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CHAPTER 3 Radioactive Decay Equations

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1 CHEMISTRY OF THE EARTH CHAPTER 3 Radioactive decay equations 3.1 The radioactive decay law As discussed in Chapter 2, the decay rate of a radioactive substance was found to decrease exponentially with time. Radioactive decay was found to follow a statistical law, that is, it is impossible to predict when any given atom will disintegrate. The rate at which a radioactive substance N1 decays is given by dN 1 = −λ N (3.1) dt 1 1 where N is the number of radioactive atoms and λ1 is called the decay constant , which represents the probability per unit time for one atom to decay. The fundamental assumption in the statistical law for radioactive decay is that this probability for decay is constant, that is, it does not depend on time or on the number of radioactive atoms present. Eq. 3.1 can be integrated with the boundary condition that the initial number of o radioactive atoms is N1 = N1 )0( . This leads to the exponential law of radioactive decay : o −λ1t N1 = N1 e (3.2) Denoting the daughter product of N1 as N2, and for the case in which N2 is a stable daughter product, the accumulation of N2 is simply given by N = N o + N o 1( − e−λ1t ) 2 2 1 (3.3) o λ1t N 2 = N 2 + N1 (e − )1 o where N 2 is the initial amount of N2 prior to the accumulation of radioactive decay products of N1. For measurement purposes, it is often convenient to express Eq. 3.2 in terms of the activity A of a radioactive substance, which is defined to be the number of decays per unit time or simply A= λ1N1. If we multiply both sides of Eq. 3.2 by λ1 , we find that dN − 1 = λ N = λ N oe−λ1t (3.4) dt 1 1 1 1 It follows that o −λt A1 = A1 e (3.5) o where A1 is the initial activity of the radioactive substance. The unit most commonly used in activity measurements is the curie (Ci), which is defined as 3.7×10 10 decays/s. The half-life of a radioactive substance is defined as the elapsed time over which the radioactive substance has decreased to half its original abundance. Thus, by letting o N1 = N1 2/ in Eq. 3.2, the half-life is given by ln 2 .0 69315 t = = (3.6) 5.0 λ λ The mean life, t , or the average amount of time a radioactive element lives, is given by Cin-Ty Lee 2 CHEMISTRY OF THE EARTH ∞ ∞ ∞ 1 1 −λ1t 1 t = − tdN = tλ1 N1dt =λ1 te dt = (3.7) ∫0 ∫0 ∫0 N o N o λ1 Thus, the mean life is simply the inverse of the decay constant. During the mean life, the abundance of a radioactive substance decreases by a factor of 1/e. 3.2 Branched decay 3.2.1 Basic equations In some cases, the parent N1 decays to two different products, a process termed branched decay . Calling these two decay modes a and b, we define the probability of each decay mode as (dN / dt ) λ = − 1 a a N 1 (3.8) (dN 1 / dt )b λb = − N1 The total decay rate of N1 is dN 1 dN 1 dN 1 − = − − = N1 ()λ ,1 a + λ ,1 b = N1λ ,1 total (3.9) dt total dt a dt b where (λ ,1 a + λ ,1 b )= λ ,1 total is the total decay constant. It follows that N1 decays according to the following law o −λ1total t N1 = N1 e (3.10) Thus, if we were to measure only the abundance or activity of the radioactive nuclide, N1, we would have no way of distinguishing between the two decay modes because we can only measure the total decay constant. However, if we were to monitor the evolution of the abundance or activity of the two daughter products of N1 as a function of time, we would be able to distinguish between the two decay modes. For example, we know that a fraction λ ,1 a / λ ,1 total of N1 decays by mode a and a fraction λ ,1 b / λ ,1 total decays by mode b. Denoting the two types of daughter products as N2,a and N2,b , the formulas describing the accumulation of each of the daughter products are N = N o + (λ / λ )N o 1( − e−λ ,1 total t ) ,2 a ,2 a ,1 a total 1 (3.11) o o −λ1total t N ,2 b = N ,2 b + ()λ ,1 b / λtotal N1 1( − e ) o o where N ,2 a and N ,2 b are the initial amounts of daughter product N2,a and N2,b present before the decay of N1. 3.2.2 Example: decay of 40 K to 40 Ar and 40 Ca An example of a branched decay is the decay of naturally occurring 40 K to stable 40 Ar and 40 Ca, the former by electron capture and the latter by β- emission (negatron emission). Electron capture decay to 40 Ar is favored by 11.2 % of 40 K atoms and beta decay to 40 Ca is favored by 88.8 %. Accordingly, the decay constants for each branching decay, λK,Ar and λK,Ca , are Cin-Ty Lee 3 CHEMISTRY OF THE EARTH λ = .0 581 ×10 −10 yr −1 K ,Ar (3.12) −10 −1 λK ,Ca = .4 962 ×10 yr -10 -1 and the total decay constant, λK,total = λK,Ar + λ K,Ca , is 5.543×10 y , corresponding to a half-life of 1.250 Gy. Referring back to Eq. 3.11, we find that the total number of 40 Ar atoms in a system with 40 K is given by 40 40 o 40 λK ,total t Ar = Ar +(λK,Ar / λK,total ) K(e −1) (3.13) where 40 Ar o is the initial number of 40 Ar atoms present prior to the decay of 40 K. Both the 40 K-40 Ar and 40 K-40 Ca isotopic systems have geologic applications. The K-Ar system, in particular, has been used extensively as a geochronometer. 3.3 Decay series 3.3.1 N1 → N 2 → N 3 In Section 3.1, we discussed the decay of a nuclide N1 to a stable daughter product N2. We now discuss the case in which N2 decays to N3. The decay rates of N1 and N2 are given by dN 1 = −λ N (3.14) dt 1 1 dN 2 = λ N − λ N (3.15) dt 1 1 2 2 where λ1 and λ2 are the decay constants of N1 and N2, respectively. Substituting Eq. 3.2 into 3.15 and rearranging yields dN 2 + λ N − λ N oe−λ1t = 0 (3.16) dt 2 2 1 1 The solution to Eq. 3.16 is λ1 o −λ1t −λ2t o −λ2t N 2 = N1 (e − e )+ N 2 e (3.17) λ2 − λ1 and can be verified by substitution into Eq. 3.16. In Eq. 3.17, the first term on the right reflects the production of N2 by decay of N1 and the loss of N2 by decay. The second term on the right represents the contribution from any initial N2 in the system. Note that if λ2 is zero, that is, N2 is a stable nuclide, then Eq. 3.17 is equivalent to Eq. 3.3. Cin-Ty Lee 4 CHEMISTRY OF THE EARTH A N1 N2 λ1N1 A N1 N2 λ2N2 N N 1 2 t B N1 N2 N3 N4 Secular equilibrium in a decay chain Figure 1. Secular Equilibrium. A better understanding of Eq. 3.17 can be had if we look at some examples. For example, let us assume that λ1 << λ2 , that is, N1 decays much slower than N2. It follows that for long times the quantity e −λ1t is much larger than e−λ2t , and therefore Eq. 3.17 simplifies approximately to λ1 o −λ1t N 2 ~ N1 e = λ1 N1 (3.18) λ2 or in other words λ1 N1 = λ2 N 2 (3.19) which essentially means that the activities (or decay rate) of N1 and N2 are equal. This asymptotic state is called secular equilibrium and is illustrated schematically in Figure X. Secular equilibrium is another way of saying a system has reached steady-state, that is, there is no longer a change in the decay rates of each nuclide. For example, let N2 represent the number of people in a restaurant. At 5 PM, the restaurant doors open. Initially, there are no people in the restaurant, but immediately after 5 PM, there is a steady stream of people entering the restaurant. Assuming each person finishes eating and leaves the restaurant in about 45 minutes, we would see that initially the number of people in the restaurant would increase, but after a while, a maximum number will be reached as people start to exit. At this point, the amount of people leaving the restaurant equals the amount entering. This situation is called steady state and is a form of secular equilibrium. At secular equilibrium , it can be shown that the stable daughter nuclide (in this case, N3) increases as o −λ1t N3 = N1 (1− e ) (3.20) Cin-Ty Lee 5 CHEMISTRY OF THE EARTH from which it can be seen that the growth of N3 depends only on the decay constant of N1. We leave the proof of Eq. 3.20 as an exercise for the student.
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