Radioactive isotope decay rate

Radiochemical methods of analysis take advantage of the decay of radioactive isotopes. A direct measurement of the rate at which a radioactive isotope decays may be used to determine its concentration in a sample. For analytes that are not naturally radioactive, neutron activation often can be used to induce radioactivity. Isotope dilution, in which a radioactively labeled form of an analyte is spiked into the sample, can be used as an internal standard for quantitative work. 

A radioactive isotope decays at a rale proportional to its concentration. If the concentration of an isotope is C (mg/L), then its rate of decay may be expressed as... 

Why are some isotopes radioactive but others are not Do all radioactive isotopes decay at the same rate Are all radioactive materials equally hazardous We address these and other questions in this section. 

The half-life (fi/z) is the time required for one-half of a given quantity of a substance to undergo change. Not all radioactive isotopes decay at the same rate. The rate of nuclear decay is generally represented in terms of the half-life of the isotope. Each isotope has its own characteristic half-life that may be as short as a few millionths of a second or as long as a billion years. Half-lives of some naturally occurring and s)mthetic isotopes are given in Table 10.2. 

The rate of decay, or activity, for a radioactive isotope follows first-order kinetics... 

From this expression, it is obvious that the rate is proportional to the concentration of A, and k is the proportionality constant, or rate constant, k has the units of (time) usually sec is a function of [A] to the first power, or, in the terminology of kinetics, v is first-order with respect to A. For an elementary reaction, the order for any reactant is given by its exponent in the rate equation. The number of molecules that must simultaneously interact is defined as the molecularity of the reaction. Thus, the simple elementary reaction of A P is a first-order reaction. Figure 14.4 portrays the course of a first-order reaction as a function of time. The rate of decay of a radioactive isotope, like or is a first-order reaction, as is an intramolecular rearrangement, such as A P. Both are unimolecular reactions (the molecularity equals 1). 

A radioactive isotope X with a half-life of 27.4 d decays into another radioactive isotope Y with a half-life of 18.7 d, which decays into the stable isotope Z. Set up and solve the rate laws for the amounts of the three nuclides as a function of time, and plot your results as a graph. 

Because exposure to radiation is a health risk, the administration of radioactive isotopes must be monitored and controlled carefully. Isotopes that emit alpha or beta particles are not used for Imaging, because these radiations cause substantial tissue damage. Specificity for a target organ is essential so that the amount of radioactive material can be kept as low as possible. In addition, an Isotope for medical Imaging must have a decay rate that is slow enough to allow time to make and administer the tracer compound, yet fast enough rid the body of radioactivity in as short a time as possible. 

Many scientists thought that Earth must have formed as long as 3.3 billion years ago, but their evidence was confusing and inconsistent. They knew that some of the lead on Earth was primordial, i.e., it dated from the time the planet formed. But they also understood that some lead had formed later from the radioactive decay of uranium and thorium. Different isotopes of uranium decay at different rates into two distinctive forms or isotopes of lead lead-206 and lead-207. In addition, radioactive thorium decays into lead-208. Thus, far from being static, the isotopic composition of lead on Earth was dynamic and constantly changing, and the various proportions of lead isotopes over hundreds of millions of years in different regions of the planet were keys to dating Earth s past. A comparison of the ratio of various lead isotopes in Earth s crust today with the ratio of lead isotopes in meteorites formed at the same time as the solar system would establish Earth s age. Early twentieth century physicists had worked out the equation for the planet s age, but they could not solve it because they did not know the isotopic composition of Earth s primordial lead. Once that number was measured, it could be inserted into the equation and blip, as Patterson put it, out would come the age of the Earth. ... 

Radioactive decay is a stochastic process that occurs at random in a large number of atoms of an isotope (see Textbox 13). The exact time when any particular atom decayed or will decay can be neither established nor predicted. The average rate of decay of any radioactive isotope is, however, constant and predictable. It is usually expressed in terms of a half-life, the amount of time it takes for half of the atoms in a sample of a radioactive isotope to decay to a stable form. 

I turn now to the expression for a radioactive isotope, for example, radiocarbon. If lambda is the decay constant, the rate of loss of radioactive atoms by decay will be lambda r m. For the sake of generality, suppose that there is a source that generates radiocarbon at a rate equal to qrc. The conservation equation becomes... 

In essence, NAA involves converting some atoms of the elements within a sample into artificial radioactive isotopes by irradiation with neutrons. The radioactive isotopes so formed then decay to form stable isotopes at a rate which depends on their half-life. Measurement of the decay allows the identification of the nature and concentration of the original elements in the sample. If analysis is to be quantitative, a series of standard specimens which resemble the composition of the archaeological artifact as closely as possible are required. NAA differs from other spectroscopic methods considered in earlier chapters because it involves reorganization of the nucleus, and subsequent changes between energy levels within the nucleus, rather than between the electronic energy levels. 

In K-Ar or zircon U-Pb dating, modeling the loss of radiogenic isotopes by volume diffusion is important. If P0 is the local concentration at t = 0 of a radioactive element decaying with constant X, a source term exists in the transport equation of the radiogenic element which is the local rate of accumulation AP0e Xt. For dual decay,... 

Any geochronometric method for estimating the age of objects based upon the generation of radioactive isotopes by cosmic radiation, followed by isotopic incorporation into the biosphere/geosphere, and their subsequent first-order decay with release of radiation and/or accumulation of daughter isotopes. These methods take advantage of the lack of any dependence of the decay rate on temperature, pressure, pH, or other physical parameters. See Radiocarbon Dating... 

A specific example of applications in the second category is the dating of rocks. Age determination is an inverse problem of radioactive decay, which is a first-order reaction (described later). Because radioactive decay follows a specific law relating concentration and time, and the decay rate is independent of temperature and pressure, the extent of decay is a measure of time passed since the radioactive element is entrapped in a crystal, hence its age. In addition to the age, the initial conditions (such as initial isotopic ratios) may also be inferred, which is another example of inverse problems. 

The Poisson distribution describes the results of experiments in which we count events that occur at random but at a definite average rate. Examples of the Poisson distribution include the number of emails we receive in a one-day period, the number of babies bom in a hospital in a two-day period, the number of decays of a radioactive isotope in a one-day period. 

The word radioactive sounds scciry, but science and medicine are stuffed with useful, friendly applications for radioisotopes. Many of these applications are centered on the predictable decay rates of various radioisotopes. These predictable rates are characterized by half-lives. The half-life of a radioisotope is simply the amount of time it tcikes for exactly half of a sample of that isotope to decay into daughter nuclei. For excimple, if a scientist knows that a sample originally contained 42 mg of a certain radioisotope and measures 21 mg of that isotope in the sample four days later, then the half-life of that radioisotope is four days. The half-lives of radioisotopes range from seconds to billions of yecirs. 

Another example is provided by the chemical fractionation of tungsten into planetary cores. Tungsten has a short-lived radioactive isotope, W, which decays into Hf. Tungsten is siderophile and hafnium is lithophile. Consequently, the daughter isotope, 182Hf, will be found either in the core or the mantle depending on how quickly metal fractionation (core formation) occurred relative to the rate of decay. The Hf- W system is used to date core formation on planetary bodies. We will discuss the details of using radioactive isotopes as chronometers in Chapters 8 and 9. 

The half-lives of 238U, 235U, and 232Th are all very much longer than those of the radioactive daughter isotopes in their decay chains. Therefore, a condition known as secular equilibrium is quickly established in which the decay rates of the daughter isotopes in the decay chain equal that of the parent isotope. In a closed system, once secular equilibrium is... 

The abundances of radioactive isotopes over time in the galaxy can be modeled based on the above considerations. With an approximately constant production rate, the abundance of a stable nuclide will grow and will be proportional to the time over which it has been produced. In contrast, the abundance of a radionuclide will reach a steady state between production and decay in about eight mean lifetimes. (We will use mean life (t) instead of... 

I he rate ot decay of a radioactive isotope is measured in terms of a charac-Jl teristic time called the half-life. This is the time it takes for half of the... 

A value of a = 1 may be obtained in Equation 15-5 under two different situations, as we have noted. First, there is the special circumstance where A is the only reactant (that is, A is unstable and decomposes without any reaction with other substances) and where B and C do not exist (thus, k = k). A common example of this situation is radioactive decay, in which a given radioactive isotope spontaneously decomposes into the isotope of another element at a rate characterized by a rate constant k. 

A reaction of this type is said to follow first-order kinetics because the rate is proportional to the concentration of a single species raised to the first power (fig. 7.2). An example is the decay of a radioactive isotope such as 14C. The rate of decay at any time (the number of radioactive disintegrations per second) is simply proportional to the amount of l4C present. The rate constant for this extremely slow nuclear reaction is 8 x 10-12 s l. Another example is the initial electron-transfer reaction that occurs when photosyn-... 

Like all radioactive isotopes, C-14 decays at a predictable rate. Its half-life of 5,730 years means that one-half the amount of C-14 normally present in a living organism is present in an organism that has been dead for 5,730 years. By suitable manipulation of the mathematics involved in half-life calculations, the approximate age of the remains of plants and animals can be determined. 

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