Nuclei | Class 12 Physics (NCERT Ch 13)

CBSE Class 12 › Unit VIII: Atoms & Nuclei

Nuclei

NCERT Chapter 13 • Binding Energy, Radioactivity, Fission & Fusion

NCERT 2025–26 Unit VIII • ~3 Marks JEE Main • 1 Question

1. Composition and Notation

A nucleus is denoted as $^A_Z X$ , where:

$Z$ = Atomic number = Number of protons
$N$ = Neutron number = Number of neutrons
$A = Z + N$ = Mass number = Total nucleons

Atomic Mass Unit (u):
$1 \, \text{u} = \dfrac{1}{12} \times \text{mass of } ^{12}\text{C atom} = 1.6605 \times 10^{-27} \, \text{kg}$

Isotopes, Isobars, Isotones:

Isotopes: Same $Z$ , different $A$ (e.g., $^{12}_6C$ , $^{14}_6C$ )
Isobars: Same $A$ , different $Z$ (e.g., $^{40}_{18}Ar$ , $^{40}_{20}Ca$ )
Isotones: Same $N$ , different $Z$ (e.g., $^{13}_6C$ , $^{14}_7N$ )

Discovery of Neutron:

James Chadwick (1932) bombarded beryllium with α-particles and observed neutral radiation that could knock out protons from paraffin wax. He concluded it was a new particle — the neutron — with mass ≈ proton.

2. Size and Density of Nucleus

Rutherford’s scattering experiments showed the nucleus is tiny (~10⁻¹⁵ m) compared to the atom (~10⁻¹⁰ m).

Radius of Nucleus:
$R = R_0 A^{1/3}$
Where $R_0 = 1.2 \times 10^{-15} \text{ m} = 1.2 \text{ fm}$ .

Nuclear Density:

Volume ∝ $R^3$ ∝ $A$ → Density = $\dfrac{\text{Mass}}{\text{Volume}} \propto \dfrac{A}{A} = \text{constant}$ .
Nuclear density ≈ $2.3 \times 10^{17} \text{ kg/m}^3$ (independent of $A$ ).

3. Mass-Energy and Binding Energy

Einstein’s mass-energy equivalence: $E = mc^2$ . The mass of a nucleus is always less than the sum of its free nucleons.

Mass Defect ( $\Delta M$ ):
$\Delta M = [Z m_p + (A-Z) m_n] - M_{\text{nucleus}}$

Binding Energy (BE):
Energy equivalent of mass defect:
$E_b = \Delta M c^2 = \Delta M \times 931.5 \text{MeV/u}$

Binding Energy per Nucleon:
$E_{bn} = \dfrac{E_b}{A}$

Graph of Binding Energy per Nucleon vs Mass Number. — The stability of a nucleus depends on its Binding Energy per Nucleon ( $E_{bn}$ ).

A range	E_bn (MeV)	Process
A < 30	<8	Fusion releases energy
30–170	~8	Stable
A > 170	<8	Fission releases energy

Key Conclusions from BE Curve:

$E_{bn}$ peaks at $A=56$ (Iron) → most stable nucleus.
$E_{bn}$ is lower for light nuclei ( $A<30$ ) and heavy nuclei ( $A>170$ ).
Energy is released when:
- Heavy nucleus splits → Fission (e.g., Uranium)
- Light nuclei fuse → Fusion (e.g., Hydrogen in Sun)

4. Nuclear Force

The strong attractive force that binds nucleons together.

Properties:

Short-range: Effective only up to ~2-3 fm.
Charge independent: Same for p-p, n-n, p-n pairs.
Stronger than Coulomb force: Overcomes proton-proton repulsion.
Repulsive core: Becomes strongly repulsive below 0.8 fm.

Graph showing variation of potential energy between two nucleons with distance. — The nuclear force is strongly attractive but becomes repulsive at extremely short distances.

5. Radioactivity

Spontaneous emission of radiation ( $\alpha, \beta, \gamma$ ) by unstable nuclei.

Derivation: Law of Radioactive Decay 3 Marks

Statement: Rate of decay ∝ number of undecayed nuclei.
$\dfrac{dN}{dt} = -\lambda N$

Integration:
$\int_{N_0}^{N} \dfrac{dN}{N} = -\lambda \int_{0}^{t} dt$
$\ln \left(\dfrac{N}{N_0}\right) = -\lambda t$

Final Result:
$N(t) = N_0 e^{-\lambda t}$

Exponential decay graph showing half-life of a radioactive substance.

Derivation: Half-Life ( $T_{1/2}$ ) 2 Marks

At $t = T_{1/2}$ , $N = N_0 / 2$ :
$\dfrac{N_0}{2} = N_0 e^{-\lambda T_{1/2}}$

$e^{\lambda T_{1/2}} = 2$ → $\lambda T_{1/2} = \ln 2$
$T_{1/2} = \dfrac{0.693}{\lambda}$

Mean Life ( $\tau$ ): $\tau = \dfrac{1}{\lambda} = 1.44 T_{1/2}$ .
Activity (R): $R = \lambda N$ (unit: Becquerel, Bq).

6. Nuclear Energy: Fission & Fusion

Q-value:
$Q = [\text{Initial mass} - \text{Final mass}] c^2$
Positive Q = exothermic (energy released).

A. Nuclear Fission

Heavy nucleus splits into lighter fragments + neutrons + energy.

$_{92}^{235}\text{U} + _0^1\text{n} \rightarrow _{56}^{144}\text{Ba} + _{36}^{89}\text{Kr} + 3 _0^1\text{n} + 200 \text{ MeV}$

Q-value: Energy released = (Initial mass – Final mass) × $c^2$ .
For fission, Q ≈ 200 MeV per event → million times more than chemical reactions.

Schematic diagram of nuclear fission of Uranium-235.

B. Nuclear Fusion

Light nuclei combine to form heavier nucleus + energy.

$_1^2\text{H} + _1^2\text{H} \rightarrow _2^3\text{He} + _0^1\text{n} + 3.27 \text{ MeV}$

Energy Source of Stars:
In the Sun, hydrogen fuses to helium via the proton-proton (p-p) cycle:
$4 _1^1\text{H} \rightarrow _2^4\text{He} + 2e^+ + 2\nu_e + 26.7 \text{ MeV}$

Thermonuclear Fusion:

Requires temperatures > 10⁷ K to overcome Coulomb barrier.

Challenge:

Confining plasma (ionized gas) at such temperatures (no material container can withstand it).

Methods: Magnetic confinement (Tokamak), inertial confinement (lasers).

Practice Time!

Solve questions on Half-life and Mass Defect: Important Numericals for Chapter 13 →

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