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Bonding in Solids
Outline
1. Ionic bonding
2. Partial covalent bonding
3. Metallic bonding
• Ions, ionic radii
• Ionic structures
• Lattice energy of ionic crystal
• Born-Haber cycle
• Coordinated polymeric structures
• Bond valence and bond
• Non-bonding electron effects
• Band theory
• Band structure of metals, insulators and semiconductors
• Band structure of inorganic solids
General idea:
• Mixed bonding type: Ionic, covalent, van der Waals, and metallic e.g. TiO and CdI2
• Ionic bonding: high symmetry with the highest possible coordination number (larger lattice energy)
• Covalent bond: directional, prefer a certain coordination environment
• Position in periodic table… high valency vs. low valency
Bonding
Ionic Bonding
Several models have been established for the calculation of ionic radii e.g. Pauling, Goldschmidt.. Shannon and Prewitt
Ionic BondingWhat’s the size of ions in crystals?
http://www.crystalmaker.com/support/tutorials/crystalmaker/AtomicRadii.html
High quality X-ray diffraction: electron density contour map
• Ions are spherical
• Central core + outer sphere of shell
• Ions are charged, elastic and polarizable
•Assignment of radii is difficult…lead to the variation of reported values
• Flexibility in ionic radii, depends on coordination environment
Ionic Bonding
LiF
• Diagonal ion have similar ionic size
Ionic Radii
Trend in ionic radii:
• Radii increase with atomic number for any vertical group (s and p-block elements)
• Isoelectronic series of cations, radii decrease with increasing charge
• Cation radius decrease with increasing oxidation state
• Cationic radius increase with increasing coordination number
• Lanthanide contraction (d and f block elements)
• Transition vs. main group
• Ions are charged, elastic and polarizable spheres
• Ionic structures are held by electrostatic force
• Highest coordination numbers (to maximize the electrostatic attraction between ions, lattice energy)
• Reduced nearest neighbor repulsive interaction (high symmetry and maximized volume)
• Local electroneutrality
Ionic Structure_General Principles
The charge of a particular ion must be balanced by equal and opposite charge on the immediately surrounding ion
Electrostatic bond strength (ebs) of cation-anion bond:
ebs = m/n
Sum of m/n = ∑ ebs = x
For cations Mm+ surround by n anions :
For anions Ox-:
For example: MgAl2O4, structure
1. Explain the structure with bond strength
2. Predict the polyhedral linkage
Local Electroneutrality
Electrostatic valence rule:
Mg
O
Al
• Cation must be in contact with its anionic neighbors. (lower limit on cation size)
• Neighboring anions may or may not be in contact
Octahedral cation site (6CN):
> 0.414 cation push the anions apart, occupy a site of larger CN
< 0.414, occupy a site of smaller CN (not stable, rattle inside the anion polyhedron)
Ratio of Rcation / Ranion
Radius Ratio Rule
(2rx)2 + (2rx)2 = [2(rM + rx)]2
2rx(√ 2) = 2(rM + rx)
rM/rx = (√ 2) – 1 = 0.414
Cubic (8CN) Tetrahedral (4CN)
Minimum rM/rx of Cubic and Tetrahedral Structures
a = 2rx
2(√ 3)rx = 2(rM + rx)
rM/rx = (√ 3) – 1 = 0.732
2(rM + rx) = cube body diagonal
(2rx)2 + (√ 2rx)2 = [2(rM + rx)]2
2rx = face diagonal
rx(√ 6) = 2(rM + rx)
rM/rx = [(√ 6) – 2]/2 = 0.225
Radius ratio rules: qualitative guide only
Borderline case:
2. Distorted structure: BaTiO3, the reversible displacement of Ti in an applied electric field, lead to Ferroelectrictiy
1. Polymorphs: contains both tetrahedral and octahedral
Predicting Trends in CN
Lattice energy U, is the net potential energy of the arrangement of charges that forms the structure.
NaCl(s) → Na+(g) + Cl-(g) ΔH = U
Ionic structure: hold together by electrostatic forces (electrostatic repulsion and attraction in crystal)
Electrostatic Forces
Coulomb’s Law:
F = Z+Z-e2/r2
Attractive force
Charge on the ions
Electric charge
Distance between the ions
Mz+ Xz-
r
Lattice Energy of Ionic Crystals
Attraction “-” and Repulsion “+”
Coulombic potential energy:
V = -(Z+Z-e2)/r
Charge on the ions
Electric charge
Distance between the ions
Born repulsion energy:
V = B / rn
constant Distance between the ions
In the range of 5 to 12
Determination of Lattice Energy
For example: NaCl
Madelung constant A, depends on crystal structure
V1 = -6(Z+Z-e2)/r
V2 = 12(Z+Z-e2)/√2r
V3 = -8(Z+Z-e2)/√3r
V = [-(Z+Z-e2)/r][6-(12/√2)+(8/√3)-(6/√4)… ]
Lattice energy is calculated by combining the net electrostatic attraction and the Born repulsion energies and finding the internuclear separation, which gives the maximum U value.
V = -(Z+Z-e2)NA/r
Coulombic potential energy
U = -(Z+Z-e2)NA/r + BN/rn
dU/dr = 0, U = [-(Z+Z-e2)NA/re](1-1/n)
Lattice energy:
More precisely: • Born repulsive term is Be(-r/ρ)
• Zero point vibration (2.25Nhν)
• van der Waals dipole interactions (-NC/r6)
Complete equation for U:
U of divalent ions >> monovalent ions (about 4 times)
Determination of Lattice Energy
U = -(Z+Z-e2)NA/re + BNe-r/ρ - NC/r6 + 2.25Nhν
(N, A, B, e, hν are constant for a particular structure)
Kapustinskii’s Equation
When CN increase, Madelung constant A and re increase, the effect are proposed to be auto-compensated
re = rc + ra, V = number of ions per formula
To predict the stable existence of several unknown compounds and the values for ionic radii
U = [-(Z+Z-e2)NA/re](1 - ρ/re)
U = [1200.5V(Z+Z-)/(rc + ra)][1 – 0.345/(rc + ra)]
(substitute ρ=0.345, A=1.745 and values for N and e)
Na+(g) + Cl-(g) → NaCl(s) ΔH = U
Na(s) + ½ Cl2(g) NaCl(s) ΔH = ΔHf
ΔHf
Cl(g) + Na+(g)
Na(g)
Cl-(g)
½ DEA
US
IP
ΔHf = S + ½ D + IP + EA + U
Lattice energy of ionic crystal is equivalent to its heat of formation, which can not be measured experimentally.
However, heat of formation can be measured relative to the reagents in their standard states.
Born-Haber Cycle
Hess’s Law
• Check the internal consistency of data
• Predict the unknown data
• Estimate the stability of unknown compound
• Evaluate the bonding (ionic vs. covalent)
• Crystal field stabilization energies of transition metals
Stability of compound: ionization potentials, lattice energies and the relative stability of elements in different oxidation states
e.g. MgCl vs. MgCl2
Uses of Born-Haber Cycle
Covalent Bonding
Partial Covalent Bond
The outer electronic charge density on an anion is polarized towards and by neighboring cation. There is some overlapping of electron density.
For example:
A transition from ionic to covalent bond as electronegativity difference between two elements decrease
AlF3 > AlCl3 > AlBr3 > AlI3
How to quantify the degree of covalent character?
Sanderson’s Method:
Regards all bonds in non-molecular structures as polar covalent. The value of partial charges on an atom can be estimated using ascale of electronegativity. The charge on the atom is ±1 in pure ionic structure.
The positive charge that would be felt by a foreign electron on arriving at the periphery of the atom.
Valence electron are not effective in screening/shielding the outside world from the positive charge on the nucleus.
Effective Nuclear Charge
Ionization potential, electron affinity and electronegativity increase, atomic radii decrease
Cation: smaller radii (nuclear charge unchange, valence electrons are removed)
Anion: larger radii
r = rc - Bδ
Atomic Radii
constant
Partial charge
Non-polar covalent bond
• Ionic radii is controversial
• Non-polar covalent bond can be measured accurately
The electronegativity of an atom is a measure of the net attractive force experienced by an outermost electron towards nucleus
S = D/Da
Principle of electronegativity equalization:
Sb = (S1S2)1/2
Electronegativity
Electron density of the atom (atomic number/atomic volume)
Electron density that would be expected for the atom by linear interpolation of the D values for the inert gas elements
When two or more atom initially different in electronegativity combine chemically, they adjust to have the same intermediate electronegativitywithin the compound
For example, NaF Sb = (SNaSF)1/2
= (0.7×5.75)1/2
= 2.006
Partial charge δ is the ratio of change in electronegativity undergone by an atom on bond formation to the change it would undergone onbecoming completely ionic with charge + or -1
δ = ΔS/ΔSc
Determination of Partial Charge and Atomic Radii
1. ΔS = S - Sb
2. ΔSc = 2.08(S)1/2
Example BaI2: SBa = 0.78 SI = 3.84
Intermediate electronegativity = Sb = (SBaSI2)1/3 = 2.26
ΔSBa = 2.26-0.78 = 1.48; ΔSI = 3.84-2.26 = 1.58
ΔSc = 1.93 (Ba); 4.08 (I)
δ = ΔS/ΔSc = 0.78 (Ba); -0.39 (I)
rBa = rc – Bδ = 1.70 Å; rI = rc – Bδ = 1.87 Å
Ba-I distance = 1.70 + 1.87 = 3.57 Å
Mooser-Pearson Plots is good for predicting the structure based on covalent character of bonds
•Quantum number n
•Difference in electronegativity
Four structural categories:
Zinc blende, wurtzite, rock salt and CsCl
Bonding Structure
Mooser-Pearson Plots
Ionicity (fi): ionic character in bonds = C2/Eg2
Optical absorption spectra of AB compounds Eg (band gap)
Eg2 = Eh
2 + C2
(C = 0 in homopolar covalent bond, and C = Eg in pure ionic bond)
Philips-Van Vechten Ionicity
For isoelectronic series of compounds, the bandgaps have contributions from: (a) homopolar bandgap, Eh and (b) a charge transfer C
Δx (Mooser-Pearson) ~ C (Philips)
n (Mooser-Pearson) ~ Eh (Philips)
A theoretical support for Mooser-Pearson Plots
Valence Bond Theory
1. Electrostatic bond strength (ebs)
Valence of a atom (Vi) = sum of bond valence between atom and neighboring atom (∑bvij)
Covalent bond
Valence bond theory is developed for molecular materials
Ionic bond
Valence sum rule:
bv inversely correlate with bond length
• To check on the correctness of a proposed structure• To locate hydrogen atom in X-ray crystallography studies• To distinguish isoelectronic atom Al3+ and Si4+ position in aluminosilicate structure
2. Formal charge of anion
Bond valence (bv)
Valence of the atom
http://wwwchem.uwimona.edu.jm:1104/courses/CFT.html
Non-bonding Electron EffectsIn transition metal compounds, the majority of the d electrons on the metal atom do not usually take part in bond formation but doinfluence the coordination environment
Crystal Field Theory
High Spin vs. Low Spin State
The magnitude Δ depends on ligand (strong vs. weak field) and also the metal (which row, 5d > 4d > 3d). It is set equal to 10Dq.
Crystal field stabilizing energies (CFSE), leads to increase lattice energy. d0, d5 and d10 do not exhibit CFSE.
Low spin, Δ > P High spin, Δ < P
Radii in Octahedral Coordination
• d0, d5 and d10: spherically symmetrical
• Electrons in t2g orbital do not shield the bonding electron from this extra charge (geometry), while electron in Eg do shielding the nuclear charge.
• For those ions with not equally occupied Eg orbital, e.g. d9, d7
(LS) and d4 (HS).
• The Eg orbital is no longer degenerate since metal ion is not a free ion but is octahedrally coordinated.
• Doubly occupied Eg orbital experience stronger repulsion will lengthen the metal-ligand bond that leads to a lowering of energy.
Jahn-Teller Distortions
Other Coordination ModesTetrahedral
• None of the d-orbitals point directly toward ligands
• dxy, dxz, dyz are closer to the ligand than other two orbitals
Other coordination structures
e.g. Cr3+ (d3) Octahedral coordination: 1.2 Δoct
Tetrahedral coordination: 0.8 Δtet
The values of CFSE may be used to predict site preferences.
Coordination preference: Spinel structures (AB2O4)
• Normal (A-tetrahedral, B-octahedral; A = M2+, B = M3+)
• Inverse (A-octahedral, B-tetrahedral and octahedral; A = M4+, B = M2+)
• Intermediate between normal and inverse
Octahedral vs. Tetrahedral
For post transition elements
e.g. Pb2+ (4f145d106s2)
Ions are repelled by the lone bond lengthen
Lone-pair Effect
Metallic Bonding
• Metal has delocalized valence electrons
• Metallic bonding is the bonding between atoms within metals. Itinvolves the delocalized sharing of free electrons among a lattice of metal atoms
Chemical approach: molecular orbital theory
Overlap of MOs, delocalized over both atoms
Formation of energy level (delocalized over all atoms in crystal)
Metallic Bonding and Band Theory
For example, Na (1s22s22p63s1):
3s and 3p bands are overlapped at interatomic distance r0, valence electron 3s1 is not confined in 3s band.
Band Structure
Free electron theory
Density of states N(E)
• Fermi level: the highest filled level at absolute zero • Work function: the energy required to remove the uppermost valence electrons from the potential wall
Physical Approach to Band Theory
Refined Band TheoryPotential inside the crystal is regarded as periodic
So the uninterrupted continuum of energy levels does not occur, only certain bands of energies are permitted for electrons
In some materials, band overlap occurs, in others, a forbidden gap exists.
Band Structure of MetalsValence band is part full, electrons in singly occupied states close to Ef are able to move
Overlapping of s and p valence bands
Band structure of insulators
• Valence band is full
• Large forbidden gap from next energy gap, e.g. 6eV
• Very small amount of electron can be promoted to empty conduction band
Band structure of semiconductors
• Valence band is full
• Small forbidden gap from next energy gap, e.g. 0.5 – 3.0eV
• (intrinsic semiconductors) number of electrons in the conduction band is governed by magnitude of bandgap and temperature
Band Structure of Insulators and Semiconductors
Band Structure of Inorganic Structures
1. III-V, II-VI and I-VII compounds
Na+: 1s22s22p6
Cl-: 1s22s22p63s23p6
e.g. NaCl
Empty cation orbitals will form conduction band, and full anion orbitals will form valence band
Promotion of a electron from valence band to conduction band is may be regarded as back transfer of charge from Cl- to Na+. Therefore, the magnitude of the bandgap is correlated to the difference in electronegativity between anion and cation.
Eg2 = Eh
2 + C2
Homopolar bandgap
Ionic energy
2. Transition metal compounds
In some case, partly filled metal d-orbitals could overlap to form d bands
a. The formal charge on the cation is small
b. The cation occurs early in the transition series
c. The cation is in the second or third transition series
d. The anion is reasonably electropositive
General guidelines:
Also related to crystal structure:
Examples please refer to book p. 120
For example: Fe3O4 (inverse spinel) vs. Mn3O4 (normal spinel)
[Fe3+]tet[Fe2+, Fe3+]octO4 [Mn2+]tet[Mn3+2]octO4
Band Structure of Inorganic Structures
3. Fullerenes and graphite
Overlap of π−π orbitals, delocalization of π electrons
Graphite:
• infinite layers of benzene molecules
• π−π* orbitals overlap by about 0.04eV in the 3D graphite structure
Band Structure of Inorganic Structures
• Narrower valence and conduction band
• Bandgap is 2.6eV
• C60 able to form a wide range of intercalation compounds in which C60 act as an electron acceptor
Fullerene (C60)