= Solution
For a simple compressible one-component equilibrium system, the fundamental thermodynamic relation is
$$
dU=T\,dS-p\,dV+\mu\,dN.
$$
The <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:
|| potential
|| definition
|| natural variables
|| differential
| <internal energy>
| $U$
| $S,V,N$
| $dU=T\,dS-p\,dV+\mu\,dN$
| <enthalpy>
| $H=U+pV$
| $S,p,N$
| $dH=T\,dS+V\,dp+\mu\,dN$
| <Helmholtz free energy>
| $F=U-TS$
| $T,V,N$
| $dF=-S\,dT-p\,dV+\mu\,dN$
| <Gibbs free energy>
| $G=U+pV-TS$
| $T,p,N$
| $dG=-S\,dT+V\,dp+\mu\,dN$
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 are
$$
\boxed{\left(\frac{\partial T}{\partial V}\right)_S
=-\left(\frac{\partial p}{\partial S}\right)_V,
\qquad
\left(\frac{\partial T}{\partial p}\right)_S
=\left(\frac{\partial V}{\partial S}\right)_p,}
$$
$$
\boxed{\left(\frac{\partial S}{\partial V}\right)_T
=\left(\frac{\partial p}{\partial T}\right)_V,
\qquad
\left(\frac{\partial S}{\partial p}\right)_T
=-\left(\frac{\partial V}{\partial T}\right)_p.}
$$
The 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>, $(\partial p/\partial T)_V=nR/V$, giving
$$
\boxed{\Delta S_{T}=\int_{V_1}^{V_2}\left(\frac{\partial p}{\partial T}\right)_VdV
=nR\ln\frac{V_2}{V_1}.}
$$
Combining the same relation with $dU=T\,dS-p\,dV$ 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. \b[<Thermodynamic potentials> organize equilibrium constraints, while <Maxwell relations> convert difficult <entropy> <derivatives> into measurable response <derivatives>.]
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