• 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

Packing

• Spheres and how they are put together

• HCP and CCP models are geometrical

constructs of how tightly we can assemble

spheres in a space

• Insertion of smaller cations into closest packed arrays yield different C.N.’s based on how big a void is created depending on arrangement

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…

Packing, Coordination, and C.N.

• Principle difference between hexagonal and cubic closest packing is repeat sequence:

– ABABAB for hexagonal

– ABCABCABC for cubic

• To classify: there are different types of hexagonal and cubic packed possibilities

• A close packed plane can yield either 3D structure depending on how it is layered, and a single type of structure does not yield a single type of site (more than one site with different C.N. is possible!)

Which is this?

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 structures

2. 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 another

Si 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

– SiO44-

Sorosilicates

– Si2O76-

Cyclosilicates

– Si6O1812-

Inosilicates

(single)

– Si2O64-

Inosilicates

(double)

– Si4O116-

Phyllosilicates

– Si2O52-

Tectosilicates

– SiO20

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.