JEE Advanced Physics Q&A – Mass-Energy Equivalence & Nuclear Reactions

Q 1 :

If a star can convert all the He nuclei completely into oxygen nuclei, the energy released per oxygen nucleus is [Mass of He nucleus is 4.0026 amu and mass of Oxygen nucleus is 15.9994 amu] [2005]

7.6 MeV
56.12 MeV
10.24 MeV
23.9 MeV

1

2

3

4

(3)

$4 {}_{2}^{4}{He} ⟶ {}_{8}^{16}O$

$B.E. = Δ m \times 931.5 MeV$

$= (4 \times 4.0026 - 15.9994) \times 931.5 = 10.24 MeV$

Q 2 :

In a radioactive decay chain reaction, ${}_{90}^{230}{Th}$ nucleus decays into ${}_{84}^{214}{Po}$ nucleus. The ratio of the number of $α$ to number of $β^{-}$ particles emitted in this process is ____. [2022]

(2)

${}_{90}^{230}{Th} ⟶ {}_{84}^{214}{Po} + n {}_{2}^{4}{He} + m {}_{- 1}^{0}e$

$\Rightarrow 230 = 214 + 4 n$ and $90 = 84 + 2 n - m$

$\Rightarrow 16 = 4 n$ and $6 = 2 n - m$

$\Rightarrow n = 4$ $\Rightarrow 6 = 8 - m$ $(∵ n = 4)$

$\Rightarrow m = 2$

So, $\frac{n}{m} = \frac{4}{2} = 2$

Q 3 :

The binding energy of nucleons in a nucleus can be affected by the pairwise Coulomb repulsion. Assume that all nucleons are uniformly distributed inside the nucleus.

Let the binding energy of a proton be $E_{b}^{p}$ and the binding energy of a neutron be $E_{b}^{n}$ in the nucleus. Which of the following statement(s) is(are) correct? [2022]

$E_{b}^{p} - E_{b}^{n}$ is proportional to $Z (Z - 1)$ where $Z$ is the atomic number of the nucleus.
$E_{b}^{p} - E_{b}^{n}$ is proportional to $A^{- \frac{1}{3}}$ where $A$ is the mass number of the nucleus.
$E_{b}^{p} - E_{b}^{n}$ is positive.
$E_{b}^{p}$ increases if the nucleus undergoes a beta decay emitting a positron.

1

2

3

4

Select one or more options

(1, 2, 4)

Binding energy of proton and neutron due to nuclear force is the same. So, the difference is due to electrostatic potential repulsion energy, and it is positive.

So $E_{b}^{p} - E_{b}^{n} =$ Electrostatic potential energy

Number of proton pairs $= {}^{Z}C_{2} = \frac{Z (Z - 1)}{2}$

So, Repulsion energy $\propto \frac{Z (Z - 1)}{2} \times \frac{1}{4 π ε_{0}} \cdot \frac{e^{2}}{R},$

$R =$ radius of the nuclei

$E_{b}^{p} - E_{b}^{n} \propto Z (Z - 1)$ $\Rightarrow (1) is correct.$

As, $R = R_{0} A^{1 / 3}$

So, $E_{b}^{p} - E_{b}^{n} \propto A^{- 1 / 3}$ $\Rightarrow (2) is correct.$

Because of electrostatic repulsion,

$E_{b}^{p} < E_{b}^{n}$ $\Rightarrow (3) is incorrect.$

Since in $β^{+}$ decay the number of protons decreases, electrostatic repulsion decreases.

$\Rightarrow E_{b}^{p} increases$ $\Rightarrow (4) is correct.$

Q 4 :

A heavy nucleus $N$ , at rest, undergoes fission $N \to P + Q,$ where $P$ and $Q$ are two lighter nuclei. Let $δ = M_{N} - M_{P} - M_{Q},$ where $M_{P}$ , $M_{Q}$ and $M_{N}$ are the masses of $P$ , $Q$ and $N$ , respectively. $E_{P}$ and $E_{Q}$ are the kinetic energies of $P$ and $Q$ , respectively. The speeds of $P$ and $Q$ are $v_{P}$ and $v_{Q}$ , respectively. If $c$ is the speed of light, which of the following statement(s) is(are) correct? [2021]

$E_{P} + E_{Q} = c^{2} δ$
$E_{P} = (\frac{M_{P}}{M_{P} + M_{Q}}) c^{2} δ$
$\frac{v_{P}}{v_{Q}} = \frac{M_{Q}}{M_{P}}$
The magnitude of momentum for $P$ as well as $Q$ is $c \sqrt{2 μ δ},$ where $μ = \frac{M_{P} M_{Q}}{(M_{P} + M_{Q})}$

1

2

3

4

Select one or more options

(1, 3, 4)

For nuclear fission reaction,

$N \to P + Q$

Energy released $= Δ M c^{2} = (M_{N} - M_{P} - M_{Q}) c^{2} = δ c^{2}$

This will be distributed as kinetic energy of $P$ and $Q$

$∴$ $E_{P} + E_{Q} = δ c^{2}$ $\dots (i)$

By conservation of momentum,

$V_{P} M_{P} = V_{Q} M_{Q}$

$∴$ $\frac{V_{P}}{V_{Q}} = \frac{M_{Q}}{M_{P}}$ $\dots (i i)$

Kinetic energy can be written as $KE = \frac{P^{2}}{2 m}$

Hence, kinetic energies are divided in the inverse ratio of masses.

$E_{P} = \frac{M_{Q}}{M_{P} + M_{Q}} δ c^{2}$ $\dots (i i i)$

By equation (i),

$\Rightarrow \frac{P^{2}}{2 M_{P}} + \frac{P^{2}}{2 M_{Q}} = δ c^{2}$

$\Rightarrow \frac{P^{2}}{2 μ} = δ c^{2}$ $\Rightarrow P = c \sqrt{2 μ δ}$ $[∵ μ = \frac{M_{P} M_{Q}}{M_{P} + M_{Q}}]$

Q 5 :

Assume that the nuclear binding energy per nucleon (B/A) versus mass number (A) is as shown in the figure. Use this plot to choose the correct choice(s) given below. [2008]

Fusion of two nuclei with mass numbers lying in the range of 1 < A < 50 will release energy.
Fusion of two nuclei with mass numbers lying in the range of 51 < A < 100 will release energy.
Fission of a nucleus lying in the mass range of 100 < A < 200 will release energy when broken into two equal fragments.
Fission of a nucleus lying in the mass range of 200 < A < 260 will release energy when broken into two equal fragments.

1

2

3

4

Select one or more options

(2, 4)

In fusion two or more lighter nuclei combine to form a comparatively heavier nucleus. When binding energy per nucleon increases for a nuclear process, energy is released. In fission, a heavy nucleus breaks into two or more lighter nuclei.

(1) For 1 < A < 50, on fusion mass number of the resulting nucleus will be less than 100.

(2) For 51 < A < 100, on fusion mass number the resulting nucleus is between 100 and 200. B/A increases, energy will be released.

(3) On fission for 100 < A < 200, the mass number for fission nuclei will be between 50 to 100. B/A decreases, no energy will be released.

(4) On fission for 200 < A < 260, the mass number for fission nuclei will be between 100 to 130. B/A will increase, energy will be released.

Topic Question Set

Q 1 :

If a star can convert all the He nuclei completely into oxygen nuclei, the energy released per oxygen nucleus is [Mass of He nucleus is 4.0026 amu and mass of Oxygen nucleus is 15.9994 amu] [2005]

Q 2 :

In a radioactive decay chain reaction, ${}_{90}^{230}{Th}$ nucleus decays into ${}_{84}^{214}{Po}$ nucleus. The ratio of the number of $α$ to number of $β^{-}$ particles emitted in this process is ____. [2022]

Q 5 :

Assume that the nuclear binding energy per nucleon (B/A) versus mass number (A) is as shown in the figure. Use this plot to choose the correct choice(s) given below. [2008]

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Topic Question Set

Q 1 : If a star can convert all the He nuclei completely into oxygen nuclei, the energy released per oxygen nucleus is [Mass of He nucleus is 4.0026 amu and mass of Oxygen nucleus is 15.9994 amu] [2005]

Q 2 : In a radioactive decay chain reaction, Th90230 nucleus decays into Po84214 nucleus. The ratio of the number of α to number of β- particles emitted in this process is ____. [2022]

Q 5 : Assume that the nuclear binding energy per nucleon (B/A) versus mass number (A) is as shown in the figure. Use this plot to choose the correct choice(s) given below. [2008]

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Q 1 :

If a star can convert all the He nuclei completely into oxygen nuclei, the energy released per oxygen nucleus is [Mass of He nucleus is 4.0026 amu and mass of Oxygen nucleus is 15.9994 amu] [2005]

Q 2 :

In a radioactive decay chain reaction, ${}_{90}^{230}{Th}$ nucleus decays into ${}_{84}^{214}{Po}$ nucleus. The ratio of the number of $α$ to number of $β^{-}$ particles emitted in this process is ____. [2022]

Q 5 :

Assume that the nuclear binding energy per nucleon (B/A) versus mass number (A) is as shown in the figure. Use this plot to choose the correct choice(s) given below. [2008]