Answer: (a)
Radius $R = R_0 A^{1/3}$ .
Ratio $\frac{R_1}{R_2} = \left(\frac{A_1}{A_2}\right)^{1/3} = \left(\frac{1}{8}\right)^{1/3} = \frac{1}{2}$ .

2. The density of a nucleus is independent of:

(a) Mass number $A$
(b) Atomic number $Z$
(c) Neutron number $N$
(d) None of the above

Answer: (a)
Nuclear density is constant ( $\approx 2.3 \times 10^{17} \text{ kg/m}^3$ ) for all nuclei, regardless of their size or Mass Number $A$ .

3. Which of the following forces is responsible for holding the nucleons together in a nucleus?

(a) Gravitational force
(b) Electrostatic force
(c) Strong Nuclear force
(d) Weak Nuclear force

Answer: (c)
The Strong Nuclear Force overcomes the electrostatic repulsion between protons to hold the nucleus together.

4. In the nuclear reaction ${}_{2}^{4}\text{He} + {}_{Z}^{A}\text{X} \rightarrow {}_{Z+2}^{A+3}\text{Y} + \text{W}$ , the particle W is:

(a) Proton
(b) Neutron
(c) Electron
(d) Positron

Answer: (b)
Conserve Mass Number: $4 + A = (A+3) + A_W \Rightarrow A_W = 1$ .
Conserve Atomic Number: $2 + Z = (Z+2) + Z_W \Rightarrow Z_W = 0$ .
Particle with $A=1, Z=0$ is a Neutron ( ${}_{0}^{1}\text{n}$ ).

5. The binding energy per nucleon is maximum for nuclei with mass number $A$ around:

(a) 2
(b) 56
(c) 100
(d) 238

Answer: (b)
The curve peaks near Iron ( $Fe^{56}$ ) with approx 8.75 MeV/nucleon, making it the most stable.

6. A radioactive substance has a half-life of 10 days. The amount of substance left after 30 days will be:

(a) 1/2
(b) 1/4
(c) 1/8
(d) 1/16

Answer: (c)
Number of half-lives $n = 30/10 = 3$ .
Remaining fraction $= (1/2)^3 = 1/8$ .

7. The SI unit of activity of a radioactive sample is:

(a) Curie
(b) Rutherford
(c) Becquerel
(d) Roentgen

Answer: (c)
1 Becquerel (Bq) = 1 decay per second.

8. During $\beta^-$ decay, a neutron inside the nucleus converts into:

(a) Proton + Electron + Antineutrino
(b) Proton + Electron + Neutrino
(c) Proton + Positron + Neutrino
(d) Neutron + Electron

Answer: (a)
$n \rightarrow p + e^- + \bar{\nu}$ (Antineutrino is released to conserve spin).

9. Energy generation in stars is primarily due to:

(a) Chemical reactions
(b) Nuclear Fission
(c) Nuclear Fusion
(d) Radioactivity

Answer: (c)
Fusion of hydrogen into helium (thermonuclear fusion) powers the sun and stars.

10. Heavy water is used in nuclear reactors as a:

(a) Fuel
(b) Coolant
(c) Moderator
(d) Shield

Answer: (c)
It slows down (moderates) fast neutrons to thermal energies to sustain the chain reaction.

Part 2: Assertion-Reason Questions (1 Mark)

Directions:
(A) Both A and R are true and R is correct explanation of A.
(B) Both A and R are true but R is NOT correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

1. Assertion (A): The density of the nucleus is very high compared to the density of the atom.
Reason (R): The nucleus contains 99.9% of the mass of the atom but occupies a negligible volume.

Answer: (A)
Correct. Since mass is high and volume is tiny ( $10^{-15}$ m scale), density is enormous ( $\approx 10^{17}$ kg/m³).

2. Assertion (A): Isotopes of an element have the same chemical properties.
Reason (R): Chemical properties depend on the electronic configuration, which is the same for isotopes.

Answer: (A)
Correct explanation. Isotopes have same $Z$ (electrons), hence same chemical behavior.

3. Assertion (A): Nuclear forces are charge independent.
Reason (R): The nuclear force between p-p, n-n, and p-n is approximately the same.

Answer: (A)
Correct. Unlike electrostatic force, nuclear force does not depend on charge.

4. Assertion (A): A free neutron is unstable.
Reason (R): A free neutron decays into a proton, an electron, and an antineutrino.

Answer: (A)
Correct. A free neutron has a mean life of about 15 minutes.

5. Assertion (A): Energy is released in nuclear fission.
Reason (R): The binding energy per nucleon of the products is greater than that of the parent nucleus.

Answer: (A)
System moves from lower stability (lower BE/nucleon) to higher stability (higher BE/nucleon), releasing energy.

6. Assertion (A): Fusion of hydrogen nuclei into helium releases energy.
Reason (R): In fusion, the mass of the product is greater than the sum of masses of reactants.

Answer: (C)
Assertion is True, Reason is False. Energy is released because mass is *lost* (mass defect). Product mass is *less* than reactants.

7. Assertion (A): Beta rays are deflected by electric and magnetic fields.
Reason (R): Beta rays consist of charged particles (electrons or positrons).

Answer: (A)
Correct. Since they carry charge, they experience Lorentz force.

8. Assertion (A): The half-life of a radioactive sample depends on the initial amount of the sample.
Reason (R): Half-life is a characteristic constant of the radioactive material.

Answer: (D)
Assertion is False. Half-life is independent of the initial amount. Reason is True.

9. Assertion (A): Nuclear density is not uniform throughout the nucleus.
Reason (R): It is maximum at the center and falls to zero at the surface.

Answer: (A)
While we approximate density as constant, in reality, it tapers off at the boundaries (“skin thickness”).

10. Assertion (A): Cadmium rods are used in nuclear reactors.
Reason (R): Cadmium is a good absorber of neutrons.

Answer: (A)
Correct. They act as control rods to regulate the reaction rate.

Part 3: Short & Long Answer Questions

1. Explain why nuclear forces are considered short-range forces. Draw a graph of potential energy vs distance between nucleons.

Nuclear forces are effective only over distances of the order of femtometers ( $10^{-15}$ m). Beyond ~2-3 fm, they drop to zero rapidly. (Draw standard potential energy curve showing attraction at $r > 0.8$ fm and repulsion at $r < 0.8$ fm).

2. Define Mass Defect and Binding Energy. Write the relation between them.

Mass Defect ( $\Delta m$ ): The difference between the sum of masses of constituent nucleons and the actual mass of the nucleus.
Binding Energy ( $E_b$ ): Energy required to separate the nucleons to infinity.
Relation: $E_b = \Delta m c^2$ .

3. Draw the Binding Energy per nucleon curve. Highlight the regions of stability, fission, and fusion.

Graph showing Binding Energy per nucleon versus Mass Number.

– Middle region ( $30 < A < 170$ ) is most stable (high BE/nucleon).
– Region $A < 30$ : Unstable, undergoes Fusion.
– Region $A > 170$ : Unstable, undergoes Fission.

4. State the Law of Radioactive Decay. Derive the relation $N = N_0 e^{-\lambda t}$ .

Rate of decay is proportional to number of undecayed nuclei: $-\frac{dN}{dt} \propto N \Rightarrow \frac{dN}{dt} = -\lambda N$ .
Integration from $N_0$ to $N$ yields $N = N_0 e^{-\lambda t}$ .

5. Distinguish between Nuclear Fission and Nuclear Fusion. Why is fusion more difficult to achieve?

Fission is splitting of heavy nuclei; Fusion is combining light nuclei.
Fusion is harder because it requires overcoming the immense electrostatic repulsion between positively charged nuclei, requiring extremely high temperatures ( $10^7$ K).

6. What is half-life ( $T_{1/2}$ )? Derive the relationship between half-life and decay constant ( $\lambda$ ).

Time taken for the number of nuclei to reduce to half its initial value.
At $t = T_{1/2}$ , $N = N_0/2$ .
$N_0/2 = N_0 e^{-\lambda T_{1/2}} \Rightarrow \ln 2 = \lambda T_{1/2} \Rightarrow T_{1/2} = \frac{0.693}{\lambda}$ .

7. Explain why heavier nuclei tend to have more neutrons than protons ( $N > Z$ ).

As $Z$ increases, electrostatic repulsion between protons increases drastically. To counterbalance this and maintain stability via the Strong Nuclear Force (which acts on all nucleons), extra neutrons are required to provide attractive force without adding repulsive charge.

8. Define ‘Activity’ of a radioactive sample. Define its SI unit.

Activity is the total decay rate of a sample: $R = -\frac{dN}{dt}$ .
SI Unit: **Becquerel (Bq)** = 1 decay/second.

9. In a nuclear reactor, what is the function of (i) Moderator, (ii) Control Rods, (iii) Coolant?

(i) Moderator (Heavy water/Graphite): Slows down fast neutrons.
(ii) Control Rods (Cadmium): Absorbs excess neutrons to control reaction rate.
(iii) Coolant (Water/Liquid Sodium): Transfers heat from core to turbine.

10. Why is the mass of a nucleus always less than the sum of the masses of its constituents?

This missing mass (Mass Defect) is converted into Binding Energy ( $E=mc^2$ ) which holds the nucleus together.

Part 4: Numericals

1. Calculate the energy released in MeV in the following reaction:
${}_{3}^{6}\text{Li} + {}_{0}^{1}\text{n} \rightarrow {}_{1}^{3}\text{H} + {}_{2}^{4}\text{He}$
Given masses: Li = 6.015126 u, n = 1.008665 u, H = 3.016049 u, He = 4.002603 u. (1 u = 931.5 MeV).

$\Delta m = (m_{Li} + m_n) - (m_H + m_{He})$
$\Delta m = (6.015126 + 1.008665) - (3.016049 + 4.002603)$
$\Delta m = 7.023791 - 7.018652 = 0.005139 \text{ u}$
Energy $Q = 0.005139 \times 931.5 = \textbf{4.78 MeV}$ .

2. The half-life of a radioactive substance is 30 days. Calculate (a) The decay constant, (b) Time taken for 3/4th of the original mass to disintegrate.

(a) $\lambda = 0.693 / 30 = 0.0231 \text{ days}^{-1}$ .
(b) If 3/4 disintegrates, remaining $N = N_0/4 = N_0 (1/2)^2$ .
This represents 2 half-lives.
Time $t = 2 \times 30 = \textbf{60 days}$ .

3. Calculate the Binding Energy per nucleon of an alpha particle ( ${}_{2}^{4}\text{He}$ ).
Given: $m_p = 1.007825$ u, $m_n = 1.008665$ u, $m_{He} = 4.002800$ u.

$\Delta m = [2(1.007825) + 2(1.008665)] - 4.002800$
$\Delta m = 4.032980 - 4.002800 = 0.03018 \text{ u}$
Total BE $= 0.03018 \times 931.5 = 28.11 \text{ MeV}$
BE/nucleon $= 28.11 / 4 = \textbf{7.03 MeV}$ .

4. A radioactive isotope has a half-life of T years. How long will it take the activity to reduce to 3.125% of its original value?

Remaining percentage = $3.125\% = \frac{3.125}{100} = \frac{1}{32}$ .
$\frac{1}{32} = \left(\frac{1}{2}\right)^5$ .
So, 5 half-lives are required.
Time $t = \textbf{5T years}$ .

5. Find the energy equivalent of 1 gram of substance.

$E = mc^2$
$E = (10^{-3} \text{ kg}) \times (3 \times 10^8)^2$
$E = 10^{-3} \times 9 \times 10^{16} = \textbf{9} \times \textbf{10}^{13} \text{ Joules}$ .

6. A nucleus with mass number 240 breaks into two fragments of mass numbers 120 each. The BE/nucleon of the unfragmented nucleus is 7.6 MeV, while that of the fragments is 8.5 MeV. Calculate the total gain in Binding Energy.

Initial BE $= 240 \times 7.6 = 1824 \text{ MeV}$ .
Final BE $= 2 \times (120 \times 8.5) = 240 \times 8.5 = 2040 \text{ MeV}$ .
Gain $= 2040 - 1824 = \textbf{216 MeV}$ .

7. The half-life of ${}^{238}\text{U}$ against $\alpha$ -decay is $4.5 \times 10^9$ years. Calculate the activity of 1g sample of ${}^{238}\text{U}$ .

$N = \frac{Avogadro}{238} = \frac{6.023 \times 10^{23}}{238} = 2.53 \times 10^{21}$ .
$\lambda = \frac{0.693}{4.5 \times 10^9 \times 365 \times 24 \times 3600} = 4.88 \times 10^{-18} s^{-1}$ .
Activity $R = \lambda N = (4.88 \times 10^{-18}) \times (2.53 \times 10^{21}) = \textbf{1.23} \times \textbf{10}^4 \text{ Bq}$ .

8. Two radioactive nuclei P and Q have disintegration constants $10\lambda$ and $\lambda$ . Initially, they have the same number of nuclei. At what time will the ratio of their nuclei be 1/e?

$\frac{N_P}{N_Q} = \frac{N_0 e^{-10\lambda t}}{N_0 e^{-\lambda t}} = e^{-9\lambda t}$ .
Given Ratio $= 1/e = e^{-1}$ .
$-9\lambda t = -1 \Rightarrow t = \textbf{1 / 9\lambda}$ .

9. Calculate the mass of ${}^{235}\text{U}$ required to produce 200 MW of power for 30 days, assuming 200 MeV is released per fission.

Total Energy $E = P \times t = 200 \times 10^6 \times (30 \times 24 \times 3600) = 5.184 \times 10^{14}$ J.
Energy per fission $= 200 \text{ MeV} = 3.2 \times 10^{-11}$ J.
Number of atoms $N = E / E_{fission} = 1.62 \times 10^{25}$ .
Mass $= \frac{N \times 235}{6.023 \times 10^{23}} = \textbf{6321 grams (approx 6.3 kg)}$ .

10. The decay constant for a radioactive nuclide is $1.5 \times 10^{-5} s^{-1}$ . What is the mean life?

Mean Life $\tau = \frac{1}{\lambda} = \frac{1}{1.5 \times 10^{-5}} = \textbf{6.67} \times \textbf{10}^4 \text{ seconds}$ .

Part 5: Case Studies

Case Study 1: Nuclear Fission
The discovery of nuclear fission of Uranium-235 by neutrons is the basis of nuclear reactors.

How much energy is released per fission of U-235?
Why is a chain reaction difficult to sustain in natural Uranium?
What is critical mass?

1. Approx 200 MeV.
2. Natural U contains mostly U-238 (99.3%) which absorbs neutrons without fission. U-235 is only 0.7%.
3. The minimum mass of fuel required to sustain a chain reaction.

Case Study 2: Binding Energy Curve
The stability of a nucleus depends on its binding energy per nucleon ( $E_{bn}$ ).

Which element has the highest $E_{bn}$ ?
Why does $E_{bn}$ decrease for very heavy nuclei?
Why is fusion energetically favorable for light nuclei?

1. Iron-56 ( $^{56}Fe$ ).
2. Due to increasing Coulomb repulsion between protons.
3. Light nuclei have low $E_{bn}$ . Fusing them creates a heavier nucleus with higher $E_{bn}$ , releasing the difference as energy.

Case Study 3: Radioactivity
Radioactivity is a random statistical process.

Does the rate of decay increase with temperature?
What is the end product of the Uranium decay series?
Which radiation has the highest penetrating power?

1. No, radioactivity is a nuclear phenomenon independent of temperature/pressure.
2. Lead ( $Pb$ ).
3. Gamma ( $\gamma$ ) rays.

Case Study 4: Carbon Dating
C-14 is a radioactive isotope used to determine the age of fossils.

What is the half-life of C-14?
Which type of decay does C-14 undergo?
Why can’t we use it for dating dinosaur bones (millions of years old)?

1. Approx 5730 years.
2. Beta decay ( $\beta^-$ ).
3. The half-life is too short; after millions of years, no detectable C-14 remains.

Case Study 5: Mass-Energy Equivalence
Einstein’s relation $E=mc^2$ revolutionized physics.

How many Joules are in 1 AMU (u)?
Is mass conserved in nuclear reactions?
What happens to mass in an exothermic nuclear reaction?

1. $1.49 \times 10^{-10}$ J (or 931.5 MeV).
2. No, mass is converted to energy.
3. A small fraction of mass is lost (converted to kinetic energy/radiation).

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Nuclei: Question Bank

Nuclei: Question Bank

Part 1: Multiple Choice Questions (1 Mark)

Part 2: Assertion-Reason Questions (1 Mark)

Part 3: Short & Long Answer Questions

Part 4: Numericals

Part 5: Case Studies

Editorial Team

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Part 1: Multiple Choice Questions (1 Mark)

Part 2: Assertion-Reason Questions (1 Mark)

Part 3: Short & Long Answer Questions

Part 4: Numericals

Part 5: Case Studies

Editorial Team

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