Maxwell Relations
#Physics
Relates second partial derivatives of thermodynamic energies with each other
Example
- $\displaystyle \mathrm{d}U=\tau \mathrm{d}\sigma-P\mathrm{d}V$
- Start with the differential of a thermodynamic energy
- $\displaystyle \frac{ \partial U }{ \partial \sigma } =\tau,\frac{ \partial U }{ \partial V } =P$
- Take partial derivatives
- $\displaystyle \frac{ \partial^2 U}{ \partial V\partial \sigma}=\frac{ \partial^2 U}{ \partial \sigma\partial V}=\left( \frac{ \partial \tau }{ \partial V } \right){\sigma}=\left( \frac{ \partial P }{ \partial \sigma } \right){V}$
- Equate the second partial derivatives
| Thermodynamic Energy | Differential Expression | Maxwell Relation |
|---|---|---|
| $\displaystyle U$ | $\displaystyle \mathrm{d}U=\tau \mathrm{d}\sigma-P\mathrm{d}V$ | $\displaystyle \left( \frac{ \partial \tau }{ \partial V } \right){\sigma}=-\left( \frac{ \partial P }{ \partial \sigma } \right){V}$ |
| $\displaystyle H$ | $\displaystyle \mathrm{d}H=\tau \mathrm{d}\sigma+V\mathrm{d}P$ | $\displaystyle \left( \frac{ \partial \tau }{ \partial P } \right){\sigma}=\left( \frac{ \partial V }{ \partial \sigma } \right){P}$ |
| $\displaystyle F$ | $\displaystyle \mathrm{d}F=-\sigma \mathrm{d}\tau-P\mathrm{d}V$ | $\displaystyle \left( \frac{ \partial \sigma }{ \partial V } \right){\tau}=\left( \frac{ \partial P }{ \partial \tau } \right){V}$ |
| $\displaystyle G$ | $\displaystyle \mathrm{d}G=-\sigma \mathrm{d}\tau+V\mathrm{d}P$ | $\displaystyle -\left( \frac{ \partial \sigma }{ \partial P } \right){\tau}=\left( \frac{ \partial V }{ \partial \tau } \right){P}$ |