Question

The main defect in stoichiometric CsI is the Schottky defect. The fraction of lattice sites that exhibit Schottky defects as a function of temperature, is as follows: 1.08 x 10exp-16 at 300 K, 1.06 x 10exp-12 at 400 K, 2.63 x 10exp-10 at 500K, 1.04 x 10exp-8 at 600 K, 4.76 x 10exp-6 at 900 K. Show whether the formation of Schottky defects is given by an Arrhenius (or Boltzmann) expression and, if so, evaluate the enthalpy of formation of one Schottky defect and 1 mol of Schottky defects.

Answer 1

fraction of lattice sites that exhibit Schottky defects                                Temperature (K)

1.08 X 10^-16                                                           300

1.06 X 10^-12                                                                                            400

2.63 X 10^-10                                                                                           500

1.04 X 10^-8                                                                                            600

4.76 X 10^-6                                                                                           900

Arrhenius expression for Schottky defects n_s = N exp (- ΔH / 2kT)

where n_s is the number of Schottky defects per unit volume, at T K, in a crystal with N cations and N anion sites per unit cell volume, and ΔHs is the enthalpy required to form one defect.

n_s/N represents the fraction of lattice sites exhibiting schottky defect

k = boltzmann constant = 1.38 X 10^-23 J/K

1.08 X 10^-16 = e^{( -} ΔH/2* 1.38 X 10^-23 * 300 )      -----------(1)

1.06 X 10^-12 = e^{( -} ΔH/2* 1.38 X 10^-23 * 400 )     ------------(2)

2.63 X 10^-10 = e^{( -} ΔH/2* 1.38 X 10^-23 * 500 )     -------------(3)

1.04 X 10^-8 = e^{( -} ΔH/2* 1.38 X 10^-23 * 600 )       ------------(4)

4.76 X 10^-6 = e^{( -} ΔH/2* 1.38 X 10^-23 * 900 )       ------------(5)

Dividing equations 1 and 2 we get

ΔH = 304.42 X 10^-21 J

Dividing equations 3 and 4 we get

ΔH = 304.42 X 10^-21 J

The fraction of schottky sites present at temperature T is obeying Arrhenius Expression

Enthalpy of formation of one Schottky defect = ΔH = 304.42 X 10^-21 J

Enthalpy of formation of 1 mol of Schottky defect = ΔH X N_A = 304.42 X 10^-21 J X 6.023 * 10²³ = 1833.522 X 10²

                                                                                        = 183.35 KJ