1. The Canonical Partition Function CHEN 689-Fall 2015

2. Some properties of the canonical partition function What is probability of finding the system in any state whose energy is E  ? The following expression gives the probability of finding the system in microstate i with E  Degeneracy of energy level  number of molecular states having that energy

3. Some properties of the canonical partition function can also be written as: The canonical partition function:

4. Relationship of the canonical partition function to thermodynamic properties But: The internal energy in thermodynamics is the average energy of the system :

5. Relationship of the canonical partition function to thermodynamic properties Then:

6. Canonical partition function for a single molecule with several independent energy modes For simplicity, assume the following: Single molecule; Two completely independent energy modes – translational and rotational; Each of these modes has only two energy states The possible energy states are:

7. Canonical partition function for a single molecule with several independent energy modes The single-particle canonical partition function is:

8. Canonical partition function for a single molecule with several independent energy modes This result can be extended to more energy modes (translational, rotational, vibrational, atomic, nuclear), as long as they are assumed to be independent, and more energy states:

9. Canonical partition function for a collection of non-interacting identical atoms First consider a single atom that can be in states 1, 2, 3, etc. Its partition function is: Now consider a collection of such atoms, assuming they are well apart so that we can neglect their potential energy of interaction Because the atoms are undistinguishable, we will define the state of the system by the number of atoms in each energy state (and not by indicating which atoms are each atomic energy state)

10. Canonical partition function for a collection of non-interacting identical atoms is the occupation number, i.e., the number of atoms in the j-th atomic state of a single atom in the i-th macroscopic state of a collection of atoms. The i-th macroscopic state of a collection of atoms is characterized by the set of occupation numbers of that state:

11. Canonical partition function for a collection of non-interacting identical atoms In each i-th macroscopic state, we have that: number of atoms in state j

12. Canonical partition function for a collection of non-interacting identical atoms Consider the occupation number of all the possible macroscopic states of a system with only two atoms, and the canonical partition function:

13. Canonical partition function for a collection of non-interacting identical atoms Consider now the square of the single atom canonical partition function:

14. Canonical partition function for a collection of non-interacting identical atoms Now, compare:

15. Canonical partition function for a collection of non-interacting identical atoms Now, compare:

16. Canonical partition function for a collection of non-interacting identical atoms Accepting this approximation, then, in the more general case, considering the occupation numbers of all the possible macroscopic states of a system with N equal atoms, and the canonical partition function:

17. Canonical partition function for a non-reacting mixture of non-interacting atoms For a non-reacting mixture of C components: because the species are distinguishable

18. Summary The canonical partition function can be related to the average system energy and to the thermodynamic internal energy; Assuming the energy modes of a single atom are independent, it is possible to obtain the partition function of a single atom as the product of the partition functions of each energy mode; We obtained expressions for the partition function of collections of non-interacting atoms: identical atoms; mixtures of different atoms.

19. HW 1- Due Wednesday, 9/9 Problems 2.1; 3.1; 3.2; 3.7