Covalent bond distance, covalent radius, and van der Waals radius are used to describe the size and distance between atoms. Covalent bond distance refers to the distance between the nuclei of two bonded atoms. Covalent radius is half of the internuclear separation between the nuclei of two single-bonded atoms of the same species (homonuclear). While van der Waals radius is used to define half of the distance between the closest approach of two non-bonded atoms of a given element.

## Introduction

Covalent bond distance, covalent radius, and van der Waals radius are all experimentally determined values that describe the distance around or between atoms.

## Covalent Bond Distance

Covalent bond distance is defined by the internuclear distance between two covalently bonded atoms and is denoted by d. It can be used as a measure of the strength of a bond and can be measured for bonded atoms that have any bond order (not just singly-bonded atoms, as is the case of rcov). Values for Covalent bond distance are determined by X-ray, electron and neutron diffraction, as well as by microwave spectroscopy. Though d is the most commonly measured value, rcov is more commonly used (see next section below). d is measured in picometers (1pm=10-12m) and ångströms (1Å=10-10m). Covalent bond distance can be used as a measure of how tightly two atoms are bonded and is often used to support calculations of bond order. Typically, the higher the bond order, the shorter the Covalent bond distance.

It should be noted that the terms 'Covalent bond distance' and 'Bond Length' are often used interchangeably. The two terms have essentially the same definition with the difference only being that Covalent bond distance can only refer to covalently bonded atoms while bond length could theoretically refer to two atoms bonded in any way.

Covalent bond radius refers to half of the distance between two singly bonded atoms of the same element ( homonuclear X-X bond) and is denoted rcov. In other words, it is defined as half the covalent bond length when the two atoms bonded are homonuclear. rcov is typically measured by X-ray diffraction. When multiple allotropes exist for a given element, rcov is determined by averaging the bond distances found in each allotrope. Covalent radius is measured in picometers (pm) or ångströms (Å) and represents one measurement that is used in determining atomic radius.

van der Waals radius is defined as half of the internuclear separation of two non-bonded atoms of the same element on their closest possible approach and is denoted rv. As shown in the following figure, the red line indicates the van der Waals radius as the blue line is the diameter. In addition to not being covalently bonded, the atoms must also not be of the same molecule. The term incorporates the intermolecular forces (dipole-dipole, dispersion) and is therefore useful in determining how molecules pack in solid. rv is measured for two non-bonded atoms in the crystal of a molecule. The name for the term comes from its relationship to van der Waals Interactions. rv is generally greater than rcov for a given element. rv is also measured in terms of picometers (pm) and ångströms (Å).

Table of Sample Values for rcov and rv
Element rcov (pm) rv (pm)
H 37 120
B 88 208
C 77 185
N 75 154
O 73 140
F 71 135
Cl 99 180
Br 114 195
I 133 215
He n/a 99

Values: Housecroft, Inorganic Chemistry, 2008

Van der Waals equation was originated from the ideal gas law:

pV = nRT (p: gas pressure, V: volume, n: amount of substance, R: ideal gas constant, T: absolute temperature).

Based on the ideal gas law above, van der Waals equation takes the molecular size and the molecular force into account. As a result, the attractive force (a/V2) is added into the pressure part. Likewise, the volume is subtracted by the molecular volume (b), which is determined by the van der Waals radius.

Therefore, the equation for one mole is written as

$\left (p + \dfrac{a}{V^2} \right ) \left (V - b \right ) = RT$

And the equation for multiple moles is written as

(p + a(n/V)2)(V - nb) = nRT (n: the number of moles).

The following figure illustrates the difference between the Covalent radius and the van der Waals radius for better understanding of two terms.

Figure: Van der Waals radius and Covalent radius (illustrated by Heesung Shim)

## References

1. Housecroft, Catherine E., Sharpe, Alan G. Inorganic Chemistry. Third ed. Pearson / Prentice Hall: Essex, UK. 2008. pp. 31, 1013, 1014
2. Chang, Raymond. Physical Chemistry for the Biosciences. First ed. University Science Books: Sausalito, CA. 2005. pp. 500
3. Oxtoby, David W., Gillis, H.P., Campion, Alan. Principles of Modern Chemistry. Sixth ed. Thomson Brooks/Cole: United States. 2008. pp. 419