Radioactive decay, law

STRATEGY The total mass of isotope in a sample is proportional to the number of nuclei of that isotope that the sample contains therefore, the time dependence of the mass follows the radioactive decay law, Eq. 3. We let m denote the total mass of the radioactive isotope at time t and m0 its initial mass. 

Radioactive decay law The mathematical description of how the amount of radioactive material diminishes over time as a result of radioactive decay. A... 

Although we have found that for internal noise the Ito-Stratonovich dilemma is undecidable for lack of a precise A(t) there are cases in which the Ito equation seems the more appropriate option. As an example we take the decay process defined in IV.6 the M-equation is (V.1.7) and the average obeys the radioactive decay law (V.1.9). As the jumps are relatively small one may hope to describe the process by means of a Langevin equation. Following the Langevin approach we guess... 

If we say that at time t = 0 we have N0 radioactive nuclei present, then integration of Equation (3.3) gives the radioactive decay law... 

An important application of the basic radioactive decay law is that of radionuclide dating. From Equation (3.6), we have... 

For every atom a random lifetime, which is distributed logarithmically according to the radioactive decay law with the half-life, Ti/2, of the nuclide is calculated ... 

Since the rate of decrease in concentration is proportional to the current (Faraday s law), and since the current is proportional to the concentration, the concentration decreases with time in accordance with a first-order law analogous to the radioactive-decay law,... 

Radionuclide concentrations are normally reported as activities. A, in units of disintegrations per minute per kilogram (dpm kg ). Nishiizumi (1987) compiled a new and comprehensive source of data for C1, A1, Be, and Mn in meteorites. Table 2 shows a few activities that are typical for stony meteorites of small to moderate size (pre-atmospheric radius less than 40 cm). To convert from dpm kg to atom g , one uses the radioactive decay law,... 

One prominent and well-known kind of nuclear component is that which is produced by the decay of naturally occurring radionuclides (see Table 1). The best- and longest-known examples are He, produced by alpha decay of the natural isotopes of uranium and Th, and " Ar, produced in one branch of the beta decay of " K. (There are several other natural radionuclides which produce He by alpha decay, but whether because of low parent abundance and/or very slow decay, only in very unusual samples is the production of He not strongly dominated by uranium and thorium.) Since radioactive decay laws are well known, the ratio of daughter to parent isotope(s) in a closed system is a simple function of time, whence this phenomenon has been long and extensively exploited as a geochronometer (e.g., see Chapter 1.16). 

As early as 1907 Bertram Boltwood had used the discovery of radioactive decay laws by Ernest Rutherford and Frederick Soddy to ascribe an age of over two billion years to a uranium mineral. In 1947 Willard Libby at the University of Chicago used the decay of to measure the age of dead organic matter. The cosmogenic radionuclide, becomes part of all living matter through photosynthesis and the consumption of plant matter. 

Stochastic aspects. The stochastic nature of reactors is illustrated by the following simple reactor model. All neutrons perform the same processes independently of each other and with probabilities that are independent of time. A neutron lives until it dies. The probability of death is difl in the time interval dt. Thus the neutron dies according to the radioactive decay law (Poisson distribution) with mean life Z. The process of death is accompanied by a prompt rebirth of k neutrons with probability pjc where... 

The same arguments may be used in the case that a neutron death is accompanied also by the prompt production of delayed neutron emitters. Each delayed neutron emitter decays according to the radioactive decay law with mean life r and emits a neutron coincident with its decay. The resulting equations are... 

The technique is made possible because when an igneous rock solidifies different minerals take up different lanthanides selectively, as reflected in their distribution coefficients. This means that different minerals can have different ratios of Sm to Nd at the time of their crystallization. As the Sm decays it will change the isotopic composition of the Nd to different extents for each different mineral. If the Sm and Nd concentrations and the Nd isotopic composition are determined for each mineral they can be related by the radioactive decay law to yield the span of time that passed since the minerals crystallized. Similar work has been done with K-Ar, Rb-Sr, U-Pb, and Th-Pb isotopic pairs. Each pair has its advantages and disadvantages. The Sm-Nd pair complements the others nicely in that both elements, being lanthanides, have very similar chemical properties, so no aberrant behavior by one member of the pair can cause a peculiar result. 

In time, we can expect every atomic nucleus of a radioactive nuclide to disintegrate, but it is impossible to predict when any one nucleus will do so. Although we cannot make predictions for a particular atom, we can use statistical methods to make predictions for a collection of atoms. Based on experimental observations, a radioactive decay law has been established. 

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... 

Using the Half-Life Concept and the Radioactive Decay Law to Describe the Rate of Radioactive Decay... 

The radioactive decay law states that the rate of decay of a radioactive material—the activity, A— is directly proportional to the number of atoms present. 

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