Question

A free neutron has a half-life of 14 minutes. Its decay constant is:
A.\[8.25 \times {10^{ - 5}}\,{{\text{s}}^{ - 1}}\]
B.\[8.25 \times {10^{ - 2}}\,{{\text{s}}^{ - 1}}\]
C.\[8.25 \times {10^{ - 4}}\,{{\text{s}}^{ - 1}}\]
D.\[8.25 \times {10^1}\,{{\text{s}}^{ - 1}}\]

Answer 1

Hint: Use the formula for the half-life of a radioactive element. This formula gives the relation between the half-life of the radioactive element and the decay constant.

Formula Used:
The half-life of a radioactive element is given by
\[{t_{\dfrac{1}{2}}} = \dfrac{{0.693}}{\lambda }\] …… (1)

Here, \[{t_{\dfrac{1}{2}}}\] is the half-life of the radioactive element and \[\lambda \] is the decay constant.

Complete step by step answer:
The half-life \[{t_{\dfrac{1}{2}}}\] of a free neutron is 14 minutes.
\[{t_{\dfrac{1}{2}}} = 14\,{\text{min}}\]

Convert the unit of half-life of the free neutron from minutes to the SI system of units.
\[{t_{\dfrac{1}{2}}} = \left( {14\,{\text{min}}} \right)\left( {\dfrac{{60\,{\text{s}}}}{{1\,{\text{min}}}}} \right)\]
\[ \Rightarrow {t_{\dfrac{1}{2}}} = 840\,{\text{s}}\]

Hence, the half-life of the free neutron is \[840\,{\text{s}}\].

Calculate the decay constant for the decay of the free neutron.

Rearrange equation (1) for the decay constant \[\lambda \].
\[\lambda = \dfrac{{0.693}}{{{t_{\dfrac{1}{2}}}}}\]

Substitute \[840\,{\text{s}}\] for \[{t_{\dfrac{1}{2}}}\] in the above equation.
\[\lambda = \dfrac{{0.693}}{{840\,{\text{s}}}}\]
\[ \Rightarrow \lambda = 0.000825\,{{\text{s}}^{ - 1}}\]
\[ \Rightarrow \lambda = 8.25 \times {10^{ - 4}}\,{{\text{s}}^{ - 1}}\]

Therefore, the decay constant is \[8.25 \times {10^{ - 4}}\,{{\text{s}}^{ - 1}}\].

Hence, the correct option is C.

Additional information:

The radioactive decay of radioactive disintegration is the process in which the nucleus of the atom spontaneously releases energy and matter.
The probability of decay per unit time is known as the decay constant of a radioactive decay.
The decay rate equation is given by
\[N = {N_0}{e^{ - \lambda t}}\]
Here, \[{N_0}\] is the initial population of radioactive elements, \[N\] is the population of the radioactive element at time and \[\lambda \] is the decay constant.
The half-life of a radioactive element is the time required for the radioactive element to decay the population of the radioactive element to half of its initial value.

Note:The unit of the half-life of the neutron is converted to second as the options for the decay constant are given in the second inverse. Otherwise, the unit of the decay constant would be minute inverse.