Maxwell relations

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Thermodynamic equations
Laws of thermodynamics
Conjugate variables
Thermodynamic potentials
Material properties
Maxwell relations
Bridgman's equations
Exact differential
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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 are a direct consequence of the fact that these expressions are exact differentials

The most well-known relations involve the following quantities:

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

When other work term besides the volume work are considered, additional relations can be derived for example including the electromotive force or the chemical potential

Ignoring these, the Maxwell relations are:

<math>

\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V </math>

<math>

\left(\frac{\partial T}{\partial P}\right)_S = +\left(\frac{\partial V}{\partial S}\right)_P </math>

<math>

\left(\frac{\partial S}{\partial V}\right)_T = +\left(\frac{\partial P}{\partial T}\right)_V </math>

<math>

\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P </math>

Each equation can be re-expressed using the relationship

<math>\left(\frac{\partial y}{\partial x}\right)_z

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

which are sometimes also known as Maxwell relations.

[edit] 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:

<math>

+T=\left(\frac{\partial U}{\partial S}\right)_V

 =\left(\frac{\partial H}{\partial S}\right)_P

</math>

<math>

-P=\left(\frac{\partial U}{\partial V}\right)_S

 =\left(\frac{\partial F}{\partial V}\right)_T

</math>

<math>

+V=\left(\frac{\partial H}{\partial P}\right)_S

 =\left(\frac{\partial G}{\partial P}\right)_T

</math>

<math>

-S=\left(\frac{\partial G}{\partial T}\right)_P

 =\left(\frac{\partial F}{\partial T}\right)_V

</math>

For a potential <math>\Phi(x,y)</math> we can define

<math>A=\left(\frac{\partial \Phi}{\partial x}\right)_y</math>

<math>B=\left(\frac{\partial \Phi}{\partial y}\right)_x</math>

Now we can use the symmetry of second derivatives to get

<math>

\left(\frac{\partial}{\partial y} \left(\frac{\partial \Phi}{\partial x}\right)_y \right)_x = \left(\frac{\partial}{\partial x} \left(\frac{\partial \Phi}{\partial y}\right)_x \right)_y </math>

This gives a Maxwell relation on the form:

<math>

\left(\frac{\partial A}{\partial y}\right)_x = \left(\frac{\partial B}{\partial x}\right)_y </math>

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