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Midterm I Results
Mean: 35.5 (out of 100 pts)Median: 33Mode: 25Max: 104Min: 2SD: 18
Compare to:2013 Mean: 59%2014 Mean: 51% ??
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Crystal Thermodynamics and Electronic Structure
Chapter 7
Monday, October 26, 2015
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Explaining Simple Ionic StructuresThe crystal structure adopted depends mostly on:
• The relative sizes of the atoms/ions• The relative number of the two types of atoms/ions• The electronic structure of the atoms/ions
The radius ratio is typically defined as r+/r–, where r+ is the radius of the cation and r– is the radius of the anion.
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Radius Ratio Rules
In order to maximize the net electrostatic attraction between ions in a structure, the Coordination Number of the Cation will be Maximized
subject to the criterion of Maintaining Cation-Anion Contact
Determined by comparison of the ratio of the ionic radii (r+/r-), with values derived from the geometric contact criterion
Radius Ratio RulesThe Radius Ratio Rules are correct ONLY 2/3 of the time!
stable unstabletransitional
_ _
_ _+
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Thermodynamics of Ionic Crystal FormationThe energetics associated with the formation of an ionic crystal can be calculated using Hess’ Law (Born-Haber cycle).
Li s Li g ∆Hsub
12
F2 g F g BDE
Li g Li g e IP
F g e– F g EA
Li g F g LiF s -∆Hlattice = -∆U - ∆nRT
Li s 12
F g LiF s ∆Hf
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Thermodynamics of Ionic Crystal FormationThe energetics associated with the formation of an ionic crystal can be calculated using Hess’ Law (Born-Haber cycle).
Li s Li g ∆Hsub
12
F2 g F g BDE
Li g Li g e IP
F g e– F g EA
Li g F g LiF s -∆Hlattice = -∆U - ∆nRT
Li s 12
F g LiF s ∆Hf
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Lattice EnergyIonic structures are held together by electrostatic forces and, therefore, are arranged
so that cations are surrounded by anions, and vice versa
Lattice Energy (U) = energy required to completely separate one mole of a crystal into a gas of its ions.
NaCl (s) → Na+ (g) + Cl- (g) , ΔHlattice ≈ U
2Z Z eVr
Electrostatic potential energy (attraction & repulsion):
Short-range Pauli repulsion:
nrBV
B: Born constantn: Born exponent (n = 5-12)UNaCl = 778 kJ/mol
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Madelung
....)46
38
2126(
2
reZZV
Madelung Constant (A):The numerical value of the series summation
• Depends only on the geometrical arrangements of ions (the crystal structure), not the lattice constant
Crystal structure Madelung constant CoordinationRock salt 1.748 6 : 6CsCl 1.763 8 : 8Zincblende 1.638 4 : 4Fluorite 5.039 8 : 4Rutile 4.816 6 : 3
Madelung Constant
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Born-Landé Equation for Lattice Energy
2
2
2 1
2 1
2
0
1(1 )
n
ne e
ne
e
Z Z e NA BNUr r
Z Z e NAdU nBNdr r r
Z Z e ArBn
Z Z e NAUr n
This is the Born-Landé equation (1918)
U is the sum of net Coulombic attraction and Pauli repulsion:
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Some Lattice Energies
• for a given crystal structure, U depends most stronglyon the charge on the ions
• the internuclear separation re has a smaller effect
Melting point: MgO: 2800°CCaO: 2572°CBaO: 1923°C
increasinglattice constant
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UBorn-Lande UBorn-Haber ΔU
AgF 920 953 33
AgCl 832 903 71
AgBr 815 895 80
AgI 777 882 105
Increasingly covalent!
The Born-Landé equation is a poor approximation to the lattice energy for compounds with significant non-ionic character
Rock Salt
CN 4structure
Limits of the Ionic Model
(calculation) (experiment)
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From MOs to Band TheoryWe’ve seen before how AOs on atoms are combined to give MOs for molecules.
Ha Hb
Ener
gy
H2
HOMO
LUMO
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From MOs to Band TheoryLook at what happens when we move from two hydrogens to four hydrogens in a chain (look familiar?).
H4
HOMO
LUMO
Ener
gy
H2
HOMO
LUMO
Ha Hb Hc Hd
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From MOs to Band TheoryEight hydrogens gives eight MOs and an even smaller HOMO-LUMO gap.
Ener
gy
H2
HOMO
LUMO
H4
HOMO
LUMO
H8
HOMO
LUMO
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Infinite 1D Chain of H Atoms
Interaction of many atoms:
N atomic orbitals always give N molecular orbitals.when N is enormous, the MOs form continuous bands (ΔE << kT)
band
bandEn
ergy
Density of States (DOS)
band is half fullfor 1D H atom chain
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Band Theory of 3D SolidsNow consider what happens in a three-dimensional solid with many, many atoms within bonding distance...
crystalline silicon
Ener
gy
Instead of a discrete MOs, we get a band of filled orbitals (valence band) and a band of empty orbitals (conduction band).
conduction band (empty)
valence band (filled)
band gap (Eg = 1.1 eV for silicon)
DOS