Real Solutions
Valentim M. B. Nunes
ESTT - IPT
April 2015
For liquid solutions is advantageous to describe the solutions as
ideal, and deviations from the ideality through excess functions.
To keep the formalism used for ideal solutions, we define the
chemical potential of a component 1 in a real solution as follows:
1   (l )  RT ln a1
*
1
were a1 is the activity of component 1 (ideal solutions: a1 = x1).
p1
a1   1 x1  sat
p1
The excess functions are thermodynamic properties of solutions
that are in excess relative to an ideal solution or ideally diluted, in
the same conditions of temperature, pressure and composition.
G  Greal(T , p, x )  Gideal(T , p, x )
E
Relationship between thermodynamic properties are preserved :
H  U  pV
E
E
G  H  TS
E
E
E
E
The partial derivatives of extensive properties are also
maintained.
 G E

 T


E
   S
 p,x
  G /T

 T
E
 G E

 p
 

 p,x
E
H
 2
T

E
  V
T , x
The partial molar excess functions are defined similarly. If M is an
extensive thermodynamic property, then the corresponding
partial molar property, mi comes:
 M
mi  
 ni
mi
E


 p ,T ,n j
 M E
 
 ni


 p ,T ,n j
By the Euler theorem:
M   ni mi
E
i
E
The activity coefficient of component i is given by:
ai
i 
xi
At constant T and p:gi real  gi ideal   RT ln fi real  ln fi ideal 
E
g i  RT ln
gi
E
f i real 
f i ideal 
ai
 RT ln  RT ln  i
xi
Fundamental results
E
g i  RT ln  i
Since,
G   ni g i  G   ni RT ln  i
E
E
E
i
then:
i
G  RT  xi ln  i
E
m
i
Symmetric convention:
 i  1 when xi  1
Anti-symmetric convention:
 1  1 when x1  1 (solvent)
 2*  1 when x2  0 (solute)
Deviations to ideality:
At low pressures:
p2
p2
*
2 
; 2 
sat
x2 p2
x2 H 2,1
Calculation of i
Binary mixtures
At moderate or low pressures (away from the critical point) the
effect of pressure in GE is low.
It is necessary to correlate GE with the composition of the
mixture. The simplest expression is:
G  Ax1 x2
E
m
A =A(T)  Empirical constant with units of energy, depending on
the temperature, but independent of the composition.
Two suffix Margules equations:
 nGmE
RT ln  i  
 ni


 p ,T ,n j
A 2
A 2
ln  1 
x2 e ln  2 
x1
RT
RT
Simple Mixtures
Similar molecules in size
Shape
Chemical structure
Variation of activity coefficients:
ln 
A/RT
0.5
x1
Redlich-Kister expansion:
Margules's equation is very simple. Generally it is required a
more complex equation to represent adequately the excess Gibbs
energy.


G  x1 x2 A  Bx1  x2   Cx1  x2   Dx1  x2   ....
E
m
2
RT ln  1  a (1) x22  b (1) x23  c (1) x24  ...
RT ln  2  a ( 2 ) x12  b ( 2 ) x13  c ( 2 ) x14  ...
a (1)  A  3B  5C  7 D ; a ( 2 )  A  3B  5C  7 D
b (1)  4B  4C  9 D  ; b ( 2 )  4( B  4C  9d )
c (1)  12C  5 D 
; c ( 2 )  12C  5 D 
d (1)  32 D
; d ( 2 )  32 D
3
The number of parameters needed to represent the activity
coefficients gives an indication of the complexity of the solution.
If the number of parameters is large (4 or more) the solution is
complex. If you only need one parameter the solution is simple.
Most frequently solutions used in chemical engineering require
two or three parameters in the expansion of Redlich-Kister.
For simple solutions, B = C = D =0
1
RT ln
 A  2 Ax1
2
ln(1/2)
0
x1
At constant p and T,
 x dm
i
i
0 e
i
 x dm
i
E
i
0
i
E
g i  RT ln  i
 x d ln 
i
i
i
0
Gibbs-Duhem
equation applied to i
Binary mixture
x1d ln  1  x2 d ln  2  0
x
d ln  2   1 d ln  1
x2
 1  x2 
d ln  1
d ln  2  
 x2 
 1  x2 
d ln  1
ln  2    
x2 
0
x1
The expansion of Wohl is a general method to express the excess
Gibbs energy in terms of parameters with physical meaning.
GmE
 2a12 z1 z2  3a112 z12 z2  3a122 z1 z22  4a1112z13 z2 
RT x1q1  x2 q2 
4a1222z1 z23  6a1122z12 z22  ...
were,
x1q1
x2 q 2
z1 
; z2 
x1q1  x2 q2
x1q1  x2 q2
Parameters q – Represents actual volumes, or sections of
molecules and measures the "sphere of influence" of molecules in
solution. Larger molecules have higher values of q's. In solutions
containing non-polar molecules:
q1 Vm,1

q2 Vm, 2
Parameters a – Represent interaction parameters. a12 represents a
1-2 interaction; a112, interaction between 3 molecules, etc.
van Laar equation
It is an application of the previous expansion. For binary solutions
of two components, not very chemically dissimilar and with
different molecular sizes (e.g. benzene and isooctane) the
coefficients a112, a122, etc., can be ignored:
E
m
G
2a12 x1 x2 q1q2

RT
x1q1  x2 q2
From the van Laar equation we obtain :
ln  1 
ln  2 
A
 A x1 
1  B x 
2

B
2
 B  x2 
1  A x 
1

were A’ = 2q1a12 and B’ = 2q2a12
2
This equation is useful for more complex mixtures. If A’ = B‘ then
we obtain the Margules's equation.
 x2 ln  2 

A'  ln  1 1 
x1 ln  1 

2

x1 ln  1 

B '  ln  2 1 
 x2 ln  2 
2
NRTL – “non-random two liquid”
UNIQUAC – “Universal quasi-chemical theory”
UNIFAC – “Universal quasi-chemical functional group activity coefficients”
VLE calculations (Bubble point, dew point and Flash)
Azeotropy
Disregarding the non-ideality in the vapor phase:

az
1
p
 sat
p1
Until now we admitted total liquid phase miscibility. Let's now
consider the cases where the liquids are only partially miscible.
At constant p and T, the condition for stability is equivalent to a
minimum Gibbs energy.
It occurs partial miscibility when,
  2Gmist 

  0
2
 x T , p
x  x1 ou x2
G E G mist  RT ( x1 ln x1  x2 ln x2 )
  2G E

2
 x1

1 1
  RT     0
T , p
 x1 x2 
For a ideal solution, GE = 0, then we never have phase separation!
Considering GE = Ax1x2
  2G E

2

x
 1

  2 A
T , p
1 1
 2 A   RT   
 x1 x2 
RT
2A 
x1 x2
The lower value that satisfies the inequality is A = 2RT
A
2
RT
The separation between stability and instability of a liquid mixture
is called incipient instability. The condition of instability is
dependent on non-ideality and temperature.
From previous equation, the critical temperature of (solution)
solubility comes:
A
T 
2R
c
By the Margules's equation Tc is always a maximum!
Theories of solutions
When two or more liquids are mixed to form a liquid solution the
purpose of theories of solutions is to express the properties of
liquid mixture in terms of Intermolecular forces and liquid
structure.
It is intended to predict the values of the coefficients of activity in
terms of meaningful, molecular properties calculated from the
properties of pure components.
Precursors: van der Waals  van Laar.
Scatchard – Hildebrand Theory
Van Laar recognized correctly that simple theories could be built if
we considered cases in which VE and SE ~ 0
Later, Hildebrand has verified experimentally that many solutions
were in agreement with those assumptions, designated regular
solutions.
It is defined cohesive energy density as follows :
U
c
Vm , L
vap
Uvap – energy of complete vaporization of saturated liquid to the ideal gas
state (infinite volume)
It is also defined, for a binary mixture, the volume fraction of 1
and 2:
x1V1
x2V2
1 
; 2 
x1V1  x2V2
x1V1  x2V2
For UE we obtain:
U  c11  c22  2c12 1 2 x1V1  x2V2 
E
For molecules for which the dominant forces are London
dispersion forces:
c12  c11c22 
1
2
Introducing the concept of solubility parameter, :
1
1  c112
1
 2  c222
We obtain:
 U
 
 Vm, L
vap




1
 U vap 

 

V
 m, L 
2
1
2
U  x1V1  x2V2 1 2 1   2 
E
2
if SE = 0, then we obtain the regular solutions equation:
RT ln  1  V1 1   2 
2
2
2
RT ln  2  V2  1   2 
2
1
2
1 e 2 are function of temperature, but 1 - 2 it’s almost
temperature independent!
The difference between the solubility parameters of a mixture
gives a measure of the non-ideality of the solution.