Question

A radioactive sample can decay by two different processes. The half-life for the first process is $T_1$ and that for the second process is $T_2$ . What will be the effective half life of the radioactive sample?

Answer 1

Use the relation between decay constant
Also half-life to substitute for decay constant in the radioactive decay equation.

Formula used:
$T_{\dfrac{1}{2}} = \dfrac{{\ln 2}}{\lambda }$
Here, $T_{\dfrac{1}{2}}$ is the half-life and $\lambda$ is the decay constant.

Complete step by step answer:
A rate of radioactive decay of a sample is directly proportional to the actual number of particles remaining in the substance at that instant of time t.
$\dfrac{{dN}}{{dt}} = - \lambda N$

Here, $\lambda$ is the proportionality constant and it is known as decay constant. The negative sign represents the substance undergoes decay over a course of time t.

The term half-life represents the time required for a half of the nuclei in the sample to undergo the decay.

The parameter half-life is related to the decay constant $\lambda$by the equation,$T_{\dfrac{1}{2}} =\dfrac{{\ln 2}}{\lambda }$

$\Rightarrow \lambda = \dfrac{{\ln 2}}{{T_{\dfrac{1}{2}} }}$

Since the radioactive sample decay by two processes, the effective rate of decay is,
$\dfrac{{dN}}{{dt}} = - \lambda N$ ...... (1)

Here, $\lambda = \lambda _1 + \lambda _2$.
Therefore,
$\dfrac{{dN}}{{dt}} = - \left( {\lambda _1 + \lambda _2 } \right)N$ ...... (2)

Compare equation (1) and (2), we can write,
$- \left( {\lambda _1 + \lambda _2 } \right)N = - \lambda N$

$\Rightarrow \left( {\lambda _1 + \lambda _2 } \right)N = \lambda N$ ...... (3)

The decay constant for the first process is,
$\lambda _1 = \dfrac{{\ln 2}}{{T_1 }}$ ...... (4)
And, the decay constant for the second process is,
$\lambda _2 = \dfrac{{\ln 2}}{{T_2 }}$ ...... (5)

The effective decay constant for the two processes is,
$\lambda = \dfrac{{\ln 2}}{T}$ ...... (6)

Substitute equations (4), (5), and (6) in equation (3).
$\left( {\dfrac{{\ln 2}}{{T_1 }} + \dfrac{{\ln 2}}{{T_2 }}} \right)N = \dfrac{{\ln 2}}{T}N$
$\Rightarrow \dfrac{1}{{T_1 }} + \dfrac{1}{{T_2 }} = \dfrac{1}{T}$

So, the correct answer is “Option B”.

Note:
sometimes students do not consider the negative sign in radioactive decay equation

\dfrac{{dN}}{{dt}} = - \lambda N$$. It will affect the answer in other problems on radioactive decay.