= Electrochemistry
<Electrochemistry> studies how <chemical reactions>, transport, and <chemical equilibrium> couple to <electric charge> and <electric potential>. In a cell, <electrons> travel through the external circuit, while mobile <ions> carry <electric current> through the <electrolyte>. At the <electrodes>, <redox> reactions connect these two forms of <electric charge> transport. This connection makes it possible both to obtain electrical <work> from a <chemical reaction> and to use electrical <work> to drive a <chemical reaction>.
An <anode> is an <electrode> where <oxidation> occurs; a <cathode> is an <electrode> where <reduction> occurs. These definitions refer to the <chemical reactions>, rather than universal positive or negative terminal signs. In a <galvanic cell> delivering electrical <work>, the <anode> is negative and the <cathode> positive. In an <electrolytic cell> driven by an external source, the <anode> is positive and the <cathode> negative. See https://doi.org/10.1515/pac-2018-0109[Pingarrón et al., Terminology of electrochemical methods of analysis (IUPAC Recommendations 2019), §§3.1 and 3.4] and https://openstax.org/books/chemistry-2e/pages/17-2-galvanic-cells[Flowers et al., Chemistry 2e, §17.2: galvanic cells].
For a worked example, consider an <electrolytic cell> containing molten <sodium chloride>, with the <anode> connected to the positive terminal and the <cathode> to the negative terminal of an external source. At the <cathode>, <sodium> <cations> gain <electrons>; at the <anode>, <chlorine> <anions> lose <electrons>:
$$
\begin{aligned}
\text{cathode, reduction:}\quad&2\mathrm{Na}^++2e^-\longrightarrow2\mathrm{Na},\\
\text{anode, oxidation:}\quad&2\mathrm{Cl}^-\longrightarrow\mathrm{Cl}_2+2e^-,\\
\text{overall:}\quad&2\mathrm{Na}^++2\mathrm{Cl}^-\longrightarrow2\mathrm{Na}+\mathrm{Cl}_2.
\end{aligned}
$$
The first line has net <electric charge> zero on each side; the second has $-2e$ on each side. Adding the lines cancels the transferred <electrons> and preserves both <chemical atom> counts and net <electric charge>. The <sodium> product is neutral, and the <chlorine> product is a diatomic <molecule>. These equations specify molten <sodium chloride>; aqueous <chemical solutions> introduce competing <water> reactions. See <Chemistry 2e by Paul Flowers and others/17.7 Electrolysis> (§17.7: electrolysis of molten sodium chloride).
<Electrochemistry> underlies <battery> charging and the manufacture of chemicals by <electrolysis>. Its particularly useful idea is that an <electron> inventory in a balanced <chemical equation> also accounts for transported <electric charge>. Transport alone is a different question from an <electrode> <chemical reaction>: an <ion> may carry <electric current> through the <electrolyte> without itself being consumed at an <electrode>.
Historically, Michael Faraday introduced the terms <electrode>, <electrolyte>, <anion>, <cation>, and <ion> in the terminology section of his 1834 paper, <Experimental Researches in Electricity, Seventh Series by Michael Faraday> (§§662–667). His §667 explicitly treats the direction assigned to <electric current> as a convention. His <ion> language predates the modern <electron> and <atomic nucleus> picture used in the charge-counting example above.
For an <electrolyte> near <chemical equilibrium>, a <Poisson-Boltzmann equation> can approximate the distribution of <ions> in an <electric potential>. Such an equilibrium model does not by itself describe a cell carrying sustained <electric current>.
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