Answer to Question #240457 in Physics for CosmoDK

Question

The radioactive isotope ²²⁹Th is an α emitter with a half-life t_{½ Th} = 7,300 y. Its daughter, ²²⁵Ra, is a β emitter with a half-life t_{½ Ra} = 14.8 d, which is much shorter than the parent's.

(a) Show that the number of ²²⁹Th nuclei obey the differential equation

dN_Ra/dt = N_Th/τ_Th - N_Ra/τ_Ra, where N_Th is the number of ²²⁹Th nuclei and N_Ra is the number of ²²⁵Ra nuclei, and τ_Th and τ_Ra are their mean lives, respectively.

(b) Suppose that at time t = 0 there are N_Th0 nuclei of the parent isotope ²²⁹Th and no nuclei of the daughter isotope ²²⁵Ra. Find N_Ra for t > 0.

(c) Show that after several years, N_Ra = N_Th τ_Ra/τ_Th.

Answer 1

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