The P-V diagram and the critical point

On the $(V,P)$ plane the Van der Waals isotherms (curves at fixed $T$) look qualitatively different above and below a characteristic temperature $T_c$, called the critical temperature:

• for $T > T_c$ every isotherm is monotonically decreasing, similar to a slightly deformed hyperbola: the gas can be compressed indefinitely without a liquid phase appearing;
• for $T < T_c$ the isotherm has an S-shaped section with a local maximum and minimum: in the region between them a single pressure corresponds to three possible volumes, and this is the signature of coexistence between liquid and vapour;
• exactly at $T = T_c$ the isotherm has a horizontal-tangent inflection point at the critical point $(V_c, P_c, T_c)$: this is where liquid and gas become indistinguishable.

For CO$_2$, $T_c \approx 304$ K $\approx 31\,°$C: at lower temperatures (a domestic refrigerator) a moderate pressure is enough to liquefy it. Above $31\,°$C, however, no pressure liquefies it: it becomes a supercritical fluid, now used industrially as a solvent for decaffeinating coffee and extracting oil from seeds.

Van der Waals isotherms in the $(V,P)$ plane. Above $T_c$ the curves are monotonic; at $T_c$ a horizontal-tangent inflection appears at the critical point; below $T_c$ the S-shaped section appears, signalling liquid-vapour coexistence.

The S shape is not a flaw

The S-shaped section of the isotherms is a manifestation of intermolecular forces, the same ones that hold a liquid together. In practice nature “cuts” the S with a horizontal segment (Maxwell construction) along which liquid and vapour coexist. Van der Waals is the bridge between the ideal gas and condensed matter.

Collegamenti

Argomenti: Teoria cinetica dei gas Concetti: Legge dei gas perfetti · Calore latente e cambiamenti di stato

Esercizi collegati: Problem — Moles and molecules in a cylinder · Problem — Temperature that doubles the pressure · Problem — Ranking of pressures by temperature