Covalent bond distance, covalent radius and van der Waals radius

Three important definitions are needed before we discuss covalent bonding. The length of a covalent bond (bond distance), d, is the internuclear separation and may be determined experimentally by  microwave spectroscopy or diffraction methods (X-ray, neutron or electron diffraction).

Fig. 1  The structure of hydrogen peroxide, H₂O₂; O atoms are shown in red.

     It is convenient to define the covalent radius, r_cov, of an atom: for an atom X, r_cov is half of the covalent bond length of a homonuclear X-X single bond. Thus, r_cov(S) can be determined from the solid state structure of S₈ (Figure 2c) determined by X-ray diffraction methods or, better still, by averaging the values of the bond distances of S-S single bonds found for all the allotropes of sulfur.

Fig. 2 Mass spectrometric traces for (a) atomic Ru and (b) molecular S8; the mass:charge ratio is m/z and in these traces z = 1. (c) The molecular structure of S₈.

For an atom X, the value of the single bond covalent radius, rcov, is half of the internuclear separation in a homonuclear X-X single bond.  The a- and b-forms of sulfur (orthorhombic and monoclinic sulfur, respectively) both crystallize with S₈ molecules stacked in a regular arrangement; the packing in the a-form (density = 2.07 g cm^-3) is more efficient than that in the b-form (density = 1.94 g cm^-3).

    Van der Waals forces operate between the molecules, and half of the distance of closest approach of two sulfur atoms belonging to different S₈ rings is defined as the van der Waals radius, r_v, of sulfur. The weakness of the bonding is evidenced by the fact that S₈ vaporizes, retaining the ring structure, without absorbing much energy. The van der Waals radius of an element is necessarily larger than its covalent radius, e.g. r_v and r_cov for S are 185 and 103pm respectively.   The van der Waals radius, r_v, of an atom X is half of the distance of closest approach of two non-bonded atoms of X.