Transcript Crystals

Crystals
Crystal Structures
Atoms (and later ions) will be viewed as hard
spheres. In the case of pure metals, the packing
pattern often provides the greatest spatial
efficiency (closest packing).
Ionic crystals can often be viewed as a closepacked arrangement of the larger ion, with the
smaller ion placed in the “holes” of the
structure.
Unit Cells
Crystals consist of repeating asymmetric
units which may be atoms, ions or molecules.
The space lattice is the pattern formed by the
points that represent these repeating structural
units.
Unit Cells
A unit cell of the crystal is an imaginary
parallel-sided region from which the entire
crystal can be built up.
Usually the smallest unit cell which exhibits
the greatest symmetry is chosen. If repeated
(translated) in 3 dimensions, the entire crystal is
recreated.
Close Packing
Since metal atoms and ions lack directional
bonding, they will often pack with greatest
efficiency. In close or closest packing, each metal
atom has 12 nearest neighbors.
The number of nearest neighbors is called
the coordination number. Six atoms surround an
atom in the same plane, and the central atom is
then “capped” by 3 atoms on top, and 3 atoms
below it.
Close Packing
If the bottom “cap” and the top “cap” are
directly above each other, in an ABA pattern,
the arrangement has a hexagonal unit cell, or is
said to be hexagonal close packed.
If the bottom and top “caps” are staggered,
the unit cell that results is a face-centered cube.
This arrangement is called cubic close packing.
Close Packing
Close Packing
Either arrangement utilizes 74% of the
available space, producing a dense arrangement
of atoms. Small holes make up the other 26%
of the unit cell.
Holes in Close Packed Crystals
There are two
types of holes created
by a close-packed
arrangement.
Octahedral holes lie
within two staggered
triangular planes of
atoms.
Holes in Close Packed Crystals
The coordination
number of an atom
occupying an
octahedral hole is 6.
For n atoms in a
close-packed
structure, there are n
octahedral holes.
Octahedral Holes
The green atoms are in
a cubic close-packed
arrangement. The small
orange spheres show the
position of octahedral
holes in the unit cell.
Each hole has a
coordination number of 6.
Octahedral Holes
The size of the octahedral hole = .414 r
where r is the radius of the cubic close-packed
atom or ion.
Holes in Close Packed Crystals
Tetrahedral holes are formed by a planar
triangle of atoms, with a 4th atom covering the
indentation in the center. The resulting hole has
a coordination number of 4.
Tetrahedral Holes
The orange spheres
show atoms in a cubic
close-packed
arrangement. The small
white spheres behind
each corner indicate the
location of the
tetrahedral holes.
Tetrahedral Holes
For a close-packed
crystal of n atoms, there
are 2n tetrahedral holes.
The size of the
tetrahedral holes = .225 r
where r is the radius of
the close-packed atom or
ion.
# of Atoms/Unit Cell

For atoms in a
cubic unit cell:
Atoms in corners are
⅛ within the cell
# of Atoms/Unit Cell
For atoms in a
cubic unit cell:

Atoms on faces are ½
within the cell
# of Atoms/Unit Cell
A face-centered
cubic unit cell
contains a total of 4
atoms: 1 from the
corners, and 3 from
the faces.
# of Atoms/Unit Cell
For atoms in a cubic unit cell:



Atoms in corners are ⅛ within the cell
Atoms on faces are ½ within the cell
Atoms on edges are ¼ within the cell
Other Metallic Crystal Structures
Body-centered cubic unit cells have an atom in
the center of the cube as well as one in each
corner. The packing efficiency is 68%, and the
coordination number = 8.
Other Metallic Crystal Structures
Simple cubic (or primitive cubic) unit cells are
relatively rare. The atoms occupy the corners of
a cube. The coordination number is 6, and the
packing efficiency is only 52.4%.
Polymorphism
Many metals exhibit different crystal
structures with changes in pressure and
temperature. Typically, denser forms occur at
higher pressures.
Higher temperatures often cause closepacked structures to become body-center cubic
structures due to atomic vibrations.
Atomic Radii of Metals
Metallic radii are defined as half the
internuclear distance as determined by X-ray
crystallography. However, this distance varies
with coordination number of the atom;
increasing with increasing coordination number.
Atomic Radii of Metals
Goldschmidt radii correct all metallic radii for a
coordination number of 12.
Coord #
12
8
6
4
Relative radius
1.000
0.97
0.96
0.88
Alloys
Alloys are solid solutions of metals. They are
usually prepared by mixing molten components.
They may be homogeneous, with a uniform
distribution, or occur in a fixed ratio, as in a
compound with a specific internal structure.
Substitutional Alloys
Substitutional alloys have a structure in which
sites of the solvent metal are occupied by solute
metal atoms. An example is brass, an alloy of
zinc and copper.
Substitutional Alloys
These alloys may form if:
1. The atomic radii of the two metals are within
15% if each other.
2. The unit cells of the pure metals are the same.
3. The electropositive nature of the metals is
similar (to prevent a redox reaction).
Interstitial Alloys
Interstitial alloys are solid solutions in which
the solute atoms occupy holes (interstices)
within the solvent metal structure. An
example is steel, an alloy of iron and carbon.
Interstitial Alloys
These alloys often have a non-metallic solute
that will fit in the small holes of the metal
lattice. Carbon and boron are often used as
solutes. They can be dissolved in a simple
whole number ratio (Fe3C) to form a true
compound, or randomly distributed to form
solid solutions.
Intermetallic Compounds
Some mixtures of metals form alloys with
definite structures that may be unrelated to the
structures of each of the individual metals. The
metals have similar electronegativities, and
molten mixtures are cooled to form compounds
such as brass (CuZn), MgZn2, Cu3Au, and
Na5Zn2.
Ionic Compounds
Since anions are often larger than cations,
ionic structures are often viewed as a closepacked array of anions with cations added, and
sometimes distorting the close-packed
arrangement.
Common Crystal Types
1. The Rock Salt (NaCl)
structureCan be viewed as a
face-centered cubic array
of the anions, with the
cations in all of the
octahedral holes, or
Common Crystal Types
1. The Rock Salt (NaCl)
structureA face-centered cubic
array of the cations with
anions in all of the
octahedral holes.
Common Crystal Types
1. The Rock Salt (NaCl)
structureThe coordination
number is 6 for both ions.
Common Crystal Types
2. The CsCl structureChloride ions occupy the
corners of a cube, with a
cesium ion in the center
(called a cubic hole) or vice
versa. Both ions have a
coordination number of 8,
with the two ions fairly
similar in size.
Common Crystal Types
3. The Zinc-blende or Sphalerite structureAnions (S2-) ions are in a face-centered cubic
arrangement, with cations (Zn2+) in half of the
tetrahedral holes.
Common Crystal Types
4. The Fluorite (CaF2) and Antifluorite structures
A face-centered cubic arrangement of Ca2+
ions with F- ions in all of the tetrahedral holes.
Common Crystal Types
4. The Fluorite (CaF2) and Antifluorite structures
The antifluorite structure reverses the
positions of the cations and anions. An example
is K2O.
Ionic Radii
Ionic radii are difficult to determine, as x-ray
data only shows the position of the nuclei, and
not the electrons.
Most systems assign a radius to the oxide ion
(often 1.26Å), and the radius of the cation is
determined relative to this assigned value.
Ionic Radii
Like metallic radii, ionic radii seem to vary
with coordination number. As the coordination
number increases, the apparent ionic radius
increases.
Ionic Radii
1. Ionic radii increase as you go down a group.
2. Radii of ions of similar charge decrease across a
period.
3. If an ion can be found in many environments,
its radius increases with higher coordination
number.
4. For cations, the greater the charge, the smaller
the ion (assuming the same coordination #).
5. For atoms near each other on the periodic
table, cations are generally smaller than anions.
Predicting Crystal Structures
General “rules” have been developed, based
on unit cell geometry, to predict crystal
structures using ionic radii.
Radius ratios, usually expressed as the (radius
of the cation)/(radius of the anion) are used.
Predicting Crystal Structures
General “rules” have been developed, based
on unit cell geometry, to predict crystal
structures using ionic radii.
Radius ratios, usually expressed as the (radius
of the cation)/(radius of the anion) are used.
This assumes that the cation is smaller than the
anion.
Predicting Crystal Structures
CN
8
r+/r≥0.70
accuracy
quite reliable
6
0.4 -0.7
moderately reliable
4
0.2 –0.4
unreliable
3
0.10 -0.20 unreliable
Energetics of Ionic Bonds
The lattice energy is a measure of the strength
of ionic bonds within a specific crystal structure.
It is usually defined as the energy change when a
mole of a crystalline solid is formed from its
gaseous ions.
M+(g) + X-(g)  MX(s)
Lattice Energy
M+(g) + X-(g)  MX(s)
∆E = Lattice Energy
Lattice energies cannot be measured directly,
so they are obtained using Hess’ Law. They will
vary greatly with ionic charge, and, to a lesser
degree, with ionic size.
1/2 bond
energy of Cl2
Electron
Affinity of Cl
Ionization
energy of K
∆Hsub of K}
∆Hf of
KCl
Lattice Energy
of KCl
Ionic charge has a
huge effect on
lattice energy.
Lattice Energy
Attempts to predict lattice energies are
generally based on coulomb’s law:
VAB = (Zae)(Zbe)
4πεorAB
Za and Zb = charge on cation and anion
e= charge of an electron (1.602 x 10-19C)
4πεo=permittivity of vacuum (1.1127 x 10-10J-1C2m-1)
rAB = distance between nuclei
Lattice Energy
Since ionic crystals involve more than 2 ions,
the attractive and repulsive forces between
neighboring ions, next nearest neighbors, etc.,
must be considered.
The Madelung Constant
The Madelung constant is derived for each
type of ionic crystal structure. It is the sum of a
series of numbers representing the number of
nearest neighbors and their relative distance
from a given ion.
The constant is specific to the crystal type
(unit cell), but independent of interionic
distances or ionic charges.
Madelung Constants
Crystal Structure
Cesium chloride
Fluorite
Rock salt (NaCl)
Sphalerite
Wurtzite
Madelung Constant
1.763
2.519
1.748
1.638
1.641
Estimating Lattice Energy
Ec = NM(Z+)( Z-) e2
4πεor
where N is Avogadro’s number, and
M is the Madelung constant (sometimes
represented by A)
This estimate is based on coulombic forces,
and assumes 100% ionic bonding.
Estimating Lattice Energy
A further modification, the Born-Mayer
equation corrects for complex repulsion within
the crystal.
Ec = NM(Z+)( Z-) e2 (1-ρ/r)
4πεor
for simple compounds, ρ=30pm
Solubility of Ionic Crystals
The dissolving of ionic compounds in water
may be viewed in terms of lattice energy and the
solvation of the gaseous ions.
MX(s)  M+(g) + X-(g)
Lattice energy
M+(g) + H2O(l)  M+(aq) Solvation
X-(g) + H2O(l)  X-(aq)
Solvation
MX(s) ) + H2O(l)  M+(aq) + X-(aq) ΔHsoln
Solubility of Ionic Crystals
Factors such as ionic size and charge,
hardness or softness of the ions, crystal structure
and electron configuration of the ions all play a
role in the solubility of ionic solids. The entropy
of solvation will also play a role in solubility.
Ionic Size
Smaller ions have a stronger coulombic
attraction for each other and also for water.
They also have less room to accommodate the
waters of hydration.
Larger ions have weaker electrostatic
attraction for each other and also for water.
They also have accommodate more waters of
hydration.
Ionic Size
The overall result of these factors result in
low solubility of salts containing two large ions
(soft-soft) or two small ions (hard-hard).
For salts containing two small ions,
especially with the same magnitude of charge,
the greater lattice energy dominates, and cannot
be easily overcome by the hydration energy of
the ions.
Ionic Size
Ionic Size
For two large ions, the hydration energies are
considerably lower, so the lattice energy
dominates the process and results in a positive
value for the enthalpy of hydration.
Ionic Size
Effect of Entropy
All ionic crystals will have an increase in
entropy upon dissolution. This increase in
entropy will increase the solubility of salts that
have an endothermic enthalpy of solution.