(a) The daughter nucleus formed in radioactive decay is

Chapter 44, Problem 34

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(a) The daughter nucleus formed in radioactive decay is often radioactive. Let N10 represent the number of parent nuclei at time t 5 0, N1(t) the number of parent nuclei at time t, and l1 the decay constant of the parent. Suppose the number of daughter nuclei at time t 5 0 is zero. Let N2(t) be the number of daughter nuclei at time t and let l2 be the decay constant of the daughter. Show that N2(t) satisfies the differential equation dN2 dt 5 l1N1 2 l2N2 (b) Verify by substitution that this differential equation has the solution N2 1t2 5 N10l1 l1 2 l2 1e 2l2t 2 e 2l1t 2 This equation is the law of successive radioactive decays. (c) 218Po decays into 214Pb with a half-life of 3.10 min, and 214Pb decays into 214Bi with a half-life of 26.8 min. On the same axes, plot graphs of N1(t) for 218Po and N2(t) for 214Pb. Let N10 5 1 000 nuclei and choose values of t from 0 to 36 min in 2-min intervals. (d) The curve for 214Pb obtained in part (c) at first rises to a maximum and then starts to decay. At what instant tm is the number of 214Pb nuclei a maximum? (e) By applying the condition for a maximum dN2/dt 5 0, derive a symbolic equation for tm in terms of l1 and l2. (f) Explain whether the value obtained in part (c) agrees with this equation.

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