JEE Mains Physics · Nuclei

Decay Modes and Half-Lives

Alpha decay takes 4 from A and 2 from Z, beta-minus adds 1 to Z, beta-plus takes 1 from Z and gamma changes neither; and after every half-life, half of what is left decays.

All Nuclei notes JEE Mains Physics strategy

Why this matters

Twenty-four PYQs, three of them numerical, and none from 2026. Fifteen count half-lives, and the last of those was in April 2023. Nine track A and Z through alpha and beta decay or balance a fission; only three of them are later than 2023 (two in April 2024, one in January 2025). Radioactivity was cut from the NCERT syllabus in 2023-24, and these dates match: this page serves older papers, the boards and NEET more than a 2026 JEE paper.

Concept 1 of 2: Alpha, beta and gamma decay: what changes

Every decay keeps two totals fixed: the number of nucleons (A) and the charge (Z). An alpha carries away 2 protons and 2 neutrons. In beta decay a neutron turns into a proton, or a proton into a neutron, so A stays and Z moves by one. Gamma is only light leaving an excited nucleus. To count a chain, use A first (only alphas change it), then fix Z with betas.

Definition

Count the alphas from the change in A: $n_\alpha = \dfrac{A_{\text{parent}} - A_{\text{daughter}}}{4}$ .
Each alpha lowers Z by 2, each beta-minus raises it by 1: $n_{\beta^-} = 2n_\alpha - (Z_{\text{parent}} - Z_{\text{daughter}})$ .
Beta-minus: $n \rightarrow p + e^- + \bar{\nu}$ (an antineutrino). Beta-plus: $p \rightarrow n + e^+ + \nu$ (a neutrino).
A free neutron can decay because it is heavier than a proton plus an electron. A free proton cannot turn into a neutron, which is heavier; inside a nucleus the energy can come from the binding energy.
A fission equation balances the same way: A and Z on both sides, counting the neutrons that are released.

Decay	What leaves the nucleus	Change in A	Change in Z	Example
Alpha ( $\alpha$ )	A helium nucleus, $^{4}_{2}\text{He}$	Falls by 4	Falls by 2	$^{226}_{88}\text{Ra} \rightarrow {}^{222}_{86}\text{Rn} + \alpha$
Beta-minus ( $\beta^-$ )	An electron and an antineutrino	No change	Rises by 1	$^{14}_{6}\text{C} \rightarrow {}^{14}_{7}\text{N} + e^- + \bar{\nu}$ A neutron becomes a proton and the partner is an ANTI-neutrino.
Beta-plus ( $\beta^+$ )	A positron and a neutrino	No change	Falls by 1	$^{22}_{11}\text{Na} \rightarrow {}^{22}_{10}\text{Ne} + e^+ + \nu$ Happens only inside a nucleus: a free proton is lighter than a neutron.
Gamma ( $\gamma$ )	A photon, as an excited nucleus drops to a lower level	No change	No change	$^{60}\text{Ni}^{*} \rightarrow {}^{60}\text{Ni} + \gamma$

Only alpha decay changes A, so count alphas from A and then betas from Z.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 26 June 2022 · Q18Moderate

Example 1 · Nuclei · Decay Modes and Half-Lives

How many alpha and beta particles are emitted when Uranium

\ _{92}U^{238}

decays to lead

\ _{82}{\text{ }Pb}^{206}

Count the alphas first

Betas do not change A, so A alone fixes the number of alphas. Starting from Z mixes two unknowns.

Beta-minus raises Z

It feels like a minus should lower Z, but losing a negative electron leaves the nucleus one charge more positive. Beta-minus: Z + 1. Beta-plus: Z − 1.

Neutrino or antineutrino

Beta-minus emits an electron and an antineutrino; beta-plus emits a positron and a neutrino. The lepton number must balance.

Drill 8 more on alpha, beta and gamma decay: what changes

Concept 2 of 2: Counting half-lives

In each half-life, half of the nuclei still present decay. So the amount left halves again and again: 1, 1/2, 1/4, 1/8, … Find n, the number of half-lives that pass, and the fraction left is (1/2)ⁿ. Mass, number of nuclei and activity all follow this same law.

Definition

$N = N_0\left(\tfrac{1}{2}\right)^{n}$ , with $n = t/T_{1/2}$ .
Fraction left $= \left(\tfrac{1}{2}\right)^{n}$ ; fraction decayed $= 1 - \left(\tfrac{1}{2}\right)^{n}$ . Read which one is asked.
Activity and mass halve on the same clock: $\dfrac{A_0}{A} = 2^{n}$ .
n need not be whole: after half a half-life the fraction left is $\left(\tfrac{1}{2}\right)^{1/2} = \dfrac{1}{\sqrt 2}$ .
A source k times above a safe level is safe after $n = \log_2 k$ half-lives.

Half-life law

N = N_0\left(\tfrac{1}{2}\right)^{t/T_{1/2}}

Worked example

A sample of 80 mg has a half-life of 6 hours. How much is left after 24 hours, and how much has decayed?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 8 Apr 2023 · Q96Moderate

Example 2 · Nuclei · Decay Modes and Half-Lives

A radio-active material is reduced to

1/8

of its original amount in 3 days. If

8 \times10^{- 3}\text{ }kg

of the material is left after 5 days. The initial amount of the material is

Decayed is not left

If 15/16 has decayed, 1/16 is left, so 4 half-lives have passed. Taking the decayed fraction as the fraction left gives a time far too short.

Halve, do not subtract halves

After two half-lives a quarter is left, not zero. Each half-life halves what remains; it does not remove half of the original.

Find n from a ratio, then the time

When a count or activity falls by a power of 2, write the ratio as

2^{n}

first. Then time = n × half-life, or half-life = time ÷ n.

Drill 14 more on counting half-lives

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Counting half-lives
Half-life law
$N = N_0\left(\tfrac{1}{2}\right)^{t/T_{1/2}}$

Reference tables (1)

Alpha, beta and gamma decay: what changes4 rows

Decay	What leaves the nucleus	Change in A	Change in Z	Example
Alpha ( $\alpha$ )	A helium nucleus, $^{4}_{2}\text{He}$	Falls by 4	Falls by 2	$^{226}_{88}\text{Ra} \rightarrow {}^{222}_{86}\text{Rn} + \alpha$
Beta-minus ( $\beta^-$ )	An electron and an antineutrino	No change	Rises by 1	$^{14}_{6}\text{C} \rightarrow {}^{14}_{7}\text{N} + e^- + \bar{\nu}$ A neutron becomes a proton and the partner is an ANTI-neutrino.
Beta-plus ( $\beta^+$ )	A positron and a neutrino	No change	Falls by 1	$^{22}_{11}\text{Na} \rightarrow {}^{22}_{10}\text{Ne} + e^+ + \nu$ Happens only inside a nucleus: a free proton is lighter than a neutron.
Gamma ( $\gamma$ )	A photon, as an excited nucleus drops to a lower level	No change	No change	$^{60}\text{Ni}^{*} \rightarrow {}^{60}\text{Ni} + \gamma$

Only alpha decay changes A, so count alphas from A and then betas from Z.

Watch out for (6)

Count the alphas first
→ Alpha, beta and gamma decay: what changes
Beta-minus raises Z
→ Alpha, beta and gamma decay: what changes
Neutrino or antineutrino
→ Alpha, beta and gamma decay: what changes
Decayed is not left
→ Counting half-lives
Halve, do not subtract halves
→ Counting half-lives
Find n from a ratio, then the time
→ Counting half-lives

Test yourself on Nuclei

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Nuclei chapter test Drill all 24 questions