# Heat capacity at constant volume and Gibbs free energy

I want to know if it is possible to derive heat capacities, in this case, in constant volume from another thermodynamic Potential which is not the Helmholtz free energy $$F$$. I am aware of the following relationship between heat capacity at constant volume, entropy, and $$F$$: $$C_v = T\left(\frac{\partial S}{\partial T}\right)_{V,N} = -T \left(\frac{\partial^2 F}{\partial T^2}\right)_{V,N}.$$

But I am trying to see if I can derive an expression for $$C_v$$ when I consider the Gibbs free energy (particle number constant $$N$$) and I want to express $$C_v = C_v(T,P,n)$$ where $$n$$ is taken as a constant and can be left out: $$G = U - TS + PV = F + PV$$

The problem I have going forwards has to do with the variables. This is what I mean: \begin{align} C_v &= -T\left(\frac{\partial S}{\partial T}\right)_{V,N} = -T\left(\frac{\partial^2 (G-PV)}{\partial T^2}\right)_{V,N}\\ C_v &= -T\left[\left(\frac {\partial^2 G}{\partial T^2}\right)_{V,N} -\left(\frac{\partial^2 PV}{\partial T^2}\right)_{V,N}\right]\\ C_v &= -T\left[\frac{\partial}{\partial T} \left(\frac{\partial G}{\partial T}\right)_{P,N}\right]_{V,N} -T\left[\frac{\partial}{\partial T} \left(\frac{\partial PV}{\partial T}\right)\right]_{V,N} \end{align}

Then, I use the following relation: $$\left(\frac{\partial P}{\partial T}\right)_{V,N} = -\left(\frac{\partial^2 G}{\partial P \partial T}\right) \left(\frac{\partial^2 G}{\partial p^2}\right)^{-1},$$ which gives me $$C_v = -T\left[\frac{\partial}{\partial T} \left(\frac{\partial G}{\partial T}\right)_{P,N}\right]_{V,N} -T\left[\frac{\partial}{\partial T} \left(-V \left(\frac{\partial^2 G}{\partial P \partial T}\right) \left(\frac{\partial^2 G}{\partial p^2}\right)^{-1} \right)\right]_{V,N}$$

And this is as far as I can get. Here is the problem. If we focus on the first term and do the derivations, I get: $$-T\left(\frac{\partial(-S)}{\partial T}\right)_{V,N}$$ Now I don't know what to do. The entropy was derived by focusing on the Gibbs equation, and now if I want to go further, I have to look at the entropy for the internal energy. But there is nothing I can do. All I can say is that the above term can be equal to: $$C_v = T\left(\frac{\partial(-S)}{\partial T}\right)_{V,N},$$ where this $$C_v$$ has nothing to do with the one in the beginning. My problem is with the 2nd term. I don't know what to do there because $$G$$ is a function of $$T$$, $$N$$, $$P$$, and we also have $$V$$, $$N$$. If I were to write it explicitly, this is what I cannot solve: $$\begin{multline} \left[ \frac{\partial}{\partial T}\left( -V(\frac{\partial^2 G}{\partial P \partial T} \right)\left( \frac{\partial^2 G}{\partial p^2} \right)^{-1} \right]_{V,N} \\ = %split here \frac{\partial}{\partial T}\left( -V\left( \frac{\partial}{\partial P}\left( \left( \frac{\partial G}{\partial T} \right)_{P,N} \right)_{T,N} \right)_{V,N}\left( \frac{\partial^2 G}{\partial p^2} \right)^{-1}_{V,N} \right) \end{multline}$$

This term: $$\frac{\partial}{\partial T}\left( -V\left( \frac{\partial}{\partial P}\left( \left( \frac{\partial G}{\partial T} \right)_{P,N} \right)_{T,N} \right)_{V,N} \right)$$

How do I try and solve this? The indexes confuse me. Can someone give me a hint?

• Welcome to Chemistry! On Chemistry mathematical and chemical expressions can be formatted using MathJax (and LaTeX Syntax). If you want to know more, please have a look here and here. We prefer to not use MathJax in the title field, see here for details. There were a couple of parenthesis/brackets missing and I) hope I didn't put them in all the wrong places. Please check. Commented Nov 7, 2021 at 23:30

What you are looking for can be trivially obtained by referring to the well known general relation between constant-volume and constant-pressure heat capacity: $$C_v= C_p -\frac{\alpha^2TV}{\chi_T}.$$ In this formula, present in almost all textbooks on thermodynamics and easily derivable by using Maxwell's relations and some partial derivative manipulation, all the quantities on the right-hand side of the equation can be represented as derivatives of the Gibbs free energy and the independent variable $$T$$: \begin{align} V &= \left(\frac{\partial G}{\partial p}\right)_{T,n};\\ C_p &= T\left(\frac{\partial S}{\partial T}\right)_{p,n}=-T\left(\frac{\partial^2 G}{\partial T^2}\right)_{p,n};\\ \alpha &=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{p,n}= \frac{1}{V}\left(\frac{\partial^2 G}{\partial T\partial p}\right);\\ \chi_T&=-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_{T,n}= -\left(\frac{\partial^2 G}{\partial p^2}\right)_{T,n}. \end{align}