द्रव्यमान क्षति कैलकुलेटर — सूत्र

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.

हल किया गया उदाहरण

10¹⁰ parent atoms (t½=24 h) decaying to a daughter (t½=6 h) after 12 hours.

  1. lambda_p = ln(2)/24 = 0.02888 h⁻¹
  2. lambda_d = ln(2)/6 = 0.11552 h⁻¹
  3. Parent: 10¹⁰ × exp(-0.02888 × 12) = 10¹⁰ × 0.7071 = 7.07 × 10⁹
  4. Daughter: 10¹⁰ × 0.02888/(0.11552-0.02888) × [0.7071 - exp(-1.3863)]
  5. Daughter = 10¹⁰ × 0.3333 × [0.7071 - 0.2500] = 1.524 × 10⁹