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|+ van der Waals radii

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! Element !! radius (Å)

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| Hydrogen || 1.2 (1.09)<br/>Values from other sources may<br/>differ significantly (see text)</small>

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The van der Waals radius, r, of an atom is the radius of an imaginary hard sphere representing the distance of closest approach for another atom.

It is named after Johannes Diderik van der Waals, winner of the 1910 Nobel Prize in Physics, as he was the first to recognise that atoms were not simply points and to demonstrate the physical consequences of their size through the van der Waals equation of state.

van der Waals volume

The van der Waals volume, V, also called the atomic volume or molecular volume, is the atomic property most directly related to the van der Waals radius. It is the volume "occupied" by an individual atom (or molecule). The van der Waals volume may be calculated if the van der Waals radii (and, for molecules, the inter-atomic distances, and angles) are known. For a single atom, it is the volume of a sphere whose radius is the van der Waals radius of the atom:<math display="block">V_{\rm w} = {4\over 3}\pi r_{\rm w}^3.</math>

For a molecule, it is the volume enclosed by the van der Waals surface.

The van der Waals volume of a molecule is always smaller than the sum of the van der Waals volumes of the constituent atoms: the atoms can be said to "overlap" when they form chemical bonds.

The van der Waals volume of an atom or molecule may also be determined by experimental measurements on gases, notably from the van der Waals constant b, the polarizability α, or the molar refractivity A.

In all three cases, measurements are made on macroscopic samples and it is normal to express the results as molar quantities.

To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N.

The molar van der Waals volume should not be confused with the molar volume of the substance.

In general, at normal laboratory temperatures and pressures, the atoms or molecules of gas only occupy about of the volume of the gas, the rest is empty space.

Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure.

Table of van der Waals radii

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! colspan=20 style="background:; padding:2px 4px;" | Van der Waals radius of the elements in the periodic table<!--

If r is the radius, then the colors follow the formula

|7= 0xff0000 + (r-110) * 0x100

Example: r = 140 for helium so:

|7= 0xff0000 + (140-110) * 0x100

|7= 0xff0000 + 0x1e * 0x100 (since 30 = 0x1e)

|7= 0xff0000 + 0x1e00

|7= 0xff1e00

-->

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Therefore, the van der Waals volume of a single atom V&nbsp;= 39.36&nbsp;Å, which corresponds to r&nbsp;= 2.11&nbsp;Å (≈ 200 picometers).

This method may be extended to diatomic gases by approximating the molecule as a rod with rounded ends where the diameter is and the internuclear distance is .

The algebra is more complicated, but the relation

<math display="block">V_{\rm w} = {4\over 3}\pi r_{\rm w}^3 + \pi r_{\rm w}^2d</math>

can be solved by the normal methods for cubic functions.

Crystallographic measurements

The molecules in a molecular crystal are held together by van der Waals forces rather than chemical bonds.

In principle, the closest that two atoms belonging to different molecules can approach one another is given by the sum of their van der Waals radii.

By examining a large number of structures of molecular crystals, it is possible to find a minimum radius for each type of atom such that other non-bonded atoms do not encroach any closer.

This approach was first used by Linus Pauling in his seminal work The Nature of the Chemical Bond.

Arnold Bondi also conducted a study of this type, published in 1964, The values of different authors are sometimes very different, so that one has to choose the ones which are closest in their physical meaning to those one wants to compare with. Here is a table with entries of four different authors. The values of Bondi from 1966 are those mostly used in crystallography:

{| class="wikitable"

|

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|r<sub>vdW</sub> / Å

|r<sub>vdW</sub> / Å

|r<sub>vdW</sub> / Å

|r<sub>vdW</sub> / Å

|-

|Element

|Atomic

number

|Bondi

2001

|Hu

2009

|Alvarez corresponding to a molar volume V&nbsp;= .

The van der Waals volume is given by

<math display="block">V_{\rm w} = \frac{\pi V_{\rm m{N_{\rm A}\sqrt{18</math>

where the factor of π/√18 arises from the packing of spheres: V&nbsp;= &nbsp;= 23.0&nbsp;Å, corresponding to a van der Waals radius r&nbsp;= 1.76&nbsp;Å.

Molar refractivity

The molar refractivity of a gas is related to its refractive index by the Lorentz–Lorenz equation:

<math display="block">A = \frac{R T (n^2 - 1)}{3p}</math>

The refractive index of helium n&nbsp;= at 0&nbsp;°C and 101.325&nbsp;kPa, which corresponds to a molar refractivity A&nbsp;= .

Dividing by the Avogadro constant gives V&nbsp;= &nbsp;= 0.8685&nbsp;Å, corresponding to r&nbsp;= 0.59&nbsp;Å.

Polarizability

The polarizability α of a gas is related to its electric susceptibility χ by the relation

<math display="block">\alpha = {\varepsilon_0 k_{\rm B}T\over p}\chi_{\rm e}</math>

and the electric susceptibility may be calculated from tabulated values of the relative permittivity ε using the relation χ&nbsp;= ε&nbsp;−&nbsp;1.

The electric susceptibility of helium χ&nbsp;= at 0&nbsp;°C and 101.325&nbsp;kPa, which corresponds to a polarizability α&nbsp;= .

The polarizability is related the van der Waals volume by the relation

<math display="block">V_{\rm w} = {1\over{4\pi\varepsilon_0\alpha ,</math>

so the van der Waals volume of helium V&nbsp;= &nbsp;= 0.2073&nbsp;Å by this method, corresponding to r&nbsp;= 0.37&nbsp;Å.

When the atomic polarizability is quoted in units of volume such as Å, as is often the case, it is equal to the van der Waals volume.

However, the term "atomic polarizability" is preferred as polarizability is a precisely defined (and measurable) physical quantity, whereas "van der Waals volume" can have any number of definitions depending on the method of measurement.

See also

  • Atomic radii of the elements (data page)
  • van der Waals force
  • van der Waals molecule
  • van der Waals strain
  • van der Waals surface

References

Further reading

  • van der Waals Radius of the elements at PeriodicTable.com
  • van der Waals Radius – Periodicity at WebElements.com