= Phase rule
{title2=$F=C-P+2$}
{wiki}
= Gibbs phase rule
{c}
{synonym}
The <phase rule> counts independent intensive controls compatible with coexistence of $P$ distinct bulk <thermodynamic phases> made from $C$ independent components:
$$
F=C-P+2.
$$
Here $P$ counts <thermodynamic phases>, not <pressure>, and $F$ does not count <phase fractions>. For the nonreacting one-component case, $C=1$ and $F=3-P$: a single <thermodynamic phase> occupies a region with $F=2$, two-<thermodynamic phase> coexistence a curve with $F=1$, and a <triple point> an isolated point with $F=0$. Fixing <pressure> externally removes one adjustable control wherever that constraint is independent.<Gibbs Phase Rule (MIT 10.213 course material)>[Gibbs Phase Rule, displayed nonreacting rule and pure-substance table]{ref}
For an elementary count without <chemical reactions>, take $C$ <chemical species>. Each <thermodynamic phase> has $C-1$ independent composition fractions, since they sum to one. Shared <temperature> and <pressure> add two controls, giving $2+P(C-1)$. Equality of each species' <chemical potential> between one reference <thermodynamic phase> and the other $P-1$ <thermodynamic phases> supplies $C(P-1)$ constraints. Subtraction gives $2+P(C-1)-C(P-1)=C-P+2$. This argument assumes the constraints are independent and treats interfaces as negligible; <chemical reactions>, external fields or critical mergers require appropriate qualifications.
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