First-order reactions radioactive decay

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. 

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... 

In the previous chapter we discussed zero order kinetics however, many reactions in nature follow a different reaction scheme, namely a first order reaction. For example the elimination of a certain drug from the metabolism might follow a first order reaction. The decay of a radioactive substance, often used as a tracer in biological experiments is always described by a first order reaction. Another example for a first order kinetics is the growth of a bacterial culture, at least during certain phases of the growth. 

Radioactive decay processes involve the emission of a particle and/or photon (a gamma ray) from the nucleus of an atom. (See Chemical Connection 5.3.8.1 Radioactive Decay—A First-Order Reaction). Alpha decay is the ejection of an alpha particle from the nucleus of the atom (Equation 5.3.8.1) and produces a daughter nucleus that has two fewer protons and a decrease of four mass units. The velocity of the alpha particle accounts for the energy range of 4-6 MeV shown in Table 5.3.8.1. While alpha radiation can cause damage to tissues, it can only do so if the source is ingested or inhaled because the energy of alpha emitters is usually very weak and can readily be stopped by a sheet of paper. 

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,... 

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,... 

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... 

We use expression (26.12), substituting the disintegration rate for the number of atoms, since we recognize that in this first-order reaction the rate is directly proportional to the amount of reactant, that is, the number of atoms. (All radioactive decay processes follow... 

Radioactive decompositions are first order reactions. The specific rate in this discipline is called the decay constant,... 

A radioactive isotope may be unstable, but it is impossible to predict when a certain atom will decay. However, if we have a statistically large enough sample, some trends become obvious. The radioactive decay follows first-order kinetics (see Chapter 13 for a more in-depth discussion of first-order reactions). If we monitor the number of radioactive atoms in a sample, we observe that it takes a certain amount of time for half the sample to decay it takes the same amount of time for half the remaining sample to decay, and so on. The amount of time it takes for half the sample to decay is the half-life of the isotope and has the symbol t1/2. The table below shows the percentage of the radioactive isotope remaining versus half-life. 

A specific example of applications in the second category is the dating of rocks. Age determination is an inverse problem of radioactive decay, which is a first-order reaction (described later). Because radioactive decay follows a specific law relating concentration and time, and the decay rate is independent of temperature and pressure, the extent of decay is a measure of time passed since the radioactive element is entrapped in a crystal, hence its age. In addition to the age, the initial conditions (such as initial isotopic ratios) may also be inferred, which is another example of inverse problems. 

There are many examples of first-order reactions. The most often encountered in geochemistry is the radioactive decay of an unstable nuclide. For example, the rate law for the decay of " Sm (Reaction 1-2) can be written as... 

A simple way to characterize the rate of a reaction is the time it takes for the concentration to change from the initial value to halfway between the initial and final (equilibrium). This time is called the half-life of the reaction. The half-life is often denoted as ti/z. The longer the half-life, the slower the reaction. The half-life is best applied to a first-order reaction (especially radioactive decay), for which the half-life is independent of the initial concentration. For example, using the decay of " Sm as an example, [ Sm] = [ Sm]o exp( kt) (derived above). Now, by definition,... 

The mean reaction time during a reaction varies as the concentration varies if the reaction is not a first-order reaction. Expressions of mean reaction time of various types of reactions are listed in Table 1-2. In practice, half-lives are often used in treating radioactive decay reactions, and mean reaction times are often used in treating reversible chemical reactions. 

In Chapter 12, the concept of half-life was used in connection with the time it took for reactants to change into products during a chemical reaction. Radioactive decay follows first order kinetics (Chapter 12). First order kinetics means that the decay rate... 

Before going on to these alternative procedures, however, we should consider a special way by which true (not pseudo) first-order reactions are often considered. In these cases,/ = k. This consideration is especially applicable to radioactive decay processes. It is common practice to describe these true first-order reactions in terms of the time required for one-half of the material to decompose (this time is called the half-life, t ). In this special circumstance [A] = i[A]0 when t = t , and Equation 15-9 becomes... 

This is similar to a first-order reaction in chemical kinetics and follows the same law as radioactive decay. The rate constant kv defined in this manner is the natural radiative rate constant which also defines the natural radiative lifetime... 

Besides its qualitative description, radioactive decay has an important quantitative description. Radioactive decay can be described as a first-order reaction, that is, the number of decays is proportional to the number of decaying nuclei present. It is described by the integrated rate law... 

Radioactive decay is what chemists refer to as a first-order reaction that is, the rate of radioactive decay is proportional to the number of each type of radioactive nuclei present in a given sample. So, if we double the number of a given type of radioactive nuclei in a sample, we double the number of particles emitted by the sample per unit time.2 This relation may be expressed as follows ... 

The constancy of the half-life for a first-order reaction is illustrated in Figure 12.7. Each successive half-life is an equal period of time in which the reactant concentration decreases by a factor of 2. We ll see in Chapter 22 that half-lives are widely used in describing radioactive decay rates. 

The rate constant for the decay can be found from (20-2). Ninety percent decay corresponds to 10% or 0.10, survival. In dealing with radioactive decay, the total population of radioactive element is used in place of its concentration. (Remember that in first-order reactions, the rate constant and the half-life, as well, are independent of the concentration units.) So, in place of the concentration ratio [A]/[A]o, put the ratio of the numbers of atoms N/Nq, or moles, or masses, of radioactive element. The mass of radioactive element encountered in the laboratory is exceedingly small a typical sample can be measured only by its activity. Since its activity is proportional to its population, the observed ratio of activities, A/Aq, can be used in place of the number ratio, N/Nq. 

A great example of a first-order reaction is the radioactive decay of unstable nuclei, where we have, for example, atoms of A decaying to certain products (Figure 4-3). If the rate of this decay is simply proportional to how much A is present, we can write a simple differential equation. We will let N be the number of unstable nuclei present at any moment (Equation 4-1), and k is, in this case, a decay constant. 

There are very few known chemical reactions which fit precisely into this scheme of two consecutive first-order reactions, but all of the well-known schemes of radioactive decay which involve two or more consecutive steps are characterized precisely by this type of behavior. 

Radioactive disintegration or decay is a random process and may be expressed as a simple first order reaction. The mathematical expression describing the rate of such a reaction is... 

This is 50,000/5730 = 8.73 half-lives. For first-order reactions (such as radioactive decay), the initial amount decreases by j during each half-life, so the fraction remaining is = 2.36 X 10 1... 

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) ... 

The solution to the first-order reaction rate equation describing the radioactive decay of carbon-14. 

Integration yields n = /j exp(-/ +/), where n and are the total number of atoms present at t and r = 0. The half-time of radioactive decay is defined by r 2 = 0.693/A+, where n = 0.5, and n =. Other examples of first-order reactions (see Section 2.7) are the oxidation of organic matter and sulfate reduction, gypsum (CaS04 2H2O) dissolution, and the oxidation of pyrite and marcasite (FeS2). 

Radium-226 in Water. Ra-226 will be considered as an example of a chemical species which is being removed from the oceanic water column by a first-order chemical reaction, radioactive decay. Other possible mechanisms of removal, such as uptake by detrital silicates and organisms, will not be discussed. The supply of Ra-226, however, from organic matter decomposing in the water column will be considered. 

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