Find the heat capacity of the system

AI Thread Summary
The discussion focuses on calculating the heat capacity of a system with three energy levels and their respective degeneracies. The partition function is defined as Z = e^(-βε) + 2e^(-2βε) + e^(-3βε), incorporating the degeneracies. The user attempted to differentiate the partition function twice to find the heat capacity but encountered issues with the accuracy of their result. They recognize the need to correctly incorporate the degeneracies into their calculations. The conversation emphasizes the importance of accurately applying statistical mechanics principles to derive the correct expression for heat capacity.
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Homework Statement



A system posesses three energy levels E1=ε, E2=2ε, E3=3ε with degeneracies g(E1)=g(E3)=1 , g(E2)=2
Find the heat capacity of the system.

Homework Equations



Z=e-βε+e-2βε+e-3βε

d2lnZ/dβ2=kT2C

The Attempt at a Solution



i got the partition function and then differenciated it twice which gives me an expression for C however its not correct and i can't see how to bring it the degeneracites i presume they are needed
 
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Z=\sum_i g_i e^{-\beta E_i}
 


thanks
 
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