Answer to Question #118557 in Atomic and Nuclear Physics for Alhassan Moammed

Question

A radioactive material A (decay constant ΠA) decays into a material B (decay constant ΠB) and then into C(decay constant ΠC) which is also radioactive. Determine the amount of C remaining after a time t.

Answer 1

$A (ПА)\to B(ПА)\to C(ПА)$

$N_A(t)=N_{A0}e^{-ПА\cdot t}$

$N_{В0}=N_{A0}-N_A(t)=N_{A0}-N_{A0}e^{-ПА\cdot t}=N_{A0}(1-e^{-ПА\cdot t})$

$N_В(t)=N_{B0}e^{-ПВ\cdot t}=N_{A0}(1-e^{-ПА\cdot t})\cdot e^{-ПВ\cdot t}$

$N_С(t)=N_{С0}e^{-ПС\cdot t}$

$N_{С0}=N_{В0}-N_В(t)=N_{A0}(1-e^{-ПА\cdot t})-N_{A0}(1-e^{-ПА\cdot t})e^{-ПВ\cdot t}=$

$=N_{A0}(1-e^{-ПА\cdot t})\cdot (1-e^{-ПВ\cdot t})$

$N_С(t)=N_{С0}e^{-ПС\cdot t}=N_{A0}(1-e^{-ПА\cdot t})\cdot (1-e^{-ПВ\cdot t})e^{-ПС\cdot t}$

$\frac{N_С(t)}{N_{A0}}=(1-e^{-ПА\cdot t})\cdot (1-e^{-ПВ\cdot t})e^{-ПС\cdot t}$ Answer