{title2=$\xi$}
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For a closed system with one specified <chemical reaction>, the <extent of reaction> $\xi$ records progress from $\xi=0$ through $n_i=n_{i,0}+\nu_i\xi$. Here $n_i$ is the <amount of substance> of species $i$, and the signed <stoichiometric vector> has negative entries for <reactants> and positive entries for <reaction products>. For the <water> <chemical equation> in the order $\mathrm{H}_2,\mathrm{O}_2,\mathrm{H}_2\mathrm{O}$, it is $(-2,-1,2)$. The normalization of $\xi$ depends on the chosen coefficients. See https://goldbook.iupac.org/terms/view/E02283[IUPAC Gold Book, extent of reaction].

With particle numbers instead of molar amounts, a molecular extent $\xi_N$ obeys $dN_i=\nu_i\,d\xi_N$, and the <chemical potentials> are measured per particle. At fixed <temperature> and <volume>, the <Helmholtz free energy> then satisfies $dF=(\sum_i\nu_i\mu_i)\,d\xi_N$. An interior <chemical equilibrium> requires $\sum_i\nu_i\mu_i=0$; the molar convention gives the same relation with chemical potentials per mole.


 Extent of reaction (source code)