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
📖 How to Use This Calculator
Steps
💡 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
Example 2 - Ra-226/Rn-222 (Secular Equilibrium)
Ra-226 (1600 yr) to Rn-222 (3.8235 day), 1 kBq Ra-226 source
Example 3 - Na-24/Mg-24 (No Equilibrium, Stable Daughter)
Na-24 (14.96 hr) decays to stable Mg-24, 500 kBq initial activity
❓ Frequently Asked Questions
🔗 Related Calculators
What is the difference between secular and transient equilibrium?
In secular equilibrium, the parent half-life is at least 100 times 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 slightly exceeds the parent at equilibrium, by the ratio t_{1/2,A}/(t_{1/2,A}-t_{1/2,B}).
What is the formula for 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}) = lambda_A / (lambda_A - lambda_B). For Mo-99/Tc-99m with half-lives 65.94 hr and 6.006 hr, the ratio is 65.94/(65.94-6.006) = 1.10. This means Tc-99m activity is 10% above Mo-99 activity at equilibrium.
How long does it take to reach secular equilibrium?
Secular equilibrium is reached after approximately 7 daughter half-lives, which corresponds to about 99.2% of the equilibrium value. For Rn-222 (t_{1/2} = 3.82 day) growing in from Ra-226, equilibrium is reached in about 26.7 days. After this time, the Rn-222 activity tracks the Ra-226 activity exactly.
What does it mean when there is no radioactive equilibrium?
No equilibrium occurs when the daughter half-life is longer than the parent half-life. In this case the daughter builds up slowly but never catches the parent activity. Both activities eventually decrease to zero, with the parent disappearing faster than the daughter.
At what time does daughter activity reach its maximum?
The daughter activity peaks at t_max = ln(lambda_B / lambda_A) / (lambda_B - lambda_A). For Mo-99/Tc-99m, this is about 22.8 hours after the generator is eluted. After this point, Tc-99m activity decreases and eventually tracks the Mo-99 decay curve from above.
Is secular equilibrium a special case of transient equilibrium?
Conceptually yes: secular equilibrium is the limiting case of transient equilibrium as the half-life ratio approaches infinity. At 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 practical threshold of ratio 100 is a convention reflecting that 99% of the maximum possible activity is reached.
What is the equilibrium activity ratio for Ra-226 and Rn-222?
Ra-226 has a half-life of 1600 yr; Rn-222 has a half-life of 3.8235 days. The half-life ratio is 1600 yr / 3.8235 days = 152,700 — well above the secular equilibrium threshold of 100. At equilibrium, A(Rn-222) = A(Ra-226). A 1 kBq Ra-226 source will be in equilibrium with 1 kBq of Rn-222 after about 27 days.
How does the Mo-99/Tc-99m generator illustrate transient equilibrium?
Mo-99 decays to Tc-99m with a half-life ratio of 65.94/6.006 = 10.98. The Tc-99m activity peaks at about 22.8 hr after elution, reaching a maximum of about 1.10 times the Mo-99 activity. After the peak, Tc-99m activity slowly decreases, maintaining a ratio of 1.10 relative to Mo-99 until the generator is exhausted.
Why does the daughter activity exceed the parent in transient equilibrium?
In transient equilibrium the daughter decays faster (shorter half-life) than the parent. To sustain a constant atom count ratio N_B/N_A = lambda_A/lambda_B, the daughter must have a higher activity A_B = lambda_B * N_B than the parent A_A = lambda_A * N_A. The ratio A_B/A_A = t_{1/2,A}/(t_{1/2,A}-t_{1/2,B}) is always greater than 1 for any transient equilibrium pair.
What practical use does the equilibrium activity ratio have in nuclear medicine?
In nuclear medicine, the equilibrium ratio tells a radiopharmacist the maximum Tc-99m activity they can elute from a Mo-99/Tc-99m generator. At equilibrium, the Tc-99m activity is 1.10 times the Mo-99 reference activity. After elution, the 6-hour Tc-99m rebuilds to equilibrium in about 42 hours (7 half-lives).
Can secular equilibrium be disrupted?
Yes. Any process that physically separates the daughter from the parent, such as chemical separation, ion exchange, or vaporization, breaks the equilibrium. After separation, the daughter decays on its own with its characteristic half-life, and if fresh parent is present, new daughter begins to grow in from zero, re-establishing equilibrium after about 7 daughter half-lives.
What is the secular equilibrium condition in terms of decay constants?
Secular equilibrium requires lambda_A much less than lambda_B, or equivalently t_{1/2,A} much greater than t_{1/2,B}. At equilibrium, the condition lambda_A * N_A = lambda_B * N_B means A_A = A_B. This is also the principle behind radioactive dating methods, where a known decay chain in secular equilibrium allows estimation of the age of a mineral sample.
How does this differ from the Bateman Equations Solver?
The Bateman Equations Solver focuses on computing time-dependent atom counts and activities for two- or three-nuclide chains given specific initial activities. This calculator adds automatic equilibrium classification, the equilibrium activity ratio formula, the time to peak daughter activity, and the time to reach 99% of equilibrium, making it the right starting point when you want to understand the equilibrium behavior first.