Beta Decay

Q.

State the law of radioactive decay. Hence derive the expression $N=N_0{e}^{-\lambda t}$ where symbols have their usual meaning

Q. The oscillating frequency of a cyclotron is $10\text{MHz}$. If the radius of its dees is $0.5\text{m}$, then the kinetic energy of a proton accelerated by the cyclotron will be:

1. $5.2\text{MeV}$
2. $7.2\text{MeV}$
3. $4.1\text{MeV}$
4. $2.3\text{MeV}$

Q.

The anion ${\mathrm{X}}^{3-}$has 18 electrons and its mass number is 31.How many neutrons are present in the nucleus ?

1. 18

2. 8

3. 31

4. 16

Q.

The neutron separation energy is defined as the energy required to remove a neutron from the nucleus. Obtain the neutron separation energies of the nuclei and from the following data:

= 39.962591 u

) = 40.962278 u

= 25.986895 u

) = 26.981541 u

Q.

A nucleus X emits a beta particle to produce a nucleus Y. If their atomic masses are $M_x$ and $M_y$ respectively. The maximum energy of the beta particle emitted is (Hint: ${\beta }^{-}$ decay)

(where $m_e$ is the mass of an electron and c is the velocity of light)

1. $(M_x-M_y-m_e)c^2$
2. $(M_x-M_y+m_e)c^2$
3. $(M_x-M_y)c^2$
4. $(M_x-M_y-2m_e)c^2$

Q.

The radionuclide ¹¹C decays according to

The maximum energy of the emitted positron is 0.960 MeV.

Given the mass values:

calculate Q and compare it with the maximum energy of the positron emitted

Q. When a nucleus with atomic number $Z$ and mass number $A$ undergoes a radioactive decay process,

$\left(1\right)$ Both $Z$ and $A$ will decrease, if the process is $\alpha$ decay.
$\left(2\right)$ $Z$ will decrease, but $A$ will not change, if the process is ${\beta }^{+}$ decay.
$\left(3\right)$ $Z$ will increase, but $A$ will not change, if the process is ${\beta }^{-}$ decay.
$\left(4\right)$ $Z$ and $A$ will remain unchanged, if the process is $\gamma$ decay.

1. $1$ & $2$ are true
2. $3$ & $4$ are true
3. $1$, $2$ & $3$ are true
4. All are true

Q. In the given nuclear reaction, A, B, C, D, E represents
${}_{92}{U}^{238}\stackrel{\alpha }{⟶}{}_{B}T{h}^A\stackrel{\beta }{⟶}{}_{D}P{a}^C\stackrel{E}{⟶}{}_{92}{U}^{234}$

1. $A=234,B=90,C=234,D=91,E=\beta$
2. $A=234,B=90,C=238,D=94,E=\alpha$
3. $A=234,B=90,C=234,D=93,E=\alpha$
4. $A=238,B=93,C=234,D=91,E=\beta$

Q. In the given reaction

${}_z{X}^A{\to }_{z+1}{Y}^A{\to }_{z-1}{K}^{A-4}{\to }_{z-1}{K}^{A-4}$
Radioactive radiations are emitted in the sequence of

1. $\alpha ,\beta ,\gamma$
2. $\beta ,\alpha ,\gamma$
3. $\gamma ,\alpha ,\beta$
4. $\beta ,\gamma ,\alpha$

Q. If a nucleus $\begin{array}{c}A\\ Z\end{array}X$ emits an $\alpha -$ particle and a $\beta -$ particle, then the daughter nucleus will have the total number of neutrons as $A-Z-n=y.$ The value of $n$ is (integer only)

Q. Why is it difficult to experimentally detect neutrinos in nuclear $\beta$-decay ?

Q.

The decomposition of the Formic acid on gold surface follows first order kinetics if the rate constant at $300\mathrm{K}$ is $1.0\times {10}^{-3}{\mathrm{s}}^{-1}$ and the activation energy${\mathrm{E}}_{\mathrm{a}}=11.488\mathrm{kJ}{\mathrm{mol}}^{-1}$, then the rate constant at $200\mathrm{K}$ is:__________×10^–5 s^–1

Q. If ${}^{\prime }{f}^{\prime }$ denotes the ratio of the number of nuclei decayed $\left(N_d\right)$ to the number of nuclei at $t=0\left(N_0\right)$ then for a collection of radioactive nuclei, the rate of change of ${}^{\prime }{f}^{\prime }$ with respect to time is given as :
[$\lambda$ is the radioactive decay constant]

1. $-\lambda e^{-\lambda t}$
2. $-\lambda (1-e^{-\lambda t})$
3. $\lambda e^{-\lambda t}$
4. $\lambda (1-e^{-\lambda t})$

Q.

The Q value of a nuclear reaction A + b → C + d is defined by

Q = [ m_A+ m_b− m_C− m_d]c² where the masses refer to the respective nuclei. Determine from the given data the Q-value of the following reactions and state whether the reactions are exothermic or endothermic.

(i)

(ii)

Atomic masses are given to be

Q. A radioactive substance is being produced at a constant rate of $200$ nuclei. The decay constant of the substance is $1{\text{s}}^{-1}.$ Assuming that initially there are no nuclei present, the time (in second) after which the number of nuclei will become $100$ is

1. $1\text{s}$
2. $\ln \left(2\right)\text{s}$
3. $\frac{1}{\ln \left(2\right)}\text{s}$
4. $2\text{s}$

Q. A certain stable nuclide after absorbing a neutron, emits $\beta -$ particle and the new nuclide formed splits spontaneously into two $\alpha -$ particles. The nuclide is-

1. ${}_3{\text{Li}}^6$
2. ${}_4{\text{Be}}^6$
3. ${}_3{\text{Li}}^7$
4. ${}_2{\text{He}}^4$

Q. The correct statement is

1. The nucleus ${}_{84}^{210}Po$ can emit a proton
2. Deuteron and alpha particle can undergo complete fusion
3. The nucleus ${}_3^{6}Li$ can emit an alpha particle
4. The nuclei ${}_{30}^{70}Zn$ and ${}_{34}^{82}Se$ can undergo complete fusion

Q.

In $\beta$-decay, the Q-value of the process is E. Then

1. KE of a $beta$-particle cannot exceed E.

2. KE of anti-neutrino emitted lies between zero and E

3. $\frac{N}{Z}$ ratio of the nucleus is altered.

4. Mass number (A) of the nucleus is altered.

Q.

A radioactive nucleus undergoes a series of decays according to the sequence.

$\mathrm{X}\stackrel{\mathrm{\beta }}{\to }{\mathrm{X}}_1\stackrel{\mathrm{\alpha }}{\to }{\mathrm{X}}_2\stackrel{\mathrm{\alpha }}{\to }{\mathrm{X}}_3\to$

If the mass number and atomic number of ${\mathrm{X}}_3$ are $132$ and $69$ respectively, what is the mass number and atomic number of$X$?

Q. Calculate the $Q$ value in the following decay using the atomic masses given in the table.
${}^{25}\text{Al}{\to }^{25}\text{Mg}+e^{+}+\nu$

Mass of ${}^{25}\text{Al}$	Mass of ${}^{25}\text{Mg}$
$24.990432u$	$24.985839u$

$[\text{Take}{m}_{electron}=0.511\frac{\text{MeV}}{c^2};c^2=931\frac{\text{MeV}}{u}]$

1. $14.816\text{MeV}$
2. $12.408\text{MeV}$
3. $11.204\text{MeV}$
4. $3.254\text{MeV}$

Q. A copper wire of cross-sectional area $0.01{\text{cm}}^2$ is under a tension of $20\text{N}$. Find the decrease in the cross-sectional area. Young's modulus and Poisson's ratio of copper are $1.1\times {10}^{11}{\text{N/m}}^2$ and $0.32$ respectively.

1. $3.16\times {10}^{-6}{\text{cm}}^2$
2. $1.16\times {10}^{-6}{\text{cm}}^2$
3. $4.16\times {10}^{-6}{\text{cm}}^2$
4. $2.16\times {10}^{-6}{\text{cm}}^2$

Q.

For the (positron) emission from a nucleus, there is another competing process known as electron capture (electron from an inner orbit, say, the K−shell, is captured by the nucleus and a neutrino is emitted).

Show that if emission is energetically allowed, electron capture is necessarily allowed but not vice−versa.

Q.

What is the role of neutrons in the nucleus, as protons are held by pi-meson, what does neutrons do?

Q.

The neutron separation energy is defined as the energy required to remove a neutron from the nucleus. Obtain the neutron separation energies of the nuclei and from the following data:

= 39.962591 u

) = 40.962278 u

= 25.986895 u

) = 26.981541 u

Q. The figure below shows a steel rod of $25m{m}^2$ cross sectional area. It is loaded at four points. K, L, M and N. Assume $E_{steel}=200GPa$. The total change in length of the rod due to loading is

1. $1\mu m$
   
2. $-10\mu m$
3. $1\mu m$
4. $-20\mu m$

Q. Which of the following rays is not deflected by a magnetic field?

1. $\alpha -$rays
2. $\beta -$rays
3. $\gamma -$rays
4. None of these

Q. Consider the beta decay

${}^{198}\text{Au}\to {}^{198}{\text{Hg}}^{\ast }+{\beta }^{-}+\overline{\nu }$

Where ${}^{198}{\text{Hg}}^{\ast }$ represents a mercury nucleus in an excited state at energy $1.088\text{MeV}$ above the ground state. What can be the maximum kinetic energy of the electron emitted? The atomic mass of ${}^{198}\text{Au}$ is $197.968233u$ and that of ${}^{198}\text{Hg}$ is $197.966760u$. $c^2=931\frac{\text{MeV}}{u}$

1. $0.28\text{MeV}$
2. $5.86\text{MeV}$
3. $9.25\text{MeV}$
4. $15.89\text{MeV}$

Q.

Write nuclear reaction equations for

(i) α-decay of (ii) α-decay of

(iii) β⁻-decay of (iv) β⁻-decay of

(v) β⁺-decay of (vi) β⁺-decay of

(vii) Electron capture of

Q. 10 grams of ${}^{57}Co$ kept in an open container beta-decays with a half-life of $270$ days. The weight of the material inside the container after $540$ days will be very nearly.

1. $10g$
2. $5g$
3. $25g$
4. $125g$

Q.

An alpha particle $\left({}_2{}^4\mathrm{He}\right)$ has a mass of $4.00300amu$. A proton has mass of $1.00783amu$ and a neutron has mass of $1.00867amu$ respectively. The binding energy of alpha particles estimated from these data is the closest to:

1. $20.4MeV$

2. $32.0MeV$

3. $24.3MeV$

4. $27.9MeV$