# 6. The solution of thermodynamic problems

With $$U$$, $$A$$, $$H$$ and $$G$$ in hand we have potentials as a functions of whichever variable pair we want: $$S$$ and $$V$$, to $$T$$ and $$P$$. Additional Legendre transforms will provide us with further potentials in case we have other variables (such as surface area $$A$$, length $$L$$, magnetic moment $$M$$, etc.).

Thermodynamic problems always involve computing a variable of interest. It may be a derivative if it is an intensive variable, or even a second derivative (higher derivatives are rarely of interest).

Example

1st order ones like

$\left( \dfrac{\partial G}{\partial P} \right)_T=V$

or 2nd order ones like

$\left( \dfrac{\partial^2 G}{\partial P^2} \right)_T= \left( \dfrac{\partial V}{\partial P} \right)_T = \kappa V$

The solution procedure is thus:

1. Select the derivative or variable to be computed;
2. Select the potential representation that makes it easiest, or corresponds to variables you already have in hand.
3. Manipulate the thermodynamic derivative you know to get the one you want.

Easy as 1-2-3! We now turn to two methods to manipulate the thermodynamic derivations: