Calculating specific heat capacity from entropy

[approx12]

Homework Statement

Given the entropy S of system, calculate the specific heat capacity C_V and C_p

Relevant Equations

$$C_P=T \left(\frac{\partial S}{\partial T}\right)_{N,P}$$ and
$$C_V=T \left(\frac{\partial S}{\partial T}\right)_{N,V}$$

Hey guys! I'm currently struggling with a specific thermodynamics problem.
I'm given the entropy of a system (where $A$ is a constant with fitting physical units): $$S(U,V,N)=A(UVN)^{1/3}$$I'm asked to calculate the specific heat capacity at constant pressure $C_p$ and at constant volume $C_V$.
I know that the two are given by the following equation:
$$C_P=T \left(\frac{\partial S}{\partial T}\right)_{N,P}$$$$C_V=T \left(\frac{\partial S}{\partial T}\right)_{N,V}$$I've tried to eleminate $U$ from the equation by calculating: $$\left(\frac{\partial S}{\partial V}\right)=\frac{P}{T}=\frac{1}{3}(NU)^{1/3}V^{-2/3}$$ Solving for $U$ and plugging it back into the original equation gives me: $$S(P,V,T)=\frac{PV}{T}$$
I don't know if my steps were correct so far but what I'm now struggling with is calculating $\left(\frac{\partial S}{\partial T}\right)_{P}$ and $\left(\frac{\partial S}{\partial T}\right)_{V}$. For me they would be both equal to $$\left(\frac{\partial S}{\partial T}\right)=-\frac{PV}{T^2}$$ But I don't think that is correct because the relationship $$C_P-C_V=\frac{TV\alpha_P^2}{\kappa_T}$$ needs to bet true.

It would be awesome if anyone could help me out with this one and point me in the right direction. Thank you!

 

[TSny]

Looks like you dropped a factor of 3 in getting to the equation $S = \frac{PV}{T}$.

I don't know if my steps were correct so far but what I'm now struggling with is calculating $\left(\frac{\partial S}{\partial T}\right)_{P}$ and $\left(\frac{\partial S}{\partial T}\right)_{V}$. For me they would be both equal to $$\left(\frac{\partial S}{\partial T}\right)=-\frac{PV}{T^2}$$

When calculating $\left(\frac{\partial S}{\partial T}\right)_{P}$ you can't treat $V$ as a constant. Likewise when calculating $\left(\frac{\partial S}{\partial T}\right)_{V}$ you can't treat $P$ as a constant.

One way to proceed is to find the equation of state that relates the 3 variables $P$, $V$, and $T$. You can then use this to write $S$ as a function of just $T$ and $P$ which then makes it straightforward to evaluate $\left(\frac{\partial S}{\partial T}\right)_{P}$. Or, you can write $S$ as a function of just $T$ and $V$ so that you can evaluate $\left(\frac{\partial S}{\partial T}\right)_{V}$. I don't know if this is the best way, but it's one way.