Thermodynamic Potentials and Maxwell Relations

Choosing the Right Energy

Internal energy is natural in the variables $S$ and $V$, but experiments rarely control entropy. Legendre transforms produce three further potentials, each natural for a different pair of controlled variables:

Their differentials follow directly:

$$dU = T\,dS - P\,dV, \quad dH = T\,dS + V\,dP$$ $$dF = -S\,dT - P\,dV, \quad dG = -S\,dT + V\,dP$$

Each says which variables the potential naturally depends on, and reading off the coefficients gives all the first derivatives at once, for instance $S = -(\partial G/\partial T)_P$ and $V = (\partial G/\partial P)_T$.

The Maxwell relations come free from the equality of mixed second derivatives. Since $\partial^2 F/\partial T\partial V$ can be taken in either order,

$$\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V$$

with three siblings from the other potentials. Their practical value is large: the left side involves entropy, which no instrument measures directly, while the right side is a straightforward $P$-$V$-$T$ measurement. Maxwell relations convert what you cannot measure into what you can.

For open systems the chemical potential enters as $\mu = (\partial G/\partial N)_{T,P}$, and each potential gains a $\mu\,dN$ term. Matter flows spontaneously from high $\mu$ to low, exactly as heat flows from high $T$ to low.

Common pitfall: treating a potential as an amount of stored energy. Only $U$ is that. $F$ and $G$ are constructed so that their decrease measures available work under particular constraints, $F$ the maximum total work at fixed $T$, $G$ the maximum non-expansion work at fixed $T$ and $P$.