Question

For a radioactive material, half-life is $10\min $ . If initially there are $600$ number of nuclei, the time taken (in minutes) for the disintegration of $450$ nuclei is?
(A) 30
(B) 20
(C) 15
(D) 10

Answer 1

Hint: Half life is denoted by \[{t_{1/2}}\] and is defined as the time required for a quantity to reduce to half of its initial value. Also, the exponential decay process of a radioactive material is given by \[N = {N_0}{\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{{{t_{1/2}}}}}}\] .

Formulas used: We will be using the formula \[N = {N_0}{\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{{{t_{1/2}}}}}}\] to find the exponential decay that the radioactive material will undergo, where $N$ is the quantity of radioactive material that remains(not yet decayed) by the end of time $t$ , ${N_0}$ is the initial amount of radioactive material that is present before the decay starts to occur, and ${t_{1/2}}$ is the halftime of the radioactive substance, which is nothing but the time required to reduce the quantity of the radioactive substance to half of its initial value.

Complete step by step answer:
Materials that contain unstable nuclei are called radioactive. Thus, the process of losing this unstable nucleus or the process by which the unstable nuclei loses its energy is called radioactive decay. The calculation of radioactive decay process is laid on the foundation of Universal law of radioactivity that states that the nucleus of a radionuclide has so much memory and hence the breaking down of such nuclei does not increase with time and instead is carried out at a constant rate.
That is exactly how we have a formula to find the exponential decaying of radioactive materials. Here we know that the number of nuclei initially present are ${N_0} = 600$ and that the half-time is ${t_{1/2}} = 10\min $, using this data we are required to find the time taken to decay $N' = 450$ nuclei.
So, let us find the number of nuclei that would be left after the disintegration, $N = N' - {N_0}$
$ \Rightarrow N = 600 - 450 = 150$ . So, there would be 150 nuclei left after the disintegration is complete. Now using the formula for finding exponential decay, \[N = {N_0}{\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{{{t_{1/2}}}}}}\] .
Substituting the terms, we get,
$ \Rightarrow 150 = 600{\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{{10}}}}$
Dividing both sides by $600$ we get,
$\dfrac{{150}}{{600}} = \dfrac{1}{4} = {\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{{10}}}}$
Now comparing the L.H.S and R.H.S,
${\left( {\dfrac{1}{2}} \right)^2} = {\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{{10}}}}$
Eliminating the bases and solving the exponents, $\dfrac{t}{{10}} = 2$, and that gives us $t = 20\min $ .
So, the time needed to decay 450 nuclei is $t = 20\min $.

So the correct answer is option B.

Note:
These properties of radioactivity are not experienced by all the elements. Only a few naturally occurring elements like uranium, thorium, plutonium and radium are radioactive and disintegrate over time. The radioactivity property is not seen in any of the other naturally occurring metals.