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Chapter 3
The Structures of Simple Solids Types of bonding in solids
● Covalent
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)
● Ionic
● Metallic
Classifications to describe crystalline solids Lattice: three dimensional infinite array of points (atoms) where each atom is surrounded in an
identical way by neighboring points
Unit cell: the simplest set of lattice points from which the entire crystal structure can be built by
purely translational displacements
The seven crystal systems:
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1. Primitive unit cell (P): has an atom at each vertex and nowhere else.
2. Body-centered unit cell (I or bcc (body-centered cubic)): has an atom at each vertex and in the
center.
3. Close packed unit cell (cp):
less wasted space
each atom will have 12 nearest neighbors
two types (hcp & fcc)
Hexagaonally close packed (hcp)
Two differently positioned layers
-layer A is set down
-layer B is in the “dimples” of layer A
-3rd layer is exactly the same as A
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Cubic close packed (ccp) aka, face centered cubic (fcc)
Three differently positioned layers
-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).
How much space is between atoms in a ccp structure????
Translational symmetry coordinates
Consider a primitive cubic cell, starting with one atom as the “origin” (location is such that
numbers are positive).
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Fractional atomic coordinates and projections To draw 2-D representations of where atoms are, use a coordinate system.
Holes in close packed structures
Octahedral hole (Oh holes):
lies between 2 planar triangles
For hcp lattice, see hexagons below:
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For ccp, Oh holes are located at midpoints of each edge of the cube AND in the center
Tetrahedral hole (Td holes): lies between a planar triangle capped with a single atom
For hcp lattices:
For ccp lattices:
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The structures of metals and alloys
Many metals adopt close packed structures—but which? hcp or ccp? Actually, there are many
polytypes.
Polytypes: structural forms in which two dimensions are the same, but the 3rd dimension aligns
the atoms differently.
See periodic table in figure 3.21, page 73, many are hcp or fcc, but not all
Body centered cubic (bcc or cubic-I)—common among metals of groups 1, 5, 6
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Primitive cubic (cubic-P)—uncommon among metals (only Po adopts this under standard
conditions)
Polymorphism: the ability to adopt different crystalline
forms at various temperatures and pressures.
Figure to the right is showing iron at various temperatures
and pressures.
Atomic radii of metals
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)
Atoms appear larger if there are more neighbors. This data applies the Goldschmidt correction.
Coordination Number 12 8 6 4
Relative Ratio 1.00 0.97 0.96 0.88
If the 12-coordinate Fe radius is 126 pm, what is the expected size of Fe at 1 bar and 298 K?
Alloys and interstitials Alloy: a blend of metallic elements prepared by mixing the molten components and then cooling
the mixture to produce a metallic solid.
● can be homogeneous
● can be made of definite compounds (definite composition and internal (crystalline) structure)
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Classification of alloys
Substitutional alloy
● atomic radii must be within 15% of size
● crystal structures of the elements must be the same
● electronegativity should be similar
Interstitial alloy (also, interstitial solid solutions)
● need one atom to be very small compared to the lattice atoms,
otherwise distortion will occur
Intermetallic compounds
● formation of a stoichiometric compound (i.e., one with a specific composition) between two or
more metals.
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Ionic Solids
Contain cations and anions in crystalline arrays
Typically these materials will have high melting points and high solubility (though
exceptions exist for both properties)
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, UC, ScN—also CaC2, CsO2, FeS2)
Consists of fcc array of anions. Cations occupy Oh holes (or vice versa!)
Coordination of each Na+? ________ each Cl-? ___________
How many NaCl units in each unit cell? ___________
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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!)
Sphalerite (aka zinc blende) structure
Named for the cubic form of a ZnS mineral; CuCl, CdS, HgS
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Wurtzite structure
Another type of ZnS mineral—hexagonal lattice instead of cubic; ZnO, AgI, AlN, SiC, NH4F
Nickel-arsenide (NiAs) structure NiS, FeS, and other sulfides adopt this structure; seen with formulas that have more polarizable
ions and smaller electronegativity differences than rock salt structured materials
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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
Rutile structure (TiO2) Others: SnO2, MgF2, NiF2, MnO2
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Perovskite structure (ABX3) Perovskite is a class of compounds, but the original was calcium titanate (CaTiO3)
Others: BaTiO3, SrTiO3
Layered Structures CdI2 and CdCl2
CdI2 structures: MgBr2, MgI2, CaI2, Mg(OH)2, many d-metal iodides
CdCl2 structures: FeCl2, CoCl2
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Spinel structure (AB2O4) MgAl2O4 is the "parent" mineral; others: Fe3O4, Co3O4, Mn3O4, Fe2GeO4, CuCr2Te4
“A” atoms occupy 1/8 of the Td holes and “B” atoms occupy half of the Oh holes of an oxide ccp lattice.
Inverse spinels also exist: B[AB]O4
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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. an ionic radius will decrease as the positive charge increases for a given cation
4. cations are smaller than anions of the same Z
5. for a given ion, a larger coordination number results in a larger radius
Radius ratio method: 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. What would the CN be for NaCl and CsCl using the radius ratio method?
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Structure maps
Empirically derived plot of versus the average principle quantum number. This is for MX
compounds (would need a different plot 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?
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Energetics of ionic bonding
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 the metal
2. ionization of Na(g)
3. dissociate the halogen
4. form Cl-(g) ions
5. bring the ions together
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.
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Calculating Lattice Enthalpy #2
Born-Meyer equation
Αd
d1
d4π
ezzNΔH
000
2
BAA
L
Where: d0 = distance between charges (in pm)
zA, zB = charges on ions
NA = Avogadro’s number
o = permittivity constant
d = constant of 34.5 pm
e = electric charge
A = Madelung constant, 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). Takes into account alternating layers of counter ions and similar ions.
ions with higher charges will form compounds with higher lattice enthalpies
ions that are smaller will form compounds with higher lattice enthalpies
The data: Ion Size(angstroms) Salt Lattice Enthalpy (kJ/mol)
Li+ 0.76 (6) LiCl 853
Mg2+ 0.72 (6) MgCl2 2524
Al3+ 0.53 (6) AlCl3 5492
For size considerations Ion Size (angstroms) Salt Lattice Enthalpy (kJ/mol)
Li+ 0.76 (6) LiCl 853 Na+ 1.02 (6) NaCl 786 K+ 1.38 (6) KCl 719
Cl- 1.67 (6) LiCl 853 Br- 1.96 (6) LiBr 815 I- 2.06 (6) LiI 757
Also may need to consider non-ionic interactions between atoms, i.e., London dispersion forces
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The value in Born-Haber and Born-Meyer is comparison 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
% error
-0.1
-2.2
-2.4
-3.6
-3.5
-7.9
-8.9
-11.9
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Stabilities of oxidation states Cations with high oxidation states are stabilized by small ions
Recall: higher charges = higher lattice energy (more electrostatic attraction)
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
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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.
o Intrinsic defects: ones that occur in a pure material
o Extrinsic defects: ones that occur due to an impurity (intentional or otherwise)
o Point defects: occur at a specific location
o Extended defects: occur in 1-, 2-, or 3-dimensional locations.
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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Color Centers
Trapped electrons can give rise to colored crystal lattices, the location of
the electron is known as an F-center (from the German word for color,
Farbe)
Non-stoichiometric compounds
Most common for metal lattices in which the metal ion can adopt multiple different oxidation
states (i.e., d- and f-metal compounds).
FeO is rarely a 1:1 ratio when in contact with O2. The O2 causes oxidation of Fe2+ to Fe3+.
The Electronic Structures of Solids Extended solids, whether metallic, covalent, or ionic can be modeled with molecular orbitals.
Metallic conductor: a substance whose electrical conductivity decreases with rising temperature
Semiconductor: a substance whose electrical conductivity increases with rising temperature
Insulators are really just a special category of semiconductors
Imagine a HUGE number of atoms forming molecular orbitals
If each atom gives 1 electron, then the orbital array should be half-full. This level is called the
Fermi level (though technically, the Fermi level should be measured at 0 K).
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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?
For semimetals, s- and p-bands just meet (for insulators, there is a gap, called the band gap)
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
Non-stoichiometric compounds can be n- or p-type depending on the metal: high oxidation state
metals tend to form n-type (Fe2O3, MnO2, CuO, WO3); p-type form with metals have a low
oxidation state (MnO, Cr2O3).
Why does heat cause these to increase conductivity?