Finding Cv(V,T) from Gibbs free energy G(P,T)

[Dai_Yue]

Homework Statement

The function G is given in the question: $$G(P,T) = \frac{-aT^2} {P},$$
where a is a positive constant.

Homework Equations

$$dG = Vdp - SdT,$$
and probably $$S(V,T) = (\frac{\partial S}{\partial T})_V dT + (\frac{\partial S}{\partial V})_T dV$$

The Attempt at a Solution

$$C_v dT = TdS,$$

∴​

$$(\frac{\partial S}{\partial T})_vdT = \frac {C_v}{T}$$
.. and that's about as far as I got.

I could find C_p by taking a partial derivative of G with respect to T and get $$(\frac{\partial S}{\partial T})_pdT = \frac {C_p}{T}$$
, which turned into something like $\frac{-2aT}{P}$ but I don't know how I would find $C_v$ without being given a starting function of V and T like Helmholt's energy. Because I'm looking for $C_v$ I'm 90% sure that the function will be a function of V & T, not P & T.

Pls help thx

 

[Chestermiller]

In order to get Cv, you are going to have to get U. U can be obtained from knowledge of H and PV. Do you know how to get H if you know G(P,T)? Do you know how to get V if you know G(P,T)?

Chet