Class 12 · Chapter 13
Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.
12 formulas across 4 topics in Nuclei.
Updated 2026-07-09 · v1.0.0
Every nucleus, however different its mass, has almost exactly the same density — a striking clue that nucleons are packed together at a fixed spacing, like incompressible drops of nuclear fluid.
R = R₀ A^(1/3) , R₀ ≈ 1.2 fm
| Symbol | Meaning | Unit | Dimension |
|---|---|---|---|
| R | nuclear radius | fm (femtometre) | [L] |
| A | mass number (total nucleons) | dimensionless | [M⁰L⁰T⁰] |
| R₀ | empirical constantR₀ ≈ 1.2 × 10⁻¹⁵ m | fm | [L] |
Valid when
ρ = mass / volume ≈ 2.3 × 10¹⁷ kg/m³ (constant, independent of A)
Mass ∝ A and volume ∝ R³ ∝ A both scale identically with A, so their ratio (density) is independent of A
| Symbol | Meaning | Unit | Dimension |
|---|---|---|---|
| ρ | nuclear density | kg/m³ | [ML⁻³] |
Valid when
Common mistakes
A nucleus weighs slightly LESS than the sum of its free, separated nucleons — that missing mass has been converted into the binding energy holding the nucleus together.
E = mc²
| Symbol | Meaning | Unit | Dimension |
|---|---|---|---|
| E | energy equivalent of mass m | J | [ML²T⁻²] |
Valid when
Δm = [Zm(p) + (A−Z)m(n)] − M(nucleus)
| Symbol | Meaning | Unit | Dimension |
|---|---|---|---|
| Δm | mass defect — the 'missing' mass | u or kg | [M] |
| m(p), m(n) | mass of a free proton and free neutron | u | [M] |
| M | actual measured mass of the nucleus | u | [M] |
Valid when
BE = Δm c² (BE in MeV = Δm in u × 931.5)
Direct application of E = mc² to the mass defect
| Symbol | Meaning | Unit | Dimension |
|---|---|---|---|
| BE | binding energy — energy needed to completely separate the nucleus into free nucleons | MeV | [ML²T⁻²] |
Valid when
BE per nucleon = BE / A
The single most important quantity for judging nuclear stability, plotted against A gives the famous BE/A curve
Valid when
Common mistakes
Both processes release energy from the SAME underlying source — moving nuclei toward the binding-energy peak near iron — but they operate at opposite ends of the periodic table.
Q = [Σm(reactants) − Σm(products)] c²
Conservation of mass-energy applied to the reactants and products of any nuclear reaction
| Symbol | Meaning | Unit | Dimension |
|---|---|---|---|
| Q | energy released (Q > 0) or absorbed (Q < 0) in the reaction | MeV | [ML²T⁻²] |
Valid when
Worth remembering
Radioactive decay is a purely random, memoryless process at the level of any single nucleus — yet a large enough sample decays with perfect statistical predictability, following one clean exponential law.
N = N₀ e^(−λt)
Rate of decay is proportional to the number of undecayed nuclei present: dN/dt = −λN, solved as an exponential decay
| Symbol | Meaning | Unit | Dimension |
|---|---|---|---|
| N | number of undecayed nuclei at time t | dimensionless (a count) | [M⁰L⁰T⁰] |
| N₀ | initial number of undecayed nuclei | dimensionless (a count) | [M⁰L⁰T⁰] |
| λ | decay constant, a property of the specific radioactive isotope | s⁻¹ | [T⁻¹] |
Valid when
T(1/2) = ln2 / λ = 0.693/λ
Time for N to fall to N₀/2 in the decay law: ½ = e^(−λT(1/2))
| Symbol | Meaning | Unit | Dimension |
|---|---|---|---|
| T(1/2) | half-life — time for half the sample to decay | s | [T] |
Valid when
τ = 1/λ = T(1/2) / ln2 ≈ 1.44 T(1/2)
| Symbol | Meaning | Unit | Dimension |
|---|---|---|---|
| τ | mean life — average lifetime of a nucleus before decaying | s | [T] |
Valid when
Common mistakes
A = λN = A₀ e^(−λt)
| Symbol | Meaning | Unit | Dimension |
|---|---|---|---|
| A | activity — number of decays per second1 Ci = 3.7 × 10¹⁰ Bq | Bq (becquerel) or Ci (curie) | [T⁻¹] |
Valid when
N/N₀ = (1/2)^(t/T(1/2))
Equivalent restatement of the decay law directly in terms of the number of elapsed half-lives
Valid when
Alpha decay reduces mass number by 4 and atomic number by 2; beta-minus decay keeps mass number the same but increases atomic number by 1 — boards frequently ask to balance a decay equation using these rules alongside the decay-law formulas.
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