• 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
<0.15
0.15
0.15-0.22
0.22
0.22-0.41
0.41
0.41-0.73
0.73
0.73-1.0
1.0
>1.0
Expected coordination
2-fold coordination
Ideal triangular
Triangular
Ideal tetrahedral
Tetrahedral
Ideal octahedral
Octahedral
Ideal cubic
Cubic
Ideal dodecahedral
dodecahedral
C.N.
2
3
3
4
4
6
6
8
8
12
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 (SO42-, NO3-, CO32-)
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
Bond Strength of Si = ½ the charge of O2-
Si4+
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
Bond Strength
= 4 (charge)/4(C.N.) = 1
O2-
Si4+
O2-
Si4+
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
– SiO44Sorosilicates
– 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.