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I want to show:

[itex] C_p - C_v = -T \big( \frac {\partial V}{\partial p} \big)_{T,n} \big( \frac {\partial p}{\partial T} \big)_{V,n}^2 [/itex]

I started by doing this:

[itex] C_p - C_v= \big( \frac {\partial H}{\partial T} \big)_{p,N} - \big( \frac {\partial U}{\partial T} \big)_{V,n} [/itex]

Applying the definitions of enthalpy and energy:

[itex] dH = TdS + V dp + \mu dn [/itex]

and

[itex]dU = TdS - p dV + \mu dn[/itex]

I can rewrite the equation like this:

[itex]= V \big(\frac {\partial p}{\partial T} \big) + p \big( \frac {\partial V}{\partial T} \big) [/itex]

(while TdS and µdn terms cancel out each other)

Now I do not know how to continue. Can anyone help :)