Maxwell Relations

Maxwell Relations

If an exaxt differential dZ can be put in the form of dZ = M.dx + N.dy, then-

is known as Euler's relation. This relation can be used to obtain the Maxwell's relations.
The four most common Maxwell relations are the equalities of the second derivatives of each of the four thermodynamic potentials (some quantity used to represent some thermodynamic state in a system), with respect to their thermal natural variable (temperature or entropy) and their mechanical natural variable (pressure or volume).
The two relations are adiabatic while the other two are isothermal.
We know that- dQ = dE + PdV
For reversible process-
dQ = TdS
So, TdS = dE + PdV
or, dE = TdS − PdV
Now compare this equation with dZ = M.dx + N.dy.
By the Euler's relation we have-

Now for the other adiabatic relation-
H = E + PV
or, dH = dE + PdV + VdP     (As TdS = dE + PdV)
or, dH = TdS + VdP
By the Euler's relation we have-

The other two Maxwell relations are isothermal and can be derived as-
We know that Helmholtz free energy is-
dA = −PdV − SdT
By the Euler's relation we have-

We know that Gibbs free energy is-
dG = VdP − SdT
By the Euler's relation we have-

Differential forms of the four thermodynamic potentials and their natural variables from which Maxwell relation derived above.

Thermodynamic Potential	Differential Form	Natural Variables
Internal Energy	dE = TdS − PdV	S and V
Enthalpy	dH = TdS + VdP	S and P
Helmholtz Free Energy	dA = −PdV − SdT	V and T
Gibbs Free Energy	dG = VdP − SdT	P and T

Mnemonic of Maxwell Relation

 Share