Van der Waals Real Gas Calculator

The ideal gas law (PV = nRT) works well at low pressures and high temperatures but fails for dense gases and near the liquid-vapor transition. Johannes van der Waals corrected it in 1873, and the result earned him the 1910 Nobel Prize.

The equation

(P + an^2/V^2)(V - nb) = nRT

Where: n = moles, R = 0.08206 L·atm/(mol·K), T in Kelvin, V in liters, P in atm.

Rearranging to solve for pressure:

P = nRT / (V - nb) - an^2 / V^2

What a and b represent

The term an^2/V^2 is the internal pressure correction, accounting for intermolecular attractive forces. Molecules attract each other, which reduces the pressure exerted on the container walls compared to an ideal gas. Large a means strong attractions (water vapor: a = 5.46; helium: a = 0.0341).

The term nb is the excluded volume correction. Molecules occupy physical space, reducing the volume available for motion. Large b means large molecules (ethane: b = 0.0638 L/mol; helium: b = 0.0237 L/mol).

When ideal gas fails badly

At high pressures (above 10 to 20 atm) and near the critical point, the corrections become significant.

Take one mole of CO2 squeezed into 0.3 litres at 50°C. The ideal gas law says PV = nRT, so P = (1 × 0.08206 × 323.15) / 0.3 = 88.4 atm. Van der Waals disagrees sharply:

P = nRT/(V − nb) − an²/V² = 26.518/0.2573 − 3.59/0.09 = 103.1 − 39.9 = 63.2 atm

That is 28% below the ideal prediction, and the split shows you where it comes from. The excluded-volume term pushed the pressure UP by about 15 atm, because the molecules only have 0.257 L of free space rather than 0.300. Then the attraction term pulled it DOWN by 39.9 atm, far more, because CO2 molecules tug on each other on their way to the wall. At this density attraction wins, which is the general rule for a gas approaching its condensation point.

Run the same gas at a large volume and the two corrections shrink to nothing. That is why the ideal gas law works so well at ordinary pressures: it is not that the corrections are absent, it is that they are small and they partly cancel.

At very high pressures (above ~300 atm), van der Waals itself becomes inaccurate because the constants a and b were derived from low-pressure measurements. More sophisticated equations of state (Peng-Robinson, Soave-Redlich-Kwong) are used in industrial process simulation.