A pure sample of radioisotope X decays to stable isotope Y with a half-life of 18.0 days
CHEMISTRY · Multi-part · US High School · Chemistry · Atomic Structure & Nuclear · Atomic Structure & Nuclear
Problem
A pure sample of radioisotope X decays to stable isotope Y with a half-life of 18.0 days. After an unknown time t, a mass-spectrometric analysis finds that the ratio of the number of atoms of Y to the number of atoms of the remaining X is 0.350.
(a) Starting from the differential rate law for a first-order process, derive an explicit expression that relates the ratio [X][Y] to the elapsed time t and the decay constant k.
(b) Use your expression from part (a) to calculate the elapsed time t that corresponds to the measured ratio 0.350. Show all algebraic steps and report t in days.
(c) Suppose isotope Y were itself radioactive with a half-life comparable to that of X. Explain qualitatively how this would alter the relationship derived in part (a) and how it would influence the time calculated in part (b).