Rates of decay

For some experiments, the solar neutrino flux and the rate of decay of the proton being extreme examples, tire count rate is so small that observation times of months or even years are required to yield rates of sufficiently small relative uncertainty to be significant. For high count rate experiments, the limitation is the speed with which the electronics can process and record the incoming infomiation. 

The friction coefficient determines the strength of the viscous drag felt by atoms as they move through the medium its magnitude is related to the diffusion coefficient, D, through the relation Y= kgT/mD. Because the value of y is related to the rate of decay of velocity correlations in the medium, its numerical value determines the relative importance of the systematic dynamic and stochastic elements of the Langevin equation. At low values of the friction coefficient, the dynamical aspects dominate and Newtonian mechanics is recovered as y —> 0. At high values of y, the random collisions dominate and the motion is diffusion-like. 

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

The rate of decay of " k is 5.543 x 10" ° yr. For any time difference ((), equation (1) relates the amount of existing now to that existing some time ago (in the present case, when the rocks were formed). 

The various chemical and physical processes that play a role ia the deterioration of art objects are not restricted to the present, even though the contemporary environment has contributed significantly to the rate of decay. Revered masterpieces have lost splendor throughout the ages. Indeed, from textual evidence, it is known how artists ia the Renaissance restored works of art from Classical times. These restorers of past centuries attempted to return the object to its original appearance. The fallacy of that idea Hes ia the fact that they could not know the exact original appearance of the work, ie, immediately after its creation therefore, they restored the object according to their subjective opinions. 

Radioactivity is equal to the rate of decay of a given radioisotope. This quantity is proportional to the number of radioactive atoms present, so that for a single isotope,... 

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). 

A promising technique is cavity ringdown laser absorption spectroscopy (307), in which the rate of decay of laser pulses injected into an optical cavity containing the sample is measured. Absorption sensitivities of 5 x 10 have been measured on a ]ls time scale. AppHcations from the uv to the ir... 

A typical shock-compression wave-profile measurement consists of particle velocity as a function of time at some material point within or on the surface of the sample. These measurements are commonly made by means of laser interferometry as discussed in Chapter 3 of this book. A typical wave profile as a function of position in the sample is shown in Fig. 7.2. Each portion of the wave profile contains information about the microstructure in the form of the product of and v. The decaying elastic wave has been an important source of indirect information on micromechanics of shock-induced plastic deformation. Taylor [9] used measurements of the decaying elastic precursor to determine parameters for polycrystalline Armco iron. He showed that the rate of decay of the elastic precursor in Fig. 7.2 is given by (Appendix)... 

Equation (A. 15) was derived with the rate of decay dependent upon F and li. We could just as well have derived expressions for D,a in terms of F and p, or F and a, or other combinations. Two of these expressions are listed... 

One example of this occurs with stress relaxation. If a polymer is deformed to a fixed strain at constant temperature the force required to maintain that strain will decay with time owing to viscous slippage of the molecules. One measure of this rate of decay or stress relaxation is the relaxation time 0, i.e. the time taken for the material to relax to 1/e of its stress on initial application of strain. 

The intermediate diphenylhydroxymethyl radical has been detected after generation by flash photolysis. Photolysis of benzophenone in benzene solution containing potential hydrogen donors results in the formation of two intermediates that are detectable, and their rates of decay have been measured. One intermediate is the PhjCOH radical. It disappears by combination with another radical in a second-order process. A much shorter-lived species disappears with first-order kinetics in the presence of excess amounts of various hydrogen donors. The pseudo-first-order rate constants vary with the structure of the donor with 2,2-diphenylethanol, for example, k = 2 x 10 s . The rate is much less with poorer hydrogen-atom donors. The rapidly reacting intermediate is the triplet excited state of benzophenone. 

D6 is MLO with the highest rate of decay heat among the three mid-loop POSs. 

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). 

The rate of decay by spontaneou.s fission increase.s with atomic number and is an important additional cause of instability in the later actinide.s (rrani-Np). 

Half-lives can be interpreted in terms of the level of radiation of the corresponding isotopes. Uranium has a very long half-life (4.5 X 109 yr), so it gives off radiation very slowly. At the opposite extreme is fermium-258, which decays with a half-life of 3.8 X 10-4 s. You would expect the rate of decay to be quite high. Within a second virtually all the radiation from fermium-258 is gone. Species such as this produce very high radiation during their brief existence. 

Because of the way in which rate of decay is measured (Figure 19.3), it is often described by the activity (A) of the sample, which expresses the number of atoms decaying in unit time. The first equation above can be written... 

Radioactivity The ability possessed by some natural and synthetic isotopes to undergo nuclear transformation to other isotopes, 513 applications, 516-518 biological effects, 528-529 bombardment reactions, 514-516 diagnostic uses, 516t discovery of, 517 modes of decay, 513-514 nuclear stability and, 29-30 rate of decay, 518-520,531q Radium, 521-522 Radon, 528 Ramsay, William, 190 Random polymer 613-614 Randomness factor, 452-453 Raoult s law A relation between the vapor pressure (P) of a component of a solution and that of the pure component (P°) at the same temperature P — XP°, where X is the mole fraction, 268... 

The sequences of radioactive decays that lead to lead are well-known and the rates of decay have been carefully measured. We shall consider the sequence based upon the relatively slow decomposition of the most abundant uranium isotope, mass 238 (natural abundance, 99%) ... 

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 ... 

Decay of Secondary Ion Concentrations. The fate of the secondary ions must now be considered. Miller (28) has observed that for C2H4/02 and C2H2/02 flames at 2 and 4 torr the rates of decay of all secondary ions, including C3H3+, are approximately the same (see, for example, Figure 1). The slow decay of the primary ion CHO+, paralleling that of H30 +, has been attributed (11) to establishment of equilibrium for Reaction 14. 

Examine the autocorrelation function. The high autocorrelations will indicate the order of the autoregressive part if any. The rate of decay of the autocorrelations will indicate a need for differencing. 

C by cosmic rays equals its rate of decay by 3 emission, so the percentage of C in the atmosphere... 

The complex has been separated by ion exchange and characterised by direct analysis . The complex has a distinctive absorption spectrum (Fig. 11), quite unlike that of Np(V) and Cr(III). The rate coefficient for the first-order decomposition of the complex is 2.32 x 10 sec at 25 °C in 1.0 M HCIO. Sullivan has obtained a value for the equilibrium constant of the complex, K = [Np(V) Cr(III)]/[Np(V)][Cr(III)], of 2.62 + 0.48 at 25 °C by spectrophotometric experiments. The associated thermodynamic functions are AH = —3.3 kcal. mole" and AS = —9.0 cal.deg . mole . The rates of decay and aquation of the complex, measured at 992 m/t, were investigated in detail. The same complex is formed when Np(VI) is reduced by Cr(II), and it is concluded that the latter reaction proceeds through both inner- and outer-sphere paths. It is noteworthy that the substitution-inert Rh(lII), like Cr(III), forms a complex with Np(V) °. This bright-yellow Np(V) Rh(III) dimer has been separated by ion-exchange... 

Another intermediate of the photolysis of TiO was observed in experiments with platinized particles (in the absence of polyvinyl alcohol). The spectrum shown in Fig. 22 is prraent immediately after the laser flash. The signal decays as shown by the inset in the figure. The rate of decay is not influenced by oxygen but is increased by oxidizable compounds such as Br ions in the solution. The broad absorption band in Fig. 22 with a maximum at 430 nm was attributed to trapped positive holes. Chemically, a trapped hole is an 0 radical anion. In homogeneous aqueous solution, 0 ... 

FIG. 12 Simulation of fluorescent decays for dye species located in the aqueous phase following laser pulses in TIR from the water-DCE interface according to Eq. (38). A fast rate constant of excited state decay (10 s ) was assumed in (a). The results showed no difference between infinitely fast or slow kinetics of quenching. On the other hand, a much slower rate of decay can be observed for other sensitizers like Eu and porphyrin species. Under these conditions, heterogeneous quenching associated with the species Q can be readily observed as depicted in (b). (Reprinted with permission from Ref 127. Copyright 1997 American Chemical Society.)... 

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