Average Time of Decay Physicists use the radioactive decay law R = R0ekt to compute the

Chapter 5, Problem 60

(choose chapter or problem)

Average Time of Decay Physicists use the radioactive decay law R = R0ekt to compute the average or mean time M until an atom decays. Let F (t) = R/R0 = ekt be the fraction of atoms that have survived to time t without decaying. (a) Find the inverse function t (F ). (b) By definition of t (F ), a fraction 1/N of atoms decays in the time interval t j N , t j 1 N Use this to justify the approximation M 1 N N j=1 t j N . Then argue, by passing to the limit as N , that M = 1 0 t (F ) dF. Strictly speaking, this is an improper integral because t (0) is infinite (it takes an infinite amount of time for all atoms to decay). Therefore, we define M as a limit M = lim c0 1 c t (F ) dF (c) Verify the formula ln x dx = x ln x x by differentiation and use it to show that for c > 0, M = lim c0 1 k + 1 k (c ln c c) (d) Show that M = 1/k by evaluating the limit (use LHpitals Rule to compute limc0 c ln c). (e) What is the mean time to decay for radon (with a half-life of 3.825 days)?