Van der Waals radius

The Van der Waals radius is the radius of an imaginary solid sphere used to model the atom and represents half the closest possible distance of approximation between two atoms or molecules in an unrelated interaction. bonding, by means of the molecular force field in the field of predominance of the forces of repulsion between the atoms of the gas.

Real gases do not behave exactly as predicted by the ideal gas model and the deviation may be considerable in some cases. Thus, for example, ideal gases do not present liquid or solid phase transitions, regardless of the drop in temperature or increase in pressure to which they are subjected.

One of the modifications of the proposed ideal gas law is the Van der Waals equation of state, which introduces two parameters a and b obtained experimentally and which They depend on the nature of the gas. The correction factor b, called exclusion volume, refers both to the proper volume of the atoms, and to the surrounding volume in which there cannot be others because at that distance the repulsion forces between the atoms predominate. gas atoms (Van der Waals forces).

Once the value of the exclusion volume is known, obtained experimentally to adjust the Van der Waals equation to the real behavior of the gas, the radius r can be obtained from the equation:

b=43π π Nar3{displaystyle b={frac {4}{3}}pi N_{a}r^{3}}

where:

• N_a It's Avogadro's number, and
• r It's Van der Waals' radio.

Table of Van der Waals radii

The following table presents the Van der Waals radii for the elements. Unless otherwise indicated, the data is provided by the Mathematica' elemental function, which is Wolfram Research, Inc.. Values are expressed in picometers (pm or 1×10⁻¹² m). The cell color goes from red to yellow with increasing radius; gray indicates missing data.

Methods of determination

Van der Waals radii can be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of the atomic spacing between pairs of unbonded atoms in crystals, or from measurements of electrical or optical properties (polarizability and molar refractivity). These various methods give values for the van der Waals radius that are similar (1–2 Å 100–200 pm) but not identical. Tabulated values of van der Waals radii are obtained by taking a weighted average of several different experimental values, and for this reason different tables will often have different values for the van der Waals radius of the same atom. In fact, 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 of State

The Van der Waals equation of state is the simplest and best-known modification of the ideal gas law to explain the behavior of real gases:

(p+a(nV~ ~ )2)(V~ ~ − − nb)=nRT{displaystyle left(p+aleft({frac {n}{tilde {V}}}}}{2}right)({tilde {V}}-nb)=nRT},

where p is the pressure, n is the number of gas moles in question and a and b depend on the particular gas,V~ ~ {displaystyle {tilde {V}}} is the volume, R is the specific gas constant in molar units and T is the absolute temperature; a correction for intermolecular forces and b corrects for finite atomic or molecular sizes; the value of b is equal to the volume of Van der Waals per gas mill. Its values vary from gas to gas.

The Van der Waals equation also has a microscopic interpretation: molecules interact with each other. The interaction is strongly repulsive at very close range, becomes slightly attractive in the intermediate range, and fades at long range. The ideal gas law must be corrected when considering the attractive and repulsive forces. For example, the mutual repulsion between molecules has the effect of excluding neighbors from a certain amount of space around each molecule. Therefore, a fraction of the total space becomes unavailable to each molecule as it executes a random movement.

In the state equation, this exclusion volume (nb) must be subtracted from the volume of the container (V), like: (V-nb). The other term introduced in the Van der Waals equation,a(nV~ ~ )2{displaystyle aleft({frac {n}{tilde {V}}}}{right)^{2}}, describes a weak force of attraction between molecules (known as Van der Waals force), which increases when n increases or V decreases and the molecules are stacked more together.

The van der Waals constant volume b can be used to calculate the Van der Waals volume of an atom or molecule with experimental data derived from measurements in gases.

For helium, b = 23.7 cm³/mol Helium is a monatomic gas and each mole of helium contains 6.022 10²³ atoms (Avogadro's constant, N_A):

Vw=bNA{displaystyle V_{rm {w}}={b over {N_{rm {A}}}}}}}

Therefore, the Van der Waals volume of a single atom V_w = 39.36 ų, corresponding to r_w = 2.11 Å (≈ 200 picometers). This method can be extended to diatomic gases by approximating the molecule as a rod with rounded ends where the diameter is 2r_w and the internuclear distance is d. The algebra is more complicated, but the relation

Vw=43π π rw3+π π rw2d{displaystyle V_{rm {w}}={4 over 3}{pi r_{rm {w}}}{3} +pi r_{rm {w}}}{2}d}

can be solved with the normal methods for cubic functions.

Crystallographic measurements

The molecules of 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 get to each other is given by the sum of their Van der Waals radii. By examining a large number of molecular crystal structures, it is possible to find a minimum radius for each type of atom so that other non-bonded atoms do not come any closer. This approach was first used by Linus Pauling in his seminal work The Nature of Chemical Bonding. Arnold Bondi also carried out such a study, published in 1964, although he also considered other methods to determine the chemical bond. van der Waals radius when arriving at their final estimates. Some of Bondi's figures are given in the table at the beginning of this article, and remain the "consensus" most commonly used for the Van der Waals radii of elements. Scott Rowland and Robin Taylor re-examined these 1964 figures in the light of more recent crystallographic data: agreement was generally very good, although they recommend a value of 1.09 Å for the Van der Waals hydrogen radius compared to Bondi.. 1.20 Å. A more recent analysis of the Cambridge Structural Database, by Santiago Álvarez, provided a new set of values for 93 natural elements.

A simple example of using 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, therefore, the distance between the nuclei can be considered equal to twice the van der Waals radius. The density of solid helium at 1.1 K and 66 atm is 0.214 g/ cm³, corresponding to a molar volume V _m =18.7 x 10⁻⁶ m³/mol. The van der Waals volume is given by

Vw=π π VmNA18{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: V_w =2.30 × 10⁻²⁹ m³ = 23.0 ų corresponding to a Van der Waals radius r_w = 1.76 Å.

Molar Refractivity

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

A=RT(n2− − 1)3p{displaystyle A={frac {RT(n^{2}-1}}{3p}}}}

The refractive index of helium n =1.0000350 at 0 °C and 101.325 kPa, which corresponds to a molar refractivity A =5.23 × 10⁻⁷ m³/mol. Dividing by Avogadro's constant gives V_w =8.685 × 10⁻³¹ m³ = 0.8685 ų corresponding to r_w = 0.59 Å.

Polarizability

The polarizability α of a gas is related to its electrical susceptibility χ_e by the relation

α α =ε ε 0kBTpχ χ e{displaystyle alpha ={epsilon _{0}k_{rm {B}T over p}chi _{rm {e}}}}}

and the electrical susceptibility can be calculated from tabulated values of the relative permittivity ε_r using the relationship χ_e = ε_r–1. The electrical susceptibility of heliumχ_e = 7 x 10^-5 at 0 °C and 101.325 kPa, which corresponds to a polarizability α = 2.307 x10^-41 cm²/V. The polarizability is related to the van der Waals volume by the relation

Vw=14π π ε ε 0α α ,{displaystyle V_{rm {w}}={1 over {4pi epsilon _{0}}}}{alpha}

so the Van der Waals volume of helium V_w = 2.073 x10^-31m³ = 0.2073 ų by this method, corresponding r_w= 0.37 Å.

When atomic polarizability is expressed in units of volume as ų as is usually the case, it is equal to the van der Waals volume. However, the term "atomic polarizability" is preferred; since polarization is a precisely defined (and measurable) physical quantity, while "Van der Waals volume" it can have any number of definitions depending on the measurement method.