Radioactive Decay Is a First-Order Kinetic Process

3 I Radioactive Decay Is a First-Order Kinetic Process 

In a collection of N identical nuclei, the number of nuclei decaying in a short time interval is proportional to the number of nuclei, which defines a first-order decay process (Section 14.2). We have 

An older unit of activity that is commonly used, especially in the United States, is the curie (Ci) (named for Marie Curie), which is defined as 3.7 X 10 Bq. The curie is the decay rate produced by 1 g of radium-226 (an isotope of one of the elements discovered by Marie Curie). 

Equation 17.2 is identical in form to the rate equation describing first-order chemical reaction kinetics (Equation 14.3), and the corresponding solution for N(t) is given by (see Equation 14.6) 

The half-lives (hence the rate constants) of radioactive isotopes vary greatly from isotope to isotope. Two extreme cases fisted in Table 17.3, for example, are uranium-238 and polonium-214  

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Radioactive decay is a first-order kinetic process. Recall that a first-order process has a characteristic half-life, which is the time required for half of any given quantity of a substance to react. (Section 14.4) Nuclear decay rates are commonly expressed in terms of half-lives. Each isotope has its own characteristic half-life. For example, the half-life of strontium-90 is 28.8 yr ( FIGURE 21.6). If we start with lO.O g of strontium-90, only 5.0 g of that isotope remains alter 28.8 yr, 2.5 g remains after another 28.8 yr, and so on. Strontium-90 decays to yttrium-90 ... 

As noted earlier, radioactive decay is a first-order kinetic process. Its rate, therefore, is proportional to the number of radioactive nuclei Nin a sample ... 

Radioactive decay is a first-order kinetics process which follows the integrated rate equation... 

Radioactive decay is a first-order process. To relate it to the first-order kinetics that we studied in Chapter 20, think of the activity as corresponding to a rate of reaction the number of atoms as corresponding to the concentration of a reactant and the decay constant. A, as corresponding to a rate constant, k. This correspondence can be carried further by writing an integrated radioactive decay law and a relationship between the decay constant and the half-life of... 

This is another example of first-order kinetic process. The process here is radioactive decay of Pu-238 isotope. In this process 1 out of every 5.5 billion Pu-238 atoms emits each second an a-particle (He" nucleus) and turns into U-234 ... 

All radioactive decay processes follow first-order kinetics. The half-life of the radioactive isotope tritium (3H, or T) is 12.3 years. How much of a 25.0-mg sample of tritium would remain after 10.9 years ... 

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 ... 

This is a radioactive decay process. Radioactive decay follows first-order kinetics. The solution to the problem simply requires the substitution of the / -value into the appropriate equation ... 

Such a reaction is described as first order and the proportionality constant k is known as the rate constant. Such first-order kinetics is observed for unimolecular processes in which a molecule of A is converted into product P in a given time interval with a probability that does not depend on interaction with another molecule. An example is radioactive decay. Enzyme-substrate complexes often react by unimolecular processes. In other cases, a reaction is pseudo-first order compound A actually reacts with a second molecule such as water, which is present in such excess that its concentration does not change during the experiment. Consequently, the velocity is apparently proportional only to [A]. 

Glusker (37, 38) attempted to prove that these processes are absent by an estimation of active chains by reaction with C14 labelled C02 or H8(T) labelled acetic acid, followed by measurements of the radioactivity of the polymer isolated. Most of the experiments were carried out with fluorenyllithium as initiator in toluene containing 10% diethyl-ether at —60°. At —78° at least 80% of the polymer chains were found to be active at the end of polymerization. The lowest fraction was appreciably less active. Similar results were obtained at —60° although no examination was made of the fractions of lowest molecular weight. Kinetic experiments indicated a first order decay of monomer concentration after an initial rapid consumption of about 3 molecules of monomer per initiator molecule. The mechanism suggested to explain these results involves rapid addition of fluorenyllithium across the vinyl double bond followed by the rapid addition of three monomer units. At this stage it is... 

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 ... 

Radioactivity is the spontaneous emission of radiation from an unstable nucleus. Alpha (a) radiation consists of helium nuclei, small particles containing two protons and two neutrons (fHe). Beta (p) radiation consists of electrons ( e), and gamma (y) radiation consists of high-energy photons that have no mass. Positron emission is the conversion of a proton in the nucleus into a neutron plus an ejected positron, e or /3+, a particle that has the same mass as an electron but an opposite charge. Electron capture is the capture of an inner-shell electron by a proton in the nucleus. The process is accompanied by the emission of y rays and results in the conversion of a proton in the nucleus into a neutron. Every element in the periodic table has at least one radioactive isotope, or radioisotope. Radioactive decay is characterized kinetically by a first-order decay constant and by a half-life, h/2, the time required for the... 

Such a chemical reaction, in which molecules are not colliding with other atoms or molecules, is called a first-order reaction because the rate at which chemical concentration changes at any instant in time is proportional to the concentration raised to the first power. Certain chemical processes, such as radioactive decay, are described by first-order kinetics. In the absence of any other sources of the chemical, first-order kinetics may lead to exponential decay or first-order decay of the chemical concentration (i.e., the concentration of the parent compound decreases exponentially with time) ... 

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 ... 

The driving force for the development of chemiluminescence-based assays (as well as any other optical or electrical detection methodology) is the replacement of radiolabels both for safety reasons and because of their intrinsic instability. Because the earliest high sensitivity immunoassays utilized antibodies with covalently attached as the label, this has served as a yardstick against which all subsequent assay technologies are measured. For this reason, it is important to understand the detection limits for I. Radioactive iodine is a y-emitter that eventually decays to a stable isotope of lead. The decay process exhibits first-order kinetics so that we can write... 

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. 

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. 

Besides chemical reactions, a number of nonchemical processes in nature also obey first-order kinetics. A particularly important example is radioactive decay, which is discussed in Chapter 17. 

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