Question

The enthalpy required to create an oxygen vacancy in CeO$_2$ is 4 eV. The number of oxygen vacancies present per mole of CeO$_2$ at 1000 K is _________. (Round off to the nearest integer) Given: $N_A = 6.02 \times 10^{23}$ mole$^{-1}$ $k_B = 8.62 \times 10^{-5}$ eV/K

Hint:

Defect concentrations in solids often follow the Boltzmann factor: $\exp(-E/k_BT)$, where $E$ is defect formation energy.

Answer 1

Correct Answer: 4232

The number of vacancies formed follows the Boltzmann distribution: \[ n = N_A \exp\left(-\frac{E}{k_B T}\right) \] Given: \[ E = 4\ \text{eV}, \qquad T = 1000\ \text{K} \] Compute the exponential term: \[ \frac{E}{k_B T} = \frac{4}{(8.62 \times 10^{-5})(1000)} = \frac{4}{0.0862} = 46.41 \] Thus: \[ \exp(-46.41) = 7.04 \times 10^{-21} \] Number of vacancies per mole: \[ n = (6.02 \times 10^{23})(7.04 \times 10^{-21}) \] \[ n = 4.24 \times 10^{3} \] \[ \boxed{4240} \]