Question

Two radioactive materials Y₁ and Y₂ have decay constants '5${\displaystyle \lambda }$' and ${\displaystyle \lambda }$ respectively. Initially they have same number of nuclei. After time 't', the ratio of number of nuclei of Y₁ to that of Y₂ is ${\displaystyle \frac{1}{\text{e}}}$, then 't' is equal to ______.

विकल्प

• ${\displaystyle \frac{\lambda }{2}}$

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

• ${\displaystyle \frac{1}{4\lambda }}$

• ${\displaystyle \lambda }$

Answer 1

Two radioactive materials Y₁ and Y₂ have decay constants '5${\displaystyle \lambda }$' and ${\displaystyle \lambda }$ respectively. Initially they have same number of nuclei. After time 't', the ratio of number of nuclei of Y₁ to that of Y₂ is ${\displaystyle \frac{1}{\text{e}}}$, then 't' is equal to ${\displaystyle \underline{\frac{1}{4\lambda }}}$

Explanation:

${\displaystyle {\text{N}}_1={\text{N}}_0{\text{e}}^{-5\lambda \text{t}}, {\text{N}}_2={\text{N}}_0{\text{e}}^{-\lambda \text{t}}}$

${\displaystyle \frac{1}{\text{e}}={\text{e}}^{-\lambda \text{t}}=\frac{1}{{\text{e}}^{4\lambda \text{t}}}}$

${\displaystyle \therefore 4\lambda \text{t}=1}$

${\displaystyle \therefore \text{t}=\frac{1}{4\lambda }}$