Question

The ratio of the number of vacancies to the number of atoms when the average energy required to create a vacancy is 0.9 eV at 400K is

• A. \( \frac{n}{N} = 4.68 \times 10^{-10} \)
• B. \( \frac{n}{N} = 4.68 \times 10^{-12} \)
• C. \( \frac{n}{N} = 4.68 \times 10^{-13} \)
• D. \( \frac{n}{N} = 4.68 \times 10^{-14} \)

Hint:

Higher temperatures increase the ratio of vacancies due to increased thermal energy overcoming the vacancy formation energy.

Answer 1

The Correct Option is C

The ratio of vacancies to atoms is calculated using:
\[\frac{n}{N} = e^{-\frac{E_v}{kT}}\]
where \(E_v = 0.9 \, \text{eV}\), \(k = 8.617 \times 10^{-5} \, \text{eV/K}\), and \(T = 400 \, \text{K}\). Substituting the values:
\[\frac{n}{N} = e^{-\frac{0.9}{8.617 \times 10^{-5} \times 400}} \approx 4.68 \times 10^{-13}\]