Quantum Physics

29.3 Radioactivity 947INTERACTIVE EXAMPLE 29.3GoalThe Activity of RadiumCalculate the activity of a radioactive substance at different times.Problem The half-life of the radioactive nucleus is 1.6 10 3 yr. If a sample initially contains 3.00 10 16 88 Rasuchnuclei, determine (a) the initial activity in curies, (b) the number of radium nuclei remaining after 4.8 10 3 yr, and(c) the activity at this later time.Strategy For parts (a) and (c), find the decay constant and multiply it by the number of nuclei. Part (b) requiresmultiplying the initial number of nuclei by one-half for every elapsed half-life. (Essentially, this is an application ofEquation 29.4b.)Solution(a) Determine the initial activity in curies.226Convert the half-life to seconds:Substitute this value into Equation 29.5 to get the decayconstant:Calculate the activity of the sample at t 0, usingR 0 N 0 , where R 0 is the decay rate at t 0 and N 0 isthe number of radioactive nuclei present at t 0:Convert to curies to obtain the activity at t 0, usingthe fact that 1 Ci 3.7 10 10 decays/s:T 1/2 (1.6 10 3 yr)(3.156 10 7 s/yr) 5.0 10 10 s 0.693T 1/20.6935.0 10 10 s 1.4 1011 s 1R 0 N 0 (1.4 10 11 s 1 )(3.0 10 16 nuclei) 4.2 10 5 decays/sR 0 (4.2 10 5 decays/s) 1 Ci3.7 10 decays/s101.1 10 5 Ci 11 Ci(b) How many radium nuclei remain after4.8 10 3 years?Calculate the number of half-lives, n:Multiply the initial number of nuclei by the number offactors of one-half:Substitute N 0 3.0 10 16 and n 3.0:(c) Calculate the activity after 4.8 10 3 yr.Multiply the number of remaining nuclei by the decayconstant to find the activity R:n 4.8 10 3 yr1.6 10 3 yr/half-lifeN N 0 1 2 nN (3.0 10 16 nuclei) 12 3.0 3.0 half-lives3.8 10 15 nucleiR N (1.4 10 11 s 1 )(3.8 10 15 nuclei) 5.3 10 4 decays/s 1.4 Ci(1)Remarks The activity is reduced by half every half-life, which is naturally the case because activity is proportional to thenumber of remaining nuclei. The precise number of nuclei at any time is never truly exact, because particles decayaccording to a probability. The larger the sample, however, the more accurate are the predictions from Equation 29.4.Exercise 29.3Find (a) the number of remaining radium nuclei after 3.2 10 3 yr and (b) the activity at this time.Answer (a) 7.5 10 15 nuclei (b) 2.8 CiPractice evaluating the parameters for the radioactive decay of various isotopes of radium by logginginto PhysicsNow at www.cp7e.com and going to Interactive Example 29.3.