The nuclide 226 88Ra decays to gaseous 222 86Rn with a t1/2 = 1600 y. The nuclide 222 86...

The nuclide ²²⁶ ₈₈Ra decays to gaseous ²²² ₈₆Rn with a t_1/2 = 1600 y. The nuclide ²²² ₈₆Rn in turn α decays with a shorter lifetime, t_1/2 = 3.82 days. If radium is originally placed in an evacuated closed container, the amount of radon gas builds up and can be measured. It is found that radon gas builds up to a constant value and that as much 222Rn is being produced as decays. This process is called secular equilibrium, and the activities are equal. (a) Show that radon builds up at the rate dN₂/dt = λ₁N₁ - λ₂N₂ where λ₁, N₁ and - λ₂, N₂ are the decay constants and number of nuclei present for radium and radon, respectively. (b) Because the decay of ²²⁶Ra is so slow, assume that N₁ is constant and show that

(c) Show that after a long time, secular equilibrium is reached.