Van der Waals radius

Last updated
van der Waals radii
Elementradius (Å)
Hydrogen 1.2 (1.09) [1]
Carbon 1.7
Nitrogen 1.55
Oxygen 1.52
Fluorine 1.47
Phosphorus 1.8
Sulfur 1.8
Chlorine 1.75
Copper 1.4
van der Waals radii taken from
Bondi's compilation (1964). [2]
Values from other sources may
differ significantly (see text)

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.

Contents

van der Waals volume

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 single atom, it is the volume of a sphere whose radius is the van der Waals radius of the atom:

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 NA.

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 11000 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 1000 times smaller than the molar volume for a gas at standard temperature and pressure.

Table of van der Waals radii

Methods of determination

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. [2]

Van der Waals equation of state

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 gases:

where p is pressure, n is the number of moles of the gas in question and a and b depend on the particular gas, 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 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 (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, , 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.

Gasd (Å)b (cm3mol–1)Vw3)rw (Å)
Hydrogen 0.7461126.6144.192.02
Nitrogen 1.097539.1364.982.25
Oxygen 1.20831.8352.862.06
Chlorine 1.98856.2293.362.39
van der Waals radii rw in Å (or in 100 picometers) 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, [5] b = 23.7 cm3/mol. Helium is a monatomic gas, and each mole of helium contains 6.022×1023 atoms (the Avogadro constant, NA):

Therefore, the van der Waals volume of a single atom Vw = 39.36 Å3, which corresponds to rw = 2.11 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 2rw and the internuclear distance is d. The algebra is more complicated, but the relation

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. [6] Arnold Bondi also conducted a study of this type, published in 1964, [2] 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 Å for the van der Waals radius of hydrogen as opposed to Bondi's 1.20 Å. [1] 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. [7]

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 or metallic bonds) 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 K and 66  atm is 0.214(6) g/cm3, [8] corresponding to a molar volume Vm = 18.7×10−6 m3/mol. The van der Waals volume is given by

where the factor of π/√18 arises from the packing of spheres: Vw = 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:

The refractive index of helium n = 1.0000350 at 0 °C and 101.325 kPa, [9] which corresponds to a molar refractivity A = 5.23×10−7 m3/mol. Dividing by the Avogadro constant gives Vw = 8.685×10−31 m3 = 0.8685 Å3, corresponding to rw = 0.59 Å.

Polarizability

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

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 = 7×10−5 at 0 °C and 101.325 kPa, [10] which corresponds to a polarizability α = 2.307×10−41 C⋅m2/V. The polarizability is related the van der Waals volume by the relation

so the van der Waals volume of helium Vw = 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.

See also

Related Research Articles

An intermolecular force (IMF) is the force that mediates interaction between molecules, including the electromagnetic forces of attraction or repulsion which act between atoms and other types of neighbouring particles, e.g. atoms or ions. Intermolecular forces are weak relative to intramolecular forces – the forces which hold a molecule together. For example, the covalent bond, involving sharing electron pairs between atoms, is much stronger than the forces present between neighboring molecules. Both sets of forces are essential parts of force fields frequently used in molecular mechanics.

The dalton or unified atomic mass unit is a non-SI unit of mass defined as 1/12 of the mass of an unbound neutral atom of carbon-12 in its nuclear and electronic ground state and at rest. The atomic mass constant, denoted mu, is defined identically, giving mu = 1/12m(12C) = 1 Da.

<span class="mw-page-title-main">Atomic radius</span> Measure of the size of an atom

The atomic radius of a chemical element is a measure of the size of its atom, usually the mean or typical distance from the center of the nucleus to the outermost isolated electron. Since the boundary is not a well-defined physical entity, there are various non-equivalent definitions of atomic radius. Four widely used definitions of atomic radius are: Van der Waals radius, ionic radius, metallic radius and covalent radius. Typically, because of the difficulty to isolate atoms in order to measure their radii separately, atomic radius is measured in a chemically bonded state; however theoretical calculations are simpler when considering atoms in isolation. The dependencies on environment, probe, and state lead to a multiplicity of definitions.

<span class="mw-page-title-main">Ideal gas law</span> Equation of the state of a hypothetical ideal gas

The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stated by Benoît Paul Émile Clapeyron in 1834 as a combination of the empirical Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. The ideal gas law is often written in an empirical form:

Physisorption, also called physical adsorption, is a process in which the electronic structure of the atom or molecule is barely perturbed upon adsorption.

van der Waals force Interactions between groups of atoms that do not arise from chemical bonds

In molecular physics and chemistry, the van der Waals force is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical electronic bond; they are comparatively weak and therefore more susceptible to disturbance. The van der Waals force quickly vanishes at longer distances between interacting molecules.

In chemistry, the molar mass of a chemical compound is defined as the ratio between the mass and the amount of substance of any sample of said compound. The molar mass is a bulk, not molecular, property of a substance. The molar mass is an average of many instances of the compound, which often vary in mass due to the presence of isotopes. Most commonly, the molar mass is computed from the standard atomic weights and is thus a terrestrial average and a function of the relative abundance of the isotopes of the constituent atoms on Earth. The molar mass is appropriate for converting between the mass of a substance and the amount of a substance for bulk quantities.

The atomic units are a system of natural units of measurement that is especially convenient for calculations in atomic physics and related scientific fields, such as computational chemistry and atomic spectroscopy. They were originally suggested and named by the physicist Douglas Hartree. Atomic units are often abbreviated "a.u." or "au", not to be confused with similar abbreviations used for astronomical units, arbitrary units, and absorbance units in other contexts.

<span class="mw-page-title-main">Van der Waals equation</span> Gas equation of state which accounts for non-ideal gas behavior

The van der Waals equation, named for its originator, the Dutch physicist Johannes Diderik van der Waals, is an equation of state that extends the ideal gas law to include the non-zero size of gas molecules and the interactions between them. As a result the equation is able to model the phase change from liquid to gas, and vice versa. It also produces simple analytic expressions for the properties of real substances that shed light on their behavior. One common way to write this dimensional equation is:

<span class="mw-page-title-main">London dispersion force</span> Cohesive force between species

London dispersion forces are a type of intermolecular force acting between atoms and molecules that are normally electrically symmetric; that is, the electrons are symmetrically distributed with respect to the nucleus. They are part of the van der Waals forces. The LDF is named after the German physicist Fritz London. They are the weakest intermolecular force.

In electromagnetism, the Clausius–Mossotti relation, named for O. F. Mossotti and Rudolf Clausius, expresses the dielectric constant of a material in terms of the atomic polarizability, α, of the material's constituent atoms and/or molecules, or a homogeneous mixture thereof. It is equivalent to the Lorentz–Lorenz equation, which relates the refractive index of a substance to its polarizability. It may be expressed as:

Polarizability usually refers to the tendency of matter, when subjected to an electric field, to acquire an electric dipole moment in proportion to that applied field. It is a property of particles with an electric charge. When subject to an electric field, the negatively charged electrons and positively charged atomic nuclei are subject to opposite forces and undergo charge separation. Polarizability is responsible for a material's dielectric constant and, at high (optical) frequencies, its refractive index.

<span class="mw-page-title-main">Collision theory</span> Chemistry principle

Collision theory is a principle of chemistry used to predict the rates of chemical reactions. It states that when suitable particles of the reactant hit each other with the correct orientation, only a certain amount of collisions result in a perceptible or notable change; these successful changes are called successful collisions. The successful collisions must have enough energy, also known as activation energy, at the moment of impact to break the pre-existing bonds and form all new bonds. This results in the products of the reaction. The activation energy is often predicted using the Transition state theory. Increasing the concentration of the reactant brings about more collisions and hence more successful collisions. Increasing the temperature increases the average kinetic energy of the molecules in a solution, increasing the number of collisions that have enough energy. Collision theory was proposed independently by Max Trautz in 1916 and William Lewis in 1918.

The atomic radius of a chemical element is the distance from the center of the nucleus to the outermost shell of an electron. Since the boundary is not a well-defined physical entity, there are various non-equivalent definitions of atomic radius. Depending on the definition, the term may apply only to isolated atoms, or also to atoms in condensed matter, covalently bound in molecules, or in ionized and excited states; and its value may be obtained through experimental measurements, or computed from theoretical models. Under some definitions, the value of the radius may depend on the atom's state and context.

Ionic radius, rion, is the radius of a monatomic ion in an ionic crystal structure. Although neither atoms nor ions have sharp boundaries, they are treated as if they were hard spheres with radii such that the sum of ionic radii of the cation and anion gives the distance between the ions in a crystal lattice. Ionic radii are typically given in units of either picometers (pm) or angstroms (Å), with 1 Å = 100 pm. Typical values range from 31 pm (0.3 Å) to over 200 pm (2 Å).

The Kelvin equation describes the change in vapour pressure due to a curved liquid–vapor interface, such as the surface of a droplet. The vapor pressure at a convex curved surface is higher than that at a flat surface. The Kelvin equation is dependent upon thermodynamic principles and does not allude to special properties of materials. It is also used for determination of pore size distribution of a porous medium using adsorption porosimetry. The equation is named in honor of William Thomson, also known as Lord Kelvin.

The van der Waals surface of a molecule is an abstract representation or model of that molecule, illustrating where, in very rough terms, a surface might reside for the molecule based on the hard cutoffs of van der Waals radii for individual atoms, and it represents a surface through which the molecule might be conceived as interacting with other molecules. Also referred to as a van der Waals envelope, the van der Waals surface is named for Johannes Diderik van der Waals, a Dutch theoretical physicist and thermodynamicist who developed theory to provide a liquid-gas equation of state that accounted for the non-zero volume of atoms and molecules, and on their exhibiting an attractive force when they interacted. van der Waals surfaces are therefore a tool used in the abstract representations of molecules, whether accessed, as they were originally, via hand calculation, or via physical wood/plastic models, or now digitally, via computational chemistry software. Practically speaking, CPK models, developed by and named for Robert Corey, Linus Pauling, and Walter Koltun, were the first widely used physical molecular models based on van der Waals radii, and allowed broad pedagogical and research use of a model showing the van der Waals surfaces of molecules.

<span class="mw-page-title-main">Gas</span> State of matter

Gas is one of the four fundamental states of matter. The others are solid, liquid, and plasma.

The Thomas–Fermi (TF) model, named after Llewellyn Thomas and Enrico Fermi, is a quantum mechanical theory for the electronic structure of many-body systems developed semiclassically shortly after the introduction of the Schrödinger equation. It stands separate from wave function theory as being formulated in terms of the electronic density alone and as such is viewed as a precursor to modern density functional theory. The Thomas–Fermi model is correct only in the limit of an infinite nuclear charge. Using the approximation for realistic systems yields poor quantitative predictions, even failing to reproduce some general features of the density such as shell structure in atoms and Friedel oscillations in solids. It has, however, found modern applications in many fields through the ability to extract qualitative trends analytically and with the ease at which the model can be solved. The kinetic energy expression of Thomas–Fermi theory is also used as a component in more sophisticated density approximation to the kinetic energy within modern orbital-free density functional theory.

In condensed matter physics and physical chemistry, the Lifshitz theory of van der Waals forces, sometimes called the macroscopic theory of van der Waals forces, is a method proposed by Evgeny Mikhailovich Lifshitz in 1954 for treating van der Waals forces between bodies which does not assume pairwise additivity of the individual intermolecular forces; that is to say, the theory takes into account the influence of neighboring molecules on the interaction between every pair of molecules located in the two bodies, rather than treating each pair independently.

References

  1. 1 2 3 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): 7384–7391. doi:10.1021/jp953141+.
  2. 1 2 3 Bondi, A. (1964). "van der Waals Volumes and Radii". J. Phys. Chem. 68 (3): 441–451. doi:10.1021/j100785a001.
  3. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Mantina, Manjeera; Chamberlin, Adam C.; Valero, Rosendo; Cramer, Christopher J.; Truhlar, Donald G. (2009). "Consistent van der Waals Radii for the Whole Main Group". The Journal of Physical Chemistry A . 113 (19): 5806–5812. Bibcode:2009JPCA..113.5806M. doi: 10.1021/jp8111556 . PMC   3658832 . PMID   19382751.
  4. "van der Waals Radius of the elements". Wolfram.
  5. Weast, Robert C., ed. (1981). CRC Handbook of Chemistry and Physics (62nd ed.). Boca Raton, FL: CRC Press. ISBN   0-8493-0462-8., p. D-166.
  6. Pauling, Linus (1945). The Nature of the Chemical Bond. Ithaca, NY: Cornell University Press. ISBN   978-0-8014-0333-0.
  7. Alvareza, Santiago (2013). "A cartography of the van der Waals territories". Dalton Trans. 42 (24): 8617–36. doi: 10.1039/C3DT50599E . hdl: 2445/48823 . PMID   23632803.
  8. Henshaw, D.G. (1958). "Structure of Solid Helium by Neutron Diffraction". Physical Review . 109 (2): 328–330. Bibcode:1958PhRv..109..328H. doi:10.1103/PhysRev.109.328.
  9. Kaye & Laby Tables, Refractive index of gases.
  10. Kaye & Laby Tables, Dielectric Properties of Materials.

Further reading