Secular and Transient Equilibrium Calculator
Enter parent and daughter half-lives to classify the equilibrium type and compute the activity ratio, peak time, and current daughter activity.
⚛️ What Is Radioactive Equilibrium?
Radioactive equilibrium is a steady-state condition in a parent-daughter decay chain where the daughter activity stops changing relative to the parent activity. It arises from the competition between two processes: the decay of the parent (which produces daughter atoms) and the decay of the daughter (which removes them). When the production rate equals the removal rate, the daughter atom count stabilizes and the two activities maintain a fixed ratio.
There are two distinct types of equilibrium depending on the ratio of parent to daughter half-lives. Secular equilibrium occurs when the parent half-life is at least 100 times longer than the daughter half-life. In this limit, the daughter activity equals the parent activity (A_B = A_A) at equilibrium, and both activities decrease together at the slow rate of the long-lived parent. Classic examples include Ra-226 (1600 yr) in equilibrium with Rn-222 (3.82 day), and U-238 (4.47 billion yr) with its numerous short-lived progeny in natural uranium minerals. Transient equilibrium occurs when the parent is longer-lived than the daughter but the ratio is less than 100. At equilibrium, the daughter activity slightly exceeds the parent by the factor t_{1/2,A}/(t_{1/2,A} - t_{1/2,B}), which is always greater than 1. The Mo-99/Tc-99m generator used in nuclear medicine scintigraphy is the most widely known example.
A third case, no equilibrium, occurs when the daughter half-life exceeds the parent half-life. In this case the daughter activity never catches the parent, and both eventually vanish. No useful generator chemistry is based on this case.
This calculator classifies any parent-daughter pair, computes the equilibrium activity ratio, finds the time at which daughter activity peaks, estimates the time to reach 99% equilibrium, and evaluates the instantaneous daughter activity at any user-specified time using the exact Bateman equation formula.
📐 Equilibrium Formulas
Classification: R = t1/2,A / t1/2,B
AB/AA|eq = t1/2,A / (t1/2,A − t1/2,B) [transient]
AB/AA|eq = 1 [secular, limit as R → ∞]
tmax = ln(λB / λA) / (λB − λA)
t99% ≈ 7 × t1/2,B
R = half-life ratio (R ≥ 100: secular; 1 < R < 100: transient; R < 1: no equilibrium)
λX = decay constant = ln(2) / t1/2,X
tmax = time of maximum daughter activity
t99% = time to reach 99.2% of equilibrium activity
Example (Mo-99/Tc-99m): R = 65.94/6.006 = 10.98 (transient); AB/AA = 1.100; tmax = 22.8 hr; t99% = 42 hr
📖 How to Use This Calculator
Steps
1
Enter parent half-life - type the half-life of nuclide A (parent) and select the time unit. Use published nuclear data values; common sources include NNDC (National Nuclear Data Center) or the IAEA nuclear data services.
2
Enter daughter half-life - type the half-life of nuclide B (daughter). Select "Stable" from the unit dropdown if B does not decay. The calculator will automatically classify the equilibrium type based on the half-life ratio.
3
Enter initial activity and evaluation time - enter the parent activity A0 at time zero with its unit, and the evaluation time t at which you want to see the current daughter activity.
4
Read the results - the calculator shows the equilibrium classification, the half-life ratio, the equilibrium A_B/A_A ratio, the time to peak daughter activity, the time to reach 99% equilibrium, and the instantaneous activities and ratio at the evaluation time.
💡 Example Calculations
Example 1 - Mo-99/Tc-99m Generator (Transient Equilibrium)
Mo-99 (65.94 hr) to Tc-99m (6.006 hr), 1000 MBq at calibration
1
Half-life ratio R = 65.94 / 6.006 = 10.98. Since 1 < R < 100, this is transient equilibrium.
2
Equilibrium ratio = 65.94 / (65.94 − 6.006) = 1.100. At equilibrium, A(Tc-99m) is 10.0% above A(Mo-99).
3
Time to max daughter activity: tmax = ln(λB/λA) / (λB − λA) = 22.8 hr. Time to 99% equilibrium: 7 × 6.006 = 42.0 hr.
Type: Transient Equilibrium | AB/AA at eq = 1.100 | tmax = 22.8 hr
Try this example →Example 2 - Ra-226/Rn-222 (Secular Equilibrium)
Ra-226 (1600 yr) to Rn-222 (3.8235 day), 1 kBq Ra-226 source
1
Half-life ratio R = (1600 × 365.25) / 3.8235 = 152,700. Since R ≥ 100, this is secular equilibrium.
2
Equilibrium ratio = 1.000 (secular limit). At equilibrium, A(Rn-222) = A(Ra-226) = 1000 Bq.
3
Time to 99% equilibrium: 7 × 3.8235 day = 26.8 day. After 26.8 days in a sealed container, all Rn-222 produced stays in and the activities equalize.
Type: Secular Equilibrium | AB/AA at eq = 1.000 | t99% = 26.8 day
Try this example →Example 3 - Na-24/Mg-24 (No Equilibrium, Stable Daughter)
Na-24 (14.96 hr) decays to stable Mg-24, 500 kBq initial activity
1
Mg-24 is stable (half-life = infinity). The daughter never decays, so no equilibrium is possible. The result type is "No Equilibrium (stable daughter)".
2
The equilibrium ratio, time to max daughter activity, and time to 99% equilibrium are all "N/A" for stable daughters.
3
At t = 24 hr, A(Na-24) = 500 × exp(−ln2 × 24/14.96) = 500 × 0.3317 = 165.8 kBq. The stable Mg-24 atom count is displayed, not an activity.
Type: No Equilibrium (stable daughter) | AB/AA = N/A
Try this example →❓ Frequently Asked Questions
What is the difference between secular and transient radioactive equilibrium?
In secular equilibrium the parent half-life is at least 100 times longer than the daughter half-life. The daughter activity equals the parent activity at equilibrium. In transient equilibrium the parent is longer-lived but by less than a factor of 100. The daughter activity at equilibrium slightly exceeds the parent by the factor t_{1/2,A}/(t_{1/2,A} - t_{1/2,B}), which is greater than 1. Both types require t_{1/2,A} greater than t_{1/2,B}.
What is the formula for the equilibrium activity ratio in transient equilibrium?
At transient equilibrium, A_B/A_A = t_{1/2,A} / (t_{1/2,A} - t_{1/2,B}). This can also be written as lambda_A / (lambda_A - lambda_B). For Mo-99/Tc-99m, the ratio is 65.94/(65.94-6.006) = 1.100, so Tc-99m activity is exactly 10.0% above Mo-99 activity at equilibrium. The formula has no maximum; as t_{1/2,B} approaches t_{1/2,A}, the ratio grows without bound.
How long does it take to reach secular equilibrium after a fresh separation?
Secular equilibrium is reached after approximately 7 daughter half-lives, corresponding to 99.2% of the equilibrium activity (1 - 2^(-7) = 0.992). For Rn-222 (3.82 day) growing into Ra-226, this takes 26.7 days. For I-131 daughter products in a reactor target, it may take only hours. The parent half-life does not affect the time to equilibrium, only the daughter half-life does.
What is the Mo-99/Tc-99m equilibrium activity ratio and why does it matter?
The Mo-99/Tc-99m equilibrium activity ratio is 1.100. This means the Tc-99m activity at equilibrium is 10.0% higher than the Mo-99 activity. Nuclear medicine physicists use this to plan generator elution: if a generator is calibrated at 10 GBq Mo-99, the maximum Tc-99m available at equilibrium is 11.0 GBq, occurring about 22.8 hours after elution. Actual eluted activity is lower because some Tc-99m is not recovered by the column.
What happens when the daughter half-life is longer than the parent half-life?
When t_{1/2,B} exceeds t_{1/2,A}, there is no radioactive equilibrium. The parent decays faster than it can build up the daughter, so the daughter activity always remains below the parent activity. Both activities eventually decrease to zero, with the parent vanishing first. The daughter activity rises, reaches a maximum when the parent is nearly exhausted, then decays on its own timeline.
At what time does the daughter activity reach its peak value?
The daughter activity peaks when its rate of production by parent decay equals its own rate of decay. Setting dA_B/dt = 0 gives t_max = ln(lambda_B / lambda_A) / (lambda_B - lambda_A). For Mo-99/Tc-99m, this equals 22.8 hr. After t_max the daughter activity decreases, following the parent decay from above (transient) or from equal activity (secular).
Is secular equilibrium a special case of transient equilibrium?
Yes. Secular equilibrium is the limiting case of transient equilibrium when the half-life ratio approaches infinity. In that limit, t_{1/2,A}/(t_{1/2,A} - t_{1/2,B}) approaches 1, so A_B/A_A approaches 1 exactly. The convention of using a ratio of 100 as the boundary between the two types is a practical threshold, not a sharp physical discontinuity, because even at a ratio of 100 the equilibrium ratio is 100/99 = 1.010, effectively indistinguishable from 1 in most measurements.
How does radioactive equilibrium apply to uranium and thorium decay series?
In an undisturbed uranium-238 ore body, U-238 (4.47 billion yr) and all 13 of its decay products from Th-234 down to Pb-206 are in secular equilibrium. This means every member of the chain has the same activity as U-238. Geochronologists exploit this: measuring the activity ratios of isotopes like Th-230/U-234 or Pa-231/U-235 reveals how long the minerals have been closed systems and hence their age.
Can equilibrium be disrupted and how quickly does it re-establish?
Equilibrium is broken whenever the daughter is physically separated from the parent, for example by chemical purification, ion exchange column elution, or gas release (as when Rn-222 escapes from Ra-226 in a soil or water sample). After separation, the daughter decays away and the parent produces new daughter from scratch. Equilibrium re-establishes after approximately 7 daughter half-lives, following exactly the Bateman N_B(t) curve.
What does the instantaneous A_B/A_A ratio tell you?
The instantaneous ratio A_B(t)/A_A(t) at the chosen evaluation time shows where the chain is in its approach to equilibrium. If it is below the equilibrium ratio, the daughter is still building up. If it equals the equilibrium ratio, the chain is at equilibrium. For transient equilibrium, the ratio overshoots just before t_max, then settles to the equilibrium value. You can verify equilibrium by checking that the instantaneous ratio matches the calculated equilibrium ratio.
How does this calculator differ from the Bateman Equations Solver?
The Bateman Equations Solver focuses on computing time-dependent atom counts and activities for two- and three-nuclide chains at a specified evaluation time. This Equilibrium Calculator focuses on classifying the equilibrium type, computing the equilibrium ratio formula, finding the time to peak daughter activity, and estimating the time to reach steady-state. Use this calculator first to understand the equilibrium behavior, then the Bateman solver for detailed time-series calculations.
Why is the time to 99% equilibrium approximately 7 daughter half-lives?
In secular equilibrium, the approach to equilibrium is dominated by the daughter's own buildup curve. The daughter activity at time t is approximately A0 * (1 - exp(-lambda_B * t)). Setting this to 0.99 gives exp(-lambda_B * t) = 0.01, so lambda_B * t = ln(100) = 4.605, meaning t = 4.605/lambda_B = 4.605 * t_{1/2,B} / ln(2) = 6.64 * t_{1/2,B}. Rounding up gives the approximation of 7 half-lives for 99.2% (since 2^7 = 128, and 1 - 1/128 = 0.9922).