Radioactive decay first-order process

Here, v denotes the mean velocity of advection, and k is a rate constant of a reaction with first order kinetics. The last term in the equation R(x) is an unspecified source or sink related term which is determined by its dependence on the depth coordinate x. Instead of R(x), one might occasionally find the expression (ERj) which emphasizes that actually the sum of different rates originating from various diagenetic processes should be considered (e.g. Berner 1980). Such reactions, still rather easy to cope with in mathematics, frequently consist of adsorption and desorption, as well as radioactive decay (first-order reaction kinetics). Sometimes even solubility and precipitation reactions, albeit the illicit simplification, are concealed among these processes of sorption, and sometimes even reactions of microbial decomposition are treated as first order kinetics. 

As pointed out in Chapter 11, radioactive decay is a first-order process. This means that the following equations, discussed on pages 294-295, apply ... 

The half-life, f1/2, of a substance is the time needed for its concentration to fall to one-half its initial value. Knowing the half-lives of pollutants such as chlorofluoro-carbons allows us to assess their environmental impact. If their half-lives are short, they may not survive long enough to reach the stratosphere, where they can destroy ozone. Half-lives are also important in planning storage systems for radioactive materials, because the decay of radioactive nuclei is a first-order process. 

For any given radionuclide, the rate of decay is a first-order process that is constant, regardless of the radioactive atoms present and is characteristic for each radionuclide. The process of decay is a series of random events temperature, pressure, or chemical combinations do not effect the rate of decay. While it may not be possible to predict exactly which atom is going to undergo transformation at any given time, it is possible to predict, on average, the fraction of the radioactive atoms that will transform during any interval of time. 

The science of kinetics deals with the mathematical description of the rate of the appearance or disappearance of a substance. One of the most common types of rate processes observed in nature is the first-order process in which the rate is dependent upon the concentration or amount of only one component. An example of such a process is radioactive decay in which the rate of decay (i.e., the number of radioactive decompositions per minute) is directly proportional to the amount of undecayed substance remaining. This may be written mathematically as follows ... 

Fig. 1 Plot of concentration remaining versus time for a first-order process (e.g., radioactive decay). 

The concentrations of the reactants change as the reaction progresses, and so the rate changes because it depends on the concentrations. An illustration of the eflfcct of time on the rate of a first-order process is the decay of a radioactive substance, considered in Sec. 22.3. 

Radioactive decay is a first-order process. See Chapter 20 for a discussion of half-lives related to nuclear reactions and other information on radioactivity. 

An ampoule of radioactive Kr-89 (half life = 76 minutes) is set aside for a day. What does this do to the activity of the ampoule Note that radioactive decay is a first-order process. 

For any given radionuclide, the rate of decay is a first-order process that depends on the number of radioactive atoms present and is characteristic for each radionuclide. The process of decay is a... 

Radioactive substances decay by first-order processes and their rate of decay is normally reported by stating their half-lives (Section 17.7). 

Radioactive decay is kinetically a first-order process (Section 12.4), whose rate is proportional to the number of radioactive nuclei N in a sample times the first-order rate constant k, called the decay constant ... 

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. 

In the present instance, though, we are given decay rates at two different times rather than values of N and N0. Nevertheless, for a first-order process like radioactive decay, in which rate = kN, the ratio of the decay rate at any time t to the decay rate at time... 

A radioactive nucleus decays by a first-order process, so that (20-1), (20-2), and (20-3) apply. The stability of the nucleus with respect to spontaneous decay may be indicated by its first-order rate constant, k, or by the half-life, ti/2-... 

If you recall, back in Chapter 5 we discussed half-life in the context of the decay of radioactive nuclei. In that chapter, we defined the half-life as the amount of time it took for one half of the original sample of radioactive nuclei to decay. Because the rate of decay only depends on the amount of the radioactive sample, it is considered a first-order process. Using the same logic, we can apply the concept of half-life to first-order chemical reactions as well. In this new context, the half-life is the amount of time required for the concentration of a reactant to decrease by one-half. The half-life equation from Chapter 5 can be used to determine the half-life of a reactant ... 

Figure 4-11. Exponential decrease with time in the number of excited states, such as can occur for the emission of fluorescence after the illumination ceases or for radioactive decay, illustrating the relationship with the lifetime (t) for a first-order process. Note that 0.37 equals 1/e, where e is the base of the natural logarithms.

Radioactive decay can be described as a first-order process. Thus, for any first-order decay process, the amount of material present declines in an exponential fashion with time. This is easy to see by integrating Equation (1.5.3) to give ... 

The decay of radioactive isotopes is a simple exponential (first-order) process. 

Figure 6-1 The decay of a radioactive isotope is a first-order process.

When a short-lived radioactive isotope is introduced into a biological system, the observed decay in radioactivity results from a combination of normal radioactive decay and biological turnover (e.g., removal of the isotope from the bloodstream by excretion or transport into tissues). If the biological turnover is a first-order process, then Aapp, the apparent first-order rate constant, is the sum of Xr (radioactive)-H A (biological). This is quite understandable since A represents the fraction of the activity present that disappears per small increment of time. Fractions can be added. The observed radioactivity at any time is given by ... 

The period t = 1/fc is sometimes referred to as the natural lifetime of species A. During time t, the concentration of A decreases to He of its original value. A second period, from r = t to f = 2t, produces an equivalent fractional decrease in concentration to 1/e of the value at the beginning of the second interval, which is (1/e) of [AJo- A more familiar example of this property of exponentials is found in the half-life tu2 of radionuclides. During a period t n, half of the atoms in a sample of a radioactive element decay to products a second period of t i2 reduces the amount of the element to one quarter of its original number, and so on for succeeding periods. Regardless of the time interval chosen, equal elapsed times produce equal fractional decreases in reactant concentration for a first-order process. 

Radionuclides have different stabilities and decay at different rates. Some decay nearly completely in a fraction of a second and others only after millions of years. The rates of all radioactive decays are independent of temperature and obey first-order kinetics. In Section 16-3 we saw that the rate of a first-order process is proportional only to the concentration of one substance. The rate law and the integrated rate equation for a first-order process (Section 16-4) are... 

The conceptual approach is particularly effective when solving problems that have half-lives that are whole number values. For more complex problems, we need to use some ideas borrowed from chemical kinetics. Radioactive decay can be described as a first order processes, which means it can be described with the following equation ... 

Zero-order reactions describe processes such as output from a pump or, in some cases, diffusion from a suspension. Radioactive decay is an example of a first-order process. 

Radioactive decay of an unstable nucleus is another example of a first-order process. For example, the half-life for the decay of uranium-235 is 7.1X10 yr. After 710 million years, a 1-kg sample of uranium-235 will contain 0.5 kg of uranium-235, and a 1-mg sample of uranium-235 will contain 0.5 mg. (We discuss the kinetics of radioactive decay thoroughly in Chapter 23.) Whether we consider a molecule or a radioactive nucleus, the decomposition of each particle in a first-order process is independent of the number of other particles present. 

Note that the activity depends only on Jf raised to the first power (and on the constant value of k). Therefore, radioactive decay is a first-order process (see Section 16.3). The only difference in the case of nuclear decay is that we consider the number of nuclei rather than their concentration. 

The decay rate (activity) of a sample is proportional to the number of radioactive nuclei. Nuclear decay is a first-order process, so the half-life does not depend on the number of nuclei. Radioisotopic methods, such as C dating, determine the ages of objects by measuring the ratio of specific isotopes in the sample. 

Understand why radioactive decay is a first-order process and the meaning of half-life convert among units of radioactivity, and calculate specific activity, decay constant, half-life, and number of nuclei estimate the age of an object from its specific activity ( 23.2) (SPs 23.4, 23.5) (EPs 23.17-23.30)... 

The relaxation is commonly described by an exponential function and so we characterize its rate by the reciprocal of the first order rate constant, a relaxation time. Even though the relaxation process may appear as an exponential process, we should not be mislead into believing that it is a real first order process similar to radioactive decay. The underlying process is generally not first order and in many cases, oftener than we like to think, this appears as a non-exponential relaxation, especially if the data were taken carefully enough. 

Knowledge Required (1) The knowledge that radioactive decay is a first-order process. (2) The integrated form of the first-order rate equation. (3) The relationship between half-life and rate constant... 

Radioactive decay is a random first-order process described by... 

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