Question

If half-life of a radioactive atom is 2.3 days, then its decay constant would be
A. 0.1
B. 0.2
C. 0.3
D. 2.3

Answer 1

Hint: In order to solve this problem we have to apply the concept of Half life. According to it is the process during which half of the radioactive substance decay. We have to use the relation between radioactive decay constant and half life of the radioactive substance.

Formula Used:
The relation between decay constant and half-life is give by
\[\lambda = \dfrac{{0.693}}{{{T_{1/2}}}}\]
Where, \[\lambda \] = decay constant or disintegration constant and \[{T_{1/2}}\] = half-life.

Complete step by step solution:
The relation between decay constant and half-life is give by
\[\lambda = \dfrac{{0.693}}{{{T_{1/2}}}}\]
Half-life of the radioactive atom is given. Which is 2.3 days. Then the decay constant will be.
\[\lambda = \dfrac{{0.693}}{{{T_{1/2}}}} \\
\Rightarrow \lambda = \dfrac{{0.693}}{{2.3}}\]
\[\therefore \lambda = 0.3\]

Hence, the correct answer is option C.

Additional Information: The law of radioactive decay states, “If a radioactive sample contains N nuclei, at a given instant the ratio of the radioactive decay (\[ - \dfrac{{dN}}{{dt}}\]) to the number of nuclei present at that instant is a constant.”
\[\dfrac{{( - \dfrac{{dN}}{{dt}})}}{N} = \lambda \\\]
\[\Rightarrow \dfrac{{dN}}{{dt}} = - \lambda N\]
The constant \[\lambda \] is called the disintegration constant. The negative sign indicates that the number of particles (N) decreases with time.

An atom that has become stable in terms of energy by emitting radiation will no longer emit radiation. The amount of radioactive nuclei decreases over time and hence the radioactivity weakens. Radioactive decay is a statistical process. The number of nuclei remaining after n half-life is equal to \[\dfrac{{{N_0}}}{{{2^n}}}\]. Where \[{N_0}\]is the initial number of nuclei present in the sample.

Half-life is the time interval during which half of the atoms of the radioactive sample decay.
\[{T_{1/2}} = \dfrac{{0.693}}{\lambda }\]
\[{T_{1/2}}\]= Half-Life
\[\lambda \]= Disintegration constant

Note: A population of radioactive atoms' size and the rate at which they are vanishing due to radioactive decay are inversely correlated and are known as the decay constant.