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The 1869 The Bakerian Lecture: On the Continuity of the Gaseous and Liquid States of Matter by Thomas Andrews describes dried carbon dioxide trapped above mercury in a sealed capillary tube. A steel screw pressurized water in a strong copper tube; movable mercury transmitted the load. A second capillary tube held air as a pressure gauge. Flowing water around the visible capillary tubes controlled temperature; a thermometer read the bath. A cathetometer read column positions, converted to volume readings using weighed-mercury calibration.^[ref]

Below the critical temperature, compression produced a visible liquid–gas boundary; above it, large density changes occurred without that boundary. Three trials gave $30.92{}^{\circ }\mathrm{C}$ as his critical temperature. Purity checks detected residual air; mercury height corrections accounted for unequal hydrostatic pressures. He explicitly left air nonideality and pressure-induced tube-capacity corrections unresolved, so his air-gauge pressures were approximate. These readouts support a liquid-gas critical point; they do not measure a homogeneous unstable loop or establish universal quantitative accuracy of the later Van der Waals equation.^[ref]

A disappearing liquid meniscus by itself has more than one possible explanation: in ordinary subcritical condensation or evaporation, one thermodynamic phase can simply run out. The stronger critical-point inference combines temperature control, compression readings and the existence of a continuous route around the endpoint of the phase coexistence curve. On that route a dilute gas can become dense without crossing a two-phase boundary. This is what continuity of liquid and gas means; it does not remove their distinct equilibrium number densities below $T_c$.

In his 12 December 1910 Nobel lecture, Johannes Diderik van der Waals explained gases and liquids through the same moving molecules, with attraction and finite size modifying their pressure. He also rejected the idea that constant $a,b$ should give universal numerical agreement. His critical-volume discussion explicitly retained derivatives of a volume-dependent $b$: the familiar $v_c=3b$ follows only when those derivatives are neglected. Thus the elegant constant-parameter Van der Waals critical point is a model result whose limitations were already part of its author's account.^[ref]

A physicist.^[ref]

The Van der Waals equation connects a dilute gas model to the possibility of a liquid-gas critical point. Its usefulness includes showing how attractions and excluded volume can change an equation of state; its simple constants do not make it an exact model for every fluid.

Johannes Diderik van der Waals received the award for an equation of state describing gases and liquids.^[ref] The Van der Waals equation connects that description to a Van der Waals critical point and Van der Waals liquid-gas coexistence.

By Johannes Diderik van der Waals. Nobel lecture delivered 12 December 1910; inspected English reprint, pp. 254–265. NTNU hosts this copy.

The Van der Waals equation is a simple real gas equation of state that can also describe a liquid-gas critical point. With particle count $N$ fixed, write the volume per particle as $v=V/N$:or equivalently $(p+a{N}^2/V^2)(V-Nb)=N{k}_{B}T$. Here $p$ is pressure, $T$ is absolute temperature, and $k_B$ is the Boltzmann constant. In SI, $v$ and $b$ have units ${\mathrm{m}}^3$ per particle, $a$ has units $\mathrm{Pa}{\mathrm{m}}^6=\mathrm{J}{\mathrm{m}}^3$, and $k_B$ has units $\mathrm{J}{\mathrm{K}}^{-1}$; the particle count is dimensionless. The pole at $v=b$ excludes smaller volume values from this model.

The $b$ correction represents excluded volume; the attraction term lowers pressure at fixed temperature and particle number density $1/v$. This is a mean-field theory, replacing detailed intermolecular structure by two parameters. At $v\gg b$ with $a/v^2\ll k_{B}T/v$, it approaches the ideal gas law $pv=k_{B}T$. It supplies a useful qualitative route from a dilute gas to phase coexistence, without a claim of quantitative accuracy for every fluid.^[ref]

Do not mix this particle convention with the molar volume convention elsewhere in this article. With amount of substance $n_{\mathrm{mol}}=N/N_A$, Avogadro constant $N_A$, and molar gas constant $R=N_A{k}_B$, one has $v_m=N_{A}v$, $b_m=N_{A}b$, and $a_m=N_A^{2}a$, so $p=RT/(v_m-b_m)-a_m/v_m^2$. The molar parameters have units ${\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$ and $\mathrm{Pa}{\mathrm{m}}^6\mathrm{mo}{\mathrm{l}}^{-2}$, respectively.

Johannes Diderik van der Waals introduced this equation of state in 1873, connecting molecular attraction with departures from the dilute ideal gas law. Its historical significance is the unified treatment of gases and liquids, rather than a universally exact fit.^[ref]