Reaction first-order behavior

Study of reversible reactions close to equilibrium. This possibility was discussed in eonnection with Scheme II and is further treated in Chapter 4. It turns out that if the displacement from equilibrium is small, the kinetics approach first-order behavior. 

In this study the reactions were followed spectrophotometrically by monitoring the loss of AX or with time. The initial concentration of hydroxy compound was at least 50 times that of the acetylating agent, and pseudo-first-order behavior was observed. This system will be discussed later. 

Remember that reaction order always must be determined by analyzing experiments. The decomposition of NO2 does not follow first-order kinetics, so a mechanism that predicts first-order behavior cannot be correct, no matter how reasonable that mechanism might appear on paper. 

Equation 1.6 is built upon the assumption that each of the removal processes that C undergoes fitllows first-order behavior. If these are chemical reactions, a first-order rate law can be written for each (individual) process in which... 

Pseudo-First-Order Behavior. In some cases, one reactant may be present in great excess over the other, and the last two equations can be simplified to their pseudo-first-order form. In Fig. 5, we illustrate the time course of a bimolecular reaction between equivalent initial concentrations of A and B. 

On the other hand, when a large excess of reactant B is used then its concentration does not change appreciably (Cg = Cgo) and the reaction approaches first-order behavior with respect to the limiting component A, or... 

It is common for many second-order reactions to exhibit pseudo-first-order behavior under conditions of nLFP. This is due to the fact that, while reactive intermediates are present in micromolar concentrations, (typically 10-50 pA/), the molecules with which they react are present in concentrations several orders of magnitude larger. As a result, the concentration of these reagents remains essentially constant during the decay of the transient species. An example is shown in Figure 18.4, where triplet benzophenone is quenched by melatonin. ... 

It is important to understand why this apparent first-order behavior is found for the course of an exchange reaction with time whatever the true kinetics of the reaction. A failure to understand this feature of exchange reactions has sometimes led to unjustifiable statements about the ratedetermining step in such reactions. It is convenient to discuss a specific example—the exchange of ethane with deuterium. Suppose that the only adsorbed species taking part in the reaction are (a) physically adsorbed... 

There are quite a few situations in which rates of transformation reactions of organic compounds are accelerated by reactive species that do not appear in the overall reaction equation. Such species, generally referred to as catalysts, are continuously regenerated that is, they are not consumed during the reaction. Examples of catalysts that we will discuss in the following chapters include reactive surface sites (Chapter 13), electron transfer mediators (Chapter 14), and, particularly enzymes, in the case of microbial transformations (Chapter 17). Consequently, in these cases the reaction cannot be characterized by a simple reaction order, that is, by a simple power law as used for the reactions discussed so far. Often in such situations, reaction kinetics are found to exhibit a gradual transition from first-order behavior at low compound concentration (the compound sees a constant steady-state concentration of the catalyst) to zero-order (i.e., constant term) behavior at high compound concentration (all reactive species are saturated ) ... 

The graphs shown in Fig. 12 are selectivity plots developed by assuming first-order behavior for all reactions. It should be mentioned that satisfactory estimation of the various rate constants requires that comparable fits must also be obtained for kinetic plots in which the product composition is plotted against reaction time using all of the rate constants obtained from the selectivity plot. The data discussed in this section satisfy this criterion, as illustrated in Fig. 13. The excellent agreement between the calculated curves and the data clearly demonstrates that all reactions exhibit pseudo-first-order kinetics under a given set of conditions. 

The application of this rate law to the simulation of electrochemical behavior requires two dimensionless input parameters ktf and KC. When these are supplied, three-dimensional chronoamperometric or chronocoulometric working surfaces [34] are generated. These working surfaces both indicate first-order behavior when KC is large and second-order behavior when KC is small. Intermediate values of KC produce the variable reaction orders between one and two that are observed experimentally when the bulk olefin concentration is varied. Appropriate curve fitting of the experimental i(t,C) data to the simulation results in the evaluation of k and K details appear in the referenced work. 

A very fast second-order disappearance of hydrated electrons with the production of molecular hydrogen has been observed in water using pulse radiolysis (9). In our system this reaction would be expected to cause large deviations from first-order behavior at high water concentrations. The absence of such deviation shows that this reaction depends strongly upon the solvent structure and not merely upon the concentration of water molecules. 

The rate of product formation will be directly proportional to [A] but inversely proportional to [Br ]. By using an excess of Br so that its initial concentration [Br ]0 does not change appreciably over the course of the reaction, pseudo-first-order behavior can be achieved with /cobs = k2Keq/[Br ]0. 

One way to test for first-order behavior is to carry out the rate determination at another initial concentration of reactant A, such as double or half the original, but preferably 10-fold or smaller (Bunnett, 1986). If the reaction is first-order, the slope according to Eq. (2.17) or (2.18) should be unchanged. It is also necessary to show that reaction rate is not affected by a species whose concentrations do not change considerably during a reaction run these may be substances not consumed in the reaction (i.e., catalysts) or present in large excess (Bunnett, 1986). 

The kinetics of the esterification of 1-octanol with hexanoic acid on zeolite BE A was studied by Nijhuis et al. [29], For the acid, a first-order behavior was found, whereas the alcohol showed a negative reaction order of -1. From the data, an Eley-Rideal mechanism was concluded. The acid adsorbs onto the surface of the catalyst and reacts with an alcohol. The adsorption of water, alcohol, ester or ether inhibits the reaction. Hoek [30] found that the adsorption constant of water is more than one order of magnitude higher than those of the other compounds. The rate law given by Nijhuis et al. [29] also includes the equilibrium limitation ... 

The rate of reaction is approximately linear with cyclohexene partial pressure, however there are systematic deviations at higher partial pressures. The first order behavior with respect to hydrogen and the slight reactant inhibition of cyclohexene suggest the following kinetic correlation ... 

Amatore and Jutand have shown that reactions of aryl bromides and iodides are accelerated by added halide, and have observed direct coordination of the halide to form an anionic Pd(0) species that undergoes oxidative addition [294, 295]. Thus, the one mechanism that is consistent with the observed first-order behavior in alkoxide and that is consistent with Amatore and Jutand s studies involves initial displacement of phosphine by tert-but-oxide to form an anionic Pd(0) species Pd[P(tBu)3](OtBu. Oxidative addition of the aryl chloride would then occur to this anionic species. These concurrent mechanisms are shown in Scheme 10. 

Many of the kinetic studies have been done at low pressures and temperatures because accurate kinetic measurements are more difficult to make at high pressures. Those measurements that have been made at high pressures ( —500 p.s.i.g.) confirm the approximate first-order relationships obtained at atmospheric pressure and below. Therefore, we believe that the kinetic information obtained at low pressures has general significance. In spite of the indicated first-order behavior, the reaction is undoubtedly complex, and the relative rates of the individual steps may change drastically with pressure. Yet there is no reason to believe that totally different mechanisms operate in the various pressure ranges. Reaction rates at atmospheric pressure and below have been determined by three different techniques ... 

A few reactor models have recently been proposed (30-31) for prediction of integral trickle-bed reactor performance when the gaseous reactant is limiting. Common features or assumptions include i) gas-to-liquid and liquid-to-solid external mass transfer resistances are present, ii) internal particle diffusion resistance is present, iii) catalyst particles are completely externally and internally wetted, iv) gas solubility can be described by Henry s law, v) isothermal operation, vi) the axial-dispersion model can be used to describe deviations from plug-flow, and vii) the intrinsic reaction kinetics exhibit first-order behavior. A few others have used similar assumptions except were developed for nonlinear kinetics (27—28). Only in a couple of instances (7,13, 29) was incomplete external catalyst wetting accounted for. 

Accordingly, a plot of In versus time gives a straight line with slope —k, as in Figure 5.6. It makes no difference whether is calculated with i = A, P, Q, or any other product, the straight lines obtained are identical. If such behavior is found experimentally, it provides a very strong indication for a network of the type of 5.15 with all reactions first order in A. 

Rice and Herzfeld [27], at a time when still little was known about free radicals and chain reactions, had tried to account for the observed first-order behavior by postulating a "mixed" termination C2H5- + H- — C2H6. However, since C2H5-outnumbers H by several orders of magnitude under typical reaction conditions, this assumption proved untenable [30]. Thereupon Kuchler and Theile [35] suggested that initiation is bimolecular in ethane provided termination occurs without a... 

The overall rate of ethane consumption then is of order one-and-a-half in ethane if the rate of C2H5 — C2H4 + H is controlled by activating collision, and of order one half if controlled by decay of the activated radical. According to Quinn, first-order behavior was observed because the reaction was studied in the "fall-off" range of pressure, that is, where rate control of C2H5 decay shifts from one step to the other. Indeed, at very low pressures the initial rate varies with (pCc) 5 [31]. 

Quinn studied initial rates—i.e., in the absence of reaction products—in a limited pressure range of 60 to 230 Torr. His hypothesis can explain the dependence on initial pressure he observed, but not what is normally defined as first-order behavior, namely, a rate proportional to the reactant concentration or partial pressure in the course of the reaction in the presence of products formed. This is because ethene (and, for that matter, almost any other molecule with the possible exception of H2) can also serve as activating collision partner. Indeed, addition of inerts has been found to boost the rate [35]. Since one mole of ethane produces approximately one mole of ethene, the concentration of potential collision partners is pc=c + pcc = pc°c and remains essentially unchanged, so that there is no effect on the form of the rate equation and the reaction order (for simplicity, this assumes ethene to be as effective a collision partner as is ethane, and H2 to be ineffective.) Nevertheless, textbooks to this day accept Quinn s explanation, if not Rice and Herzfeld s. 

First-order behavior (as normally defined) at any pressure can be rationalized if the first propagation step is made reversible. This is not unreasonable because the step in the forward direction is strongly endothermic (+159 kJ mol 1), so its reverse should make itself felt long before the reverse of the overall reaction becomes noticeable. The rate of this reverse step is proportional to a product, so that the retardation it exerts increases with progressing conversion. This translates into a higher apparent reaction order. Quantitatively, the mechanism 9.38 with termination 9.39 and reversible first step gives a rate equation of the form... 

Rates of thermal cracking are first-order in good approximation for propane, butane and still higher hydrocarbons [21], This is remarkable because chain mechanisms with initiation by break-up of a reactant normally result in reaction orders of one half or one-and-a-half, depending on which radical is consumed by termination. First-order behavior can result from "mixed" termination, which, however, can in most cases be ruled out as dominant mechanism (see Section 9.3). A more probable explanation is a combination of effects that key hydrocarbon radicals participate in several steps of different molecularities, that some steps are reversible, and that some unimolecular ones require collision partners. 

Another possible correlation between coal structure and pyrolysis behavior is indicated by the temperature dependence of the evolution of pyrolytic water being strikingly different for the two coals. Figure 5 shows pyrolytic water evolution data for experiments in which the sample was heated at 1000°C/sec to the peak temperature indicated on the abscissa and then immediately allowed to cool at around 200°C/sec. The smooth curves are based on a single reaction, first-order decomposition model (7,8) and on the stated temperature-time history. Parameters used for the lignite have been published (8) while for the bituminous coal the Arrhenius frequency factor and activation energy were taken as 1013 sec"1 and 35 kcal/mol, respectively, with the yield of pyrolytic water ultimately attainable estimated from experimental measurements as 4.6 wt % of the coal (as-received). 

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