Rate equation, overall

Overall Rate Equations with Diffusional Resistances. 23-52... 

OVERALL RATE EQUATIONS WITH DIFFUSIONAL RESISTANCES... 

The adsorption of carbon monoxide retards the reduction reaction with the rate constant k, followed by the desorption reaction with a rate constant k in the overall rate equation... 

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... 

Figure 3-7 gives plots of Equations 3-54 and 3-58, respeetively. Consider the seeond order reaetion 2A-I-B—produets, whieh is first order with respeet to both A and B, and therefore seeond order overall. The rate equation is ... 

In developing mathematieal expressions for seleetivities, knowledge of the rate equations are required. This is beeause the instantaneous seleetivity is defined in terms of the rate ratios. The parameters that affeet the instantaneous and the overall seleetivities are exaetly the same as those influeneing the reaetion rates, namely, the eoneentration, temperature, aetivation energy, time of reaetion (residenee time in flow reaetors), eatalysts, and the fluid meehanies. 

It is otherwise for complex reactions, for which the rate equation may or may not be simply related to the overall stoichiometric reaction. For example, the rate equation for the alkaline hydrolysis of ethyl acetate, which is a complex reaction (see Section 1.2),... 

The power a is called the order of reaction with respect to reactant A, b is the order with respect to B, and the sum (a + b. ..) is the overall order of the reaction. Many rate equations are of forms different from Eq. (1-11)—for example, concentration teims may appear in the denominator—and then the concept of reaction order is not applicable. 

The isolation technique showed that the reaction is first-order with respect to cin-namoylimidazole, but treatment of the pseudo-first-order rate constants revealed that the reaction is not first-order in amine, because the ratio k Jc is not constant, as shown in Table 2-2. The last column in Table 2-2 indicates that a reasonable constant is obtained by dividing by the square of the amine concentration hence the reaction is second-order in amine. For the system described in Table 2-2, we therefore find that the reaction is overall third-order, with the rate equation... 

Consider further Scheme XIV and rate equations (3-139) to (3-141). Evidently Cb will be small relative to (Ca + Cc) if ( -i + 2) i- Then B plays the role of a reactive intermediate in the overall reaction A— C. This is the usual condition that is taken as a warrant for the application of the steady-state approximation. If Cb is small, it is reasonable that Cb will be small throughout most the reaction, so it is set equal to zero. As Wong (53) has pointed out, however, the condition Cb = 0 is a sufficient but unnecessary condition for Eq. (3-142) to hold. Erom Eq. (3-140) we obtain... 

The currently popular approach is to compare the rate of the intramolecular reaction with the rate of an intermolecular reaction in which the reacting groups are closely similar. The intermolecular reaction will usually be overall second-order, in accordance with the rate equation... 

Given this evaporation rate, the overall mass transfer coefficient may then be calculated from Equation 4-391,... 

In certain circumstances even the parabolic rate law may be observed under conditions in which the oxide is porous and permeated by the oxidising environment". In these cases it has been shown that it is diffusion of one or other of the reactants through the fluid phase which is rate controlling. More usually however the porous oxide is thought to grow on the surface of a lower oxide which is itself growing at a parabolic rate. The overall rate of growth is then said to be paralinear - and may be described by the sum of linear and parabolic relationships (see equations 1.197 and 1.198). 

In all calculations [RCOOH] is a variable parameter and the final rate equation is a function of [RCOOH] and of K n is the overall reaction order when the reaction is carried out with stoichiometric amounts of add and alcohol. However, it is important to mention that it is the global acidity x of the medium and not [RCOOH] which is measured ... 

In the modified Ridd mechanism for region B the deprotonation of the A-nitroso-anilinium ion Ar —NH2NO in Scheme 3-23 is rapid, and therefore does not influence the overall rate. However, the second-order term in the rate equation for region C (Scheme 3-25) is consistent with a mechanism in which the deprotonation of the A-nitrosoanilinium ion (Scheme 3-24) and of the C-nitrosoanilinium dication (Scheme 3-22) belongs to the rate-determining part of the reaction. 

Derive the rate equation for the formation of the stable double helix and express the rate constant of the overall reaction in terms of the rate constants of the individual steps. 

This reaction cannot be elementary. We can hardly expect three nitric acid molecules to react at all three toluene sites (these are the ortho and para sites meta substitution is not favored) in a glorious, four-body collision. Thus, the fourth-order rate expression 01 = kab is implausible. Instead, the mechanism of the TNT reaction involves at least seven steps (two reactions leading to ortho- or /mra-nitrotoluene, three reactions leading to 2,4- or 2,6-dinitrotoluene, and two reactions leading to 2,4,6-trinitrotoluene). Each step would require only a two-body collision, could be elementary, and could be governed by a second-order rate equation. Chapter 2 shows how the component balance equations can be solved for multiple reactions so that an assumed mechanism can be tested experimentally. For the toluene nitration, even the set of seven series and parallel reactions may not constitute an adequate mechanism since an experimental study found the reaction to be 1.3 order in toluene and 1.2 order in nitric acid for an overall order of 2.5 rather than the expected value of 2. 

Substituting for F(x) in Equation 7 results in a differential equation which can be solved for x, and can then be used in the overall polymerization rate (Equation 3), yielding... 

Kinetic analysis based on the Langmuir-Hinshelwood model was performed on the assumption that ethylene and water vapor molecules were adsorbed on the same active site competitively [2]. We assumed then that overall photocatalytic decomposition rate was controlled by the surface reaction of adsorbed ethylene. Under the water vapor concentration from 10,200 to 28,300ppm, and the ethylene concentration from 30 to 100 ppm, the reaction rate equation can be represented by Eq.(l), based on the fitting procedure of 1/r vs. 1/ Ccm ... 

In particular, reactions in heterogeneous catalysis are always a series of steps, including adsorption on the surface, reaction, and desorption back into the gas phase. In the course of this chapter we will see how the rate equations of overall reactions can be constructed from those of the elementary steps. 

The activation energy of the overall reaction equals that of the first step, a,i-Note that fast elementary steps following the one that limits the rate become kine-tically insignificant, whereas fast steps before the rate-determining step do enter the rate equation, as they directly affect the concentration of the intermediate that is converted in the rate-determining step. 

Although the two cases represent entirely different reaction mechanisms, the overall rate of reaction maintains the same form with respect to its dependence on reactant concentration. Measurements of the kinetics would in both cases reveal the reaction to be first order in [R]. In general, it is not possible to prove that a mechanism is correct on the basis of kinetic measurements, as one can almost always find a modified mechanism leading to the same behavior of the rate equation. It is often possible, however, to exclude certain mechanisms on the basis of kinetic measurements. 

Using Eqns. 5.4-47 and 5.4-48, the unknowns 9>, Oa, and 6b can be expressed as functions of the rate constants and concentrations and then substituted in the overall net rate equation (r/t = n = n= rO which finally yields the following reaction rate expression ... 

In the above theory, the interfacial concentrations Coi and Cli are not measurable directly and are therefore of relatively little immediate use. In order to overcome this apparent difficulty, overall mass transfer rate equations are defined by analogy to the film equations. These are based on overall... 

Simple algebra, based on a combination of the film and overall mass transfer rate equations, lead to the following equations, relating the respective overall mass transfer coefficients and the coefficients for the two films... 

For a non-linear equilibrium relationship, in which the slope of the equilibrium curve varies with concentration, the magnitudes of the overall mass transfer coefficients will also vary with concentration, even when the film coefficients themselves remain constant. The use of overall mass transfer coefficients in mass transfer rate equations should therefore be limited to the case of linear equilibria or to situations in which the mass transfer coefficient is known to be... 

Note that the transfer rate equation is based on an overall concentration driving force, (X-X ) and overall mass transfer coefficient, Kl. The two-film theory for interfacial mass transfer shows that the overall mass transfer coefficient, Kl, based on the L-phase is related to the individual film coefficients for the L and G-phase films, kL and ko, respectively by the relationship... 

In general the rate equation for a heterogeneous reaction accounts for more than one process. The present consideration is directed to the general problem of combining the rates for processes of different kinds. Let r1( r2,..., rn be the rates of changes for the individual processes that are to be accounted for by an overall rate. If the changes occur by parallel paths, then the overall rate will be greater than the rate for any individual path. In fact, if the different parallel paths are independent of each other, the overall rate will be simply the sum of all the individual rates, or... 

Step 3. Equation 17.11 can now be derived from the overall reaction rate equation, Equation 17.10, using the expressions derived in Step 2 for the concentrations of the six enzyme species. 

Operation with an excess of ammonia in the reactor has the effect of increasing the rate due to the C fHl term. However, operation with excess ammonia decreases the concentration of ethylene oxide, and the effect is to decrease the rate due to the CEO term. Whether the overall effect is a slight increase or decrease in reaction rate depends on the relative magnitude of a and b. Consider now the rate equations for the by product reactions ... 

A certain element of confusion is to be met with both in textbooks, and in the literature, over the use and meaning of the terms order (cf. p. 39) and molecularity as applied to reactions. The order is an experimentally determined quantity, the overall order of a reaction being the sum of the powers of the concentration terms that appear in the rate equation ... 

A kinetic distinction between the operation of the SN1 and SN2 modes can often be made by observing the effect on the overall reaction rate of adding a competing nucleophile, e.g. azide anion, N3e. The total nucleophile concentration is thus increased, and for the SN2 mode where [Nu ] appears in the rate equation, this will result in an increased reaction rate due to the increased [Nut]. By contrast, for the Stfl mode [Nut] does not appear in the rate equation, i.e. is not involved in the rate-limiting step, and addition of N3e will thus be without significant effect on the observed reaction rate, though it will naturally influence the composition of the product. 

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