Chapter 2 Structures of Solids

8/12/2019 Chapter 2 Structures of Solids

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Chapter 2 Slide 2 of 85

States of Matter Compared


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Some Characteristics of Crystalline Solids



Network Covalent Solids

These substances contain a network of covalent bonds that extendthroughout a crystalline solid, holding it firmly together.

In material science, polymorphism is the ability of a solid material to

exist in more than one form or crystal structure. Diamond, graphite andthe Buckyball are examples of polymorphs of carbon. -ferrite,

austenite, and -ferrite are polymorphs of iron. When found inelemental solids the condition is also called allotropy.

The allotropes of carbonprovide a good example

1. Diamond has each carbon bonded to four other carbons in atetrahedral arrangement using sp3 hybridization.

2. Graphite has each carbon bonded to three other carbons in thesame plane using sp2 hybridization.

3. Fullerenes and nanotubes are roughly spherical and cylindricalcollections of carbon atoms using sp2 hybridization.



Crystal Structure of Diamond

Covalent bond


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Crystal Structure of Graphite

Covalent bond

van der Waals

force


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Structure of a Buckyball

sp2 hybrid Carbon



sp2 hybrid Carbon

Carbon Nano-tube

A nanotube (also known as a buckytube) is a member of the fullerene structural

family, which also includesbuckyballs. Whereas buckyballs are spherical in shape,a nanotube is cylindrical, with at least one end typically capped with a hemisphere

of the buckyball structure. Their name is derived from their size, since the

diameter of a nanotube is on the order of a few nanometers , while they can be up

to several centimeters in length. There are two main types of nanotubes: single-

walled nanotubes (SWNTs) and multi-walled nanotubes (MWNTs).



Experimental Determination

of Crystal Structures

Braggs Law

2 d sin = n


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X-Ray Diffraction Image & Pattern

Single crystal Powder


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Crystal Lattices

To describe crystals, three-dimensional views must be used.

The repeating unit of the lattice is called the unit cell.

The simple cubic cell (primitive cubic) is the simplest unit

cell and has structural particles centered only at its corners.

The body-centered cubic (bcc) structure has an additional

structural particle at the center of the cube.

The face-centered cubic (fcc) structure has an additionalstructural particle at the center of each face.


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Unit Cells In

Cubic Crystal Structures

simple cubic

(primitive cubic)

bcc fcc


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Face-centered cubicBody-centered cubicPrimitive cubic


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Occupancies per Unit Cells

Primitive cubic: a = 2r

1 atom/unit cell

occupancy = [4/3(r3)]/a3 = [4/3(r3)]/(2r)3= 0.52 = 52%

Body-centered cubic: a = 4r/(3)1/2

2 atom/unit celloccupancy = 2 x [4/3(r3)]/a3 = 2 x [4/3(r3)]/[4r/(3)1/2]3

= 0.68 = 68%

Face-centered cubic: a = (8)1/2 r

4 atom/unit celloccupancy = 4 x [4/3(r3)]/a3 = 4 x [4/3(r3)]/[(8)1/2 r]3

= 0.74 = 74%

Closest packed


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a

ab

aba abc

unoccupied holes

Closest packed


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abab

abcabc

Cubic close-packed structure

= Face-centered cubic

Hexagonal close-packed structure

polytypes

C l S f M l


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Crystal Structures of Metals


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Interionic Forces of Attraction

E= (Z+Z-e2)/4r e = 1.6 x 10-19 C

In vacuum, 0 = 8.85 x 10-12 C2m-1J-1

In water, H2O = 7.25 x 10-10 C2m-1J-1= 82 0In liquid ammonia, NH3 = 2.2 x 10-10 C2m-1J-1= 25 0

U it C ll f R k S lt


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Unit Cell of Rock-Salt

(Sodium Chloride)

Cl- at fcc

Na+ at Oh holes

Coord. #: Na+: 6; Cl-: 6

atom/ unit cell

Na: Cl= 4: 4 = 1: 1 NaCl


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Unit Cell of Cesium Chloride

Cl- at primitive cubicCs+ at Cubic holes

Coord. #: Cs+: 8; Cl-: 8

atom/ unit cell

Cs: Cl= 1: 1 CsCl


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Unit Cell of Cubic Zinc Sulfide

(Sphalerite or Zinc blende)

Coord. #: Zn2+: 4; S2-: 4

atom/ unit cell

Zn: S = 4: 4 = 1: 1 ZnS

S2- at fcc

Zn2+ at Td holes

A B H b C l t C l l t


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A Born-Haber Cycle to Calculate

Lattice Energy

1/2D(Cl-Cl)

sublimation (Na)

IE(Na) -EA(Cl)

lattice energy U0

= -f

0(NaCl) + sublimation

(Na) + 1/2D(Cl-Cl)

+ IE(Na) - EA(Cl)

= (+411 +107 +122 + 496 349) kJ/mol

= +787 kJ/mol

f0(NaCl)

U0NaCl(s) Na+(g) + Cl-(g)

Na(g) + Cl(g)Na(s) + 1/2Cl2(g)


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Lattice Energy & Madelung Constant

( ) ( ) ( ) 0U-H 0


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Electrostatic energy in a crystal lattice, between a pair

of ions

For NaCl crystal, Z+ =Z- =1

r = d

( )r

eA

0

2

c 4

ZZE

+

= A : Madelung Constant


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The value of Madelung constant is determined only by the

geometry of the lattice, and independent of the ionic

radius and charge.

Born equation


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electrostatic energy repulsion of close shells

B: constant

n: Born exponent

Born equation

12Xe, Au+

10Kr, Ag+

9Ar, Cu+

7Ne

5He

nIon configuration

At minimum PE, attractive and repulsive forces are balanced,


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Let U0= -(PE)0N N: Avogadro s number

Born-Lande equation

, p ,

and d= d0.0

)(=

d

PE


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For NaCl

A = 1.74756

Z1= 1, Z2= -1

d0= rNa++ rCl-= 2.81 x10-10mn = (nNa++ nCl-) /2 = (7+9)/2 = 8

U0= 755.2 kJ/mol

experimental U0= 770 kJ/mol

Modif ication of Born-Lande equation


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Modif ication of Born-Lande equation

)345.0

1(4

ZZ

000

2

0dd

eNAU =

+

Unit in ? , 10-10 m

2. Kapustinski i equation

Improving the repulsion

)345.0

1(ZZ

00

0 dd

nU

=

+

= 1.21 MJ. ? mol-1

n= ions per formulae.g. n= 2 for NaCl

n= 5 for Al2O3

A ? crystal lattice ? r+/r- ? r 0A/n ~constant

1. Born-Mayer equation

( ) ( ) ( )g2ss3 COMOMCO +


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+218.1+183.8+130.4+48.3G0 (kJ/mol)

13001100840300T (?C)

+172.1+171.0+160.6+175.0S0 (J/K mol)

+269.3+234.6+178.3+100.6H0 (kJ/mol)

BaSrCaMgData at 298K

( ) ( ) ( )g2ss3 COMOMCO +

A small cation increases the lattice enthalpy of the oxide

more than that of a carbonate.

0

0

000

Tiondecomposit

T-

S

H

SHG

=

=

Solubility of ionic compounds


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< 0 > 0

y p

0000T-T- SLSHG ==

Enthalpy of solutionHydration enthalpy

Free energy of solution


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Hydration enthalpy

+

+

+=+=

+

+

rrHHH

rrU

XMhydration11

1

0

= 1

12

2

r

Z

Hhyd

In general, difference in ionic size favors solubility in water.

Ionic compound MX tends to be most soluble when rX-rM > 0.8?


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Correlation between Hsolution and the differences between the hydration

enthalpy of the ions


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Fajan's rules

Fajan's Rule: the degree of covalent character of ionic

bond

Polarization effects: (a) idealized ion pair with no polarization; (b)

mutually polarized ion pair; (c) polarization sufficient to form

covalent bond. Dashed lines represent hypothetical unpolarized ions.

In 1923, Fajan suggested rules to predict the degree of covalent


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, j gg p g

character in ionic compounds:

Thepolarization of an ionic bond and thus the degree ofcovalency is high if :

1) the charges on the ions are high.

eg., Al3+ ; Ti4+ ----- favors covalent character.

Na

+

; K

+

----- favors ionic character.

2) the cation is small.

e.g., Na+ ion is larger than that of Al3+

Thus, Al3+ favors covalent character.

3) the anion is large.

e.g. F- ionic radius : 0.136 nm favors ionic character.

I- ionic radius : 0.216 nm favors covalent character.


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4) An incomplete valence shell electron configuration

Noble gas configuration of the cation better shielding and less

polarizing powere.g. Hg2+ (r = 102 pm) is more polarizing than Ca 2+ (r = 100 pm)

HgO decomposed at 500 ?C

CaO m.p. 2613 ?C

783

742

775

1418

m.p. (?C)

hexagonal

rhom.

cubic

cubic

Crys str

259

236

276

645 (dec)

m.p. (?C)

tetragonal

rhombohedral

orthorhombic

cubic

Crys str

CaI2

CaBr2

CaCl2

CaF2

Halides

HgI2

HgBr2

HgCl2

HgF2

Halides

Examples:


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a) Charge factor

i) Cations

Na+ Mg2+ Al3+

Examples:

For ions with noble gas (ns2 np6 ) structure, the only factor that

have to be considered are size and charge factor.

ii) Anions

N3- O2- F-

Increasing charge : increase in polarizing power

Increase in ionic charge, more ready to be polarized


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1.5 x 10-5183sublimationAlCl3

57-70SiCl4

1410

1440

boiling point

/C

715

800

melting point

/C

Conductance in

molten state

Chlorides

29MgCl2

133NaCl

AlCl3 has covalent character. In fact, AlCl3 exists as dimer Al2Cl6at room conditions.

ovalency

b) Size factor


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)

i) Cations

Be2+ Mg2+ Ca2+ Sr2+ Ba2+

The smaller the cation, the higher is its polarizing power

52774CaCl2

56870SrCl2

Conductance in

molten state

melting point /CChlorides

955BaCl2

29715MgCl2

0.056404BeCl2

covalency

ii) Anions


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F- Cl- Br- I-

The larger the anion, the more polarizable is the anion.

eg.,

NaF NaCl NaBr NaI

melting point /C 990 800 755 651

covalency

( ) ( ) ( )COMOMCO + Some covalent


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( ) ( ) ( )g2ss3 COMOMCO +

1360BaCO3

1289SrCO3

900CaCO3

540MgCO3

250BeCO3

Decomp. Temp. (?C)Carbonates

1923BaO

2430SrO

2613CaO

2826MgO

2530BeO

m.p. (?C)Oxides

Ionic compounds

ovalency

wurtzite

structure

NaClstructure

Some covalent

character

Close-packing of Spheres


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Close packing of Spheres

in Three Dimensions


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Octahedral holesTetrahedral holes Cubic holes

Close packed structure


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rh/r = 0.156

rh/r = 0.225

rh/r = 0.414


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Considering

anion-anion

repulsion

Unit Cell of Rock-Salt


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(Sodium Chloride)

Cl-

at fccNa+ at Oh holes

Coord. #: Na+: 6; Cl-: 6

atom/ unit cell

Na: Cl= 4: 4 = 1: 1 NaCl


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Rock-salt structure

Unit Cell of NiAS


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Unit Cell ofNiAS

As3- at hcp

Ni3+ at Oh holesPolariable cation

& anion

U i C ll f C i Chl id


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Unit Cell of Cesium Chloride

Cl- at primitive cubicCs+ at Cubic holes

Coord. #: Cs+: 8; Cl-: 8atom/ unit cell

Cs: Cl= 1: 1 CsCl

Unit Cell of Cubic Zinc Sulfide


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(Sphalerite or Zinc blende)

Coord. #: Zn2+: 4; S2-: 4

atom/ unit cell

Zn: S = 4: 4 = 1: 1 ZnS

S2- at fcc

Zn2+ at Td holes

Unit Cell of Hexagonal Zinc


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Sulfide (Wurtzite)

S2- at hcp

Zn2+ at Td holesPolymorph of ZnS

PtS


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Pt2+

at fccS2- at Td holes

PtS4 unit is planar Pt-S more covalent

Unit Cell of Fluorite Structure


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(Calcium Fluoride)

Ca2+ at fcc

F- at Td holesCoord. #: Ca2+: 8; F-: 4

atom/ unit cell

Ca: F= 4: 8 = 1: 2 CaF2

Fluorite Structure- MX2


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Large M

Antifluorite structure- M2XLarge X, e.g. Na2O

Unit Cell of Rutile TiO2


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Unit Cell of Rutile TiO2

O2- at hcp

Ti4+ at Oh holesCoord. #: Ti: 6; O: 3atom/ unit cell

Ti: O= 2: 4 = 1: 2


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Increasing covalency


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A Structural Map for compounds of MX


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Increasing covalency


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A Structural Map for compounds of MX2

Salts of highly polarizing cations and easily

polarizable anions have layered structures.


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Salts of highly polarizing cations and easily



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CdCl2

Cl- at ccp

Cd 2+ at Oh sites

(alternate O layers)

van der Waals force


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CdI2

I-

at hcpCd 2+ at Oh sites

(alternate O layer)

298pm


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MoS2Mo4+ at hcp

S2- at Td sites

366pm

Unit Cell of Perovskite

C TiO


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CaTiO3

AIIBIVO3

AIIIBIIIO3

Coord. #: A: 12; B: 6atom/ unit cell

A: B: O= 1: 1: 3

A and O together at ccp

B at 1/4 Oh holes

Unit Cell of ReO3


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Perovskite structure CaTiO3 without Ca

Unit Cell of Spinel

M Al O


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MgAl2O4

Normal Spinel

AII[BIII]2O4, AIV[BII]2O4 , A

VI[BI]2O4e.g. NiCr

2

O4

, Co3

O4

, Mn3

O4

Inverse SpinelB[AB]O4e.g. Fe3O4

O 2- at fcc

A at 1/8 Td holes

B at 1/2 Oh holes


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YBa2Cu3O7


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Quartz SiO2


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Mineral Ideal formulab CECc

(meq/100 g)

Dioctahedral minerals

Pyrophyllite Al2(Si4O10)(OH)2 0

Montmorillonite Nax(Al2-xMgx)(Si4O10) 60 - 120


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(OH)2.zH2O

Beidellite Mx(Al2)(AlxSi4-xO10)

(OH)2.zH2O

60 - 120

Nontronite Mx(Fe3+,Al)2(AlxSi4-xO10)

(OH)2.zH2O

60 - 120

Trioctahedral minerals

Talc Mg3(Si4O10)(OH)2 0

Hectorite (Na2Ca)x/2(LixMg3-x)(Si4O10)(OH)2.zH2O

60 - 120

Saponite Cax/2Mg3(AlxSi4-xO10)

.zH2O

60 - 120

Sauconite Mx(Zn,Mg)3(AlxSi4-xO10)

.zH2O

a: Only major cations are shown.

b: x depends on the origin of the mineral; montmorillonites can

show a degree of substitution x in the octahedral sheet in the range

0.05- 0.52. Natural samples generally show substitutions in both

octahedral and tetrahedral sheets, which renders the real situations

more complex.

c: Cation-exchange capacity.


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Chapter 2 Structures of Solids