Question

starting from A=-kTlnZ. Derive the statistical mechanical formulas for S,P,U

Answer 1

On Partial Differentiation Of Equation A=-kTlnZ

we get the Statistical Mechanical Formula Of S is as follows

where k=Boltzmann Constant

Now we derive Statistical Mechanical Formulas For P & U in terms of Canonical Partition Function Of A

we know that Gibbs free energy is G=A+PV--------->1

A=G-PV--------------------------------------------------->2

dU=TdS-Pdv------------------------------------------------->3

Now On Partial Differentiation and Equating 2 & 3 equations

where -------------------------->4

=Chemical Potential of Species Or substance B

From Equation 3 we get on solving

we know that U=A+TS

on solving and combining the equations of 2,3,4

All these Statistical Mechanical Formulas For S,U,P are dependent on the Canonical partition function of A.