Quantum Physics

946 Chapter 29 Nuclear PhysicsA general decay curve for a radioactive sample is shown in Active Figure 29.6. Itcan be shown from Equation 29.2 (using calculus) that the number of nuclei presentvaries with time according to the equationThe hands and numbers of this luminouswatch contain minute amountsof radium salt. The radioactive decayof radium causes the phosphors toglow in the dark.N 012 N 014 N 0N(t)T 1/2N =N 0 e – t2T 1/2ACTIVE FIGURE 29.6Plot of the exponential decay law forradioactive nuclei. The vertical axisrepresents the number of radioactivenuclei present at any time t, and thehorizontal axis is time. The parameterT 1/2 is the half-life of the sample.Log into PhysicsNow at www.cp7e.comand go to Active Figure 29.6, whereyou can observe the decay curves fornuclei with varying half-lives.TIP 29.2 Two Half-Lives Don’tMake a Whole-LifeA half-life is the time it takes for halfof a given number of nuclei to decay.During a second half-life, half theremaining nuclei decay, so in twohalf-lives, three-quarters of theoriginal material has decayed,not all of it.t© Richard Megna/Fundamental PhotographsN N 0 e t[29.4a]where N is the number of radioactive nuclei present at time t, N 0 is the numberpresent at time t 0, and e 2.718 . . . is Euler’s constant. Processes that obeyEquation 29.4a are sometimes said to undergo exponential decay. 1Another parameter that is useful for characterizing radioactive decay is the halflifeT 1/2 . The half-life of a radioactive substance is the time it takes for half of agiven number of radioactive nuclei to decay. Using the concept of half-life, it canbe shown that Equation 29.4a can also be written asN N 0 1 2 n[29.4b]where n is the number of half-lives. The number n can take any non-negative valueand need not be an integer. From the definition, it follows that n is related to timet and the half-life T 1/2 byn Setting N N 0 /2 and t T 1/2 in Equation 29.4a givesN 02 N 0e T 1/2[29.4c]Writing this in the form eT 1/2 2 and taking the natural logarithm of both sides,we getT 1/2 ln 2 0.693[29.5]This is a convenient expression relating the half-life to the decay constant. Notethat after an elapsed time of one half-life, N 0 /2 radioactive nuclei remain (by definition);after two half-lives, half of these will have decayed and N 0 /4 radioactivenuclei will be left; after three half-lives, N 0 /8 will be left; and so on.The unit of activity R is the curie (Ci), defined as1 Ci 3.7 10 10 decays/s[29.6]This unit was selected as the original activity unit because it is the approximateactivity of 1 g of radium. The SI unit of activity is the becquerel (Bq):1Bq 1 decay/s [29.7]Therefore, 1 Ci 3.7 10 10 Bq. The most commonly used units of activity are themillicurie (10 3 Ci) and the microcurie (10 6 Ci).Quick Quiz 29.1tT 1/2What fraction of a radioactive sample has decayed after two half-lives have elapsed?(a) 1/4 (b) 1/2 (c) 3/4 (d) not enough information to sayQuick Quiz 29.2Suppose the decay constant of radioactive substance A is twice the decay constant ofradioactive substance B. If substance B has a half life of 4 hr, what’s the half life ofsubstance A? (a) 8 hr (b) 4 hr (c) 2 hr (d) not enough information to say1 Other examples of exponential decay were discussed in Chapter 18 in connection with RC circuits and in Chapter 20in connection with RL circuits.