Solution Thermodynamics
· This week we will be principally reviewing the interrelationships present between the various properties we use to describe mixtures
· Towards the end of class there should also be time to review some solutions to problems from the textbook
1. Fugacity and Partial Molar Properties
We given definitions for both component and mixture fugacity:
If we integrate both from the ideal-gas state (where 
and 
) to the real-gas state:
If we take the partial molar property of this latter equation we get:
There are two possible rearrangements of this:
Or:
From the first rearrangement we find that:
From the second rearrangement we find that:
If these quantities are partial molar properties, it thus follows that:
And that:
2. Shape of Fugacity vs Composition Curve
One example of the significance of the Gibbs-Duhem equation is the insight it gives us as to the shape of the fugacity curve:
For a binary mixture (
), we get:
Or:
If we take the limit as x
1® 1, x2® 0 we find:
Or:
Thus the fugacity vs. composition curve is tangent to the Lewis-Randall line (
) as x
3. Dependence of f and f on T and P
We begin with the definition of fugacity:
and integrate from the ideal-gas state to real-gas state:
But by the Gibbs-Helmholtz Equation:
Thus:
For the pressure dependence, we again use the definition of fugacity:
Giving:
Or:
Thus:
· These represent the fundamental property relations for f and f
·
Unfortunately, we're generally more interested in 
and 
than we are in f and f
4. Dependence of 
and 
on T and P
Consider a constant-composition mixture:
We take the partial molar property of each term:
But 
, and 
is constant:
Similarly,
Leads to:
Or:
5. Property Change of Mixing
Is defined as:
where
M = property of mixture (at some T and P)
= property of pure i (in its standard state)
Notes
·
Since 
·
This suggests that 
. Can verify:
Thus we can write for the various properties:
But we recall the following relationships:
Thus:
6. Activity
Obviously, the quantity (
) is important. We can evaluate it using the definition of fugacity:
We now define the activity as this fugacity ratio:
We can once again re-write the expressions for the property changes of mixing:
7. Ideal Solution
We now define the concept of an ideal solution:
For an ideal solution we have:
8. Excess Properties
Mixture properties are usually evaluated by reference to the property of the ideal mixture:
where M = the actual property of mixture
M
id = the property of an ideal mixtureM
E, M and Mid are all at the same T,P & xNote that M
id, and thus ME, depends on choice ofstandard states
The quantity "M" could refer to any intensive thermodynamic property, including a property change of mixing:
Thus we conclude that the excess property change of mixing and the excess property are exactly the same quantity:
However, D M and D M
id are easier to evaluate in most cases, thus in practice we use the relation: 
Recall that D M
id=0 for U, H, V, CP, and CV, thus:
Thus we can write:
Or:
Similarly, we write for the other excess properties:
9. Activity Coefficient
Clearly the ratio of the activity to the mole fraction is an important quantity; we thus define it as the activity coefficient:
Of course, each of the excess property expressions can now be re-written in terms of the activity coefficient:
Of these excess properties, we are particularly interested in the excess Gibbs energy. Let's find the partial molar property corresponding to it:
But, using the definition of component fugacity:
Thus:
Or:
We see that 
is the partial molar property of G
10. Excess Gibbs Energy and Fugacity
As we have seen,
Since the first term is of the form 
:
We find that:
And since:
11. Activity as a Partial Molar Property
Among our relations for excess properties we saw:
Thus:
But:
So:
12. Gibbs-Duhem Equations for D G and G
EWe can now write a fundamental property relation for D G:
Or:
And the corresponding Gibbs-Duhem Equation is:
In similar fashion, a general property relation for G
E is: 
And the corresponding Gibbs-Duhem Equation is:
13. Dependence of Activity Coefficient on T & P
Note that these last expressions give, in the style of the Maxwell Relations:
And:
Main Points Reviewed This Week
1. Fugacity and Partial Molar Properties
2. Shape of Fugacity vs Composition Curve
3. Dependence of f and f on T and P
4. Dependence of 
and 
on T and P
5. Property Change of Mixing
6. Activity
7. Ideal Solution
8. Excess Properties
9. Activity Coefficient
10. Excess Gibbs Energy and Fugacity
11. Activity as a Partial Molar Property
12. Gibbs-Duhem Equations for D G and G
E13. Dependence of Activity Coefficient on T & P