How Do You Derive U, T, P, C_v, and V for This Thermodynamic System?

[BurrowK]

Given S=A*(N*V*U)^{1/3} where A is a constant.
a) What is U, T, and P for the system?
b) what are C_v and V?
c) 2 identical bodies consisting of a material obeying the equations found in A. N and V are the same for both and they are initially at temps T_1 and T_2. The two bodies are brought together, what is the final temp?

I know that dS=1/T dU+P/T dV-(mu)/T dN, but I'm not sure how this helps.

I'm unsure where to start for this, my prof forgot to lecture on the subject before giving us this question. ANy help is appreciated.

 

[Mapes]

Hi BurrowK, welcome to PF. From your dS equation it looks like

$$\frac{1}{T}=\left(\frac{dS}{dU}\right)_{V,N}$$

which should help your figure out T (and P can be calculated similarly). It may also help that

$$U=TS-PV+\mu N$$

$$C_V=T\left(\frac{dS}{dT}\right)_{V,N}$$

$$V=\left(\frac{dH}{dP}\right)_{S,N}$$

where the enthalpy $H=U+PV$. Does this get you started?