= Solution
Usable biological <Gibbs free energy>[free energy] has several interconvertible forms. At approximately fixed <temperature> and pressure, <Chemical reaction>[chemical reactions] are described by <Chemical potential>[chemical potentials], $\mu_i=\mu_i^\circ+RT\log a_i$, and $\Delta_rG=\sum_i\nu_i\mu_i$. <ATP hydrolysis>, for example, has $\Delta_rG=\Delta_rG^{\circ\prime}+RT\log(a_{\rm ADP}a_{\rm Pi}/a_{\rm ATP})$ under the specified biochemical standard-state convention. The <Thermodynamic activity>[activities] and <proton>/<ion> convention matter. Breaking a bond alone costs <energy>; the net free-energy change includes the products, solvent and all bond changes, not a store released simply by snapping one bond.
<Redox>[Redox reactions] provide <Gibbs free energy>[free energy] when electrons move between donors and acceptors with a favourable potential difference: $\Delta G=-nF\Delta E$. Electron-transfer chains can couple that drop to <Membrane transport>[ion transport], building an <electrochemical gradient>. For transfer from an outer to an inner reservoir,
$$
\Delta\widetilde\mu=RT\log\frac{a_{\rm in}}{a_{\rm out}}+zF(V_{\rm in}-V_{\rm out}).
$$
Both <concentration> difference and <Electric potential>[electrical potential] contribute. <ATP synthase> couples downhill <ion> flow to nucleotide <Gibbs free energy>[free energy]; driven pumps can perform the reverse conversion and maintain gradients used by transporters and excitability.
<Work>[Mechanical work] appears as $F\,dx$ or torque times angle, and can be stored elastically, for example $\kappa x^2/2$. Molecular motors convert <Gibbs free energy>[chemical free energy] into movement or <force>; mechanically driven reactions can conversely reshape chemical transitions. Electrical <energy> can also be stored in <membrane capacitance>, while bulk ionic <concentration> gradients store additional <Gibbs free energy>[chemical free energy]. Light supplies photon <energy> $h\nu$, which photosynthetic machinery can convert into redox and <chemical potential> differences.
Coupling is essential: an overall favourable reaction can drive an unfavourable one only when their processes are linked, rather than merely present in the same cell. At fixed conditions the extracted work is bounded by the consumed free-energy decrease. Dissipation produces <heat> and <entropy>, so <energy> conservation does not mean all <energy> remains available for work. <Thermal fluctuation>[Thermal fluctuations] can drive random motion, but a persistent directional cycle requires a nonequilibrium free-energy source; an equilibrium asymmetric structure alone is not a free motor.
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