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Question: The rate of radioactive disintegration at an instant for a radioactive sample of half life \[2.2\tim...

The rate of radioactive disintegration at an instant for a radioactive sample of half life 2.2×109s2.2\times {{10}^{9}}sis 1010s{{10}^{10}}s. The number of radioactive atoms in that sample at that instant is?
A. 3.17×10193.17\times {{10}^{19}}
B. 3.17×10203.17\times {{10}^{20}}
C. 3.17×10173.17\times {{10}^{17}}
D. 3.17×10183.17\times {{10}^{18}}

Explanation

Solution

Radioactivity refers to the phenomenon in which the substance decays by emission of radiation. Half-life is defined as the time taken by the radioactive substance in which the number of atoms undecayed becomes half of the atoms that were taken initially. A material containing unstable nuclei is considered to be a radioactive material.

Complete step by step answer:
Given the half life, T1/2{{T}_{1/2}}= 2.2×109s2.2\times {{10}^{9}}s
We know the relationship between half life and decay constant is T1/2λ=0.693{{T}_{1/2}}\lambda =0.693
T1/2λ=0.693 ⇒2.2×109×λ=0.693 ⇒λ=0.6932.2×109 ⇒λ=3.15×10−10s−1 {{T}_{1/2}}\lambda =0.693 \\\ \Rightarrow 2.2\times {{10}^{9}}\times \lambda =0.693 \\\ \Rightarrow \lambda =\frac{0.693}{2.2\times {{10}^{9}}} \\\ \Rightarrow \lambda =3.15\times {{10}^{-10}}{{s}^{-1}} \\\
Now activity of a radioactive substance is related to the half life as: R=λNR=\lambda N
Here R is the disintegrations per second taking place of the radioactive material.
Putting the given values,
1010=3.15×10−10×N ∴N=3.17×1019{{10}^{10}}=3.15\times {{10}^{-10}}\times N \\\ \therefore N=3.17\times {{10}^{19}}

So, the correct option is A.

Additional Information:
Half-life is the time for half the radioactive nuclei in any sample to undergo radioactive decay. For example, after 2 half-lives, there will be one fourth the original material remains, after three half-lives one eight the original material remains, and so on. Half-life is a convenient way to assess the rapidity of a decay

Note: Activity and number of disintegrations per second are both the same thing.Half life is measured in seconds or years depending upon the nature of the material. The law of radioactivity can also be stated as: N=N0e−λtN={{N}_{0}}{{e}^{-\lambda t}}
Where the number of initial atoms is N0{{N}_{0}}. Then the number of undecayed atoms is N.