Question

The initial number of atoms in a radioactive element is $16 \times {10^{20}}$ and its half-life is 8 hours the number of atoms that disintegrated in 24 hours and the energy liberated are
$a){\text{ }}1.4 \times {10^{20}},{\text{ }}5.6 \times {10^7}J$
$b){\text{ }}14 \times {10^{20}},{\text{ }}56 \times {10^7}J$
$c){\text{ }}14 \times {10^8},{\text{ }}56 \times {10^8}J$
$d){\text{ }}14 \times {10^7},{\text{ }}56 \times {10^7}J$

Answer 1

In order to solve the question, we will first of all use the Relation between half-life and decay constant to find the decay constant then we will use the Integrated equation of radioactivity decay to find the number of atoms that disintegrated and for energy will use the relation between energy planck's constant. speed of light and decay constant.

Formula Used:
$\dfrac{{dN}}{{dt}} = - \lambda N$
N is the size of a population of radioactive atoms at a given time t
dN is the amount by which the population decreases in time dt
$\lambda$ is the decay constant
Integrated equation of radioactivity decay
$N = {N_0}{e^{ - \lambda t}}$
${N_0}$ is the initial number of atoms in a radioactive element
Relation between half-life and decay constant
${T_{\dfrac{1}{2}}} = \dfrac{{0.693}}{\lambda }$
${T_{\dfrac{1}{2}}}$= Half-life of radioactive element
Energy formula
$E = \dfrac{{hc}}{\lambda }$
E is the energy
h is planck's constant
c is speed of light

Complete step-by-step solution:
In the question we are given
The initial number of atoms in a radioactive element (${N_0}$) = $16 \times {10^{20}}$
Half-life of radioactive element ( ${T_{\dfrac{1}{2}}}$ ) = 8 hours
And we have to find the number of atoms that disintegrated in 24 hours and the energy liberated
We will use the Relation between half-life and decay constant to find the decay constant
${T_{\dfrac{1}{2}}} = \dfrac{{0.693}}{\lambda }$
Substituting the value of half life
$8 = \dfrac{{0.693}}{\lambda }$
Taking decay constant on left hand side
$\lambda = 0.087$
Now we will use the Integrated equation of radioactivity decay to find the number of atoms that disintegrated in 24 hours
$N = {N_0}{e^{ - \lambda t}}$
Substituting the values
$N = 16 \times {10^{20}} \times {e^{ - 0.087 \times 24}}$
After calculation we get number of atoms that disintegrated in 24 hours
$N = 14 \times {10^{20}}$
Now we will find the energy using the formula
$E = \dfrac{{hc}}{\lambda }$
Plank’s constant $h = 6.6 \times {10^{ - 34}}{m^2}kg{\kern 1pt} {s^{ - 1}}$
Speed of light $c = 3 \times {10^8}m{s^{ - 1}}$
Substituting the values
$E = \dfrac{{6.6 \times {{10}^{ - 34}} \times 3 \times {{10}^8}}}{{0.08}}$
Hence solving the calculation, we get value of energy equals to
$E = 56 \times {10^7}J$
Hence, the correct option $b){\text{ }}14 \times {10^{20}},{\text{ }}56 \times {10^7}J$

Note: Many of the people will make the mistake by not using the integrated radioactive decay equation instead of that using original equation using that can’t be possible for this question which can be realized after the first step as it does not consist both the initial amount and the decay amount of radioactive element.