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What percentage of the initial number of active nuclei will decay during one mean lifetime in a sample of a radioactive material?

Last updated date: 14th Jun 2024
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Hint: The mean life time of a radioactive sample is the average time taken by the nuclei to disintegrate. We can use the law of radioactive decay to determine the percentage of nuclei disintegrated from time zero to the mean life time (\[\lambda \]) of the sample.

Formula Used:
The Law of Radioactive Decay is used for finding the solution:
  & \dfrac{dN}{dt}=-\lambda N \\
 & N={{N}_{0}}{{e}^{-\lambda t}} \\
 N is the number of nuclei left after time ‘t’,
\[{{N}_{0}}\]is the initial number of radioactive nuclei,
\[\lambda \] is the disintegration constant

Complete answer:
Let us consider that there were \[{{N}_{0}}\] numbers of radioactive nuclei in the sample. At \[t=0,N={{N}_{0}}\], i.e., at the initial time, the number of radioactive nuclei left is equal to the number of given samples.
Using the law of Radioactive decay, we can understand that at a time ‘t’ the number of nuclei left is ‘N’, which is given by,
\[N={{N}_{0}}{{e}^{-\lambda t}}\]
We need to find the number of nuclei that decayed, which can be given by,
  & \text{ }{{N}^{*}}={{N}_{0}}-N \\
 & \text{ }{{N}^{*}}={{N}_{0}}-{{N}_{0}}{{e}^{-\lambda t}} \\
 & \Rightarrow \text{ }{{N}^{*}}={{N}_{0}}(1-{{e}^{-\lambda t}})\text{ ------(1)} \\
Where, \[{{N}^{*}}\]is the number of nuclei that got decayed in the mean time ‘t’.
We know that, the disintegration constant (\[\lambda \]) is the reciprocal of the mean time ‘t’.\[\Rightarrow \text{ }\] \[\lambda \text{=}\dfrac{1}{t}\text{ ------(2)}\]
Therefore, the number of nuclei decayed can be given by –
  & \Rightarrow {{N}^{*}}={{N}_{0}}(1-{{e}^{-\lambda t}}) \\
 & \Rightarrow {{N}^{*}}={{N}_{0}}(1-{{e}^{-\dfrac{1}{t}\times t}}) \\
 & \Rightarrow {{N}^{*}}={{N}_{0}}(1-{{e}^{-1}}) \\
 & \Rightarrow {{N}^{*}}={{N}_{0}}(0.63) \\
The percentage of nuclei decayed in mean time ‘t’ is 0.63 of the sample, I.e., \[63%\].

The answer is given by option C.

The mean lifetime is the average lifetime for the total radioactive sample to decay. It is the reciprocal of the disintegration or decay constant. Half-life time is the time taken for the half the number of the nuclei in the sample to be decayed. It is not equal to the mean lifetime.
The mean life is 1.44 times the half life time of a sample.