Maxwell relations

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Maxwell's relations are a set of equations in thermodynamics which are derivable from the definitions of the four thermodynamic potentials. They involve the following quantities:

V is volume
T is temperature
P is pressure
S is entropy

Ignoring the chemical potential, they are:

[\left(\frac\right)_S =-\left(\frac\right)_V]

[\left(\frac\right)_S =+\left(\frac\right)_P]

[\left(\frac\right)_T =+\left(\frac\right)_V]

[\left(\frac\right)_T =-\left(\frac\right)_P]

Each equation can be re-expressed using the relationship

[\left(\frac\right)_z=1\left/\left(\frac\right)_z\right.]

which are sometimes also known as Maxwell relations.

Derivation of Maxwell's relations

From the theory of the thermodynamic potentials, it is known that the following relationships are true for a single phase simple fluid with a constant number of particles:

[+T=\left(\frac\right)_V =\left(\frac\right)_P]

[-P=\left(\frac\right)_S =\left(\frac\right)_T]

[+V=\left(\frac\right)_S =\left(\frac\right)_T]

[-S=\left(\frac\right)_P =\left(\frac\right)_V]

For a potential [\Phi(x,y)] we can define

[A=\left(\frac\right)_y]

[B=\left(\frac\right)_x]

Now we can use the symmetry of second derivatives to get

[\left(\frac\left(\frac\right)_y\right)_x=\left(\frac\left(\frac\right)_x\right)_y]

This gives a Maxwell relation on the form:

[\left(\frac\right)_x=\left(\frac\right)_y]

which are just Maxwell's relations. For example, for the potential [U] we have [T=(\partial U/\partial S)_V] and [-P=(\partial U/\partial V)_S] so that [(\partial T/\partial V)_S = -(\partial P/\partial S)_V]

Maxwell relations

Derivation of Maxwell's relations

See also