Question

The rate of radioactive disintegration at an instant for a radioactive sample of half life $2.2\times {{10}^{9}}s$ is ${{10}^{10}}{{s}^{-1}}$. The number of radioactive atoms in that sample at that instant is:
A. $3.17\times {{10}^{19}}$
B. $3.17\times {{10}^{20}}$
C. $3.17\times {{10}^{17}}$
D. $3.17\times {{10}^{18}}$

Answer 1

Here we can find the number of radioactive atoms from the Law of Radioactive Decay, which states that, when the radioactive material undergoes decay, the number of nuclei undergoing the decay, per unit time, is proportional to the total number of nuclei In the sample material.

Formulas used:
$R=\lambda N$
Where R is the decay rate
$\lambda$ is the radioactive decay constant or disintegration constant
And N is the radioactive nuclei at time t.

Complete step by step solution:
In our question we are given the half-life of the radioactive sample, that is ${{t}_{1/2}}=2.2\times {{10}^{9}}s$
We are also given the radioactive disintegration rate or the decay rate R, that is ${{10}^{10}}{{s}^{-1}}$
From the formula, we know that $R=\lambda N$
Hence we can modify it as
$\begin{aligned} & N=\dfrac{R}{\lambda }=\dfrac{R}{0.693}\times {{t}_{1/2}} \\\ & \\\ \end{aligned}$
$\Rightarrow N=\dfrac{{{10}^{10}}\times 2.2\times {{10}^{9}}}{\begin{aligned}& 0.693 \\\ & \\\ \end{aligned}}$
$\therefore N=3.17\times {{10}^{19}}atoms$

Hence, the number of radioactive atoms in that sample at that instant is $3.17\times {{10}^{19}}$atoms.Therefore we can conclude that option A is the correct answer among the four given options.

Note: Here, If the given value is the mean life of the sample.
We can find the radioactive decay constant by using the formula
$\tau =\dfrac{1}{\lambda }$
Where $\tau$is the mean life of the sample
$\lambda$ is the radioactive decay constant or disintegration constant.
The mean life is usually 1.443 times longer than its half life. It is the average life of a radioactive substance where it lives as itself without further disintegration.