Maxwell Relations: Where They Come From and How to Use Them
Maxwell relations look like four formulas to memorise. They are actually one idea applied four times, and deriving them beats remembering them.
The one idea underneath all four
For any well-behaved function of two variables, the order of mixed partial differentiation does not matter:
Thermodynamic potentials are state functions, so their differentials are exact and this condition holds. Every Maxwell relation is that statement applied to one potential. Once you see this, there is nothing to memorise — only a derivation to reproduce, which takes about thirty seconds.
The four potentials and their natural variables
| Potential | Differential | Natural variables |
|---|---|---|
| Internal energy U | dU = T dS − P dV | S, V |
| Enthalpy H = U + PV | dH = T dS + V dP | S, P |
| Helmholtz A = U − TS | dA = −S dT − P dV | T, V |
| Gibbs G = H − TS | dG = −S dT + V dP | T, P |
Notice that each is obtained from another by a Legendre transform, which swaps one variable for its conjugate. That is why the four differentials look so similar and why their signs alternate in a regular way.
Deriving one relation, completely
Take the Gibbs energy, whose natural variables are T and P:
Comparing with the general form dG = (∂G/∂T)P dT + (∂G/∂P)T dP gives
Now differentiate each again with respect to the other variable and set the results equal:
That is the Maxwell relation from G. The same three steps applied to U, H and A give the other three.
All four, for reference
Two standard applications
Entropy change on compression
To find how entropy varies with pressure at constant temperature, use the Gibbs relation. For an ideal gas, V = nRT/P, so (∂V/∂T)P = nR/P and
Integrating gives ΔS = −nR ln(P2/P1) — the familiar isothermal result, now derived rather than recalled.
The internal pressure of a gas
The quantity (∂U/∂V)T measures how internal energy responds to volume at fixed temperature. Using the relation from A gives
For an ideal gas, P = nRT/V makes T(∂P/∂T)V exactly equal to P, so the internal pressure is zero. That is the formal proof that ideal gas internal energy depends only on temperature — a result usually quoted and rarely derived, and therefore a good discriminating question.
Frequently asked questions
Do I have to memorise all four?
No, and it is safer not to. Memorise the four differentials, which are short, and derive whichever relation the question needs. Memorised relations get sign errors under pressure; derived ones do not.
How do I keep the signs straight?
Take them from the differential you are working with. In dG = −S dT + V dP the entropy term carries a minus sign, and that minus propagates into the resulting relation. Never try to remember signs independently of the differential they came from.
What are natural variables and why do they matter?
They are the pair for which the potential's differential takes its simplest form. Expressed in those variables, the potential contains complete thermodynamic information; in other variables it does not. This is also why G, whose natural variables T and P are the ones usually controlled in a laboratory, is the most used potential in chemistry.
Which exam asks this most?
CSIR-NET and GATE both use Maxwell relations in derivation and numerical questions. IIT-JAM tends to stay with the differentials themselves and simpler applications.
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