Maxwell relations explained

400px|right|thumb|Flow chart showing the paths between the Maxwell relations.

P is pressure, T temperature, V volume, S entropy, \alpha coefficient of thermal expansion, \kappa compressibility, C _{V} heat capacity at constant volume, C _{P} heat capacity at constant pressure.

Maxwell's relations are a set of equations in thermodynamics which are derivable from the symmetry of second derivatives and from the definitions of the thermodynamic potentials. These relations are named for the nineteenth-century physicist James Clerk Maxwell.

Equations

See also: symmetry of second derivatives. 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 $\backslash frac\; n(n-1)$ possible Maxwell relations where

n is the number of natural variables for that potential.

The four most common Maxwell relations

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$$\backslash left(\backslash frac\backslash right)\_z=1\backslash left/\backslash left(\backslash frac\backslash right)\_z\backslash right.$$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 \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.

Proposition:

\left( (\partial w)/(\partial y) \right) _{z} = \left( (\partial w)/(\partial x) \right) _{z} \left( (\partial x)/(\partial y) \right) _{z}

Proof: This is just the chain rule.

Proposition:

\left( (\partial x)/(\partial y) \right) _{z} \left( (\partial y)/(\partial z) \right) _{x} \left( (\partial z)/(\partial x) \right) _{y} = -1

Proof. We can ignore

w . Then locally the surface is just ax + by + cz + d = 0 . Then \left( (\partial x)/(\partial y) \right) _{z} =- (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 havedE = - pdV + TdS .

Now,

\partial _{V,S} E = \partial _{S,} E since the surface is C ^{2} , that is,$$\backslash left(\backslash frac\backslash right)\_\; =\; \backslash left(\backslash frac\backslash right)\_$$which yields the result.

Another derivation

Based on.^[2]

Since

dU = TdS - PdV , around any cycle, we have$$0\; =\; \backslash oint\; dU\; =\; \backslash oint\; TdS\; -\; \backslash oint\; PdV$$Take the cycle infinitesimal, we find that (\partial(P, V))/(\partial(T, S)) = 1 . That is, the map is area-preserving. By the chain rule for Jacobians, for any coordinate transform (x, y) , we have$$\backslash frac\; =\; \backslash frac$$Now setting (x, y) to various values gives us the four Maxwell relations. For example, setting (x, y) = (P, S) gives us \left( (\partial T)/(\partial P) \right) _{S} = \left( (\partial V)/(\partial S) \right) _{P}

Extended derivations

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 based on Jacobians

If we view the first law of thermodynamics,$$dU\; =\; T\; \backslash ,\; dS\; -\; P\; \backslash ,\; dV$$as a statement about differential forms, and take the exterior derivative of this equation, we get$$0\; =\; dT\; \backslash ,\; dS\; -\; dP\; \backslash ,\; dV$$since

d(dU) = 0 . This leads to the fundamental identity$$dP\; \backslash ,\; dV\; =\; dT\; \backslash ,\; dS.$$

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$$\backslash frac\; =\; 1.$$

The Maxwell relations now follow directly. For example,$$\backslash left(\backslash frac\; \backslash right)\_T\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash left(\backslash frac\; \backslash right)\_V,$$The critical step is the penultimate one. The other Maxwell relations follow in similar fashion. For example,$$\backslash left(\backslash frac\; \backslash right)\_S\; =\; \backslash frac\; =\; \backslash frac\; =\; -\; \backslash left(\backslash frac\; \backslash right)\_V.$$

General Maxwell relationships

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:

$$\backslash left(\backslash frac\backslash right)\_\; =\backslash left(\backslash frac\backslash right)\_\backslash qquad=\backslash frac$$

\Omega(\mu, V, T) yields:^[3]

See also

• Table of thermodynamic equations
• Thermodynamic equations

Notes and References

1. Book: Pippard, A. B. . Elements of Classical Thermodynamics:For Advanced Students of Physics . 1957-01-01 . Cambridge University Press . 978-0-521-09101-5 . 1st . Cambridge . English.
2. Ritchie . David J. . 2002-02-01 . Answer to Question #78. A question about the Maxwell relations in thermodynamics . American Journal of Physics . en . 70 . 2 . 104–104 . 10.1119/1.1410956 . 0002-9505.
3. Web site: Thermodynamic Potentials . live . https://web.archive.org/web/20221219112005/https://www.oulu.fi/tf/statfys/lectures_old/english/therpot.pdf . 19 December 2022 . University of Oulu.