 Solution

For a simple compressible one-component equilibrium system, the fundamental thermodynamic relation isThe internal energy has natural variables $(S,V,N)$: its derivatives give $T$, $-p$ and chemical potential $\mu$. The natural variables of a thermodynamic potential identify the independent arguments whose conjugates appear directly in a potential's differential. A Legendre transform exchanges a variable for its conjugate, giving potentials suited to different controlled conditions:

For fixed composition and the appropriate closed-system constraints, equilibrium minimizes $U$ at fixed $S,V$, $H$ at fixed $S,p$, $F$ at fixed $T,V$, or $G$ at fixed $T,p$, with other allowable internal variations understood. Enthalpy is useful for constant-pressure heat exchange with only pressure–volume work. Helmholtz free energy is suited to fixed-temperature, fixed-volume systems, and Gibbs free energy to fixed-temperature, fixed-pressure phase or chemical equilibrium. Coexisting phases of one substance have equal chemical potentials.

Where a potential is twice continuously differentiable, equality of mixed derivatives gives Maxwell relations. At fixed $N$, the four usual forms areThe fixed subscripts and signs follow from the chosen natural variables. Derivatives through a nonsmooth phase boundary require separate phase descriptions rather than automatic use of the smooth single-phase formulas.

As an application, the Helmholtz relation turns pressure data into an isothermal entropy change. For an ideal gas, $(\partial p/\partial T{)}_V=nR/V$, givingCombining the same relation with $dU=TdS-pdV$ yields $(\partial U/\partial V{)}_T=T(\partial p/\partial T{)}_V-p$, which vanishes for an ideal gas and reveals attractive-energy effects for a van der Waals gas. Thermodynamic potentials organize equilibrium constraints, while Maxwell relations convert difficult entropy derivatives into measurable response derivatives.