Question

A certain radioactive material can undergo three different types of decay, each with a different decay constant $\lambda ,2\lambda$ and $3\lambda$. Then, the effective decay constant ${\lambda }_{eff}$ is

• A. $6\lambda$
• B. $4\lambda$
• C. $2\lambda$
• D. $3\lambda$

Answer 1

The correct option is A $6\lambda$

Writing the radioactive decay law for each of the three individual equations:
$\frac{d{N}_1}{dt}=-\lambda N$

$\frac{d{N}_2}{dt}=-2\lambda N$

$\frac{d{N}_3}{dt}=-3\lambda N$

Adding the three equations and the radioactive decay law.
$\frac{d(N_1+N_2+N_3)}{dt}=-6\lambda N$

$\frac{dN}{dt}=-6\lambda N$

Hence the effective constant is $6\lambda$.