Phase Transitions and Critical Phenomena

Coexistence and Its End

At a phase transition two phases coexist, and coexistence has a precise condition: equal temperature, equal pressure, and equal chemical potential. The last is what actually selects the transition point, the stable phase at given $T$ and $P$ is whichever has the lower $\mu$.

Ehrenfest classified transitions by which derivative of $G$ first jumps. In a first-order transition the first derivatives jump, so volume and entropy change discontinuously and latent heat $L = T\Delta S$ is absorbed. Melting and boiling are the everyday examples: heat goes in and the temperature does not move. In a continuous (second-order) transition the first derivatives are smooth and the second ones diverge, no latent heat, but a diverging heat capacity. The ferromagnetic transition at the Curie point is the standard case.

Along a first-order coexistence line the Clausius-Clapeyron equation fixes the slope:

$$\frac{dP}{dT} = \frac{L}{T\Delta V}$$

The sign of $\Delta V$ therefore sets the direction the line leans. Water is the famous anomaly: ice is less dense than liquid water, so $\Delta V < 0$ on melting and the solid-liquid line slopes backwards: which is why pressure melts ice.

For a real gas the van der Waals isotherms below $T_c$ contain an unphysical region where $(\partial P/\partial V)_T > 0$, which would be mechanically unstable. The Maxwell construction replaces that loop with a horizontal tie-line placed so the two enclosed areas are equal, and that line is the coexistence region.

Raising the temperature shrinks the coexistence region until, at the critical point, the two phases become identical and the distinction disappears. Near it, properties follow power laws with critical exponents that are strikingly universal, wildly different substances share the same exponents, because near the critical point the microscopic details stop mattering.

Common pitfall: assuming heat always raises temperature. During a first-order transition every joule goes into the latent heat of rearrangement, and the temperature is pinned until the change is complete, which is exactly what makes an ice bath a reliable fixed point.