Maxwell relations

The Maxwell relations in thermodynamics can be derived from the symmetry of second derivatives and the definitions of the thermodynamic potentials, or from Jacobian determinants. The most common Maxwell relations involve the potential functions <math>U</math> (the total internal energy), <math>H</math> (enthalpy), <math>A</math> (Helmholtz free energy), and <math>G</math> (Gibbs free energy), and the functions of state <math>P</math> (pressure), <math>T</math> (absolute temperature), <math>V</math> (volume), and <math>S</math> (entropy).

They are named for the physicist James Clerk Maxwell, who first presented them in his text Theory of Heat (1872). Maxwell's relations are useful in problem-solving quantities that are difficult to measure to those that are easier to work with.

Equations

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 <math>x_i</math> and <math>x_j</math> are two different natural variables for that potential,

where the partial derivatives are taken with all other natural variables held constant. For every thermodynamic potential there are <math display="inline">\frac{1}{2} n(n-1)</math> possible Maxwell relations where <math>n</math> is the number of natural variables for that potential.

The four most common Maxwell relations

400px|right|thumb|Flow chart showing the paths between the Maxwell relations. <math>P</math> is pressure, <math>T</math> temperature, <math>V</math> volume, <math>S</math> entropy, <math>\alpha</math> [[coefficient of thermal expansion, <math>\kappa</math> compressibility, <math>C_V</math> heat capacity at constant volume, <math>C_P</math> heat capacity at constant pressure.

]]

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 <math>T</math>, or entropy and their mechanical natural variable (pressure <math>P</math>, or volume

where the potentials as functions of their natural thermal and mechanical variables are the internal energy <math>U(S, V)</math>, enthalpy <math>H(S, P)</math>, Helmholtz free energy <math>F(T, V)</math>, and Gibbs free energy <math>G(T, P)</math>. 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 reciprocal relation<math display="block">\left(\frac{\partial y}{\partial x}\right)_z

1\biggl/\left(\frac{\partial x}{\partial y}\right)_z.</math>

Derivations

First derivation

For a given set of four real variables <math>(x, y, z, w)</math>, restricted to move on a 2-dimensional <math>C^2</math> surface in <math>\R^4</math>. Knowing two of them enables the remaining two to be determined. In particular, one may take any two variables as the independent variables, and let the other two be the dependent variables, then take all these partial derivatives.

This derivation exploits the reciprocal relation

<math> \left(\frac{\partial w}{\partial y}\right)_{z} = \left(\frac{\partial w}{\partial x}\right)_{z} \left(\frac{\partial x}{\partial y}\right)_{z}, </math>

and the cyclical relation

<math> \left(\frac{\partial x}{\partial y}\right)_z \left(\frac{\partial y}{\partial z}\right)_x \left(\frac{\partial z}{\partial x}\right)_y = -1. </math>

There are four real variables <math>(T, S, P, V)</math>, 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 <math>V, S</math> as the independent variables, and <math>U</math> as the dependent variable. Start with the thermodynamic identity <math> dU = -PdV + TdS </math>.

Now, <math>\partial_{V,S}U = \partial_{S, V}U</math> since the surface is <math>C^2</math>, that is,<math display="block"> \left[\frac{\partial}{\partial V} \left(\frac{\partial U}{\partial S}\right)_{V} \right]_{S} = \left[\frac{\partial}{\partial S} \left(\frac{\partial U}{\partial V}\right)_{S} \right]_{V}, </math>which yields the result.

Derivation in terms of Jacobians

This derivation is as follows.

<math display="block"> \begin{align}

\left(\frac{\partial N}{\partial V}\right)_{\mu, T} &=& \left(\frac{\partial P}{\partial \mu}\right)_{V,T} &=& -\frac{\partial^2 \Omega }{\partial \mu \partial V}\\

\left(\frac{\partial N}{\partial T}\right)_{\mu, V} &=& \left(\frac{\partial S}{\partial \mu}\right)_{V,T} &=& -\frac{\partial^2 \Omega }{\partial \mu \partial T}\\

\left(\frac{\partial P}{\partial T}\right)_{\mu, V} &=& \left(\frac{\partial S}{\partial V}\right)_{\mu,T} &=& -\frac{\partial^2 \Omega }{\partial V \partial T}

\end{align}</math>