Embibe Experts Solutions for Chapter: Nuclear Physics, Exercise 3: Exercise - 3

Match list-$\mathrm{I}$ of the nuclear processes with a list-$\mathrm{II}$ containing parent nucleus and one of the end products of each process and then select the correct answer using the codes given below the lists

List-$\mathrm{I}$		List-$\mathrm{II}$
$\left(\mathrm{P}\right)$	Alpha decay	$\left(1\right)$
$\left(\mathrm{Q}\right)$	${\beta }^{+}$-decay	$\left(2\right)$
$\left(\mathrm{R}\right)$	Fission	$\left(3\right)$
$\left(\mathrm{S}\right)$	Proton emission	$\left(4\right)$	${}_{\mathbf{94}}{}^{\mathbf{239}}\mathit{P}\mathit{u}\mathbf{\to }{}_{\mathbf{57}}{}^{\mathbf{140}}\mathit{L}\mathit{a}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}$

A fission reaction is given by ${}_{92}{}^{236}\mathrm{U}\to {}_{54}{}^{140}\mathrm{Xe}+{}_{38}{}^{94}\mathrm{Sr}+x+y$, where $x$ and $y$ are two particles. Considering ${}_{92}{}^{236}\mathrm{U}$ to be at rest, the kinetic energies of the products are denoted by $K_{\mathrm{Xe}}$, $K_{\mathrm{Sr}}$, $K_{\mathrm{x}} \left(2 \mathrm{MeV}\right)$ and $K_{\mathrm{y}} \left(2 \mathrm{MeV}\right)$, respectively. Let the binding energies per nucleon of ${}_{92}{}^{236}\mathrm{U}$, ${}_{54}{}^{140}\mathrm{Xe}$ and ${}_{38}{}^{94}\mathrm{Sr}$ be $7.5 \mathrm{MeV}$, $8.5 \mathrm{MeV}$ and $8.5 \mathrm{MeV}$, respectively. Considering different conservation laws, the correct option(s) is (are)

An accident in a nuclear laboratory resulted in the deposition of a certain amount of radioactive material of half-life $18 \mathrm{days}$ inside the laboratory. Tests revealed that the radiation was $64 \mathrm{times}$ more than the permissible level required for the safe operation of the laboratory. What is the minimum number of days after which the laboratory can be considered safe for use?

The electrostatic energy of $Z$ protons uniformly distributed throughout a spherical nucleus of the radius $R$ is given by $E=\frac{3}{5}\frac{Z\left(Z-1\right)e^2}{4\mathrm{\pi }{\epsilon }_{0}R}$.

The measured masses of the neutron, ${}_1{}^1\mathrm{H}$, ${}_7{}^{15}\mathrm{N}$ and ${}_8{}^{15}\mathrm{O}$ are $1.008665 \mathrm{u}$, $1.007825 \mathrm{u}$, $15.000109 \mathrm{u}$ and $15.003065 \mathrm{u}$, respectively. Given that the radii of both the ${}_7{}^{15}\mathrm{N}$ and ${}_8{}^{15}\mathrm{O}$ nuclei are the same, $1\mathrm{u}=931.5 \mathrm{MeV} {\mathrm{c}}^{-2}$ ($c$ is the speed of light) and ${\mathrm{e}}^2/\left(4\mathrm{\pi }{\epsilon }_0\right)=1.44 \mathrm{MeV} \mathrm{fm}$. Assuming that the difference between the binding energies of ${}_7{}^{15}\mathrm{N}$ and ${}_8{}^{15}\mathrm{O}$ is purely due to the electrostatic energy, the radius of either of the nuclei is $\left(1 \mathrm{fm}={10}^{–15} \mathrm{m}\right)$

In a radioactive decay chain, ${}_{90}{}^{232}\mathrm{Th}$ nucleus decays to ${}_{90}{}^{212}\mathrm{Pb}$ nucleus. Let $N_{\alpha }$ and $N_{\beta }$ be the number of $\alpha$ and ${\beta }^{-}$ particles, respectively, emitted in this decay process. Which of the following statements is (are) true?

Assume that a neutron breaks into a proton and an electron. The energy released during this process is (mass of the neutron$=1.6725\times {10}^{–27} \mathrm{kg}$, Mass of proton$=1.6725\times {10}^{–27} \mathrm{kg}$, the mass of an electron$=9\times {10}^{–31} \mathrm{kg}$)

Half-lives of two radioactive elements $A$ and $B$ are $20 \mathrm{minutes}$ and $40 \mathrm{minutes}$, respectively. Initially, the samples have an equal number of nuclei. After $80 \mathrm{minutes}$, the ratio of decayed numbers of $A$ and $B$ nuclei will be:

A radioactive nucleus $A$ with a half-life $T$, decays into a nucleus $B$. At $t=0$, there is no nucleus $B$. At some time $t$, the ratio of the number of $B$ to that of $A$ is $0.3$. Then, $t$ is given by,