What are Bateman equations?

Bateman equations describe the time evolution of daughter isotope activities in a decay chain. For a two-step chain, daughter activity A₂(t) = A₀ × [λ₂/(λ₂−λ₁)] × (e^(−λ₁t) − e^(−λ₂t)), where λ = ln2/t½.

What is secular equilibrium?

Secular equilibrium occurs when the parent half-life is much longer than the daughter (T₁ >> T₂). After several daughter half-lives, all daughter activities equal the parent activity. Example: U-238 (4.5 Gyr) and its short-lived daughters.

What is transient equilibrium?

Transient equilibrium occurs when the parent half-life is only moderately longer than the daughter (T₁ ~ 10× T₂). The ⁹⁹Mo/⁹⁹mTc generator is the classic example — peak ⁹⁹mTc occurs ~23 hours after calibration.

Why is the Mo-99/Tc-99m generator important?

Tc-99m is the most widely used nuclear medicine diagnostic isotope (15–20 million procedures/year worldwide). It is produced via ⁹⁹Mo (t½=65.9h) decay, with peak ⁹⁹mTc (t½=6h) yield around 23 hours after calibration.

Half-Life Decay Chain Calculator

Calculate parent and daughter isotope activities in multi-step radioactive decay chains using Bateman equations. Covers U-238, Th-232, Cs-137, and the ⁹⁹Mo → ⁹⁹mTc nuclear medicine generator.

Decay Chain

Initial Parent Activity A₀ (Bq)

Elapsed Time

Time Unit

Parent Activity (Bq)

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Daughter 1 Activity (Bq) —

Equilibrium Status —

Extended More scenarios, charts & detailed breakdown ▾

Initial U-238 Activity A₀ (Bq) ?

Time (years)

U-238 Activity (Bq)

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Th-234 Activity (Bq) —

Equilibrium Status —

Professional Full parameters & maximum detail ▾

Initial ⁹⁹Mo Activity (GBq) ?

Time Since Calibration (hours)

Elution Efficiency (%)

⁹⁹Mo Activity (GBq)

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⁹⁹mTc Available (GBq) —

Elutable ⁹⁹mTc after Efficiency (GBq) —

Peak ⁹⁹mTc Time Note —

U-238 Chain / Radon Hazard Note —

How to Use This Calculator

Select a decay chain preset (U-238, Th-232, Cs-137, or Mo-99/Tc-99m).
Enter initial parent activity in Bq.
Enter elapsed time and time unit.
Results show parent and first daughter activities and equilibrium status.
Use Extended tabs for specific chain analysis or custom two-step chains.

Formula

Parent: A₁(t) = A₀e^(−λ₁t)

Daughter (Bateman): A₂(t) = A₀ × [λ₂/(λ₂−λ₁)] × (e^(−λ₁t) − e^(−λ₂t))

λ = ln(2)/t½

Example

⁹⁹Mo (A₀=10 GBq, t½=65.9h) → ⁹⁹mTc (t½=6h). At t=24h: ⁹⁹Mo ≈ 9.0 GBq, ⁹⁹mTc ≈ 7.8 GBq (transient equilibrium approaching).

Frequently Asked Questions

Bateman equations describe the time evolution of daughter isotope activities in a decay chain. For a two-step chain, daughter activity A₂(t) = A₀ × [λ₂/(λ₂−λ₁)] × (e^(−λ₁t) − e^(−λ₂t)), where λ = ln2/t½.
Secular equilibrium occurs when the parent half-life is much longer than the daughter (T₁ >> T₂). After several daughter half-lives, all daughter activities equal the parent activity. Example: U-238 (4.5 Gyr) and its short-lived daughters.
Transient equilibrium occurs when the parent half-life is only moderately longer than the daughter (T₁ ~ 10× T₂). The ⁹⁹Mo/⁹⁹mTc generator is the classic example — peak ⁹⁹mTc occurs ~23 hours after calibration.
Tc-99m is the most widely used nuclear medicine diagnostic isotope (15–20 million procedures/year worldwide). It is produced via ⁹⁹Mo (t½=65.9h) decay, with peak ⁹⁹mTc (t½=6h) yield around 23 hours after calibration.

Related Calculators

Sources & References (5) ▾

NIST Nuclear Data – Radioactive Decay — NIST
IAEA – Nuclides Database and Decay Data — IAEA
NNDC NuDat – National Nuclear Data Center — National Nuclear Data Center, BNL
OpenStax University Physics Vol. 3 Ch. 10 – Nuclear Physics — OpenStax
Knoll G F – Radiation Detection and Measurement, 4th Ed. — Wiley