1 Suppose we consider a large number of containers each containing initially 10000 atoms of a radioactive material with a half life of 1 yr . After 1 yr,

The gravitational force between a H -atom and another particle of mass $m$ will be given by Newton's law

$$F=G \frac{M \cdot m}{r^2}, \text { where } r \text { is in } \mathrm{km} \text { and }$$

When a nucleus in an atom undergoes a radioactive decay, the electronic energy levels of the atom

$M_x$ and $M_y$ denote the atomic masses of the parent and the daughter nuclei respectively in radioactive decay. The $Q$-value for a $\beta^{-}$decay is $Q_1$ and that for a $\beta^{+}$decay is $Q_2$. If $m_e$ denotes the mass of an electron, then which of the following statements is correct?

Tritium is an isotope of hydrogen whose nucleus triton contains 2 neutrons and 1 proton. Free neutrons decay into $p+\bar{e}+\bar{n}$. If one of the neutrons in Triton decays, it would transform into $\mathrm{He}^3$ nucleus. This does not happen. This is because

Heavy stable nuclei have more neutrons than protons. This is because of the fact that

In a nuclear reactor, moderators slow down the neutrons which come out in a fission process. The moderator used have light nuclei. Heavy nuclei will not serve the purpose, because

Fusion processes, like combining two deuterons to form a He nucleus are impossible at ordinary temperatures and pressure. The reasons for this can be traced to the fact

Samples of two radioactive nuclides $A$ and $B$ are taken $\lambda_A$ and $\lambda_B$ are the disintegration constants of $A$ and $B$ respectively. In which of the following cases, the two samples can simultaneously have the same decay rate at any time?

The variation of decay rate of two radioactive samples $A$ and $B$ with time is shown in figure. Which of the following statements are true?

$\mathrm{He}_2^3$ and $\mathrm{He}_1^3$ nuclei have the same mass number. Do they have the same binding energy?

Draw a graph showing the variation of decay rate with number of active nuclei.

Which sample A or B shown in figure has shorter mean-life?

Which one of the following cannot emit radiation and why? Excited nucleus, excited electron.

In pair annihilation, an electron and a positron destroy each other to produce gamma radiations. How is the momentum conserved?

Why do stable nuclei never have more protons than neutrons?

Consider a radioactive nucleus $A$ which decays to a stable nucleus $C$ through the following sequence

$$A \rightarrow B \rightarrow C$$

Here $B$ is an intermediate nuclei which is also radioactive. Considering that there are $N_0$ atoms of $A$ initially, plot the graph showing the variation of number of atoms of $A$ and $B$ versus time.

A piece of wood from the ruins of an ancient building was found to have a ${ }^{14} \mathrm{C}$ activity of 12 disintegrations per minute per gram of its carbon content. The ${ }^{14} \mathrm{C}$ activity of the living wood is 16 disintegrations per minute per gram. How long ago did the tree, from which the wooden sample came, die? Given half-life of ${ }^{14} \mathrm{C}$ is 5760 yr .

Are the nucleons fundamental particles, or do they consist of still smaller parts? One way to find out is to probe a nucleon just as Rutherford probed an atom. What should be the kinetic energy of an electron for it to be able to probe a nucleon? Assume the diameter of a nucleon to be approximately $10^{-15} \mathrm{~m}$.

A nuclide 1 is said to be the mirror isobar of nuclide 2 if $Z_1=N_2$ and $Z_2=N_1$.(a) What nuclide is a mirror isobar of ${ }_{11}^{23} \mathrm{Na}$ ? (b) Which nuclide out of the two mirror isobars have greater binding energy and why?

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

$${ }^{38} \text { Sulphur } \xrightarrow[=2.48 \mathrm{~h}]{\text { half-life }}{ }^{38} \mathrm{Cl} \xrightarrow[=0.62 \mathrm{~h}]{\text { half-life }}{ }^{38} \mathrm{Ar} \text { (stable) }$$

Assume that we start with $1000{ }^{38} \mathrm{~S}$ nuclei at time $t=0$. The number of ${ }^{38} \mathrm{Cl}$ is of count zero at $t=0$ 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 $B=2.2 \mathrm{MeV}$. 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 be $E$ be for such a process to happen.

The deuteron is bound by nuclear forces just as 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 a coulomb potential but with an effective charge $e^{\prime}$

$$F=\frac{1}{4 \pi \varepsilon_0} \cdot \frac{e^{\prime 2}}{r}$$

estimate the value of $\left(e^{\prime} / e\right)$ given that the binding energy of a deuteron is 2.2 MeV .

Before the neutrino hypothesis, the beta decay process was throught to be the transition.

$$n \rightarrow p+\bar{e}$$

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 result found are tabulated as follows

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

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

Nuclei with magic number of proton $Z=2,8,20,28,50,52$ and magic number of neutrons $N=2,8,20,28,50,82$ and 126 are found to be very stable.

(i) Verify this by calculating the proton. separation energy $S_p$ for ${ }^{120} \operatorname{Sn}(Z=50)$ and ${ }^{121} \mathrm{Sb}(Z=51)$.

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

$$\begin{aligned} S_p & =\left(M_{Z-1, N}+M_{\mathrm{H}}-M_{Z, N}\right) c^2 . \\ \text{Given,}\quad { }^{119} \mathrm{In} & =118.9058 \mathrm{u},{ }^{120} \mathrm{Sn}=199.902199 \mathrm{u}, \\ { }^{121} \mathrm{Sb} & =120.903824 \mathrm{u},{ }^1 \mathrm{H}=1.0078252 \mathrm{u} . \end{aligned}$$

(ii) What does the existence of magic number indicate?