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Correct option :  (3)

Explanation :

Energy is released in a nuclear (fission or fusion) reaction, if the binding energy per nucleon of the products, is more than that for reactants. This is so only far statement (3). In all the other statements, B.E/nucleon, of products, is less than that of the reactants. In such a process energy, has to be absorbed.

The unified atomic mass unit is an accepted but non-SI unit of mass. It is defined to be equal to 1/12 of the mass of a free atom of the isotope of carbon with mass number 12 which is at rest and in its ground state. It is denoted by u.

Its value in SI unit is obtained experimentally. 1 u = 1.660538782 (83) × 10^-27 kg, with the standard uncertainty in the last two digits given in v parenthesis.

Taking, 1 u = 1.660538782 × 10^-27 kg

c = 2.99792458 × 10⁸ m/s,

e = 1.602176462 × 10^-19 C and using the relation E = me² { ≡ m ≡ E / c² ),

we get, 1 u = 1.49241783 × 10^-10 J/c²

≅ 1.492 × 10^-10 J/c²

and 1 u = 931.494042 MeV / c²

≅ 931.5 MeV/c²

For t = nT, 

N = N₀/ 2ⁿ.

In this case, 

n = 2.

∴ f = 2T 

= 2 × 4 = 8 days is the required time.

Nature of β-particles : A β-particle is an electron or a positron. 

Properties of β-particles :

1. β-particles have a moderate ionizing power. It is about 100 times less than that of α-particles, but 100 times more than that of γ-rays. 

2. They have a moderate penetrating power. It is about 100 times more than that of α-particles, but 100 times less than that of γ-rays.

3. They are deflected by electric and magnetic fields. Their deflection is more than the deflection of α-particles in the same field but in the opposite direction.

4. They affect photographic plates. 

5. They cause fluorescence in a fluorescent material such as zinc sulphide.

6. Their energies and speeds are very high. Their speed is of the order of 10 m/s. Some β-particles have speeds of the order of 0.99 c, where c is the speed of light in free space. 

7. They cause more biological damage than α-particles, because they have more penetrating power.

[Note : When a nucleus emits an electron, one of its neutrons changes to a proton; the electron is accompanied by a neutral and almost massless particle called antineutrino \(\vec v_e\) with which the electron shares its energy and momentum. Hence, β-particles are emitted with speeds ranging from about 0 to 0.99 c

The properties of a positron are identical to those of an electron except that it carries a positive charge of the same magnitude as an electron. A positron emission in a β⁺ decay is accompanied by a neutrino v_e. A positron is an anti-particle of an electron, and an antineutrino is an antiparticle of neutrino.

The existence of a small neutral particle, emitted simultaneously with the electron in β -decay, was proposed in 1931 by Wolfgang Pauli (1900 -1958), Austrian-US theoretical physicist. It was confirmed experimentally in 1956 by Frederic Reines and Clyde Lorrain Cowan, Jr., US physicists. This neutral particle, that appears in β⁺ -decay, was called the neutrino. It travels with a speed very close to that of light in free space. The existence of positron (in 1928) and other antiparticles was predicted by Paul Adrien Maurice Dirac (1902 – 84), British theoretical physicist. All these predictions were eventually confirmed experimentally; the positron was discovered in 1932 by Carl David Anderson, US physicist.]

Correct option is (B) 1 – \(\cfrac1e\)

Nature of γ-rays :

1. γ-rays are electromagnetic waves of very short wavelength (about 10^-12 m to 10^-14 m). 

2. They are uncharged.

Properties of γ-rays :

1. γ-rays produce feeble ionization. Their ionizing power is 104 times less than that of a-particles and 100 times less than that of β-particles.

2. They have the maximum penetrating power. It is about 100 times that of βparticles and 10⁴ times that of αparticles.

3. They are not deflected by electric and magnetic fields as they are uncharged.

4. They affect photographic plates.

5. They cause fluorescence in a fluorescent material such as zinc sulphide.

6. Their speed in free space is 3 × 108 m/s( the same as that of light waves and X-rays in free space).

7. γ-rays can be diffracted by crystals. In recent times, γ-ray diffraction has emerged as a powerful tool in structural and defect studies of crystals.

8. They destroy living cells and tissues and are used for destroying cancer cells.

[Note : γ-rays (not established clearly in Berquerel’s work) were discovered in 1900 by Paul Villard (1860-1934), French physicist.]

A given nucleus does not emit α- and βparticles simultaneously. However, on emission of α or β-particles, most nuclei are left in an excited state. A nucleus in an excited state emits a γ-ray photon in a transition to the lower energy state. Hence, α- and βparticle emissions are often accompanied by γ-rays.

(a) A radioactive transformation in which an aparticle is emitted is called α-decay. In an α-decay, the atomic number of the nucleus decreases by 2 and the mass number decreases by 4.

Example : \(^{38}_{92}U\) → \(^{234}_{90}Th\) + \(^4_2α\)

Q = [m_U – m_Th – m_α]c²

(b) A radioactive transformation in which a βparticle is emitted is called β-decay. In a β -decay, the atomic number of the nucleus increases by 1 and the mass number remains unchanged.

Example : \(^{23}_{90}Th\) → \(^{234}_{91}Pa\) + \(^0_{-1}e\) + \(\bar v_e\)

where \(\bar v_e\) is the antineutrino emitted to conserve the momentum, energy and spin. Q = [m_Th – m_pa – m_e]c²

In a β⁺ -decay, the atomic number of the nucleus decreases by 1 and the mass number remains unchanged.

Example : \(^{30}_{15}P\) → \(^{30}_{14}Si\) + \(^0_{+1}e\) + v_e

where v_e is the neutrino emitted to conserve the momentum, energy and spin.

Q = [m_p – m_Si – m_e]c²

[Note : The term f_i particle refers to the electron (or positron) emitted by a nucleus.]

The decay constant or disintegration constant of a radioactive element is defined as the ratio of the disintegration rate at an instant to the number of undecayed nuclei of the element present at that instant.

Let N₀ be the number of nuclei of a radioactive element present at time t = 0 and N, the number of undecayed nuclei at time t. From the radioactive law,

N = N₀e^-λt

where λ is the decay constant. At t = λ^-1, the fraction of undecayed nuclei is 

\(\cfrac{N}{N_0}\) = e^{-λ × λ-1} = e^-1 = \(\cfrac1e\)

Since e ≅ 2.718,

\(\cfrac{N}{N_0}\) = \(\cfrac1{2.718}\) = 0.3679

Therefore, about 36.79% ≈ 37% of the original nuclei remains undecayed after a time λ^-1 Since λ is the probability that a nucleus of the element will decay in one second, λ^-1 gives the mean-life or the mean life time τ of the radioactive element measured in second; τ = λ^-1.

1. Number of α-particles emitted = 32/4 = 8 

2. Number of β-particles emitted = 16 – 8 = 8.

Data : R₀ = 1.2 × 10^-15 m

Nuclear radius, R=R₀A^1/3

(i) For ²⁰⁶Pb, A = 206

∴ R = (1.2 × 10^-15) (206)^1/3

= 1.2 × 10^-15 × 5.906 

= 7.087 × 10^-15 m = 7.087 fm

(ii) For ²⁰⁸Pb, A = 208

∴ R = (1.2 × 10^-15)(208)^1/3

= 1.2 × 10^-15 × 5.925 

= 7.11 × 10^-15 m = 7.11 fm

A reaction between the nucleus of an atom and a bombarding particle leading to the production of a new nucleus and, in general, the ejection of one or more particles is known as a nuclear reaction.

Example : \(^4_2\alpha\) + \(^{27}_{13}AI\) \(\longrightarrow\) \(^{30}_{15}P\) + \(^4_0n\) + Q

Here, \(^4_2\alpha\) is the bombarding particle, \(^{27}_{13}AI\) is the target nucleus, \(^{30}_{15}P\) is the product nucleus \(^4_0n\) is the energy released in the reaction. It is carried (usually) by the ejected particle.

The total momentum, energy, spin, charge and number of nucleons are conserved in a nuclear reaction.

Bohr proposed :

(i) quantised of energy states related to the transition of electrons from one orbit to another. 

(ii) quantisation of orbit or angular momentum.