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CHEM 511 Chapter 6 page 1 of 20
Chapter 6 Structures and energetics of metallic and ionic solids
Types of bonding
● Metallic
● Ionic (non-directional bonding)
● Covalent (directional bonding)
Significant sharing of electrons between atoms. Can form vast arrays (e.g. C—diamond, graphite;
SiO2—quartz, cristobalite) or molecular solids (e.g. CO2, SO2, H2O)
Two cases for covalent:
amorphous
crystalline
Packing of spheres
Lattice: three dimensional infinite array of points (atoms) where each atom is surrounded in an
identical way by neighboring points
Unit cell: the smallest repeating unit in a solid state lattice from which the entire crystal structure
can be built by purely translational displacements
There are seven basic crystal systems that are described by lengths (a,b,c) and angles (α,β,γ)—(this
information is not in your book)
CHEM 511 Chapter 6 page 2 of 20
Close packed unit cell:
less wasted space
each atom will have 12 nearest neighbors
Types of cp unit cells
Hexagaonally close packed (hcp)
layer A is set down
layer B is in the “dimples” of layer A
3rd layer is exactly the same as A
CHEM 511 Chapter 6 page 3 of 20
Cubic close packed (ccp) aka, face centered cubic (fcc)
set down layer A
layer B is placed in “dimples” of layer A
layer C is placed in “dimples” of layer B, but not directly above atoms in the A layer
Often, atoms can be squeezed in the empty spaces between atoms (holes).
Calculate the volume of space occupied by the atoms in a ccp structure.
Holes in close packed structures
Octahedral holes (Oh holes):
lie between 2 planar triangles
CHEM 511 Chapter 6 page 4 of 20
For ccp, Oh holes are located at midpoints of each edge of the cube AND in the center
Tetrahedral hole (Td): lies between a planar triangle capped with a single atom
Non-close-packed structures Body centered cubic (bcc or cubic-I): atoms at each corner and in the center
Simple or primitive cubic (cubic-P): atoms only at corners
CHEM 511 Chapter 6 page 5 of 20
Packing of spheres applied to the elements We will focus mainly on the metallic elements as shown below. Atoms typically crystallize in the
hcp or ccp structure type, but not exclusively. More importantly, multiple crystallization phases
may be possible—depending on the temperature and pressure.
Polymorphism: the ability to adopt different crystalline forms at various temperatures and
pressures.
This is the phase diagram for Fe.
What do you notice about the packing of higher pressure forms? Does this make sense?
Typically a transition to a higher temperature would result in less close-packed structures. Is this
borne out on the figure?
(1 bar = 0.987 atm)
CHEM 511 Chapter 6 page 6 of 20
Metallic radii rmetal: one half the distance between the nearest-neighbor atoms in a solid state metallic lattice
This value is dependent on the coordination number (i.e., the nearest neighbor atoms)
Coordination Number 12 8 6 4
Relative Ratio 1.00 0.97 0.96 0.88
The periodic table on previous page shows rmetal for 12-coordinate species. What is the CN for a
bcc lattice?
So if the tabulated value for the rmetal of Na is 191 pm, what would the sodium radius in the bcc
lattice be at 1 bar and 298 K?
What is the periodic trend for size on going down a group? <look at periodic table for values>
What is the correlation between lattice type and melting point?<look at periodic table for values>
Alloys and intermetallic compounds Alloy: an intimate mixture or compound of two or more metals or metals and nonmetals.
Properties will be different that the elements separately.
● can be homogeneous
● can be made of definite compounds (definite composition and internal (crystalline)
structure)
CHEM 511 Chapter 6 page 7 of 20
Classification of alloys
(a) Substitutional alloy
atomic radii must be within 15% of size
crystal structures of the elements must be the same
electronegativity should be similar
(b) Interstitial alloy (also, interstitial solid solutions)
need one atom to be very small compared to the lattice atoms,
otherwise distortion will occur
(c) Intermetallic compounds
formation of a stoichiometric compound (i.e., one with a specific composition) between two
or more metals.
CHEM 511 Chapter 6 page 8 of 20
Bonding in metals and semiconductors Extended solids, whether metallic, covalent, or ionic can be modeled with molecular orbitals.
Metallic conductor: a substance whose electrical conductivity decreases with rising temperature (or
it can be said conversely: the resistivity increases with rising temperature)
Semiconductor: a substance whose electrical conductivity increases with rising temperature (or:
the resistivity decreases with rising temperature)
Insulators are just a special category of semiconductors
To understand this, imagine forming a molecular orbital system for a collection of lithium atoms.
Band: a group of MOs in which the energy difference is so small that the system behaves as if a
continuous, non-quantized variation of energy is possible
If each atom gives 1 electron, then the orbital array should be half-full. This level is called the
Fermi level (technically, this level is measured at absolute zero, but it is impossible to reach this
temperature).
Ge
CHEM 511 Chapter 6 page 9 of 20
Which letter in the figure links the band diagram to the appropriate type of conductor?
For metals, electrons are filled to the Fermi level and thermal energy can promote the electron to
allow them to conduct around the metal. So why will an increase in temperature decrease the
conductivity?
Semiconductors Intrinsic semiconductors (no doping necessary): small band gap, therefore thermal energy used to
promote electrons to the conduction band (upper band)
Extrinsic semiconductors (doping necessary)-results in p- or n-type semiconductors
CHEM 511 Chapter 6 page 10 of 20
Why does heat cause these to increase conductivity?
Ionic radius What happens to the size of an atom when it loses an electron? Why?
What happens to the size of an atom when it gains an electron? Why?
Measuring the size of an ion is complicated—and made more complicated since coordination
number will change the size of an ion. One system uses oxygen as a standard and measures other
ions against it.
Ionic Solids
Contain cations and anions in crystalline arrays
Often one ion will be in fcc or hcp and the other ion fills in Oh or Td holes.
Rock Salt structure
Named for NaCl, but many ionic compounds conform to this crystal structure
(LiCl, KBr, RbI, AgCl, AgBr, MgO, CaO, TiO, NiO, BaS, UC, ScN)
Consists of fcc array of anions. Cations occupy Oh holes (or vice versa!)
CHEM 511 Chapter 6 page 11 of 20
Cesium Chloride structure
Named for CsCl, but many ionic compounds conform to this crystal structure
(CsBr, CsI, NH4Cl, NH4Br, TlCl, TlBr, and some intermetallics: CuZn, CuPd, AuMg)
Each anion occupies a vertex and the cation is in the center of the box (or vice versa !)
Fluorite structure
Named for the mineral CaF2
(others: BaCl2, HgF2 , PbO2, ThO2, CeO2, PrO2, UO2, ZrO2, HfO2, NpO2, PuO2, AmO2)
Ca occupy fcc array and F occupy both types of Td holes
Anti-fluorite structure
Has basically the same structure as fluorite, but cations and anions switched positions
K2O, Na2O, Na2S, K2S
CHEM 511 Chapter 6 page 12 of 20
Sphalerite (aka zinc blende) structure
Named for the mineral ZnS. Other compounds that adopt this structure: CuCl, CdS, HgS
Wurtzite structure
Another type of ZnS mineral
ZnO, AgI, SiC, NH4F
CHEM 511 Chapter 6 page 13 of 20
Rutile structure (TiO2) Others: SnO2, MgF2, NiF2, MnO2
Perovskite structure (ABX3) Perovskite is a class of compounds, but the prototype is calcium titanate (CaTiO3)
Others: BaTiO3, SrTiO3
CHEM 511 Chapter 6 page 14 of 20
Rationalizing Structures
Ionic radii
As noted earlier, a reference value is needed. Usually oxygen is assumed to be 140 pm
Trends are:
1. ionic radii increase going down a group (lanthanide contraction notwithstanding)
2. the radii of ions of the same charge decreases across a period
3. for a given ion, a larger coordination number results in a larger radius
4. an ionic radius will decrease as the positive charge increases for a given cation
Radius ratio: taking a ratio of the ions' sizes, you can “predict” the coordination of the ions
As the difference in size gets to be larger, the large ions will get closer together (small ions aren't
there to keep them apart). Thus, like charges get close together and there is repulsion!
EX. Predict the CN for NaCl and CsCl using the radius ratio method. Appendix 6 has ionic radii.
Structure maps
Empirically derived plot of versus the average principle quantum number (this figure is for MX
solids only, a different diagram is used for MX2).
EX. Given that the electronegativity of Ag is 1.9
and Br is 2.8, what would you predict for CN of
AgBr? What does the radius ratio predict (Ag+
r=126 pm)?
CHEM 511 Chapter 6 page 15 of 20
Lattice Energy: estimates from an electrostatic model The book gives multiple equations to calculate the lattice energy, ΔHL or ΔU (i.e., the energy
associated with the formation of an ionic lattice compared to gas phase ions (though usually, this is
defined as the reverse process)).
What do take away from all of these equations:
(1) the basic equation describes the attraction between the primary ions
r4π
ezzΔH
0
2-
L
z+, z - are the charges on the ions
e is the charge on an electron (1.602 10-19
C)
o = permittivity constant (8.854 10-12
F/m)
r = distance between ions (in m)
Consider NaCl. From the Na’s perspective, what are the nearest
neighbors? the next nearest neighbors? the next, next nearest neighbors?
the next, next, next nearest neighbors? How does this affect the attraction
of the ion in the lattice?
Madelung constants: depends on the arrangement of ions (strictly, it is a value representing the
coulomb energy of an ion pair in a crystal relative to the coulomb energy of an isolated ion pair).
Structure Type A
NaCl 1.7476
CsCl 1.7627
α-ZnS (wurtzite) 1.6413
Β-ZnS (sphalerite) 1.6381
CaF2 2.5194
CHEM 511 Chapter 6 page 16 of 20
Besides Coulombic interactions, there are other factors to consider: electron-electron repulsion,
nuclear-nuclear repulsion, finite sizes of the ions, etc. This gives rise to the Born forces, which
have a repulsive value in the equation (n in the table below).
All of these factors refine to the Born-Meyer equation: r
1r4π
ezzLAΔH
0
2BA
L
Where: L = Avogadro’s number
A = Madelung constant
z+, z - = charges on ions
e = electric charge
o = permittivity constant
r = distance between charges
ρ = constant of 35 pm (listed as 34.5 pm in some books) for alkali halides
The data: Ion Size(angstroms) Salt Lattice Enthalpy (kJ/mol)
Li+ 0.76 (6) LiCl 852
Mg2+
0.72 (6) MgCl2 2524
Al3+
0.54 (6) AlCl3 5492
For size considerations Ion Size (angstroms) Salt Lattice Enthalpy (kJ/mol)
Li+ 0.76 (6) LiCl 852
Na+ 1.02 (6) NaCl 787
K+ 1.38 (6) KCl 715
Cl- 1.81 (6) LiCl 852
Br- 1.96 (6) LiBr 815
I- 2.20 (6) LiI 761
CHEM 511 Chapter 6 page 17 of 20
Born-Haber Cycle Imagine the reaction between Na and Cl2 , normalized to make one mole of product.
If we break this into a series of steps and calculate the energy needed for each step we can
determine how stable the ionic lattice is.
The steps:
1. sublime (atomize) the metal
2. ionization of Na(g)
3. dissociate the halogen
4. form Cl-(g) ions
5. bring the ions together
CHEM 511 Chapter 6 page 18 of 20
The Born-Haber cycle is useful for predicting if a solid is largely ionic or not. If the measured
value for Hf is close to the calculated value, the solid is largely ionic.
The significance of the Born-Haber cycle and Born-Meyer equation is when the values are
compared to experimental data. If calculations are close, the system is largely ionic; if the
calculations deviate from experimental data, then some covalent character may be present.
Thermal stabilities of ionic solids
In general, large cations stabilize large anions (and vice versa)
Consider the decomposition of carbonates. Salt Decomposition
Temperature (oC)
MgCO3 300 CaCO3 840 SrCO3 1100 BaCO3 1300
As M2+
gets larger, decomposition temperature increases. Why? We must compare lattice enthalpy
of reactants and products.
CHEM 511 Chapter 6 page 19 of 20
Stabilities of oxidation states Cations with high oxidation states are stabilized by small ions
Recall: higher charges = higher lattice energy (more electrostatic attraction)
Due to the small size of F- (relative to other anions), it can stabilize certain cations that other halides
cannot.
EX. AgF2, CoF3, MnF4
As size increases (F-<Cl
-<Br
-<I
-) ions can’t get as close—what is the effect on stabilization?
Solubility
A compound made of different-sized ions tends to be more water soluble that a compound made of
similar-sized ions. Species Solubility
(g/100 mL) Solubility
(Molarity) Mg(OH)2 0.0009 0.0002 Ca(OH)2 0.185 0.025 Sr(OH)2 0.41 0.034 Ba(OH)2 3.05 0.178
To dissolve, MX(s) M+(aq) + X
-(aq)
Hydration enthalpy is inversely proportional to individual atom radii
Lattice enthalpy is dependent on the distance between ions
CHEM 511 Chapter 6 page 20 of 20
Defects in Crystal Structures Throughout this chapter we have discussed structures of crystalline materials—how did we define
crystalline?
Sometimes, however, imperfections can cause a crystal lattice to have defects.
Schottky Defect
In essence, the equivalent of a formula unit (MX, MX2, or ABX3, etc) is missing from the lattice.
See below at the sodium chloride lattice.
Frenkel Defect
The migration of cations and/or anions to holes not normally containing those ions. See below for a
AgBr lattice with a silver ion moved.
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