= Redox electron balance
{title2=$n_{e,\mathrm{lost}}=n_{e,\mathrm{gained}}$}
Consider the reacting <zinc> and <copper> species in the <zinc-copper displacement experiment>. Split the net <chemical equation> into two <half-reactions>:
$$
\begin{aligned}
\text{oxidation:}\quad&\mathrm{Zn}(s)\longrightarrow\mathrm{Zn}^{2+}(aq)+2e^-,\\
\text{reduction:}\quad&\mathrm{Cu}^{2+}(aq)+2e^-\longrightarrow\mathrm{Cu}(s).
\end{aligned}
$$
The <zinc> <oxidation state> rises from $0$ to $+2$; the <copper> <oxidation state> falls from $+2$ to $0$. Add the <half-reactions>: two released <Electrons> match two accepted <Electrons>. Cancelling $2e^-$ from both sides gives<Chemistry 2e by Paul Flowers and others>[§4.2, “Balancing Redox Reactions via the Half-Reaction Method”, steps 6–7: matching and cancelling exchanged electrons]{ref}
$$
\boxed{\mathrm{Zn}(s)+\mathrm{Cu}^{2+}(aq)\longrightarrow\mathrm{Zn}^{2+}(aq)+\mathrm{Cu}(s).}
$$
Thus the closed net <chemical reaction> has zero net production of free <Electrons>. Cancellation does not destroy <Electrons>: it removes a matched internal exchange from the written <chemical equation>. The <half-reactions> alone are incomplete descriptions of the whole reacting system. In a <galvanic cell>, the exchange can pass through a wire; in the <zinc-copper displacement experiment>, the reacting materials meet at the same metal surface. The equations do not require a reservoir of free <Electrons> in the bulk <chemical solution>.
\Image[/redox-coupled-half-reaction-ledger.png]
{title=<Electron> bookkeeping with coupled <half-reactions> for <zinc> and <copper>}
{id=image-redox-coupled-half-reaction-ledger}
{description=Original algebraic diagram: both <half-reactions> exchange two <Electrons>, so their sum has matching $2e^-$ terms. The cancellation indicates internal accounting, not an <Electron> trajectory or an <elementary reaction>. No cell geometry or terminal polarity is implied.}
Check both kinds of balance. The <oxidation> line has net <electric charge> $0$ on the left and $(+2-2)e=0$ on the right; the <reduction> line has $(+2-2)e=0$ on the left and $0$ on the right. The net <chemical equation> contains one <zinc> and one <copper> <chemical atom> on each side and has net <electric charge> $+2e$ on each side, where $e$ is the positive <elementary charge>. The written reacting species need not be electrically neutral as a group: the bulk <chemical solution> also contains compensating <anions>.
There is an even more literal check. <Zinc> has <atomic number> $30$ and <copper> $29$, so the initial pair contains $30+(29-2)=57$ <Electrons>. The final pair contains $(30-2)+29=57$ <Electrons>. Only their distribution has changed. Conserved <chemical element> counts give the usual chemical <mass conservation> inventory; changing the counts of neutral and charged species does not create or remove their <atomic nuclei>.
\Image[/redox-zinc-copper-conserved-counts.png]
{title=Conserved <chemical atom>, <electric charge> and total <Electron> counts in <zinc>/<copper> <redox>}
{id=image-redox-zinc-copper-conserved-counts}
{description=Original exact count table for one net <chemical reaction> as written. The <Electron> row counts all <Electrons> assigned to the reacting species, not only <valence electrons>. The charge row is in units of the positive <elementary charge>. Omitted <anions> would make the bulk <chemical solution> neutral. These are counts, not measured amounts or a drawing of localized <Electrons>.}
Matching <Electrons> is necessary but does not replace balancing every <chemical element>. The <acidic permanganate-iron balance> illustrates unequal one- and five-<Electron> exchanges, with <water> and <protons> also needed. Conversely, a balanced <chemical equation> alone establishes neither a <reaction rate> nor a detailed mechanism. Its usefulness is that it fixes the <stoichiometry> even when those additional questions remain open.
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