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I have a question about changing variables in the context of thermodynamics, but I suppose this would extend to any set of variables that have defined and nonzero partial derivatives on a given set of points. First I should define the variables. $T$ is temperature, $U$ is internal energy, $H$ is enthalpy, $F$ is Helmholtz free energy, $G$ is Gibbs free energy, $P$ is pressure, $V$ is volume, $S$ is entropy, $C_V$ is heat capacity at constant volume, and $C_P$ is heat capacity at constant pressure.

The four functions $U$,$H$,$F$, and $G$ are called thermodynamic potentials or thermodynamic functions. Each one has a set of variables, called natural variables, from which you can derive other thermodynamic variables through partial differentiation in a much cleaner way than using with variables other than these natural variables as the starting point.

For pure substances, the natural variables for $U$ are $S$ and $V$, the natural variables for $H$ are $S$ and $P$, the natural variables of $F$ are $T$ and $V$, and the natural variables of $G$ are $T$ and $P$.

$U=U(S,V);H=H(S,P);F=F(T,V);G=G(T,P)$

So, for a pure substance, the natural variables of the internal energy $U$ are entropy $S$ and volume $V$. The total differential of $U$ is then $$dU(S,V)=TdS-PdV=\left (\frac{\partial U }{\partial S} \right )_VdS+\left ( \frac{\partial U}{\partial V} \right )_SdV=\frac{\partial U}{\partial S}(S,V)dS+\frac{\partial U}{\partial V}(S,V)dV$$

The question that I have came to mind when I was proving the relations

$$C_V=\left ( \frac{\partial U}{\partial T} \right )_V=T\left ( \frac{\partial S}{\partial T} \right )_V$$

$$C_P=\left ( \frac{\partial H}{\partial T} \right )_P=T\left ( \frac{\partial S}{\partial T} \right )_P$$

When I worked through this, I simply started from the thermodynamic function that was most convenient. For the relation for $C_V$, I started with $U(S,V)$ and did the following:

$$dU(S,V)\rightarrow dU(S(T,V),V)=TdS(T,V)-PdV=T\left [ \left ( \frac{\partial S}{\partial T} \right )_VdT+\left ( \frac{\partial S}{\partial V} \right )_TdV \right ]-PdV$$

$$dV=0\Rightarrow dU(S(T,V),V)=T\left ( \frac{\partial S}{\partial T} \right )_VdT=\left ( \frac{\partial U}{\partial T} \right )_VdT\Rightarrow C_V=T\left ( \frac{\partial S}{\partial T} \right )_V$$

For the relation with $C_P$, I did a similar thing. The natural variables of the enthalpy $H$, for a pure substance, is entropy $S$ and pressure $P$. The total differential for enthalpy is

$$dH(S,P)=TdS+VdP=\left ( \frac{\partial H}{\partial S} \right )_PdS+\left ( \frac{\partial H}{\partial P} \right )_SdP$$

To derive the relation for $C_P$, I went from $dH(S,P)$ to $dH(S(T,P),P)$, where

$$dH(S(T,P),P)=TdS(T,P)+VdP$$

After that I did essentially the same procedure as the one I did for $C_V$.

What I'm wondering now is, how would I do a two-variable change of coordinates starting from the natural variables of a thermodynamic function? In each of the above examples, I sort of used entropy $S$ as a dummy variable and made it a function of a variable that I wanted to write the thermodynamic function in terms of, and a variable that the thermodynamic function was already written in terms of, that happened to be one of its natural variables.

But what if I wanted to write the total differential for internal energy, $dU(S,V)$, in terms of another set of variables outside of the natural variables of $U$, like $dU(T,P)$?

Would I have to do something like this? $$dU(S,V)\rightarrow dU(S(T,P),V(T,P))=TdS(T,P)-PdV(T,P)$$ $$=T\left [ \left ( \frac{\partial S}{\partial T} \right )_PdT+\left ( \frac{\partial S}{\partial P} \right )_TdP \right ]-P\left [ \left ( \frac{\partial V}{\partial T} \right )_PdT+\left ( \frac{\partial V}{\partial P} \right )_TdP \right ]$$

What if I wanted to write the thermodynamic function in terms of another thermodynamic function/functions with their own set of natural variables? An example would be going from $U(S,V)$ to $U(H,F)$, where, in terms of natural variables, $U=U(S,V)$, $H=H(S,P)$, and $F=F(T,V)$? How would that work? I know this part might not make physical sense whatsoever, but I want to know just for the sake of the mathematics.

Would I write something like

$$dU(S,V)\rightarrow dU(S(H(S,P),F(T,V)),V(H(S,P),F(T,V)))$$

$$=TdS(H(S,P),F(T,V))-PdV(H(S,P),F(T,V))$$

$$=T\left [ \left ( \frac{\partial S}{\partial H} \right )_FdH(S,P)+\left ( \frac{\partial S}{\partial F} \right )_HdF(T,V) \right ]-P\left [ \left ( \frac{\partial V}{\partial H} \right )_FdH(S,P)+\left ( \frac{\partial V}{\partial F} \right )_HdF(T,V) \right ]$$

and continue to expand the $dH(S,P)$ and $dF(T,V)$? Sorry for the long post, but this has been bugging me for a while.

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    $\begingroup$ Can you clarify what is the question? Is it about if they can be expressed in other variables? Or is there any meaningful derivative? $\endgroup$
    – Greg
    Oct 15, 2021 at 8:23
  • $\begingroup$ @Greg I'm essentially confused about natural variables somewhat. I'm wondering if you can do a change of variables from the natural variables of a thermodynamic potential; for example $dU(S,V)$ to $dU(T,P)$, by writing $dU(S(T,P),V(T,P))$. Is this allowed? $\endgroup$ Oct 15, 2021 at 17:29

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