Equation for S from state func and C

[gulsen]

Heat capacity of a liquid is $$C=T^4$$ and the state function is $$V(T,P) = Aexp(aT-bP)$$
Derive an equation for entropy. Use the relevant Maxwell relations.



$$dU = T dS - PdV$$
$$\frac{\partial U}{\partial T}_V = C = T^4 \Rightarrow U = \frac{T^5}{5} + f(V)$$
Since it's a liquid, and there're no separate $$C_V$$ and $$C_P$$, I assumed that expansion can be ignored, so $$dU \approx TdS$$ and
$$dS = \frac{dU}{T} = T^3 dT$$

but it's unlikely to be true since I haven't used the state function or Maxwell relation at all. My assumption is probably wrong. Anyone solve the problem?

 

[gulsen]

Solved it.That equation I wrote will have an integration factor $$f(V)$$.
Using $$\frac{\partial S}{\partial V}_T = \frac{\partial P}{\partial T}_V$$, we have the solution for S, this time with an integration factor of T. By compraison of these two statements of S, it's perfectly defined.