On this page
"How many α and β particles are emitted when uranium-238 becomes lead-206?" looks like it needs the whole decay chain. It does not. Two lines of arithmetic on the start and end nuclides give the answer, and two quick checks tell you whether it is right. This post explains the method, works it on a real chain, and sets out which exams still ask it.
Which exams still include this?
Check this first, because the topic has moved.
JEE (Advanced). The 2026 information brochure lists "Atomic nucleus; α, β and γ radiations; Law of radioactive decay" under Modern Physics.
JEE (Main). The NTA syllabus for 2026 covers composition and size of the nucleus, mass defect, binding energy, and fission and fusion under Atoms and Nuclei, but not radioactive decay.
NEET (UG). The NEET (UG) 2026 syllabus has the same Atoms and Nuclei list, also without decay.
CBSE Class 12. The CBSE physics curriculum for 2026-27 lists composition and size of the nucleus, nuclear force, mass defect, binding energy, fission and fusion for the Nuclei chapter.
The current NCERT chapter Nuclei (2026-27 reprint) still has a short section 13.6 naming the three decays: α-decay emits a helium nucleus, β-decay emits electrons or positrons, and γ-decay emits high-energy photons. So this is mainly a JEE (Advanced) skill, and you need the conservation rules more than the textbook text. Check your exam's current syllabus.
What does each decay do to A and Z?
Write a nuclide as mass number A over atomic number Z. Charge and nucleon number are both conserved.
α-decay. A helium nucleus (A = 4, Z = 2) leaves. A falls by 4 and Z by 2.
β⁻-decay. An electron (charge −1, A = 0) leaves as a neutron becomes a proton. A is unchanged and Z rises by 1.
γ-decay. A photon leaves. Neither A nor Z changes, so γ rays never enter the count.
(β⁺-decay, a positron, lowers Z by 1. Series problems in exams almost always mean β⁻.)
The two-line method
Only α-decay changes A, so:
number of α decays = (A₁ − A₂) ÷ 4
Each α lowers Z by 2 and each β⁻ raises it by 1, so Z₂ = Z₁ − 2nα + nβ, which rearranges to:
number of β decays = 2nα − (Z₁ − Z₂)
Worked example 1: uranium-238 to lead-206
The US Environmental Protection Agency's page on radioactive decay says the decay chain that begins with uranium-238 culminates in lead-206, after forming intermediates such as uranium-234, thorium-230, radium-226 and radon-222. Uranium has Z = 92 and lead Z = 82.
α count: (238 − 206) ÷ 4 = 32 ÷ 4 = 8.
β count: 2 × 8 − (92 − 82) = 16 − 10 = 6.
Neutron check: uranium-238 has 238 − 92 = 146 neutrons; lead-206 has 206 − 82 = 124. The chain loses 22. Eight α particles carry off 16 neutrons, and six β⁻ decays turn 6 neutrons into protons: 16 + 6 = 22.
Worked example 2: pieces of the same chain
The intermediates EPA lists let you check sections of the chain. Thorium is Z = 90, radium 88 and radon 86.
Uranium-238 to uranium-234. α: 4 ÷ 4 = 1. β: 2 × 1 − 0 = 2. One α and two β⁻ bring the nucleus back to the same element, as an isotope four mass units lighter.
Thorium-230 to radium-226. α: 1. β: 2 − 2 = 0.
Radium-226 to radon-222. α: 1. β: 2 − 2 = 0.
Which quick checks catch errors?
A must drop by a multiple of 4. β and γ do not change A. If A₁ − A₂ is not divisible by 4, you have the wrong nuclide or a misprint.
A ÷ 4 leaves the same remainder. Because every change in A is a multiple of 4, a chain stays in one family. 238 = 4 × 59 + 2 and 206 = 4 × 51 + 2: both leave remainder 2.
A negative β count is a warning. It means the question involves β⁺ or an arithmetic slip.
The neutron check. Neutrons lost = 2nα + nβ for β⁻ decays.
Which mistakes cost marks?
Subtracting atomic numbers directly and calling it the β count. Ten is not the answer for uranium to lead; six is.
Forgetting that β⁻ raises Z. Students who subtract instead get 2 × 8 + 10 = 26.
Counting γ rays as changing anything.
Mixing mass number with atomic mass. Use the integers 238 and 206, not 238.05 u.
Assuming the order. Two chains with the same endpoints can step through different nuclides.
A practice routine
Use these pairs. Answers follow; cover them first.
Thorium-232 (Z = 90) to lead-208 (Z = 82).
Uranium-235 (Z = 92) to lead-207.
Thorium-228 to bismuth-212 (Z = 83).
Your own: pick any nuclide from the EPA chain above and count the decays to lead-206.
Answers: (1) 6 α, 4 β; (2) 7 α, 4 β; (3) 4 α, 1 β. Run the remainder and neutron checks on each.
If you get a count wrong and cannot find the slip, the Fermi Problem Library has a step-by-step solver that explains the reasoning rather than printing an answer. For the decay law, half-life and mean life, read our post on radioactive decay and half-life, and for the energy side, see nuclear binding energy problems. To see how the two JEE papers differ, read JEE Main vs JEE Advanced.
Before your next mock, do pair 1 in under a minute, with both checks.
