Non-Ideal Solutions
This development is patterned after that found in Molecular
Themodynamics by D. A. McQuarrie and John D. Simon. Consider a
molecular model of a regular binary solution in which similarly sized solute
and solvent molecules are randomly distributed throughout the solution. Let
the potential energy of this solution be given by the expression:
E = N11 e11 + N22 e22 + N12 e12
Here N12 is the number of neighboring pairs of solvent and solute molecules
and e12 is the energy of interaction between a solute and solvent molecule
and is negative (why?).
What do N11 and e11 represent?
Since the molecules are randomly distributed , the probability that two
solvent molecules are neighbors is given by:
(N1 - 1) / (N1 + N2 - 1) ~ N1 / (N1 + N2) = X1
or the mole fraction of solvent molecules in the solution. If z is the
coordination number, i.e., the number of nearest neighbors for a given
molecule, then on average a given solvent molecule will have:
X1 z
solvent molecules as nearest neighbors and, since there are N1 solvent
molecules in the solution, the number of solvent - solvent neighboring pairs
will be:
N11 = N1 X1 z / 2
Dividing by two prevents us from counting each solvent - solvent
interaction twice.
42C.1
The total energy of interaction between all the solute and solvent molecules is
thus:
E = (N1 z X1 / 2 ) e11 + (N2 z X2 / 2 ) e22 + (N2 z X1) e12
Could you justify the terms in this expression?
Here the coordination number has been assumed to be the same for both
solute and solvent molecules, since we have assumed the solvent and solute
molecules to be of similar size. Using the definitions of the mole fractions
we can write:
E = [ (N12 z / 2 ) e11 + (N22 z / 2 ) e22 + (N1 N2 z) e12 ] / (N1 + N2)
We now define w as a measure of non-ideality as:
w = 2 e12 - e11 - e22
What is w for an ideal solution?
Using w to eliminate e12 from the expression for E gives:
E = (N1 z / 2 ) e11 + (N2 z / 2 ) e22 + N1 N2 z / [2 (N1 + N2)] w
Could you derive this result?
Since w = 0 for an ideal solution, the 1st two terms represent the contribution
to the total energy of interaction for an ideal solution and the last term
represents the contribution from non-ideality:
E = Eideal + ( N1 N2 z / [2 (N1 + N2)] ) w
Sustituting into the definition of the Gibb’s free energy we have:
G = H-TS = E+PV-TS
= Eideal + P V - T S + ( N1 N2 z / [2 (N1 + N2)] ) w
42C.2
Since the solute and solvent molecules were assumed to be
randomly distributed and of similar size, V and S are the same as for
an ideal solution and we can write:
G = Gideal + ( N1 N2 z / [2 (N1 + N2)] ) w
We can convert the numbers of molecules in this expression to
moles by dividing by avogadros number No:
G = Gideal + ( [n1 n2 z No / [2 (n1 + n2)] ) w
The chemical potential of the solvent is defined as:
u1 = ( G /  n1 )T, P, n2 = ( Gideal /  n1 )T, P, n2
+ [ n2 / (n1 + n2) - n1 n2 / (n1 + n2)2 ] z No w / 2
= u1, ideal + ( X2 - X1 X2 ) z No w / 2
= u1, ideal + X22 z No w / 2
Can you justify these last few steps?
Using the definition of the chemical potential for an ideal solution:
u1, ideal = u1pure + R T ln X1
we can write for the chemical potential for our non-ideal solution:
u1 = u1pure + R T ln X1 + X22 z No w / 2
= u1pure + R T ln ( X1 e + X22
Can you justify these steps?
42C.3
z No w / ( 2 R T )
)
Comparing this expression with the expression for the chemical potential
of the solvent in a non-ideal solution derived in the notes on Partial Molar
Variables, Chemical Potential, Fugacities, Activities, and Standard States:
u1 = u1pure + R T ln a1
identifies the activity of the solvent in this non-ideal solution as:
o
a1 = f1 / f1 = f1 / f1pure
= X1 e + X2
2 zN w/(2RT)
o
= X1 e + e X2
2/(RT)
where e = z No w / 2.
Could you reproduce these derivations for the solute or for a multicomponent solution?
This equation will predict how the fugacity of the solvent above a non-ideal
solution will vary with mole fraction.
Under what conditions will this equation reproduce Raoult’s law?
Fugacity of Solute in a Non-Ideal Solution
1.00
f2 / f2 (pure)
0.80
postive deviation from
Raoult's law, w > 0
0.60
0.40
0.20
0.00
0.00
Raoult's law, w = 0
0.20
0.40
0.60
m ole fraction solute, X2
42C.4
0.80
1.00
What does a positive deviation from Raoult’s law imply about the relative
interaction energies, e11, e22, and e12?
How would a curve representing a negative deviation from Raoult’s law
appear on this plot?
The molar Gibb’s free energy of mixing to form a non-ideal solution is given
by:
DGmix = RT [ X1 ln a1 + X2 ln a2 ]
= R T [ X1 ln g1 X1 + X2 ln g2 X2 ]
= R T [ X1 ln X1 + X2 ln X2 + X1 ln g1 + X2 ln g2 ]
gi is the activity coefficient of species i, which in an ideal solution is equal to
one, hence:
DGmix, ideal = R T [ X1 ln X1 + X2 ln X2 ]
The excess Gibb’s free energy of mixing per mole of solution is defined as
the difference between the Gibb’s free energy of mixing of a non-ideal
solution and that of an ideal solution:
DGmix, excess = DGmix - DG mix, ideal = R T [ X1 ln g1 + X2 ln g2 ]
Using the result we just derived
a1 =
g1 X1
= X1 e + e X22 / ( R T )
we can write for our regular solution:
DGmix, excess =
e [ X1 X22
+ X2 X12 ] =
Can you justify this derivation?
42C.5
e X1 X2
In our regular solution in which the solute and solvent molecules are
randomly distributed we have assumed that the entropy of mixing to form and
ideal solution is the same as the entropy of mixing to form a non-ideal
solution. Derive an expression for the excess enthalpy of mixing, DHmix, excess.
What is DHmix, excess if the solution is ideal?
It is instructive to plot:
DGmix / e = ( DG mix, ideal + DG mix, excess ) / e
= ( R T / e ) [ X1 ln X1 + X2 ln X2 ] + X1 X2
versus the mole fraction of one of the components, say X2 for different values
of (R T / e):
Note that for either a temperature low enough or a deviation of
interaction energies from idealilty high enough, DGmix shows two minima
indicating that the solution has separated into two phases.
42C.6
Plot R T / e versus the mole fractions of the solute corresponding to
the minima in the plots of DGmix / e versus mole fraction of solute
to give the liquid - liquid phase diagram shown below for a regular
solution:
Note that for a given solute and solvent in a non-ideal regular
solution there will be an upper consulate temperature above which
thermal motion will prevent phase separation.
42C.7