= Triple point
A triple point is a <temperature>–<pressure> pair at which three distinct bulk <thermodynamic phases> of a one-component substance coexist in <thermodynamic equilibrium>. In the familiar <solid>–<liquid>–<gas> case, their three <phase coexistence curves> meet. Other combinations, such as two <crystal polymorphs> and a <liquid>, also qualify: “triple” counts <thermodynamic phases>, rather than states-of-matter categories.
For the same substance, the <chemical potential> in each <thermodynamic phase> must agree:
$$
\mu_s(T_t,p_t)=\mu_l(T_t,p_t)=\mu_g(T_t,p_t).
$$
There are two independent equality conditions for two controls, <temperature> and <pressure>. Generically they select an isolated pair $(T_t,p_t)$; the third pairwise equality follows from the first two. Equivalently, the <phase rule> gives $F=1-3+2=0$. This counts independent intensive controls, not <phase fractions>: changing the amounts of the three <thermodynamic phases> need not change the ideal bulk <temperature> and <pressure>, provided all three remain present. Different sets of <crystal polymorphs> can give different triple points for the same substance.<Gibbs Phase Rule (MIT 10.213 course material)>[Gibbs Phase Rule, three-phase count and pressure–temperature diagram discussion]{ref}
At a <liquid-gas critical point>, the two fluid <thermodynamic phases> instead merge, and their <mass density> difference and <latent heat> vanish. It is the endpoint of the <liquid>–<gas> <phase coexistence curve>, rather than a meeting of three distinct <thermodynamic phases>. The regular independent-equality count used for a triple point cannot explain this merger. A triple point can retain finite <mass density> and <entropy> differences between all three <thermodynamic phases>.<Statistical Physics by David Tong>[§5.1.2, printed p. 141: vanishing liquid–gas discontinuity at the critical point; Figure 41 separates it from the three-phase junction]{ref}
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