# Radioactive Decay

> Radioactive decay for JEE Physics; exponential decay law, half-life and mean life, decay types and nuclear binding energy.

- Canonical URL: https://prepelephant.com/topics/jee/physics/radioactive-decay
- Exam / course: JEE · Subject: Physics
- Publisher: PrepElephant (https://prepelephant.com) — Prepared and reviewed by the PrepElephant Academic Review Team
- First published: 2026-10-02
- Last updated: 2026-10-02
- How to cite: "Radioactive Decay", PrepElephant, https://prepelephant.com/topics/jee/physics/radioactive-decay

## Direct answer

Radioactive decay is spontaneous disintegration in which each nucleus of a species has a fixed probability lambda (the decay constant) of decaying per unit time, whatever the temperature, pressure or chemical state. The number of nuclei and the activity therefore fall exponentially: N = N0 e^(−lambda t) and A = lambda N, with half-life t(1/2) = 0.693/lambda and mean life 1/lambda. Decays conserve mass number and charge, releasing energy set by the mass defect through E = delta m c^2.

## What you must remember

- Decay law: N = N0 e^(−lambda t); activity A = lambda N = A0 e^(−lambda t); the becquerel is one decay per second, and the curie is 3.7 × 10^10 becquerels.
- Half-life t(1/2) = 0.693/lambda; after n half-lives N = N0/2^n — two half-lives leave one quarter, never zero.
- Mean life = 1/lambda = t(1/2)/0.693, about 1.44 half-lives; lambda itself is the per-nucleus decay probability per second.
- Decay modes: alpha — helium nucleus, charge +2e, most ionising and least penetrating; beta minus — electron plus antineutrino from a neutron becoming a proton (A unchanged, Z up by one); gamma — uncharged photon, most penetrating.
- Nuclear equations balance mass number and charge on both sides: U-238 to Th-234 plus He-4.
- Binding energy = mass defect × 931.5 MeV per atomic mass unit; the binding energy per nucleon peaks near 8.8 MeV around iron-56, so fission of heavy nuclei and fusion of light nuclei both release energy.
- The beta electron is created within the nucleus during the decay; it is not one of the atom's orbital electrons.

## Common confusion

The stubborn misconception is that half-life depends on sample size or on time — it is a fixed property of the nuclide, with the same fraction vanishing in every interval. The second error is linear thinking: expecting the sample gone after two half-lives, forgetting exponential survival leaves a quarter. In balancing beta decay, remember the emitted electron comes from the nucleus and the antineutrino carries the energy that once made beta spectra look non-conserving.

## Exam-focused takeaway

JEE Main tests fraction remaining after stated half-lives, activity arithmetic and equation balancing — numerical-value questions where exponent bookkeeping decides everything. JEE Advanced adds lambda from a given fraction and time, half-life–mean-life conversion, parent-daughter growth chains, binding energies from atomic masses (mind the electrons), and Q-value reasoning for fission and fusion. Anchor the arithmetic in N = N0 e^(−lambda t) and balance A and Z line by line.

## Frequently asked questions

### Why is half-life independent of the initial amount?

Each nucleus decays with fixed probability per unit time, so halving the nuclei halves the decays per second; the fractional rate — and hence the half-life — is unchanged.

### What remains after two half-lives?

One quarter of the nuclei and one quarter of the activity; exponential decay approaches zero but never reaches it exactly.

### What distinguishes alpha, beta and gamma rays?

Alpha particles are helium nuclei, heavy and stopped by paper; beta particles are nuclear-born electrons, stopped by thin metal; gamma rays are uncharged photons, the most penetrating, needing lead or concrete.

### Where does the beta-decay electron come from?

From the nucleus: a neutron converts to a proton, emitting an electron and an antineutrino; the atomic number rises by one with the mass number unchanged.

### Why do both fission and fusion release energy?

Binding energy per nucleon peaks near iron-56, so heavy nuclei splitting and light nuclei merging both form more tightly bound products, releasing the difference per nucleon.

### How is binding energy computed?

Sum the free nucleon masses, subtract the nuclear mass, and convert the defect through 1 u = 931.5 MeV/c^2.