1. Generalized van der Waals Partition Function

2. Statistical mechanical background
We begin with the canonical partition function of a pure fluid of N identical molecules in a volume V at temperature T (Chapter 7):
De Broglie wavelength

3. Thermodynamic properties from the configuration integral
The Helmholtz energy is:
The pressure is:

4. Statistical mechanical background
Let us now define residual properties (res) as the difference between the property of the fluid and that of the ideal gas of same temperature and density.
Note this definition is different from the definition usually adopted in classical solution thermodynamics.

5. Statistical mechanical background
Now assume pairwise additivity:
Under this assumption, the following expression was derived for the average configuration energy in Chapter 11:

6. Statistical mechanical background
Next, we note that interactions between real molecules have a hard core part in which the energy is infinite and the radial distribution function is zero.
In the limit in which the temperature goes to infinity, kT is much large than the interaction energies, giving 1 for the Boltzmann factor, except in the hard core region.
With these comments in mind, let us now integrate:

7. Statistical mechanical background
Then:
non-hard core part of the potential

8. Statistical mechanical background
Let us define the mean potential per atom over the temperature range of the integration:

9. Statistical mechanical background
Using this definition:
Hard core part

10. Statistical mechanical background
To evaluate:
we consider a single hard sphere test particle moving through a volume V with all other particles at fixed positions. An overlap means infinite interaction energy and the Boltzmann factor is zero. If there is no overlap, the interaction energy is zero (hard spheres).
The volume accessible to the center of the test particle, which is referred to as free volume:

11. Statistical mechanical background
In summary:
Generalized van der Waals partition function

12. Statistical mechanical background
In summary:
Energetic part
Entropic part

13. Statistical mechanical background
We will also define the coordination number as the number of molecules in the soft part of the potential of a central molecule:
Range of the potential

14. Application to pure fluids
Developing a model in the context of the generalized van der Waals theory requires expressions for the free volume and for the mean potential.
For non-interacting point molecules the free volume is equal to the total volume and the mean potential is equal to zero. Therefore, we have that:
This is the partition function of a pure ideal gas.

15. Application to pure fluids
The Free Volume
The simplest model (which is used in the van der Waals equation of state and others such as Peng-Robinson) assumes that around each molecule there is a region of excluded volume
where the centers of other molecules cannot be present. When this analysis is made, one molecule is taken to be the central molecule with (N-1) neighbors. Also, to avoid double counting, it is necessary to divide this excluded volume by 2.

16. Free volume and excluded volume for HS at low and moderate density

17. Application to pure fluids
The Free Volume
The free volume is the total volume minus the excluded volume:
The entropic contribution to the pressure then is:

18. Application to pure fluids
The Free Volume
This model fails at high densities as it overestimates the excluded volume.
In Chapter 12, we discussed the Percus-Yevick solution of the Ornstein-Zernicke equation that leads to the Carnahan-Starling equation of state. The corresponding free volume expression is:
And the entropic contribution to the pressure is:

19. Application to pure fluids
The Free Volume
Simpler than Carnahan-Starling , but less accurate, is the free volume expression of Kim, Lin, and Chao:
And the entropic contribution to the pressure is:

20. free volume as a function of reduced density
simulation and Carnahan-Starling
van der waals
Kim-Lin Chao

21. Application to pure fluids
The Mean Potential
The simplest model (which is used in the van der Waals equation of state) considers that the interactions follow the square-well potential:
In this case:

22. Application to pure fluids
The Mean Potential
We now apply:

23. Application to pure fluids
The Mean Potential
The inner integral is the coordination number:
Therefore:

24. Application to pure fluids
The Mean Potential
To proceed, it is necessary to make assumptions to obtain an expression for the coordination number. Assuming the radial distribution function is equal to 1 in the range of the potential:
With this assumption, the coordination number is a linear function of density.

25. Application to pure fluids
The Mean Potential
The mean potential becomes:

26. Application to pure fluids
The van der Waals equation of state
Combining the free volume and mean potential results of the previous slides:

27. Application to pure fluids
The van der Waals equation of state
The corresponding pressure is:

28. Application to pure fluids
The van der Waals equation of state
The corresponding pressure is:
For one mol:

29. Application to pure fluids
The van der Waals equation of state
Note:
Parameters a and b in the van der Waals equation of state have a molecular-based physical interpretation even though, in practice, they are generally obtained from the critical properties of pure substances.

30. Application to pure fluids
van der Waals EOS with temperature-dependent parameters
Developing the van der Waals equation of state, we assumed the radial distribution function was equal to 1 in the range of the potential.
Let us step back and reconsider this assumption. We will use the solution for the radial distribution function in dilute systems of spherical molecules, derived in Chapter 11:

31. Application to pure fluids
van der Waals EOS with temperature-dependent parameters
For the square-well potential:

32. Application to pure fluids
van der Waals EOS with temperature-dependent parameters
The coordination number becomes:

33. Coordination number for the SW fluid
(Rsw = 1.5) as a function of the reduced
inverse temperature
Points are MC simulation;
dotted line: Van der Waals;
dashed line: RK
dashed-dot line: PR

34. Coordination number for the SW fluid
(Rsw = 1.5) as a function of the reduced inverse
temperature
Points are MC simulation
two phase
region

35. Application to pure fluids
van der Waals EOS with temperature-dependent parameters
The mean potential becomes:

36. Application to pure fluids
van der Waals EOS with temperature-dependent parameters
The derivation of the van der Waals EOS can be redone, leading to:

37. Application to pure fluids
Generalized van der Waals theory for arbitrary pure fluids
Using this equation for model development involves assuming the density and temperature dependence of the:
free volume
coordination number

38. Application to pure fluids
Generalized van der Waals theory for arbitrary pure fluids
Many of the models commonly used in chemical engineering design were developed empirically. An interesting “exercise” is to work backwards from these expressions and find out their underlying assumptions.

39. Mixtures
Generalized van der Waals theory for mixtures

40. Mixtures
Generalized van der Waals theory for mixtures

41. Mixtures Generalized van der Waals theory for mixtures
This framework has two applications, namely, developing:
Equations of state
Excess Gibbs energy models for liquid solutions

42. Mixtures Generalized van der Waals theory for mixtures
To derive an equation of state, the pressure is obtained using:

43. Mixtures
Consider again interactions following the square-well potential. Then:
We now define a symbol to represent the average number of i molecules around a central j molecule, i.e., the coordination number of i molecules around a central j molecule:

44. Mixtures
Then:

45. Mixtures The Free Volume
The simplest model (which is used in the van der Waals equation of state and others such as Peng-Robinson) is:

46. Mixtures
The Free Volume
For hard spheres:

47. Mixtures
The Free Volume
The entropic contribution to the pressure is:

48. Mixtures Species-species coordination number
For the square-well potential in the low density limit:

49. Mixtures Species-species coordination number
An important quantity in model development is the ratio of the coordination number of unlike species and of like species, e.g.:

50. Mixtures Species-species coordination number We find that:
Divide the numerator and denominator of the term in the left-hand side by the total number of molecules around a central molecule of type j to find a ratio of mole fractions around this central molecule:

51. Mixtures Species-species coordination number
We observe that the ratio of the local mole fractions is generally different from the ratio of the global mole fractions:
This is called local composition effect. Molecular simulations (Monte Carlo and molecular dynamics) also predict this phenomenon.

52. Ratios of local compositions
to bulk compositions as
functions of the reduced density
for the SW fluid with
e22 = 1.2 kT = 2 e11
Symbols: simulation results
solid, dashed-dotted, and dotted
lines are models

53. Mixtures Species-species coordination number
According to this expression, the local composition effect depends on energy differences and on size differences. At infinite temperature:

54. Mixtures Species-species coordination number
In a binary mixture, the total coordination number of a given species, e.g., species 1 is given by:
This shows that the total coordination number of a species depends on temperature, density, and mole fractions.

55. total coordination numbers Nc1 (unfilled) and Nc2 (filled) symbols as functions of density
and mole fraction from simulation. The symbols represent different reduced densities

56. Mixtures Species-species coordination number
However, the simplest expression for the species-species coordination number results from assuming random mixture:
In this case:

57. Mixtures
Mean potential
We will combine the following three equations:

58. Mixtures
Mean potential
The result is:

59. Mixtures The van der Waals equation of state
Combining the free volume and mean potential results of the previous slides:

60. Mixtures
The van der Waals equation of state

61. Mixtures Alternative local composition models Wilson
Whiting and Prausnitz

62. Mixtures
Alternative local composition models
Hu et al.
Sandler

63. Mixtures Comparison with molecular simulation data.
None of these local composition models is highly accurate, even though they try to predict the phenomenon using molecular-based properties (size, shape, and intermolecular interactions).
Models used in engineering often circumvent these theoretical shortcoming by fitting parameters to experimental data of real pure substances and mixtures.
As theory and computational power advance, the expectation is that newer models may appear, capable of predicting properties from first principles of quantum and statistical mechanics.