Understanding the Relationship between Heat Capacity and Internal Energy

[mhellstrom]

Hi all,

I have to show that the heat capacity can be expressed as

Cv = Nk(1+1/45(Om/T)^2 + ...)

where the internal energy is given as

E = NkT*(1-(Om/(3T)-1/45(Om/T)^2)

Normally I would just differentiate but if I do this I get something completely different - how to proceed any hints appreciated thanks in advance

Best regards

Magnus

 

[tiny-tim]

Hi Magnus!

I have to show that the heat capacity can be expressed as

Cv = Nk(1+1/45(Om/T)^2 + ...)

where the internal energy is given as

E = NkT*(1-(Om/(3T)-1/45(Om/T)^2)

Normally I would just differentiate*…

erm … NkT*(1 - Om/(3T) - 1/45(Om/T)^2) = Nk(T - Om/3 - 1/45(Om/T))

 

[mhellstrom]

yes of course... Thanks for the help. The partition function is also given in the exercise

$$q_{rot} = \frac{T}{\omega}*(1+\frac{1}{3}(\frac{\omega}{T}+\frac{1}{15}(\frac{\omega}{T})^2+...)$$

I presume the internal energy is given as

E = -N(dLn Zrot / d beta)

I would really like to know how to get from the partition function to the internal energy.
The problem for me is how manage the differentiation of the serie. Any help or advise appreciated. Thanks in advance.

Best

Magnus

 

[tiny-tim]

durrr … honestly no idea what that's all about …

i thought this was a straightforward calculus problem!

i think you'd better start a new thread, so as to get someone else to answer