Free energy

Free energy is to a constant $T$ system what $E$ is to a mechanical system.
Helmholtz free energy is the available energy to do work at constant $T$ and $V$ .
In a system kept at constant $T$ and $V$ , interacting with the surroundings only through an exchange of heat (i.e. no work), the Helmholtz free energy never increases.
In an isolated system at constant $T$ and $V$ , Helmholtz free energy is minimized in equilibrium.
Free energy refers to the free energy of the system only $F = F_{system}$ .

Four thermodynamic potentials, each natural for a different set of held-fixed variables:

Potential	Definition	Natural variables
Helmholtz $F$	$E - T S$	$T, V, N$
Enthalpy $H$	$E + P V$	$S, P, N$
Gibbs $G$	$E + P V - T S$	$T, P, N$
Grand $Ω$	$E - T S - μ N$	$T, V, μ$

See Euler equation and Gibbs-Duhem for the underlying structure.

Helmholtz free energy $F = E - T S$

Differential (using $d E = T d S - P d V + μ d N$ and chain rule on $T S$ ):

d F = d E - T d S - S d T = - P d V + μ d N - S d T

$d S$ drops out — $F = F (V, N, T)$ . This is the Legendre transform replacing $S$ with $T$ .

Maxwell relations:

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

Why it's useful: $V, N, T$ are easily measurable. Contrast with $E (S, V, N)$ and $S (E, V, N)$ where entropy is hard to measure directly.

Free energy and work

For an isolated system at constant $T$ and $V$ , interacting with surroundings only by heat exchange — see Helmholtz work inequality derivation:

W \leq - Δ F

equality iff reversible. Free energy is literally the energy free to do work.

When $W = 0$ : $Δ F \leq 0$ — Helmholtz free energy never increases at constant $T, V$ .

Equilibrium = state of minimum $F$ (for isolated system at constant $T, V$ ).

Example — two gases separated by a partition at constant $T$ , minimising $F$ :

d F = - P_{1} d V_{1} - P_{2} d (V - V_{1}) = (P_{2} - P_{1}) d V_{1} = 0 ⟹ P_{1} = P_{2}

Same result as maximising total entropy — the two are equivalent.

Connection to partition function

From the canonical ensemble, $S = ⟨ E ⟩ / T + k_{B} \ln Z$ , so:

F = - k_{B} T \ln Z

Equivalently: $e^{- β F} = Z = \sum e^{- β E}$ — free energy equals energy when there is only one microstate.

For monatomic ideal gas:

F = - N k_{B} T (\ln \frac{V}{N} + \frac{3}{2} \ln \frac{2 π m k_{B} T}{h^{2}} + 1)

Checks: $- {(\frac{\partial F}{\partial T})}_{V, N} = S$ ✓ and $- {(\frac{\partial F}{\partial V})}_{T, N} = \frac{N k_{B} T}{V} = P$ ✓ (ideal gas law)

Spring and gas example

See Spring-gas equilibrium derivation. Piston+gas system in heat bath — equilibrium via $\frac{\partial F}{\partial x} = 0$ :

\frac{\partial F}{\partial x} = \underset{= F_{piston} = - k x}{\underset{⏟}{\frac{\partial E}{\partial x}}} - T \frac{\partial S}{\partial x} = F_{piston} - F_{gas} = 0

Minimising $F$ is equivalent to balancing forces.

Energy (non)-minimisation

Total energy is conserved — it never minimises. What minimises is free energy:

F = - T [S_{sys} + \frac{- E}{T}] = - T [S_{sys} + S_{surr}]

Minimising $F$ = maximising total entropy = 2nd law. "Energy minimisation" (ball rolling downhill, spring with friction) is always free energy minimisation driven entirely by entropy.

Free energy is a property of the ensemble, not of a single microstate. $F = F_{e} x t s y s t e m$ only — minimise the system's $F$ , ignore the bath. Use $F$ for constant $T, V$ ; use $G$ for constant $T, P$ .

Gibbs free energy $G = E + P V - T S$

d G = d E + d (P V) - d (T S) = V d P - S d T + μ d N

$G = G (P, N, T)$ . Maxwell relations:

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

Natural potential for chemistry and biology: reactions at constant $T, P$ are spontaneous iff $Δ G \leq 0$ .

Grand free energy $Ω = E - T S - μ N$

Natural for the grand canonical ensemble (constant $T, V, μ$ ; $N$ fluctuates).

Helmholtz free energy F=E−TS