Calculateur de Défaut de Masse
Calculez le défaut de masse d'un noyau atomique.
Parent Atoms Remaining
7,071,067,811.865
Parent Atoms Remaining vs Elapsed Time
Formule
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.
Exemple Résolu
10¹⁰ parent atoms (t½=24 h) decaying to a daughter (t½=6 h) after 12 hours.
- 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
Questions Fréquentes
What is secular equilibrium?
When the parent half-life is much longer than the daughter, the daughter activity builds up until it equals the parent activity. They then decay together at the parent rate. Example: Ra-226 (1600 yr) → Rn-222 (3.8 days).
What is transient equilibrium?
When the parent half-life is longer but not vastly so, the daughter activity rises above the parent activity before both decay at the parent rate. Example: Mo-99 (66 h) → Tc-99m (6 h) in medical generators.
What about longer decay chains?
Natural decay series (U-238, U-235, Th-232) have many steps. The Bateman equations extend to n members but become algebraically complex. In secular equilibrium, all members have the same activity.
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