= Helmholtz free energy
{c}
{title2=$F$}
{wiki}
= Helmholtz energy
{c}
{synonym}
The <Helmholtz free energy> is
$$
F=U-TS,
$$
where $U$ is <internal energy>, also denoted $E$ elsewhere here, $T$ is absolute <temperature>, and $S$ is <entropy>. For the <closed system> of fixed composition and only <pressure>-<volume (thermodynamics)> <work> described above, spontaneous relaxation at fixed <temperature> and <volume (thermodynamics)> cannot increase $F$. Stable <thermodynamic equilibrium> is a <global minimum> over the permitted alternatives, including different <thermodynamic phases>. A rigid vessel fixes the total <volume (thermodynamics)>; it does not require each coexisting <thermodynamic phase> to retain its own <volume (thermodynamics)>.
For a constructed comparison at $300\,\mathrm K$, suppose one candidate has $1.5\,\mathrm{kJ}$ more <internal energy> but $6\,\mathrm{J\,K^{-1}}$ more <entropy> than another at the same <temperature> and <volume (thermodynamics)>. Its difference is
$$
\Delta F=1500-300(6)=-300\,\mathrm J.
$$
Thus the candidate with higher <internal energy> has lower <Helmholtz free energy>. This is the practical reason for retaining the <entropy> term when selecting a state.
For equilibrium states within a smooth branch of fixed composition, the <first law of thermodynamics> gives $dU=T\,dS-p\,dV$, hence
$$
dF=-S\,dT-p\,dV,\qquad
S=-\left(\frac{\partial F}{\partial T}\right)_V,\qquad
p=-\left(\frac{\partial F}{\partial V}\right)_T.
$$
These <derivatives> extract <entropy> and <pressure>; the differential is not an evolution equation for relaxation. If <chemical species> amounts $n_i$ vary, add $\sum_i\mu_i\,dn_i$, with molar <chemical potentials> $\mu_i$. A <closed system> can still change <chemical species> amounts through <chemical reactions>; the fixed-composition simplification then needs the conserved-inventory constraints of <chemical equilibrium>. See https://ocw.mit.edu/courses/5-60-thermodynamics-kinetics-spring-2008/e763a06aa3b5de09cd6bfea6280f219c_5_60_lecture14.pdf[MIT 5.60, Lecture 14, fundamental equations and chemical potentials].
The minimum principle has a short explanation. At fixed <volume (thermodynamics)>, no boundary <work> occurs, so $Q=\Delta U$. A <thermal reservoir> at positive <temperature> $T$ loses <entropy> $Q/T$. Consequently
$$
\Delta S_{\rm total}=\Delta S-\frac{\Delta U}{T}=-\frac{\Delta F}{T}\geq0,
$$
by the <Second law of thermodynamics>. Here both endpoint states share the imposed <temperature>. See https://ocw.mit.edu/courses/5-60-thermodynamics-kinetics-spring-2008/64efc6f359b20c43dab61fc77791d744_5_60_lecture12.pdf[MIT 5.60, Lecture 12, criteria for spontaneous change].
The microscopic counterpart is $F=-k_BT\ln Z$ in the <canonical ensemble>, with <Boltzmann constant> $k_B$ and <partition function> $Z$. The <partition function> examples compute the balance between <internal energy> and <entropy>; the <chemical equilibrium> example already displays an ideal-mixture <Helmholtz free energy> minimum.
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