How Can I Calculate Internal Energy Using Grand Canonical Ensemble Method?

[mkbh_10]

Homework Statement

To calculate internal energy for a system of non interacting S.H.O's in 1 dimension at constant T & u(chem potential) using grand partition function

The Attempt at a Solution

L(Grnd prtition fn)= Summation(z^N)*[Z(TVN)] where z = fugacity , Z = partition fn of canonical ensemble . for Harmonic oscc E = (n+1/2)hw , so i substituted this in the grnd prtition fn , let it vary from n=0 to K , and N also varies from 1 to N , then i apply log form which i get a series of odd numbers for energy and (N1+N2+------+N) for the fugacity term and then differentiate with respct to beta which is 1/KT , and i dnt hve any term involving B , so i can't evaluate Cv , where is the problem .

I am taking U(internal energy) = -(d/dbeta)ln(L)

 

[BigJDubs]

, and taking different terms for N (N1,N2,---Nk) , so i am getting a series of odd numbers for energy, but no term involving beta , so how to calculate Cv , which should be Cv = (du/dT).