NCERT Solutions for Chapter: Nuclei, Exercise 4: LA

Sometimes a radioactive nucleus decays into a nucleus which itself is radioactive. An example is:

${ }^{38}\mathrm{Sulphur} {\underset{=2.48 \mathrm{h}}{\overset{\mathrm{half} \mathrm{life}}{\to }}}^{38}\mathrm{Cl}{\underset{0.62 \mathrm{h}}{\overset{\mathrm{half} \mathrm{life}}{\to }}}^{38}\mathrm{Ar}\left(\mathrm{stable}\right)$

Assume that we start with $1000$${}^{38}\mathrm{S}$ nuclei at time $\mathrm{t} = 0$. The number of ${}^{38}\mathrm{Cl}$ is of count zero at$\mathrm{t}=0$ and will again be zero at $t=\infty$. At what value of $t,$ would the number of counts be a maximum?

Deuteron is a bound state of a neutron and a proton with a binding energy $\mathrm{B}=2.2\mathrm{MeV}$. A $\gamma -$ray of energy $E$is aimed at a deuteron nucleus to try to break it into a (neutron+proton) such that the $n$and $p$ move in the direction of the incident $\gamma$-ray. If $E = B$, show that this cannot happen. Hence, calculate how much bigger than $B$ must $E$ be for such a process to happen.

The deuteron is bound by nuclear forces just as $\mathrm{H}$-atom is made up of $p$ and $e$ bound by electrostatic forces. If we consider the force between neutron and proton in deuteron as given in the form of a Coulomb potential but with an effective charge $e\text{'}$

$F=\frac{1}{4\pi {\epsilon }_0}\frac{e^2}{r}$

Estimate the value of $\left(e^{\text{'}}/e\right)$, given that the binding energy of a deuteron is $2.2\mathrm{MeV}$

Before the neutrino hypothesis, the beta decay process was thought to be the transition,
$n\to p+\overline{e}$
If this was true, show that if the neutron was at rest, the proton and electron would emerge with fixed energies and calculate them. Experimentally, the electron energy was found to have a large range.

The activity $R$ of an unknown radioactive nuclide is measured at hourly intervals. The results found are tabulated as follows:

$\begin{array}{|lccccc|}\hline t\left(\mathrm{h}\right)& 0& 1& 2& 3& 4\\ R\left(\mathrm{MBq}\right)& 100& 35.36& 12.51& 4.42& 1.56\\ \hline\end{array}$

Plot the graph of $R$ versus $t$ and calculate half-life from the graph.

The activity $R$of an unknown radioactive nuclide is measured at hourly intervals. The results found are tabulated as follows:

$\begin{array}{|lccccc|}\hline t\left(\mathrm{h}\right)& 0& 1& 2& 3& 4\\ R\left(\mathrm{MBq}\right)& 100& 35.36& 12.51& 4.42& 1.56\\ \hline\end{array}$

Plot the graph of In  $\left(\frac{R}{R_0}\right)$versus $t$ and obtain the value of half-life from the graph.

Nuclei with magic no. of proton $Z=2,8,20,28,50,52$ and magic no. of neutrons $N=2,8,20,28,50,82\text{ and }126$ are found to be very stable,

 Verify this by calculating the proton separation energy $S_p$for ${}^{120}Sn(Z=50){\text{ and }}^{121}Sb=(Z=51)$. The proton separation energy for a nuclide is the minimum energy required to separate the least tightly bound proton from a nucleus of that nuclide. It is given by

${\mathrm{S}}_{\mathrm{p}}=\left(M_{Z-1}{,}_N+M_H-M_{Z,N}\right)c^2$

Given, ${}^{119}\mathrm{In}=118.9058\mathrm{u},{ }^{120}\mathrm{Sn}=119.902199\mathrm{u}{}^{121}\mathrm{Sb}=120.903824\mathrm{u}{,}^1\mathrm{H}=1.0078252\mathrm{u}$

Nuclei with magic no. of proton $Z=2,8,20,28,50,52$ and magic no. of neutrons $N=2,8,20,28,50,82\text{ and }126$ are found to be very stable,

 What does the existance of magic number indicate?