1. Other Partition Functions

2. Motivation
So far, we considered the statistics of systems of constant volume and mole numbers (or initial mole numbers in the case of reacting systems) kept at constant temperature.
By considering this setup, we derived the canonical partition function, Q, for a several ideal gas situations (monoatomic, diatomic, non-reacting and reacting mixtures):
We also established connections to several properties of macroscopic systems, e.g.:

3. Motivation
The structure of classical thermodynamics is such that if one knows:
it is possible to derive all thermodynamic properties of a macroscopic system.

4. Motivation
But classical thermodynamics is also rich in the possibility of adopting independent variables other than temperature, volume, and mole numbers. For example, if one knows the Gibbs energy as function of temperature, pressure, and mole numbers:
it is possible to derive all thermodynamic properties of a macroscopic system.

5. Motivation
Can we develop the statistical thermodynamics of systems subject to other specifications?

6. Microcanonical Ensemble for a Pure Fluid
In the microcanonical ensemble, the system is subject to the following specifications:
constant number of molecules (in non-reactive systems) or constant number of initial molecules (in reactive systems);
constant volume;
constant energy.
According to the first postulate of statistical thermodynamics:
All microstates of the system of volume V that have the same energy and number of particles are equally probable.

7. Microcanonical Ensemble for a Pure Fluid
For simplicity, let us consider the case of a pure substance that does not undergo any chemical reaction. The system has N molecules, energy E, and volume V. The number of microstates accessible to the system is its degeneracy at this energy level:
Each of these microstates is equally probable according to the first postulate of statistical thermodynamics. The probability of observing one of them is:

8. Microcanonical Ensemble for a Pure Fluid
Therefore:
In Chapter 3, the following expression was derived:

9. Microcanonical Ensemble for a Pure Fluid
Applying it to the microcanonical ensemble:
But all the states have the same energy and the sum of the probabilities equals 1: 1

10. Microcanonical Ensemble for a Pure Fluid

11. Microcanonical Ensemble for a Pure Fluid
Let us now recall some results of classical thermodynamics:

12. Microcanonical Ensemble for a Pure Fluid
Therefore:

13. Grand canonical Ensemble for a Pure Fluid
In the grand canonical ensemble, the system is subject to the following specifications:
constant chemical potential (by being in contact with an infinite reservoir of constant chemical potential of the species of interest);
constant volume;
constant temperature.
This ensemble is often used in studies of adsorption, assuming the chemical potential in the bulk (non-adsorbed) phase is constant.

14. Grand canonical Ensemble for a Pure Fluid
The grand canonical partition function is:
This ensemble is often used in studies of adsorption, assuming the chemical potential in the bulk (non-adsorbed) phase is constant

15. Grand canonical Ensemble for a Pure Fluid
The average number of molecules in the system is:

16. Grand canonical Ensemble for a Pure Fluid
Also:

17. Grand canonical Ensemble for a Pure Fluid
Also:

18. Grand canonical Ensemble for a Pure Fluid
The average energy U in the system is:

19. Grand canonical Ensemble for a Pure Fluid
Also:

20. Grand canonical Ensemble for a Pure Fluid
Also:

21. Grand canonical Ensemble for a Pure Fluid
The total derivative of the natural logarithm of the grand canonical partition function is:
Using the results of the previous slides, we have that:

22. Grand canonical Ensemble for a Pure Fluid
We also have that:
Combining these two equations:

23. Grand canonical Ensemble for a Pure Fluid
Note that:
Combining these two equations:

24. Grand canonical Ensemble for a Pure Fluid
Compare now with the expression for dU for a pure substance from classical thermodynamics:
They can only be equal if the coefficients of the differential forms are equal:

25. Grand canonical Ensemble for a Pure Fluid
But a few slides before, we showed that:
Then:

26. Grand canonical Ensemble for a Pure Fluid
But a few slides before, we showed that:
Then:

27. Grand canonical Ensemble for a Pure Fluid
Let us now use another expression that comes from classical thermodynamics: the Euler relation for a pure substance:
In the previous slides, we obtained expressions for the several properties that appear in the Euler relation. The next step is to replace them in Euler’s relation.

28. Grand canonical Ensemble for a Pure Fluid
Then:

29. Grand canonical Ensemble for a Pure Fluid
Using that:
The expression for the pressure then becomes:
There is a simple connection between the product PV and the partition function of the grand canonical ensemble.

30. Grand canonical Ensemble for a Pure Fluid
It is also possible to show that for a system of fixed volume V (refer to the textbook for details):

31. Isothermal-Isobaric Ensemble for a Pure Fluid
In the isothermal-isobaric ensemble, the system is subject to the following specifications:
constant mole numbers;
constant pressure;
constant temperature.
This ensemble involves specifications very common in chemical engineering design problems

32. Isothermal-Isobaric Ensemble for a Pure Fluid
The isothermal-isobaric partition function is:
fixed number of molecules in a container with flexible walls that allow
heat transfer from an infinite bath of fixed T and P

33. Isothermal-Isobaric Ensemble for a Pure Fluid
There is a direct connection between the Gibbs energy and the partition function in the isothermal-isobaric ensemble:

34. Restricted Grand or Semi-Grand Canonical Ensemble for a Pure Fluid
In this ensemble, the system is subject to the following specifications:
constant mole numbers of all substances numbered from 2 and upwards;
constant chemical potential of substance 1
constant volume;
constant temperature.
rigid, thermally
conductive
walls, permeable
only to species 1
m1, N2, N3, N4…,
T, P, m1
This ensemble involves specifications common in osmotic equilibrium problems

35. Restricted Grand or Semi-Grand Canonical Ensemble for a Pure Fluid
The semi-grand canonical partition function is:

36. Restricted Grand or Semi-Grand Canonical Ensemble for a Pure Fluid
It can be shown that: