Chapter 2 Structures of Solids - National Taiwan …sfcheng/inchem94/Chapter 2 Structures of...

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Chapter Two

Structures of Solids


States of Matter Compared



Some Characteristics of Crystalline Solids


Network Covalent Solids• These substances contain a network of covalent bonds that extend

throughout 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 and the Buckyball are examples of polymorphs of carbon. α-ferrite, austenite, and δ-ferrite are polymorphs of iron. When found in elemental solids the condition is also called allotropy.

• The allotropes of carbon provide a good example1. Diamond has each carbon bonded to four other carbons in a

tetrahedral arrangement using sp3 hybridization.

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

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


Crystal Structure of Diamond

Covalent bond

Crystal Structure of Graphite

Covalent bond

van der Waalsforce

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 includes buckyballs. 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 Determinationof Crystal Structures

Bragg’s Law2 d sinθ = n λ



X-Ray Diffraction Image & Pattern

Single crystal Powder


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 additional

structural particle at the center of each face.


Unit Cells InCubic Crystal Structures

simple cubic(primitive cubic)

bcc fcc


Face-centered cubicBody-centered cubicPrimitive cubic


Occupancies per Unit CellsPrimitive cubic: a = 2r

1 atom/unit celloccupancy = [4/3(πr3)]/a3 = [4/3(πr3)]/(2r)3

= 0.52 = 52%Body-centered cubic: a = 4r/(3)1/2

2 atom/unit celloccupancy = 2 x [4/3(πr3)]/a3 = 2 x [4/3(πr3)]/[4r/(3)1/2]3

= 0.68 = 68%Face-centered cubic: a = (8)1/2 r

4 atom/unit celloccupancy = 4 x [4/3(πr3)]/a3 = 4 x [4/3(πr3)]/[(8)1/2 r]3

= 0.74 = 74%Closest packed


a

ab

aba abcunoccupied holes

Closest packed


abab

abcabc

Cubic close-packed structure

= Face-centered cubic

Hexagonal close-packed structure

polytypes


Crystal Structures of Metals

Interionic Forces of Attraction

E = (Z+Z-e2)/4πεr e = 1.6 x 10-19 C

In vacuum, ε0 = 8.85 x 10-12 C2m-1J-1

In water, εH2O = 7.25 x 10-10 C2m-1J-1= 82 ε0

In liquid ammonia, εNH3 = 2.2 x 10-10 C2m-1J-1= 25 ε0


Unit Cell of Rock-Salt(Sodium Chloride)

Cl- at fcc

Na+ at Oh holesCoord. #: Na+: 6; Cl-: 6atom/ unit cellNa: Cl= 4: 4 = 1: 1 NaCl


Unit Cell of Cesium Chloride

Cl- at primitive cubicCs+ at Cubic holes

Coord. #: Cs+: 8; Cl-: 8atom/ unit cellCs: Cl= 1: 1 CsCl


Unit Cell of Cubic Zinc Sulfide(Sphalerite or Zinc blende)

Coord. #: Zn2+: 4; S2-: 4atom/ unit cellZn: S = 4: 4 = 1: 1 ZnS

S2- at fccZn2+ at ½ Td holes


A Born-Haber Cycle to CalculateLattice Energy

1/2D(Cl-Cl)

∆Ηsublimation (Na)

IE(Na) -EA(Cl)

lattice energy U0 = - ∆Ηf0(NaCl) + ∆Ηsublimation (Na) + 1/2D(Cl-Cl)

+ IE(Na) - EA(Cl)

= (+411 +107 +122 + 496 –349) kJ/mol

= +787 kJ/mol

∆Ηf0(NaCl)

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

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


Lattice Energy & Madelung Constant

( ) ( ) ( ) 0 U- H 0 <=∆→+ −+sgg MXXM

U0: lattice energy

Factors contributed to Lattice energy•electrostatic energy ~90%•repulsion of close shells ~8%•dispersion forces ~1% •zero-point energy (lattice vibration at 0K)•correction for heat capacity ~1%


Electrostatic energy in a crystal lattice, between a pair of ions

For NaCl crystal, Z+ =Z- =1r = d

( )reA

0

2

c 4ZZ

Eπε

−+

= A : Madelung Constant


The value of Madelung constant is determined only by the “geometry of the lattice”, and independent of the “ionic radius” and “charge”.


electrostatic energy repulsion of close shells

B: constantn: Born exponent

Born equation

12Xe, Au+

10Kr, Ag+

9Ar, Cu+

7Ne

5He

nIon configuration


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

Born-Lande equation

At minimum PE, attractive and repulsive forces are balanced, and d= d0.

0)(

=∂

∂d

PE


For NaCl

A = 1.74756Z1 = 1, Z2 = -1d0 = rNa+ + rCl- = 2.81 x10-10 mn = (nNa+ + nCl-) /2 = (7+9)/2 = 8

⇒ U0= 755.2 kJ/mol

experimental U0= 770 kJ/mol


Modification of Born-Lande equation

)345.0

1(4

ZZ

000

2

0 ddeNA

U −=−+

πε

Unit in ? , 10-10 m

2. Kapustinskii equation

Improving the repulsion

)345.0

1(ZZ

00

0 ddn

U −−

=−+κ

κ = 1.21 MJ. ? mol-1

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

n= 5 for Al2O3

A ? crystal lattice ? r+/r- ? r0A/n ~constant

1. Born-Mayer equation


+218.1+183.8+130.4+48.3∆G0 (kJ/mol)

13001100840300T (?C)+172.1+171.0+160.6+175.0∆S0 (J/K mol)

+269.3+234.6+178.3+100.6∆H0 (kJ/mol)

BaSrCaMgData at 298K

( ) ( ) ( )g2ss3 CO MO MCO +→

A small cation increases the lattice enthalpy of the oxide more than that of a carbonate.

0

0

000

Tion decomposit

T-

SH

SHG

∆∆

=

∆∆=∆


< 0 > 0

Solubility of ionic compounds

0000 T- T- SLSHG ∆=∆∆=∆

Enthalpy of solution Hydration enthalpy

Free energy of solution


Hydration enthalpy

−+

−+

+∞=∆+∆=∆

+∞

−+

rrHHH

rrU

XMhydration11

1 0

−−=∆

ε1

12

2

rZ

Hhyd

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

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


Correlation between ∆Hsolution and the differences between the hydration enthalpy of the ions


Fajan's rules

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

Polarization effects: (a) idealized ion pair with no polarization; (b) mutually polarized ion pair; (c) polarization sufficient to formcovalent bond. Dashed lines represent hypothetical unpolarized ions.


In 1923, Fajan suggested rules to predict the degree of covalent character in ionic compounds:

The polarization of an ionic bond and thus the degree of covalency 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.


4) An incomplete valence shell electron configurationNoble gas configuration of the cation better shielding and less polarizing power e.g. Hg2+ (r = 102 pm) is more polarizing than Ca 2+ (r = 100 pm)

HgO decomposed at 500 ?CCaO 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


a) Charge factori) 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) AnionsN3- O2- F-

Increasing charge : increase in polarizing power

Increase in ionic charge, more ready to be polarized


1.5 x 10-5183sublimationAlCl3

57-70SiCl4

14101440

boiling point /°C

715800

melting point /°C

Conductance in molten state

Chlorides

29MgCl2

133NaCl

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

covalency


b) Size factor

i) CationsBe2+ 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

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


( ) ( ) ( )g2ss3 CO MO MCO +→

1360BaCO3

1289SrCO3

900CaCO3

540MgCO3

250BeCO3

Decomp. Temp. (?C)Carbonates

1923BaO

2430SrO2613CaO

2826MgO

2530BeO

m.p. (?C)Oxides

Ionic compounds

covalency

wurtzitestructure

NaClstructure

Some covalent character


Close-packing of Spheresin Three Dimensions


Octahedral holesTetrahedral holes Cubic holes

Close packed structure


rh/r = 0.156

rh/r = 0.225

rh/r = 0.414




Considering anion-anion

repulsion


Unit Cell of Rock-Salt(Sodium Chloride)

Cl- at fcc

Na+ at Oh holesCoord. #: Na+: 6; Cl-: 6atom/ unit cellNa: Cl= 4: 4 = 1: 1 NaCl


Rock-salt structure


Unit Cell of NiAS

As3- at hcp

Ni3+ at Oh holesPolariable cation& anion


Unit Cell of Cesium Chloride

Cl- at primitive cubicCs+ at Cubic holes

Coord. #: Cs+: 8; Cl-: 8atom/ unit cellCs: Cl= 1: 1 CsCl


Unit Cell of Cubic Zinc Sulfide(Sphalerite or Zinc blende)

Coord. #: Zn2+: 4; S2-: 4atom/ unit cellZn: S = 4: 4 = 1: 1 ZnS

S2- at fccZn2+ at ½ Td holes


Unit Cell of Hexagonal Zinc Sulfide (Wurtzite)

S2- at hcp

Zn2+ at ½ Td holesPolymorph of ZnS


Pt2+ at fccS2- at ½ Td holesPtS4 unit is planar Pt-S more covalent

PtS


Unit Cell of Fluorite Structure(Calcium Fluoride)

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

atom/ unit cellCa: F= 4: 8 = 1: 2 CaF2


•Fluorite Structure- MX2 Large M

•Antifluorite structure- M2XLarge X, e.g. Na2O


Unit Cell of Rutile TiO2

O2- at hcpTi4+ at ½ Oh holes

Coord. #: Ti: 6; O: 3atom/ unit cellTi: O= 2: 4 = 1: 2




A Structural Map for compounds of MX

Increasing covalency






A Structural Map for compounds of MX2

Increasing covalency


Salts of highly polarizing cations and easily polarizable anions have layered structures.


Salts of highly polarizing cations and easily polarizable anions have layered structures.

CdCl2Cl- at ccpCd 2+ at ½ Oh sites

(alternate O layers)

van der Waals force


CdI2I- at hcpCd 2+ at ½ Oh sites

(alternate O layer)


MoS2Mo4+ at hcpS2- at Td sites

298pm

366pm


Unit Cell of PerovskiteCaTiO3

AIIBIVO3

AIIIBIIIO3

Coord. #: A: 12; B: 6atom/ unit cellA: B: O= 1: 1: 3

A and O together at ccpB at 1/4 Oh holes


Unit Cell of ReO3

Perovskite structure CaTiO3 without Ca


Unit Cell of SpinelMgAl2O4

Normal SpinelAII[BIII]2O4, AIV[BII]2O4 , AVI[BI]2O4

e.g. NiCr2O4, Co3O4 , Mn3O4

Inverse SpinelB[AB]O4e.g. Fe3O4

O 2- at fccA at 1/8 Td holesB at 1/2 Oh holes



YBa2Cu3O7


Quartz SiO2






Mineral Ideal formulab CECc (meq/100 g)

Dioctahedral minerals Pyrophyllite Al2(Si4O10)(OH)2 0 Montmorillonite Nax(Al2-xMgx)(Si4O10)

(OH)2.zH2O 60 - 120

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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