edited by JOHN W. MOORE
Bateman Equations Simplified for Computer Usage Jack G. Kay' National Center for Atmospheric Resear~h,~ Boulder, CO 80307
For a sequence of successive first-order reactions, as in a radioactive decay series, it sometimes is necessary to calculate the amount of eachsubstance present after thereactious have continued for a specified period of time. For a typical problem, consider the analysis of radon, a naturally occurring radioactive substance that is, itself, produced by radioactive decav of radium. Radium and radon are members of a radioactive decay series beginning with uranium; therefore, these radioisotoves arr distributed throughout the earth's crust, in the rocks and soil, wherever uran&n is found. S p e c i f i c a l l ~ , ~ ~(tlI2 ~ R= n 3.8 d) is produced by alpha decay of 226Ra(tI12 = 1600 y), which, in turn, traces its ancestry back to 23W (tIl2 = 4.5 X lo9 y). The main products of radon decay consist of a series of radioisotopes of polonium, lead, and bismuth, terminating in %06Pb, which is stable. Figure 1 shows the main branch of the decay scheme leading from 226Rato 206Pb. It is often necessary to determine the concentration of 222Rnin air, either because it is a pollutant and potential health hazard in the home or work area or because it is useful as R nonreactive tracer of atmospheric movcmrnt. One method of analysis invol\.es the srparation of radon from air. using charcoa1;and subsequent transfer of the radon into a scintillation cell for counting. The zinc sulfide scintillator senses the alpha particles emitted by 222Rn,2 1 8 P ~and , 214P~, converting their signals into flashes of light the intensity of which is proportional to the alpha energy deposited in the scintillator. Counting is done using a photomultiplier and avvrovriate electronic circuitrv. .. . If counting is delayed for fo;r hours, or so, after the cell is filled with radon, then theshort-bed polonium isotopes will he in secular equilibrium with the radon, i.e., the decay rates of 2'8Po and 214P0 each become equal to the decay rate of 222Rn.Under these conditions, neglecting the differences in the scintillator response to different alpha emitters, the radon count rate is taken to be 113 of the observed count rate. On the other hand, if the relationship between the amount of each daughter isotope present and the amount ofZz2Rn originally transferred is known, then counting can begin soon after filling the cell and the results used to determine the radon. With the microcomputers now readily available, it is a simple task to model exactly the radioactive decay series so as to use the observed count rate at any time after the cell is filled to determine the radon in the cell. In 1910, Bateman3 derived the appropriate solutions to the differential equations needed for modeling radioactive decay series of any length. The Bateman equations have continued to be used in equations for radioactive decay kinetics presented in more recent textbooks and reference^.^ The form in which these equations are presented by Bateman is not particularly well suited for easy entry into a computer program, however. In 1977, Fegan5 used a transition matrix approach in developing amore efficient computer code for such activity calculations. 970
Journal of Chemical Education
Figure 1. 22%adecay series (major branch). a designates alpha decay mode; 8- designates beta decay mode.
I have found that the solutions to the appropriate differential equations can be presented in a simple algebraic form requiring only short statements to be entered into the computer program. Since the equations are written in simpler form than originally presented by Bateman, i t is easier to check the program for errors before using it. The equations entered into the computer program retain their simplicity even as the number of steps in the decay series increases. The derivation and results are as follows: A series of first-order reactions occurring in sequence is represented by the following radioactive decay series:
In this sequence, Ni represents the number of atoms of type i decaying with decay constant Xi. The decay constant, A, is related to the half-life, tllz, by the equation X = In 2/tIl2. The relevant differential equations are
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Permanent address: Department of Chemistry, Drexel University, Philadelphia, PA 19104. The National Center for Atmospheric Research is sponsored by the National Science Foundation. Bateman, H. Proc. Cambridge Phil. Soc. 1910, 15,423. See. for examoie. la)Evans. R. D. The AtomicNucleus: McGraws HM: hew Yor6.1955:dhapter 15. lo) Evans. R. D. ~ e a l l h ~ h y1980, 38, 1173. (c)Friedlander. G : Kennedy., W.; Mac as. E. S.: M I er. J. M., Nuclear and Radiochemisrry. 3rd ed.: W ley: New Yo,
