• Nuclear reactions determine element abundance…

• Is the earth homogeneous though?

• Is the solar system??

• Is the universe???

Earth = anion balls with cations in the spaces…

• View of the earth as a system of anions packed together By size and abundance, Si and O are the most important

• If we consider anions as balls, then their arrangement is one of efficient packing, with smaller cations in the interstices

• Closest packed structures are ones in which this idea describes atomic arrangement – OK for metals, sulfides, halides, some oxides

Closest Packing

• Coordination number (C.N) - # of anions bonded to a cation larger cation, higher C.N.

• Anions are much larger than most cations anion arrangements in 3 dimensions = packing

• Hexagonal Closest Packed (HCP) - spheres lie atop each other– vertical sequence ABABAB

• Cubic closest packed (CCP) – spheres fill in gaps of layer below – vertical sequence ABCABC

• Exceptions to closest packing – Body centered cubic (BCC), polyhedra, and others…

Pauling’s Rules for ionic structures

1. Radius Ratio Principle – • cation-anion distance can be calculated from

their effective ionic radii• cation coordination depends on relative radii

between cations and surrounding anions• Geometrical calculations reveal ideal Rc/Ra ratios

for selected coordination numbers• Larger cation/anion ratio yields higher C.N. as

C.N. increases, space between anions increases and larger cations can fit

• Stretching a polyhedra to fit a larger cation is possible

C.N. calculations

• Application of pythagorean theorem: c2=a2+b2

• End up with ranges of Rc/Ra values corresponding to different C.N.

Rc/Ra Expected coordination C.N.

<0.15 2-fold coordination 2

0.15 Ideal triangular 3

0.15-0.22 Triangular 3

0.22 Ideal tetrahedral 4

0.22-0.41 Tetrahedral 4

0.41 Ideal octahedral 6

0.41-0.73 Octahedral 6

0.73 Ideal cubic 8

0.73-1.0 Cubic 8

1.0 Ideal dodecahedral 12

>1.0 dodecahedral 12

Pauling’s Rules for ionic structures2. Electrostatic Valency Principle

– Bond strength = cation valence / C.N.– Sum of bonds to a ion = charge on that ion– Relative bond strengths in a mineral containing

>2 different ions:• Isodesmic – all bonds have same relative strength• Anisodesmic – strength of one bond much stronger

than others – simplify much stronger part to be an anionic entity (SO4

2-, NO3-, CO3

2-)• Mesodesmic – cation-anion bond strength =

½ charge, meaning identical bond strength available for further bonding to cation or other anion

Bond strength – Pauling’s 2nd Rule

Si4+

Bond Strength = 4 (charge)/4(C.N.) = 1

Bond Strength of Si = ½ the charge of O2-

O2- has strength (charge) to attract either anotherSi or a different cation – if it attaches to another Si, the bonds between either Si will be identical

O2-Si4+ Si4+O2-

Mesodesmic subunit – SiO44-

• Each Si-O bond has strength of 1

• This is ½ the charge of O2-

• O2- then can make an equivalent bond to cations or to another Si4+ (two Si4+ then share an O)

• Reason silicate can easily polymerize to form a number of different structural configurations (and why silicates are hard)

Nesosilicates – SiO4

4-

Sorosilicates– Si2O7

6-

Cyclosilicates – Si6O18

12-

Inosilicates (single) – Si2O6

4-

Inosilicates (double) – Si4O11

6-

Phyllosilicates – Si2O5

2-

Tectosilicates – SiO2

0

Pauling’s Rules for ionic structures

3. Sharing of edges or faces by coordinating polyhedra is inherently unstable– This puts cations closer together and they will

repel each other

Pauling’s Rules for ionic structures

4. Cations of high charge do not share anions easily with other cations due to high degree of repulsion

5. Principle of Parsimony – Atomic structures tend to be composed of only a few distinct components – they are simple, with only a few types of ions and bonds.