Question

The decay constant of a radioactive element is defined as the reciprocal of the time interval after which the number of atoms of the radioactive element falls to nearly
A. 50% of its original number.
B. 36.8% of its original number.
C. 63.2% of its original number.
D. 75% of its original number.

Answer 1

Hint: We have a radioactive element that undergoes a radioactive decay process. We need to find the number of atoms lost during the process. By radioactive decay law we have an equation for the number of decayed atoms and in the question it is said that the decay constant is the reciprocal of the time interval. By equating this we will get the total number of atoms lost.
Formula used:
Number of decayed atoms,
\[N={{N}_{0}}{{e}^{-\lambda t}}\]

Complete answer:
In the question it is said that the decay constant of a radioactive element is defined as the reciprocal of the time interval when the number of atoms of the radioactive elements falls.
We need to find the number of atoms that fall during the radioactive decay process of the element.
From the law of radioactive decay we have the equation for number of decayed elements,
\[N={{N}_{0}}{{e}^{-\lambda t}}\], were ‘N’ is the total number of decayed atoms after a time ‘t’, ‘${{N}_{0}}$’ is number of atoms in the element before the radioactive decay process, ‘$\lambda $’ is the decay constant and ‘t’ is the time period.
From the question, we know that the decay constant,
$\lambda =\dfrac{1}{t}$
Applying this in the equation, we get
\[\begin{align}
  & N={{N}_{0}}{{e}^{-\dfrac{1}{t}\times t}} \\
 & N={{N}_{0}}{{e}^{-1}} \\
\end{align}\]
Therefore we get the value of ‘N’ as,
$N=\dfrac{{{N}_{0}}}{e}$
We know the value of ‘e’
$e=2.718$
Therefore we get ‘N’ as,
$\begin{align}
  & N=\dfrac{1}{2.718}{{N}_{0}} \\
 & N=0.3679{{N}_{0}} \\
\end{align}$
From this we get that the number of decayed atoms is 0.3679 times the original number.
Converting this into percentage, we get
The total number of elements after the decay is 36.8% of the original number.

Hence the correct answer is option B.

Note:
Radioactive decay is the process of spontaneous breaking down of atomic nuclei that result in the release of energy and radiation from the nucleus.
Alpha decay, beta decay and gamma decay are the three types of radioactive decay.
According to the radioactive law the probability per unit time that a nucleus will decay is a constant.
This constant is the decay constant given by ‘$\lambda $’.