When the ideal gas law is not enough
The ideal gas law, PV = nRT, treats gas molecules as points that never interact. That works well for air at room conditions, but it breaks down when a gas is compressed hard or cooled toward the point where it liquefies. In 1873 the Dutch physicist Johannes Diderik van der Waals added two simple corrections, and his equation became the first successful model of real gases and of the transition to a liquid.
The equation
(P + a × n² ÷ V²) × (V − n × b) = n × R × T
- a accounts for the attraction between molecules, which pulls them inward and lowers the pressure on the walls.
- b is the volume taken up by one mole of the molecules themselves, which is not available to move in.
- R = 0.083145 L·bar/(mol·K), T is the absolute temperature in kelvins.
Solved for pressure: P = nRT ÷ (V − nb) − a n² ÷ V².
Constants for common gases
| Gas | a (L²·bar/mol²) | b (L/mol) |
|---|---|---|
| Helium | 0.0346 | 0.0238 |
| Hydrogen | 0.2476 | 0.02661 |
| Nitrogen | 1.370 | 0.0387 |
| Oxygen | 1.382 | 0.03186 |
| Methane | 2.283 | 0.04278 |
| Carbon dioxide | 3.640 | 0.04267 |
| Ammonia | 4.225 | 0.0371 |
| Water vapor | 5.536 | 0.03049 |
Gases with strong attractions between molecules, such as water and ammonia, have large values of a.
Worked example
1 mol of carbon dioxide in a 1 L container at 77 °F (298.15 K):
- Ideal gas pressure: 0.083145 × 298.15 ÷ 1 = 24.79 bar.
- Van der Waals pressure: 24.79 ÷ (1 − 0.04267) − 3.640 = 22.25 bar, about 10% lower.
- Compressibility factor: Z = 22.25 × 1 ÷ 24.79 = 0.898: attractions dominate.
- Predicted critical point: Tc = 8a ÷ 27Rb = 304 K (31 °C) and Pc = a ÷ 27b² = 74 bar, very close to the measured values for CO₂.
The critical point
Above its critical temperature, a gas cannot be liquefied by pressure alone. Van der Waals’ equation predicts a critical point for every gas from its a and b values, and below that temperature it produces the S-shaped curves that, with Maxwell’s construction, describe the coexistence of liquid and vapor. CO₂ above 31 °C and 74 bar becomes a supercritical fluid, used to decaffeinate coffee.
Limits
The van der Waals equation captures the trends but is not very accurate near the critical point or at very high pressures. Engineers use more refined equations, such as Redlich–Kwong, Peng–Robinson or tabulated data, for design work. As a teaching tool and a quick estimate, it shows why real gases deviate from ideal behavior and in which direction.