Radioactive Decay Equilibrium

When a radionuclide decays, it does not disappear but is transformed into a new nuclear species of higher binding energy and often differing Z, A, J, tt, and so on. The equations of radioactive decay discussed so far have focused on the decrease of the parent radionuclides but have ignored the formation (and possible decay) of daughter, granddaughter, and so forth, species. It is the formation and decay of these children that is the focus of this section. 

Let us begin by considering the case when a radionuclide 1 decays with decay constant A.1 forming a daughter nucleus 2, which in turn decays with decay constant 2. Schematically, we have 

Rate of change of 2 = rate of production — rate of decay of nuclei present of 2 2 

This is a first-order linear differential equation and can be solved using the method of integrating factors that we show below. Multiplying both sides by e1 2, we have 

These two equations, (3.21) and (3.22), are the general expressions for the number of daughter nuclei and the daughter activity as a function of time, respectively. 

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The radon concentration indoors in terms of radiation dose exposure is expressed in WL (Working Level, the radiation level of 100 pCi per litre or 3700 Bq per m of Rn in equilibrium with its decay products). Effects of radon are given in terms of WLM (Working Level Months), which is the exposure at 1 WL for one working month, or 170 h. Since there are 365 X 24 = 8760 h per year and 80% of them (7000 h) are spent indoors, the annual time of exposure is 7000/170 = 41 working months . A world-wide representative value adopted by the United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR, 1977) is that 1 pCi per litre or 37 Bq per m is the average Rn concentration indoors, with an equilibrium factor (ratio of Rn decay product concentration to their concentration in radioactive decay equilibrium with Rn gas) of 0.5. This corresponds to an average concentration of 0.005 WL or 5 mWL. The annual indoor exposure is then 0.005 WL x 41 WM = 0.205 WLM. 

Carbon-14 (C-14) with a half-life of 5730 years decays to nitrogen-14 (N-14). A sample of carbon dioxide containing carbon in the form of C-14 only is sealed in a vessel at 1.00-atmosphere pressure. Over time, the CO2 becomes NO2 through the radioactive decay process. The following equilibrium is established ... 

Figure 1. (a) Schematic representation of the evolution by radioactive decay of the daughter-parent (N2/N1) activity ratio as a function of time t after an initial fractionation at time 0. The initial (N2/Ni)o activity ratio is arbitrarily set at 2. Time t is reported as t/T2, where T2 is the half-life of the daughter nuclide. Radioactive equilibrium is nearly reached after about 5 T2. (b) Evolution of (N2/N1) activity ratios for various parent-daughter pairs as a function of time since fractionation (after Williams 1987). Note that the different shape of the curves in (a) and (b) is a consequence of the logarithmic scale on the x axis in (b). 

Although there are three Rji isotopes in the U- and Th-decay series, only is sufficiently long lived tm= 3.8 days) to be a useful estuarine tracer. Radioactive decay of Ra continuously produces Rn, which because of its short half-life is generally in secular equilibrium in seawater. Being chemically non-reactive except for very weak Van der Waals bonding makes this isotope a unique marine tracer in that it is not directly involved in biogeochemical cycles. 

The abundance of each element is fixed by its binding energy, which characterises its strength as an entity, and the temperature and density of free neutrons and protons attacking the nucleus (Fig. A3.1). If, as is usually the case, nuclear equilibrium is reached before a significant number of radioactive decays have had the time to occur, an auxiliary constraint can be imposed the total number density of protons and neutrons, both free and bound, must preserve the mean n/p ratio. 

In the environment, thorium and its compounds do not degrade or mineralize like many organic compounds, but instead speciate into different chemical compounds and form radioactive decay products. Analytical methods for the quantification of radioactive decay products, such as radium, radon, polonium and lead are available. However, the decay products of thorium are rarely analyzed in environmental samples. Since radon-220 (thoron, a decay product of thorium-232) is a gas, determination of thoron decay products in some environmental samples may be simpler, and their concentrations may be used as an indirect measure of the parent compound in the environment if a secular equilibrium is reached between thorium-232 and all its decay products. There are few analytical methods that will allow quantification of the speciation products formed as a result of environmental interactions of thorium (e.g., formation of complex). A knowledge of the environmental transformation processes of thorium and the compounds formed as a result is important in the understanding of their transport in environmental media. For example, in aquatic media, formation of soluble complexes will increase thorium mobility, whereas formation of insoluble species will enhance its incorporation into the sediment and limit its mobility. 

A simple way to characterize the rate of a reaction is the time it takes for the concentration to change from the initial value to halfway between the initial and final (equilibrium). This time is called the half-life of the reaction. The half-life is often denoted as ti/z. The longer the half-life, the slower the reaction. The half-life is best applied to a first-order reaction (especially radioactive decay), for which the half-life is independent of the initial concentration. For example, using the decay of " Sm as an example, [ Sm] = [ Sm]o exp( kt) (derived above). Now, by definition,... 

In some studies it is desirable to do constant infusion to achieve a steady state or equilibrium condition which is a function of input, extraction rate, tissue washout, and radioactive decay (23). Figure 6 shows the yield of Rb-82 at various elution rates to a steady-state condition. At the faster flow rate of 5.33 ml/min, there is 24% yield of Rb-82 and at the slower flow rate of 2.15 ml/min there is about 1% yield of Rb-82. The lower yield at the slower flow rate is mostly accounted for in decay during transit through the line to the patient. 

Because plants take in carbon dioxide as long as they live, any carbon-14 lost to decay is immediately replenished with fresh carbon-14 from the atmosphere. In this way, a radioactive equilibrium is reached where there is a constant ratio of about 1 carbon-14 atom to every 100 billion carbon-12 atoms. When a plant dies, replenishment of carbon-1.4 stops. Then the percentage of carbon-14 decreases at a constant rate given by its half-life, but the amount of carbon-12 does not change because this isotope does not undergo radioactive decay. The longer a plant or other organism is dead, therefore, the less carbon-14 it contains relative to the constant amount of carbon-12. 

In a reactor, the energy per fission, including the energy of the delayed neutrons and of the fission products, is 200 MeV. To produce 1 MW thermal energy, 3.1 x 1016 fissions per second are required. If the half-life of the fission product is short compared with the duration of operation of the reactor, its activity comes into equilibrium when creation by fission equals radioactive decay. Assuming a constant level of power for a duration of Tsecs, the activity is 3.1 x 104/(1 — exp—AT) TBq per MW. Some fission products themselves absorb neutrons (the socalled reactor poisons) and for them the calculation of activity is more complicated. Figure 2.2 shows the combined activity of 1 g of fission products formed in an instantaneous burst of fission and also from 1 g of fission products formed over a period of a year (Walton, 1961). The activity from a short burst decays approximately as t-1 2. 

For the most part, the ratio of 235U to 238U is constant at an atom/atom ratio of 1/138 (0.72%), which corresponds to a radioactive decay ratio of 0.047. For the ratio of 234U to 238U, variations are observed due to geochemical processes, but - because uranium-234 is a progeny of 238U - at radioactive equilibrium, the atom ratio is 5.4 xlO-5, while the radioactive decay ratio is 1.00. 

Because of interference from the radioactive decay of other nuclides (which are typically formed with much higher yields), extraction systems with relatively high decontamination factors from actinides, Bi, and Po must be chosen, and the transactinide activity can only be measured in the selectively extracting organic phase. For this reason, measurement of distribution coefficients is somewhat difficult. By comparing the Rf or Db detection rate under a certain set of chemical conditions to the rate observed under chemical control conditions known to give near 100% yield, distribution coefficients between about 0.2 and 5 can be determined. If the control experiments are performed nearly concurrently, many systematic errors, such as gas-jet efficiency and experimenter technique, are cancelled out. Additionally, extraction systems which come to equilibrium in the 5-10 second phase contact time must be chosen. 

Then, having broached the subject of the relaxation of the ion s atmosphere—its taking up a dissymmetric shape when the ion moves—we went on to tackle the subject of relaxation quite generally. For example, if an electric field is suddenly applied to a solution, it would orient the solvent dipoles therein. A new equilibrium would then be set up. The relaxation time is a measure of the time it takes to set up this new equilibrium. At first it seems peculiar that one should call it a measure of and not the time itself. However, the situation is similar to that of radioactive decay because in changing from state 1 to state 2, the concentration of aradioactive nucleus decreases exponentially with time, taking an infinite time to disappear completely. Since this is not a practical measure, we agree to use another measure of the rate of decay—the time to decline by 63%. 

Helium is the second most abundant element in the universe. In the Earth, it is continuously formed by radioactive decay, mostly of uranium and thorium. Its present concentration in the atmosphere is probably the equilibrium concentration between the amount being released from the Earth s crust and the amount of hehum escaping from the atmosphere into space. The atmosphere represents the major source for neon, argon, krypton, and xenon. They are produced as by-products during flactional distillation of liquid air. Radon is obtained from the radioactive decay of radium. 

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