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

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 $$\frac{dQ}{dP}\Big\rvert_T=-(C_P-C_V)\frac{\kappa}{\beta}$$

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.

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).
• your hint helps a lot. I have derived the required relation. But I have a doubt that this problem is given in Dittman's book in chapter First Law of Thermodynamics (before the concept of entropy). So, can we derive this without writing $dQ=TdS$?
– Iti
Apr 23, 2021 at 16:47
• Yeah! I think so. Substitute $dS=dQ/T$ and see if that make sense. Apr 23, 2021 at 18:08
• But to find $\frac{dQ}{dP}\Big\rvert_T$ which is equal to $\frac{TdS}{dP}\Big\rvert_T=\frac{\partial S}{\partial V}\Big\rvert_T\frac{\partial V}{\partial P}\Big\rvert_T$, we have to consider $S(T,V)$. I think without considering entropy at all, we can't derive this identity?
– Iti
Apr 23, 2021 at 18:32