Decay Chain Calculator Formula
Understand the math behind the decay chain calculator. Each variable explained with a worked example.
Formulas Used
Parent Atoms Remaining
parent_remaining = n0_parent * exp(-lp * time)Daughter Atoms Present
daughter_atoms = n0_parent * lp / (ld - lp) * (exp(-lp * time) - exp(-ld * time))Variables
| Variable | Description | Default |
|---|---|---|
n0_parent | Initial Parent Atoms (N0) | 10000000000 |
hl_parent | Parent Half-Life(hours) | 24 |
hl_daughter | Daughter Half-Life(hours) | 6 |
time | Elapsed Time(hours) | 12 |
lp | Derived value= log(2) / hl_parent | calculated |
ld | Derived value= log(2) / hl_daughter | calculated |
How It Works
Two-Member Decay Chain (Bateman Equation)
When a parent isotope decays to a radioactive daughter, the daughter builds up and then decays itself. The Bateman equations describe this coupled system.
Formulas
N_parent(t) = N0 × exp(-lambda_p × t)
N_daughter(t) = N0 × lambda_p/(lambda_d - lambda_p) × [exp(-lambda_p × t) - exp(-lambda_d × t)]
The daughter activity initially rises, peaks, then decreases. If the parent is much longer-lived, secular equilibrium is reached where daughter activity equals parent activity.
Worked Example
10¹⁰ parent atoms (t½=24 h) decaying to a daughter (t½=6 h) after 12 hours.
n0_parent = 10000000000hl_parent = 24hl_daughter = 6time = 12
- 01lambda_p = ln(2)/24 = 0.02888 h⁻¹
- 02lambda_d = ln(2)/6 = 0.11552 h⁻¹
- 03Parent: 10¹⁰ × exp(-0.02888 × 12) = 10¹⁰ × 0.7071 = 7.07 × 10⁹
- 04Daughter: 10¹⁰ × 0.02888/(0.11552-0.02888) × [0.7071 - exp(-1.3863)]
- 05Daughter = 10¹⁰ × 0.3333 × [0.7071 - 0.2500] = 1.524 × 10⁹
Ready to run the numbers?
Open Decay Chain Calculator