First order decay, radioactivity

Any geochronometric method for estimating the age of objects based upon the generation of radioactive isotopes by cosmic radiation, followed by isotopic incorporation into the biosphere/geosphere, and their subsequent first-order decay with release of radiation and/or accumulation of daughter isotopes. These methods take advantage of the lack of any dependence of the decay rate on temperature, pressure, pH, or other physical parameters. See Radiocarbon Dating... 

RADIOACTIVE DECAY. Many atomic nuclei have unstable neutron-to-proton ratios and undergo spontaneous first-order decay through the emission of a, I3, or (3 particles or gamma rays. 

Equation (20.3) is in the form of first-order decay equation which describes a first-order chemical reaction or radioactive decay. It es-... 

Glusker (37, 38) attempted to prove that these processes are absent by an estimation of active chains by reaction with C14 labelled C02 or H8(T) labelled acetic acid, followed by measurements of the radioactivity of the polymer isolated. Most of the experiments were carried out with fluorenyllithium as initiator in toluene containing 10% diethyl-ether at —60°. At —78° at least 80% of the polymer chains were found to be active at the end of polymerization. The lowest fraction was appreciably less active. Similar results were obtained at —60° although no examination was made of the fractions of lowest molecular weight. Kinetic experiments indicated a first order decay of monomer concentration after an initial rapid consumption of about 3 molecules of monomer per initiator molecule. The mechanism suggested to explain these results involves rapid addition of fluorenyllithium across the vinyl double bond followed by the rapid addition of three monomer units. At this stage it is... 

Like all first-order processes, radioactive decay is characterized by a half-life, f]/2, the time required for the number of radioactive nuclei in a sample to drop to half its initial value (Section 12.5). For example, the half-life of iodine-131, a radioisotope used in thyroid testing, is 8.02 days. If today you have 1.000 g of I, then 8.02 days from now you will have only 0.500 g of remaining because one-half of the sample will have decayed (by beta emission), yielding 0.500 g of MXe. After 8.02 more days (16.04 total), only 0.250 g of will remain after a further 8.02 days (24.06 total), only 0.125 g will remain and so on. Each passage of a half-life causes the decay of one-half of whatever sample remains, as shown graphically by the curve in Figure 22.2. The half-life is the same no matter what the size of the sample, the temperature, or any other external condition. 

Radioactivity is the spontaneous emission of radiation from an unstable nucleus. Alpha (a) radiation consists of helium nuclei, small particles containing two protons and two neutrons (fHe). Beta (p) radiation consists of electrons ( e), and gamma (y) radiation consists of high-energy photons that have no mass. Positron emission is the conversion of a proton in the nucleus into a neutron plus an ejected positron, e or /3+, a particle that has the same mass as an electron but an opposite charge. Electron capture is the capture of an inner-shell electron by a proton in the nucleus. The process is accompanied by the emission of y rays and results in the conversion of a proton in the nucleus into a neutron. Every element in the periodic table has at least one radioactive isotope, or radioisotope. Radioactive decay is characterized kinetically by a first-order decay constant and by a half-life, h/2, the time required for the... 

Storage stability over time under fixed conditions of temperature, pH value and concentration of additives often can be expressed by a first-order decay law (analogous to radioactive decay) [Eq. (2.19)]. 

The only reactions that are strictly first order are radioactive decay reactions. Among chemical reactions, thermal decompositions may seem first order, but an external energy source is generally required to excite the reaction. As noted earlier, this energy is usually acquired by intermolecular collisions. Thus, the reaction rate could be written as... 

But, enough of this dalliance in other fields, let s get back to Ostwald and the order of a reaction. We ve illustrated a first-order decay process, but if we were talking about a chemical reaction, rather than radioactive decay, we would use concentration in moles per liter (mol I/1) rather than using the number of molecules or moles of a material in our differential equation. This is usually indicated by putting square brackets around the symbol for the reacting group, where k is now called the rate constant (Equation 4-3). 

Radioactive decay can be described as a first-order process. Thus, for any first-order decay process, the amount of material present declines in an exponential fashion with time. This is easy to see by integrating Equation (1.5.3) to give ... 

At zero-time, B is made radioactive by injecting or feeding the organism a very small amount of high specific activity B. What will we observe if C is indeed an intermediate behveen B and D We observe an immediate rise in the specific activity of B followed by a first-order decay as labeled B is converted to C and unlabeled A is converted to B. The total pool of B remains constant because for every mole of B converted to C, a mole of A is converted to B. Similarly, under this steady-state condition, the total pools of C, D, E, 5ind so on, remain constant. Only the radioactivity in B, C, and so on, changes with time. The specific activity of an intermediate increases when labeled molecules enter the pool. The specific activity of the intermediate is unaffected by its further metabolism (labeled and unlabeled molecules leave the pool at a constant ratio). The specific activity of an intermediate decreases (by dilution) when the specific activity of the precursor entering the pool is lower than the specific activity of the intermediate in the pool. 

Kinetics of radioactive decay first order decay with half-life tyx = ln2/ = 0.693/k. 

Such a chemical reaction, in which molecules are not colliding with other atoms or molecules, is called a first-order reaction because the rate at which chemical concentration changes at any instant in time is proportional to the concentration raised to the first power. Certain chemical processes, such as radioactive decay, are described by first-order kinetics. In the absence of any other sources of the chemical, first-order kinetics may lead to exponential decay or first-order decay of the chemical concentration (i.e., the concentration of the parent compound decreases exponentially with time) ... 

In Equation 58, the time-dependent terms between the braces contain the decay constant A. Therefore, the rate of change in Ra-226 concentration at any depth (dC/dt) depends on the decay rate constant. Thus, in the case of a first-order reaction (radioactive decay), the rate of change in concentration depends on the reaction rate constant, whereas it has been shown in the preceding section that for a zero-order reaction (oxygen consumption), the rate of change in concentration (dC/dt) is independent of its rate constant. 

From the large number of mathematical models for the transport of transformation products with kinetic reactions that can be considered in the Rockflow system we have chosen a first-order chemical nonequilibrium model to simulate the sorption reaction. It can be described by the governing solute transport equation with rate-limited sorption and first-order decay in aqueous and sorbed phases. This model includes the processes of advection, dispersion, sorption, biological degradation or radioactive decay of the contaminant in the aqueous and/or sorbed phases. Figure 6.1 illustrates the conceptual model for sequential decay of a reactive species. 

Figure 3.4 shows the first order decay of nd the exponential curve is typical of any simple radioactive decay process. A characteristic feature is that the time taken for the number of nuclides present at time t, A, to decrease to... 

As a result of this first-order behavior, radioactive decay is a process that can be described mathematically by an exponential decrease in the number of parent nudides Nas a function of time (Figure 1.3) ... 

So, we see as a laboratory source of alpha particles the supply would be pretty constant over a long period of time. Another consideration is that radium is in the same column of the periodic chart as Ca and so biologically it might have similar chemistry to Ca and become trapped in bone tissue where it would be radioactive for a long time. Thus, this interlude regarding the fact that first-order decay is a useful model for nuclear processes has provided an opportunity to discuss some aspects of nuclear chemistry. Considering the crossover of physics and chemistry in the work of the Curies (Marie, Pierre, and Irene) and information in the popular domain regarding nuclear chemistry, we think this brief discussion is justified as an essential part of physical chemistry. 

In the case of many dynamical processes, the rate of change of a quantity is linearly proportional to the same quantity. Such processes include first-order decays in chemistry or radioactive decays in physics. The change of the quantity can then be described by an exponential function as shown in Eq. (6.1), and therefore the rate of change can be characterised by the time period needed to decrease the original quantity by e, where e is the basis of the natural logarithm having an approximate value of 2.71828. 

G(t) is often, but not necessarily, a simple function decaying exponentially to zero on either side of t = 0 with a time constant x, the correlation time for the motion under consideration. For example, random jumps of otherwise fixed interactions will give G(t) the same first-order decay as occurs with random radioactive fission of nuclei. Random angular diffusion will achieve the same. In the exponential case, for any frequency co,... 

The rate of decay, or activity, for a radioactive isotope follows first-order kinetics... 

From this expression, it is obvious that the rate is proportional to the concentration of A, and k is the proportionality constant, or rate constant, k has the units of (time) usually sec is a function of [A] to the first power, or, in the terminology of kinetics, v is first-order with respect to A. For an elementary reaction, the order for any reactant is given by its exponent in the rate equation. The number of molecules that must simultaneously interact is defined as the molecularity of the reaction. Thus, the simple elementary reaction of A P is a first-order reaction. Figure 14.4 portrays the course of a first-order reaction as a function of time. The rate of decay of a radioactive isotope, like or is a first-order reaction, as is an intramolecular rearrangement, such as A P. Both are unimolecular reactions (the molecularity equals 1). 

Perhaps the most important first-order reaction is that of radioactive decay, in which an unstable nucleus decomposes (Chapter 2). Letting X be the amount of a radioactive isotope present at time t,... 

As pointed out in Chapter 11, radioactive decay is a first-order process. This means that the following equations, discussed on pages 294-295, apply ... 

The half-life, f1/2, of a substance is the time needed for its concentration to fall to one-half its initial value. Knowing the half-lives of pollutants such as chlorofluoro-carbons allows us to assess their environmental impact. If their half-lives are short, they may not survive long enough to reach the stratosphere, where they can destroy ozone. Half-lives are also important in planning storage systems for radioactive materials, because the decay of radioactive nuclei is a first-order process. 

All radioactive decay processes follow first-order kinetics. The half-life of the radioactive isotope tritium (3H, or T) is 12.3 years. How much of a 25.0-mg sample of tritium would remain after 10.9 years ... 

As in a unimolecular chemical reaction, the rate law for nuclear decay is first order. That is, the relation between the rate of decay and the number N of radioactive nuclei present is given by the law of radioactive decay ... 

In this context, k is called the decay constant. The law tells us that the activity of a radioactive sample is proportional to the number of atoms in the sample. As we saw in Section 13.4, a first-order rate law implies an exponential decay. It follows that the number N of nuclei remaining after a time t is given by... 

State whether the following statements are true or false. If false, explain why. (a) The dose equivalent is lower than the actual dose of radiation because it takes into account the differential effects of different types of radiation, (b) Exposure to 1 X 1 ()x Bq of radiation would be much more hazardous than exposure to 10 Ci of radiation, (c) Spontaneous radioactive decay follows first-order kinetics, (d) Fissile nuclei can undergo fission when struck with slow neutrons, whereas fast neutrons are required to split fissionable nuclei. 

Since S/t has units of moles per volume per time and a has units of moles per volume, the rate constant for a first-order reaction has units of reciprocal time e.g., s. The best example of a truly first-order reaction is radioactive decay for example,... 

Radioactive decay provides splendid examples of first-order sequences of this type. The naturally occurring sequence beginning with and ending with ° Pb has 14 consecutive reactions that generate a or /I particles as by-products. The half-lives in Table 2.1—and the corresponding first-order rate constants, see Equation (1.27)—differ by 21 orders of magnitude. 

The important phenomenon of exponential decay is the prototype first-order reaction and provides an informative introduction to first-order kinetic principles. Consider an important example from nuclear physics the decay of the radioactive isotope of carbon, carbon-14 (or C). This form of carbon is unstable and decays over time to form nitrogen-14 ( N) plus an electron (e ) the reaction can be written as... 

C15-0058. Radioactive isotopes decay according to first-order kinetics. For one particular isotope, 1.00 nmol registers 1.2x10 decays in 1.00 min. (a) How many decays will occur in 1.00 min if 5.00 nmol of this isotope are present (b) What fraction of the isotope decays per minute in each case (c) Explain the relationship between your answers to (a) and (b). 

For any given radionuclide, the rate of decay is a first-order process that is constant, regardless of the radioactive atoms present and is characteristic for each radionuclide. The process of decay is a series of random events temperature, pressure, or chemical combinations do not effect the rate of decay. While it may not be possible to predict exactly which atom is going to undergo transformation at any given time, it is possible to predict, on average, the fraction of the radioactive atoms that will transform during any interval of time. 

---