NucleiNEET Physics · Class 12 · NCERT Chapter 13

Introduction

The nucleus is tiny (~10⁻¹⁵ m), positive, and holds nearly all of the atom's mass. Understand it and you understand where the Sun's energy comes from, why Chernobyl was a disaster, and how we know the age of ancient bones. This chapter is short on heavy maths but heavy on facts you must memorise: the BE curve, the decay law, half-life vs mean life, and 1 u = 931.5 MeV.

Expect 1 to 2 NEET questions every year. Common asks: half-life, mean life, alpha and beta decay equations, binding energy from mass defect, position of Fe-56 on the BE curve, and energy released in fission or fusion.

Composition: Z, N, A

A nucleus has Z protons and N = A - Z neutrons. A is the mass number (total nucleons). Notation: ᴬ_Z X. E.g. ¹⁴₆C means 6 protons, 8 neutrons. Isotopes: same Z, different N (chemistry identical, mass different). Isobars: same A, different Z. Isotones: same N.

Nuclear size and density

Nuclear volume scales as A. Mass scales as A. So density is essentially the same for every nucleus, ~ 2.3 × 10¹⁷ kg/m³. A teaspoon of nuclear matter would weigh half a billion tonnes.

Nuclear radius scales as the cube root of A: R = R_0 A^(1/3). Nuclear density is nearly the same for every nucleus (~ 2.3 × 10¹⁷ kg/m³).

Mass-energy equivalence

Einstein's relation E = m c². In nuclear units, the natural mass unit is the atomic mass unit:

So even tiny mass differences correspond to large nuclear energies.

Mass defect and binding energy

If you weigh the protons and neutrons separately and add up, the total exceeds the actual mass of the nucleus. The difference is the mass defect:

Multiplied by c², it is the binding energy of the nucleus:

The BE per nucleon = BE / A. Roughly 8 MeV for most stable nuclei.

Mass defect = constituent masses - actual nuclear mass. Multiply by c² (using 1 u = 931.5 MeV) to get binding energy. Divide by A for BE per nucleon.

BE per nucleon curve

Plot BE/A against A. Three regions:

• Steep climb from H to about A = 30 (more nuclear-force partners → tighter binding).
• Peak at Fe-56 with BE/A ≈ 8.79 MeV (most stable).
• Slow decline beyond Fe (Coulomb repulsion of many protons starts to dominate).

Consequence: nuclei lighter than Fe gain energy when they fuse (move up the curve). Nuclei heavier than Fe gain energy when they split (also moving towards Fe).

Hover or click a dot to see that nucleus. Curve peaks near Fe-56 at ~8.8 MeV per nucleon. Light → fusion gains energy (climbing). Heavy → fission gains energy (falling back to peak).

Nuclear force

• Short range: ~ 1 fm. Beyond a few fm, it vanishes.
• Strongly attractive at the nuclear range; repulsive at very small distances (~0.4 fm).
• Charge-independent: same between p-p, n-n, p-n.
• Saturated: each nucleon interacts only with its neighbours, not all others.
• Spin-dependent (deuteron is bound only in spin-aligned state).

Radioactivity: alpha, beta, gamma

Some nuclei are unstable and decay spontaneously by emitting:

• Alpha (α): a He-4 nucleus. ΔZ = -2, ΔA = -4. e.g. ²³⁸₉₂U → ²³⁴₉₀Th + α.
• Beta-minus (β⁻): an electron + antineutrino. n → p + e⁻ + ν̄. ΔZ = +1, ΔA = 0.
• Beta-plus (β⁺): a positron + neutrino. p → n + e⁺ + ν. ΔZ = -1, ΔA = 0.
• Gamma (γ): a high-energy photon. No change in Z or A.

Compare the three radioactive decay channels. NEET tests the rules ΔZ, ΔA and the example chain.

Radioactive decay law

Each nucleus has a constant probability per unit time to decay. So:

Activity (number of disintegrations per second):

SI unit: becquerel (Bq) = 1 disintegration per second. Older unit: curie (Ci) = 3.7 × 10¹⁰ Bq.

Number of undecayed nuclei drops exponentially. Every half-life, half of what is left disappears.

Half-life, mean life, activity

Time for the number to drop to half:

Mean life:

Both N and A reduce by the same factor: after n half-lives, fraction left = (1/2)ⁿ.

Three quantities, fully determined by any one. Pick which to type.

Application: carbon dating

Living things absorb C-14 at a fixed ratio with C-12 (set by cosmic rays in the atmosphere). After death the ratio falls exponentially with t_1/2 = 5730 years. Measure the C-14 fraction left, get the age.

Living things absorb C-14 at the same ratio as the atmosphere. After death, no more C-14 is added; what remains decays with t_1/2 = 5730 years. Measure the C-14 left, get the age.

Nuclear fission

A heavy nucleus splits into two medium-mass nuclei plus neutrons:

BE/n rises from 7.6 MeV (U-235) to ~8.5 MeV (products). Difference × A ≈ 200 MeV released as kinetic energy. The released neutrons can trigger more fissions (chain reaction). Controlled in nuclear reactors via control rods that absorb neutrons; uncontrolled in nuclear weapons.

Nuclear fusion

Two light nuclei combine into a heavier one:

Powers stars including the Sun (where four hydrogens fuse via the proton-proton chain). On Earth, controlled fusion is hard because temperatures of ~10⁷ K are needed to overcome Coulomb repulsion.

Energy released in nuclear reactions = (BE_after − BE_before) of the system. Both fission of heavy nuclei and fusion of light ones move products closer to Fe-56, releasing the difference.

Worked NEET problems

Summary cheat sheet

• Size: $R=R_0{A}^{1/3}$, R_0 = 1.2 fm.
• Density: ~ 2.3 × 10¹⁷ kg/m³ (constant).
• Mass-energy: 1 u = 931.5 MeV.
• Mass defect: $\Delta M=Z{m}_p+(A-Z)m_n-M$.
• BE: $\text{BE}=\Delta M\times 931.5\text{MeV}$.
• Curve peak: Fe-56 at ~8.79 MeV/n.
• α decay: ΔZ = -2, ΔA = -4. β⁻: ΔZ = +1, ΔA = 0.
• Decay law: $N=N_0{e}^{-\lambda t}$.
• Half-life: $t_{1/2}=\ln 2/\lambda$. Mean life: $\tau =1/\lambda \approx 1.44t_{1/2}$.
• Activity: $A=\lambda N$.
• Fission U-235: ~200 MeV. Fusion 4 H → He: ~26.7 MeV.

Next: try the interactive widgets for binding energy, half-life and fission/fusion, or work through the 30 NEET PYQs with full solutions. To time yourself, take the free 10-question mock test.