Helmholtz Free Energy · aiwiki.page

Helmholtz free energy is a thermodynamic potential defined as a system’s internal energy minus the product of its absolute temperature and entropy. Usually written $F$ in physics and $A$ in chemistry, it provides an equilibrium criterion for systems maintained at constant temperature and volume. It also connects macroscopic thermodynamics with microscopic statistical mechanics through the partition function. (old.goldbook.iupac.org)

Definition and physical meaning

The defining equation is

F=U-TS,

where $U$ is internal energy, $T$ is absolute temperature, and $S$ is entropy. IUPAC calls this quantity “Helmholtz energy.” Like $U$ , it has the dimensions of energy; its SI unit is the joule, with temperature expressed in kelvins. It is a state function: its change depends on the initial and final states rather than the path connecting them. (old.goldbook.iupac.org)

The term “free” refers to energy available for conversion into work under specified conditions, not to an additional form of stored energy. At a fixed temperature, the expression balances energetic and entropic contributions: a state with higher internal energy can nevertheless have lower free energy if its entropy is sufficiently greater. Minimizing energy alone therefore does not generally predict finite-temperature equilibrium. (damtp.cam.ac.uk)

Natural variables and thermodynamic relations

For a simple compressible system containing one chemical species, the equilibrium fundamental relation is

dU=T\,dS-p\,dV+\mu\,dN,

where $p$ denotes pressure, $V$ volume, $N$ particle number, and $\mu$ chemical potential. Substitution into the definition gives

dF=-S\,dT-p\,dV+\mu\,dN.

Consequently, the natural variables are $T,V,N$ . The transformation from $U(S,V,N)$ to $F(T,V,N)$ is a Legendre transform, replacing entropy with its conjugate variable, temperature. For mixtures, the last term becomes $\sum_i\mu_i\,dN_i$ ; additional work modes require additional conjugate-variable terms. (ocw.mit.edu)

The corresponding partial derivatives recover measurable properties:

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

Thus a complete free-energy expression determines both thermal and mechanical behavior, including an equation of state. Differentiating again yields response functions, such as the constant-volume heat capacity:

C_V=-T\left(\frac{\partial^2F}{\partial T^2}\right)_{V,N}.

These identities apply within the equilibrium description and with the indicated variables held fixed. (damtp.cam.ac.uk)

Equilibrium and available work

For a closed thermodynamic system held at fixed temperature and volume, with no externally supplied non-expansion work, spontaneous relaxation cannot increase Helmholtz free energy:

\Delta F\leq0.

Stable thermodynamic equilibrium corresponds to the minimum accessible free energy with respect to unconstrained internal variables. This criterion follows from the second law of thermodynamics applied to the system together with its thermal reservoir. It is not a universal rule that $F$ decreases: changing the external constraints or supplying work can increase it. (damtp.cam.ac.uk)

For a process exchanging heat only with a reservoir at temperature $T$ , and whose equilibrium endpoints have that temperature, the total work delivered by the system obeys

W_{\mathrm{out}}\leq-\Delta F.

Equality is attained in the reversible limit. At constant volume, expansion work vanishes, so this bound concerns other work modes. If volume changes, expansion work is included in the total. The bound concerns a free-energy difference, not the absolute numerical value of $F$ . (damtp.cam.ac.uk)

Statistical-mechanical formulation

In the canonical ensemble, temperature, volume, and particle number are fixed while energy fluctuates through contact with a thermal reservoir. The partition function is

Z(T,V,N)=\sum_r e^{-\beta E_r}, \qquad \beta=\frac{1}{k_BT},

where $E_r$ labels microscopic energy levels and $k_B$ is the Boltzmann constant. Their equilibrium probabilities follow the Boltzmann distribution, $P_r=e^{-\beta E_r}/Z$ . Helmholtz free energy is then

F=-k_BT\ln Z.

This equation makes free energy a bridge between microscopic state counting and macroscopic thermodynamic properties. The sum includes each accessible state, including distinct states with equal energy. (damtp.cam.ac.uk)

For a dilute, noninteracting, monatomic ideal gas in the classical regime,

Z=\frac{1}{N!}\left(\frac{V}{\lambda^3}\right)^N, \qquad \lambda=\frac{h}{\sqrt{2\pi m k_BT}},

where $h$ is the Planck constant and $m$ is particle mass. For large $N$ ,

F\simeq Nk_BT \left[\ln\left(\frac{N\lambda^3}{V}\right)-1\right].

Its volume derivative yields $pV=Nk_BT$ . The factorial accounts for particle indistinguishability; omitting it gives an incorrect entropy and free-energy dependence on particle number. The classical approximation requires $N\lambda^3/V\ll1$ . (damtp.cam.ac.uk)

Relation to Gibbs free energy

Gibbs free energy is related by

G=F+pV=H-TS,

where $H=U+pV$ is enthalpy. Helmholtz free energy is suited to fixed-temperature, fixed-volume constraints; Gibbs free energy is suited to fixed-temperature, fixed-pressure constraints. The distinction is therefore about controlled variables, rather than competing definitions of energy. At fixed temperature and pressure, the decrease in $G$ bounds reversible non-expansion work, whereas the decrease in $F$ bounds total isothermal work under the conditions stated above. (damtp.cam.ac.uk)