Radioactive decay equations general

Radioactive Decay Equations General Decay Equations... 

The relationship between the decay constant X and the half-life tll2 can be derived from the general radioactive decay equation... 

The rubidium-strontium geochronometer used in the Rb-Sr geochronological method is based on the radioactive 3 -decay of Rb to Sr. The growth of radiogenic Sr in a Rb rich mineral can be described by the following Equation (9.6). In the rubidium-strontium age dating method, the radioactive Rb isotope with a natural isotope abundance of 27.85 % and a half-hfe of 4.88 X 10 ° years is fundamental to the 3 decay to the isobar Sr. The equation for the Rb-Sr method can be derived from the general equation of radioactive decay (Equation 8.8 in Section 8.8) ... 

Exponential functions are useful in modeling many scientific phenomena. For example, scientists use exponential functions to describe bacterial growth and radioactive decay. The general form of exponential equations is f(x) = Cax (C is a constant). Consider the following problem involving bacterial growth. 

Equations (5.1) define a direction vector at each point (t,y) of the n+1 dimensional space. Fig. 5.1 shows the field of such vectors for the radioactive decay model (5.2). Any function y(t), tangential to these vectors, satisfies (5.2) and is a solution of the differential equation. The family of such curves is the so called general solution. For (5.2) the general solution is given by... 

The counting efficiency (e) of the proportional detector is calculated as the ratio of the net count rate, in s, to the activity (A), in Bq, of this standard radionuclide solution. The net count rate is the standard s gross count rate (RG) minus the detector s background count rate (RB). The reported disintegration rate (A) is the product of the radionuclide concentration, in Bq L 1, and the amount of counted sample, in L, adjusted for the radioactive decay of the radionuclide between standardization and measurement. Equation 2A.1 is the general form of this equation. 

The equations and solutions for closed-system radioactive decay chains have been known since Bateman (1910). To understand the behavior of these systems, however, it is useful to express them as a linear system of ordinary differential equations and use some basic results from linear algebra to discuss the general solutions. This treatment helps to elucidate the ideas of secular equilibrium and relaxation to equihbrium. 

Equations (2.8) and (2.9) describe the time development of the number of nuclei of any isotope i in a radioactive decay series by means of n coupled linear inhomogeneous differential equations. The general solution of any of these equations is the summation of the general solution of the homogeneous equation... 

If the number of nuclei present at some original time r = 0 is designated as Nq, (4.40a) upon integration becomes the general equation for simple radioactive decay ... 

Beta particles are high-energy electrons which are ejected from the nucleus. Since there are not normally any electrons in the nucleus, the beta particles must have been produced in the nucleus during radioactive decay. It is now known that during this process, a neutron changes into a proton and an electron. From this it follows that the atomic number (the number of protons) of the nuclide increases by one, while the mass number (the number of protons and neutrons) remains the same. The electron escapes from the nucleus and is now called a beta particle. The general equation is - e-... 

Calculations using Equations 17.3 and 17.4 are similar to those that we encountered in Section 14.3 for first-order chemical reactions. One major difference is that although it is the rate constant that is generally provided for chemical reactions, the half-life is more commonly given for nuclear reactions, fii addition, in chemical reactions, we generally measure the concentration as a function of time, whereas in radioactive decay, it is the rate (or activity) that is measured. 

The most important class of first-order reactions is the radioactive decomposition of atomic nuclei. Each nucleus of radium 226 or other radio-nucleide has a probability of decomposition in unit time that is independent of the concentration (in general, of the presence of other particles), and in consequence the process of radioactive decay is represented by Equations 10-1 and 10-4. 

As it is easier to determine precisely the ratio of two isotopes relative to each other than the absolute concentration of an individual isotope, both sides of the equation above are generally divided by a stable isotope of the element to which the daughter belongs. In the case of the radioactive decay of Rb to Sr, for instance, everything is normalized to Sr, so the equation becomes... 

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... 

Nuclei which are radioactively unstable usually decay by the emission of one of three particles from the nucleus, traditionally labelled a, p and y particles. The largest, slowest and least penetrating of these are the a particles, which turn out to be the nucleus of the helium atom - i.e., two protons and two neutrons, with an overall charge of + 2. Decay by a emission is restricted to the heavier elements, and can be summarized in the following general equation ... 

Since we do not weigh these streaks, we assumed that the Ag + Cu + Zn + As + Sb + Au = 100%. By comparing the radioactivity levels in the streaks with those in the standard alloy streaks (each corrected for decay to a common reference time) we can then calculate the percent ratios Cu/Ag, Au/Ag, Zn/Ag, Sb/Ag, and As/Ag. The combination of these ratio data and the above equation allows us to calculate the individual percent values. Various chemical analyses reported in the literature for silver coins and art objects indicate that other elements such as lead and tin (which we do not detect) are usually present at less than 2%. There are notable exceptions however. Some types of coins from certain periods contain up to 10-15% Pb + Bi. As a result, we have always performed direct neutron Howitzer silver analysis on at least a few coins of each general type that are analyzed by streak analysis. The silver data for the Howitzer analysis are invariably lower than those for the streak analysis, but this is to be expected for two reasons the... 

In most practical situations, the number of radioactive atoms present is exceedingly high and the probability of detection very small. This means that the number of decays detected n decays or C counts) is very much smaller than the number of radioactive atoms present N). (Exceptions to this general situation, when the efficiency of detection and probability of particle emission are very high and when the count period is comparable to the half-life of the nuclide, are discussed in Section 5.7.) In fact, if we assume the detection efficiency to be subsumed into / , it makes no difference to the statistics whether we consider number of decays or number of counts detected and from now on we can take n and C as equivalent. Under these circumstances, various mathematical approximations can be made to Equation (5.18) which lead to a new form for the probability distribution. 

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