What is the Van der Waals equation?

It is a modification of the ideal gas law that accounts for the finite volume of molecules (b) and the attractive forces between them (a): (P + n²a/V²)(V − nb) = nRT.

What do a and b mean?

a corrects for intermolecular attractions and is larger for polar or strongly interacting molecules like H₂O. b corrects for the volume occupied by molecules themselves and is larger for bigger molecules.

When should I use Van der Waals instead of PV = nRT?

Use it at high pressures (gas compressed to low molar volume) and/or low temperatures (near condensation), where real-gas deviations matter.

Johannes Diderik van der Waals published the equation in his 1873 doctoral thesis, winning the 1910 Nobel Prize in Physics for the work.

Where do the a and b constants come from?

They are measured experimentally for each substance and tabulated. Common gases like H₂, O₂, N₂, CO₂, He, and H₂O have well-known values.

Is the Van der Waals equation exact?

No. It is a two-constant approximation that improves on the ideal gas law but still misses some real gas behavior, especially very close to the critical point. More elaborate equations of state (Redlich-Kwong, Peng-Robinson) trade simplicity for accuracy in industrial applications.

Van der Waals Equation Calculator

A real-gas equation of state: (P + n²a/V²)(V − nb) = nRT. Pick a preset gas or supply your own a and b.

Van der Waals Equation Calculator

(P + n²a/V²)(V − nb) = nRT. R = 0.0821 L·atm/(mol·K).

Gas (preset)

a (L²·atm/mol²)

b (L/mol)

n (mol)

V (L)

T (K)

Pressure (real gas)

0.9953 atm

P = nRT/(V−nb) − n²a/V² = 0.9953 atm | Ideal: 1.001 atm

Beyond the Ideal Gas Law

The Ideal Gas Law PV = nRT assumes molecules are point particles with no mutual attraction. Real molecules occupy space and attract one another. The Van der Waals equation introduces two empirical constants, a for attractions and b for excluded volume, to keep the predicted pressures and volumes accurate over a much wider range of conditions.

The equation form (P + n²a/V²)(V − nb) = nRT reduces to PV = nRT when a and b go to zero. The correction P + n²a/V² adds back the pressure "lost" to intermolecular attractions, while V − nb subtracts the volume already occupied by the molecules themselves.

Where Van der Waals Matters Most

Use this equation when the gas is dense: compressed natural gas in cylinders, the high-pressure side of a refrigeration cycle, or any system near a phase transition. At ambient laboratory conditions for nitrogen or oxygen, the simpler Charles' and Boyle's laws give answers within a fraction of a percent of Van der Waals.

Van der Waals Constants for Common Gases

Gas	a (L²·atm/mol²)	b (L/mol)
Hydrogen (H₂)	0.2476	0.02661
Oxygen (O₂)	1.378	0.03183
Nitrogen (N₂)	1.370	0.0387
Carbon dioxide (CO₂)	3.640	0.04267
Helium (He)	0.0346	0.0238
Water vapour (H₂O)	5.536	0.03049
Methane (CH₄)	2.283	0.04278
Ammonia (NH₃)	4.225	0.03707

These are the same preset values the calculator above loads when you pick a gas from the dropdown. Notice that small, non-polar molecules like helium and hydrogen have the smallest a values (weak attraction, close to ideal), while polar water vapour has the largest.

Worked Example 1: CO₂ at High Pressure

1 mol of CO₂ in 0.5 L at 300 K. Ideal gas law: P = nRT/V = (1 × 0.08206 × 300) / 0.5 = 49.2 atm. Van der Waals, using a = 3.64 and b = 0.04267 for CO₂: P = nRT/(V − nb) − n²a/V² = 24.62/(0.5 − 0.04267) − 3.64/0.5² = 53.8 − 14.6 = 39.2 atm. The real gas pressure is about 20% lower than the ideal prediction, because CO₂'s intermolecular attraction pulls the molecules inward and reduces the force they exert on the container wall.

Worked Example 2: Helium at Standard Conditions

1 mol He at 273.15 K and 22.4 L. Ideal gas law predicts P = nRT/V = (1 × 0.08206 × 273.15) / 22.4 ≈ 1.00 atm. Van der Waals, using a = 0.0346 and b = 0.0238 for helium, also predicts about 1.00 atm. Helium is so close to an ideal gas at this density that the two corrections almost exactly cancel out, which is why helium is often the reference gas chemistry students meet first.