Radioactive decay equations calculation

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 ... 

For each sample, the pj and P2 ratios were calculated using the radioactive decay equations for U, U and Th (equations (1) and (2)). 

Any radionuclide, whatever the type of radiation emitted, is characterized by its half-life T (Table 17.1), which is the time taken for half of corresponding atom population in the sample to decompose (from initial time, t = 0). Calling A the radioactive decay constant, the law of radioactivity decay allows calculation of the number of atoms N present after time t for a population containing Nq atoms initially. The integrated form of this law is written as equation 17.4 ... 

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 ... 

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. 

The usual procedure for radiocarbon dating is to bum a tiny sample of the object to be dated, collect the CO2 that is produced, and compare its rate of radioactive decay with that of a fresh CO2 sample. The ratio of counts gives Nq jN, which can then be substituted into Equation to calculate t. Mass spectroscopic isotope analysis can also be used to obtain the Nq jN value, as Example illustrates. 

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... 

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 ... 

To calculate t, the length of time since the radioactive decay commenced (i.e. since the fabric precursor died), we again insert values into the integrated form of the first-order rate equation, Equation (8.33). We then insert our previously calculated value of k ... 

The concentration of the radioactive nuclide (reactant, such as Sm) decreases exponentially, which is referred to as radioactive decay. The concentration of the daughter nuclides (products, including Nd and He) grows, which is referred to as radiogenic growth. Note the difference between Equations l-47b and l-47c. In the former equation, the concentration of Nd at time t is expressed as a function of the initial Sm concentration. Hence, from the initial state, one can calculate how the Nd concentration would evolve. In the latter equation, the concentration of Nd at time t is expressed as a function of the Sm concentration also at time t. Let s now define time t as the present time. Then [ Nd] is related to the present amount of Sm, the age (time since Sm and Nd were fractionated), and the initial amount of Nd. Therefore, Equation l-47b represents forward calculation, and Equation l-47c represents an inverse problem to obtain either the age, or the initial concentration, or both. Equation l-47d assumes that there are no other ot-decay nuclides. However, U and Th are usually present in a rock or mineral, and their contribution to " He usually dominates and must be added to Equation l-47d. 

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. 

Geochronological measurements (isochrone methodology) are based on the radioactive decay of the parent nuclide to the daughter nuclide using the fundamental Equation (8.8) for calculating the ages of minerals. 

You know that radioactive decay is first-order, so it is necessary only to find the rate constant for decay, from which the half-life may be calculated by means of Equation 15-11. To make the first-order plot, first convert cpm to log cpm to get... 

Radioactive decay with emission of particles is a random process. It is impossible to predict with certainty when a radioactive event will occur. Therefore, a series of measurements made on a radioactive sample will result in a series of different count rates, but they will be centered around an average or mean value of counts per minute. Table 1.1 contains such a series of count rates obtained with a scintillation counter on a single radioactive sample. A similar table could be prepared for other biochemical measurements, including the rate of an enzyme-catalyzed reaction or the protein concentration of a solution as determined by the Bradford method. The arithmetic average or mean of the numbers is calculated by totaling all the experimental values observed for a sample (the counting rates, the velocity of the reaction, or protein concentration) and dividing the total by the number of times the measurement was made. The mean is defined by Equation 1.1. 

Consider the nuclei 15C, 15N, and 150. Which of these nuclei is stable What types of radioactive decay would the other two undergo Calculate the binding energy difference between 15N and 150. Assuming this difference comes from the Coulomb term in the semiempirical binding energy equation, calculate the nuclear radius. 

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 2 Th age equation, calculated assuming (i) initial Th/ U = 0 and (ii) all changes in isotope ratios are the result of radioactive decay and ingrowth (no chemical/diagenetic shifts in isotope ratios), is... 

Here A represents the amount of decaying radionuclide of interest remaining after some time t, and Aq is the amount present at the beginning of the observation. The k is the rate constant, which is different for each radionuclide. Each atom decays independently of the others, so the stoichiometric coefScient a is always 1 for radioactive decay. We can therefore drop it from the calculations in this chapter and write the integrated rate equation as... 

Use of these equations is illustrated for the fission-product decay chain of mass number 92 considered in Sec. 3.2. Assume production of Sr, the first nuclide of the chain, at a constant rate P = 1/h for a period of 3 h (T = 3 h), followed by several hours of radioactive decay with P = 0. The amounts of Sr and Y, calculated by applying Eqs. (2.37) and (2.38), respectively, are shown in Fig. 2.9. The amount of stable Zr during the period of F=0 is obtained from the material-balance equation ... 

In solids and liquids the total path length for a-particles from radioactive decay is quite short. However, in gases at standard temperature and pressure the paths are several centimeters long (Table 6.2). The range in air for a-particles with an initial energy MeV can be calculated by the enq>irical equation (Pgj = 1.293 kg m ) ... 

Plan Lead-206 is the product of the radioactive decay of uranium-238. We will assume that the only source of lead-206 in the rock is from the decay of uranium-238, with a known half-life. To apply first-order kinetics expressions (Equations 21.19 and 21.20) to calculate the time elapsed since the rock was formed, we first need to calculate how much initial uranium-238 there was for every 1 mg that remains today. 

All radioactive decays occur with first-order kinetics, with the exception of electron capture, which is a two-particle collision. The differential rate law for radioactive decay is given by Equation (2.7). After integration, an alternative and more useful form of the rate law is shown by Equation (2.8). The half-life of radioactive decay is defined as the length of time it takes for the number of unstable nuclides to decrease to exactly one-half of their original value. The half-life, t can be calculated using Equation (2.9), where k is the first-order rate constant... 

The calculation was based on a simplified form of the Bateman equation - see Eq. (7.42) in Chap. 7, Vol. 1 - for closed system radioactive decay to describe the °Po content of degassed lavas as a function of time. Using appropriate initial conditions for solving O Eq. (7.36) in Chap. 7, Vol. 1, one obtains the following relationship for the activities ... 

In chronometry, the age of the sample is defined not in terms of the decay of a parent nuclide, but rather as the in-growth of a daughter activity. Radionuclides that are linked to one another by the processes of radioactive decay have relative concentrations that can be calculated with the Bateman equations, which express the simple laws of radioactive decay and ingrowth. If there exists a time at which all the descendant radionuclides have been removed from the mother material, that time can be determined through the measurement of the relative concentrations of the mother and daughter nuclides at a later time. The time interval between the purification of the sample and the subsequent analysis of the sample is defined as the age of the material at the analysis time. The technique does not apply when the half-life of the daughter nuclide involved in the determination is significantly shorter than the elapsed time. 

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 unstable carbon atoms eventually form CO2, which mixes with ordinary CO2 in the air. Because carbon-14 in the atmosphere is continually produced (by neutron bombardment) and destroyed (by radioactive decay), a dynamic equilibrium is reached in which the ratio of carbon-14 to carbon-12 in the atmosphere remains constant over time at a value of 1.3 X 10 . Thus, in a 1-g sample of carbon from the atmosphere, we should find 1.3 X 10 g of carbon-14. Using Equation 17.2 we can calculate the expected activity of one gram of atmospheric carbon ... 

For measurement results giving a Poisson distribution. as in radioactive decay, the equation u.sed to calculate the standard deviation square root of the number of counts recorded when the number is large. The net signal 5 is obtained as the difference between the measured signal (5-i-J ) minus the background B (e.g.. Compton continuum in gamma spectrometry). Thus the statistically derived standard deviation is given as... 

There is a limit to the negative period that can be developed in a reactor by negative reactivity additions. Soon after the insertion of a large amount of negative reactivity such as a scram, the prompt neutron population decreases to a low level. Neutron population is predominantly the result of delayed neutrons which are produced by fission product decay. Within a short time, 2-3 minutes, all of the short lived delayed neutron precursors have decayed away. At this point, and from this point on, the core neutron population is sustained by decay of the longest lived fission product precursor, bromine-87, with a half life of = 55.72 seconds. Since the rate at which core neutron population decreases is determined by radioactive decay of bromine-87, an effective reactor period can be calculated by setting equations (2,9) and (4.7) equal. Neutron population, N will be used to replace activity, A, and power, P, respectively in the two equations. 

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