A long-lived isotope M (half-life 5.0 yr) undergoes radioactive decay to isotope N, which has a half-life of 2.0 yr. Isotope N then decays to stable isotope O. A freshly purified 1.00 mol sample of M is sealed in a container at t = 0 with no N or O present.
(a) Write an explicit expression for the number of O atoms as a function of time t. Your expression should involve only t, the initial amount of M, and any needed decay constants.
(b) Using your result from part (a), determine the time at which the number of O atoms equals 90 % of the initial number of M atoms.
(c) Discuss qualitatively whether neglecting the small mass difference between the isotopes would significantly affect the time calculated in part (b).