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The activity of a freshly prepared radioactive sample is ${{10}^{10}}$ disintegration per second, whose mean life is ${{10}^{9}}s$. The mass of an atom of this radio-isotope is ${{10}^{-25}}$ kg. The mass (in mg) of the radioactive sample is
A. 1
B. 2
C. 3
D. 4

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Last updated date: 14th Jun 2024
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Answer
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Hint: Radioactive emission is a process by which an unstable nucleus becomes stable through emission. Find the expression for the number of nuclei after the radioactive emission for a certain interval of time. Obtain the expression for the activity of a radioactive substance. Put the given values to find the number of atoms and from this find the mass of the sample.

Complete answer:
Radioactivity is a process where particles or electromagnetic radiation are emitted from the nucleus of an atom with unstable nuclei. The radioactive nuclei emit radioactive emission to go to the stable nuclei state.
The number of original nuclei, N, remaining due to radioactive emission after a time t from the original sample of ${{N}_{0}}$ nuclei is,
$N={{N}_{0}}{{e}^{-\dfrac{t}{\tau }}}$
The activity of the radioactive sample can be defined as the number of disintegrations per second. Mathematically we can express that,
$\begin{align}
  & A=\dfrac{dN}{dt}=-\dfrac{1}{\tau }{{N}_{0}}{{e}^{-\dfrac{t}{\tau }}} \\
 & A=-\dfrac{1}{\tau }N \\
\end{align}$
Where, A is the activity of the radioactive sample, $\tau $ is the mean life of the radioactive sample,
The activity of the sample is given as, $\dfrac{dN}{dt}={{10}^{10}}$
Taking the magnitude only, we can write,
$\dfrac{N}{\tau }={{10}^{10}}$
The mean life of the sample of is $\tau ={{10}^{9}}\operatorname{s}$
Putting this value,
$\begin{align}
  & \dfrac{N}{{{10}^{9}}}={{10}^{10}} \\
 & N={{10}^{19}} \\
\end{align}$
Now, the mass of an atom of this radio-isotope is ${{10}^{-25}}$ kg.
Total number of atoms of the radio-isotope is $N={{10}^{19}}$
So, the mass of the radioactive sample is,
$\begin{align}
  & M={{10}^{19}}\times {{10}^{-25}}kg \\
 & N={{10}^{-6}}kg \\
 & N=1mg \\
\end{align}$
So, the mass of the radioactive sample is 1mg.

The correct option is (A).

Note:
We have three types of radioactive decay which are – alpha decay, beta decay and the gamma decay. The law of radioactive decay states that the probability per unit time that a nucleus will decay is a constant which is independent of time.