Maxwell relations

Maxwell relations show that various other quantities also vanish.

Important examples are the Maxwell relations and the relations between heat capacities.

From these we get the Maxwell relations.

The four most common Maxwell relations are:

The Maxwell relations are statements of equality among the second derivatives of the thermodynamic potentials.

Maxwell relations are equalities involving the second derivatives of thermodynamic potentials with respect to their natural variables.

With the substitution of two of the Maxwell relations and the definition of chemical potential, this is transformed into:

Furthermore (Maxwell relations):

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.

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.

It is seen that for every thermodynamic potential there are n(n 1)/2 possible Maxwell relations where n is the number of natural variables for that potential.

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.

The experimental usefulness of these functions is restricted to conditions where certain variables (T, and V or external p) are held constant, although they also have theoretical importance in deriving Maxwell relations.

A mnemonic used by students to remember the Maxwell relations (in thermodynamics) is "Good Physicists Have Studied Under Very Fine Teachers", which helps them remember the order of the variables in the square, in clockwise direction.

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):

James Clerk Maxwell makes public the thought experiment which will become known as Maxwell's demon in the philosophy of thermal and statistical physics, in his book Theory of Heat and establishes Maxwell relations, statements of equality among the second derivatives of the thermodynamic potentials with respect to different thermodynamic variables.