The Maxwell relations are derived from Euler’s reciprocity relation. The relations are expressed in partial differential form. The Maxwell relations consists of the characteristic functions: internal energy U, enthalpy H, Helmholtz free energy F, and Gibbs free energy G and thermodynamic parameters: entropy S, pressure P, volume V, and temperature T. Following is the table of Maxwell relations for secondary derivatives:
\(\begin{array}{l}+(\frac{\partial T}{\partial V})_{S}=-(\frac{\partial P}{\partial S})_{V}=\frac{\partial^2 U}{\partial S\partial V}\end{array} \) |
\(\begin{array}{l}+(\frac{\partial T}{\partial P})_{S}=+(\frac{\partial V}{\partial S})_{P}=\frac{\partial^2 H}{\partial S\partial P}\end{array} \) |
\(\begin{array}{l}+(\frac{\partial S}{\partial V})_{T}=+(\frac{\partial P}{\partial T})_{V}=\frac{\partial^2 F}{\partial T\partial V}\end{array} \) |
\(\begin{array}{l}-(\frac{\partial S}{\partial P})_{T}=+(\frac{\partial V}{\partial T})_{V}=\frac{\partial^2 G}{\partial T\partial P}\end{array} \) |
What are Maxwell’s relations?
These are the set of thermodynamics equations derived from a symmetry of secondary derivatives and from thermodynamic potentials. These relations are named after James Clerk Maxwell, who was a 19th-century physicist.
Derivation of Maxwell’s relations
Maxwell’s relations can be derived as:
And
From
T=
And
Common forms of Maxwell’s relations
| Function | Differential | Natural variables | Maxwell Relation |
| U | dU = TdS – PdV | S, V | \(\begin{array}{l}(\frac{\partial T}{\partial V})_{S}=-(\frac{\partial P}{\partial S})_{V}\end{array} \) |
| H | dH = TdS + VdP | S, P | \(\begin{array}{l}(\frac{\partial T}{\partial P})_{S}=(\frac{\partial V}{\partial S})_{P}\end{array} \) |
| F | dF = -PdV – SdT | V, T | \(\begin{array}{l}(\frac{\partial P}{\partial T})_{V}=(\frac{\partial S}{\partial V})_{T}\end{array} \) |
| G | dG = VdP – SdT | P, T | \(\begin{array}{l}(\frac{\partial V}{\partial T})_{P}=-(\frac{\partial S}{\partial P})_{T}\end{array} \) |
Where,
T is the temperature
S is the entropy
P is the pressure
V is the volume
U is the internal energy
H is the entropy
G is the Gibbs free energy
F is the Helmholtz free energy
With respect to pressure and particle number, enthalpy and Maxwell’s relation can be written as:
Solved Examples
Example 1:
Prove that
Solution:
Combining first and second laws:
dU = TdS – pdV
Diving both the sides by dV
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