chem 511 chapter 6 page 1 of 20 chapter 6 - usc upstate: …faculty.uscupstate.edu/cbender/web page...

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)


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


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


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


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)


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)


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.


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


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


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!)


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


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


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


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)?


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


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


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


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.


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


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.

chem 511 chapter 6 page 1 of 20 chapter 6 - usc upstate: …faculty.uscupstate.edu/cbender/web page...

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