Thermodynamic Potentials and Maxwell Relations

The first law supplies an energy that changes by heat and work; the second law supplies an entropy and a temperature. Combined for a reversible process, they collapse into a single differential relation among state functions, the fundamental relation. Everything measurable about a simple system — its equation of state, its heat capacities, its response to fields — is a partial derivative of that relation or of one of its Legendre transforms. This lesson builds the four standard potentials, reads off the natural variables of each, and derives the Maxwell relations that make otherwise inaccessible derivatives computable.

The fundamental relation

For a reversible process the second law gives $δ Q_{rev} = T d S$ and the first law gives $δ W = - P d V$ for a simple fluid, plus $μ d N$ if particles are exchanged. Substituting into $d U = δ Q + δ W$ ,

d U = T d S - P d V + μ d N .

Though derived along a reversible path, this is a relation among state functions and their differentials, so it holds between any two neighboring equilibrium states however they are connected. It is the fundamental relation in the energy representation, and it identifies the intensive variables as partial derivatives of $U (S, V, N)$ :

T = (\frac{\partial U}{\partial S})_{V, N}, P = - (\frac{\partial U}{\partial V})_{S, N}, μ = (\frac{\partial U}{\partial N})_{S, V} .

The variables $(S, V, N)$ are the natural variables of $U$ : given $U$ as a function of them, every other thermodynamic quantity follows by differentiation, with no integration constant left undetermined. A relation like $U (S, V, N)$ that contains the complete thermodynamics is a fundamental equation; an equation of state such as $P V = N k_{B} T$ is only a partial derivative of one and does not by itself determine the rest.

Because $U$ , $S$ , $V$ , $N$ are all extensive, $U (λ S, λV, λ N) = λ U (S, V, N)$ . Applying Euler's theorem for first-order homogeneous functions to this scaling gives the Euler relation

U = T S - P V + μ N,

and differentiating it and subtracting the fundamental relation yields the Gibbs–Duhem relation $S d T - V d P + N d μ = 0$ , which shows the three intensive variables $T$ , $P$ , $μ$ are not independent: fixing two determines the third.

Legendre transforms and the four potentials

The energy $U (S, V, N)$ has entropy among its natural variables, but $S$ is neither directly controlled nor directly measured in the laboratory, where temperature is the accessible variable. A Legendre transform exchanges a natural variable for its conjugate derivative without losing information, replacing $S$ by $T = \partial U / \partial S$ .

Carrying out the transform on each conjugate pair generates the standard potentials. Each subtracts a product of conjugates from $U$ and thereby swaps one extensive natural variable for its intensive partner.

Internal energy $U (S, V, N)$ , with $d U = T d S - P d V + μ d N$ .
Helmholtz free energy $F = U - T S$ , natural variables $(T, V, N)$ , with $d F = - S d T - P d V + μ d N .$
Enthalpy $H = U + P V$ , natural variables $(S, P, N)$ , with $d H = T d S + V d P + μ d N .$
Gibbs free energy $G = U - T S + P V$ , natural variables $(T, P, N)$ , with $d G = - S d T + V d P + μ d N .$
Grand potential $Φ = U - T S - μ N = F - μ N$ , natural variables $(T, V, μ)$ , with $d Φ = - S d T - P d V - N d μ .$

From the Euler relation $G = μ N$ , so the Gibbs energy per particle is the chemical potential, and $Φ = - P V$ , so the grand potential is minus the pressure times the volume. These identities are used constantly in the ensemble lessons: $F = - k_{B} T ln Z$ ties the Helmholtz energy to the canonical partition function, and $Φ = - k_{B} T ln Ξ$ ties the grand potential to the grand partition function.

The four potentials linked by Legendre transforms; each arrow swaps one conjugate pair,

S \leftrightarrow T

subtracting

T S

and

V \leftrightarrow P

adding

P V

, so opposite corners differ by both swaps.

Natural variables and minimum principles

Each potential is the quantity minimized at equilibrium when its natural intensive variables are held fixed by reservoirs. The entropy-maximum principle $d S \geq 0$ for an isolated system translates, by Legendre transform, into a minimum principle for each potential under its matching constraints.

Potential	Symbol	Natural variables	Differential	Minimized at fixed
Internal energy	$U$	$S, V, N$	$T d S - P d V + μ d N$	$S, V, N$
Helmholtz	$F$	$T, V, N$	$- S d T - P d V + μ d N$	$T, V, N$
Enthalpy	$H$	$S, P, N$	$T d S + V d P + μ d N$	$S, P, N$
Gibbs	$G$	$T, P, N$	$- S d T + V d P + μ d N$	$T, P, N$
Grand potential	$Φ$	$T, V, μ$	$- S d T - P d V - N d μ$	$T, V, μ$

The Helmholtz energy is minimized by a system at fixed temperature and volume, which is the setting of the canonical ensemble; the Gibbs energy at fixed temperature and pressure, the setting of most chemistry; the grand potential at fixed temperature, volume, and chemical potential, the setting of the grand-canonical ensemble. Each minimum expresses the same competition: $F = U - T S$ is lowered either by reducing energy or by raising entropy, and temperature sets the exchange rate between the two. At low $T$ the energy term dominates and the system orders; at high $T$ the entropy term dominates and it disorders.

The Helmholtz free energy

F = U - T S

starts at

U

when

T = 0

and falls below it as temperature rises; the gap between the two curves is the entropy term

T S

, so higher

T

favors the disordered (high-

S

) state.

Maxwell relations

Each potential's differential has the form $d Ψ = X d y + Y d z$ with $X$ and $Y$ the first partial derivatives. Because $Ψ$ is a state function, its mixed second partials are equal, $\partial^{2} Ψ/ \partial y \partial z = \partial^{2} Ψ/ \partial z \partial y$ , which forces a relation between the cross-derivatives of $X$ and $Y$ . These are the Maxwell relations. From the four potentials (at fixed $N$ ):

(\frac{\partial T}{\partial V})_{S} = - (\frac{\partial P}{\partial S})_{V} (U), (\frac{\partial S}{\partial V})_{T} = (\frac{\partial P}{\partial T})_{V} (F),

(\frac{\partial T}{\partial P})_{S} = (\frac{\partial V}{\partial S})_{P} (H), (\frac{\partial S}{\partial P})_{T} = - (\frac{\partial V}{\partial T})_{P} (G) .

A mnemonic organizes all four. Arrange the four natural variables $S, T, P, V$ on the sides of a square and the four potentials on its corners, ordered so that each potential sits between its two natural variables. The relations are read by following the corners and edges; the two diagonal arrows fix the signs.

The thermodynamic square: each potential sits on the edge between its two natural variables (

U

between

S, V

;

F

between

V, T

;

G

between

T, P

;

H

between

P, S

); conjugate pairs

S

–

T

and

V

–

P

sit on opposite corners, and the diagonal arrows fix the Maxwell-relation signs.

Extracting inaccessible derivatives

The practical value of the Maxwell relations is that entropy is not directly measurable, but the relations trade an entropy derivative for a derivative of the equation of state, which is. The Helmholtz relation is the workhorse example.

The same maneuver gives the general energy equation. From $d U = T d S - P d V$ at fixed $T$ ,

(\frac{\partial U}{\partial V})_{T} = T (\frac{\partial S}{\partial V})_{T} - P = T (\frac{\partial P}{\partial T})_{V} - P,

using the Helmholtz relation again. For an ideal gas the right-hand side is $T \cdot N k_{B} / V - P = 0$ , confirming $(\partial U / \partial V)_{T} = 0$ without appeal to the Joule experiment — the temperature-only dependence of the ideal-gas energy follows from its equation of state alone. For a van der Waals gas the same formula gives a nonzero $(\partial U / \partial V)_{T} = a N^{2} / V^{2}$ , the internal-pressure term that makes a real gas cool on free expansion.

A Maxwell relation converts an unmeasurable entropy derivative on the left into a derivative of the equation of state on the right, which follows directly from

P (T, V)

Combined with the response-function identities of the next lesson, the Maxwell relations reduce the entire thermodynamics of a simple substance to three measured inputs: the equation of state $P (T, V)$ , one heat capacity as a function of temperature, and a single reference entropy. Everything else is a derivative.

Summary

The fundamental relation $d U = T d S - P d V + μ d N$ combines the first and second laws; $U (S, V, N)$ is a fundamental equation containing the whole thermodynamics, with $T, P, μ$ its first derivatives. The Euler relation $U = T S - P V + μ N$ and Gibbs–Duhem $S d T - V d P + N d μ = 0$ follow from extensivity.
Legendre transforms swap conjugate pairs to give $F = U - T S$ , $H = U + P V$ , $G = U - T S + P V$ , and $Φ = F - μ N$ , each with its own natural variables and each minimized at equilibrium under those variables. $G = μ N$ and $Φ = - P V$ .
Equality of mixed second partials of each potential yields the four Maxwell relations, summarized by the thermodynamic square.
The Maxwell relations convert unmeasurable entropy derivatives into equation-of- state derivatives; $(\partial S / \partial V)_{T} = (\partial P / \partial T)_{V}$ gives the ideal-gas $N k_{B} / V$ and the general energy equation $(\partial U / \partial V)_{T} = T (\partial P / \partial T)_{V} - P$ .