How Do You Prove Key Thermodynamic Identities?

[thekenw]

Can Anyone Show Me the Steps to solving for these equalities; the proofs for them as it were:

Prove that, (dT/dP)s=TV((alpha)p/Cp)

and, Cp/Cv=Kt/Ks=gamma

and, Cp(Kt-Ks)= TV((alpha)p)^2

(alpha)p = coefficiant of thermal expansion, Kt= isothermal compressibility, Ks= adiabatic compressibility. (dx/dy)z means that z is held constant, like s in the first equation (dT/dP)s

Im really lost here and would really appreciate it if anyone could show me the steps to prove any or all three of the above equalities. Thank You

 

[Mapes]

Hi thekenw, welcome to PF. The partial derivative identity

$$\left(\frac{\partial x}{\partial y}\right)_z=-\left(\frac{\partial y}{\partial z}\right)^{-1}_x \left(\frac{\partial z}{\partial x}\right)^{-1}_y$$

might come in handy here.