Bonding in Solids - National Chiao Tung...

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Bonding in Solids

• What is the chemical bonding?• Bond types:

– Ionic (NaCl vs. TiC ?)– Covalent– Van der Waals– Metallic

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Ions and Ionic Radii

LiCl

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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.

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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.

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Ions in Crystal

1.Similar to Pauling & Goldschmidt’s table.2. Based on r(O2-) and r(F-).3. Obtained from X-ray electron density map.

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Ionic Radiia) 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 Structurea) 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’selectrostatic valence rule.)

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Rutile

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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 = ½Al3+ (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

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• 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

2.25

2.16

2.16

2.19

2.56

2.56

2.27

2.27

2.10

Orthorhombic, Pnam

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PbTiO3

Tetragonal, P4mm

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Lattice Energy

NaCl(s) Na+(g) + Cl-(g)ΔH = U

a) Electrostatic forces attractionV = - (Z+Z-e2)/r

b) Short-range repulsive forcesV = B/rn

reZZFdrV

r 2−+

∞−== ∫

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Madelung constant

N : M constant

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2.6 Lattice Energy

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nrBN

rNAeZZU +−= −+

2

Therefore,1

2

+−+ −= nr

nBNr

NAeZZdr

dU

when 0=dr

dU

then

nNAreZZB

n 12 −−+=

and therefore

)11(2

nrNAeZZU

e

−−= −+

Lattice Energy I

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Van der Waals or London forces, zero-point energy, correction for heat capacity

Lattice Energy II

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Kapustinskii’s eq.1-molkJ 345.015.1200

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+

= −+

acac rrrrZVZU

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 (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 HgOsp hybridization?

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AlF3, AlCl3, AlBr3, AlI3 ionic – covalent

AlF3 AlCl3

AlBr3 (Al2Br3 unit) AlI3 (Al2I3 unit)

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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

006.2== FNab SSS

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Partial Charge

• The ratio of the change in electronegativityundergone 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/SBaSI

2 = 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

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• The radii of the partially charged atom:

Ba: rBa = rc - Bδ = 1.98 – 0.348x0.78 = 1.71 Å

rI = 1.87 Åd(Ba-I) = 3.58 Å (exp = 3.59 Å)

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• It is unrealistic and misleading to assign a radius to the chloride ion which is constant for all solid chlorides.

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• Calculations show that the actual charge carried by an oxygen never exceed -1 and is usually much less than -1.

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Mooser-Pearson plots and Ionicities

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bcc

fcc

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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.

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• Bond valences are defined empirically.• Valence rule:

Vi: valence of atom Ibvij: bond valence between atom i and j

∑=j

iji bvV

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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

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Non-bonding electron effects• d-electrons in transition metal compounds

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Ca2+ Mn2+ Zn2+

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Crystal Field Stabilization Energy

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Jahn-Teller distortions

d9 (Cu2+), d7 (LS) and d4 (HS, Cr2+)

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• Cu2+(d9), Cr2+ (d4)-MO oxides:Ti, V, Mn, Fe, Co, Ni: NaCl-typeCuO: distorted CuO6 octahedralCrO: NA

• MF2:-Ti, V, Mn, Fe, Cu, Ni, Zn: rutile-Cr, Cu: distorted rutile

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Square plannar coordination

• d8 ions: Ni2+, Pd2+, Pt2+

• Square planar coordination is more common with 4d and d transition elements.

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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. Inert pair effect

• Example: Pb+2 environment in PbTiO3

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PbTiO3

Tetragonal, P4mm

Pb

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Non Linear Optic Materials

Pb6Ti2Nb8O30Cm2m

Chem. Mater. 2004, 16, 3616-3622

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