The structure of Maxwell relations is a statement of equality among the second derivatives for continuous functions. It follows directly from the fact that the order of differentiation of an analytic function of two variables is irrelevant (Schwarz theorem). In the case of Maxwell relations the function considered is a thermodynamic potential and $x_{i))$ and $x_{j))$ are two different natural variables for that potential, we have
where the partial derivatives are taken with all other natural variables held constant. For every thermodynamic potential there are ${\textstyle {\frac {1}{2))n(n-1)}$ possible Maxwell relations where $n$ is the number of natural variables for that potential.
The four most common Maxwell relations
The four most common Maxwell relations are the equalities of the second derivatives of each of the four thermodynamic potentials, with respect to their thermal natural variable (temperature$T$, or entropy$S$) and their mechanical natural variable (pressure$P$, or volume$V$):
where the potentials as functions of their natural thermal and mechanical variables are the internal energy$U(S,V)$, enthalpy$H(S,P)$, Helmholtz free energy$F(T,V)$, and Gibbs free energy$G(T,P)$. The thermodynamic square can be used as a mnemonic to recall and derive these relations. The usefulness of these relations lies in their quantifying entropy changes, which are not directly measurable, in terms of measurable quantities like temperature, volume, and pressure.
Each equation can be re-expressed using the relationship
which are sometimes also known as Maxwell relations.
Derivations
Short derivation
This section is based on chapter 5 of.^{[1]}
Suppose we are given four real variables $(x,y,z,w)$, restricted to move on a 2-dimensional $C^{2))$ surface in $\mathbb {R} ^{4))$. Then, if we know two of them, we can determine the other two uniquely (generically).
In particular, we may take any two variables as the independent variables, and let the other two be the dependent variables, then we can take all these partial derivatives.
Proof. We can ignore $w$. Then locally the surface is just $ax+by+cz+d=0$. Then $\left({\frac {\partial x}{\partial y))\right)_{z}=-{\frac {b}{a))$, etc. Now multiply them.
Proof of Maxwell's relations:
There are four real variables $(T,S,p,V)$, restricted on the 2-dimensional surface of possible thermodynamic states. This allows us to use the previous two propositions.
It suffices to prove the first of the four relations, as the other three can be obtained by transforming the first relation using the previous two propositions.
Pick $V,S$ as the independent variables, and $E$ as the dependent variable. We have
$dE=-pdV+TdS$.
Now, $\partial _{V,S}E=\partial _{S,V}E$ since the surface is $C^{2))$, that is,
Maxwell relations are based on simple partial differentiation rules, in particular the total differential of a function and the symmetry of evaluating second order partial derivatives.
Derivation
Derivation of the Maxwell relation can be deduced from the differential forms of the thermodynamic potentials:
The differential form of internal energy U is
Since we also know that for functions with continuous second derivatives, the mixed partial derivatives are identical (Symmetry of second derivatives), that is, that
The other two Maxwell relations can be derived from differential form of enthalpy $dH=T\,dS+V\,dP$ and the differential form of Gibbs free energy $dG=V\,dP-S\,dT$ in a similar way. So all Maxwell Relationships above follow from one of the Gibbs equations.
Extended derivation
Combined form first and second law of thermodynamics,
as a statement about differential forms, and take the exterior derivative of this equation, we get
$0=dT\,dS-dP\,dV$
since $d(dU)=0$. This leads to the fundamental identity
$dP\,dV=dT\,dS.$
The physical meaning of this identity can be seen by noting that the two sides are the equivalent ways of writing the work done in an infinitesimal Carnot cycle. An equivalent way of writing the identity is
${\frac {\partial (T,S)}{\partial (P,V)))=1.$
The Maxwell relations now follow directly. For example,
The above are not the only Maxwell relationships. When other work terms involving other natural variables besides the volume work are considered or when the number of particles is included as a natural variable, other Maxwell relations become apparent. For example, if we have a single-component gas, then the number of particles N is also a natural variable of the above four thermodynamic potentials. The Maxwell relationship for the enthalpy with respect to pressure and particle number would then be:
where μ is the chemical potential. In addition, there are other thermodynamic potentials besides the four that are commonly used, and each of these potentials will yield a set of Maxwell relations. For example, the grand potential$\Omega (\mu ,V,T)$ yields:^{[2]}