Geos 306, Fall 2011, Lecture 6
Pauling's Rules



The "Radius Ratio" Concept

Radius Ratio Rules


rC/rA
Coord. #
Angle
Config.
Image
Example
0.000-0.155
2
180°
Linear

(HF2)-1
0.155-0.225
3
120°
Trigonal planar
(CO3)-2
0.225-0.414
4
109.47°
Tetrahedral
(SiO4)-4
0.414-0.732
4
90°
Square planar
(CuO4)-6
0.414-0.732
6
90°
Octahedral
(NaCl6)-5
0.732-1.000
8
70.53°
Square -bipyramid
(CsCl8)-7
1.000
12
60°
Closest-packed
(KO12)-23

Pauling's Rules

1. A coordination polyhedron of anions is formed about each cation, the cation-anion distance equaling the sum of their characteristic packing radii and their radius ratio determining both the nature of the coordination polyhedron and therefore the coordination number of the cation.

2. An ionic structure will be stable to the extent that the sum of the strengths, S, of the electrostatic bonds that reach an anion from adjacent cations equals the charge, Z, on that anion.

ZA = S Si

(The strength of an electrostatic bond is defined as the cation's valence divided by its coordination number, S = ZC/CN).

3. The sharing of edges and particularly of faces by two anion polyhedra decreases the stability of an ionic crystal structure.

4. In a crystal structure containing different cations, those of high valency and small coordination number tend not to share polyhedron elements with each other.

5. The number of essentially different kinds of constituents in a crystal tends to be small.

From Bloss, Crystallography and Crystal Chemistry

  1. Prewitt's addendum: Given that the chemical formula for a crystal is charge balanced, then the sum of the coordination numbers of the cations must equal the sum of the coordination numbers of the anions.


S=ZC/CN
If the charge on a cation is ZC and the coordination number is CN, then the number of electrons per bond is what Pauling considered to be the bond strength. It seems obvious that Si with 4 valence electrons can make a stronger bond than Mg with 2 valence electrons. Question: If you were to melt topaz, which bonds would break first?



Reading:
  • Wenk and Bulakh: finish Chapters 1 and 2.
  • Klein: p 64-80
  • Nesse: Chapter 4