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Class XII · Chapter 13

Nuclei

Why a nucleus weighs slightly less than the sum of its parts — and how that tiny missing mass powers the sun, nuclear reactors, and radioactive decay.

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1Composition of the Nucleus

A nucleus is built from protons and neutrons, together called nucleons. It's fully described by two numbers: Z, the atomic number (number of protons, which fixes the element), and A, the mass number (total protons + neutrons).

  • Isotopes: same Z, different A (same element, different neutron count) — e.g. carbon-12 and carbon-14.
  • Isobars: same A, different Z.
  • Isotones: same number of neutrons (N = A − Z), different Z.

2Size and Density of the Nucleus

Nuclear Radius
R = R₀ A1/3, R₀ ≈ 1.2 fm
1 fm (femtometre) = 10⁻¹⁵ m — nuclei are roughly 10,000 times smaller than the atom containing them.
Key Insight Since volume scales as R³ ∝ A, and mass scales as A too, nuclear density is essentially the same for every nucleus, light or heavy — around 2.3×10¹⁷ kg/m³. This constancy is itself an important clue about how nucleons are packed and interact (see Nuclear Force, below).

3Mass Defect and Binding Energy

Weigh a nucleus carefully, and it always comes out slightly lighter than the sum of its individual, separated protons and neutrons. That missing mass — the mass defect — hasn't vanished; by Einstein's mass-energy equivalence, it was converted into the energy that holds the nucleus together.

Mass Defect
Δm = [Z mp + (A − Z) mn] − Mnucleus
Binding Energy
Eb = Δm c² = Δm × 931.5 MeV (Δm in atomic mass units, u)
This is exactly the energy you'd need to supply to pull the nucleus apart into free, separated nucleons — a direct measure of how tightly bound it is.

4Binding Energy per Nucleon

Dividing total binding energy by A gives binding energy per nucleon — a far more useful measure of nuclear stability than total binding energy alone, since it lets you compare nuclei of different sizes fairly.

The Curve That Explains Nuclear Energy Plotted against A, binding energy per nucleon rises sharply for light elements, peaks around iron and nickel (A ≈ 56), then slowly declines for heavier elements. This single curve explains why both fission (splitting heavy nuclei) and fusion (combining light nuclei) release energy: either process moves nuclei toward that peak, releasing the difference in binding energy.

5Nuclear Force

Protons packed together should repel each other electrically (Coulomb's Law, Chapter 1) — so something far stronger must hold a nucleus together. That's the nuclear force, distinct from gravity or electromagnetism.

  • Short-range: effective only over nuclear distances (a few fm); essentially zero beyond that.
  • Strongly attractive at these ranges — far stronger than the electric repulsion between protons.
  • Charge-independent: acts the same way between proton-proton, proton-neutron, and neutron-neutron pairs.
  • Saturated: each nucleon interacts only with its nearest neighbours, not with every other nucleon in the nucleus — this is exactly why nuclear density and binding energy per nucleon stay roughly constant regardless of nucleus size.

6Radioactivity and the Decay Law

Some nuclei are unstable and spontaneously transform, emitting radiation in the process — radioactivity. Decay is a random process for any single nucleus, but for a large sample, it follows a precise statistical law.

Law of Radioactive Decay
N = N₀ e−λt
N₀ = initial number of undecayed nuclei, λ = decay constant (probability of decay per unit time, characteristic of the isotope). Activity, the decay rate, is A = λN = −dN/dt.

7Half-Life and Mean Life

Half-Life
T½ = ln 2 / λ ≈ 0.693 / λ
The time for exactly half of any sample to decay — and this fraction is constant no matter how much (or little) sample you start with.
Mean Life
τ = 1/λ = T½ / 0.693 ≈ 1.44 T½
The average lifetime of a nucleus before it decays — always somewhat longer than the half-life.

8Alpha, Beta and Gamma Decay

  • Alpha decay: nucleus emits a helium nucleus (2 protons + 2 neutrons). A decreases by 4, Z decreases by 2. Alpha particles are heavy and slow, easily stopped (even by paper), but highly ionising at close range.
  • Beta-minus decay: a neutron converts to a proton, emitting an electron and an antineutrino. A stays the same, Z increases by 1.
  • Beta-plus decay: a proton converts to a neutron, emitting a positron and a neutrino. A stays the same, Z decreases by 1.
  • Gamma decay: a nucleus in an excited energy state drops to a lower state, emitting a high-energy photon. Neither A nor Z changes — it's pure de-excitation, no transmutation.

9Nuclear Fission and Fusion

Fission

A heavy nucleus (like uranium-235) splits into two lighter nuclei plus a few neutrons, releasing roughly 200 MeV per event. The released neutrons can trigger further fissions in nearby nuclei — a chain reaction, the basis of both nuclear reactors (controlled) and nuclear weapons (uncontrolled).

Fusion

Two light nuclei (like hydrogen isotopes) combine into a heavier one, also releasing energy — per unit mass, considerably more than fission. This is what powers the Sun and every other star, though it requires extreme temperature and pressure to force nuclei close enough to overcome their mutual electric repulsion.

Formula Summary

Nuclear Radius
R = R₀A^(1/3)
Binding Energy
E_b = Δm × 931.5 MeV
Decay Law
N = N₀e^(−λt)
Half-Life
T½ = 0.693/λ
Mean Life
τ = 1/λ
Alpha Decay
A→A−4, Z→Z−2
Beta-Minus Decay
A same, Z→Z+1

Solved Examples

Example 1 · Binding Energy

A nucleus has a mass defect of 0.03 u. Find its binding energy.

Solution: Eb = Δm × 931.5 MeV = 0.03 × 931.5 ≈ 27.9 MeV.

Example 2 · Half-Life

A radioactive sample has a half-life of 10 days. What fraction of the original sample remains after 30 days?

Solution: 30 days = 3 half-lives. Remaining fraction = (1/2)³ = 1/8 of the original sample.

Quick Check

1. The nuclear force is best described as:
Long-range and weak
✓ Short-range, strongly attractive, charge-independent (correct)
Only acting between protons
Identical in strength to gravity
2. In beta-minus (β⁻) decay, the mass number A:
Decreases by 1
✓ Remains unchanged (correct)
Increases by 1
Decreases by 4
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