Maxwell Relations

Relates second partial derivatives of thermodynamic energies with each other

Example

• $\mathrm{d}U=\tau \mathrm{d}\sigma -P\mathrm{d}V$
  • Start with the differential of a thermodynamic energy
• $\frac{\partial U}{\partial \sigma }=\tau ,\frac{\partial U}{\partial V}=P$
  • Take partial derivatives
• $\frac{{\partial }^{2}U}{\partial V\partial \sigma }=\frac{{\partial }^{2}U}{\partial \sigma \partial V}={\left(\frac{\partial \tau }{\partial V}\right)}_{\sigma }={\left(\frac{\partial P}{\partial \sigma }\right)}_V$
  • Equate the second partial derivatives

Thermodynamic Energy	Differential Expression	Maxwell Relation
$U$	$\mathrm{d}U=\tau \mathrm{d}\sigma -P\mathrm{d}V$	${\left(\frac{\partial \tau }{\partial V}\right)}_{\sigma }=-{\left(\frac{\partial P}{\partial \sigma }\right)}_V$
$H$	$\mathrm{d}H=\tau \mathrm{d}\sigma +V\mathrm{d}P$	${\left(\frac{\partial \tau }{\partial P}\right)}_{\sigma }={\left(\frac{\partial V}{\partial \sigma }\right)}_P$
$F$	$\mathrm{d}F=-\sigma \mathrm{d}\tau -P\mathrm{d}V$	${\left(\frac{\partial \sigma }{\partial V}\right)}_{\tau }={\left(\frac{\partial P}{\partial \tau }\right)}_V$
$G$	$\mathrm{d}G=-\sigma \mathrm{d}\tau +V\mathrm{d}P$	$-{\left(\frac{\partial \sigma }{\partial P}\right)}_{\tau }={\left(\frac{\partial V}{\partial \tau }\right)}_P$