- #1

cwill53

- 220

- 40

##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.