Chapter 2 discusses the structures of solids, focusing on crystalline solids and their characteristics, including the concept of polymorphism and allotropy in materials like carbon and iron. It elaborates on different crystal structures such as cubic lattices and their variations (simple cubic, body-centered cubic, and face-centered cubic), as well as techniques like x-ray diffraction for determining crystal structures. Additionally, the chapter addresses factors affecting lattice energy and covalent character in ionic compounds, illustrating through various examples such as sodium chloride and calcium fluoride.

1. Chapter 2 Slide 1 of 85
Chapter Two
Structures of Solids

2. Chapter 2 Slide 2 of 85
States of Matter Compared

3. Chapter 2 Slide 3 of 85

4. Chapter 2 Slide 4 of 85
Some Characteristics of Crystalline Solids

5. Chapter 2 Slide 5 of 85
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 example
1. 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.

6. Chapter 2 Slide 6 of 85
Crystal Structure of Diamond
Covalent bond

7. Crystal Structure of Graphite
Covalent bond
van der Waals
force

8. Structure of a Buckyball
sp2 hybrid Carbon

9. Chapter 2 Slide 9 of 85
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).

10. Chapter 2 Slide 10 of 85
Experimental Determination
of Crystal Structures
Bragg’s Law
2 d sinθ = n λ

11. Chapter 2 Slide 11 of 85

12. Chapter 2 Slide 12 of 85
X-Ray Diffraction Image & Pattern
Single crystal Powder

13. Chapter 2 Slide 13 of 85
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.

14. Chapter 2 Slide 14 of 85
Unit Cells In
Cubic Crystal Structures
simple cubic
(primitive cubic)
bcc fcc

15. Chapter 2 Slide 15 of 85
Face-centered cubicBody-centered cubicPrimitive cubic

16. Chapter 2 Slide 16 of 85
Occupancies per Unit Cells
Primitive cubic: a = 2r
1 atom/unit cell
occupancy = [4/3(πr3)]/a3 = [4/3(πr3)]/(2r)3
= 0.52 = 52%
Body-centered cubic: a = 4r/(3)1/2
2 atom/unit cell
occupancy = 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 cell
occupancy = 4 x [4/3(πr3)]/a3 = 4 x [4/3(πr3)]/[(8)1/2 r]3
= 0.74 = 74%
Closest packed

17. Chapter 2 Slide 17 of 85
a
ab
aba abc
unoccupied holes
Closest packed

18. Chapter 2 Slide 18 of 85
abab
abcabc
Cubic close-packed structure
= Face-centered cubic
Hexagonal close-packed structure
polytypes

19. Chapter 2 Slide 19 of 85
Crystal Structures of Metals

20. 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

21. Chapter 2 Slide 21 of 85
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

22. Chapter 2 Slide 22 of 85
Unit Cell of Cesium Chloride
Cl- at primitive cubic
Cs+ at Cubic holes
Coord. #: Cs+: 8; Cl-: 8
atom/ unit cell
Cs: Cl= 1: 1 CsCl

23. Chapter 2 Slide 23 of 85
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 ½ Td holes

24. Chapter 2 Slide 24 of 85
A Born-Haber Cycle to Calculate
Lattice Energy
1/2D(Cl-Cl)
∆Ηsublimation (Na)
IE(Na) -EA(Cl)
lattice energy U0 = - ∆Ηf
0(NaCl) + ∆Ηsublimation (Na) + 1/2D(Cl-Cl)
+ IE(Na) - EA(Cl)
= (+411 +107 +122 + 496 –349) kJ/mol
= +787 kJ/mol
∆Ηf
0(NaCl)
U0
NaCl(s) Na+(g) + Cl-(g)
Na(g) + Cl(g)Na(s) + 1/2Cl2(g)

25. Chapter 2 Slide 25 of 85
Lattice Energy & Madelung Constant
( ) ( ) ( ) 0U-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%

26. Chapter 2 Slide 26 of 85
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
ZZ
E
πε
−+
= A : Madelung Constant

27. Chapter 2 Slide 27 of 85
The value of Madelung constant is determined only by the
“geometry of the lattice”, and independent of the “ionic
radius” and “charge”.

28. Chapter 2 Slide 28 of 85
electrostatic energy repulsion of close shells
B: constant
n: Born exponent
Born equation
12Xe, Au+
10Kr, Ag+
9Ar, Cu+
7Ne
5He
nIon configuration

29. Chapter 2 Slide 29 of 85
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

30. Chapter 2 Slide 30 of 85
For NaCl
A = 1.74756
Z1 = 1, Z2 = -1
d0 = rNa+ + rCl- = 2.81 x10-10 m
n = (nNa+ + nCl-) /2 = (7+9)/2 = 8
⇒ U0= 755.2 kJ/mol
experimental U0= 770 kJ/mol

31. Chapter 2 Slide 31 of 85
Modification of Born-Lande equation
)
345.0
1(
4
ZZ
000
2
0
dd
eNA
U −=
−+
πε
Unit in ? , 10-10 m
2. Kapustinskii equation
Improving the repulsion
)
345.0
1(
ZZ
00
0
dd
n
U −
−
=
−+
κ
κ = 1.21 MJ. ? mol-1
n= ions per formula
e.g. n= 2 for NaCl
n= 5 for Al2O3
A ? crystal lattice ? r+/r- ? r0
A/n ~constant
1. Born-Mayer equation

32. Chapter 2 Slide 32 of 85
+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 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
∆
∆
=
∆∆=∆

33. Chapter 2 Slide 33 of 85
< 0 > 0
Solubility of ionic compounds
0000
T-T- SLSHG ∆=∆∆=∆
Enthalpy of solution Hydration enthalpy
Free energy of solution

34. Chapter 2 Slide 34 of 85
Hydration enthalpy
−+
−+
+∞=∆+∆=∆
+
∞
−+
rr
HHH
rr
U
XMhydration
11
1
0






−−=∆
ε
1
1
2
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?

35. Chapter 2 Slide 35 of 85
Correlation between ∆Hsolution and the differences between the hydration
enthalpy of the ions

36. Chapter 2 Slide 36 of 85
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.

37. Chapter 2 Slide 37 of 85
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.

38. Chapter 2 Slide 38 of 85
4) An incomplete valence shell electron configuration
Noble 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 ?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

39. Chapter 2 Slide 39 of 85
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

40. Chapter 2 Slide 40 of 85
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 Al2Cl6
at room conditions.
covalency

41. Chapter 2 Slide 41 of 85
b) Size factor
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

42. Chapter 2 Slide 42 of 85
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

43. Chapter 2 Slide 43 of 85
( ) ( ) ( )g2ss3 COMOMCO +→
1360BaCO3
1289SrCO3
900CaCO3
540MgCO3
250BeCO3
Decomp. Temp. (?C)Carbonates
1923BaO
2430SrO
2613CaO
2826MgO
2530BeO
m.p. (?C)Oxides
Ionic compounds
covalency
wurtzite
structure
NaCl
structure
Some covalent
character

44. Chapter 2 Slide 44 of 85
Close-packing of Spheres
in Three Dimensions

45. Chapter 2 Slide 45 of 85
Octahedral holesTetrahedral holes Cubic holes
Close packed structure

46. Chapter 2 Slide 46 of 85
rh/r = 0.156
rh/r = 0.225
rh/r = 0.414

47. Chapter 2 Slide 47 of 85

48. Chapter 2 Slide 48 of 85

49. Chapter 2 Slide 49 of 85
Considering
anion-anion
repulsion

50. Chapter 2 Slide 50 of 85
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

51. Chapter 2 Slide 51 of 85
Rock-salt structure

52. Chapter 2 Slide 52 of 85
Unit Cell of NiAS
As3- at hcp
Ni3+ at Oh holesPolariable cation
& anion

53. Chapter 2 Slide 53 of 85
Unit Cell of Cesium Chloride
Cl- at primitive cubic
Cs+ at Cubic holes
Coord. #: Cs+: 8; Cl-: 8
atom/ unit cell
Cs: Cl= 1: 1 CsCl

54. Chapter 2 Slide 54 of 85
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 ½ Td holes

55. Chapter 2 Slide 55 of 85
Unit Cell of Hexagonal Zinc
Sulfide (Wurtzite)
S2- at hcp
Zn2+ at ½ Td holesPolymorph of ZnS

56. Chapter 2 Slide 56 of 85
Pt2+ at fcc
S2- at ½ Td holes
PtS4 unit is planar Pt-S more covalent
PtS

57. Chapter 2 Slide 57 of 85
Unit Cell of Fluorite Structure
(Calcium Fluoride)
Ca2+ at fcc
F- at Td holesCoord. #: Ca2+: 8; F-: 4
atom/ unit cell
Ca: F= 4: 8 = 1: 2 CaF2

58. Chapter 2 Slide 58 of 85
•Fluorite Structure- MX2
Large M
•Antifluorite structure- M2X
Large X, e.g. Na2O

59. Chapter 2 Slide 59 of 85
Unit Cell of Rutile TiO2
O2- at hcp
Ti4+ at ½ Oh holes
Coord. #: Ti: 6; O: 3
atom/ unit cell
Ti: O= 2: 4 = 1: 2

60. Chapter 2 Slide 60 of 85

61. Chapter 2 Slide 61 of 85

62. Chapter 2 Slide 62 of 85
A Structural Map for compounds of MX
Increasing covalency

63. Chapter 2 Slide 63 of 85

64. Chapter 2 Slide 64 of 85

65. Chapter 2 Slide 65 of 85

66. Chapter 2 Slide 66 of 85

67. Chapter 2 Slide 67 of 85
A Structural Map for compounds of MX2
Increasing covalency

68. Chapter 2 Slide 68 of 85
Salts of highly polarizing cations and easily
polarizable anions have layered structures.

69. Chapter 2 Slide 69 of 85
Salts of highly polarizing cations and easily
polarizable anions have layered structures.
CdCl2
Cl- at ccp
Cd 2+ at ½ Oh sites
(alternate O layers)
van der Waals force

70. Chapter 2 Slide 70 of 85
CdI2
I- at hcp
Cd 2+ at ½ Oh sites
(alternate O layer)

71. Chapter 2 Slide 71 of 85
MoS2
Mo4+ at hcp
S2- at Td sites
298pm
366pm

72. Chapter 2 Slide 72 of 85
Unit Cell of Perovskite
CaTiO3
AIIBIVO3
AIIIBIIIO3
Coord. #: A: 12; B: 6
atom/ unit cell
A: B: O= 1: 1: 3
A and O together at ccp
B at 1/4 Oh holes

73. Chapter 2 Slide 73 of 85
Unit Cell of ReO3
Perovskite structure CaTiO3 without Ca

74. Chapter 2 Slide 74 of 85
Unit Cell of Spinel
MgAl2O4
Normal Spinel
AII[BIII]2O4, AIV[BII]2O4 , AVI[BI]2O4
e.g. NiCr2O4, Co3O4 , Mn3O4
Inverse Spinel
B[AB]O4
e.g. Fe3O4
O 2- at fcc
A at 1/8 Td holes
B at 1/2 Oh holes

75. Chapter 2 Slide 75 of 85

76. Chapter 2 Slide 76 of 85
YBa2Cu3O7

77. Chapter 2 Slide 77 of 85
Quartz SiO2

78. Chapter 2 Slide 78 of 85

79. Chapter 2 Slide 79 of 85

80. Chapter 2 Slide 80 of 85

81. Chapter 2 Slide 81 of 85

82. Chapter 2 Slide 82 of 85
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.

83. Chapter 2 Slide 83 of 85

84. Chapter 2 Slide 84 of 85

85. Chapter 2 Slide 85 of 85