Quantum Physics

948 Chapter 29 Nuclear PhysicsEXAMPLE 29.4GoalRadon GasCalculate the number of nuclei after an arbitrary time and the time required for a given number of nuclei to decay.22286 RnProblem Radon is a radioactive gas that can be trapped in the basements of homes, and its presence in highconcentrations is a known health hazard. Radon has a half-life of 3.83 days. A gas sample contains 4.00 10 8 radonatoms initially. (a) How many atoms will remain after 14.0 days have passed if no more radon leaks in? (b) What is theactivity of the radon sample after 14.0 days? (c) How long before 99% of the sample has decayed?Strategy The activity can be found by substitution into Equation 29.5, as before. Equation 29.4a (or Eq. 29.4b)must be used to find the number of particles remaining after 14.0 days. To obtain the time asked for in part(c), Equation 29.4a must be solved for time.Solution(a) How many atoms will remain after 14.0 days have passed?Determine the decay constant from Equation 29.5: 0.693 0.693T 1/2 3.83 days 0.181 day1Now use Equation 29.4a, taking N 0 4.0 10 8 , and thevalue of just found to obtain the number N remainingafter 14 days:N N 0 e t (4.00 10 8 atoms)e (0.181 day1 )(14.0 days) 3.17 10 7 atoms(b) What is the activity of the radon sample after14.0 days?Express the decay constant in units of s 1 :From Equation 29.3 and this value of , compute theactivity R: (0.181 day 1 ) 1 day8.64 10 4 s 2.09 106 s 1R N (2.09 10 6 s 1 )(3.17 10 7 atoms) 66.3 decays/s 66.3 Bq(c) How much time must pass before 99% of the samplehas decayed?Solve Equation 29.4a for t, using natural logarithms:Substitute values:ln(N ) ln(N 0 e t ) ln(N 0 ) ln(e t )ln(N ) ln(N 0 ) ln(e t ) tt ln(N 0) ln(N ) ln(N 0/N )t ln(N 0/0.01 N 0 )2.09 10 6 s 1 2.20 106 s 25.5 daysRemarks This kind of calculation is useful in determining how long you would have to wait for radioactivity at agiven location to fall to safe levels.Exercise 29.4(a) Find the activity of the radon sample after 12.0 days have elapsed. (b) How long would it take for 85.0% of thesample to decay?Answer (a) 95.3 Bq (b) 9.08 10 5 s 10.5 days29.4 THE DECAY PROCESSESAs stated in the previous section, radioactive nuclei decay spontaneously via alpha,beta, and gamma decay. As we’ll see in this section, these processes are very differentfrom each other.