Chapter 3 The Structures of Simple Solids

CHM 511 chapter 3 page 1 of 25

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:


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


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).


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:


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:


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


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)


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.


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? ___________


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


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


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


From 2012 NIH study of TiO2 in food


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


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


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?


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?


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.


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, 8.854 10-12 F/meter

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


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


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


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.


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).

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?

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Chapter 3 The Structures of Simple Solids