Van der Waals Equation — Real Gases & Phase Transitions
The ideal gas law PV = nRT is a triumphant simplification — but it breaks down badly at high pressures and low temperatures. Johannes Diderik van der Waals earned the 1910 Nobel Prize in Physics for a two-parameter correction that accounts for the finite size of molecules and the attractive forces between them. His equation predicts not just gas behaviour but the entire liquid-gas phase transition, including the critical point where gas and liquid become indistinguishable.
1. Where the Ideal Gas Law Fails
The ideal gas law PV = nRT rests on two key assumptions: molecules have zero volume (point particles), and molecules exert no forces on each other between collisions. At low pressures and high temperatures — where molecules are far apart and moving fast — these simplifications work well. But at higher pressures or lower temperatures, real departures emerge:
- Positive deviations (PV/nRT > 1): occur at very high pressures where molecular volume becomes significant — molecules take up a non-negligible fraction of the container volume, leaving less space for motion.
- Negative deviations (PV/nRT < 1): occur at moderate pressures where intermolecular attractions are important — the attractive pull between molecules reduces the pressure they exert on the walls compared to the ideal prediction.
- Condensation: the ideal gas law predicts gas behaviour at all temperatures, but real gases liquefy when cooled — a phenomenon entirely absent from the ideal model.
The compressibility factor Z = PV/nRT deviates from unity for all real gases; only hydrogen and helium (at room temperature) remain close to ideal over a wide pressure range due to their tiny size and weak interactions.
2. The Van der Waals Equation
Van der Waals proposed two corrections to the ideal gas law, one for each failure. The resulting equation, for n moles of gas in volume V at temperature T, is:
The two correction terms have clear physical interpretations:
The Pressure Correction: + a/V²
Molecules near the wall of a container experience a net inward pull from their neighbours — a surface tension-like effect. This reduces the momentum transferred to the wall, so the actual pressure P is less than the ideal pressure. The ideal pressure must therefore be corrected upward by an amount proportional to the square of the molecular number density (∝ 1/V²), giving the term n²a/V².
The parameter a (units: Pa·m⁶/mol² or L²·atm/mol²) measures the strength of intermolecular attractions — larger a means stronger attractive forces.
The Volume Correction: − nb
Real molecules have a finite hard-core volume. The volume available for motion is not V but V minus the excluded volume occupied by the molecules themselves. The parameter b (units: m³/mol or L/mol) is approximately four times the actual volume of one mole of molecules (due to the excluded-volume geometry of spheres).
3. Van der Waals Constants: a and b
The constants a and b are determined experimentally by fitting the equation of state to P-V-T measurements, or equivalently from the critical point data. Selected values:
| Gas | a (L²·atm/mol²) | b (L/mol) | T_c (K) | Character |
|---|---|---|---|---|
| He | 0.034 | 0.0237 | 5.2 | Very weak interactions, tiny |
| H₂ | 0.244 | 0.0266 | 33.2 | Small, weakly interacting |
| N₂ | 1.39 | 0.0391 | 126.2 | Moderate |
| CO₂ | 3.59 | 0.0427 | 304.2 | Polar, larger molecule |
| H₂O | 5.54 | 0.0305 | 647.1 | Strong H-bonds, small |
| C₆H₆ (benzene) | 18.8 | 0.1193 | 562.2 | Large, polarisable |
The trend is clear: larger, more polarisable molecules have higher a (stronger London dispersion forces), and molecules with polar or hydrogen-bonding character also show elevated a. The b parameter scales roughly with molecular volume.
4. The Critical Point: T_c, P_c, V_c
The critical point is the unique temperature and pressure at which the distinction between liquid and gas phases disappears. Above the critical temperature T_c, no amount of pressure can liquefy the gas — it becomes a supercritical fluid with properties intermediate between gas and liquid.
Mathematically, the critical point is the inflection point of the P-V isotherm:
Note that V_c = 3b implies the critical molar volume is three times the excluded volume, and the compressibility factor at the critical point is:
Real gases show Z_c in the range 0.23–0.29, somewhat below 0.375, reflecting the limitations of the van der Waals equation near the critical point. More sophisticated equations of state (Peng-Robinson, Soave-Redlich-Kwong) improve the prediction.
Critical Opalescence
Near the critical point, density fluctuations occur on length scales comparable to visible light wavelengths, causing intense light scattering — the fluid appears milky white. This beautiful phenomenon, called critical opalescence, was famously studied by Marian Smoluchowski and Albert Einstein.
5. The Reduced Equation of State
A profound insight of van der Waals was the law of corresponding states: all gases behave similarly when their properties are expressed in terms of reduced variables — dimensionless ratios relative to the critical point:
Substituting into the van der Waals equation using the critical constants:
This reduced van der Waals equation contains no material-specific constants! Nitrogen at T_r = 1.5, P_r = 2 behaves identically (in reduced variables) to carbon dioxide or water at the same reduced conditions. This universality is exploited in engineering to estimate properties of poorly characterised fluids from known critical data.
6. Maxwell Construction and Phase Coexistence
Below the critical temperature, the van der Waals equation predicts isotherms with an unphysical region where (∂P/∂V)_T > 0 — pressure increasing as volume increases. This "van der Waals loop" or "S-shaped" curve cannot represent stable equilibrium states, because it would imply a mechanically unstable fluid (a compression that reduces pressure).
James Clerk Maxwell resolved this in 1875 with the equal-area construction: the true liquid-gas coexistence pressure at a given temperature is found by drawing a horizontal line through the S-curve such that the two enclosed areas are equal:
This condition is equivalent to requiring equal chemical potentials (Gibbs free energies per mole) of the coexisting liquid and gas phases — a fundamental requirement for thermodynamic equilibrium:
The horizontal line at P_eq represents the two-phase region — a mixture of liquid (molar volume V_liq) and gas (molar volume V_gas). The relative proportions of liquid and gas are given by the lever rule:
7. Liquefaction of Gases
A gas can only be liquefied by compression if the temperature is below T_c. Above T_c, no pressure will cause condensation. This explains why early attempts to liquefy "permanent gases" (H₂, N₂, O₂) failed — experimenters were working above the critical temperatures without knowing it.
The Joule-Thomson effect is the key practical tool: when a gas at high pressure passes through a porous plug into lower pressure, it cools (if below the Joule-Thomson inversion temperature) or warms (above). For most gases at room temperature, the inversion temperature is well above 300 K, so throttling causes cooling.
The Joule-Thomson coefficient is:
For nitrogen: T_inv ≈ 621 K — well above room temperature, so N₂ cools on throttling at 300 K. For hydrogen: T_inv ≈ 204 K — below room temperature, so H₂ must be pre-cooled (using liquid nitrogen) before throttling will liquefy it. This is the Linde-Hampson liquefaction cycle.