1. Gibbs-Duhem/Partial Molar Properties

2. Gibbs-Duhem Equation Useful for defining equilibrium concepts
Integrate from state 1 to state 2
State 1: no mass
(U,S,V,N)=0
State 2: some mass
(U,S,V,N)<>0

3. Define U

4. Gibbs-Duhem
Plug it in!
When we are at equilbrium

5. Partial Molar Properties
Consider equilibrium
M(T,P,ni)
M is any property (V, S, G, etc..)
Partial molar property

6. Notation and Definition***
Look at derivative for definition
M – extensive property
m – intensive property
To evaluate, just take the derivative

7. Terminology****
Example: mix 50 mL of water with 50 mL of ethanol, what happens? -
Vi –
V –

8. Derive contribution of partial molar properties

9. Derive contribution of partial molar properties

10. Simplifications System size can be anything
System size can change by any amount
Independent and arbitrary
Both “brackets” are 0

11. Derive contribution of partial molar properties
Gibbs-Duhem Eq

12. Derive for special case of binary system (from SVNA)
Gibbs-Duhem Eq

13. Derive for special case of binary system (from SVNA)

14. Derive for special case of binary system (from SVNA)

15. Derive for special case of binary system (from Koretsky)
Gibbs-Duhem Eq

16. Analytical Analysis
From virial EOS, determine v1, v2, v, vIS, Δvmix, , and
T=333 K
P = 10 bar
B11=-910 cc/mol
B22=-1330 cc/mol
B12=-2005 cc/mol
y1=0.2

17. Analytical Analysis For v, just plug it in
For v1 and v2, what should you do?

18. Analytical Analysis
For v, just plug it in

19. Analytical analysis
Ideal solution -

20. Analytical Analysis
Binary system

21. Alternative Analytical Analysis
If this WASN’T a binary system

22. Summary Gibbs-Duhem Defined partial molar properties
Developed formula for binary mixtures
Analytical solution
Other than partial molar properties, plug and/or chug
Remember formula for ideal solution

23. Summary For binary partial molar properties
Convert all y2y1 (y2=1-y1)
Take derivative
Plug derivate and mixture property into formula
simplify

24. Summary Partial molar properties (general)
Convert from intensive to extensive
Multiply by total moles
Convert mole fractions to ni
Take derivative
Remember nT is a function of n1
n2 is NOT a function of n1
Simplify