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Class 12 Physics Notes

Nuclei Class 12 Notes

Complete, exam-ready notes on the nucleus: composition and size of the nucleus, mass defect and binding energy, radioactivity with the exponential decay law, half-life and mean life, and nuclear fission and fusion — written for CBSE boards, JEE and NEET revision.

Class12SubjectPhysicsCoversCBSE · JEE · NEET

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

What is the nucleus in one line?

The nucleus is a tiny, dense core of protons and neutrons (nucleons) held together by the strong nuclear force; its energy is tied to the mass defect, released in radioactivity, fission and fusion.

Nuclear Composition and Size

Nucleons

Protons and neutrons inside a nucleus. Atomic number Z counts protons, mass number A = Z + N counts total nucleons. Nuclei with the same Z but different N are isotopes; the same A are isobars; the same N are isotones.

  • Nuclear radius:
  • R=R0A1/3R = R_0 A^{1/3}
  • with
  • R0≈1.2 fmR_0 \approx 1.2\,\text{fm}
  • .
  • Nuclear density is nearly constant for all nuclei, about
  • 2.3×1017 kg/m32.3\times 10^{17}\,\text{kg/m}^3
  • — far denser than ordinary matter.
  • The strong nuclear force binds nucleons and is short-range, stronger than electric repulsion between protons.
R=R0A1/3,R0≈1.2 fmR = R_0 A^{1/3},\qquad R_0 \approx 1.2\,\text{fm}
Nuclear radius

Mass Defect and Binding Energy

Mass defect

Δm\Delta m

The nucleus is lighter than its separate nucleons: Δm=[Zmp+(A−Z)mn]−Mnucleus\Delta m = [Zm_p + (A-Z)m_n] - M_{\text{nucleus}}. This mass is converted into the binding energy holding the nucleus together, Eb=Δm c2E_b = \Delta m\,c^2.

  • 1 atomic mass unit (u) = 1.66 × 10⁻²⁷ kg ≈ 931.5 MeV of energy.
  • Binding energy per nucleon peaks near iron-56 (~8.8 MeV), the most stable nucleus.
  • Nuclei lighter than iron gain energy by fusing; nuclei heavier than iron gain energy by fission.
Δm=[Zmp+(A−Z)mn]−M,Eb=Δm c2\Delta m = [Zm_p + (A-Z)m_n] - M,\quad E_b = \Delta m\,c^2
Mass defect and binding energy

Radioactivity and Decay Laws

Radioactive decay

The spontaneous disintegration of an unstable nucleus. α decay emits a helium nucleus (A−4, Z−2), β⁻ decay converts a neutron into a proton (A same, Z+1) with an antineutrino, and γ decay releases excess energy with no change of Z or A.

  • Decay law:
  • N=N0 e−λtN = N_0\,e^{-\lambda t}
  • where λ is the decay constant.
  • Activity:
  • A=λNA = \lambda N
  • , in becquerel (1 Bq = 1 decay/s).
  • Mean life:
  • τ=1λ\tau = \frac{1}{\lambda}
  • .
N=N0e−λt,A=λN,τ=1λN = N_0e^{-\lambda t},\qquad A = \lambda N,\qquad \tau = \frac{1}{\lambda}
Decay law, activity and mean life

Half-Life and Mean Life

Half-life

T1/2T_{1/2}

The time for half the nuclei in a sample to decay: T1/2=ln⁡2λ=0.693λT_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}. After n half-lives the fraction remaining is (1/2)n(1/2)^n.

  • Carbon-14 dating uses the 5730-year half-life to date once-living material.
  • Mean life is the average lifetime of a nucleus:
  • τ=1.443 T1/2\tau = 1.443\,T_{1/2}
  • .
  • If the fraction remaining is f, the age is
  • t=ln⁡(1/f)λt = \frac{\ln(1/f)}{\lambda}
  • .
T1/2=0.693λ,τ=1λ=1.443 T1/2T_{1/2} = \frac{0.693}{\lambda},\quad \tau = \frac{1}{\lambda} = 1.443\,T_{1/2}
Half-life and mean life

Nuclear Fission and Fusion

Fission

A heavy nucleus (e.g. Uranium-235) splits after absorbing a slow neutron, releasing energy and 2–3 more neutrons. A chain reaction results; a nuclear reactor controls it with a moderator and control rods.

  • Fusion joins light nuclei (e.g. deuterium + tritium → helium + neutron) releasing enormous energy — the source of the Sun's power.
  • Fusion needs very high temperatures (thermonuclear) and is not yet harnessed for power plants.
  • Energy released:
  • Q=(minitial−mfinal)c2Q = (m_{\text{initial}} - m_{\text{final}})c^2
  • .

Stability rule

Energy is released whenever the products are closer to iron on the binding-energy curve: fission for heavy nuclei, fusion for light ones. This single idea answers most energy questions.

Solved Examples

Example: The half-life of a radioactive substance is 5 years. What fraction of a sample remains after 15 years?

Solution: 15 years = 3 half-lives, so the fraction remaining is (12)3=18\left(\tfrac{1}{2}\right)^3 = \frac{1}{8}. Equivalently, using N=N0e−λtN = N_0e^{-\lambda t} with λ=0.693/5\lambda = 0.693/5 gives the same result.

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

Nuclear radius

R=R0A1/3R = R_0 A^{1/3}

Mass defect

Δm=[Zmp+(A−Z)mn]−M\Delta m = [Zm_p + (A-Z)m_n] - M

Binding energy

Eb=Δm c2E_b = \Delta m\,c^2

Decay law

N=N0e−λtN = N_0e^{-\lambda t}

Half-life

T1/2=0.693λT_{1/2} = \frac{0.693}{\lambda}

Activity

A=λNA = \lambda N

Mean life

τ=1λ\tau = \frac{1}{\lambda}

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

  • Nuclear radius R = R₀A^(1/3); density is constant across all nuclei.
  • Binding energy per nucleon peaks at iron-56.
  • 1 u = 931.5 MeV.
  • Half-life T½ = 0.693/λ; mean life τ = 1/λ = 1.443·T½.
  • α decay: A−4, Z−2; β⁻ decay: Z+1, A same; γ: no change in Z or A.
  • Fission of heavy nuclei and fusion of light nuclei both release energy (toward iron).

FAQ

Common questions

What binds the nucleus together?

The strong nuclear force, which acts between nucleons at very short range and overcomes the electric repulsion between protons. It is much stronger than the electrostatic force but falls off rapidly beyond a femtometre or so.

What is the difference between α, β and γ rays?

α rays are helium nuclei (charge +2, A−4, Z−2), β⁻ rays are fast electrons (Z+1, same A), and γ rays are high-energy photons that carry away excess energy with no change to the nucleus.

How does carbon dating work?

Living tissue maintains a constant carbon-14 proportion; after death it decays with a 5730-year half-life. Measuring the remaining C-14 tells how long ago the organism died.

Why is iron the most stable nucleus?

Iron-56 has the highest binding energy per nucleon (~8.8 MeV), so it sits at the peak of the binding-energy curve — nuclei on either side can release energy by moving toward it.

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