Pseudo-exponential decay rate

Since P in general can range from 0.5 to 0.8 for coil polymers, no single expression can be given for the integral in the equation. However, since it represents an average proportional to , the decay rate will be lower than for This observation is bom out in the inset of Figure 13.10, where rjjit) is plotted and the pseudo-exponential decay rate is found to be 73% of the pseudo-exponential decay rate. 

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... 

Figure 1. Competition kinetics for the Ru(NH2)62y reduction of Co([14 aneNk)-(0H,)0 Reactions at 25°C, pH 2, and n = 0.1(NaClO,). Individual pseudo-first-order rate constants were determined from the exponential (to four half-lives) decay of Co([14]aneN,)(OH2)022 absorbance at 360 nm. Reactions were performed by mixing a solution containing Ru(NH2)62 and Co([14]aneNh)(OHt) -(1 X I 3 M) with a solution saturated in 02(1.2 X 10 3 in an Aminco stopped-...

Pseudo-first-order rate constants for carbonylation of [MeIr(CO)2l3]" were obtained from the exponential decay of its high frequency y(CO) band. In PhCl, the reaction rate was found to be independent of CO pressure above a threshold of ca. 3.5 bar. Variable temperature kinetic data (80-122 °C) gave activation parameters AH 152 (+6) kj mol and AS 82 (+17) J mol K The acceleration on addition of methanol is dramatic (e. g. by an estimated factor of 10 at 33 °C for 1% MeOH) and the activation parameters (AH 33 ( 2) kJ mol" and AS -197 (+8) J mol" K at 25% MeOH) are very different. Added iodide salts cause substantial inhibition and the results are interpreted in terms of the mechanism shown in Scheme 3.6 where the alcohol aids dissociation of iodide from [MeIr(CO)2l3] . This enables coordination of CO to give the tricarbonyl, [MeIr(CO)3l2] which undergoes more facile methyl migration (see below). The behavior of the model reaction closely resembles the kinetics of the catalytic carbonylation system. Similar promotion by methanol has also been observed by HP IR for carbonylation of [MeIr(CO)2Cl3] [99]. In the same study it was reported that [MeIr(CO)2Cl3]" reductively eliminates MeCl ca. 30 times slower than elimination of Mel from [MeIr(CO)2l3] (at 93-132 °C in PhCl). 

That is, A decays exponentially with time determined by (kl7[B]0), as if it were a first-order reaction. Thus under these so-called pseudo-first-order conditions, a plot of ln[A] against time for a given value of [B]0 should be linear with a slope equal to ( — I7[B]0). These plots are carried out for a series of concentrations of [B](l and the values of the corresponding decays determined. Finally, the absolute rate constant of interest, kl7, is the slope of a plot of the absolute values of these decay rates against the corresponding values of [B] . Some examples are discussed below. 

Fig. 2.4. Computed concentration.histories for autocatalytic model with rate constants given exactly as in Table 2.1 (a) exponential decay of precursor (b) intermediate concentrations a(t) and 6(r), showing initial pseudo-stationary-state behaviour but subsequent development of an oscillatory period of finite duration, 1752 s < t < 3940 s.

All experiments were carried out under pseudo-first order conditions with DMS in large excess over OH. Exponential OH decays were observed under all experimental conditions investigated. Plots of k (the pseudo-first order OH decay rate) versus DMS concentration were linear. Values for k0bs were obtained from linear least squares determinations of the slopes of kf versus [DMS] plots. Measured values for kQ s as a function of temperature, pressure, and O2 concentration are summarized in Table I. 

A spectroscopic investigation of the formation of THF-Cu"Cl2 complexes has been described, and irradiation of [Cu(Dto)2] (Dto = dithiooxalate) has been found to induce an intramolecular Dto Cu two-electron transfer with cleavage of the C-C bond in the Dto ligand and the formation of SCO. The kinetics of photo-oxidation of pyrene by Cu" in SDS micelles have been measured, but oxidants such as Eu" and Hg do not produce pyrene cations. A non-exponential decay of fluorescence is observed, and this is interpreted in terms of a model due to Tachiya which restricts the numbers of quenchers in a micelle. Transient Cu"-alkyl species are formed on flash photolysis of Cu -bis(amino-acid) complexes such as those of serine and valine, and pseudo first order rate constants for the decay of the transients have been obtained. 

The direct coupling of P430 with MV was further supported by the acceleration ofthe decay rate ofthe former that is observed with increasing MV concentration. The kinetic plots showed that P430 decay was exponential at all MV concentrations examined. The decay ty, vs. MV concentration fitted a pseudo-first-order reaction, giving an estimated rate constant of 9.6-10 M - at 22 °C. ... 

Figure 38B represents a similar stimulation in which k Jk4 = 1. In this case, the contribution of HIn(c) to the measured dynamics, noted as the biexponential decay, is appreciable much earlier than in Figure 38A. Figure 38C (kin/k4 =10) depicts the dynamics when the two populations HIn(s) and HIn(c) are in fast equilibrium, and HIn decays as a single exponential function, characterized by apparent pseudo first-order rate constants. 

Bc2 produced by flash photolysis of seawater decays by parallel first- and second-order reactions. The environmentally important exponential decay is a pseudo first-order reaction of Br2 with the carbonate/ bicarbonate system in seawater. A chemical speciation model for the free ions and ion-pairs in seawater and in solutions at seawater ionic strength allowed us to measure the dependence of the pseudo first-order rate term, a, on individual C02-containing species. A predictive equation based on reaction of Br2 with free C03 and the hgC03°, NaC03 and CaC03 ion pairs accounts for the mean seawater a at pH 8.1 within experimental uncertainty. The reaction productfs) are unknown. 

In excess of dissolved O2, the concentration of [Fe(CN)5NO] decays exponentially with a pseudo-first-order rate constant feobs that correlates... 

As a result, the signals obtained in this sort of experiment are almost always analyzed as if they are a first-order exponential decay. In other words, the logarithm of the signal s intensity is plotted versus time the slope of this plot is the measured parameter. This slope, which has the units of reciprocal time, is a pseudo-first-order rate constant. These slopes are rarely reported instead, their reciprocals, called relaxation times x, are the data given. 

These time-dependent or pseudo-steady-state results are shown in Fig. 1.9. The numerical values correspond to the data in Table 1.1, but the curves have the same qualitative form for any values of the rate constants, etc. b(t) falls exponentially throughout the reaction whilst d(t) builds up to a maximum before decaying. 

Our a - Ki (in Reference 1 ) our B - K2 (in Reference ). a is an exponential rate constant characterizing true or pseudo first-order processes and B - K(2)/c Is a second-order rate constant divided by molar extinction coefficient characterizing radical-radical decay processes important at high intensities, but not under low-intensity illumination conditions. Aq initial optical density. 

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