Lecture 6： Imperfect Gases Statistical Thermodynamics Lecture 6： Imperfect Gases Dr. Ronald M. Levy ronlevy@temple.edu

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  remainder 1 -0.00064 0.00000 100 -0.06754 +0.02127 -0.00036 1000 -0.38404 +0.68788 +0.37237 2

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        Pair potential     Mayer F function 6

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

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

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

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

Square well potential                     For square well potential:       11

Lennard-Jones potential.                     12

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