Equations of State & Gases · Concept

Van der Waals Equation

Reading time: 9 minConfidence: HighLast verified: October 2026

The van der Waals equation, (p + an²/V²)(V − nb) = nRT, is an equation of state proposed by Johannes Diderik van der Waals in 1873. The parameter a accounts for attractive forces between molecules, which reduce the pressure on the walls, and b for the volume excluded by the molecules themselves. It was the first equation of state to describe both gas and liquid and to predict a critical point, earning van der Waals the 1910 Nobel Prize in Physics.

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One sentence that captures itThe van der Waals equation fixes the ideal gas law by admitting two facts: molecules take up space, and they pull on each other.

The ideal gas law treats molecules as dimensionless points that ignore one another. That works at low density but cannot explain why gases condense into liquids. Van der Waals made two small corrections: subtract the volume the molecules themselves occupy, and reduce the pressure because neighbouring molecules attract and slow each other down before they hit the wall.

With just these two changes, a single equation describes gas, liquid and the transition between them, including the critical point, above which liquid and gas become indistinguishable. The weak attractions it introduced are now called van der Waals forces (see Johannes Diderik van der Waals).

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Worked example — carbon dioxideFor CO₂, tabulated parameters are a = 0.3640 Pa m⁶ mol⁻² and b = 4.267×10⁻⁵ m³/mol. One mole in 1.00 L at 300 K: the ideal gas law gives 24.9 bar; van der Waals gives p = RT/(V−b) − a/V² ≈ 22.4 bar, about 10% lower because attraction dominates. The critical constants predicted are T_c = 8a/27Rb ≈ 304 K and p_c = a/27b² ≈ 7.40 MPa, matching measured values (304.1 K, 7.38 MPa) closely, but only because tabulated a and b are usually fitted to the critical point. The real test is the critical compressibility factor: van der Waals predicts Z_c = 3/8 = 0.375, while CO₂'s measured value is about 0.274. (Computed in Python.)

Gas	a (Pa m⁶ mol⁻²)	b (10⁻⁵ m³/mol)
Helium	0.00346	2.38
Nitrogen	0.137	3.87
Carbon dioxide	0.364	4.27
Water	0.554	3.05

Common mistake — Commonly ConfusedThe van der Waals equation is qualitatively right but quantitatively poor near the critical point and for liquids. Its isotherms below T_c contain an unphysical region where pressure increases with volume; real systems instead follow a flat line of liquid–vapour coexistence, located by Maxwell's equal-area construction (see Maxwell Relations).

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Primary sourcevan der Waals, J.D. (1873). Over de Continuïteit van den Gas- en Vloeistoftoestand (On the Continuity of the Gaseous and Liquid States). Doctoral thesis, Leiden University.

Corresponding states. Writing the equation in reduced variables p_r = p/p_c, V_r = V/V_c, T_r = T/T_c gives a universal form with no substance-specific parameters, (p_r + 3/V_r²)(3V_r − 1) = 8T_r. Van der Waals's law of corresponding states (1880) holds approximately for many simple fluids and guided Kamerlingh Onnes in liquefying helium (see Heike Kamerlingh Onnes).

Critical behaviour. Near the critical point, van der Waals theory is a mean-field theory and predicts critical exponents such as β = 1/2 for the difference between liquid and gas densities. Experiments give β ≈ 0.326, the same as the three-dimensional Ising model, a universality explained by renormalisation-group theory (Wilson, Nobel Prize 1982). Modern engineering uses cubic equations of state such as Peng–Robinson, or reference equations fitted to extensive data.

Virial expansion. At low density the van der Waals equation gives a second virial coefficient B(T) = b − a/RT, which is negative at low temperature (attraction dominates) and positive at high temperature, crossing zero at the Boyle temperature T_B = a/Rb (see Kinetic Theory of Gases).

Concept Relationship Map

Requires: ideal gas law, intermolecular forces
Enables: critical point, phase transitions, corresponding states
Commonly confused with: an accurate engineering equation of state

Sources

van der Waals, J.D. (1873). Doctoral thesis, Leiden University.
The Nobel Foundation (1910). The Nobel Prize in Physics 1910. nobelprize.org.
Atkins, P., de Paula, J. (2014). Atkins' Physical Chemistry (10th ed.). Oxford University Press.

Last verified: October 2026 · Source: van der Waals (1873); Atkins (2014) · Confidence: high

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