the van der Waals equation
/ vahn der VAHLS /
The ideal gas law pretends molecules are dimensionless points that never attract one another. That fiction cannot condense a gas into a liquid, so it can never describe boiling or a critical point. The van der Waals equation is the simplest patch that fixes both flaws: it gives molecules a finite size and a mutual attraction, and in doing so it becomes the first single equation of state to contain a liquid, a gas and the critical point between them.
Per mole it reads (p + a/v^2)(v - b) = R T, where v is the molar volume. The constant b is the excluded volume that finite-sized molecules cannot overlap into (replacing v by v - b), and a encodes the attractive pull, which lowers the pressure by a/v^2. Setting the first and second volume derivatives of p to zero locates a critical point at v_c = 3b, T_c = 8a/(27 R b), p_c = a/(27 b^2). Written in reduced variables p/p_c, v/v_c, T/T_c the equation becomes universal — the law of corresponding states.
Below T_c a van der Waals isotherm develops an unphysical S-shaped wiggle with a region where (dp/dV)_T > 0 (mechanically unstable); the physical flat coexistence plateau is restored by the Maxwell equal-area construction, yielding liquid-vapour coexistence. Honest caveat: the model is mean-field and only qualitatively correct — it gets the existence of the transition right but predicts the wrong critical exponents, because it ignores the long-range fluctuations that dominate near T_c.
For carbon dioxide the van der Waals constants are about a = 0.364 Pa m^6/mol^2 and b = 4.27 x 10^-5 m^3/mol. Plugging into T_c = 8a/(27 R b) gives roughly 304 K and p_c = a/(27 b^2) about 7.3 MPa — close to the measured critical point of CO2, from a two-parameter model.
Two molecular constants, a and b, already predict CO2's critical temperature and pressure to within a few percent.
The van der Waals equation captures condensation and a critical point qualitatively, but as a mean-field theory it gives incorrect critical exponents (e.g. beta = 1/2 rather than the measured ~0.33); the S-shaped loop below T_c is unphysical and must be cut by the Maxwell construction.