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MODULE 7.1 Solution
Radioactive Chains—Never the Same Again
Prerequisite: Module 2.2, “Unconstrained Growth and Decay.”
Introduction
The mass Q(t) of a radioactive substance decays at a rate proportional to the mass of the substance (see the section “Unconstrained Decay” in Module 2.2, “Uncon strained Growth and Decay”). Thus, for positive disintegration constant,or decay constant,r,we have the following differential equation:
dQ/dt = –rQ(t)
and its difference equation counterpart:
∆Q = –rQ(t – ∆t)∆t
In this module, we model the situation where one radioactive substance decays into another radioactive substance, forming a chain of such substances. For example, radioactive bismuth-210 decays to radioactive polonium-210, which in turn decays to lead-206. We consider the amounts of each substance as time progresses.
Modeling the Radioactive Chain
If a radioactive substance, substanceA,decays into substance substanceB,we say that substanceA is the parent of substanceB and that substanceB is the child of sub stanceA.If substanceB is also radioactive, substanceB is the parent of another sub stance, substanceC,and we have a chain of substances. Figure 7.1.1 depicts the situ ation where A,B,and C are the masses of radioactive substances, substanceA, substanceB,and substanceC,respectively; and different disintegration constants, decay_rate_of_A (a) and decay_rate_of_B (b), exist for each decay.
234 Module 7.1 A B C
decay B to C
decay A to B
decay rate of A decay rate of B
Figure 7.1.1 Chain of decays
Quick Review Question 1
Suppose A and B are the masses of substanceA and substanceB,respectively, at time t; ∆A and ∆B are the changes in these masses; and a and b are the positive disintegra tion constants.
a. Using these constants and variables along with arithmetic operators, such as minus and plus, give the difference equation for the change in the mass of substanceA,∆A.
b. Through disintegration of substanceA, substanceB’s mass increases, while some of substanceB decays to substanceC.Give the difference equation for the change in the mass of substanceB,∆B.
c. In Figure 7.1.1, where A,B,and C are the masses of three radioactive sub stances, give the formula as it appears in a system dynamics tool’s equation for the flow decay_A_to_B.
d. Give the formula as it appears in a systems dynamics tool’s equation for the flow decay_B_to_C.
The mass of substanceA that decays to substanceB is aA.Thus, in Figure 7.1.1, the flow decay_A_to_B contains the formula decay_rate_of_A * A.What substan ceA loses, substanceB gains. However, substanceB decays to substanceC at a rate proportional to the mass of substanceB,bB.Consequently, in Figure 7.1.1, the flow decay_B_to_C contains the mass that flows from one stock to another, decay_rate_ of_B * B.The total change in the mass of substanceB consists of the gain from sub stanceA minus the loss to substanceC with the result multiplied by the change in time, ∆t:
∆B = (aA – bB) ∆t
We consider the initial amounts of substanceB and substanceC to be zero.
Additional System Dynamics Projects 235
Projects
1. a. With a system dynamics tool or a computer program, develop a model for a radioactive chain of three elements, from substanceA to substanceB to substanceC.Allow the user to designate constants. Generate a graph and a table for the amounts of substanceA, substanceB, and substanceC versus time. Answer the following questions using this model.
b. Explain the shapes of the graphs.
c. As the decay rate of A,a,increases from 0.1 to 1, describe how the time of the maximum total radioactivity changes. The total radioactivity is the sum of the change from substanceA to substanceB and the change from substanceB to substanceC,or the total number of disintegrations. Why?
d. (The verification in Part d requires calculus.) With b being the decay rate of B, in several cases where a < b, observe that eventually we have the following approximation:
B A
≈
a
b a −
− −
With the ratio of the mass of substanceB (B) to the mass of substanceA (A) being almost constant, a/(b - a), we say the system is in transient equilibrium.Eventually, substanceA and substanceB appear to decay at the same rate. Using the following material, verify this approximation:
Find the exact solution to the differential equation for the rate of change of A with respect to time, dA/dt = –aA (see the section “Analytic
Solution” in Module 2.2, “Unconstrained Growth and Decay”).
Verify
aA
that B =
− −
is the initial mass of sub
( ), where A0
stanceA, is a solution to the differential equation
for the rate of change of
b ae b
0
at bt
B with respect to time (see the difference equation for ∆B). What number
does e-at approach as t goes to infinity? For a < b,which is smaller, e-at or
-bt
B is approximately equal to what?
e ? Thus, for large t,
e. Using your model from Part a, observe in several cases where a > b that the ratio of the mass of substanceB to the mass of substanceA does not approach a number. Thus, transient equilibrium (see Part d) does not occur in this case.
f. (Requires calculus) Verify the observation from Part e analytically using
work similar to that in Part d.
and B ≈
g. If a is much smaller than b, we have A ≈ A0
aA 0
−
. With the two
b a
amounts being almost constant, we have a situation called secular equi
librium.Observe this phenomenon for the radioactive chain from ra
226
222
218
dium-226 to radon-222 to polonium-218: Ra
→
Rn →
Po , where
b,
226
222
the decay rate of Ra
, a,is 0.00000117/da and the decay rate of Rn ,
is 0.181/da. Using your work from Part a, run the simulation for at least one year.
236 Module 7.1
h. (Requires calculus) Show analytically that the approximations from Part g hold.
i. In the radioactive chain Bi210 → Po210 → Pb206(bismuth-210 to polo nium-210 to lead-206), the decay rate of Bi210, a,is 0.0137/da and the decay rate of Po210, b,is 0.0051/da. Assuming the initial mass of Bi210is 10–8g and
using your model from Part a, find, approximately, the maxi mum mass of Po210and when the maximum occurs.
aA
j. (Requires calculus) In Part d, we verified that B =
e b 0at bt
b a
− − −− ( ).Using
this result, find analytically the maximum of mass of substanceB and when this maximum occurs.
k. Check your approximations of Part i using your solution to Part j. l. For the chain in Part g, use your solution to Part j to find when the largest mass of
Rn222occurs.
m. For the chain in Part g, use your simulation of Part a to approximate the
time when the largest mass of Rn222 occurs. How does your approximation compare with the analytical solution of Part l?
2. Develop a model for a chain of four elements. Perform simulations, observa tions, and analyses similar to those before. Discuss your results
Answers to Quick Review Question
1. a. ∆A = –aA ∆t
b. ∆B = (aA – bB)∆t
c. decay_A_to_B = decay_rate_of_A * A
d. decay_B_to_C = decay_rate_of_B * B
Reference
Horelick, Brindell, and Sinan Koont. 1979, 1989. “Radioactive Chains: Parents and Children.” UMAP Module 234.COMAP, Inc.
