D n
u u u
= + (2.2.2)
where,
u s | = , and
2
z
s
Dr
c
= , for r <a, and
2
(1 )
[ ]
(1 )
ka
kr
z e kr
s
Dr e ka
c +
=
+
, for r >a
Where r is the distance from the centre of the ion and a is the closest distance
of approach of ions. The solution of these equations with the appropriate relations
substituted for r and a gives D as a function of r, the distance from the centre of
the ion.
Electrolytic ions entering into the solvent, which is a dielectric continuum of the
dielectric constant D, create an internal field around the central ion of the ionic
atmosphere and the average effect of all such ions is to reduce the effective value of D
to

of the solvent of the electrolytic solution, in which

is the required mean value

and

amounts to the lowering in the dielectric constant. The method is indicated in

the fig given below in figure 2.2.1, treating the solvent as a continuously non uniform
system having spatial distribution of the ions. For a volume fraction lying between to
(+ d) ,
where =
3
8
( )
3
Nr t

and
2
2 (4 ) d N r dr | t =

here r is the distance of the point of calculation from the center of the reference ion.
The integration of this within suitable limits yield:
[ / log(1 )] [3 ( )/(2 )] dD d D D D D D
| |
| = +

In this equation D

is the local value of D for the volume fraction between and (+ d
).
84

is the required mean value. This fundamental equation for lowering of D is to be
cumulatively used, layer after layer, by a process of integration to get the final value of
the dielectric constant. The integration of this equation gives the result
= (2.2.3)
This result is relevant to the case of disperse volume fraction of (
2

1
) of
dielectric constant D (=D
1
) which is being added to a volume fraction (1-
2
)
corresponding to a value of D (=D
2
), further if the changes in D are very small i.e.,
is much less than D
2
, to get,
(2.2.4)
The above process of integration should start at a point representing the
continuous phase. The final value then is the required mean value . At each spot the
value of D given by the equation represents the mean value of D over the volume
fraction treated so far .It is in the nature of the equation that the final result will be only
slightly dependent on the exact dielectric constant of the ion itself, if the volume fraction
concerned is very small.

Fig: 2. 2.1 : Basic Principle of the Glueckauffs Calculation of lowering of
r

85


Fig: 2. 2. 2. Ionic atmosphere structure in presence of an external field as per the
conception of Debye and Huckel and Falkenhagen
2. 2. 2: Contribution of the Glueckauffs deppression in the dielectric constant, and
tracing the information about Eigen and Tamms Ion- Pairs:
According to Glueckauffs model, the variation of the concentration of the
electrolyte that is added to the solvent continuously alters the relative permitivity of the
solvent when the solvent is treated as a dielectric continuum. This leads to a continuous
change of the relative permitivity of the medium in which the ions of the electrolyte are
located. Consequently the limiting law with the modified electric data would not be
linear. The corresponding plots, between relevant ordinates chosen, reveal changes in the
86

gradients, significant of the three types of the ion pairs envisaged by Eigen and Tamm
mechanism. A

and B

are the two important locations of the basic D- H equations,

involving the impregnation of the variation of the dielectric constant data given in this
thesis, where information about the Eigen and Tomm mechanism is obtained.
2.3: Eigen and Tamms Ion pair Mechanism Study [ETIPM]:
The mechanism has been established on vigorous experimental data, obtained by
several workers. However the fundamental concept of ion pairs has the origin from the
basic Coulomb interaction. Hence the dielectric continuum is responsible to organize the
ion solvent interactions and the successive stages of the mechanism as envisaged from
the fundamental concepts and the equation developed by Bjerrum. Ions of opposite
charge are naturally attracted to each other by the electrostatic force given by coulombs
law. The most important factor to determine the extent of ion- association is the dielectric
constant of the solvent. The concept has been worked out earlier by Bjerrum, and by
other authors including Fuoss et.al., all of whom failed to realise the findings of
Glueckauff. Eigen and Tamm proposed the Ion Pair Formation Mechanism. Eigen [123]
was awarded Nobel Prize in Chemistry for the year 1967 for the study of fast ionic
reactions along with Leo de Maeyer using nanosecond ultrasound absorption and thermal
conductivity techniques and was the first to bring out the existence of the mechanism on
rigorous experimental basis. Generally ion- pairs are formed when a cation and an anion
come together. Details of the mechanism will be explained in the chapter Results and
Discussions, under the article: 4.2.1.
Presently, three distinct types of ion- pairs on the basis of solvation of the two
ions are recognized. The mechanism can be broadly viewed as a three step process, fig: 2.
3. 1. For the purpose of this presentation, secondary solvation is ignored. When both ions
have a complete primary solvation sphere, the ion-pair may be termed as fully solvated
or solvent- separated. When there is one solvent molecule between cation and anion, the
ion-pair may be termed as solvent- shared. Finally when the ions are in contact with
each other the ion-pair is termed a contact ion- pair. Even in a contact ion- pair, the ions
retain most of their solvation shell. The nature of this solvation shell is generally not
known with any certainty. In aqueous solutions and in other donor solvents, metal cations
87

are surrounded by four and nine solvent molecules in the primary solvation shell, but the
nature of solvation of anions is mostly unknown.
An alternative name for the solvent- separated and solvent- shared ion- pairs
together is an outer- sphere complex and denotes a complex between a solvated metal
cation and an anion. Similarly a contact ion-pair may be termed an inner-sphere
complex. The essential difference between the three types is the closeness with which
the ions approach each other: fully solvated >solvent-shared >contact. With fully
solvated and solvent shared ion- pairs the interaction is primarily electrostatic, but in a
contact ion-pair there will also be some covalent character in the bond between cation
and anion. An ion- triplet may be formed from one cation and two anions, or from one
anion and two cations [139]. Higher aggregates, such as a tetramer may be formed.
Ternary ion-associates involve the association of three species [140]. Another type,
named intrusion ion- pair has also been characterized. Several details are available in
literature about study of the kinetics of ion-pair formation, and its technological
applications. To mention some of them are the dielectric relaxation studies by Buchner
and Hefter, activity coefficient and compressibility studies by Brahmajirao,
thermodynamic studies of alcoholic electrolyte solutions by Gunavardhan Naidu et.al.,
hromatography with non- polar stationary phases by Csaba Horvath, ion pair formation
from Photo dissociation by Devinder Kaur et.al.

Fig: 2. 3. 1. Three stages of ion- pair formation mechanism
88

Buchner [141] used Dielectric Relaxation Spectroscopy (DRS) as a probe, using the
fact that for electrolyte solutions, polarization essentially originates from the orientation
fluctuations of permanent dipoles, from intra molecular polarizability and from ion
motion. It can be investigated either in the time domain or as a function of the frequency
of a harmonic field. In the latter case, the response is conveniently expressed in terms of
the complex total (relative) permittivity. The attractive interactions between cations and
water are dominated by electrostatic forces. Buchner established in his DRS that possible
effects of dissolved ions on the solvent spectrum are
(i) a shift of solvent relaxation time, monitoring the influence of the ions on
structure and dynamics of the bulk solvent;
(ii) a reduction of its dispersion amplitude, S
s
and
(iii) the emergence of a new solvent relaxation.
The effects (ii) and (iii) essentially reflect solvation, i.e., direct ion- solvent
interactions whereas (i) and (ii) are always observed, albeit with considerably differing
magnitude depending on the solvent and the electrolyte [142,143]. So far new solvent
relaxations have only been observed for solutions of large hydrophobic ions in water
[144,145 and 146]. Buchner, Hefter and Wachter established in their studies that in the
case of H- bonding solvents, DRS essentially probe the cooperative rearrangement of the
H- bond system.
Further they found that depending on the relative strength of ion- solvent versus
solvent- solvent interactions and on the structure of the H- bond system, dissolved ions
(i) may be neutral toward the dynamics of the bulk solvent, (ii) act as defects
catalysing dipole reorientation, (iii) even act as shield against the attack of defects for
the solvent molecules in their primary solvation shell.
Chandrika Akhilan [132, 133, 134], presented an explanation viable to the ion
pair formation of Eigen and Tamm in her work of dielectric relaxation on copper
sulphate solutions. She concluded after a detailed investigation that the presence of
solvent- separated ion pairs was detected partially by U-V visible is spectro- photometry.
89

Further, the findings revealed the presence of three ion- pair types: double solvent-
separated, solvent-shared and contact ion pairs. This was confirmed in the present
studies. The central concept in the development of the Debye- Huckel model of ionic
solutions is that of an ionic atmosphere, a time-averaged, spherical haze around a central
ion that has a net charge equal in magnitude but opposite in sign to that of the central ion.
The ionic atmosphere causes the electric potential to decay with distance more sharply
than the Coulomb potential when the charge is scattered around the central ion with the
ionic atmosphere absent. It has several implications and more about the system is getting
revealed. Solutes of electrolytes are non- ideal, more so at relatively low concentrations.
2. 3. 1: Pseudo- Lattice Theory by Bahe and Varela:
Robinson and Stokes included experimental evidence of the existence of a lattice
arrangement in concentrated NaCl solutions and a short description of a Pseudo- Lattice
model in their classical book on electrolytes. Gouy- Chapman theory of inhomogeneous
ionic solutions assumes the electrical double layer in contrast to central concepts in the
Debye- Huckel theory of homogeneous electrolyte solutions. These inhomogeneities in
the distribution of charge in the bulk solution are associated to local electro neutrality and
they provide the structural tools for the understanding of these systems in the dilute
concentration regime.
The result which constitutes the basis of the contemporary pseudo lattice theory of
ionic solutions is due to Bahe [147]. He reported a qualitatively and quantitatively correct
interpretation of the thermodynamic properties of concentrated electrolyte solutions in
terms of the existence of a statistical arrangement of the ions in the bulk, in agreement
with X-ray data induced by long range Coulomb interactions and dielectric constant
gradients. He provided correct deduction of the cube- root law which was known to fit
activity coefficient data at very low molal concentrations taking into account Coulomb
forces and the interactions between the electric fields of the ions and the gradient of
dielectric constant near the surface of ions in solution induced by their electric fields.
Varela et al.,[148] after extensive work on these aspects, adopted the suggestion
of Frank and Thompson, that to make the Poisson Boltzmann distribution viable even at
90

reasonably higher concentrations of electrolytic solutions. Starting with the experimental
evidence of Robinson and Stokes, the Pseudo- Lattice model was developed by Bahe
introducing short- range ion- ion interactions in the Pseudo- Lattice formalism, modelling
them by a hard- core potential with an attractive tail representing Vander Waals
interactions. The resulting theory is capable of describing the thermodynamic and
transport properties of highly concentrated solutions in a satisfactory manner. The Debye
constant corresponding to an ionic cloud model is replaced by the average distance
between ions, in the Pseudo- lattice formalism. The Pseudo- lattice theory of electrolyte
solutions is capable of accounting for the thermodynamic properties of binary and ternary
mixtures of ionic liquids.
As the concentration of the electrolyte solution increases, the average distance
between ions decreases, so the short-range interactions between them become
progressively more important with respect to their Coulomb counterparts. Assuming a
Restricted Primitive Model (RPM) description, all the ions were treated as having the
same hard-core radius,
r
m
=(r
+
+r
-
)/ 2 (2. 3. 1)
where r+ and r
-
are the radii of cations and anions, respectively.
Starting from the energy of interaction between ions i and j separated by a
distance, the attractive tail of the potential was evaluated. Attractive dispersive London
forces and Debye and Keesom dipole- dipole interaction forces were included in this.
From the theoretical perspective, one expects that a pseudo- lattice model is particularly
well adapted to the peculiarities of ionic liquids. Bahe-Varela pseudo-lattice theory of
electrolyte solutions is capable of accounting for the thermodynamic properties of binary
and ternary mixtures of ionic liquids. The free energy of the system could be estimated
and Bahes approach completed by Varela et al., concerning structured electrolyte
solutions with short- range interactions was successfully applied by Bou Malham, et.al,
[149], to understand volumetric properties of several aqueous and non-aqueous mixtures
of ionic liquids. Consequently, the pseudo-lattice theory as for the first time proved to be
valid for mixtures of conventional molecular solvent.
91

The surface tension of the pseudo- lattice ionic systems could be calculated
adapting the treatment of Shuttle Worth [150]. This surface area can be related to the
pseudo-lattice cell size. It may be assumed to start with that in pure ionic liquids ion-
isolvent interactions are absent and that the short- range repulsive term is expected to be
dominant due to the great number of ionic contacts. It reduces to the simple relation that
the surface tension is directly proportional to the fourth power of density i.e.,
4
. This
result was verified for ionic liquids by Deetlefs et. al., [151]. In this theory, the ions were
assumed to move randomly between the cells in a statistical lattice representing the
cooperatively rearranging regions in the liquid.
Classical hopping over energy barriers in between adjacent cells are the main
mechanism by which the charge transport takes place. On average, only two different
types of ionic environments of an ion were supposed to occur: i) -cells with no ions in
nearest neighbour cells, on which an ion experiences an energy barrier , and ii) -
cells associated to ions in its neighbourhood, with a well depth . The first ones are
mainly associated to solvent molecules in the lattice or even to low density regions in the
bulk (hole-like regions), and the latter to other ions.
2. 4: Ultrasonic, densimetry and Refractive Index studies:
The study of propagation behaviour of ultrasonic waves in liquid system and
solids is now rather well established as an effective means for examining certain physical
properties of the materials. It is particularly well adapted for examining changes in
physical properties while they occur. The data obtained from ultrasonic propagation
parameters viz., ultrasonic velocity and attenuation and its variation with concentration of
one of the components helps to understand the nature of molecular interaction in terms of
these physical parameters. Owing to the sensitivity to very low population densities at
high energy states, ultrasonic methods have been preferred and are reported to be the
complementary to the other techniques like dielectric relaxation, infrared spectroscopy,
nuclear magnetic resonance etc. Several empirical and semi empirical formulae have
been developed correlating velocity and density with very useful thermodynamic
parameters.
92

2. 4. 1: Flickering Cluster model of liquid water:
According to this model, the water is supposed to consist of hydrogen-bonded
clusters and unbonded water molecules. The molecules in the interior of the clusters are
quadruplet bonded (ice like) and unbounded water molecules are supposed to occupy the
space in between the clusters. The clusters are sometimes referred as open structure'
water and the dense mono- meric fluid is referred to as 'closed structure water. The
mixture is a dynamic mixture and the breakdown of clusters is a cooperative process.
When one hydrogen bond breaks in the cluster, the whole cluster breaks down resulting
of an increase in the close packed structure of water. The flickering cluster model of
liquid water is given in figure below. When the organic and inorganic solutes are
dissolved in water, it splits them into positive and negative ions, whose net structure
breaking property would first disturb the open structure of water thereby increasing free
monomer population, an action that would be followed by structural reorganization
leaving the molecules in closely fitted helical cavities. Such an increase in the closed
packed structure of water results in increased cohesion of water molecules leading to an
increase in the ultrasonic velocity and viceversa. Rajarajan, in his thesis [152], discussed
in detail about the application of this model to paramagnetic substances.

Fig: 2. 3. 2. Flickering Cluster model of liquid water


93

2. 4. 2: Ultrasonic velocity:
Ultrasonic velocity and related data of liquid mixtures are found to be the most
powerful tool in testing the theories of liquid state due to its ability of characterizing
physiochemical behaviour of the medium or interacting components. The ion- dipole
attractions vary directly as the size of the ion, charge and magnitude of the dipole.
The displacement () and the velocity(c) of the waves are related by the wave equation,
2 2
2 2 2
1
x c t
c c
=
c c
(2.4.1)
assuming that the compressions and rarefactions are both reversible and adiabatic.
This assumes the form
2 2
2 2 s
x t

|
c c
=
c c
(2.4.2)
In terms of adiabatic or isentropic compressibility s, and density of the medium ,
giving the primary relation
2
1
s
c
|
=
(2.4.3)

however when the strain generated by the applied sound wave during the process of
propagation is compression. a negative sign gets added, to get
2
1
s
c
|
=
(2.4.4)

hence the numerical value of the s is chosen in calculation.
The variation of ultrasonic velocity in a solution depends upon the increase or
decrease of intermolecular free length after mixing the component.


94

2. 4. 3: Adiabatic compressibility and Apparent molal compressibility :
It is fractional decrease of volume per unit increase of pressure, when no heat
flows in or out and again it is inversely proportional to square of the ultrasonic velocity.
The compressibility data indicate an ordering interaction leading to the formation of ionic
complex. The deviation in this data can be explained by considering loss of dipolar
association , difference in size and shape of the component molecules which leads to
decrease in velocity and increase in compressibility. Dipole- dipole interaction between
unlike molecules leads to increase in velocity and decrease of compressibility.
Adiabatic compressibility of a solution was calculated using the relation,

s
=1/c
2
(2.4.5)
In general the apparent molar compressibility is given by the equation
0
0
0
0
1000
[ ] [ ]
k o
M
m
|
| | |

= +
(2.4.6)
in which ,
0
are the adiabatic compressibilities and ,
0
, are densities of the
solution and solvent , respectively.
2. 4. 4: Jacobsons Inter-molecular Free Length Theory:
It is the distance between the surfaces of the neighbouring molecules. It indicates
significant interaction between solute and solvent molecules due to which the structural
arrangements in the neighbourhood of the constituent ions is considerably affected.
The Lewi Tonks equation of state [153] for liquid is given by
a g
PV R T =

(2.4.7)
where R
g
is universal constant, V
a
is the available volume which is the difference
between the molar volume V and the geometric closest packing of the molecules V
o
, i.e.,
V
a
+V
O =
V (2.4.8)
95

The ultrasonic velocity in a liquid is given by the relation
2
P
C

c
=
c
(2.4.9)
where P is the change in pressure due to the sound wave and is the corresponding
change in density of the medium under adiabatic conditions. From these equations Kittel
[154] showed that the ultrasonic velocity (c) in a liquid is given by
1
2
3
[ ]
g
a liq
R T
V
C
V MV
=

(2.4.10)
in which
liq
V is the ratio of specific heats at constant pressure to constant volume.
Equation (2.4.10) indicates a linear relation between Va and
liq
V Jacobson [155]
suggested that the adiabatic compressibility of a liquid can be understood in terms of the
intermolecular free length, L
f
.
L
f
is related to the available volume Va and the surface area per molecule Y through the
relation
2
a
f
V
L
Y
=

(2.4.11)
where,
1
2
9
0
[36 ] Y NnV =

N is Avagadro Number and V
0
is the volume at a temperature of absolute zero. The
intermolecular free length depends on the type of packing and the extent of association in
a given liquid.
From the study of adiabatic compressibility and L
f
in the case of non associated
organic liquids, Jacobson has shown that the plot of log
s
Vs log L
o
is a linear one
suggesting a relation of the type,
s f
KL
_
| =

(2.4.12)
96

where K and x are constants and the values of the constants X and log K are 2.082
and 7.097 C.G.S units at 20Cin the case of non - associated liquids. The free length L
f

calculated using the relation (2.4.11) agrees with the free length calculated using the
relation (2.4.12) and the maximum deviation being approximately 3% and this is
attributed to uncertainties of the parameters in (2.4.11).
The computation of the free length for mixtures of liquids from molar volumes
has been extended by Jacobson by revising the definition L
f
= [2 Va / Y] , as
2[ ( )]
( )
A B
M A O B O
f
A A B B
V X V X V
L
X Y X Y
+
=
+

(2.4.13)
in which
A
X and
B
X
A
O
V and
B
O
V represent the mole fractions and volumes at absolute zero
of liquid components A and B ,and
M
V is the molar volume of the mixture,
A
Y and
B
Y
represent the surface areas of each of the molecules belonging to the component.
2. 4. 5: Relative Association:
Relative association is a parameter used to assess the association in any solution
relative to the association existing in water or solvent at 0C. It is estimated using the
following relation.
1
0 3
0
. ( )[ ]
C
R A
C

=
(2.4.14)

Where and C are the density and ultrasonic velocity of the solution at any
temperature, and

0
, and C
0
, are density and ultrasonic velocity of water (solvent) at OC.
The relative association is influenced by the breaking up of the solvent molecules on
addition of solute to it and subsequent salvation of ions by the free solvent molecules.
The former effect results a decrease while the latter increase the values of relative
association.
97

2. 4. 6: Acoustic impedance:
Several studies on the intermolecular interactions in the liquid mixtures are based
on acoustic impedance data.
The specific characteristic impedance or acoustic impedance, Z is given as the
ratio of the acoustic pressure p in the media to the associated particle speed as
z
C

=
(2.4.15)
And for plane waves this ratio becomes,
z
C

= (2.4.16)
In terms of Ultrasonic velocity and density using the earlier discussed relations to get
Z = U (2.4.17)
With c being the sound speed of a medium, and with the sign dependent upon the
direction of the propagation. The acoustic impedance can be a complex or a real quantity
dependent upon the visco- elastic properties of the media. For plane waves, reflection
coefficient R can be related to the acoustic impedance of the propagation media as
1 2 2 2 1 1
1 2 2 2 1 1
z z C C
R
z z C C



= =
+ +
(2.4.18)
This is a very useful relation, since it relates all important fundamental parameters.
2. 4. 7: Surface Tension:
The property of a liquid on account of which it tends to keep minimum number of
molecules in its free surface is defined as surface tension. The property of surface tension
is caused by cohesion of similar molecules, and is responsible for many of the behaviours
of liquids. This parameter is used to study the surface composition of aqueous solution of
the mixtures. The surface tension of a liquid mixture is not a simple function of the
surface tension of the pure liquids. At the interface, there is migration of the species
98

having the lowest surface tension, or free energy per unit area, at the temperature of the
system. This migration results in liquid phase and vapour phase rich in the components
with highest and lowest surface tensions respectively. A variation of surface tension
shows the attractive interactions between the two solutions.
Surface tension can be calculated in terms of ultrasonic velocity and density using
the relation,
= (6.310
-4
) c
3/2
(2.4.19)
2. 4. 8: Density:
The density of a solution is the sum of mass concentrations of the components of
that solution.
Mass concentration of each given component
i
in a solution sums to density of
the solution.

Expressed as a function of the densities of pure components of the mixture and
their volume participation, it reads:

Density is the measure of solvent-solvent and ion-solvent interactions.
2. 4. 9: Apparent molal volume:
The apparent and partial molal volumes of electrolyte solutions have proved a
very important tool in elucidating the structural interactions i.e. ion-ion, ion-solvent and
solute-solvent interactions occurring in solution.
If the concentration of solute is expressed as the number of moles of solute per
1000 grams of solvent, then apparent molal volume can be expressed as
[1000(
0
-)/m
0
] + M/
0
(2.4.20)
99

where
0
is density of solvent, is density of solution , m is the molality of solution and
M is molecular weight of the solute.
2. 4. 10: Refractive index:
The knowledge of refractive index property at different temperatures of liquid
mixtures is an important step for their structure characterization. Along with other
thermodynamic data, refractive index values are also useful for practical purposes in
engineering calculations. Refractive index is useful to assess purity of substances, to
calculate the molecular electronic polarizability, to estimate the boiling point with
Meissner method or to estimate other properties such as viscosity and other
thermodynamic properties. The studied systems have industrial utility. The aqueous
organic type solvents are frequently used as chemical and biochemical reaction media.
The refractive index data can be used to study mixing rules of Lorentz-Lorenz, Wiener,
Heller, Gladstone- Dale, Arago- Biot and Edwards to predict the refractive index of
binary and ternary mixtures using refractive indices of pure components.


100

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