Radioactive Decay
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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.