The enthalpy of a gas is defined by $\displaystyle H=U+pV$ where U satisfies $\displaystyle dU=TdS-pdV$ Determine a relationship between the differentials of H, S and p. Hence show that

$\displaystyle \left ( \frac{\partial V}{\partial S} \right )_p= \left ( \frac{\partial T}{\partial p} \right )_S $

By regarding U as a function of p and V show that

$\displaystyle (\frac{\partial S}{\partial V})_p (\frac{\partial T}{\partial p})_V - (\frac{\partial S}{\partial p})_V (\frac{\partial T}{\partial V})_p = 1

$