Helmholtz free energy

Definition and physical meaning

For a thermodynamic system with internal energy $U$, temperature $T$, and entropy $S$, the Helmholtz free energy (often just “free energy”) is the state function

The term $TS$ represents the part of energy that is “bound up” as thermal disorder when the system is at temperature $T$. In contact with a thermal reservoir that fixes $T$, decreases in $F$ quantify how much energy can be converted into useful work while respecting the second law of thermodynamics .

Differential form and natural variables

For a simple compressible single-component system,

with pressure $p$, volume $V$, chemical potential $\mu$, and particle number $N$.

Thus $F$ is naturally a function of $(T,V,N)$, mixing intensive variables (like $T$) and extensive variables (like $V$ and $N$), and it generates familiar response relations:

• $S = -\left(\partial F/\partial T\right)_{V,N}$,
• $p = -\left(\partial F/\partial V\right)_{T,N}$,
• $\mu = \left(\partial F/\partial N\right)_{T,V}$.

Cross-differentiation yields a Maxwell relation , for example

Work interpretation at fixed temperature

For a system held at fixed temperature by a thermal reservoir, changes in $F$ bound the amount of work that can be extracted. With the standard sign convention for work , in an isothermal process at fixed $(T,V,N)$ the maximum work the system can deliver (i.e. “useful work,” excluding $p\,dV$ expansion work) satisfies

with equality for a reversible process .

Relation to Legendre transforms and ensembles

Starting from a fundamental relation $U(S,V,N)$, $F$ is obtained by a Legendre transform that trades the extensive variable $S$ for the conjugate temperature $T$.

In statistical mechanics (using the canonical ensemble convention and natural logarithm convention ),

where $k_B$ is the Boltzmann constant and $Z$ is the canonical partition function.