= Chemiosmosis
{wiki}
<Chemiosmosis> couples transmembrane <ion> movement down an <electrochemical gradient> to a chemical transformation. For mitochondrial <ATP> synthesis, <protons> return from the <intermembrane space> to the <mitochondrial matrix> through <ATP synthase>.<Molecular Biology of the Cell, 4th edition by Bruce Alberts and others>[Chapter 14, Electron-Transport Chains and Their Proton Pumps, Summary: inward proton flow driving ATP synthesis]{ref}
An original thermodynamic example separates the electrical and chemical contributions. At fixed <temperature> $T$, use ideal dilute <thermodynamic activities> $a$ and the <electric potential> $\psi$ on each side. The molar <electrochemical potential> difference for inward transfer of a singly charged <proton> is
$$
\Delta\widetilde\mu_{\rm in}
=RT\ln\!\left(\frac{a_{\rm matrix}}{a_{\rm outside}}\right)
+F(\psi_{\rm matrix}-\psi_{\rm outside}).
$$
Here $R$ is the <molar gas constant>, $F$ the <Faraday constant>, and $\ln$ the <natural logarithm>. The sign convention is destination minus origin: a negative difference makes inward transfer favourable. Both lower <proton> activity and a more negative <electric potential> inside favour inward movement.
Stipulate $T=300\,\mathrm K$, $a_{\rm matrix}/a_{\rm outside}=0.1$, $\psi_{\rm matrix}-\psi_{\rm outside}=-0.12\,\mathrm V$, $R=8.314\,\mathrm{J\,mol^{-1}\,K^{-1}}$ and $F=96485\,\mathrm{C\,mol^{-1}}$. These illustrative inputs give approximately $-5.743$ and $-11.578\,\mathrm{kJ\,mol^{-1}}$ for the chemical and electrical terms, hence $-17.321\,\mathrm{kJ\,mol^{-1}}$ per transferred <proton>. For a hypothetical device coupling three inward <protons> per <ATP> to an <ATP>-formation cost of $45\,\mathrm{kJ\,mol^{-1}}$,
$$
\Delta G_{\rm coupled}=45+3\Delta\widetilde\mu_{\rm in}
\approx-6.964\,\mathrm{kJ\,mol^{-1}}.
$$
The sign follows by adding changes in the <state function> <Gibbs free energy>. It establishes thermodynamic feasibility for this chosen model, not its <reaction rate>. Three is a stipulated coupling count, not a universal <ATP synthase> <stoichiometry>; transport costs and leaks can also consume the available <free energy>.
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