1. SOLUTION THERMODYNAMICS:
ERT 206: Thermodynamics
Miss Anis Atikah Ahmad
Tel:
anis

2. OUTLINE Fundamental Property Relation
Chemical Potential & Phase Equilibra
Partial Properties
The Ideal-Gas Mixture Model
Fugacity & Fugacity Coefficient: Pure Species
Fugacity & Fugacity Coefficient: Species in Solution
Generalized Correlations for Fugacity Coefficient

3. 1. Fundamental Property Relation
Total Gibbs energy change of any CLOSED system:
*applied to a single phase fluid in a closed system wherein no
chemical reactions occurs & the composition is constant.
Therefore;
No of moles of all chemical species are held constant.

4. 1. Fundamental Property Relation
For single phase, OPEN system (material pass into & out of the system),
*where ni is the no of moles of species i.
* nj indicates that all mole no except the ith are held constant.
Defined as chemical potential, μi

5. 1. Fundamental Property Relation
Since , ,
total change of Gibbs energy for OPEN system becomes:
This equation is a fundamental property relation for SINGLE-PHASE FLUID SYSTEM of variable mass & composition

6. 1. Fundamental Property Relation
For special case of one mol of solution, n=1 & ni = xi ,
becomes
G as a function of T, P and xi
At constant T & x:
At constant P & x:
At constant P & x:
from

7. 2. The Chemical Potential & Phase Equilibra
For a CLOSED system consisting of TWO phases in equilibrium, each individual phase is OPEN to other (mass transfer between phases may occur).
phase
Presumption: At equilibrium, T & P are the same in all phases.

8. 2. The Chemical Potential & Phase Equilibra
The change in total Gibbs energy of the two-phase system is the sum of these equations: +
Reduced to this form:

9. 2. The Chemical Potential & Phase Equilibra
At equilibrium,
Therefore,
The changes of &
result from mass transfer btween the phases.

10. 2. The Chemical Potential & Phase Equilibra
Since
Therefore, or
Multiple phases at the same T & P are in equilibrium when the chemical potential of each species is the SAME in all phases.

11. 3. Partial Properties
Partial molar property, of species i in solution,
is a measure of the response of total property nM
to the addition at constant T & P of differential
amount of species i to a finite amount of
solution.

12. 3. Partial Properties
Properties
example
Solution properties Partial properties Pure-species properties

13. 3. Partial Properties
Since Therefore, or
Because:

14. 3.1 Molar Properties & Partial Molar Properties
The total thermodynamic properties of a homogeneous phase are functions of T, P, and the no of moles of individual species. The total differential of nM is:

15. 3.1.1 Molar Properties from Partial Molar Properties
The total differential of nM is:
At constant n,
*x denotes differentiation at constant composition

16. 3.1.1 Molar Properties from Partial Molar Properties
Because , According to product rule,

17. 3.1.1 Molar Properties from Partial Molar Properties
Factorizing n and dn terms, Mathematically,
x n

18. 3.1.1 Molar Properties from Partial Molar Properties
summability equation
Comparing
Gibbs/Duhem equation
(at constant T & P)

19. 3.2 Partial Properties in Binary Solutions
For binary solution, the summability relation, becomes: At constant T & P, Gibbs/Duhem eq, Thus, for binary solution, Gibbs/Duhem eq. can be written as,
(A)
(B)
becomes
(C)

20. 3.2 Partial Properties in Binary Solutions
Substituting into differential of summability relation: Because x1+x2=1, x1=1-x2 and dx1=-dx2,
(B)
(D)

21. 3.2 Partial Properties in Binary Solutions
The summability relation, can also be written as:

22. 3.2 Partial Properties in Binary Solutions
Gibbs/Duhem equations, may be written in derivative forms:

23. 3.3 Relations among Partial Properties
The change of Gibbs energy, Application of the criterion of exactness yields:
Recall:

24. Also applies to other property that are independent of pressure
4. The Ideal-Gas Mixture
Gibbs Theorem: Thus, for general partial property; The enthalpy of an ideal gas is independent of pressure;
A partial molar property (other than volume) of a constituent species in an ideal gas mixture is equal to the corresponding molar property of the species as a pure ideal gas at the mixture temperature but at pressure equal to its partial pressure in the mixture
Also applies to other property that are independent of pressure
[1]

25. 4. The Ideal-Gas Mixture
The entropy of an ideal gas does depend on pressure;
(const T)

26. 4. The Ideal-Gas Mixture Since Thus, or [2]
A partial molar property ideal gas mixture is equal to the molar property of the species as a pure ideal gas at the mixture temperature but at pressure equal to its partial pressure in the mixture
Since
Thus, or
[2]

27. 4. The Ideal-Gas Mixture For Gibbs energy of an ideal-gas mixture,
the partial properties is:
Recall:
Thus, or
[3]
Recall:

28. 4. The Ideal-Gas Mixture So far, we have derive:
Substituting [1], [2] & [3] into the summability relation,
[1]
[2]
[3]
[4]
[5]
[6]

29. 4. The Ideal-Gas Mixture
The properties of mixture for enthalpy, entropy and Gibbs energy:
Rearranging [4] & [5] give enthalpy & entropy change of mixing;
[4]
[5]
[6]
For ideal gas, mixing at constant T & P is not accompanied by heat transfer

30. Integration constant at constant T
4. The Ideal-Gas Mixture
Alternatively, in eq ( ) can be expressed by giving its T and P dependence:
Integration gives;
Thus eq becomes:
[3]
(const T)
Integration constant at constant T
[7]
[3]
[8]

31. 4. The Ideal-Gas Mixture Substituting eq into summability relation,
[8]
Substituting eq into summability relation,
Thus, Gibbs energy of an ideal-gas mixture, becomes:
[8]

32. 5. Fugacity & Fugacity Coefficient: Pure Species
Fugacity of pure species i, with units of pressure
For real fluid;
Substracting eq , from eq ;
[8]
[7]
[8]
Fugacity: pressure of an ideal gas which has the same chemical potential as the real gas.
Residual
Gibbs energy
Fugacity coefficient

33. 5. Fugacity & Fugacity Coefficient: Pure Species
From previous chapter,
Thus,
(const T)

34. 5. Fugacity & Fugacity Coefficient: Pure Species
From previous chapter, OR
Thus, becomes:
(const T)

35. 5. Fugacity & Fugacity Coefficient: Pure Species
For vapor/liquid equilibrium;
By difference:
At equilibrium,
Thus,
For saturated vapor
For saturated liquid

36. 5. Fugacity & Fugacity Coefficient: Pure Species
Therefore,
For a pure species coexisting liquid & vapor phases, they are in equilibrium when they have the same temperature, pressure & fugacity.

37. 5.Fugacity & Fugacity Coefficient: Pure Species
For pure liquid, A B C

38. 5. Fugacity & Fugacity Coefficient: Pure Species
For pure liquid, A B C

39. 5. Fugacity & Fugacity Coefficient: Pure Species
For pure liquid, A B C
From
From
(at constant T)

40. 5. Fugacity & Fugacity Coefficient: Pure Species
Ratio C becomes,
Substituting A, B, and C terms;

41. 6. Fugacity & Fugacity Coefficient: Species in Solution
For a species i in a mixture of real gas or in a solution of liquids;
At equilibrium;
Fugacity of species i in solution
Multiple phases at the same T & P are in equilibrium when the fugacity of each constituent species is the same in all species

42. 6. Fugacity & Fugacity Coefficient: Species in Solution
For a species i in a mixture of real gas or in a solution of liquids;
Partial residual Gibbs energy
Fugacity coefficient of species i in solution

43. 6. 1 Fugacity Coefficient from the Virial Equation of State
For n mol of gas mixture:
the bimolecular interaction between molecule i and j

44. 6. 1 Fugacity Coefficient from the Virial Equation of State
Differentiation with respect to n1:
Thus becomes;

45. 6. 1 Fugacity Coefficient from the Virial Equation of State
For binary mixture:
OR can be written as: OR
with

46. 6. 1 Fugacity Coefficient from the Virial Equation of State
Multiplying by n; Substituting ; By differentiation;

47. 6. 1 Fugacity Coefficient from the Virial Equation of State
Substituting into ; Similarly;

48. 6. 1 Fugacity Coefficient from the Virial Equation of State
For multicomponent gas mixtures;
where
and
with

49. 7. Generalized Correlation for Fugacity Coefficient
For pure gases;
Hence,at P=Pr , becomes;
From Pitzer correlation;
At constant Tr

50. 7. Generalized Correlation for Fugacity Coefficient
Substitution of , becomes; OR
Where;

51. 7. Generalized Correlation for Fugacity Coefficient
can also be written as:

52. 7. Generalized Correlation for Fugacity Coefficient
can also be simplified by substituting:
Thus, OR

53. 7. Generalized Correlation for Fugacity Coefficient
For gas mixtures;
Where;
and

54. 7. Generalized Correlation for Fugacity Coefficient
The calculation for (proposed by Prausnitz et al. );
Interaction parameter (specific to i-j molecule pair)

55. 7. Generalized Correlation for Fugacity Coefficient
EXAMPLE: Estimate & for an equimolar mixture of
methyl ethyl ketone (1)/toluene(2) at 50°C and 25kPa. Set all kij=0 ij
Tcij /K
Pcij/bar
Vcij/cm3mol-1
Zcij
ωij 11
535.5
41.5
267
0.249
0.323 22
591.8
41.1
316
0.264
0.262

56. 7. Generalized Correlation for Fugacity Coefficient
EXAMPLE – Step 1: Calc Tcij, Vcij, Zcij &ωcij for ij=12 ij
Tcij /K
Pcij/bar
Vcij/cm3mol-1
Zcij
ωij 11
535.5
41.5
267
0.249
0.323 22
591.8
41.1
316
0.264
0.262 12
563.0
41.3
291
0.256
0.293

57. 7. Generalized Correlation for Fugacity Coefficient
EXAMPLE- Step 2: Calc Trij, B0, B1 & Bij for all ij ij
Trij /K B0 B1
Bij/cm3mol-1 11
0.603
-0.865
-1.300
-1,387 22
0.546
-1.028
-2.045
-1,860 12
0.574
-0.943
-1.632
-1,611

58. 7. Generalized Correlation for Fugacity Coefficient
EXAMPLE: Step 3:Estimate &