1. Lecture 6： Imperfect Gases
Statistical Thermodynamics
Lecture 6： Imperfect Gases
Dr. Ronald M. Levy

2. 1

3. remainder 1
100
1000 2

4. 3

5. 4

6. 5

7. Pair potential
Mayer F function 6

8. Mayer F function
In terms of Mayer f function, the first 2 Virial coefficient: 7

9. Cluster diagram and star diagram
Numbered circles represent particles whose coordinates we integrate over the bond between pairs of circles means particles are connected by a Mayer f function.
All the Virial coefficients can be expressed in terms of a special set of cluster, These are called doubly connected, or star diagram. 3 1 2 1 2
Cluster diagrams 3 1 2 3 1 2 3 1 2 3 1 2
Doubly connected
Star diagram
Not doubly connected
Not star diagram
Note: There are 2 topologically distinct cluster diagrams for 3 particles, there us 1 star diagram and 3 which is not 8

10. Star diagram for 4 particles:
All star diagram for j+1 particles.
(The sum of all products of f functions that connect 1,2,…,j+1 molecules such that the cluster are connected so that the removal of any point, along with all lines to that point, still results in a connected graph)
Star diagram for 4 particles: 1 2 4 3 1 4 2 3 1 4 2 1 2 4 3 9

11. Hard sphere potential:
Hard sphere potential:
Very important theoretically, can get exact results! Starting point for ##### very important for liquids
𝜎/2 10

12. Square well potential
For square well potential: 11

13. Lennard-Jones potential.12

14. 12