The Equation for Radioactive Decay

Let N = number of radioactive atoms currently present, = initial number of radioactive atoms, tj/2 = half-life of the isotope, and t = elapsed time. Then the equation used is the following  

Depending upon what variables are already known, these equations can be solved for N, N, t, orv 

In the equation for radioactive decay, N can stand for any unit of quantity-numbers or moles of atoms, or numbers of grams. 

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One can apply the formalism discussed above to a wide variety of systems to produce a radiometric date. In this book, we will use the word date to mean the time calculated from the ratio of a radioactive isotope and its daughter isotope using the equation for radioactive decay. An age is the time between a natural event and the present. A date becomes a valid age when the conditions described in the previous paragraph are met. This terminology, suggested by Faure (1986), is not always used in the literature, where age and date are often used interchangeably. But there is value to the distinction because it helps a reader understand which numbers are significant. 

A more accurate technique is based on the use of a short-lived radionuclide, e.g. ""Tc or "In. The count rate is then measured a number of times during at least one half-life with the source left untouched in position all the time. When the background count rate can be neglected (which is usually the case) combining (8.8a) with the equation for radioactive decay gives after some algebra... 

Note that this equation looks similar to the equation for radioactive decay, equation (2.5), except for the additional term. Because of that term, the solution is somewhat... 

Note the similarity between these equations and the equations for radioactive decay, neutron activation, and power change. We would expect the solutions for X and I to be exponentials. The solutions should be most similar to neutron activation because equation (8.2) is nearly identical in form to the activation equation. Thus xenon and iodine concentrations start at zero and build up exponentially to equilibrium values. 

If the ratio of the rate of radioactive decay from the excavation site to the rate of decay from a fresh tree is 0.75, the archaeologists would use the equation for radioactive decay (above) to conclude that the tree that was used to build the house was chopped down about 2,400 years ago, giving the archaeologists an estimate of the age of the house, and perhaps a village of which the house was a part. (Ages are rounded off to allow for experimental error. A key assumption in the calculation is that the concentration of "C in the atmosphere has been constant since the tree died, which may or may not be strictly true.)... 

The equation of radioactive decay for the Sm-Nd system in the sediment reads... 

Exercise. Apply these considerations to the M-equation for radioactive decay and find in this way the absorbing states. 

Gascoyne, 1992b). Surface waters, therefore, have very low Th/ U ratios. This condition is examined more fully in a subsequent section, but for the moment we can assume that a growing speleothem includes U into its crystal lattice but incorporates negligible °Th. If the crystal lattice remains a closed system with respect to the loss or gain of U and Th, the equations for radioactive production and decay of U, and Th govern the geochemical evolution of the system as follows ... 

Equation [2-26], the basic equation for radioactive decay, can be used ... 

The equations of Sec. 6.2 give the number of atoms of each fission product after a reactor has been run at stated conditions for a specified time. If the reactor is then shut down, the fission products build up and decay in accordance with the laws of simple radioactive decay, which were outlined in Sec. 3. If the nuclides in the decay chain are removed orJy by radioactive decay during reactor operations, the equations of Sec. 3 describe the changes with time of the number of atoms of any nuclide in the decay chain. If a member of a fission-product decay chain or its precursors in the decay chain are removed by neutron absorption, equations for the amount of each nuclide present at time t after shutdown may be obtained by applying the equations of radioactive decay to the amount present at shutdown. 

Aii radioactive decay processes foiiow first-order kinetics. What does this mean What happens to the rate of radioactive decay as the number of nuciides is haived Write the first-order rate law and the integrated first-order rate law. Define the terms in each equation. What is the half-life equation for radioactive decay processes How does the half-life depend on how many nuclides are present Are the half-life and rate constant k directly related or inversely related ... 

Supported is based on the average measurement of °Pb activity determined by measuring the activity of a °Pb as a decay product such as Pb in the lowest section of the sediment profile where °Pb activity is constant. Sedimentation or accretion rates are estimated from the excess (unsupported) °Pb profiles in the sediment profile using the constant initial concentration method (Goldberg et al., 1977). Linear regression analysis is used to solve for (X/s) in the log-transformed equation for radioactive decay ... 

The rate equation for radioactive decay has the same form as the rate law for a first-order chemical reaction. Indeed, radioactive decay is a first-order rate process, and the mathematical relationships used in Chapter 14 for first-order reactions apply here also. 

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 ... 

If we consider this pair of radioactive isotopes for time scales greater than six half-lives of N2, Equation (3b) can be simplified. Because each decay series starts with a long-lived parent, it is commonly the case that A,2. In this case, after six half lives, e approaches zero and can be removed from the equation. For time scales such that 6T2 [Pg.6]

The age equation. Because of extremely low initial °Th/ U ratios in surface corals, we first present the version of the °Th age equation calculated assuming an initial condition of °Th/ U = 0. Below, we present tests that indicate that this assumption holds for most surface corals. We then present a variant of this equation, which relaxes the criterion that initial °Th/ U = 0, but requires some knowledge of initial °Th/ Th values. It may be necessary to employ this second equation in unusual cases involving surface corals, with deep-sea corals, and in some other marine and lacustrine carbonates. The °Th age equation, calculated assuming (1) initial 230Th/238u = ("2) all changes in isotope ratios are the result of radioactive decay and... 

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. ... 

If we choose a much larger than 1 (thin samples d<0.5L) or h pL (thick samples d>>L), the final steady-state exhalation deviates very little from the free exhalation rate and we do not need to know the reshaping time or use Equation 2 for corrections. An air grab sample taken at any time (and corrected for radioactive decay if necessary) after closure, will yield the free exhalation rate to a good approximation, provided that the can is perfectly radon-tight. 

For these reasons, since the pioneering work of Libby the measurement technique used has been different from mass spectrometry and has exploited the characteristics of the process of the radioactive decay. From the law of radioactive decay, the activity of a sample, namely the number of decays per unit time, is proportional to the number of radioactive isotopes (i.e. their concentration in the sample). Indeed, by differentiation, Equation (16.1) becomes ... 

It has been hoped [20,21] that a method could be developed which would directly detect the radioatoms that are present in nature by an efficient ultra-sensitive mass spectrometer technique which would not itself depend upon the fact that the atoms being investigated are radioactive. The advantage of an efficient mass spectrometer system for long-lived radioisotopes can be seen from the equation for calculating the number of atoms present in a sample from its measured radioactive decay rate ... 

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. 

Equation (9.6) is the basic equation describing the decay of all radioactive particles, and, when plotted out, gives the familiar exponential decay curve. The parameter X is characteristic of the parent nucleus, but is not the most readily visualized measure of the rate of radioactive decay. This is normally expressed as the half life (7/ 2). which is defined as the time taken for half the original amount of the radioactive parent to decay. Substituting N = Na/2 into the Equation (9.6) gives ... 

The equation for the growth of a stable daughter from a radioactive parent can be easily derived from Equation (9.6) above, which is the familiar radioactive decay curve. We can write that ... 

Except for radioactive decays, other reaction rate coefficients depend on temperature. Hence, for nonisothermal reaction with temperature history of T(t), the reaction rate coefficient is a function of time k(T(t)) = k(t). The concentration evolution as a function of time would differ from that of isothermal reactions. For unidirectional elementary reactions, it is not difficult to find how the concentration would evolve with time as long as the temperature history and hence the function of k(t) is known. To illustrate the method of treatment, use Reaction 2A C as an example. The reaction rate law is (Equation 1-51)... 

Because a diffusion profile does not end abruptly (except for some special cases), it is necessary to quantify the meaning of diffusion distance. To do so, examine Equation 3-40a. Define the distance at which the concentration is halfway between Co and to be the mid-distance of diffusion, Xmid- The concept of Xmid is similar to that of half-life ti/2 for radioactive decay. From the definition, Xmid can be solved from the following ... 

In the rubidium-strontium age dating method, radioactive 87Rb isotope with a natural isotope abundance of 27.85 % and a half-life of 4.8 x 1010 a is fundamental to the 3 decay to the isobar 87 Sr. The equation for the Rb-Sr method can be derived from Equation (8.9) ... 

Any radionuclide is characterised by its half-life r whose value is independent of the type of decay products that are created. Half-life is defined as the time required (from initial time t = 0) for the decomposition of half the atoms in the sample. The law of radioactive decay allows calculation of the number of atoms N left at time t in a population with N0 atoms initially. The integrated form of this law is given by the following equation ... 

Exercise. In the radioactive decay process the state n = 0 is an absorbing state. Show that the equations for the remaining pn (n = 1,2,...) constitute a master equation with absorbing boundary. The state n = 0 functions as limbo state. 

A problem not mentioned in Chapter 15 is one that is very special for radioactive decay when the elapsed time given in the problem is insignificant in comparison with the half-life. Under such circumstances, Equation 26-2 is totally inappropriate, and the proper equation to use is Equation 26-1. In this case, consider —dN to be the number of atoms that disintegrate in a finite period of time df, which is negligible compared to q consider also that A remains constant during this same period of time. The following problem shows this application of Equation 26-1. 

This is the same model process that we described above for radioactive decay of if we substitute decay constants by rate constants, and amount of substance by concentration, and assume that [A]0 = 1 mol dm-3, we can adapt equation (7.40) derived in Problem 7.5(c) to describe how [B] varies with time ... 

The first term on the right-hand side of equation (2.14), termed the rate factor, represents the direct contamination of herbage by fallout during the growing season. The second term, lag rate factor, is the contribution from the previous year s fallout. This includes the contribution of uptake from the surface soil and matt and also the effects of carry-over of silage and other feeding stuffs from one year to the next. The third term, soil factor, represents the contribution of root uptake, allowing for radioactive decay and reduced availability as nuclides move down the soil profile and become fixed to clay minerals. 

In a balanced equation, the sum of the subscripts (atomic numbers), written or implied, must be the same on the two sides of the equation. The sum of the superscripts (mass numbers), written or implied, must also be the same on the two sides of the equation. For example, the equation for the first step in the radioactive decay of 226Ra is ... 

A) Identify the type of radioactive decay that oxygen-14 will undergo, and write a balanced nuclear equation for the process. 

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