6.4: How It's Done in Thermodynamics

To see how, we'll briefly review a very similar situation in thermodynamics: recall the expression that naturally arises for incremental energy, say for the gas in a heat engine, is

\begin{equation}
d E(S, V)=T d S-P d V
\end{equation}

where \(S\) is the entropy and \(\begin{equation}
T=\partial E / \partial S
\end{equation}\) is the temperature. But \(S\) is not a handy variable in real life -- temperature \(\begin{equation}
T
\end{equation}\) is a lot easier to measure! We need an energy-like function whose incremental change is some function of \(\begin{equation}
d T, d V \text { rather than } d S, d V
\end{equation}\) The early thermodynamicists solved this problem by introducing the concept of the free energy,

\begin{equation}
F=E-T S
\end{equation}

so that \(\begin{equation}
d F=-S d T-P d V
\end{equation}\). This change of function (and variable) was important: the free energy turns out to be more practically relevant than the total energy, it's what's available to do work.

So we've transformed from a function \(\begin{equation}
E(S) \text { to a function } F(T)=F(\partial E / \partial S) \text { (ignoring } P, V
\end{equation}\) which are passive observers here).