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On my professor's lecture notes he claims that $c_V>0$. Intuitively, this would have been my guess as well but he mentions how you can prove this by constructing a setup:

"Construct an isolated system, which is divided into two parts with internal energies $U_1$ and $U_2$ and volumes $V_1$ and $V_2$"

He then argues that maximising entropy at fixed volume will require $c_V>0$.

Here's my attempt at this, but I couldn't get very far. Since the system is isolated then $U=U_1+U_2=\textit{const}$. Also, $V_1$ and $V_2$ must be constant so then I get the total entropy: $$dS =\dfrac{1}{T_1}dU_1+\dfrac{1}{T_2}dU_2 = \left(\dfrac{1}{T_1}-\dfrac{1}{T_2} \right)dU_1 = 0$$ and so $T_1=T_2$. This doesn't seem to help and I'm not really sure how to proceed from here. Any ideas?

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Yes, this is on the right track. By considering two systems like this, you establish that the system is at equilibrium when the two temperatures are equal, where $$ \frac{\partial S}{\partial U}=\frac{1}{T} $$ In other words, transferring a small amount of energy $\delta U$ from one system to the other results in zero change in the total entropy, to first order in $\delta U$. This is what you expect if the total entropy is at a maximum. But it would also apply if the total entropy were at a minimum!

To distinguish between these two cases, you need to consider how the total entropy would change to second order in $\delta U$. You can do this by differentiating the above equation one more time with respect to $U$. This will give an expression on the right hand side involving the temperature and the heat capacity. If the entropy is a maximum (with respect to small transfers of energy between the two systems), and if you are happy that temperature is positive (I'm aware of a few cases where some people argue it could be negative, but normally we take it as positive) then it should follow that the heat capacity is positive.

I won't give the full answer because I believe that this falls into the homework-and-exercises category of this site, but that should be enough for you to derive the answer along the lines suggested by your professor.

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