Heat capacity at constant volume and density

[Silviu]

Hello! I encountered in a problem the terms heat capacity at constant volume and density ($n$) and heat capacity at constant chemical potential ($\mu$) and volume and I need to prove a relation between them. What is their definition? I thought that for the first one it would be $C_n=\frac{\partial U}{\partial T}|_{n,V}$ and for the second one $C_\mu=\frac{\partial U}{\partial T}|_{\mu,V}$ and I can take $U$ from $U=TS+\mu N-pV$ and derive each term with respect to temperature, keeping the corresponding variables fixed. Is this logic correct?

 

[Charles Link]

Problem was solved and had an interesting solution. See Specific heats