= Phase diagram
{wiki}
A phase diagram partitions a space of thermodynamic variables into regions occupied by different equilibrium <thermodynamic phases> and records the <phase coexistence curves> between them. For a pure, one-component substance, a <pressure>–<temperature> <phase diagram> gives one point for each pair $(T,p)$. In a single-<thermodynamic phase> region both controls can vary independently. On a two-<thermodynamic phase> <phase coexistence curve>, they obey one relation; at a <triple point>, both are fixed. These statements assume bulk <thermodynamic equilibrium>, fixed composition and no additional applied fields.<Gibbs Phase Rule (MIT 10.213 course material)>[Gibbs Phase Rule, single-, two- and three-phase counts and final pressure–temperature diagram paragraphs]{ref}
At fixed <temperature> and <pressure>, the stable bulk <thermodynamic phase> minimizes <Gibbs free energy> for a fixed <amount of substance>. For a pure substance this is a comparison of <chemical potentials> on a common normalization. Two candidate <thermodynamic phases> coexist where $\mu_\alpha(T,p)=\mu_\beta(T,p)$ and neither is exceeded in stability by a third <thermodynamic phase>. Differentiating their equality gives the <Clapeyron equation>. Thus a <phase coexistence curve> is a locus of equal stability, rather than a trajectory that every heated sample must follow.<Statistical Physics by David Tong>[§5.1.1, equations (5.3)–(5.4), printed p. 137: chemical-potential equality; §5.1.2, equation (5.6), printed pp. 140–141: coexistence slope]{ref}
The following original cartoon uses dimensionless positive coordinates $\theta=T/T_*$ and $\pi=p/p_*$, with arbitrary reference scales $T_*,p_*$. It represents no particular substance and contains no measured data. Stipulate a <triple point> at $(1,1)$, a <liquid-gas critical point> at $(3,5)$, and these drawing rules:
$$
\begin{aligned}
\pi_{\rm sub}(\theta)&=e^{\theta-1},&&0<\theta\le1,\\
\pi_{\rm vap}(\theta)&=2\theta-1,&&1\le\theta\le3,\\
\theta_{\rm melt}(\pi)&=1+0.1(\pi-1),&&\pi\ge1.
\end{aligned}
$$
They specify <sublimation>, <liquid>–<gas> <phase coexistence> and <melting> boundaries for the picture. They are not an <equation of state>, do not predict <latent heat> or <mass density>, and do not supply a quantitative critical theory. In particular, the <melting> slope is merely an illustrative choice.
\Image[/generic-pressure-temperature-coexistence.png]
{title=Generic dimensionless <pressure>–<temperature> <phase diagram>: curves, <triple point> and <liquid-gas critical point>}
{description=Original cartoon with stipulated drawing rules. Three distinct bulk <thermodynamic phases> coexist at $(1,1)$; the <liquid>–<gas> boundary ends at $(3,5)$. The fluid region beyond that endpoint is connected: no extra boundary is implied by the fading background. These coordinates are not <water> data.}
For example, at fixed $\pi=3$ the <melting> boundary is at $\theta=1.2$ and the <liquid>–<gas> boundary is at $\theta=2$. Heating from $(\theta,\pi)=(1.6,3)$ to $(3.6,3)$ therefore starts in <liquid>, crosses <phase coexistence> at $(2,3)$, and ends in <gas>. In an equilibrium conversion at that crossing, added <heat> can change the <phase fraction> at fixed <temperature> through <latent heat>. The <pressure>–<temperature> point alone does not specify how much of each <thermodynamic phase> is present: for that one needs an additional amount or <volume (thermodynamics)> constraint.
Now connect exactly the same endpoints by raising $\pi$ from $3$ to $6$ at $\theta=1.6$, moving to $\theta=3.6$ at $\pi=6$, then lowering $\pi$ to $3$. The first leg stays on the <liquid> side because $\theta_{\rm melt}$ ranges only from $1.2$ to $1.5$ and $\pi_{\rm vap}(1.6)=2.2<3$. The second leg lies above the entire <liquid>–<gas> boundary, whose largest drawn <pressure> coordinate is $5$; the third lies beyond its largest <temperature> coordinate, $3$. No leg crosses a <first-order phase transition> boundary. The middle leg passes through a <supercritical fluid> region, with $\theta>3$ and $\pi>5$, and the final leg returns to a gaslike state continuously.
\Image[/liquid-to-gas-crossing-and-critical-detour.png]
{title=The same <liquid> and <gas> endpoints: crossing <phase coexistence> or going around the <liquid-gas critical point>}
{description=Original dimensionless cartoon. The orange constant-<pressure> path crosses the <liquid>–<gas> boundary at $(2,3)$. The blue path follows $(1.6,3)$ to $(1.6,6)$ to $(3.6,6)$ to $(3.6,3)$ and avoids every drawn <phase coexistence curve>. Dashed coordinate guides locate the critical endpoint without representing additional boundaries.}
Avoiding a <first-order phase transition> does not mean that no <heat> or <work> is exchanged. Along the detour, <mass density>, <enthalpy> and <heat capacity> may change substantially but continuously. The <liquid-gas critical point> itself has singular behavior, so the detour goes around it rather than through it. This connected fluid region is the striking lesson of a <phase diagram>: a familiar change from <liquid> to <gas> need not involve boiling across a sharp boundary.
An equilibrium <phase diagram> also does not predict how rapidly a conversion occurs. <Metastability> and a barrier described by <classical nucleation theory> can delay the observed <phase transition>. Nor should a binary-composition <phase diagram> be read as the same picture: its coexistence regions and <lever rule> encode composition and <phase fractions>, and can occupy an area even though pure-substance two-<thermodynamic phase> coexistence occupies a curve in the <pressure>–<temperature> plane.
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