Question

The decay constant of radioactive substances is \[4.33 \times {10^{ - 4}}\] per year. Calculate its half life period.

Answer 1

Hint Radioactive decay is a process in which spontaneous breakdown of an atomic nucleus of a radioactive substance occurs which results in the emission of radiation from the nucleus.
Half-Life is basically the time needed by a radioactive substance (or one half the atoms) to disintegrate or decay into another substance.
It is given by ${t_{{1{\left/
 { {1 2}} \right.
} 2}}} = \dfrac{{\ell n2}}{\lambda }$ where $\lambda $ is the decay constant of the radioactive substance.
Complete step by step answer Let us first discuss the process of radioactive decay.
Radioactive decay is a process in which spontaneous breakdown of an atomic nucleus of a radioactive substance occurs which results in the emission of radiation from the nucleus. The radioactive nucleus which undergoes decay in a radioactive process is known as parent nucleus and this parent nucleus produces a daughter nucleus in the radioactive process. It is given by the formula
$N = {N_0}{e^{ - \lambda T}}$ where ${N_0}$ is the initial amount of the radioactive substance, $N$ is the amount of that substance remains after time $T$ and $\lambda $ is the decay constant of the radioactive substance.
Now, Half-Life is basically the time needed by a radioactive substance (or one half the atoms) to disintegrate or decay into another substance.
So, if we put $N = \dfrac{{{N_0}}}{2}$ in the above equation of radioactive decay, we will get the expression for half life period. Therefore we get
${t_{{1{\left/
 { {1 2}} \right.
} 2}}} = \dfrac{{\ell n2}}{\lambda }$
Now, according to the question, decay constant of radioactive substance is given as \[\lambda = 4.33 \times {10^{ - 4}}{\text{ per year}}\] . So, substituting this value in the above equation we have
${t_{{1{\left/
 { {1 2}} \right.
} 2}}} = \dfrac{{0.693}}{{4.33 \times {{10}^{ - 4}}}} = 1.6 \times {10^3}{\text{ years}}$

Hence, half life period of the substance is $1.6 \times {10^3}{\text{ years}}$ .

Note The radioactive decay is a random process, i.e. we are not able to predict the decay of individual atoms. Radioactive isotopes normally decay to harmless substances. Some isotopes decay in hours or even minutes but others decay very slowly and some decay in years.
The principle of the half-life period was first discovered by Ernest Rutherford in 1907.