Nuclear Decay Calculations

A radioactive isotope may be unstable, but it is impossible to predict when a certain atom will decay. However, if a statistically large enough sample is examined, some trends become obvious. The radioactive decay follows first-order kinetics (see Chapter 14 for a more in-depth discussion of first-order reactions and equations). If the number of radioactive atoms in a sample is monitored, it can be determined that it takes a certain amount of time for half the sample to decay it takes the same amount of time for half the remaining sample to decay and so on. The amount of time it takes for half the sample to decay is called the half-life of the isotope and is given the symbol r1/2. The table below shows the percentage of radioactive isotope remaining versus half-life. 

HALF-LIFE, t1/2 PERCENT RADIOACTIVE ISOTOPE REMAINING 

As a general rule, the amount of radioactivity at the end of 10 half-lives drops below the level of detection and the sample is said to be safe.  

Half-lives may be very short, 4.2 x 10-6 seconds for Po-213, or very long, 4.5 x 109 years for U-238. The long half-lives of some waste products is a major problem with nuclear fission reactors. Remember, it takes 10 half-lives for the sample to be safe. 

If only multiples of half-lives are considered, the calculations are very straightforward. For example, 1-131 is used in the treatment of thyroid cancer and has a tu2 of eight days. How long would it take to decay to 25% of its original amount Looking at the chart, you see that 25% decay would occur at two half-lives or 16 days. However, since radioactive 

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Example 17.4 shows a typical nuclear decay calculation. 

Decay Schemes. Eor nuclear cases it is more useful to show energy levels that represent the state of the whole nucleus, rather than energy levels for individual atomic electrons (see Eig. 2). This different approach is necessary because in the atomic case the forces are known precisely, so that the computed wave functions are quite accurate for each particle. Eor the nucleus, the forces are much more complex and it is not reasonable to expect to be able to calculate the wave functions accurately for each particle. Thus, the nuclear decay schemes show the experimental levels rather than calculated ones. This is illustrated in Eigure 4, which gives the decay scheme for Co. Here the lowest level represents the ground state of the whole nucleus and each level above that represents a different excited state of the nucleus. 

Exotic Decays. In addition to the common modes of nuclear decay, two exotic modes have been observed. These decay modes are of theoretical interest because theh long half-Hves place strict constraints on the details of any theory used to calculate them. 

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. 

The -y-ray photons emitted by the nuclear decay of a technetium-99 atom used in radiopharmaceuticals have an energy of 140.511 keV. Calculate the wavelength of these "y-rays. 

Chemical effects of nuclear decay have been studied in Germanium through the use of Ge and Ge. Ge decays to Ga with a 275 day half-life by 100% electron capture with no y quanta emitted. Ge is a P emitter which decays to As with a 11.3 h half-life, by three jS transitions having maximum energies of 710 keV (23%), 1379 keV (35%) and 2196 keV (42%). From this are calculated maximum recoil energies of 1.7 eV, 4.5 eV and 10.2 eV, respectively. 

As shown in Example, Equation is used to find a nuclear half-life from measurements of nuclear decays. Equation is used to find how much of a radioactive substance will remain after a certain time, or how long it will take for the amount of substance to fall by a given amount. Example provides an illustration of this t q)e of calculation. In Section 22-1. we show that Equation also provides a way to determine the age of a material that contains radioactive nuclides. 

Figure 7.12 Proton-nucleus potential for the semiclassical calculation of the 151Lu partial proton half-life. [From S. Hofmann, In D. N. Poenaru (Ed.), Nuclear Decay Modes, Copyright 1996 by IOP Publishing. Reprinted by permission of IOP Publishing.]...

Nuclear decay (Chapter 26) is a very important first-order process. Exercises at the end of that chapter involve calculations of nuclear decay rates. 

Experimental and theoretical studies have been performed with respect to the ions arising from protonation of spiro[2.2]pentane 72. Early semiempirical computations and mass spectrometric evidence pointed to cation 73 as the initial protonated spiro[2.2]pentane (Scheme 5.5). This would rearrange to form pyramidal intermediate 74, which would open up to yield cyclopentyl cation 75. Subsequently, using various experimental methods (nuclear decay, radiolysis, FT-ICR MS) in combination with ab initio calculations (HF/6-31G and MP2/6-31G levels), Schleyer and coworkers have found evidence that the corner-protonated cation 76 is a relatively long-lived intermediate in the gas phase. Edge-protonated isomeric cation 77, in turn, is 35.5 kcal mol less stable than ion 76. 

In addition to the above tools, we are developing other tools related to Galactic chemical evolution. These include a Nuclear Reactions Tool, a Nuclear Network Tool, and a Stellar Ejecta Tool. The Nuclear Reactions Tool will help users calculate nuclear reaction rates and help organize, view, and sort many of the common parameters need for these calculations. The Nuclear Network Tool will provide an easy way to evolve a system of species through time for a given environment s temperature and pressure. The features of the Stellar Ejecta Tool are designed to help a user understand the isotopic anomalies found in primitive meteorites or presolar grains. The Stellar Ejecta Tool will provide an easy way to view the isotopic abundance of a star s ejecta, run a nuclear decay network on this material, and then mix it with a second distribution of isotopic abundances. In this way it can simulate systems such as a late injection of material into the early solar nebula. When these tools are released, we will announce them over the Webnucleo mail list (see below). 

Although a richness of information has been obtained, a number of open questions still remain. For elements which were chemically identified, like Rf or Sg, a more detailed study, both theoretical and experimental, should follow. Elements 109, 110 and 111 are still to be studied experimentally the prerequisites for their successful experimental studies should be similar to those of the lighter transactinides. These include the existence of isotopes long enough for chemical studies, knowledge of their nuclear decay properties, so that they can be positively identified, synthesis reactions with the highest possible cross sections and suitable techniques for their separation. For those elements, predictions of the chemical behaviour are a matter of future research. Especially difficult will be the accurate prediction of adsorption of the heaviest elements on various surfaces, or their precipitation from aqueous solutions by determining electrode potentials. For that, further developments in accurate calculational schemes are needed. More sophisticated methods are needed to treat weak interactions, which are important for physisorption processes. 

The higher stability of the alkynediazonium ion towards dediazoniation, relative to that of the alkenediazonium ion, is consistent with structure calculations obtained by Glaser (1987, 1989 see Sect. 5.3). It is unlikely, therfore, that alkyne cations can be obtained by dediazoniation of alkynediazonium ions. An alkynyl cation was formed, however, by spontaneous nuclear decay in l,4-bis(tritioethynyl)benzene, as found by Angelini et al. (1988). 

Strontium-90, a radioactive isotope, is a major product of an atomic homb explosion. It has a half-life of 28.1 yr. (a) Calculate the first-order rate constant for the nuclear decay, (b) Calculate the fraction of Sr that remains after 10 half-lives, (c) Calculate the number of years required for 99.0 percent of °Sr to disappear. 

Decay properties of transuranium nuclides lead to the understanding of proton excess heavy nuclei verification of the proton drip line, nuclear structure of large deformed nuclei such as octupole and hexadecapole deformation, and fission barrier heights. There are several textbooks and review articles on nuclear decay properties of transuranium nuclei (e.g., Hyde et al. 1964 Seaborg and Loveland 1985 Poenaru 1996). Theoretical nuclear models of heavy nuclei are presented by Rasmussen (1975) and the nuclear structure with a deformed single-particle model is discussed by Chasman et al. (1977). Radioactive decay properties of transuranium nuclei are tabulated in the Table of Isotopes (Firestone and Shirley 1996). Recent nuclear and decay properties of nuclei in their ground and isomeric states are compiled and evaluated by Audi et al. (1997), while the calculated atomic mass excess and nuclear ground-state deformations are tabulated by MoUer et al. (1995). 

For nuclear disintegration such as a decay, calculation of recoil energy ( r) of the residual nucleus is as follows ... 

This type of motion, where the electron is chemically forbidden in certain regions of space, resembles the tunneling mechanism in physics, first discovered in radioactive nuclear decays. Since the term tunneling has been an accepted name for decades, there is no reason to adopt another name. However, the original tunneling model is a qualitative model. In ordinary quantum chemical calculations, there is no simple way to calculate tunneling barriers for the electron and there is also no reason to do so. Quantitative results can be obtained with the help of molecular orbital (MO) methods. The electron tunneling model is based on the overlap between the D and A wave functions and this still holds true in MO models. 

Building blocks, isotopes, nuclear reactions (particularly neutron induced), neutron sources, radioactive decay, calculation of radiation level from radioisotope ionization of gases, and ionization in neutron detectors by secondary actions. 

Klapdor, H.V. and Metzinger, J., Predictions of the Decay Heat of Nuclear Reactors by Microscopic Beta Decay Calculations., Proc. Int. Conf. on Nuclear Data for Science and Technology - Mito, Japan (1988). 

Na = (1000gal)e (3 ) = 208.19 gal, so assuming conservation of the initial 1000 gal (neglecting evaporation) we can calculate the volume in tank C as whatever is not in tank A or tank B. The volume in tank C = (1000—208.19 555.16) = 236.65 gal at r = 6.7936 days. We also realize that eventually all the solution will end up in tank C with tanks A and B completely empty. If this was a problem in nuclear decay, we might have so few individual atoms starting from a number like 1000 atoms that we would have to round the values of Na, Nb, and Nc to integer values... 

Comphcated theoretical calculations, based on filled shell (magic number) and other nuclear stabiUty considerations, have led to extrapolations to the far transuranium region (2,26,27). These suggest the existence of closed nucleon shells at Z = 114 (proton number) and N = 184 (neutron number) that exhibit great resistance to decay by spontaneous fission, the main cause of instabiUty for the heaviest elements. Eadier considerations had suggested a closed shell at Z = 126, by analogy to the known shell at = 126, but this is not now considered to be important. 

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 ... 

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