Editing Van der Waals radius (section)

== Methods of determination ==
{{refimprove science|date=June 2015}}
Van der Waals radii may be determined from the [[mechanics|mechanical]] properties of gases (the original method), from the [[critical point (thermodynamics)|critical point]], from measurements of atomic spacing between pairs of unbonded atoms in [[crystal]]s 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&nbsp;[[angstrom|Å]], 100–200&nbsp;[[picometre|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.<ref name="Bondi1964"/>

=== Van der Waals equation of state ===
{{main|Van der Waals equation}}

The van der Waals equation of state is the simplest and best-known modification of the [[ideal gas law]] to account for the behaviour of [[real gas]]es:
<math display="block">\left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT,</math>
where {{mvar|p}} is pressure, {{mvar|n}} is the number of moles of the gas in question and {{mvar|a}} and {{mvar|b}} depend on the particular gas, <math>\tilde{V}</math> is the volume, {{mvar|R}} is the specific gas constant on a unit mole basis and {{mvar|T}} the absolute temperature; {{mvar|a}} is a correction for intermolecular forces and {{mvar|b}} corrects for finite atomic or molecular sizes; the value of {{mvar|b}} equals the van der Waals volume per mole of the gas. 
[[van der Waals constants (data page)|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 a very short distance, becomes mildly attractive at the intermediate range, and vanishes at a 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 ({{math|''nb''}}) should be subtracted from the volume of the container ({{mvar|V}}), thus: ({{math|''V'' - ''nb''}}). 
The other term that is introduced in the van der Waals equation, <math display="inline">a\left (\frac{n}{\tilde{V}}\right )^2</math>, describes a weak attractive force among molecules (known as the [[van der Waals force]]), which increases when {{mvar|n}} increases or {{mvar|V}} decreases and molecules become more crowded together.

{| class="wikitable floatright" align=right
|-
! Gas
! ''d'' ([[Ångström|Å]])
! ''b'' (cm{{sup|3}}mol{{sup|–1}})
! ''V''{{sub|w}} (Å{{sup|3}})
! ''r''{{sub|w}} (Å)
|-
| [[Hydrogen]]
| 0.74611
| align=center | 26.61
| 34.53
| 2.02
|-
| [[Nitrogen]]
| 1.0975
| align=center | 39.13
| 47.71
| 2.25
|-
| [[Oxygen]]
| 1.208
| align=center | 31.83
| 36.62
| 2.06
|-
| [[Chlorine]]
| 1.988
| align=center | 56.22
| 57.19
| 2.39
|-
| colspan=5 | <small>van der Waals radii ''r''{{sub|w}} in Å (or in 100 picometers) calculated from the [[van der Waals constant]]s<br/>of some diatomic gases. Values of ''d'' and ''b'' from Weast (1981).</small>
|}

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]],<ref name="CRC">{{RubberBible62nd}}, p. D-166.</ref> ''b''&nbsp;= 23.7&nbsp;cm{{sup|3}}/mol. Helium is a [[monatomic gas]], and each mole of helium contains {{val|6.022|e=23}} atoms (the [[Avogadro constant]], ''N''{{sub|A}}):
<math display="block">V_{\rm w} = {b\over{N_{\rm A}}}</math>
Therefore, the van der Waals volume of a single atom ''V''{{sub|w}}&nbsp;= 39.36&nbsp;Å{{sup|3}}, which corresponds to ''r''{{sub|w}}&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 {{math|2''r''{{sub|w}}}} and the internuclear distance is {{mvar|d}}. 
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 force]]s rather than [[chemical bond]]s. 
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''.<ref name="Pauling1945">{{cite book | first = Linus | last = Pauling | author-link = Linus Pauling | year = 1945 | title = The Nature of the Chemical Bond | location = Ithaca, NY | publisher = Cornell University Press | isbn = 978-0-8014-0333-0| title-link = The Nature of the Chemical Bond }}</ref> 
Arnold Bondi also conducted a study of this type, published in 1964,<ref name="Bondi1964"/> although he also considered other methods of determining the van der Waals radius in coming to his final estimates. 
Some of Bondi's figures are given in the table at the top of this article, and they remain the most widely used "consensus" values for the van der Waals radii of the elements. 
Scott Rowland and Robin Taylor re-examined these 1964 figures in the light of more recent crystallographic data: on the whole, the agreement was very good, although they recommend a value of 1.09&nbsp;Å for the van der Waals radius of [[hydrogen]] as opposed to Bondi's 1.20&nbsp;Å.<ref name="RowlandRS1996"/> A more recent analysis of the [[Cambridge Structural Database]], carried out by Santiago Alvarez, provided a new set of values for 93 naturally occurring elements.<ref name="Alvarez2013">{{cite journal |last=Alvarez |first=Santiago |year=2013 |title=A cartography of the van der Waals territories |journal=[[Dalton Trans.]] |volume=42 |issue=24 |pages=8617–36 |doi=10.1039/C3DT50599E |pmid=23632803 |doi-access=free |hdl-access=free |hdl=2445/48823}}</ref> The values of different authors are sometimes very different, so that one has to chose 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 valuse of Bondi from 1966 are those mostly used in crystallography:
{| class="wikitable"
|
|
|''r''<sub>vdW</sub> / Å
|''r''<sub>vdW</sub>  / Å
|''r''<sub>vdW</sub>  / Å
|''r''<sub>vdW</sub>  / Å
|-
|'''Element'''
|'''Atomic'''
'''number'''
|'''Bondi'''<ref name="Bondi1964" />    
1966
|'''Batsanov'''<ref>{{Cite journal |last=Batsanov |first=S. S. |date=2001 |title=[No title found] |url=https://link.springer.com/10.1023/A:1011625728803 |journal=Inorganic Materials |volume=37 |issue=9 |pages=871–885 |doi=10.1023/A:1011625728803}}</ref> 
2001
|'''Hu'''<ref>{{Cite journal |last=S. Z. Hu, Z. H. Zhou, B. E. Robertson |date=2009 |title=Consistent approaches to van der Waals radii for the metallic elements |url= |journal=Z. Kristallogr. |volume=224 |issue=8 |pages=375–383|doi=10.1524/zkri.2009.1158 |bibcode=2009ZK....224..375H }}</ref>           
2009
|'''Alvarez'''<ref name="Alvarez2013" /> 
2014
|-
|H
|1
|1.2
|1.0
|1.08
|1.2
|-
|Li
|3
|1.82
|2.15
|2.14
|2.12
|-
|Be
|4
|
|1.85
|1.69
|1.98
|-
|B
|5
|
|1.75
|1.68
|1.91
|-
|C
|6
|1.7
|1.7
|1.53
|1.77
|-
|N
|7
|1.55
|1.6
|1.51
|1.66
|-
|O
|8
|1.52
|1.55
|1.49
|1.5
|-
|F
|9
|1.47
|1.45
|1.48
|1.46
|-
|Na
|11
|2.27
|2.45
|2.38
|2.5
|-
|Mg
|12
|1.73
|2.15
|2.00
|2.51
|-
|Al
|13
|
|2.05
|1.92
|2.25
|-
|Si
|14
|2.1
|2.05
|1.93
|2.19
|-
|P
|15
|1.8
|1.95
|1.88
|1.9
|-
|S
|16
|1.8
|1.8
|1.81
|1.89
|-
|Cl
|17
|1.75
|1.8
|1.75
|1.82
|-
|Se
|17
|1.90
|1.9
|1.92
|1.82
|-
|K
|19
|2.75
|2.85
|2.52
|2.73
|-
|Ca
|20
|
|2.45
|2.27
|2.62
|-
|Sc
|21
|
|2.25
|2.15
|2.58
|-
|Ti
|22
|
|2.10
|2.11
|2.45
|-
|V
|23
|
|2.05
|2.07
|2.42
|-
|Cr
|24
|
|2.0
|2.06
|2.45
|-
|Mn
|25
|
|2.0
|2.05
|2.45
|-
|Fe
|26
|
|2.0
|2.04
|2.44
|-
|Ni
|28
|1.63
|1.95
|1.97
|2.4
|-
|Cu
|29
|1.40
|1.9
|1.96
|2.38
|-
|Zn
|30
|1.39
|2.0
|2.01
|2.39
|-
|Ga
|31
|1.87
|2.05
|2.03
|2.32
|-
|Ge
|32
|
|2.05
|2.05
|2.29
|-
|As
|33
|1.85
|2.05
|2.08
|1.88
|-
|Br
|35
|1.85
|1.9
|1.9
|1.86
|-
|Rb
|37
|
|3.0
|2.61
|3.21
|-
|Sr
|38
|
|2.6
|2.42
|2.84
|-
|Y
|39
|
|2.4
|2.32
|2.75
|-
|Zr
|39
|
|2.3
|2.23
|2.52
|-
|Nb
|41
|
|2.15
|2.18
|2.56
|-
|Mo
|42
|
|2.1
|2.17
|2.45
|-
|Tc
|43
|
|2.1
|2.16
|2.44
|-
|Ru
|44
|
|2.05
|2.13
|2.46
|-
|Rh
|45
|
|2.0
|2.1
|2.44
|-
|Pd
|46
|1.63
|2.05
|2.1
|2.15
|-
|Ag
|47
|1.72
|2.05
|2.11
|2.53
|-
|Co
|47
|
|1.95
|2
|2.4
|-
|Cd
|48
|1.62
|2.2
|2.18
|2.43
|-
|In
|49
|1.93
|2.25
|2.21
|2.43
|-
|Sn
|50
|2.17
|2.2
|2.23
|2.42
|-
|Sb
|51
|
|2.25
|2.24
|2.47
|-
|Te
|52
|2.06
|2.15
|2.11
|1.99
|-
|I
|53
|1.98
|2.1
|2.09
|2.04
|-
|Cs
|55
|
|3.15
|2.75
|3.48
|-
|Ba
|56
|
|2.7
|2.59
|3.03
|-
|La
|57
|
|2.5
|2.43
|2.98
|-
|Hf
|72
|
|2.25
|2.23
|2.63
|-
|Ta
|73
|
|2.2
|2.22
|2.53
|-
|W
|74
|
|2.15
|2.18
|2.57
|-
|Re
|75
|
|0.21
|2.16
|2.49
|-
|Os
|76
|
|2.0
|2.16
|2.48
|-
|Ir
|77
|
|2.0
|2.13
|2.41
|-
|Pt
|78
|1.72
|2.05
|2.13
|2.32
|-
|Au
|79
|1.66
|2.0
|2.14
|2.32
|-
|Hg
|80
|1.70
|2.1
|2.23
|2.45
|-
|Tl
|81
|1.96
|2.25
|2.27
|2.47
|-
|Pb
|82
|2.02
|2.3
|2.37
|2.6
|-
|Bi
|83
|
|2.35
|2.38
|2.54
|-
|Th
|90
|
|2.45
|2.45
|2.93
|-
|U
|91
|1.86
|2.4
|2.41
|2.71
|}
A simple example of the use of crystallographic data (here [[neutron diffraction]]) is to consider the case of solid helium, where the atoms are held together only by van der Waals forces (rather than by [[Covalent bond|covalent]] or [[metallic bond]]s) and so the distance between the nuclei can be considered to be equal to twice the van der Waals radius. 
The density of solid helium at 1.1&nbsp;K and 66&nbsp;[[Atmosphere (unit)|atm]] is {{val|0.214|(6)|u=g/cm3}},<ref name="Henshaw1958">{{cite journal | first = D.G. | last = Henshaw | year = 1958 | title = Structure of Solid Helium by Neutron Diffraction | journal = [[Physical Review]] | volume = 109 | issue = 2 | pages = 328–330 | doi = 10.1103/PhysRev.109.328|bibcode = 1958PhRv..109..328H }}</ref> corresponding to a [[molar volume]] ''V''{{sub|m}}&nbsp;= {{val|18.7|e=-6|u=m3/mol}}. 
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 [[Sphere packing|packing of spheres]]: ''V''{{sub|w}}&nbsp;= {{val|2.30|e=-29|u=m3}}&nbsp;= 23.0&nbsp;Å{{sup|3}}, corresponding to a van der Waals radius ''r''{{sub|w}}&nbsp;= 1.76&nbsp;Å.

=== Molar refractivity ===

The [[molar refractivity]] {{mvar|A}} of a gas is related to its [[refractive index]] {{mvar|n}} by the [[Lorentz–Lorenz equation]]:
<math display="block">A = \frac{R T (n^2 - 1)}{3p}</math>
The refractive index of helium ''n''&nbsp;= {{val|1.0000350}} at 0&nbsp;°C and 101.325&nbsp;kPa,<ref>Kaye & Laby Tables, [http://www.kayelaby.npl.co.uk/general_physics/2_5/2_5_7.html Refractive index of gases].</ref> which corresponds to a molar refractivity ''A''&nbsp;= {{val|5.23|e=-7|u=m3/mol}}. 
Dividing by the Avogadro constant gives ''V''{{sub|w}}&nbsp;= {{val|8.685|e=-31|u=m3}}&nbsp;= 0.8685&nbsp;Å{{sup|3}}, corresponding to ''r''{{sub|w}}&nbsp;= 0.59&nbsp;Å.

=== Polarizability ===

The [[polarizability]] ''α'' of a gas is related to its [[electric susceptibility]] ''χ''{{sub|e}} 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 [[Permittivity|relative permittivity]] ''ε''{{sub|r}} using the relation ''χ''{{sub|e}}&nbsp;= ''ε''{{sub|r}}&nbsp;−&nbsp;1. 
The electric susceptibility of helium ''χ''{{sub|e}}&nbsp;= {{val|7|e=-5}} at 0&nbsp;°C and 101.325&nbsp;kPa,<ref>Kaye & Laby Tables, [http://www.kayelaby.npl.co.uk/general_physics/2_6/2_6_5.html Dielectric Properties of Materials].</ref> which corresponds to a polarizability ''α''&nbsp;= {{val|2.307|e=-41|u=C⋅m<sup>2</sup>/V}}. 
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''{{sub|w}}&nbsp;= {{val|2.073|e=-31|u=m3}}&nbsp;= 0.2073&nbsp;Å{{sup|3}} by this method, corresponding to ''r''{{sub|w}}&nbsp;= 0.37&nbsp;Å.

When the atomic polarizability is quoted in units of volume such as Å{{sup|3}}, 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.