Question

If the half-lives of a radioactive element for $\alpha$ - decay and $\beta$ - decay are 4 years and 12 years respectively. What percent would its total activity be of its initial activity after 12 years?

• A. 50
• B. 25
• C. 12.5
• D. 6.25

Answer 1

The correct option is D 6.25

We know that,

$\lambda =\frac{\ln 2}{halflife}$

so decay constant for $\alpha$- decay $=\frac{\ln 2}{4}$

and decay constant for $\beta$- decay $=\frac{\ln 2}{12}$

So, total decay constant$=\frac{\ln 2}{4}$+$\frac{\ln 2}{12}$

$\Rightarrow \lambda =\frac{\ln 2}{3}$

Also we know that,

$R=R_o{e}^{-\lambda t}$

after 12 years,

$\frac{R}{R_o}=e^{-\frac{\ln 2}{3}\times 12}$

$\Rightarrow \frac{R}{R_o}=\frac{1}{16}$

for converting in percentage multiply by 100 both side,

%R=$\frac{1}{16}\times 100=6.25$%