Question

If the half lives of a radioactive element for $\alpha$ and $\beta$ decay are 4 years and 12 years respectively, the ratio of its inital activity and that after 12 years will be:

• A. 6.25%
• B. 12.5%
• C. 25%
• D. 50%

Answer 1

Answer: (A)

6.25%

The decay constant for $\alpha$ decay $=k_1=\frac{ln2}{4}$

The decay constant for $\beta$ decay $=k_2=\frac{ln2}{12}$
The radioactive element decays both ways simultaneously.

Thus, $\frac{dN}{dt}=-(k_1+k_2)N$

$⟹N=N_0{e}^{-(k_1+k_2)t}$

The ratio of the activities at $t=12$ and at $t=0$ is $\frac{(k_1+k_2)N}{(k_1+k_2)N_0}$$=\frac{N}{N_0}=e^{-(k_1+k_2)\left(12\right)}$$=0.0625=6.25$%