Finding the entropy from the heat capacity

[Coffee_]

Let's say that we have some canonical ensemble where I know that the heat capacity is given by

$C_{V}=\alpha(N,V) T^{n}$

Since $C_{V}=T\frac{\partial S(T,V)}{\partial T}$ I know that

$S(V,T)=\frac{1}{n} \alpha(N,V) T^{n} + f(N,V)$

Where the function $f(N,V)$ has to do with the fact that I'm only taking the derivative wrt. to T and can lose such additional terms in general.

I also know that $S(V,0)=0$ which means that $f(N,V,)=0$ which means I can always use this trick to find the entropy if the heat capacity is known. Obviously if $C_{V}$ is an uglier function of T we'd have to integrate and so on.

$QUESTION$: When is it okay to do such a reasoning and when isn't it?

 

[Chestermiller]

For an ideal gas, assuming C_V is constant, $S = N(C_vln(T) + R ln(V)+C)$. So this doesn't even work for an ideal gas. For real substances, there are phase changes that occur prior to getting to absolute zero that need to be included also.