One-Step First-Order Reactions

Consider a simple one-step first-order reaction. 

If P absorbs more strongly than R at X, then Equation 7.15 can lead to Equation 7.25 with [Rq] = [Aq]. 

The linearized forms of Equation 7.24 and Equation 7.25 have been very often used to calculate k either by graphical procedure (i.e., the plot of In (A bs - A ) or In (A., - A bs) vs. t) or by using the linear least-squares technique. The 

The use of the linearized form of the nonlinear equation Equation 7.24 or Equation 7.25 in the determination of the rate constant k requires the exact value of A, under a typical reaction condition of kinetic run, which is sometimes difficult to obtain, especially when the rate of reaction is very slow or the reaction under investigation is not a simple one-step irreversible reaction. There is no perfect, decisive, and completely error-free method to determine an exact value of A,. Some experimental approaches have been described by Jencks for the determination of a reliable value of A. The necessary and basic requirement in these approaches is that the reaction must obey the first-order rate law within the reaction period of at least 10 half-lives (i.e., time required for 99.9% completion of the reaction). This requirement is difficult to achieve with complete certainty even with moderately slow reactions. 

/ The Iterative Method and Linear Least-Squares Regression Analysis of Kinetic Data 

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In Figure 4, the calculated mass losses for cellulose at different constant heating rates and initial sample masses are compared to experimental TGA results. The TGA curves at heating rates of 0.14 K/min and O.S K/min had been used to evaluate the kinetic parameters for the one step first order reaction model which was incorporated into the model to calculate the sample temperature distribution. Since the temperature gradients in those samples are nearly zero, the results of the heat transport reaction model represent simultaneously the best fit for the assumed reaction model. At a heating rate of 108 K/min, the initial sample mass influences the temperature at which a given mass loss is attained. Cellulose samples with mo = I - 3 mg are affected only to a minor... 

No steady-state theory for kinetically controlled heterogeneous IT has been developed for micropipettes. However, for a thin-wall pipette (e.g., RG < 2) the micro-ITIES is essentially uniformly accessible. When CT occurs via a one-step first-order heterogeneous reaction governed by Butler-Volmer equation, the steady-state voltammetric response can be calculated as [8a]... 

The FDS5 pyrolysis model is used here to qualitatively illustrate the complexity associated with material property estimation. Each condensed-phase species (i.e., virgin wood, char, ash, etc.) must be characterized in terms of its bulk density, thermal properties (thermal conductivity and specific heat capacity, both of which are usually temperature-dependent), emissivity, and in-depth radiation absorption coefficient. Similarly, each condensed-phase reaction must be quantified through specification of its kinetic triplet (preexponential factor, activation energy, reaction order), heat of reaction, and the reactant/product species. For a simple charring material with temperature-invariant thermal properties that degrades by a single-step first order reaction, this amounts to -11 parameters that must be specified (two kinetic parameters, one heat of reaction, two thermal conductivities, two specific heat capacities, two emissivities, and two in-depth radiation absorption coefficients). 

Since first-order kinetics with respect to sulfate ion is indicated under some conditions for bacterial sulfate reduction, it is useful to consider the isotopic behaviour of a simple one-step first-order conversion (Fig. 6.2.4a). The term kinetic isotope effect describes the competing reactions (1) and (2),... 

In the normal process ( ), step (J) occurs very rapidly and step (/) is the rate-determining step, whereas in the inhibition process (B), step (3) occurs very slowly, generally over a matter of days, so that it is rate determining. Thus it has been demonstrated with AChE that insecticides, eg, tetraethyl pyrophosphate and mevinphos, engage in first-order reactions with the enzyme the inhibited enzyme is a relatively stable phosphorylated compound containing one mole of phosphoms per mole of enzyme and as a result of the reaction, an equimolar quantity of alcohoHc or acidic product HX is hberated. 

First-order reaction (Section 11.4) A reaction whose rate-limiting step is unimolecular and whose kinetics therefore depend on the concentration of only one reactant. 

The rate law of a reaction is an experimentally determined fact. From this fact we attempt to learn the molecularity, which may be defined as the number of molecules that come together to form the activated complex. It is obvious that if we know how many (and which) molecules take part in the activated complex, we know a good deal about the mechanism. The experimentally determined rate order is not necessarily the same as the molecularity. Any reaction, no matter how many steps are involved, has only one rate law, but each step of the mechanism has its own molecularity. For reactions that take place in one step (reactions without an intermediate) the order is the same as the molecularity. A first-order, one-step reaction is always unimolecular a one-step reaction that is second order in A always involves two molecules of A if it is first order in A and in B, then a molecule of A reacts with one of B, and so on. For reactions that take place in more than one step, the order/or each step is the same as the molecularity for that step. This fact enables us to predict the rate law for any proposed mechanism, though the calculations may get lengthy at times." If any one step of a mechanism is considerably slower than all the others (this is usually the case), the rate of the overall reaction is essentially the same as that of the slow step, which is consequently called the rate-determining step. ... 

A distinction between "molecularity" and "kinetic order" was deliberately made, "Mechanism" of reaction was said to be a matter at the molecular level. In contrast, kinetic order is calculated from macroscopic quantities "which depend in part on mechanism and in part on circumstances other than mechanism."81 The kinetic rate of a first-order reaction is proportional to the concentration of just one reactant the rate of a second-order reaction is proportional to the product of two concentrations. In a substitution of RY by X, if the reagent X is in constant excess, the reaction is (pseudo) unimolecular with respect to its kinetic order but bimolecular with respect to mechanism, since two distinct chemical entities form new bonds or break old bonds during the rate-determining step. 

In a first-order reaction, the rate-determining step involves a transformation where one reactant reacts to give one product, that is, A — B. In first-order reactions, there is an exponential decrease in the reactant concentration, so that at any given time, the transformation rate is dependent on the corresponding concentration of the reactant at the same time. This can be expressed in the following way ... 

In first-order reactions, the rate expression depends upon the concentration of only one species, whereas second-order reactions show dependence upon two species, which may be the same or different. The molecularity, or number of reactant molecules involved in the rate-determining step, is usually equivalent to the kinetic reaction order, though there can be exceptions. For instance, a bimolecular reaction can appear to be first order if there is no apparent dependence on the concentration of one of the... 

The basic premise of the original kinetic description of inhibition was that, for a reaction to proceed on a surface, one or more of the reactants (A) must be adsorbed on that surface in reversible equilibrium with the external solution, having an equilibrium adsorption constant of KA, and the adsorbed species must undergo some transformation involving one or more adsorbed intermediates (n) in the rate-limiting step, which leads to product formation. The product must desorb for the reaction cycle to be complete. If other species in the reaction mixture (I) can compete for the same adsorption site, the concentration of the adsorbed reactant (Aad) on the surface will be lower than when only pure reactant A is present. Thus, the rate of conversion will depend on the fraction of the adsorption sites covered by the reactant (0A) rather than the actual concentration of the reactant in solution, and the observed rate coefficient (fcobs) will be different from the true rate coefficient (ktme). In its simplest form the kinetic expression for this phenomenon in a first-order reaction can be described as follows ... 

In single step voltammetry, the existence of chemical reactions coupled to the charge transfer can affect the half-wave potential Ey2 and the limiting current l. For an in-depth characterization of these processes, we will study them more extensively under planar diffusion and, then, under spherical diffusion and so their characteristic steady state current potential curves. These are applicable to any electrochemical technique as previously discussed (see Sect. 2.7). In order to distinguish the different behavior of catalytic, CE, and EC mechanisms (the ECE process will be analyzed later), the boundary conditions of the three processes will be given first in a comparative way to facilitate the understanding of their similarities and differences, and then they will be analyzed and solved one by one. The first-order catalytic mechanism will be described first, because its particular reaction scheme makes it easier to study. 

Strange as it seems, there are one-step reactions in nature, e.g. first-order reactions of monomolecular decomposition... 

A catalytic cycle is a sequence of steps. When one step is much slower than the others, we say that this step is rate determining and we ignore, for kinetics purposes, the other (fast) steps. Nevertheless, sometimes there are two slow steps with similar rates, and sometimes the rate of a specific step changes in the course of the reaction or under different conditions. One common situation is that of two consecutive first-order reactions, as in Eq. (2.44). 

Because the rate-determining step involves just one molecule, the rate equation shows rate = fc[R] CO Cl], and the reaction is called a first-order reaction as the rate is proportional to just one concentration. A first-order reaction involves the unimolecular decomposition of something in the rate-deter mining step. 

Rate laws are employed to evaluate reaction mechanisms in soil-water systems. To accomplish this, kinetics are used to elucidate the various individual reaction steps or elementary reactions. Identifying and quantifying the elementary steps of a complex process allow one to understand the mechanism(s) of the process. For example, unimolecular reactions are generally described by first-order reactions bimolecular reactions are described by second-order reactions,... 

Kinetics of Consecutive Reactions Reactions which take place in two or more steps, one after the other are called consecutive reactions. Their characteristic features are illustrated with an example consider two consecutive first order reaction. Thus, the sequence... 

A simple first order reaction following reversible charge transfer is one of the few cases for which an analytical solution to the diffusion-kinetic differential equations can be obtained. For reactions (1) and (2) under diffusion-controlled charge-transfer conditions after a potential step, the partial differential equations which must be solved are (18) and (19). After Laplace transforma-... 

Stirred reactor theory reveals a fixed maximum mass loading rate for a fixed reactor volume and pressure. Any attempts to overload the system will quench the reaction. Attempts have been made to determine chemical kinetic parameters from stirred reactor measurements however, the usefulness of such measurements is limited. First, the analysis is based on the assumption that a hydrocarbon-air system can be represented by a simple one-step overall order kinetic expression. Recent evidence indicates that such an assumption is not realistic. Second, the analysis is based on the assumption of complete instantaneous mixing, which is impossible to achieve experimentally. 

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