Maxwell relations
From Free net encyclopedia
Template:Thermodynamic equations 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:
- <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.
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>
If, for any potential <math>\Phi</math> we have
- <math>A=\left(\frac{\partial \Phi}{\partial x}\right)_y</math>
- <math>B=\left(\frac{\partial \Phi}{\partial y}\right)_x</math>
But we know that:
- <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>
which gives:
- <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>