Radioactive decay curve

The equation for the growth of a stable daughter from a radioactive parent can be easily derived from Equation (9.6) above, which is the familiar radioactive decay curve. We can write that ... 

Exponential decay, where log Y = A(B x), such as a radioactive decay curve. 

The rate of radioactive decay is by convention expressed as the half-life, T1/2, defined as the time span during which a given concentration of the radioactive element atoms decays to half their initial value. T1/2 of tritium is 12.3 years. Thus, after 12.3 years one-half the initial concentration of tritium atoms is left, after 24.6 years only one-quarter is left, and so on. A radioactive decay curve of tritium is given in Fig. 10.1. Using the decay curve it is possible to determine, for example, how many years it takes for a given amount of tritium to decay to 20% of the initial amount. The answer, obtained from Fig. 10.1, is 29 years. Similarly, one can determine what percentage of an initial amount of tritium will be left after 20 years. The answer is 32% (read from Fig. 10.1). 

Fig. 10.1 Radioactive decay curve of tritium. After 12.3 years 50% of an initial concentration is left after 24.6 years 25% is left, etc.

The radioactive decay curve of 14C is given in Fig. 11.1. What fraction of an initial concentration of 14C is left after 10,000 years The answer, read from Fig. 11.1, is 27%. 

Fig. 12.1 A radioactive decay curve of 36C1. P36 is the percentage of atmospheric 36C1 left in the sample. The time axis is expressed in half-lives and in 105-year units. The marked point of 56% P36, taken from the Lachlan case study, reveals water with an age of about 290,000 years. The mode of defining the P36 is explained in later sections.

Answer 12.1 About 700,000 years, as read from the radioactive decay curve given in Fig. 12.1. 

Evidently, fluorescers with decay times much longer than the lamp pulse characteristics can be analyzed in much the same way as radioactive decay curves. A semilogarithmic plot of fluorescence intensity against time is linear, with a slope proportional to the decay time and the ordinate intercept providing a quantitative measure of the amount of fhiorophore. If the lamp pulse time and the decay time of the fhiorophore are comparable, the fluorophore s decay charac-... 

Decay of a radioisotope that has a reasonably short fj/2 is experimentally determined by following its activity as a function of time. Graphing the results produces a radioactive decay curve as shown in Figure 10.1. 

Fig. 1. Plasma radioactivity decay curves of apolipoproteins C-I, C-II, and C-in (compared with apo A-I) as a function of time...

TLA was developed in the United Kingdom [20,27]. The surface of a sample (which can be a coupon or a section or tube or pipe) is irradiated to make it slightly radioactive. Two samples are irradiated. One sample is protected from corrosion the second is exposed to the corrosive fluid, in this case the water system. The radioactivity of each of two samples is monitored 2is a function of time. There will be a radioactive decay curve characteristic of the protected sample and a second decay curve for the exposed sample. The difference between the two curves indicates metal loss, which can be converted to corrosion rate. This technique has not been broadly used, but some field use has been reported [27,22]. It shows promise of being an automated coupon which provides a continuous record of corrosion. Figure 4 shows a pipe specimen with an activated area, and Fig. 5 shows data generated in an inhibited cooling tower system. 

Fl9Ure 10 3 A radioactive decay curve (bars give values at 1, 2, and 3 half-lives). 

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). 

Figure 23. Measured ( °Th/ Th) ratios in basalts from Piton de la Fournaise (Reunion Island) as a function of their eraption ages deduced from mineral isochrons. These ratios decrease with increasing emption ages as a result of post-eraptive radioactive decay. The curve shows the theoretical evolution by radioactive decay for a rock with a Th/U ratio of 3.95 and a ( °Th/ Th) ratio of 0.93, similar to the values measured in presently erapted lavas. An approximate age can thus be obtained from the measured ( °Th/ Th) ratio of an old sample. Part of the dispersion around the theoretical curve are due to small source heterogeneities (slightly variable ( °Th/ rh) and Th/U ratios), also evidenced by Sr/ Sr ratios (Condomines et al. 1988, and unpublished results).

When both ( " U/ U) and ( °Th/ U) activity ratios are studied, discussion and interpretation of the data are often presented as a plot of ( " U/ U) against ( °Th/ U) activity ratios (Thiel et al. 1983 Osmond and Ivanovich 1992) (Fig. 16). In such a diagram, instantaneous U gains and losses are represented by straight-line vectors and the radioactive decays by curved lines. Due to the relative decay constants of the nuclides, a... 

In the previous discussion of radioactive decay, it was noted that the rate of decay is directly proportional to the amount of undecayed substance remaining. In a solution of a radioactive substance, a similar relationship would hold for the concentration of undecayed substance remaining. If a solution of a radioactive substance were allowed to decay and a plot were constructed of the concentration remaining versus time, the plot would be a curve such as that shown in Fig. 1. 

Equation (9.6) is the basic equation describing the decay of all radioactive particles, and, when plotted out, gives the familiar exponential decay curve. The parameter X is characteristic of the parent nucleus, but is not the most readily visualized measure of the rate of radioactive decay. This is normally expressed as the half life (7/ 2). which is defined as the time taken for half the original amount of the radioactive parent to decay. Substituting N = Na/2 into the Equation (9.6) gives ... 

As shown in figure 11.8B, because the ratio A1/A2 becomes constant, the slopes of the combined decay curves of the two radionuclides attain a constant value corresponding to the half-life of the longer-lived term (curves a and b in figure 11.8B). Moreover, assuming identical detection coefficients for the two species, their radioactivity ratio also attains a constant value of... 

Figure 11,8 Composite decay curves for (A) mixtures of independently decaying species, (B) transient equilibrium, (C) secular equilibrium, and (D) nonequilibrium, a composite decay curve b decay curve of longer-lived component (A) and parent radio nuclide (B, C, D) c decay curve of short-lived radionuclide (A) and daughter radionuclide (B, C, D) d daughter radioativity in a pure parent fraction (B, C, D) e total daughter radioactivity in a parent-plus-daughter fraction (B). In all cases, the detection coefficients of the various species are assumed to be identical. From Nuclear and Radiochemistry, G. Friedlander and J. W. Kennedy, Copyright 1956 by John Wiley and Sons. Reprinted by permission of John Wiley and Sons Ltd.

Assuming identical detection coefficients for the two species, the radioactivity ratio obviously reduces to 1. This condition, known as secular equilibrium, is illustrated in figure 11.8C for ty2, = °° and ti/2 2 = 0.8 hr. Secular equilibrium can be conceived of as a limiting case of transient equihbrium with the angular coefficient of decay curves progressively approaching the zero slope condition attained in figure 11.8C. 

Equations (5.1) define a direction vector at each point (t,y) of the n+1 dimensional space. Fig. 5.1 shows the field of such vectors for the radioactive decay model (5.2). Any function y(t), tangential to these vectors, satisfies (5.2) and is a solution of the differential equation. The family of such curves is the so called general solution. For (5.2) the general solution is given by... 

This suggests that the heat source had actually been mixed into outer layers and its effect began to dominate the light curve from t 26 d. From the observational side, Phillips (1988) noted that the changes and kinks of the color started from t = 25 d, which may indicate the appearance of heat flux due to radioactive decays. 

Fig. 8 - The observed bolometric light curve (Catchpole et aJ. 1987 and Hamuy et a1. 1987) compared to that which would result from 100% optical conversion and escape of energy from the radioactive decay of 56Ni and 56Co. The upper curve is for 0.20 M of mass 56 and the lower curve is for 0.07 M .

Like all first-order processes, radioactive decay is characterized by a half-life, f]/2, the time required for the number of radioactive nuclei in a sample to drop to half its initial value (Section 12.5). For example, the half-life of iodine-131, a radioisotope used in thyroid testing, is 8.02 days. If today you have 1.000 g of I, then 8.02 days from now you will have only 0.500 g of remaining because one-half of the sample will have decayed (by beta emission), yielding 0.500 g of MXe. After 8.02 more days (16.04 total), only 0.250 g of will remain after a further 8.02 days (24.06 total), only 0.125 g will remain and so on. Each passage of a half-life causes the decay of one-half of whatever sample remains, as shown graphically by the curve in Figure 22.2. The half-life is the same no matter what the size of the sample, the temperature, or any other external condition. 

Figure 4. Decay curves for radioactivity induced in beef processed to a 5-megarad dose (9)...

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