Waalor

Meet Waalor, a crowd of gas particles who take up room and tug gently on each other.

The ideal gas law treats particles as points that never attract. Real particles have size, and they pull on each other slightly. Johannes van der Waals added two corrections. The b term removes the space the particles fill, and the a term accounts for the attraction.

For one mole of carbon dioxide in 1.0 L at 300 K, the ideal law gives about 2.49 × 10⁶ Pa. Using a = 0.364 Pa·m⁶/mol² and b = 4.27 × 10⁻⁵ m³/mol, the van der Waals equation gives about 2.24 × 10⁶ Pa, roughly 10 percent lower.

At low pressure the two answers agree, because particles are far apart. Squeeze the gas hard and the difference grows.

The equation also predicts that gases turn to liquid when cooled and compressed enough, which earned van der Waals the 1910 Nobel Prize. At low pressure and high temperature the corrections shrink, and the ideal gas law works again.