A laboratory begins with a pure sample of isotope X that undergoes β-decay to isotope Y, which in turn undergoes γ-decay to stable isotope Z. The half-life of X is 12 h, and the half-life of Y is 4.0 h. Technicians record the activities (decays per second) of X and Y for many hours. They find that the activity of Y rises from zero at t = 0 to a maximum at about t = 6 h and then decreases gradually thereafter.
(a) Describe qualitatively why the activity of isotope Y increases during the first several hours even though Y itself is radioactive.
(b) Explain why there must be a specific instant at which the activities of X and Y are equal. Identify the competing physical processes that make this equality inevitable.
(c) Describe the qualitative time dependence of the number of stable Z nuclei from t = 0 to very long times. Justify your description in terms of the observed behavior of X and Y.