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Concept
Maxwell relations
Summary
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.
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 and 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 possible Maxwell relations where is the number of natural variables for that potential.
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 , or entropy ) and their mechanical natural variable (pressure , or volume ):
where the potentials as functions of their natural thermal and mechanical variables are the internal energy , enthalpy , Helmholtz free energy , and Gibbs free energy . 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.
This section is based on chapter 5 of.
Suppose we are given four real variables , restricted to move on a 2-dimensional surface in . Then, if we know two of them, we can determine the other two uniquely (generically).
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