Radioactive Decay Series
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Direct answer
Uranium-238 is a chain, not an event: fourteen successive decays — eight alpha and six beta — walk it down to stable lead-206, and every mass number in the chain is 4n + 2 because each alpha removes exactly four nucleons' worth of mass while betas leave A unchanged. Nature sorts all heavy nuclides this way: the 4n series (thorium-232 → lead-208), the 4n + 2 series (uranium-238 → lead-206), the 4n + 3 series (uranium-235, the actinium series → lead-207) and the 4n + 1 series (neptunium-237, extinct in nature because its 2.14 million-year half-life is far shorter than Earth's age). When a parent decays into a radioactive daughter, the daughter's population first grows, may reach a maximum, and in the long-time limit settles into secular equilibrium λ1N1 = λ2N2, where the daughter's activity equals the parent's.
What you must remember
- Series classification: A mod 4 names the series — 4n thorium (Th-232 → Pb-208), 4n+2 uranium (U-238 → Pb-206), 4n+3 actinium (U-235 → Pb-207), 4n+1 neptunium (Np-237 → Bi-209, gone from nature).
- Counting rule: from (A, Z) to the stable end, number of alphas = (A_initial − A_final)/4 and betas follow from charge balance; U-238 → Pb-206 needs 8 α and 6 β.
- Daughter growth: N2(t) = λ1N1(0)/(λ2 − λ1) × (e^−λ1t − e^−λ2t); the daughter starts at zero, peaks, and its activity approaches the parent's when the parent is long-lived.
- Secular equilibrium: when λ1 << λ2 (parent much longer-lived), after a few daughter half-lives λ1N1 = λ2N2 — equal activities, daughter atoms held at N2 = (λ1/λ2)N1.
- Transient equilibrium: when parent is only moderately longer-lived, the daughter's activity peaks above the parent's and then decays with the parent's half-life.
- No equilibrium: when the daughter is longer-lived than the parent (λ2 < λ1), no steady state forms; the daughter simply outlives the parent and dominates the activity at late times.
- Radon-222: the t1/2 = 3.82-day alpha emitter in the U-238 chain that seeps into basements — the practical face of series decay and indoor radiation hazard.
Secular equilibrium worked through
Radium-226 (half-life 1600 years) decays to radon-222 (3.82 days): λ_Ra << λ_Rn comfortably. Seal pure radium in a vial and wait a month — about eight radon half-lives. Radon builds up until its decay rate matches its production rate: λ_Rn N_Rn = λ_Ra N_Ra, equal activities. The equilibrium radon population is N_Rn = (λ_Ra/λ_Rn)N_Ra — far fewer atoms, each decaying far faster, cancelling in the activity.
Extend to the whole chain: an old uranium ore body has every one of the fourteen daughters in secular equilibrium with U-238, each with the same activity as the parent. That is why uranium ore is far more radioactive per kilogram than refined uranium, and why escaping radon is the chain's public-health ambassador. If instead the parent is short-lived and the daughter long-lived — the no-equilibrium case — the early activity belongs to the parent and the entire long tail to the daughter, which is how Mo-99 → Tc-99m generators, nuclear medicine's workhorse, are milked from a parent that dies away in days.
How the series is examined
The arithmetic trap leads: "how many alpha and beta decays carry U-235 to Pb-207?" — alphas = (235 − 207)/4 = 7, and beta count comes from charge bookkeeping (92 − 2×7 = 78 needed versus Z_Pb = 82, so 4 betas). Second, the mod-4 classification is a free mark mislaid: any isotope's series follows from A mod 4, e.g. Ac-225 (225 = 4×56 + 1) belongs to the neptunium series, not the actinium series. Third, secular equilibrium is misremembered as equal atom numbers N1 = N2; it is equal activities, and the atom ratio is the inverse half-life ratio. Fourth, the neptunium series' absence from nature is a reason-based favourite: Np-237's half-life is so short geologically that it has long since decayed away. Main tests classification and counting; Advanced sets the growth curve or asks for activity-versus-time sketches.
Frequently asked questions
Why do all naturally occurring decay series have mass numbers 4n, 4n+2 or 4n+3?
Because alpha decay lowers A by exactly 4 and beta decay leaves A unchanged, so a chain's mass numbers stay on one residue class of 4 forever; 4n+1 exists but its neptunium parent has decayed away on Earth.
How many alpha and beta decays link U-238 to stable Pb-206?
Eight alpha decays (reducing A by 32) and six beta decays to correct the charge, taking Z from 92 down to 82.
What is secular equilibrium in a decay chain?
The steady state where the daughter's activity equals the parent's, λ1N1 = λ2N2, reached when the parent is much longer-lived and a few daughter half-lives have passed.
Are the numbers of parent and daughter atoms equal in secular equilibrium?
No — only the activities match; the atom populations are inversely proportional to the decay constants, N2/N1 = λ1/λ2.
Why is radon-222 the hazard everyone mentions from the uranium series?
It is a noble gas with a 3.82-day half-life formed continuously from radium in soil and rock, so it escapes into air and can accumulate in poorly ventilated buildings before decaying.