Pauling's rules
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Pauling's rules are five rules published by Linus Pauling in 1929[1] for determining the molecular structures of complex crystals.
1. A coordinated polyhedron of anions is formed about each cation, the cation-anion distance determined by the sum of ionic radii and the coordination number (C.N.) by the radius ratio.
| Radius ratio | C.N. | polyhedron |
|---|---|---|
| .225 | 4 | tetrahedron |
| .414 | 6 | octahedron |
| .592 | 7 | capped octahedron |
| .645 | 8 | square antiprism (anti cube) |
| .732 | 8 | cube |
| .732 | 9 | tri-capped trigonal prism |
| 1.00 | 12 | cube octahedron |
2. An ionic structure will be stable to the extent that the sum of the strengths of the electrostatic bonds that reach an anion equal the charge on that anion. ("The electrostatic valence rule".)
- i.e. A stable ionic structure must be arranged to preserve Local Electroneutrality.
This is expressed mathematically as:
| ξ = | ∑ | si |
| i |
- where ξ is the charge of the anion and the summation is over the adjacent cations.
Some examples are:
| Cation | Radius ratio | CN | Electrostatic bond strength |
|---|---|---|---|
| Li+ | 0.34 | 4 | 0.25 |
| Mg2+ | 0.47 | 6 | 0.33 |
| Sc3+ | 0.60 | 6 | 0.5 |
3. The sharing of edges and particularly faces by two anion polyhedra decreases the stability of an ionic structure.
- Effect is largest for cations with high charge and low C.N (especially when r+/r- approaches the lower limit of the polyhedral stability).
4. In a crystal containing different cations, those of high valency and small coordination number tend not to share polyhedron elements with one another.
5. The number of essentially different kinds of constituents in a crystal tends to be small.("The rule of parsimony".)
- Similar environments for chemically similar atoms.
