Overall rate expression

Kinetic data provide information only about the rate-determining step and steps preceding it. In the hypothetical reaction under consideration, the final step follows the rate-determining step, and because its rate will not affect the rate of the overall reaction, will not appear in the overall rate expression. The rate of the overall reaction is governed by the second step, which is the bottleneck in the process. The rate of this step is equal to A2 multiplied by the molar concentration of intermediate C, which may not be directly measurable. It is therefore necessary to express the rate in terms of the concentrations of reactants. In the case under consideration, this can be done by recognizing that [C] is related to [A] and [B] by an equilibrium constant ... 

A kinetic expression which is equivalent to that for general acid catalysis also occurs if a prior equilibrium between reactant and the acids is followed by rate-controlling proton transfer. Each individual conjugate base will appear in the overall rate expression ... 

Generally, all praetieal reaetions oeeur by a sequenee of elementary steps that eolleetively eonstitute the meehanism. The rate equation for the overall reaetion is developed from the meehanism and is then used in reaetor design. Although there are eases where experimental data provide no information about intermediate ehemieal speeies, experimental data have provided researehers with useful guidelines in postulating reaetion meehanisms. Information about intermediate speeies is essential in identifying the eorreet meehanism of reaetion. Where many steps are used, different meehanisms ean produee similar forms of overall rate expression. The overall rate equation is the result... 

The following describes many types of reaction mechanisms with a view toward developing their overall rate expressions. 

The following mechanism has been postulated for the decomposition of ethane into ethylene and hydrogen. The overall rate expression is first order in ethane. 

In microkinetics, overall rate expressions are deduced from the rates of elementary rate constants within a molecular mechanistic scheme of the reaction. We will use the methanation reaction as an example to illustrate the... 

Using the concept of the rate-determining step significantly simplifies the overall rate expression. Therefore, it is widely used in the analysis of kinetie data, especially in the field of heterogeneous catalysis. 

For mechanisms that are more complex than the above, the task of showing that the net effect of the elementary reactions is the stoichiometric equation may be a difficult problem in algebra whose solution will not contribute to an understanding of the reaction mechanism. Even though it may be a fruitless task to find the exact linear combination of elementary reactions that gives quantitative agreement with the observed product distribution, it is nonetheless imperative that the mechanism qualitatively imply the reaction stoichiometry. Let us now consider a number of examples that illustrate the techniques used in deriving an overall rate expression from a set of mechanistic equations. 

Because of the variety of chain reaction mechanisms that exist, it is difficult to develop completely general rate equations for these processes. However, some generalizations with regard to the overall rate expression can be made for certain classes of reactions. 

A reaction rate expression that is proportional to the square root of the reactant concentration results when the dominant termination step is reaction (4c), that is, the termination reaction occurs between two of the radicals that are involved in the unimolecular propagation step. The generalized Rice-Herzfeld mechanism contained in equations 4.2.41 to 4.2.46 may be employed to derive an overall rate expression for this case. 

In order for the overall rate expression to be 3/2 order in reactant for a first-order initiation process, the chain terminating step must involve a second-order reaction between two of the radicals responsible for the second-order propagation reactions. In terms of our generalized Rice-Herzfeld mechanistic equations, this means that reaction (4a) is the dominant chain breaking process. One may proceed as above to show that the mechanism leads to a 3/2 order rate expression. 

For Rice-Herzfeld mechanisms the mathematical form of the overall rate expression is strongly influenced by the manner in which the chains are broken. It can also be shown that... 

Although the above Rice-Herzfeld mechanisms lead to simple overall rate expressions, do not get the impression that this is always the case. More detailed discussions of these types of reactions may be found in textbooks (31-34) and in the original literature. 

Since the branching parameter a is greater than unity (usually it is 2), it is conceivable that under certain circumstances the denominator of the overall rate expression could become zero. In principle this would lead to an infinite reaction rate (i.e., an explosion). In reality it becomes very large rather than infinite, since the steady-state approximation will break down when the radical concentration becomes quite large. Nonetheless, we will consider the condition that Mol - 1) is equal to (fst T fgt) to be a valid criterion for an explosion limit. 

However, as the pressure is decreased, one eventually reaches a point where the rate of the decomposition reaction becomes much larger than the collisional deactivation process, so that /c3 /c2[A]. In this situation the overall rate expression becomes second-order in A. 

Discuss the acid-base catalysis and suggest an overall rate expression for the acid-base catalysis. 

From the overall rate expression obtain the following ... 

The Mallard-Le Chatelier development for the laminar flame speed permits one to determine the general trends with pressure and temperature. When an overall rate expression is used to approximate real hydrocarbon oxidation kinetics experimental results, the activation energy of the overall process is found to be quite high—of the order of 160kJ/mol. Thus, the exponential in the flame speed equation is quite sensitive to variations in the flame temperature. This sensitivity is the dominant temperature effect on flame speed. There is also, of course, an effect of temperature on the diffusivity generally, the dif-fusivity is considered to vary with the temperature to the 1.75 power. 

These equations could be generalized even further (see Ref. 7) by simply writing JChJthj instead of II, where If- is the heat of formation per unit mass at the base temperature of each species j. However, for notation simplicity— and because energy release is of most importance for most combustion and propulsion systems—an overall rate expression for a reaction of the type which follows will suffice... 

An important characteristic of a catalyst s action is that the mechanism of the reaction is altered in a manner that allows for a lower activation energy. In a number of nonenzymatic examples, a reactant or product can act as a catalyst as well, and the definition must be altered to include substances that appear in the overall rate expression with a power higher than the corresponding... 

Fromm and Rudolph have discussed the practical limitations on interpreting product inhibition experiments. The table below illustrates the distinctive kinetic patterns observed with bisubstrate enzymes in the absence or presence of abortive complex formation. It should also be noted that the random mechanisms in this table (and in similar tables in other texts) are usually for rapid equilibrium random mechanism schemes. Steady-state random mechanisms will contain squared terms in the product concentrations in the overall rate expression. The presence of these terms would predict nonhnearity in product inhibition studies. This nonlin-earity might not be obvious under standard initial rate protocols, but products that would be competitive in rapid equilibrium systems might appear to be noncompetitive in steady-state random schemes , depending on the relative magnitude of those squared terms. See Abortive Complex... 

To get an overall rate expression, write the individual rate steps on the same basis (unit surface of burning particle, unit volume of fermenter, unit volume of cells, etc.). 

Dilute A diffuses through a stagnant liquid film onto a plane surface consisting of B, reacts there to produce R which diffuses back into the mainstream. Develop the overall rate expression for the L/S reaction... 

Problems of Combining Resistances. Suppose that we have found the correct mechanism and resultant rate equation for the surface phenomenon. Combining this step with any of the other resistance steps, such as pore of film diffusion, becomes rather impractical. When this has to be done, it is best to replace the multiconstant rate equation by an equivalent first-order expression, which can then be combined with other reaction steps to yield an overall rate expression. 

The Overall Rate Expression. Since materials in the two separate phases must contact each other before reaction can occur, both the mass transfer and the chemical rates will enter the overall rate expression. 

Now the overall rate expression for the reaction will have to account for the mass transfer resistance (to bring reactants together) and the resistance of the chemical reactions step. Since the relative magnitude of these resistances can vary greatly we have a whole spectrum of possibilities to consider. 

For SR of higher hydrocarbons, Rostrup-Nielsen " and Tottrup " postulated a Langmuir-Hinshelwood-Houghen-Watson (LHHW) kinetic model. It was assumed that the hydrocarbon chemisorbs on a dual catalytic site, followed by successive a-scission of the C-C bond. The resulting Ci species react with adsorbed steam to form H2 and CO. The expressions were lit to data for SR of n-Cv on a Ni/MgO catalyst at 500°C the overall rate expression is " " ... 

For a first-order reaction, in either reactant, the combination of these equations and the elimination of the surface and liquid-phase concentrations lead to the formulation of an overall rate, expressed as a function of the bulk gas-phase concentration. This procedure is essentially the same as the one presented analytically in Section 3.1.2 for the derivation of an overall rate in three-phase systems. 

Note that here the gas-phase concentration is constant and thus its inlet concentration is present in the overall rate expression. 

In the absence of reactant hydrogen, these two steps describe self-hydro-genation. To the best of our knowledge, only one kinetic parameter has been reported in the literature for the overall rate expression of ethene self-hydrogenation (381). This is the activation energy, which has been estimated from TPD data to be 18 kcal mol-1. 

The actual overall rate expression for a given chemical reaction can be determined reliably only by experiment. It often is a very complicated equation, appearing to bear no relation to the overall chemical equation in the way that Equation 15-2 (the rate equation) is related to Equation 15-1 (the chemical equation). 

It is important to know the overall rate expression for a reaction, because this permits you to control the reaction and to predict the reaction times needed for different conditions. This expression usually can be determined experimentally without any knowledge of the reaction mechanism itself in fact, it is a useful aid to working out the mechanism. One objective of the material that follows is an explanation of the manner in which the empirical (experimentally determined) rate expression is obtained for a great many reactions. 

The overall rate expression for this second order reaction is... 

Series (1) tells us that the reaction is second-order with respect to NO, and series (2) that the reaction is first-order with respect to 02. The superiority of l/Pso versus t for series (1) is much more apparent from the graphs (the first-order plot has a very pronounced curve, whereas the second-order plot is extremely Straight) than from the correlation coefficients (0.9999 for the second versus 0.9908 for the first). We conclude, therefore, that we need the rate constant for the overall rate expression... 

The two series agree on the value of k that should be used in the overall rate expression. 

Under some circumstances there will be a resistance to the transport of material from the bulk fluid stream to the exterior surface of a catalyst particle. When such a resistance to mass transfer exists, the concentration CA of a reactant in the bulk fluid will differ from its concentration CAi at the solid-gas interface. Because CAi is usually unknown it is necessary to eliminate it from the rate equation describing the external mass transfer process. Since, in the steady state, the rates of all of the steps in the process are equal, it is possible to obtain an overall rate expression in which CM does not appear explicitly. 

We see that, in principle, the overall reaction rate can be expressed in terms of coefficients such as the reaction rate constant and the mass transfer coefficient. To be of any use for design purposes, however, we must have knowledge of these parameters. By measuring the kinetic constant in the absence of mass transfer effects and using correlations to estimate the mass transfer coefficient we are really implying that these estimated parameters are independent of one another. This would only be true if each element of external surface behaved kinetically as all other surface elements. Such conditions are only fulfilled if the surface is uniformly accessible. It is fortuitous, however, that predictions of overall rates based on such assumptions are often within the accuracy of the kinetic information, and for this reason values of k and hD obtained independently are frequently employed for substitution into overall rate expressions. 

Were reversal of Equation 9.99 to compete with the subsequent removal of a hydrogen atom from 40, the rate of the hydrogen removal step would enter the overall rate expression the reaction would then show an isotope effect. (See Section 7.4, p. 385.) In some instances, for example, when the benzoyl radical attacks benzene, the initial addition is apparently reversible,175 and an isotope effect is found.176... 

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