A radioactive isotope decays at a rate proportional to its concentration. If the concentration of an isotope is C (mg/L), then its rate of decay may he expressed as rd [mg/ (L∙s)] = kC where k is a constant.

(a) A volume V (L) of a solution of a radioisotope whose concentration is C0 (mg/L) is placed in a closed vessel. Write a balance on the isotope in the vessel and integrate it to prove that the half-life t1/2 of the isotope—by definition, the time required for the isotope concentration to decrease to half of its initial value—equals (In 2)/k.

(b) The half-life of 56Mn is 2.6 h. A batch of this isotope that was used in a radio tracing experiment has been collected in a holding tank. The radiation safety officer declares that the activity (which is proportional to the isotope concentration) must decay to 1% of its present value before the solution can be discarded. How long will this take?