Partial derivatives of enthelpy and Maxwell relations

[Like Tony Stark]

Homework Statement

Write the following partial derivatives in terms of heat capacity ($c$), compressibility ($\kappa$) and coefficient of thermal expansion ($\alpha$).

$(\frac{\partial H}{\partial V})_U$
$(\frac{\partial H}{\partial V})_S$

With $H$: enthalpy, $U$: internal energy, $V$: volume, $S$: entropy and the subscript denotes a magnitud held constant in differentiation.

Relevant Equations

$dH=dU+Pdv+vdP$
$dU=TdS-PdV$

I've attached images showing my progress. I have used Maxwell relations and the definitions of $\alpha$, $\kappa$ and $c$, but I don't know how to continue. Can you help me?

 

[Chestermiller]

Your picture is unreadable.

 

[stevendaryl]

You have that $H = U + PV$. And you also have that $dU = T dS - P dV$. So when $U$ is constant, you have: $dH = d(PV) = PdV + V dP$. And you also have when $U$ is constant, $TdS - PdV = 0$. So we conclude:

$\frac{\partial H}{\partial V}|_U = P + V \frac{\partial P}{\partial V}|_U$

Now, if you think of $P$ as a function of $S$ and $V$, then

$\frac{\partial P}{\partial V}|_U = \frac{\partial P}{\partial V}|_S + \frac{\partial P}{\partial S}|_V \frac{\partial S}{\partial V}|_U$

At this point, you can use $TdS - PdV = 0$ (when $dU = 0$) to rewrite $\frac{\partial S}{\partial V}|_U$, and you can use one of the Maxwell relations to rewrite $\frac{\partial P}{\partial S}|_V$.