Doubt in deriving a relation for internal energy as a function of $T$ and $P$

Question

I am solving a problem given in Dittman and Zemansky's Heat and thermodynamics. The problem is given below-

Regarding internal energy of a hydrostatic system to be a function of $T$ and $P$, derive the following equation-
$\Big(\frac{\partial U}{\partial P}\Big)_T=PV\kappa -(C_P-C_V)\frac{\kappa}{\beta}$
where $\kappa$- isothermal compressibility $\Big(-\frac{1}{V}\frac{\partial V}{\partial P}\Big\rvert_T\Big)$ and.
$\beta$- isobaric expansion coefficient $\Big(\frac{1}{V}\frac{\partial V}{\partial T}\Big\rvert_P\Big)$

I am solving it as
From first law of thermodynamics,
$$dU=dQ-PdV$$
Differentiating the above equation by $dP$ gives

$\frac{\partial U}{\partial P}\Big\rvert_T \frac{dP}{dP}+\frac{\partial U}{\partial T}\Big\rvert_P \frac{dT}{dP}=\frac{dQ}{dP}-P\frac{dV}{dP}$
At constant temperature the above equation becomes
$\frac{\partial U}{\partial P}\Big\rvert_T =\frac{dQ}{dP}\Big\rvert_T-P\frac{\partial V}{\partial P}\Big\rvert_T$
$\implies \frac{\partial U}{\partial P}\Big\rvert_T =\frac{dQ}{dP}\Big\rvert_T-P\frac{\partial V}{\partial P}\Big\rvert_T\Big(\frac{-V}{-V}\Big)$
$\implies \frac{\partial U}{\partial P}\Big\rvert_T =\frac{dQ}{dP}\Big\rvert_T+PV\kappa$

I am not able to evaluate $\frac{dQ}{dP}\Big\rvert_T$. We write $dQ=\bigg(\frac{dQ}{dT}\bigg) dT$
This $\frac{dQ}{dT}$ is either at constant pressure or constant volume depending on the process.
But how to solve $\frac{dQ}{dP}\Big\rvert_T=-(C_P-C_V)\frac{\kappa}{\beta}$

Please help, I am very confused.

Answer 1

Here I'll prove $$\left.\frac{\partial Q}{\partial p}\right|_T=-(C_p-C_V)\frac{\kappa }{\beta}$$ or $$T\left.\frac{\partial S}{\partial p}\right|_T=-(C_p-C_V)\frac{\kappa }{\beta}$$ or $$T\left.\frac{\partial V}{\partial T}\right|_T=(C_p-C_V)\frac{\kappa }{\beta}\Rightarrow C_p-C_V=\frac{VT\beta^2}{\kappa}$$

You can find the proof for the above in Thermal Physics Blundell Exercise $16.5$ I'm just giving the steps if you want to do your own.

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1. Consider $S=S(T,V)$ and write $dS$.
2. Differentiate with respect to $T$ at constant $p$. Identify $C_p$ and $C_V$ and substitute.
3. Use different technique to show the rest of the terms can be written in term $\beta $ and $\kappa$ (it's not that hard).