Question

What fraction ($\frac{n}{N}$) of the lattice sites are vacant at $298K$ for a crystal for which the energy required to make a defect?

($1eV=1.0602\times {10}^{-19}$)

Answer 1

$\frac{n}{N}=e^{-\frac{E}{2KT}}$

$n$= Number of defect per $c{m}^3$

$N$= Total number of atom per $c{m}^3$

$E$= Energy required to create defect

$K$= Boltzmann constant

$T$= Temperature in Kelvin

$\frac{n}{N}=e^{-\frac{E}{2KT}}=e^{\frac{-1.0602\times {10}^{-19}J}{2\times 1.38\times {10}^{-23}J{K}^{-1}\times 298K}}$

$=e^{-0.00128\times {10}^4}$

$\Rightarrow \frac{n}{N}=e^{-12.8}$

$\therefore ln\frac{n}{N}=ln{e}^{-12.8}=-12.8$

$\Rightarrow \log \frac{n}{N}=\frac{-12.8}{2.303}=-5.55$

$\therefore \frac{n}{N}$= Antilog$\left(\overline{5}.55\right)=2.8\times {10}^{-6}$