How To Calculate A Coordination Number

Certain kinds of atoms form regular three-dimensional repeating structures when they bond with other elements. These repeating patterns are called crystal lattices, and they are characteristic of ionic solids, or compounds that contain ionic bonds, such as table salt (much more below).

These crystals have tiny repeating sections that boast cations, or positively charged atoms, at their center. This central atom is geometrically associated with a certain number of anions via one of a number of familiar patterns. Each anion in turn can be envisioned as sitting at the center of its own repeating unit and associating with a certain number of cations, which may be the same number or a different number as in the cation-at-center example.

This number, called the coordination number or ligancy, applies to ions rather than "native" atoms and determines the greater three-dimensional shape of the solid in predictable ways that relate to fundamental atomic architecture. It also determines the color owing to specific and unique distances between electrons and other components of the crystal lattice.

If you happen to have access to three-dimensional models of common crystal lattice patters, you can visually inspect one "unit" from the perspective of both the anion and the cation and see how many "arms" reach out to the ion of opposite charge. In most cases, however, you will have to rely on a combination of online research and using molecular formulas.

Example: The formula for the ionic compound sodium chloride, or table salt, is NaCl. This means that every cation should have exactly one anion associated with it; in the language of ligancy, this means that the cation Na⁺ and the anion Cl⁻ have the same coordination number.

Upon inspection, the structure of NaCl shows each Na⁺ ion having a Cl⁻ neighbor above and below, to the left and to the right, and ahead and behind. The same is true from the Cl⁻ perspective. The coordination number for both ions is 6.

Cations and anions present a 1:1 molecular ratio in a crystal, which means that they have the same coordination number, but this does not mean that the number is fixed at 6. The number 6 is a convenient number in three-dimensional space because of the up-down-right-left-forward-backward symmetry. But what if these "connections" were oriented diagonally, as if pointing away from the center of a cube toward all its corners?

In fact, this is how the lattice of cesium chloride, or CsCl, is arranged. Cesium and sodium have the same number of valence electrons, so in theory NaCl and CsCl might exhibit similar crystals. However, a cesium ion is far more massive than a sodium ion, and because it takes up more space, it is better accommodated with a coordination number of 8. Now, neighboring ions are found purely along diagonals; they are more distant than in NaCl, but also more numerous.

Because cesium and chlorine exist in a 1:1 ratio in this compound, the coordination number for the chloride ion in this instance is 8.

Titanium oxide (TiO₂) is an example of a crystal structure containing anions and cations in a 2:1 ratio. So, the fundamental unit of the lattice is tetrahedral: Each Ti⁴⁺ cation is amid six O^2-ions, while every O^2-ion has three immediate Ti⁴⁺ neighbors.

The coordination number for Ti⁴⁺ is 6, while that of the O^2-ion is 3. This makes chemical sense since the formula TiO₂ implies that twice as many oxygen ions exist in this compound as do titanium ions.