Paulings Rules

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Crystal Chemistry Nesse, Chapter 3 Klein
Hurlbut, Chapter 4
EPSC210 Introductory Mineralogy
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2nd Mineral identification test this
Friday. Make sure you show up for the right
session. People who filled the 2nd session last
time have first pick if they wish to register for
the 1st session this time. No wooden model this
time... Only straight mineral identification
(mineral name and formula).
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Sulfur (Z16) 1s2 2s2 2p6 3s2 3p4 In minerals,
the common valence states of S are S6 1s2 2s2
2p6 3s 3pemptied as in neon S-2 1s2 2s2 2p6
3s2 3p6filled as in argon The difference in
electronegativity between S (2.5) and O (3.5)
gives the bond S-O a fairly covalent
character. In SO42- the 3s and 3p orbitals of
sulfur are hybridized into a tetrahedral
arrangement sp3. This enables electron sharing
among 4 identical bonds with a tetrahedron of
four oxygen ions. Less common sulfite SO3-2,
S4 1s2 2s2 2p6 3s2 hyposulfite (S2O4-2), S3
1s2 2s2 2p6 3s2 3p1.
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The bonding of ions with similar
electronegativity values have hardly any ionic
character. If their electronegativity values are
both high, the bonding will be dominantly
covalent. If their electronegativiy values are
both low, the bonding will be dominantly
metallic. Valence electrons will travel freely
throughout the crystal structure rather than
being held by a single atom or a pair of atoms.
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The degree of ionic character for a chemical bond
depends mostly on the difference in
electronegativity between chemical elements.
Paulings scale of ionic character 1 e0.25
(Xa-Xc)squared
lt- Si-O bond
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Structures determined by X-ray diffraction
suggest that hybridized orbitals (sp2, sp3,
sp3d2) occur in crystalline structures.
Tetrahedral and octahedral coordinations are
particularly common.
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sp3 hybridization -- and tetrahedral coordination
with angles of 109.5 degrees-- are common C-C
(diamond) Si-O (silicate minerals) S-O
(SO4 2- in sulfate minerals)
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C (Z6 1s 2s22p2can form four (4) identical bonds
to other carbon atoms if it promotes one electron
from its 2s orbital to a 2p orbital... and mixes
s and p orbitals into four hybridized sp3
orbitals.
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Recognize a fcc cell? The coordination (no of
nearest neighbors) is 4. Very hard (H10) but
cleavable along (111).
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The pair of electron not used in sp2 forms pi
bonding... Electrons can move among C atoms
parallel to the sheets... Some metallic character.
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The C sheets are electrically neutral... and held
together by the much weaker forces of Van der
Waals bonding.
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The 8 most abundant elements in the Earths
crust Element Weight Atom Volume Weight
in crust
total Earth 0 46.6 62.5 91.7
29.5 Si 27.7 21.2 0.2
15.2 Al 8.1 6.5 0.5
1.1 Fe 5.0 1.9 0.5
34.6 Ca 3.6 1.9 1.5
1.1 Na 2.8 2.6 2.2
0.6 K 2.6 1.4
3.1 0.1 Mg 2.1 1.8
0.4 12.7
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The bonding character of most mineral structures
can be described as nearly ionic. We can use the
valence state of anions and cations to calculate
how many neighbours each ion should have in a
structure, and compare bond strenths among the
different anions by considering their valence and
the distances between their nuclei.
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Given the predominance of oxygen and its high
electronegativity (3.5), it is the main anion in
the Earths crust. As a result, most minerals
consist of Si-O and Me-O bonds (Me metal, i.e.
element with electronegativity lt 2.5) , and are
dominantly ionic in character.
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• Properties of ionic solids
• ionic bonds are more soluble in water
• the ions tend to pack as hard spheres
• packing tends to be highly symmetrical
• brittle when deformed (sliding one part of the
  crystal over another would bring cations against
  cations and anions against anions)
• good cleavage

halite NaCl fluorite CaF2
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Both structures have a face-centered cell. The
ions are shrunk for visibility their nearly
spherical electronic cloud actually occupies all
the available space.
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halite, NaCl a 5.64, Z 4, H 2.5
cubic cleavage fluorite, CaF2 a 5.46, Z
4, H 4 octahedral cleavage
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How are the ionic radii (values in angstroms)
affected by increasing atomic number within a
single family of the periodic table?
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Relative sizes of 1s and 2s orbitals in lithium.
The n value of the last shell occupied is an
indicator of the relative size of the electronic
cloud.
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The ionic character decreases with increasing
size of either the cation or the anion. Large
ions are polarizable, i.e. their electronic cloud
is deformed. Solids with the same structure may
show a transition between ionic character to
slightly metallic character.
Replace Na by Ag... what do you get?
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halite NaCl
galena PbS
Same structure, cubic cleavage but metallic
luster, opaque streak, less brittle and nearly
insoluble
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Each chemical formula must be electrically
neutral. Some ions can substitute for each other
in a crystalline structure, but they must be a
good match for each other in term of size and
charge. Some minerals are always nearly pure
(diamond, quartz)... Others are members of
compositional series called solid solutions. In
others, substitution among ions can occur but is
quite limited (happens only in small amounts).
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Paulings Rule 1 Around every cation, a
coordination polyhedron of anions forms, in which
the cation-anion distance is determined by the
radius sums and the coordination number (C.N.)
is determined by the radius ratio.
Radius ratio Rx/Rz where Rx radius of cation
Rz radius of anion C.N. no of nearest
neighbours
C.N. 4
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How are ionic radii affected by increasing atomic
number along a row of the periodic table? NOTE
Most ions occur in more than one C.N.
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Are these ionic radii physically
realistic? Well, not necessarily. Look at the
values of ionic radii compiled for C4 and N5.
Do you notice anything odd? The radii are the
space left between oxygen anions, whose precise
position was determined by X-rays. The size of
oxygen anions is calculated from quantum
mechanics.
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When they form covalent bonds, oxygens are drawn
closer to the cation than predicted by ionic
radii because covalent bonding requires that
some orbitals from cation and anion overlap.
For C-O and N-O bonds, the overlap of oxygen ions
is counted as a negative space left for the
cation radius.
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2 ways of packing spheres of equal size I)
hexagonal close packing layer A has B and C
voids layer B covers B voids layer A aligned
with 1st layer layer B aligned with 2nd
layer II) cubic closest packing layer A has B
and C voids layer B covers B voids in layer
A layer C covers B voids in layer B
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Why is this first type of closest packing called
hexagonal close packing?
When seen from above, the spheres are in a
pattern with a 6-fold rotation axis perpendicular
to each layer. Each sphere has exactly 12
neighbours.
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Why is this second type of closest packing called
cubic close packing?
When seen from a different angle, the spheres are
packed within a cube. Each sphere has exactly 12
neighbours.
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Many metals crystallize with hcp or ccp
structures...
but very few of them are minerals. Why?
?
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Structures based on CCP are often isometric,
F-cell.
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• In some minerals, like fluorite and halite, the
  large anions are cubic close-packed, in a
  face-centered cell.
• In fluorite, the smaller Ca2 cations occupy
  the B voids left between these large anions.
• In hcp, the cationic sites could be
• - B voids surrounded by 4 anions (3 from layer A,
  3 from layer B)
• C voids surrounded by 6 anions (3 from layer A, 3
  from layer b)
• NOTE a void is a hole...

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One can prove geometrically that the smallest ion
that could sit in that hole without rattling
must have a radius of 0.732 or more, if the ions
around have a radius of 1.0 (units are
unspecified).
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A cation surrounded by six larger anions of
radius 1, must have a radius between 0.414 (to
avoid rattling) and 0.732 (or the anions will be
too far apart to shield it.)
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Rx usually the smaller cation
Rz usually the larger anion
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The radius ratio rule is a generalization, rather
than a firm prediction. Its usefulness lies in
the fact that exceptions, in specific crystal
structures, are a sign of structures stable at
unusually high temperature or pressure conditions.
In Al2SiO5 polymorphs, Al occurs as either AlO4,
(Rx/Rz0.27) AlO5 (Rx/Rz0.34) AlO6
(Rx/Rz0.39) polyhedra.
viAl2SiO5
ivAlviAlSiO5
vAlviAlSiO5
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The effect of pressure is to favour
higher-than-usual coordination around small
cations. It is as if pressure reduced the size
of anions (here the O2- ions) faster than the
size of the cations (already smaller, less
compressible electronic cloud). Therefore, more
anions than usual can be packed around a small
cation. Among the aluminosilicates, this is
kyanite, viAl2SiO5 , with Al in 6-fold
coordination.
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Rule 3 The sharing of edges, and particularly
the sharing of faces of two anion polyhedra in a
crystal structure decreases its stability.
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Sharing of only the corners of polyhedra
places the positively charged cations at the
greatest distance from each other. This minimizes
the repulsive forces among the cations.
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Example of Rule 4 In a crystal structure
containing several cations, those of high valency
and small coordination number tend not to share
polyhedral elements.
In kyanite, Al2SiO5, SiO4 polyhedra do not share
corners with similar polyhedra. AlO6 tetrahedra
share edges with each other, and corners with
SiO4.
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Could carbonate ions ever share oxygen anions and
form layered structures? e.v. strength of a
C4-O2- bond is calculated by taking charge on
cation / C.N. 4/3
An oxygen anion, shared by two C4 ions, would
receive twice 4/3 valence units 8/3 (or 2.67),
which far exceeds its need (2) This is never
observed.
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Higher temperatures favour crystal structures of
lower density. Chemical bonds stretch only
slightly at higher temperatures. Another way to
decrease density without stretching bonds, is to
modify the linkage among polyhedra. Polyhedra
with smaller C.N. tend to share less edges and
more corners. This leads to less efficient
packing. Among polymorphs, higher temperature
favours structures built of smaller-than-usual
coordination polyhedra.
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Andalusite, Alvi Alv SiO5 G 3.2, low pressure
Sillimanite, Alvi Aliv SiO5, G 3.23, opposite
effects of pressure and temperature.
Kyanite, Alvi Alvi SiO5 G 3.6, high pressure
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Rule 2, The Electrostatic Valency Principle An
ionic structure will be stable to the extent that
the sum of the strengths of the electrostatic
bonds that reach an ion equal the charge on that
ion.
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There are many polymorphs of SiO2, stable at
different ranges of temperature and pressure.
Most of them are built of SiO4 tetrahedra.
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We must first define electrostatic valency,
e.v. e.v Charge on the ion / C.N. C.N.
coordination number (number of nearest
neighbours)
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Si4 (valence 4), by bonding to four oxygen ions
(4 Si-O bonds), satisfies exactly half of the
valence need of O2- (valence2). This Si-O
bonding is said to be mesodesmic.
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Isodesmic structures all bonds of same e.v.
strength
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Rule 5, The Principle of Parsimony. The number
of different kinds of constituents in a crystal
tends to be small.
A typical hornblende may include 14 metallic
elements but they occupy no more than 5 types of
sites.
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Throughout the rest of this course we will see
how these principles apply to minerals,
particularly the silicate minerals.