The van der Waals radius, rw, 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.
1 Van der Waals volume 2 Methods of determination
Van der Waals equation
3.1 Further reading
4 External links
Van der Waals volume
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The van der Waals volume, Vw, 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 spherical single atom, it is the volume of a sphere whose radius is the van der Waals radius of the atom:
displaystyle V_ rm w = 4 over 3 pi r_ rm w ^ 3
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
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Van der Waals radii may be determined from the mechanical properties
of gases (the original method), from the critical point, from
measurements of atomic spacing between pairs of unbonded atoms in
crystals or from measurements of electrical or optical properties (the
polarizability and the molar refractivity). These various methods give
values for the van der Waals radius which are similar (1–2 Å,
100–200 pm) but not identical. Tabulated values of van der
Waals radii are obtained by taking a weighted mean of a number of
different experimental values, and, for this reason, different tables
will often have different values for the van der Waals radius of the
same atom. Indeed, there is no reason to assume that the van der Waals
radius is a fixed property of the atom in all circumstances: rather,
it tends to vary with the particular chemical environment of the atom
in any given case.
Van der Waals equation
p + a
− n b ) = n R T
displaystyle left(p+aleft( frac n tilde V right)^ 2 right)( tilde V -nb)=nRT
where p is pressure, n is the number of moles of the gas in question and a and b depend on the particular gas,
displaystyle tilde V
is the volume, R is the specific gas constant on a unit mole basis and T the absolute temperature; a is a correction for intermolecular forces and b corrects for finite atomic or molecular sizes; the value of b equals the Van der Waals volume per mole of the gas. 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 neighbors from a certain amount of space 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,
displaystyle aleft( frac n tilde V right)^ 2
, describes a weak attractive force among molecules (known as the van der Waals force), which increases when n increases or V decreases and molecules become more crowded together.
Gas d (Å) b (cm3mol–1) Vw (Å3) rw (Å)
Hydrogen 0.74611 26.61 44.19 2.02
Nitrogen 1.0975 39.13 64.98 2.25
Oxygen 1.208 31.83 52.86 2.06
Chlorine 1.988 56.22 93.36 2.39
Van der Waals radii calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981).
The van der Waals constant b volume can be used to calculate the van
der Waals volume of an atom or molecule with experimental data derived
from measurements on gases.
For helium, b = 23.7 cm3/mol.
displaystyle V_ rm w = b over N_ rm A
Therefore, the van der Waals volume of a single atom Vw = 39.36 Å3, which corresponds to rw = 2.11 Å. This method may be extended to diatomic gases by approximating the molecule as a rod with rounded ends where the diameter is 2rw and the internuclear distance is d. The algebra is more complicated, but the relation
displaystyle V_ rm w = 4 over 3 pi r_ rm w ^ 3 +pi r_ rm w ^ 2 d
can be solved by the normal methods for cubic functions.
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
displaystyle V_ rm w = frac pi V_ rm m N_ rm A sqrt 18
where the factor of π/√18 arises from the packing of spheres: Vw = 6971229999999999999♠2.30×10−29 m3 = 23.0 Å3, corresponding to a van der Waals radius rw = 1.76 Å. Molar refractivity The molar refractivity A of a gas is related to its refractive index n by the Lorentz–Lorenz equation:
R T (
− 1 )
displaystyle A= frac RT(n^ 2 -1) 3p
The refractive index of helium n =
7000100003500000000♠1.0000350 at 0 °C and 101.325 kPa,
which corresponds to a molar refractivity A =
6993523000000000000♠5.23×10−7 m3/mol. Dividing by the
displaystyle alpha = epsilon _ 0 k_ rm B T over p chi _ rm e
and the electric susceptibility may be calculated from tabulated values of the relative permittivity εr using the relation χe = εr–1. The electric susceptibility of helium χe = 6995700000000000000♠7×10−5 at 0 °C and 101.325 kPa, which corresponds to a polarizability α = 6959230700000000000♠2.307×10−41 cm2/V. The polarizability is related the van der Waals volume by the relation
displaystyle V_ rm w = 1 over 4pi epsilon _ 0 alpha ,
so the van der Waals volume of helium Vw = 6969207300000000000♠2.073×10−31 m3 = 0.2073 Å3 by this method, corresponding to rw = 0.37 Å. When the atomic polarizability is quoted in units of volume such as Å3, 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. References
^ Rowland RS, Taylor R (1996). "Intermolecular nonbonded contact
distances in organic crystal structures: comparison with distances
expected from van der Waals radii". J. Phys. Chem. 100 (18):
^ a b c Bondi, A. (1964). "Van der Waals Volumes and Radii". J. Phys.
Chem. 68 (3): 441–451. doi:10.1021/j100785a001.
^ Weast, Robert C., ed. (1981). CRC Handbook of Chemistry and Physics
(62nd ed.). Boca Raton, FL: CRC Press. ISBN 0-8493-0462-8. ,
^ Pauling, Linus (1945). The Nature of the Chemical Bond. Ithaca, NY:
Cornell University Press. ISBN 0-8014-0333-2.
^ Rowland, R. Scott; Taylor, Robin (1996). "Intermolecular Nonbonded
Contact Distances in Organic
Huheey, James E.; Keiter, Ellen A.; Keiter, Richard L. (1997). Inorganic Chemistry: Principles of Structure and Reactivity (4th ed.). New York: Prentice Hall. ISBN 0-06-042995-X.