Question

The number of atoms of a radioactive substance of half-life T is $N_0$ at t = 0. The time necessary to decay from $N_0/2$ atoms to $N_0/10$ atoms will be

• A. $\frac{5}{2}T$
• B. $Tln5$
• C. $Tln\left(\begin{array}{c}\frac{5}{2}\end{array}\right)$
• D. $T\frac{ln5}{ln2}$

Answer 1

The correct option is D $T\frac{ln5}{ln2}$

Using first order decay, $\frac{\ln 2}{T_{1/2}}t=\ln \frac{N_o}{N}$

$⟹$ $\frac{\ln 2}{T_{1/2}}(t_2-t_1)=\ln \frac{N_1}{N_2}$ where $T_{1/2}=T$

Given : $N_1=N_o/2$ $N_2=N_o/10$

$\therefore$ $\frac{\ln 2}{T}\Delta t=\ln \frac{N_o/2}{N_o/10}$

OR $\frac{\ln 2}{T}\Delta t=\ln 5$

$⟹$ $\Delta t=T\frac{\ln 5}{\ln 2}$