The rate of nuclear decay

One of the most important properties of a radioactive nuclide is its lifetime. At present it is not possible to predict theoretically when any particular nucleus in a sample will decay. However, the number of nuclides in a sizeable sample that will decompose in a given time can be measured, and it is found that this rate of decay is characteristic of a given isotope. In fact, the rate of decay of an isotope is constant and unvarying. That is, if a fraction of a radioactive nuclide decays in a certain time interval f, then the same fraction of the remainder will decay in another increment of time f, irrespective of external conditions. Nuclear reactions are not affected by outside influences such as temperature and pressure and it is not possible to significantly alter the constant rate of radioactive decay. For example, radioactive strontium-90, an important 

It is found that it will take 28.5 years for half of the sample to decay and another 28.5 years for half of the remaining strontium-90 to decay and so on  

The disintegration of atomic nuclei can be expressed by the rate equation  

To relate the concentration No that exists at time f = 0 to the amount present at any later time, N, Equation (16.1) must be integrated, to give  

The value of the rate constant, k, can be determined from a plot of In N against t, the slope of the straight-line graph being k. 

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Notice that both the electric charge and the total number of nuclear particles (nucleons) are conserved in the nuclear decomposition. Careful study of the rate of this nuclear decay shows that in a given period of time a constant fraction of the nuclei present will undergo decomposition. This observation allows us to characterize or describe the rate of nuclear decay in a very simple manner. We simply specify the length of time it takes for a fixed fraction of the nuclei initially present to decay. Normally we pick the time for... 

The equation for the decay of a nucleus (parent nucleus - daughter nucleus + radiation) has exactly the same form as a unimolecular elementary reaction (Section 13.7), with an unstable nucleus taking the place of a reactant molecule. This type of decay is expected for a process that does not depend on any external factors but only on the instability of the nucleus. The rate of nuclear decay depends only on the identity of the isotope, not on its chemical form or temperature. 

The rate of nuclear decay depends only on the identity of the nucleus (isotope), not on its chemical form or temperature. 

The rate of nuclear decay is proportional to the number of radioactive nuclei N ... 

The half-life of a radioactive element is the time it takes for one-half of the nuclei in a sample to decay. The rate of nuclear decay can be used to date fossils and artifacts. 

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 radioactive decay of an unstable nucleus is a random process. In any given interval of time, there is a well-defined probability that a given nucleus will decay. This probability is independent of time and is the same for aU nuclei of a given type, but is different for different isotopes. The number of nuclei decaying per unit time is the rate of nuclear decay (or activity), which can be measured using devices, such as the Geiger-Mueller counter (Figure 17.4). 

Measuring the Rate of Nuclear Decay Activity of a sample... 

There are many potential advantages to kinetic methods of analysis, perhaps the most important of which is the ability to use chemical reactions that are slow to reach equilibrium. In this chapter we examine three techniques that rely on measurements made while the analytical system is under kinetic rather than thermodynamic control chemical kinetic techniques, in which the rate of a chemical reaction is measured radiochemical techniques, in which a radioactive element s rate of nuclear decay is measured and flow injection analysis, in which the analyte is injected into a continuously flowing carrier stream, where its mixing and reaction with reagents in the stream are controlled by the kinetic processes of convection and diffusion. 

The basic concepts of nuclear structure and isotopes are explained Appendix 2. This section derives the mathematical equation for the rate of radioactive decay of any unstable nucleus, in terms of its half life. 

Because radioactive decay is a nuclear process, the rate of radioactive decay is totally unaffected by any external factors. Unlike chemical reactions, therefore, there is no dependency on temperature, or pressure, or any of the other environmental factors which affect the rate at which normal chemical reactions occur. This is the reason why radioactive decay chronometers, such as 14C, Ar-Ar, and U-series methods, are so important in geology and archaeology - they provide an absolute clock . 

All nuclear transformations proceed spontaneously at rates that are not altered by ordinary chemical or physical processes. For any population of unstable atoms, the rate of nuclear transformation or radioactive decay is first order that is, proportional to the number, N, of decomposing nuclei present ... 

One of the most basic questions in nuclear astrophysics is How do the nuclei heavier than iron get produced This question was first answered by Burbidge, Burbidge, Fowler and Hoyle in 1957 [35]. They proposed that these elements are produced through the slow (s) and rapid (r) neutron-capture processes. The words rapid/slow refer to the rate of neutron capture compared to the rate of /3-decay in the astrophysical conditions. Figure 13 shows the nuclei involved in the r- and s-processes. The s-process path stays close to the valley of stability whereas the r-process path moves staying close to the drip line. The figure also shows the nuclei involved in the rp-process these are proton rich nuclei where capture of protons are involved and that the rate is compared to the / + rates. 

We shall begin the process of constructing a mathematical model of the nuclear reactor by considering the behaviour of the precursor groups in more detail. Let C, be the concentration of nuclei per m of the /th precursor group at time, t. The rate of radioactive decay, Ri, will be proportional to the existing concentration. 

A radioactive nucleus which emits a particle to become transformed to another nucleus is described as decaying to that nucleus. Such a radioactive event is called radioactive decay. Radionuclides decay at different rates. Some can decay in millionths of a second, others take millions of years. Decay is independent of all the variables which affect chemical reactions such as temperature, pressure, and concentration. This poses particular difficulty with regard to the disposal of nuclear wastes. The rate of radioactive decay is characterized by the loss of a constant percent per unit time, not a constant number of moles per unit time. We therefore characterize the decay rate by specifying the time required for 50 percent of the original material to decay. This period of time is called the half-life, given the symbol, tj/j- The constant percent change means that 50 percent will be lost during the first half-life, 50 percent of what is left after the first half-life will decay over the second half-life, etc. 

Equation 11.8 relates the half-life of any first-order reaction to its rate constant. Because k does not depend on the amount of substance present, neither does t. The half-life is most often used to describe the kinetics of nuclear decay. All... 

Measurement, in time domain, of the rate of nuclear magnetization decay towards equilibrium following the excitation by... 

A radiation counter can be used to measure the rate of nuclear disintegrations in a radioactive material. The activity of a radioactive source is the number of nuclear disintegrations per unit time occurring in a radioactive material. A curie (Ci) is a unit of activity equal to 3.700 X 10 disintegrations per second. For example, a sample of technetium having an activity of 1.0 X 10 Ci is decaying at the rate of... 

Neutron Activation Analysis Few samples of interest are naturally radioactive. For many elements, however, radioactivity may be induced by irradiating the sample with neutrons in a process called neutron activation analysis (NAA). The radioactive element formed by neutron activation decays to a stable isotope by emitting gamma rays and, if necessary, other nuclear particles. The rate of gamma-ray emission is proportional to the analyte s initial concentration in the sample. For example, when a sample containing nonradioactive 13AI is placed in a nuclear reactor and irradiated with neutrons, the following nuclear reaction results. 

When irradiation is complete, the sample is removed from the nuclear reactor, allowed to cool while any short-lived interferences that might be present decay to the background, and the rate of gamma-ray emission is measured. 

Neutron-rich lanthanide isotopes occur in the fission of uranium or plutonium and ate separated during the reprocessing of nuclear fuel wastes (see Nuclearreactors). Lanthanide isotopes can be produced by neutron bombardment, by radioactive decay of neighboring atoms, and by nuclear reactions in accelerators where the rate earths ate bombarded with charged particles. The rare-earth content of solid samples can be determined by neutron... 

The analysis of steady-state and transient reactor behavior requires the calculation of reaction rates of neutrons with various materials. If the number density of neutrons at a point is n and their characteristic speed is v, a flux effective area of a nucleus as a cross section O, and a target atom number density N, a macroscopic cross section E = Na can be defined, and the reaction rate per unit volume is R = 0S. This relation may be appHed to the processes of neutron scattering, absorption, and fission in balance equations lea ding to predictions of or to the determination of flux distribution. The consumption of nuclear fuels is governed by time-dependent differential equations analogous to those of Bateman for radioactive decay chains. The rate of change in number of atoms N owing to absorption is as follows ... 

Strategy Nuclear decays are first-order reactions. Use the first-order rate calculation to find k. Part (b) differs from part (c) in that (b) relates concentration and time, while (c) relates concentration and rate. For nuclear decay, concentration can be expressed in moles, grams, or number of atoms. 

What Do We Need to Know Already Nuclear processes can be understood in terms of atomic structure (Section B and Chapter 1) and energy changes (Chapter 6). The section on rates of radioactive decay builds on chemical kinetics (particularly Sections 13.4 and 13.5). 

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