Radioactive Equilibrium Calculator

Calculate secular and transient equilibrium for radioactive parent-daughter systems.

Ra-226: 1600yr
Rn-222: 3.82 days=0.01046yr
1 μCi = 37,000 Bq
In daughter half-lives
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Three Regimes, Set by the Half-Life Ratio

When a radioactive parent decays into a radioactive daughter, the daughter's activity depends entirely on how the two half-lives compare.

ConditionRegimeDaughter behaviour
t½(parent) » t½(daughter)SecularDaughter activity equals parent's
t½(parent) > t½(daughter)TransientDaughter slightly exceeds parent, then decays with parent's half-life
t½(parent) < t½(daughter)No equilibriumParent vanishes first; daughter decays independently
Transient: Ad/Ap = λd/(λd − λp)

In secular equilibrium the parent barely decays over many daughter half-lives, so daughter production is effectively constant and its activity climbs to match the parent's exactly. Radium-226 with radon-222 is the classic case: 1,600 years against 3.82 days.

Why Mo-99/Tc-99m Generators Work

The medical isotope supply chain depends on transient equilibrium. Molybdenum-99 has a 66-hour half-life — long enough to manufacture and ship — and decays to technetium-99m at 6 hours, which is ideal for imaging but far too short to transport.

The generator holds Mo-99 on an alumina column. Tc-99m grows in and is eluted with saline, and because equilibrium re-establishes over about 24 hours, the same generator can be milked daily for around two weeks. Maximum Tc-99m activity occurs roughly 23 hours after the previous elution.

Worked Examples

Example 1: Ra-226/Rn-222: t1=1600yr, t2=3.82d
Secular equilibrium: t1>>t2
Result: At equilibrium: A_Rn = A_Ra (equal activities)
Basis for radon in buildings from radium in soil
Example 2: Mo-99/Tc-99m: t1=2.75d, t2=6.01hr
Transient equilibrium: t1/t2~11
Result: A_Tc = 0.957×A_Mo at equilibrium
Medical generator — milked every 24hr
Example 3: Generator elution timing
Mo-99/Tc-99m generator, previous elution 24 hours ago
Result: Near-maximum Tc-99m available
Activity peaks around 23 hours after elution. This is why hospital nuclear medicine departments elute each morning.
Example 4: Why the daughter slightly exceeds the parent
λd/(λd − λp) for Mo-99/Tc-99m
Result: Ratio about 1.1 in principle
In practice only about 86% of Mo-99 decays route through Tc-99m, so measured generator output is lower than the simple ratio suggests.

Common Mistakes

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Confusing secular with transient equilibrium

Secular requires the parent half-life to be vastly longer — typically more than a hundredfold. Transient applies when it is merely longer.

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Assuming daughter activity never exceeds parent

In transient equilibrium the daughter slightly exceeds the parent, by the factor λd/(λd − λp).

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Expecting equilibrium when the daughter is longer-lived

If the daughter outlives the parent, no equilibrium forms. The parent disappears and the daughter simply decays on its own schedule.

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Eluting a generator too frequently

Tc-99m needs time to grow in. Eluting hourly yields little each time; roughly 24-hour intervals give near-maximum activity.

Frequently Asked Questions

Why are Mo-99/Tc-99m generators useful?
Mo-99 (t½=66hr) produced in reactor. Ships to hospital. Tc-99m (t½=6hr) grows in from Mo-99. 'Milk' Tc-99m every 24hr. Perfect for medical imaging: short enough to minimize patient dose, long enough for scan procedures. 80% of nuclear medicine uses Tc-99m.
Secular vs transient equilibrium?
Secular: t½(parent)>>t½(daughter): activities become equal. Transient: t½(parent)>t½(daughter) by factor 10-1000: daughter activity exceeds parent by factor λ₂/(λ₂-λ₁). No equilibrium: t½(parent)
What is secular equilibrium?
When the parent half-life is vastly longer than the daughter's, so parent decay is effectively constant and daughter activity rises to equal it.
What is transient equilibrium?
When the parent half-life is longer but not vastly so. The daughter reaches a slightly higher activity than the parent, then decays with the parent's half-life.
Why are Mo-99/Tc-99m generators used?
Mo-99 at 66 hours can be shipped, while Tc-99m at 6 hours cannot. The generator lets hospitals produce fresh Tc-99m on site daily.
How often can a generator be eluted?
Roughly daily. Tc-99m activity rebuilds to near maximum about 23 hours after the previous elution, and a generator lasts about two weeks.

Formula Explorer connections

Interpretation: This formula describes how reactants, products, ions or phases distribute when opposing processes reach equilibrium. Assumption: Use equilibrium rather than initial concentrations, correct stoichiometric exponents, and the specified temperature; activities may replace concentrations in nonideal systems.