Question

A radioactive substance decays to $1/{16}^{th}$ of its initial mass in $40$ days. The half life of the substance, in days, is?

• A. $20$
• B. $10$
• C. $5$
• D. $2.5$

Answer 1

The correct option is B $10$

activity of radioactive substance at any time t is given by

$\Rightarrow N=N_0{e}^{-\lambda t}$

given that $\frac{N}{N_0}=\frac{1}{16}$ at time $t=40days$

$\Rightarrow \frac{1}{16}=e^{-40\lambda }\Rightarrow e^{40\lambda }=16\Rightarrow \lambda =\frac{\ln 16}{40}$

we know that

$t_{\frac{1}{2}}=\frac{\ln 2}{\lambda }$

after putting the value of $\lambda$ in above equation

$\Rightarrow t_{\frac{1}{2}}=\frac{\ln 2}{\ln 16}\times 40=10$

so the correct option will be $\left(B\right)$.