Bonding in Solids
• What is the chemical bonding?
• Bond types:
– Ionic (NaCl vs. TiC ?)
– Covalent
– Van der Waals
– Metallic
1
Ions and Ionic Radii
LiCl
2
Ions
(a) Ions are essentially spherical.
(b) Ions may be regarded as composed of two
parts: a central core in which most of the
electron density is concentrated and an outer
sphere of influence which contains very little
electron density.
(c) Assignment of radii to ions is difficult; even
for ions which are supposedly in contact, it is not
obvious where one ion ends and another begins.
3
Where is the border bet. M+ and N- ?
1. Ions are charged, but they cannot be regard
as hard sphere.
2. The e- density does not decrease abruptly to
zero.
4
Ions in Crystal
1.Similar to Pauling &
Goldschmidt’s table.
2. Based on r(O2-)
and r(F-).
3. Obtained from Xray electron density
map.
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Ionic Radii
a) s- and p-block elements: radii increase with
atomic number for vertical group
b) For isoelectronic cations, radii decrease with
increassing charge.
c) For element which can have > +1 oxidation
state, the radius decreases with increasing
oxidation state, e.g. V2+, V3+, V4+, V5+.
d) For an element which can have various
coordination numbers, the cationic radius
increases with increasing coordination number.
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Ionic Radii cont.
e) Lanthanide contraction (due to ineffective
shielding of the nuclear charge by the d and,
especially, f electrons), e.g. La3+ (1.20 A) --Eu3+
(1.09 A)--Lu3+ (0.99 A).
f) The radius of a particular transition metal ion is
smaller than that of the corresponding main
group ion for the reasons given in (e), e.g.
octahedral radii, Rb+(1.63 A) and Ag+ (1.29 A)
or Ca2+ (1.14 A) and Zn2+ (0.89 A).
g) Diagonal relationship: Li+ (0.88 A) and Mg+2
(0.86 A).
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2.3 Ionic Structure
a) Ions are charged, elastic and polarizable
spheres.
b) Ionic structures are held together by
electrostatic forces and are arranged so that
cations are surrounded by anions, and vice
versa.
c) To maximize the net electrostatic attraction
between ions (i.e. the lattice energy),
coordination numbers are as high as possible,
provided that the central ion 'maintains contact'
with its neighboring ions of opposite charge.
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Ionic Structure cont.
(d) Next nearest neighbor interactions are of the
anion-anion and cation - cation type and are
repulsive. Like ions arrange themselves to be as
far apart as possible and this leads to structures
of high symmetry with a maximized volume.
(e) The valence of an ion is equal to the sum of the
electrostatic bond strengths between it and
adjacent ions of opposite charge. (Pauling’s
electrostatic valence rule.)
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Rutile
10
Electrostatic bond strength
• For a cation Mm+ surrounded by n anions,
Xx-
ebs = m/n
∑(m/n) = x
• MgAl2O4:Mg2+ (Td site) ebs = 2/4 = ½
Al3+ (Oh site) ebs = 3/6 = ½
Oxygen charge ∑ebs(3Al3+ + 1Mg2+) = 2
(Each O atom is surrounded by three Al3+
and one Mg2+ cations, the observed
charge is equal to oxygen’s charge)
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• SiO4 cannot share a common corner in
silicate structures:
• Si4+: ebs = 4/4 = 1
• Two Td corner: ebs = 2 (O atom connect
to two Si atoms)
• Three Td corner: ebs = 3 (unreasonable)
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The radius ratio rule
(2rx)2 + (2rx)2
=[2(rM+rx)]2
2rx√2 = 2 (rM+rx)
rM/rx = 0.414
14
15
• A cation must be in contact with its anionic
neighbors.
• Neighboring anions may or may not be in
contact.
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Borderline Radius & Distorted Structures
V2O5
Orthorhombic, Pnma
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ZrO2
2.27
2.16
2.56
2.19
2.25
2.10
2.27
2.16
2.56
Orthorhombic, Pnam
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PbTiO3
Tetragonal, P4mm
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Lattice Energy
NaCl(s) Æ Na+(g) + Cl-(g)
ΔH = U
a) Electrostatic forces attraction
2
r
Z
Z
e
V = - (Z+Z-e2)/r V = ∫ Fdr = − + −
∞
r
b) Short-range repulsive forces
V = B/rn
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Madelung constant
N : M constant
22
2.6 Lattice Energy
23
Lattice Energy I
Z + Z − e 2 NA
BN
U = −
+ n
r
r
dU
Therefore,
dr
when
then
dU
dr
Z + Z − e 2 NA
nBN
=
−
r
r n +1
= 0
Z + Z − e 2 NAr
B=
n
n −1
and therefore
1
Z + Z − e 2 NA
(1 − )
U =−
re
n
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Lattice Energy II
Van der Waals or London forces, zero-point energy, correction for heat capacity
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Kapustinskii’s eq.
1200.5VZ + Z −
U=
rc + ra
⎛
0.345 ⎞
⎜⎜1 −
⎟⎟ kJ mol -1
rc + ra ⎠
⎝
V: # of ions per formula unit
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Born-Haber Cycle
ΔHf
= S + (1/2)D + IP + EA + U (could be estimated)
= 109 + 121 + 493.7 + (-356) + (-764.4)
= -410.9 kJ mol-1
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The Synthesis of XePtF6
• XePtF6 was first synthesized by Barlett at
1962.
• The idea for this compound is from the
formation of O2PtF6 Æ (O2)+(PtF6)• The 1st IE of O2 (1169 kj/mol) and Xe
(1176 kj/mol) are similar!
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Stabilities of Real and Hypothetical ionic compounds
S
0.5D
IP
EA
U
ΔHf (calc)
NaCl
-764.4 kJ mol-1
KCl
-701.4
ArCl
0
121
1524
-356
-754
544
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Partial Covalent Bonding
SrO
HgO
sp hybridization?
32
AlF3, AlCl3, AlBr3, AlI3
ionic – covalent
AlF3
AlBr3 (Al2Br3 unit)
AlCl3
AlI3 (Al2I3 unit)
33
Sanderson’s Model
• To calculate partial charge of atom
• To evaluate ionic and covalent bonding for
total energy of ionic compounds
– Effective nuclear charge
– Atomic radii (r = rc - Bδ; δ = ΔS/Sc)
– Electronegativity and charged atoms
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Effective Nuclear Charge
• The positive charge that would be felt by a
foreign electron on arriving at the
periphery of the atom.
• The valence electrons are not very
effective in shielding the outside world
from the positive charge on the nucleus.
Therefore, any incoming e- feels a positive,
attractive charge.
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Screening constants
• The value of screening constants in
different elements could be obtained
theoretically.
• The valence electron experience an
increasingly strong attraction to the
nucleus on going from sodium (Na) to
chlorine (Cl).
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Atomic Radius
• Atomic radii vary considerably for a
particular atom depending on bond type
and CN.
• With increasing amount of partial positive
charge, the radii become smaller
• Sanderson’s model:
r = rc - Bδ
rc: non-polar covalent radius; δ: partial
charge (estimated)
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Electronegativity and Partially Charged Atoms
• The magnitude of the partial charge
depends on the initial difference in
electronegativity between the two atoms.
• Sanderson’s model for electronegativity:
S = D/Da
D: electron density of the atom (atomic
number/atomic volume)
Da: expected electron density
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Electronegativity Equalization
• When two or more atom initially different in
electronegativity combine chemically, they
adjust to have the same intermediate
electronegativity within the compound.
• NaF: geometric mean of their χ
Sb = S Na S F = 2.006
40
Partial Charge
• The ratio of the change in electronegativity
undergone by an atom on bond formation
to the change it would have undergone on
becoming completely ionic with charge +
or -1.
• NaF: Assume 75% ionic
∆Sc = 2.08/S (changes in χ)
Partial charge δ = ∆S/ ∆Sc
∆S = S - Sb
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Example: BaI2
• SBa = 0.78; SI = 3.84
• Sb = 3/SBaSI2 = 2.26
•
•
•
•
•
•
For Ba, ∆S = 2.26 – 0.78 = 1.48
For iodine ∆S = 3.84 – 2.26 = 1.58
∆Sc: 1.93 (Ba), 4.08(I) (from tab. 2.10)
δBa = 1.48/1.93 = 0.78
δI = 1.58/4.08 = -0.39
The result suggest BaI2 is ~ 39% ionic
42
• The radii of the partially charged atom:
Ba: rBa
= rc - Bδ = 1.98 – 0.348x0.78
= 1.71 Å
rI = 1.87 Å
d(Ba-I) = 3.58 Å (exp = 3.59 Å)
43
• It is unrealistic and misleading to assign a
radius to the chloride ion which is constant
for all solid chlorides.
44
• Calculations show that the actual charge
carried by an oxygen never exceed -1 and
is usually much less than -1.
45
Mooser-Pearson plots and Ionicities
46
bcc
fcc
47
Bond Valence and Bond Length
• Most molecular materials may be
described satisfactorily using valence
bond theory.
• For non-molecular inorganic materials, the
VBT is not always fit.
• Pauling, Brown, Shannon, Donnay et. al. :
Bond order (bond valence) in a structure.
48
• Bond valences are defined empirically.
• Valence rule:
Vi = ∑ bvij
j
Vi: valence of atom I
bvij: bond valence between atom i and j
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Applications
•
•
•
•
To check the valence state of cation atom.
To locate the position of H+
To distinguish between Al3+ and Si4+
Transition metals in oxide compounds
51
Non-bonding electron effects
• d-electrons in transition metal compounds
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53
54
Ca2+ Æ Mn2+ Æ Zn2+
55
Crystal Field Stabilization Energy
56
Jahn-Teller distortions
d9 (Cu2+), d7 (LS) and d4 (HS, Cr2+)
57
• Cu2+(d9), Cr2+ (d4)
-MO oxides:
Ti, V, Mn, Fe, Co, Ni: NaCl-type
CuO: distorted CuO6 octahedral
CrO: NA
• MF2:
-Ti, V, Mn, Fe, Cu, Ni, Zn: rutile
-Cr, Cu: distorted rutile
58
Square plannar coordination
• d8 ions: Ni2+, Pd2+, Pt2+
• Square planar coordination is more
common with 4d and d transition elements.
59
Tetrahedral Coordination
The magnitude of the splitting is generally less in a tetrahedral field.
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Inert Pair Effect
• Heavy, post-transition metals: Tl, Sn, Pb,
Sb, Bi
• These metals usually exhibit a valence
state that is two less than the group
valence. Æ Inert pair effect
• Example: Pb+2 environment in PbTiO3
62
PbTiO3
Pb
Tetragonal, P4mm
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Non Linear Optic Materials
Pb6Ti2Nb8O30
Cm2m
Chem. Mater. 2004, 16, 3616-3622
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