Enter Decay Chain Inputs
Example Data Table
| Scenario | Parent Half-Life | Daughter Half-Life | Elapsed Time | Expected Trend |
|---|---|---|---|---|
| Lab source A | 12 hours | 3 hours | 10 hours | Transient equilibrium approach |
| Tracer source B | 40 days | 4 days | 18 days | Strong secular tendency |
| Short chain C | 8 days | 6 days | 5 days | Weak buildup, no strong equilibrium |
Formula Used
The calculator uses decay constants from half-life values. For each isotope, the decay constant is λ = ln(2) / T1/2. Parent quantity after time t is N1(t) = N10e-λ₁t.
Daughter quantity includes surviving initial daughter material and new daughter atoms formed from parent decay. The combined daughter solution is N2(t) = N20e-λ₂t + [λ₁N10 / (λ₂ - λ₁)](e-λ₁t - e-λ₂t).
Activity is computed as A = λN. Equilibrium behavior is judged by comparing parent and daughter half-lives. When the parent half-life is much longer than the daughter half-life, secular equilibrium becomes the dominant long-term pattern.
How to Use This Calculator
Enter the initial parent quantity and any starting daughter quantity. Then provide both half-lives using the same unit system. Choose the elapsed time and keep all time values consistent.
Set an equilibrium target percentage if you want an estimate for when the daughter activity approaches the parent activity. A value like 95% is useful for practical lab planning.
Press the calculate button to display the result block above the form. Use the CSV option for spreadsheet records and the PDF option for printable summaries.
FAQs
1. What does radioactive equilibrium mean?
It describes a decay chain condition where daughter activity rises and approaches the parent activity pattern. This usually appears when the parent isotope decays more slowly than the daughter isotope.
2. What is secular equilibrium?
Secular equilibrium occurs when the parent half-life is far longer than the daughter half-life. After enough time, daughter activity becomes nearly equal to parent activity and follows it closely.
3. What is transient equilibrium?
Transient equilibrium appears when the parent half-life is longer, but not dramatically longer, than the daughter half-life. Daughter activity can briefly exceed parent activity before both decline together.
4. Why must both half-lives use one unit?
All decay equations depend on consistent time scaling. If one half-life is entered in hours and the other in days, the calculated decay constants and activity ratios become physically incorrect.
5. Can I enter zero initial daughter quantity?
Yes. That case represents a fresh sample where daughter material is absent at the start. The calculator will then model daughter buildup only from the parent isotope.
6. Why is time to target sometimes undefined?
If the daughter does not decay faster than the parent, the usual equilibrium approach formula does not apply. In that situation, a clear equilibrium target time cannot be estimated.
7. Does the calculator return activity or atom count?
It returns both. The tool calculates remaining parent and daughter quantities, then converts them into activities using each isotope’s decay constant for better physical interpretation.
8. Is this suitable for classroom and lab work?
Yes. It is helpful for teaching two-member decay chains, checking buildup timing, and comparing activity ratios. It should still be verified against your course notes or lab procedures.