First-order radioactive decay

Half-life (first order, radioactive decay)... 

Radioactive decay of unstable atomic nuclei is a first-order process. The half-life for the (first-order) radioactive decay of C is 5730 a (1 a is the SI unit 1 annum, for 1 year the nuclide emits p particles, high-energy electrons, with an energy of 0.16 MeV). 

Radioactive decay of nuclei is a first-order reaction decay rate (activity A) is therefore dependent on the concentration (content) of the radionuclide and is the product of this concentration (more precisely, the number of atoms of radionuclide N) and the decay constant X (in s" ) A =-(dN/dt) = X.N. The basic unit of activity, according to the System International (SI system) is the Bq (becquerel). One Bq (in s ) is defined as the activity of a quantity of radioactive material in which one nucleus decays per second. Previously, the frequently used unit was the Ci (curie) defined as 3.7 x 10 decays per second. For conversion, the following relationship can be used 1 Ci = 3.7 x lO Bq. Number of radionuclide atoms transformed in time t is f T=NQ.e", where Nq is the initial number of atoms of the radionuclide at the time t=0. During conversion, the number of radioactive atoms of the radioactive nuclide is continuously decreasing. Combining both equations we get the relation expressing the dependence of activity on time A = -(dN/dt) The... 

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. 

The science of kinetics deals with the mathematical description of the rate of the appearance or disappearance of a substance. One of the most common types of rate processes observed in nature is the first-order process in which the rate is dependent upon the concentration or amount of only one component. An example of such a process is radioactive decay in which the rate of decay (i.e., the number of radioactive decompositions per minute) is directly proportional to the amount of undecayed substance remaining. This may be written mathematically as follows ... 

Fig. 1 Plot of concentration remaining versus time for a first-order process (e.g., radioactive decay). 

The concentrations of the reactants change as the reaction progresses, and so the rate changes because it depends on the concentrations. An illustration of the eflfcct of time on the rate of a first-order process is the decay of a radioactive substance, considered in Sec. 22.3. 

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,... 

Equation (8.33) suggests the half-life is independent of the amount of material initially present, so radioactive decay follows the mathematics of first-order kinetics. 

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 ... 

When species i disappears by either radioactive decay or chemical reaction with first-order kinetics, the mass balance equation must be changed according to... 

Radioactive decay is a first-order process. See Chapter 20 for a discussion of half-lives related to nuclear reactions and other information on radioactivity. 

This radioactive decay process follows first-order kinetics. Substitute the value of k into the appropriate equation ... 

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. 

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