Question

A radioactive sample decays in two modes. In one mode its half life is $t_1$ and in the other mode its half life is $t_2$. Find the overall half life

• A. $t_1+t_2$
• B. $\frac{t_1+t_2}{2}$
• C. $\frac{t_1{t}_2}{t_1+t_2}$
• D. $\frac{t_1{t}_2}{t_1-t_2}$

Answer 1

Answer: (C)

$\frac{t_1{t}_2}{t_1+t_2}$

Let the decay rate of two processes be ${\lambda }_1=ln2/t_1$ and ${\lambda }_2=ln2/t_2$.

So the decay rate of quantity $N$ is given by:

$-dN\left(t\right)/dt=N{\lambda }_1+N{\lambda }_2$

$\Rightarrow N\left(t\right)=N_o{e}^{-({\lambda }_1+{\lambda }_2)t}$

$\Rightarrow N\left(t\right)=N_o{e}^{-(\frac{t_1+t_2}{t_1{t}_2})ln2t}$

So overall half life is $\frac{t_1{t}_2}{t_1+t_2}$