• $summary_{type}$: application
• $word_{count}$: 150

In a decay chain A -> B -> C, the rate equation for B is $dN_B$/dt = $lambda_A$$N_A$ - $lambda_B$$N_B$. In secular equilibrium (when t_$\frac{1}{2}$ of A >> t_$\frac{1}{2}$ of B): $lambda_A$$N_A$ = $lambda_B$$N_B$, meaning activities are equal throughout the chain. The natural radioactive series: U-238 series (A = 4n+2, ends at Pb-206, 14 decays: 8 alpha + 6 beta), U-235 series (A = 4n+3, ends at Pb-207), Th-232 series (A = 4n, ends at Pb-208). To find the number of alpha and beta decays: alpha count = $\frac{A_initial - A_final}{4}$, then beta count is determined by balancing Z. For the U-238 -> Pb-206 series: alpha count = $\frac{238-206}{4}$ = 8, Z change from alpha alone = -16, actual Z change = 82-92 = -10, so beta-minus count = -16-(-10) = 6.