Deriving the TdS Equation for Thermal Physics

[Elzair]

Homework Statement

Derive the following equation

Homework Equations

$$TdS = C_{V} \left( \frac{\partial T}{\partial P} \right)_{V}dP + C_{P} \left( \frac{\partial T}{\partial V} \right)_{P}dV$$

The Attempt at a Solution

$$dU = \delta Q - \delta W$$

$$\delta Q = TdS$$ for a closed system

$$C_{P} = T \left( \frac{\partial S}{\partial T} \right)_{P}$$

$$C_{V} = T \left( \frac{\partial S}{\partial T} \right)_{V}$$

I am not sure where to go from here.

 

[siddharth]

I think the trick here is playing around with the relations.

If you consider S as a function of p and v as independent variables, then
$$dS = \left(\frac{\partial S}{\partial P}\right)_V dP + \left(\frac{\partial S}{\partial V}\right)_P dV$$

But, $$\frac{\partial S}{\partial P}_V = \left(\frac{\partial S}{\partial T}\right)_V \left(\frac{\partial T}{\partial P}\right)_V$$, and so on.