2. TiO₂ (rutile) is a wide band gap semiconductor that can use ultraviolet light to catalyze the decomposition of water. TiO₂ has a melting point of 1843 ^oC, has a conductivity comparable to very lightly doped silicon, and is insoluble in water.
a) Consider an ionic model. Estimate the lattice energy using the Born-Landé equation. The Born exponent can be estimated using the electron configuration (see the table below) and a weighted average for each ion in the lattice. Calculate the experimental lattice energy using a Born-Haber cycle (the sublimation energy of Ti is 473 kJ/mole, the heat of formation of TiO₂ is -944.0 kJ/mol, and EA₂ for oxygen is -779.6 kJ/mol). How well do the experimental and theoretical estimates match? Estimate the solvation energy in water for each ion (ε = 78). Is this consistent with the solubility of TiO₂ in water? Why or why not? How can an ionic model account for the semiconducting properties?

electron configuration	n
[He]	5
[Ne]	7
[Ar]	9
[Kr]	10
[Xe]	12

b) Consider a covalent model. Write the Lewis dot structure for TiO₂. Is this an appropriate bonding description? Why or why not? Within this model, which orbitals are responsible for creating the valence band? The conduction band?

c) Which model better accounts for the properties of TiO₂? Defend your decision.

a) the Born-Landé equation is

N₀ = 6.022×10²³
M = 2.408 for the rutile structure
Z⁺ = +4
Z^- = -2
e = 1.602×10^-19 C
π = 3.14159
ε₀ = 8.854×10^-12 C²J^-1m^-1
r₀ ~ r₊ + r_- = 74.5 + 126 pm = 2.005×10^-10 m
n ~ [9 + 2(7)]/3 = 7.67 (Ti⁴⁺ has the [Ar] configuration and O^2- has the [Ne] configuration)

E_lat = 11610 kJ/mol

A Born-Haber cycle requires the following reactions:

Ti(s) + O₂(g) → TiO₂(s) ΔH^o_f = -944.0 kJ/mol
Ti(s) → Ti(g) S = +473kJ/mol
Ti(g) → Ti⁴⁺(g) I = IP₁+IP₂+IP₃+IP₄= (6.82+13.58+27.491+43.266)×96.4869 = 8795 kJ/mol
O₂(g) → 2O(g) BDE = 493.59 kJ/mol
2O(g) → 2O^2-(g) E = 2(141.0-779.6) = -1277.2 kJ/mol
Ti⁴⁺(g) + 2O^2-(g) → TiO₂(s) E_lat

-ΔH^o_f + S + I + BDE -E - E_lat = 0
E_lat = -(-944.0) + 473 + 8795 + 493.59 -(-1277.2) = 11983 kJ/mol

Solvation energy is estimated by 

For Ti⁴⁺

For O^2-

By this estimate, there is more than enough solvation energy to overcome the lattice energy so the observed solubility is not consistent with the calculation. The problem is that Ti⁴⁺ reacts with water to form oxo and hydroxo species.

Within an ionic model, semiconduction must be attributed to lattice defects such as reduction of a few of the Ti⁴⁺ ions to Ti³⁺ (with the appropriate number of vacancies to account for charge balance). The extra electron found on the Ti³⁺ predicts that TiO₂ would be an n-type semiconductor.

b) Lewis structure:

This is probably a reasonable description of the bonding in gas phase TiO₂ but does not account for the high stability in the solid phase. Within this description, the filled valence band would probably be constructed from π bonding orbitals and the empty conduction band from π* antibonding orbitals. This predicts TiO₂ to be an intrinsic semiconductor.

c) Given the high melting point, the ionic description is more apt. The low solubility clearly indicates covalent contributions, however. The fact that the band gap is in the UV (a large value) implies that as an intrinsic semiconductor the conductivity would be much lower than that of silicon. Thus, the ionic description is favored to account for the semiconductor properties.