Rates of Decay and Half-Life

Radionuclides have different stabilities and decay at different rates. Some decay nearly completely in a fraction of a second and others only after millions of years. The rates of all radioactive decays are independent of temperature and obey first-order kinetics. In Section 16-3, we saw that the rate of a first-order process is proportional to the concentration of only one substance. The rate law and the integrated rate equation for a first-order process (see Section 16-4) are 

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 variable k is the rate constant, which is different for each radionuclide. Each atom deca) independently of the others, so the stoichiometric coefficient 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 

Because Aq/A is a ratio, Aq and A can represent either molar concentrations of a reactant or masses of a reactant. The rate of radioactive disintegrations follows first-order kinetics, so it is proportional to the amount of A present we can write the integrated rate equation in terms of N, the number of disintegrations per unit time  

In nuclear chemistry, the decay rate is usually expressed in terms of the half-life, ti/2, of the process. This is the amount of time required for half of the original sample to react. For a first-order process, fi/2 is given by the equation 

Because all radioactive decay follows first-order kinetics, a similar plot for any radionuclide shows the same shape of exponential decay curve. About ten half-lives (288 years for Sr) must pass for any radionuclide to lose 99.9% of its radioactivity. 

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Nuclei with Atomic Number Greater Than 83 Detection of Radiation Rates of Decay and Half-Life Disintegration Series Uses of Radionuclides Artificial Transmutations of Elements Nuclear Fission Nuclear Fission Reactors Nuclear Fusion... 

I hulcr pseudo-first-order conditions, reactant A exhibits simple first-order decay. The rate of decay and half-life are determined by the magnitude of kapp, the apparent first-order rale constant for loss of A. Since /c, f B] and k2 [C] are in units of reciprocal time, the following half-lives can he defined ... 

Detection of Rad iation 22-10 Rates of Decay and Half-Life 22-11 Decay Series 22-12 Uses of Radionuclides 22-13 Artificial Transmutations of Elements 22-14 Nuclear Fission 22-15 Nuclear Fission Reactors 22-16 Nuclear Fusion... 

In addition to X, another term used for characterizing rate of decay is half-life (ty2), the time required for half of the initial number of atoms to decay. If we substitute t = ty2 and N = Nq 2 into equation 7.5 we obtain the following equation ... 

The amount of time required for one-half of the atoms of a radionuclide to transform is called its radioactive half-life. The rate of decay, and thus the half-life, for each radionuclide is unique. The half-life of U is very long, 4.5x10 years the half-lives of U and U are orders of magnitude lower,... 

The units of the decay constant are (time) , the same as for any first-order rate constant. Integrated rate law for nuclear decay and half-life... 

In this problem, we are asked to calculate the molar mass of a radioactive isotope. Grams of sample are given in the problem, so if we can find moles of sample we can calculate the molar mass. The rate constant can be calculated from the half-life. Then, from the rate of decay and the rate constant, the number of radioactive nuclei can be calculated. The number of radioactive nuclei can be converted to moles. 

Each radioisotope formed in an element subject to a nuclear particle bombardment is uniquely characterized by its half-life (rate of decay) and the types and energies of the radiations it emits as it decays. Therefore, a positive identification of the radioelement is possible. The amount of element in the bombarded or activated sample can be determined directly from a measurement of the radionuclide s radioactivity because the induced radioactivity is directly proportional to the number of atoms of the stable isotope in the sample and to the intensity (flux) of the nuclear particles interacting with the stable nuclei. 

The rate of decay of any radioactive isotope can be represented by its characteristic half-life, the period required for one-half of the radioactive material originally present to undergo radioactive decay. Short half-lives are the results of high rates of decay, and long half-lives are the results of low rates of decay. 

Decay products of the principal radionuclides used in tracer technology (see Table 1) are not themselves radioactive. Therefore, the primary decomposition events of isotopes in molecules labeled with only one radionuclide / molecule result in unlabeled impurities at a rate proportional to the half-life of the isotope. Eor and H, impurities arising from the decay process are in relatively small amounts. Eor the shorter half-life isotopes the relative amounts of these impurities caused by primary decomposition are larger, but usually not problematic because they are not radioactive and do not interfere with the application of the tracer compounds. Eor multilabeled tritiated compounds the rate of accumulation of labeled impurities owing to tritium decay can be significant. This increases with the number of radioactive atoms per molecule. 

Approximately 25—30% of a reactor s fuel is removed and replaced during plaimed refueling outages, which normally occur every 12 to 18 months. Spent fuel is highly radioactive because it contains by-products from nuclear fission created during reactor operation. A characteristic of these radioactive materials is that they gradually decay, losing their radioactive properties at a set rate. Each radioactive component has a different rate of decay known as its half-life, which is the time it takes for a material to lose half of its radioactivity. The radioactive components in spent nuclear fuel include cobalt-60 (5-yr half-Hfe), cesium-137 (30-yr half-Hfe), and plutonium-239 (24,400-yr half-Hfe). 

Measurement of specific activity. The half-life of a nuclide can be readily calculated if both the number of atoms and their rate of decay can be measured, i.e., if the activity A and the number of atoms of P can be measured, then X is known from A = XP. As instrumentation for both atom counting and decay counting has improved in recent decades, this approach has become the dominant method of assessing half-lives. Potential problems with this technique include the accurate and precise calibration of decay-counter efficiency and ensuring sufficient purity of the nuclide of interest. This technique provides the presently used half-lives for many nuclides, including those for the parents of the three decay chains, U, U (Jaffey et al. 1971), and Th. 

The SI unit of activity is the becquerel (Bq) 1 Bq = that quantity of radioactive material in which there is 1 transformation/second. Since activity is proportional to the number of atoms of the radioactive material, the quantity of any radioactive material is usually expressed in curies, regardless of its purity or concentration. The transformation of radioactive nuclei is a random process, and the number of transformations is directly proportional to the number of radioactive atoms present. For any pure radioactive substance, the rate of decay is usually described by its radiological half-life, TR, i.e., the time it... 

Radioactive decay is a stochastic process that occurs at random in a large number of atoms of an isotope (see Textbox 13). The exact time when any particular atom decayed or will decay can be neither established nor predicted. The average rate of decay of any radioactive isotope is, however, constant and predictable. It is usually expressed in terms of a half-life, the amount of time it takes for half of the atoms in a sample of a radioactive isotope to decay to a stable form. 

The best sealed-in minerals are zircons, zirconium silicate minerals which are formed when melted lava on the flanks of volcanoes solidifies. When the zircons crystallize out, they incorporate radioactive uranium (in particular 238U), which decays in several steps, leading Anally to the lead isotope 208Pb. The rate of decay is very low, as the half-life of uranium-238 is 4.5 x 109 years. Thus, the U-Pb-zircon method for age determination of Precambrian rock is very important. The fossils studied by Schopf were sandwiched between two lava layers (Schopf, 1999). The volcanic layers were dated to 3.458 0.0019 x 109 years and 3.471 0.005 x 109 years the age of the fossil layer (Apex chert) was thus determined to be about 3.465xlO9 years. 

The lifetime of a spin adduct is thus seen to be dependent on both its rate of formation and its decay, meaning that the half-life of a particular spin adduct under the conditions of its generation is a somewhat uncertain quantity, as frequently noticed in the literature. The intrinsic stability of a particular spin adduct should preferably be determined on isolated specimens, as for example reported for the trichloromethyl adduct of PBN, which was shown to survive almost unchanged in benzene solution after 90 days in the dark (Janzen et al., 1993). 

The time (symbolized by t) needed for a concentration of a molecular entity to decrease, in a first-order decay process to e of its initial value. In this case, the lifetime (sometimes called mean lifetime) is equal to the reciprocal of the sum of rate constants for all concurrent first-order decompositions. If the process is not first-order, the term apparent lifetime should be used, and the initial concentration of the molecular entity should be provided. The terms lifetime and half-life should not be confused. See Half-Life Fluorescence... 

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