Question

A radioactive element disintegrates for an interval of time equal to its mean lifetime. The fraction that has disintegrated is ______

Options

• ${\displaystyle \frac{1}{\text{e}}}$

• ${\displaystyle \frac{\text{e}-1}{\text{e}}}$

• ${\displaystyle \frac{\text{e}}{\text{e}-1}}$

• ${\displaystyle \frac{\text{e}}{2}}$

Answer 1

A radioactive element disintegrates for an interval of time equal to its mean lifetime. The fraction that has disintegrated is ${\displaystyle \underline{\frac{\text{e}-1}{\text{e}}}}$.

Explanation:

Mean life time ${\displaystyle \tau =\frac{1}{\lambda }}$

Now, N = ${\displaystyle {\text{N}}_0{\text{e}}^{-\lambda \text{t}}}$

∴ For t = τ = ${\displaystyle \frac{1}{\lambda }}$

${\displaystyle \frac{\text{N}}{{\text{N}}_0}={\text{e}}^{-\lambda \times \frac{1}{\lambda }}=\frac{1}{\text{e}}}$

∴ The fraction that has disintegrated is given as,

${\displaystyle 1-\frac{\text{N}}{{\text{N}}_0}=1-\frac{1}{\text{e}}=\frac{\text{e}-1}{\text{e}}}$