Second-order half-time

Particle Concentration. The effect of particle concentration was essentially that which can be expected from a second order reaction (time for half-agglomeration was inversely proportional to the number of particles originally present). However, at TS > 35% and higher viscosities and TS < 15% of base latex, the reaction was faster or slower, respectively, than expected. 

It was pointed out that a bimolecular reaction can be accelerated by a catalyst just from a concentration effect. As an illustrative calculation, assume that A and B react in the gas phase with 1 1 stoichiometry and according to a bimolecular rate law, with the second-order rate constant k equal to 10 1 mol" see" at 0°C. Now, assuming that an equimolar mixture of the gases is condensed to a liquid film on a catalyst surface and the rate constant in the condensed liquid solution is taken to be the same as for the gas phase reaction, calculate the ratio of half times for reaction in the gas phase and on the catalyst surface at 0°C. Assume further that the density of the liquid phase is 1000 times that of the gas phase. 

Figure 2-1 is a plot of Eq. (2-10) from n = 0 to = 4. Note that equal time irrcrements result in equal fractional decreases in reactant concentration thus in the first half-life decreases from 1.0 to 0.50 in the second half-life it decreases from 0.50 to 0.25 in the third half-life, from 0.25 to 0.125 and so on. This behavior is implicit in the earlier observation that a first-order half-life is independent of concentration. 

Half life The time required to convert half of the original amount of reactant to product, 294 first-order, 294 second-order, 296... 

Second-order kinetics, (a) Derive expressions for the half-time and lifetime of A if the rate law for its disappearance is v = fc[A]2 (b) calculate t]/i and t for the data presented in Section 2.2 (c) calculate the second half-life, t /i(2), i.e., the time elapsed between 50 percent and 75 percent completion, for the same reaction (d) compare fj/2(l) and fi/>(2), and contrast this result with that from first-order kinetics. 

The first of these reactions was carried out in 1,4-cyclohexadiene over a temperature range of 39 to 100 °C. It is fairly slow the half-times were 20 h and 3.4 min at the extremes. Reaction (7-11) is quite fast the second-order rate constant, kn, was evaluated over the range 6.4 to 47.5 °C. Values of feio and fen are presented in Table 7-1. The temperature profiles are depicted in Fig. 7-1 from their intercepts and slopes the activation parameters can be obtained. A nonlinear least-squares fit to Eq. (7-1) or... 

A second-order reaction has a long tail of low concentration at long reaction times. The half-life of a second-order reaction is inversely proportional to the concentration of the reactant. 

The half-life for the second-order reaction of a substance A is 50.5 s when [A] = 0.84 mol-L. Calculate the time needed for the concentration of A to decrease to (a) one-sixteenth (b) one-fourth (c) one-fifth of its original value. 

The reduction of Co(lll) by Fe(II) in perchloric acid solution proceeds at a rate which is just accessible to conventional spectrophotometric measurements. At 2 °C in 1 M acid with [Co(IlI)] = [Fe(II)] 5 x 10 M the half-life is of the order of 4 sec. Kinetic data were obtained by sampling the reactant solution for unreacted Fe(Il) at various times. To achieve this, aliquots of the reaction mixture were run into a quenching solution made up of ammoniacal 2,2 -bipyridine, and the absorbance of the Fe(bipy)3 complex measured at 522 m/i. Absorbancies of Fe(III) and Co(lll) hydroxides and Co(bipy)3 are negligible at this wavelength. With the reactant concentrations equal, plots of l/[Fe(Il)] versus time are accurately linear (over a sixty-fold range of concentrations), showing the reaction to be second order, viz. 

The half-life of a reactant is the time required for half of that reactant to be converted into products. For a first order reaction, the half-life is independent of concentration so that the same time is required to consume half of any starting amount or concentration of the reactant. On the other hand, the half-life of a second-order reaction does depend on the starting amount of the reactant. 

Figure 1 shows typical first-order plots (log m versus time) for the reaction at 0 °C. The plots are curved towards the time-axis, indicating an acceleration. This acceleration only becomes evident at, or sometime after, the first half-life. The first-order rate-constants k1 in Table 1 correspond to the slopes of the rectilinear portion of the first-order plots. The kv is directly proportional to the initial concentration of the acid, as shown in Figure 2. The slope of this plot gives the second-order rate constant k2 in the rate expression ...

It is not uncommon to find the persistence of a spin adduct quantified in terms of half-life . This is a dangerous practice unless the experimental conditions are precisely defined, or it is known that the nitroxide decays by a unimolecular process. Decay may depend on reaction with a reducing agent present in the system, in which case the concentration of this species will influence the half-life. More commonly, decay will be second order (p. 5), in which case the time for disappearance of 50% of the spin adduct will show a profound dependence on its absolute concentration. The possibility of bimolecular association of nitroxides has been recognized for many years, but only very recently has it been suggested that this may be a complication under experimental conditions employed for spin trapping. Whilst the problem, which was encountered with the important [DMPO-HO ] system (Bullock et al., 1980), seems unlikely to be widespread, it is one which should always be borne in mind in quantitative studies. 

Very rarely are measurements themselves of much use or of great interest. The statement "the absorption of the solution increased from 0.6 to 0.9 in ten minutes", is of much less use than the statement, "the reaction has a half-life of 900 sec". The goal of model-based analysis methods presented in this chapter is to facilitate the above translation from original data to useful chemical information. The result of a model-based analysis is a set of values for the parameters that quantitatively describe the measurement, ideally within the limits of experimental noise. The most important prerequisite is the model, the physical-chemical, or other, description of the process under investigation. An example helps clarify the statement. The measurement is a series of absorption spectra of a reaction solution the spectra are recorded as a function of time. The model is a second order reaction A+B->C. The parameter of interest is the rate constant of the reaction. 

Al MAS NMR has been demonstrated to be an invaluable tool for the zeoHte sdentist It provides a simple and direct way to quantify the proportions of A1 in four [Al(4)j, five [Al(5)j and six [Al(6)j coordinations. Quantitative determination of these species is an important issue in catalysis, and major effort is devoted on this topic. As mentioned already, for A1 only the central transition (-i-half to —half selective exdtation ) is detected. The central transition is unaffected by first order quadmpolar interaction, but the presence of second order effects causes broadening and complicates the quantitation of the A1 species. Usually hydrated samples and short radiofrequency pulses are employed for quantitative determination of framework and extra framework aluminum species. It is uncertain whether hydration changes the coordination of A1 species. Certain extra framework A1 can have very large quadmpolar interactions resulting in very broad lines ( NMR invisible ) [155, 202]. Unlike Si NMR, Al has a short relaxation time due to its quadmpolar nature, and the Al NMR spectrum with good signal to noise can be obtained in a relatively short time. 

The thermal decomposition of FCIO 2 in Monel was studied by Macheteau and Gillardeau (183). Decomposition to CIF and O2 was observed at 100°C (2.5% in 144 hr) and 200°C (10% in 235 hr), but a temperatme >250°C was required for rate measurements. It was found that the decomposition is of first order and monomolecular at temperatures up to 285°C. At 300°C the reaction becomes second-order. The calculated rate constants and half-life times are summarized in Table XVI. The... 

V = V max [S]// m- A reaction of higher order is called pseudo-first-order if all but one of the reactants are high in concentration and do not change appreciably in concentration over the time course of the reaction. In such cases, these concentrations can be treated as constants. See Order of Reaction Half-Life Second-Order Reaction Zero-Order Reaction Molecularity Michaelis-Menten Equation Chemical Kinetics... 

The concentration evolution curves of Figures 2-la and 2-lb may be used to estimate the half-life or mean reaction time. When Figures 2-la and 2-lb are compared, the mean reaction time is found to differ by four orders of magnitude Hence, for second-order reactions, the timescale to reach equilibrium in general depends on the initial conditions. This is in contrast to the case of first-order reactions, in which the timescale to reach equilibrium is independent of the initial conditions. 

Radical decay kinetics have been shown to be 3/2 order, falling to first order, and also second order, falling with time deviations are apparently due to side reactions of 118. Radical half-lives are strongly influenced by the nature of the aryl substituents, being particularly short for ortho-substituted Ar because of inhibited delocalization. The corresponding compounds 39 have, accordingly, enhanced thermal stability, a factor useful in some commercial thermo- and photographic processes. 

Assuming first order reaction kinetics, the sorption rate that was determined for adsorption and desorption was 0.187 sec . A reaction rate of 0.187 sec implies a half time of reaction of 3.7 seconds. 

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