1. Solution thermodynamics theory
Chapter 11

2. topics Fundamental equations for mixtures Chemical potential
Properties of individual species in solution (partial properties)
Mixtures of real gases
Mixtures of real liquids

3. A few equations For a closed system
Total differential form, what are (nV) and (nS)
Which are the main variables for G??
What are the main variables for G in an open system of
k components?

4. G in a mixture (open system)

5. G in a mixture of k components at T and P
How is this equation reduced if n =1

6. 2 phases (each at T and P) in a closed system
Apply this equation to each phase
Sum the equations for each phase, take into account that
In a closed system:

7. We end up with
How are dnia and dnib related at constant n?

8. For 2 phases, k components at equilibrium
Thermal equilibrium
Mechanical equilibrium
Chemical equilibrium
For all i = 1, 2,…k

9. In order to solve the PE problem
Need models for mi in each phase
Examples of models of mi in the vapor phase
Examples of models of mi in the liquid phase

10. Now we are going to learn about
Partial molar properties
Because the chemical potential is a partial molar property
At the end of this section think about this
What is the chemical potential in physical terms
What are the units of the chemical potential
How do we use the chemical potential to solve a PE (phase equilibrium) problem

11. Partial molar property
Solution property
Partial property
Pure-species property

12. example Open beaker: ethanol + water, equimolar Total volume nV
T and P
Add a drop of pure water, Dnw
Mix, allow for heat exchange, until temp T
Change in volume ?

13. Total vs. partial properties
See derivation page 384

14. Derivation of Gibbs-Duhem equation

15. Gibbs-Duhem at constant T&P
Useful for thermodynamic consistency tests

16. Binary solutions
See derivation page 386

17. Obtain dM/dx1 from (a)

18. Example 11.3
We need 2,000 cm3 of antifreeze solution: 30 mol% methanol in water.
What volumes of methanol and water (at 25oC) need to be mixed to obtain 2,000 cm3 of antifreeze solution at 25oC
Data:

19. solution Calculate total molar volume
We know the total volume, calculate the number of moles required, n
Calculate n1 and n2
Calculate the volume of each pure species

20. Note curves for partial molar volumes

21. From Gibbs-Duhem:
Divide by dx1, what do you conclude respect to the slopes?

22. Read and work example 11.4
Given H=400x1+600x2+x1x2(40x1+20x2) determine partial molar enthalpies as functions of x1, numerical values for pure-species enthalpies, and numerical values for partial enthalpies at infinite dilution
Also show that the expressions for the partial molar enthalpies satisfy Gibbs-Duhem equation, and they result in the same expression given for total H.

23. Now we are going to start looking at models for the chemical potential of a given component in a mixture
The first model is the ideal gas mixture
The second model is the ideal solution
As you study this, think about the differences, not only mathematical but also the physical differences of these models

24. The ideal-gas mixture model
EOS for an ideal gas
Calculate the partial molar volume for an ideal gas in an ideal gas mixture

25. For an ideal gas mixture

26. For any partial molar property other than volume, in an ideal gas mixture:

27. Partial molar entropy (igm)

28. Partial molar Gibbs energy
Chemical potential of
component i in an
ideal gas mixture
*******************************************************************************
This is m for a pure component !!!

29. Problem
What is the change in entropy when 0.7 m3 of CO2 and 0.3 m3 of N2, each at 1 bar and 25oC blend to form a gas mixture at the same conditions? Assume ideal gases.
We showed that:

30. solution
n = PV/RT= 1 bar 1 m3/ [R x 278 K]
DS = J/K

31. Problem
What is the ideal work for the separation of an equimolar mixture of methane and ethane at 175oC and 3 bar in a steady-flow process into product streams of the pure gases at 35oC and 1 bar if the surroundings temperature Ts = 300K?
Read section 5.8 (calculation of ideal work)
Think about the process: separation of gases and change of state
First calculate DH and DS for methane and for ethane changing their state from
P1, T1, to P2T2
Second, calculate DH for de-mixing and DS for de-mixing from a mixture of ideal
gases

32. solution Wideal = DH – Ts DS = -2484 J/mol = -7228 J/mol
= -15,813 J/mol K
Wideal = DH – Ts DS = J/mol

33. Now we introduce a new concept: fugacity
When we try to model “real” systems, the expression for the chemical potential that we used for ideal systems is no longer valid
We introduce the concept of fugacity that for a pure component is the analogous (but is not equal) to the pressure

34. We showed that: Pure component i, ideal gas Component i in a mixture
of ideal gases
Let’s define:
For a real fluid, we define
Fugacity of pure species i

35. Residual Gibbs free energy
Valid for species i
in any phase and
any condition

36. Since we know how to calculate residual properties…
Zi from an EOS, Virial, van der Waals, etc

37. examples
From Virial EOS
From van der Waals EOS

38. Fugacities of a 2-phase system
One component, two phases:
saturated liquid and saturated vapor at Pisat and Tisat
What are the equilibrium conditions for a pure component?

39. Fugacity of a pure liquid at P and T

40. Fugacity of a pure liquid at P and T

41. example
For water at 300oC and for P up to 10,000 kPa (100 bar) calculate values of fi and fi from data in the steam tables and plot them vs. P
At low P, steam is an ideal gas => fi* =P*
Get Hi* and Si* from the steam tables at 300oC and the lowest P, 1 kPa
Then get values of Hi and Si at 300oC and at other pressure P and calculate fi (P)

43. Problem
For SO2 at 600 K and 300 bar, determine good estimates of the fugacity and of GR/RT.
SO2 is a gas, what equations can we use to calculate f = f/P
Find Tc, Pc, and acentric factor, w, Table B1, p. 680
Calculate reduced properties: Tr, Pr
Tr=1.393 and Pr=3.805

45. High P, high T, gas: use Lee-Kessler correlation
From tables E15 and E16 find f0 and f1
f0 = 0.672; f1 = 1.354
f = f0 f1w = 0.724
f = f P = x 300 bar = bar
GR/RT = ln f =

46. Problem
Estimate the fugacity of cyclopentane at 110oC and 275 bar. At 110 oC the vapor pressure of cyclopentane is bar.
At those conditions, cyclopentane is a high P liquid

47. Find Tc, Pc, Zc,, Vc and acentric factor, w, Table B1, p. 680
Calculate reduced properties: Tr, Prsat
Tr = and Prsat = 0.117
At P < Prsat we can use the virial EOS to calculate fisat

48. fisat = 0.9
P-correction term:
Get the volume of the saturated liquid phase, Rackett equation
Vsat = cm3/mol
f = bar

49. Generalized correlations: fugacity coefficient
Tables E13 to E16
Lee-Kessler

50. Now we introduce the concept of fugacity for a given component in a mixture
Fugacity of component i in a mixture plays an analogous role to the partial pressure of the same component i in an ideal mixture
At low pressure, the fugacity of i in the mixture tends to be the partial pressure of i.
This means that the fugacity coefficient of i in the mixture tends to 1 at low pressures

51. We showed that: Pure component i, ideal gas Component i in a mixture
of ideal gases
Let’s define:
For a real fluid, we define
Fugacity of pure species i

52. Now lets consider component i in solution
Component i in a mixture
of ideal gases
Now lets consider component i in solution
Component i in a real
solution
Fugacity of component i in solution
We can also define a fugacity coefficient of i in solution

53. We can also relate fugacity (in solution) to a residual property

54. Residual partial Gibbs free energy

55. How to calculate fugacity coefficients in solution

56. How to calculate fugacity coefficients in solution
Valid for mixtures of gases at
low and
moderated pressures

57. How to calculate fugacity coefficients in solution
From the Virial EOS (truncated after the second term)

58. You can show that
Derivation is in page 406, but try to do it by yourself first

59. problem
For the system methane (1)/ethane (2)/propane (3) as a gas, estimate
at T = 100oC, P = 35 bar, y1 =0.21, and y2 =0.43

60. solution 1) Get Tc, Pc, and Vc for each of the components
2) Calculate mixture properties:
wij, TCij, PCij, Zcij and Vcij equations to 11.74
For example we will have T11, T22, T33, T12, T13, T23
From this equation get Bij that we need to
calculate dik

61. For example Results of d in cm3/mol d11 = 0, d22 =0, d33 =0

62. results

63. When we deal with mixtures of liquids or solids
We define the ideal solution model
Compare it to the ideal gas mixture, analyze its similarities and differences

64. The ideal solution model
Component i in a mixture
of ideal gases
That is obtained by using
In the first term
of this equation
Now we define
Ideal solution model

65. Other thermodynamic properties for the ideal solution: partial molar volume

66. partial molar entropy in the ideal solution

67. partial molar enthalpy in the ideal solution

68. Chemical potential ideal solution
Chemical potential component i
in a Real solution
Chemical potential
Pure component i
Subtracting:
For the ideal solution

69. Lewis-Randall rule
Lewis-Randall rule

70. When is the ideal solution valid?
Mixtures of molecules of similar size and similar chemical nature
Mixtures of isomers
Adjacent members of homologous series

71. problem
For the system methane (1)/ethane (2)/propane (3) as a gas, estimate
at T = 100oC, P = 35 bar, y1 =0.21, and y2 =0.43
Assume that the mixture is an ideal solution
Obtain reduced pressures, reduced temperatures, and calculate

72. Results: methane (1) ethane (2) propane (3)
Virial model
Ideal solution

73. Now we want to define a new type of residual properties
Instead of using the ideal gas as the reference, we use the ideal solution

74. Excess properties
The most important excess function is the excess Gibbs free energy GE
Excess entropy can be calculated from the derivative of GE with respect to T
Excess volume can be calculated from the derivative of GE with respect to P
And we can also define partial molar excess properties

76. Definition of activity coefficient

77. Summary

78. Summary

82. HW # 3, Due Monday, September 17
Problems 11.2, 11.5, 11.8, 11.12, 11.13
HW # 4, Due Monday, September 24
Problems 11.18, (b), 11.21, (a), 11.24(a), 11.25