Question

A radioactive sample has initial activity of 28 dpm, 30 minutes later its activity is 14 dpm. How many atoms of nuclide were present initially?
A.2800
B.1212
C.528
D.2802

Answer 1

Hint: To answer this question we must have the knowledge of radioactive equations. Radioactive equations follow first order kinetics. Using the given data we will calculate the half life of radioactive and hence using initial activity we will calculate the initial number of nuclei.

Formula used: \[\lambda = \dfrac{{0.693}}{{{{\text{t}}_{1/2}}}}\] where \[\lambda \] is Decay constant \[{{\text{t}}_{1/2}}\] is half life of radioactive.
\[{\text{Activity = }}\lambda {{\text{N}}_{\text{o}}}\] Here \[{{\text{N}}_{\text{o}}}\] is initial number of nuclide.

Complete step by step solution:
Half life of a radioactive substance is defined as the time when half of the reactant had been reacted. Since it is given to us in the question that alter 30 minutes the activity of the radioactive sample changes from 28 dpm to 14 dpm that is reduced to half. So half life is 30 minutes.
Using the half like we can calculate the decay constant:
\[\lambda = \dfrac{{0.693}}{{{{\text{t}}_{1/2}}}}\]
Substituting the value of half life,
$\Rightarrow$ \[\lambda = \dfrac{{0.693}}{{30{\text{ min}}}}\]
After doing the calculation we get, \[\lambda = 0.0231{\text{ mi}}{{\text{n}}^{ - 1}}\].
Now the initial activity is given to us, and we have calculated the decay constant using the formula we can calculate the initial number of nuclei as:
\[{\text{Activity = }}\lambda {{\text{N}}_{\text{o}}}\]
$\Rightarrow$\[{\text{28 dpm = 0}}{\text{.0231}}{{\text{N}}_{\text{o}}}\]
Rearranging the equation as:
$\Rightarrow$\[{{\text{N}}_{\text{o}}} = \dfrac{{28{\text{ dpm }}}}{{0.0231{\text{ mi}}{{\text{n}}^{ - 1}}}}\]
We will get the initial number of nuclei as: 1212.

Hence the correct option is B.

Note: Actually the radioactive decay is the same as the first order reaction that we do in chemical kinetics. The only change is rate constant is replaced by decay constant and rate of reaction is replaced by activity. DPM is the abbreviation for disintegration per minute. It defines the amount of radioactive disintegrations per minute.