Van der Waals radius

The van der Waals radius of an atom is the radius of an imaginary hard sphere which can be used to model the atom for many purposes. Van der Waals radii are determined from measurements of atomic spacing between pairs of unbonded atoms in crystals.

The van der Waals radius is named after Johannes Diderik van der Waals, winner of the 1910 Nobel Prize in Physics.

Real gases do not behave exactly as predicted. In some cases the deviation can be extremely large. For example, ideal gases could never become liquids or solids, no matter how much they were cooled or compressed. Modifications of the ideal gas law, $$P\tilde{V} = nRT$$, were therefore proposed. Particularly useful and well known is the van der Waals equation of state: $$\left (P + a(\frac{n}{\tilde{V}})^2\right ) (\tilde{V} - nb) = nRT$$, where a and b are adjustable parameters determined from experimental measurements carried out on actual gases. Their values vary from gas to gas.

The van der Waals equation also has a microscopic interpretation. Molecules interact with one another. The interaction is strongly repulsive at very short distance, becomes mildly attractive at intermediate range, and vanishes at long distance. The ideal gas law must be corrected when attractive and repulsive forces are considered. For example, the mutual repulsion between molecules has the effect of excluding neighbours from a certain amount of territory around each molecule. Thus, a fraction of the total space becomes unavailable to each molecule as it executes random motion. In the equation of state, this volume of exclusion (nb) should be subtracted from the volume of the container (V), thus: (V - nb). The other term that is introduced in the van der Waals equation, $$a(\frac{n}{\tilde{V}})^2$$, describes a weak attractive force among molecules, which increases when n increases or V decreases and molecules become more crowded together.

Van der Waals volume
The van der Waals volume of an atom is the volume of a sphere with the Van der Waals radius of the atom.

Two atoms which are not chemically bonded have a minimum distance between their centers, which is equal to the sum of their Van der Waals radii. However, if the atoms are bonded by a covalent bond, the distance between their centers is smaller.

Therefore, the van der Waals volume of a molecule with covalent bonds is smaller than the sum of the van der Waals volumes of the atoms.

The van der Waals volume of a system of molecules is the sum of their van der Waals volumes.