Exponential law of decay

Note that what has been just derived is the exponential law of decay (see O Eq. (7.6)) justifying the notation A (reserved for the decay constant throughout this chapter) for the proportionality constant in Eq. (7.9). 

From the exponential law of decay applied to (specific) activities one obtains... 

The number e is a special number that allows one to describe the exponential laws of growth and decay. If f(t) > 0 is the amount of a substance present at time t, then the rate of change of / at time t is proportional to f(t) and can be written as / (0 = kf t) for some constant k and every time t in some interval. We integrate this differential equation as follows ... 

That means for large t the correlation function decays towards the static average of the quantity Z. For systems exhibiting simple dynamics the decay from (Z2) to (Z)2 can be described by a single exponential law of the form... 

The solution of Eq. (7.5) yields the formula generally referred to as the exponential law of radioactive decay ... 

Note that the 1/e rule expressed by Eq. (7.20) is not a general criterion for mean times it is a specific feature of the exponential distribution associated with the exponential law of simple decays. (See also the comment after Eq. (7.44).)... 

In the case of the lifetime distribution of radionuclides, excited states, etc., the expected value t is called the mean life, while the median Ti,2 is referred to as the half-life. The explanation for the name half-life is given in the next subsection on the exponential law of radioactive decay. It will be shown that the half-life is independent of the time elapsed, which is an obvious proof of the agelessness of radionuclides. Note that physicists often use the term lifetime not only in the sense it is used in this chapter, but also in the sense mean life. Fortunately, in the really important cases, i.e., when quantitative statements are made (e.g., the lifetime of the radionuclide is 10 s ), the ambiguity is removed and the reader can be sure that such a statement actually refers to the mean life. 

One can easily recognize the exponential law of radioactive decay in the above formula. The correspondence (N)(f) N means that the exponential law applies to the expected number of atoms rather than to the concrete numbers that are measured. The latter show a fluctuation about those expectations according to the standard deviation of the binomial distribution... 

The exponential laws of radioactive-series decay and growth of radionuclides were first formulated by Rutherford and Soddy in 1902, to explain their results (Rutherford and Soddy 1902,1903) on the thorium series of radionuclides. In 1910, Bateman (Bateman 1910) derived generalized mathematical expressions that were used to describe the decay and growth of the naturally occurring actinium, uranium, and thorium series until the discovery of nuclear fission and other new radioactive decay series were found in the 1940s. For the description of half-lives and decay constants, activities and number of radionuclides involved in the decay of two radionuclides, Friedlander et al. (1981) have given a representative overview (see also O Chap. 5 in Vol. 1). 

The conditions chosen make the reaction appear to be first-order overall, although the reaction is really not first-order overall, unlessjy and happen to be 2ero. If a simple exponential is actually observed over a reasonable extent (at least 90—95%) of decay the assumptions are considered vaUdated and is obtained with good precision. The pseudo-first-order rate constant is related to the k in the originally postulated rate law by... 

The law of radioactive decay implies that the number of radioactive nuclei decreases exponentially with time with a characteristic half-life. Radioactive isotopes are used to determine the ages of objects. 

In real systems, a distribution in the characteristic time may lead to a stretched exponential decay. In the thermally activated regime where the relaxation of the magnetization is due to the Orbach mechanism, the temperature dependence of the relaxation time may be described by an Arrhenius law of the form ... 

Eigenvalues of the operator Qr are real while the largest of them, Af, equals unity by definition. As a result, in the limit n-> oo all items in the sum (Eq. 38), excluding the first one, Q Q f = Xr/Xfh will vanish. In this case, chemical correlators will decay exponentially along the chain on the scale n 1/ In AAt values n < n the law of the decay of these correlators differs, however, from the exponential one even for binary copolymers. This obviously testifies to non-Markovian statistics of the sequence distribution in molecules (see expression Eq. 11). The closer is to unity, the greater are the values of n. The situation when n 1 corresponds to proteinlike copolymers. 

If now we consider a large number of molecules N0, the fraction still in the excited state after time t would be N/N0 — e kt where N is the number unchanged at time t. This exponential law is familiar to chemists and biological scientists as the first-order rate law and by analogy fluorescence decay is a first-order process—plots of fluorescence intensity after an excitation event are exponential and each type of molecule has its own characteristic average lifetime. 

In order to check the proposed model of et photobleaching, in refs. 40 and 62 the kinetics of et photobleaching in the presence of acceptor additives in vitreous water-alkaline and water-ethylene glycol matrices at 77 K was studied. Typical curves for photobleaching are presented in Fig. 31. The addition of acceptors is shown to result in an essential increase in the rate of e,r. photobleaching, the kinetics of e,r decay in the presence of additives being described by an exponential law in accordance with eqn. (28). 

Edmond Becquerel (1820-1891) was the nineteenth-century scientist who studied the phosphorescence phenomenon most intensely. Continuing Stokes s research, he determined the excitation and emission spectra of diverse phosphors, determined the influence of temperature and other parameters, and measured the time between excitation and emission of phosphorescence and the duration time of this same phenomenon. For this purpose he constructed in 1858 the first phosphoroscope, with which he was capable of measuring lifetimes as short as 10-4 s. It was known that lifetimes considerably varied from one compound to the other, and he demonstrated in this sense that the phosphorescence of Iceland spar stayed visible for some seconds after irradiation, while that of the potassium platinum cyanide ended after 3.10 4 s. In 1861 Becquerel established an exponential law for the decay of phosphorescence, and postulated two different types of decay kinetics, i.e., exponential and hyperbolic, attributing them to monomolecular or bimolecular decay mechanisms. Becquerel criticized the use of the term fluorescence, a term introduced by Stokes, instead of employing the term phosphorescence, already assigned for this use [17, 19, 20], His son, Henri Becquerel (1852-1908), is assigned a special position in history because of his accidental discovery of radioactivity in 1896, when studying the luminescence of some uranium salts [17]. 

In Table II are reported the values of v0, and rR obtained for different temperatures as well as the experimental and calculated wavenumber v of the peak of the stationary spectrum. Figure 2.21, where the solid lines represent calculated decays, shows that the experimental results can well be accounted for by the expressions (2.37) and (2.38). These results indicate that the relaxation of the electronic energy of the TICT state of DMABN due to interaction with the polar medium can well be described by a single exponential law not only for the n-butyl chloride solution but also for the solutions in alcohols. This relaxation process, leading to final states having an electronic energy markedly lower than that of the unrelaxed charge-transfer states, is responsible for the presence of an intramolecular potential barrier for the reverse reaction to the locally excited B state the barrier is made evident by the... 

The experimental decays iB(t) of the 350 nm band have been compared with curves calculated (solid lines in Fig. 5.1) by adjusting the parameters t" and r° in Eqs. (4.218) and (4.219) the spontaneous decay rate kr has been approximated by the value kB = kf + kB measured in a nonpolar solvent. It should be noted that with the photon-counting detection method the investigation of the fast initial nonexponential decay is hindered at low viscosity by poor resolution and only the exponential part of the decay is observable. At high viscosities (i7>100cp) the deviation from an exponential law is clearly visible. For the streak camera measurements the observations are opposite to those previously mentioned at high viscosities the semilogarithmic plot of iB(f) appears linear, whereas at low viscosities the decay shows nonexponential behavior. In Fig. 5.2 are represented the actual B decays calculated with the best fit values of the two relaxation times t° and r". Their variation with the temperature has also been examined Fig. 5.3 shows that they follow well those of -q/T and 17, respectively, as expected from the expressions (4.216) and (4.220) ... 

To convert an optical signal into a concentration prediction, a linear relationship between the raw signal and the concentration is not necessary. Beer s law for absorption spectroscopy, for instance, models transmitted light as a decaying exponential function of concentration. In the case of Raman spectroscopy of biofluids, however, the measured signal often obeys two convenient linearity conditions without any need for preprocessing. The first condition is that any measured spectrum S of a sample from a certain population (say, of blood samples from a hospital) is a linear superposition of a finite number of pure basis spectra Pi that characterize that population. One of these basis spectra is presumably the pure spectrum Pa of the chemical of interest, A. The second linearity assumption is that the amount of Pa present in the net spectrum S is linearly proportional to the concentration ca of that chemical. In formulaic terms, the assumptions take the mathematical form... 

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