Concentration rate laws

Because partial pressures are proportional to concentrations, rate laws for gas-phase reactions can also be expressed in terms of partial pressures, for instance, as... 

Isolating isoprene as the only reactant whose concentration changes simplifies the general rate law to a form that can be tested against the single concentration rate laws ... 

Complex metal hydrides, hydrogen storage and, 13 851 Complex salts krypton, 17 333 radon, 17 335 xenon, 17 326-330 Compomers, 8 334-335 Component concentrations, rate law and, 14 609... 

Because partial pressures are proportional to concentrations, rate laws for gas-phase reactions can also be expressed in terms of partial pressures, for instance, as rate = kPx for a first-order reaction of a gas X. What are the units for the rate constants when partial pressures are expressed in Torr and time is expressed in seconds for (a) zero-order reactions ... 

Which rate law we choose to determine by experiment often depends on what types of data are easiest to collect. If we can conveniently measure how the rate changes as the concentrations are changed, we can readily determine the differential (rate/concentration) rate law. On the other hand, if it is more convenient to measure the concentration as a function of time, we can determine the form of the integrated (concentration/time) rate law. We will discuss how rate laws are actually determined in the next several sections. 

We now must determine the ratio F/Fq a function of volume V or the catalyst weight, W to account for pressure drop. We then can combine the concentration, rate law, and design equation. However, whenever accounting for the effects of pressure drop, the differential form of the mole balance (design equation) must be used. 

Rather than combining the concentrations, rate laws, and mole balances to write everything in terms cf the molar flow rate as we did in the past, it is more convenient here to write our computer solution (either POLYMATH or our own program) using equations for FA, and so on. Consequently, we shall write Equations (E6-8.9)through (E6-8.12)and (E6-8.19) through (E6-8.25)as individual lines and let the computer combine them to obtain a solution. 

Thus the overall concentration rate law in the solution is finally expressed as in Eq. (156) in a general case. 

In reaction kinetics the absorbance of just one reactant in the solution to be examined is a trivial case. Then eq. (4.1) can be used directly to transform the concentration in the rate equation to a relationship containing the absorbances. Taking a single step photoreaction according to eq. (3.35) instead of the concentration rate law (using decadic units)... 

A rate law tells us how the rate of a reaction changes at a particular temperature as we change reactant concentrations. Rate laws can be converted into equations that tell us what the concentrations of fhe reacfanfs or producfs are af any time during fhe course of a reaction. The mafhematics required involves calculus. We don f expect you to be able to perform fhe calculus operations however, you should be able to use fhe resulting equations. We will apply fhis conversion to fwo of fhe simplesf rate laws fhose fhaf are firsf order overall and fhose fhaf are second order overall. 

Addition of Electrophiles.—Cr + and CrCHCla + react in perchloric acid solution to give CrCH2Cl + and equal quantities of Cr + and CrCl +. At high acid concentrations rate law (1) is followed which suggests that the first and rate-determining step is... 

The Langmuir-Hinshelwood picture is essentially that of Fig. XVIII-14. If the process is unimolecular, the species meanders around on the surface until it receives the activation energy to go over to product(s), which then desorb. If the process is bimolecular, two species diffuse around until a reactive encounter occurs. The reaction will be diffusion controlled if it occurs on every encounter (see Ref. 211) the theory of surface diffusional encounters has been treated (see Ref. 212) the subject may also be approached by means of Monte Carlo/molecular dynamics techniques [213]. In the case of activated bimolecular reactions, however, there will in general be many encounters before the reactive one, and the rate law for the surface reaction is generally written by analogy to the mass action law for solutions. That is, for a bimolecular process, the rate is taken to be proportional to the product of the two surface concentrations. It is interesting, however, that essentially the same rate law is obtained if the adsorption is strictly localized and species react only if they happen to adsorb on adjacent sites (note Ref. 214). (The apparent rate law, that is, the rate law in terms of gas pressures, depends on the form of the adsorption isotherm, as discussed in the next section.)... 

The course of a surface reaction can in principle be followed directly with the use of various surface spectroscopic techniques plus equipment allowing the rapid transfer of the surface from reaction to high-vacuum conditions see Campbell [232]. More often, however, the experimental observables are the changes with time of the concentrations of reactants and products in the gas phase. The rate law in terms of surface concentrations might be called the true rate law and the one analogous to that for a homogeneous system. What is observed, however, is an apparent rate law giving the dependence of the rate on the various gas pressures. The true and the apparent rate laws can be related if one assumes that adsorption equilibrium is rapid compared to the surface reaction. 

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. 

A general fonn of the rate law , i.e. the differential equation for the concentrations is given by... 

It is clear from figure A3.4.3 that the second-order law is well followed. Flowever, in particular for recombination reactions at low pressures, a transition to a third-order rate law (second order in the recombining species and first order in some collision partner) must be considered. If the non-reactive collision partner M is present in excess and its concentration [M] is time-independent, the rate law still is pseudo-second order with an effective second-order rate coefficient proportional to [Mj. 

The system of coupled differential equations that result from a compound reaction mechanism consists of several different (reversible) elementary steps. The kinetics are described by a system of coupled differential equations rather than a single rate law. This system can sometimes be decoupled by assuming that the concentrations of the intennediate species are small and quasi-stationary. The Lindemann mechanism of thermal unimolecular reactions [18,19] affords an instructive example for the application of such approximations. This mechanism is based on the idea that a molecule A has to pick up sufficient energy... 

The Landolt reaction (iodate + reductant) is prototypical of an autocatalytic clock reaction. During the induction period, the absence of the feedback species (Irere iodide ion, assumed to have virtually zero initial concentration and fomred from the reactant iodate only via very slow initiation steps) causes the reaction mixture to become kinetically frozen . There is reaction, but the intemiediate species evolve on concentration scales many orders of magnitude less than those of the reactant. The induction period depends on the initial concentrations of the major reactants in a maimer predicted by integrating the overall rate cubic autocatalytic rate law, given in section A3.14.1.1. 

It is convenient to analyse tliese rate equations from a dynamical systems point of view similar to tliat used in classical mechanics where one follows tire trajectories of particles in phase space. For tire chemical rate law (C3.6.2) tire phase space , conventionally denoted by F, is -dimensional and tire chemical concentrations, CpC2,- are taken as ortliogonal coordinates of F, ratlier tlian tire particle positions and velocities used as tire coordinates in mechanics. In analogy to classical mechanical systems, as tire concentrations evolve in time tliey will trace out a trajectory in F. Since tire velocity functions in tire system of ODEs (C3.6.2) do not depend explicitly on time, a given initial condition in F will always produce tire same trajectory. The vector R of velocity functions in (C3.6.2) defines a phase-space (or trajectory) flow and in it is often convenient to tliink of tliese ODEs as describing tire motion of a fluid in F with velocity field/ (c p). 

Because of tire underlying dissipative nature of tire chemical systems tliat tire ODEs (C3.6.2) represent, tliey have anotlier important property any volume in F will shrink as it evolves. For a given set of initial chemical concentrations tire time evolution under tire chemical rate law will approach arbitrarily close to some final set of points in... 

For tliis model tire parameter set p consists of tire rate constants and tire constant pool chemical concentrations l A, 1 (Most chemical rate laws are constmcted phenomenologically and often have cubic or otlier nonlinearities and irreversible steps. Such rate laws are reductions of tire full underlying reaction mechanism.)... 

A simple kinetic order for the nitration of aromatic compounds was first established by Martinsen for nitration in sulphuric acid (Martin-sen also first observed the occurrence of a maximum in the rate of nitration, occurrii for nitration in sulphuric acid of 89-90 % concentration). The rate of nitration of nitrobenzene was found to obey a second-order rate law, first order in the concentration of the aromatic and of nitric acid. The same law certainly holds (and in many cases was explicitly demonstrated) for the compounds listed in table 2.3. 

Chloroanisole and p-nitrophenol, the nitrations of which are susceptible to positive catalysis by nitrous acid, but from which the products are not prone to the oxidation which leads to autocatalysis, were the subjects of a more detailed investigation. With high concentrations of nitric acid and low concentrations of nitrous acid in acetic acid, jp-chloroanisole underwent nitration according to a zeroth-order rate law. The rate was repressed by the addition of a small concentration of nitrous acid according to the usual law rate = AQ(n-a[HN02]atoioh) -The nitration of p-nitrophenol under comparable conditions did not accord to a simple kinetic law, but nitrous acid was shown to anticatalyse the reaction. 

By using higher concentrations of nitrous acid, and reducing that of nitric acid, the nitration of both compounds was brought under the control of the following rate law ... 

First-order nitrations. The kinetics of nitrations in solutions of acetyl nitrate in acetic anhydride were first investigated by Wibaut. He obtained evidence for a second-order rate law, but this was subsequently disproved. A more detailed study was made using benzene, toluene, chloro- and bromo-benzene. The rate of nitration of benzene was found to be of the first order in the concentration of aromatic and third order in the concentration of acetyl nitrate the latter conclusion disagrees with later work (see below). Nitration in solutions containing similar concentrations of acetyl nitrate in acetic acid was too slow to measure, but was accelerated slightly by the addition of more acetic anhydride. Similar solutions in carbon tetrachloride nitrated benzene too quickly, and the concentration of acetyl nitrate had to be reduced from 0-7 to o-i mol 1 to permit the observation of a rate similar to that which the more concentrated solution yields in acetic anhydride. 

The effects of added species. The rate of nitration of benzene, according to a rate law kinetically of the first order in the concentration of aromatic, was reduced by sodium nitrate, a concentration of io mol 1 of the latter retarding nitration by a factor of about Lithium nitrate... 

In a rate law, the proportionality constant between a reaction s rate and the concentrations of species affecting the rate k). 

A second requirement is that the rate law for the chemical reaction must be known for the period in which measurements are made. In addition, the rate law should allow the kinetic parameters of interest, such as rate constants and concentrations, to be easily estimated. For example, the rate law for a reaction that is first order in the concentration of the analyte. A, is expressed as... 

The integrated form of the rate law for equation 13.4, however, is still too complicated to be analytically useful. We can simplify the kinetics, however, by carefully adjusting the reaction conditions. For example, pseudo-first-order kinetics can be achieved by using a large excess of R (i.e. [R]o >> [A]o), such that its concentration remains essentially constant. Under these conditions... 

Direct-Computation Rate Methods Rate methods for analyzing kinetic data are based on the differential form of the rate law. The rate of a reaction at time f, (rate)f, is determined from the slope of a curve showing the change in concentration for a reactant or product as a function of time (Figure 13.5). For a reaction that is first-order, or pseudo-first-order in analyte, the rate at time f is given as... 

A rate law describes how the rate of a reaction is affected by the concentration of each species present in the reaction mixture. The rate law for reaction A5.1 takes the general form of... 

Several important points about the rate law are shown in equation A5.4. First, the rate of a reaction may depend on the concentrations of both reactants and products, as well as the concentrations of species that do not appear in the reaction s overall stoichiometry. Species E in equation A5.4, for example, may represent a catalyst. Second, the reaction order for a given species is not necessarily the same as its stoichiometry in the chemical reaction. Reaction orders may be positive, negative, or zero and may take integer or noninteger values. Finally, the overall reaction order is the sum of the individual reaction orders. Thus, the overall reaction order for equation A5.4 isa-l-[3-l-y-l-5-l-8. 

Demonstrating that a reaction obeys the rate law in equation A5.11 is complicated by the lack of a simple integrated form of the rate law. The kinetics can be simplified, however, by carrying out the analysis under conditions in which the concentrations of all species but one are so large that their concentrations are effectively constant during the reaction. For example, if the concentration of B is selected such that [B] [A], then equation A5.11 simplifies to... 

A variation on the use of pseudo-ordered reactions is the initial rate method. In this approach to determining a reaction s rate law, a series of experiments is conducted in which the concentration of those species expected to affect the reaction s rate are changed one at a time. The initial rate of the reaction is determined for each set of conditions. Comparing the reaction s initial rate for two experiments in which the concentration of only a single species has been changed allows the reaction order for that species to be determined. The application of this method is outlined in the following example. 

The order of the rate law with respect to the three reactants can be determined by comparing the rates of two experiments in which the concentration of only one of the reactants is changed. For example, in experiment 2 the [H+] and the rate are approximately twice as large as in experiment 1, indicating that the reaction is first-order in [H+]. Working in the same manner, experiments 6 and 7 show that the reaction is also first-order with respect to [CaHeO], and experiments 6 and 8 show that the rate of the reaction is independent of the [I2]. Thus, the rate law is... 

The foregoing conclusion does not mean that the rate of the reaction proceeds through Table 5.1 at a constant value. The rate of reaction depends on the concentrations of reactive groups, as well as on the reactivities of the latter. Accordingly, the rate of the reaction decreases as the extent of reaction progresses. When the rate law for the reaction is extracted from proper kinetic experiments, specific reactions are found to be characterized by fixed rate constants over a range of n values. 

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