Second-order mixed case reaction

Therefore, the reaction is first-order in both A and B and second-order overall. Such a reaction is referred to as a second-order mixed case, since the... 

The distinction between the SnI and jSat2 mechanisms is not necessarily always a sharp one, and if the attacking group Y can facilitate the departure of X, an intennediate case may occur. Such intermediate cases seem to arise in reactions in which the nucleophilic reagent is a solvent molecule, when, unfortunately, kinetic order with respect to solvent is almost impossible to clarify. It is important to note that where the attacking group Y is an ion such as a halide, X"", or Oil or RO the displacement reaction usually follows fairly clean second-order mixed kinetics. The confusion that arises when Y is a solvent molecule is readily understood when we consider that the mechanism of ionization will involve very strong ion-solvent interactions. In fact ionization is not possible without such interactions. 

In Sec. 2.1, we worked through the details of a second-order mixed reaction, which is first-order in each of two components. We wiU consider briefly here the various third-order cases (those involving reactants of multiple types are worked out in detail by Benson (I960)). The simplest case involves only one reactant for which the rate law can be written as... 

One final case needs consideration. In Chapter 2 an equation was derived for the concentration of A when the rate law for mixed first and second orders is v = k [A] + 2A-i[A]2. The first term may represent k [A][B], in the case where [B]o [A]0, and the reaction scheme is... 

From a mathematical standpoint the various second-order reversible reactions are quite similar, so we will consider only the most general case—a mixed second-order reaction in which the initial system contains both reactant and product species. 

In Fig. 11.10 the ratio of c(t)/cq is plotted against Da for well-mixed BR and LCFR with half-, first-, and second-order kinetics systems. Again, we observe that although each case has different concentration histories and flow conditions, we can have the following simple rule-of-thumb analysis for complex reactive processing systems reactions are roughly half complete at Da = 1 they are practically complete at Da = 10 and the systems are essentially unreacted at Da = 0.1. The entire dynamic state of the reaction is in the region 10 1 < Da < 10 this is a similar conclusion to that on Fig. 11.7 earlier in this section. 

Figure 13 shows how the steady-state exit conversion X[— 1 — Cnm(z — 1)/C 5jii m] varies with the Damkohler number Da for different values of the dimensionless mixing time t](= tmix/r). The figure shows how non-uniform feeding could significantly reduce the conversion as compared to premixed feed for the case of a bimolecular second-order reaction (e.g. by a factor of 2 for the case of f] — 0.1), when mixing limitations are present in the system. 

Two further cases of opposing reactions which can be easily resolved by present methods may be mentioned. They are opposing reactions of second order and of mixed order. 

Case 2. A case of mixed first- and second-order reaction can be represented by... 

The utility of the model to predict the effects of interdroplet mixing on extent of reaction was demonstrated for the case of a solute diffusing from the dispersed phase and undergoing second-order reaction in the continuous phase. For this comparison the normalized volumetric dispersed-phase concentration distribution is deflned as fv(y) dy equal to the fraction of the total volume of the dispersed phase with dimensionless concentration in the range y to y -i- dy, where y = c/cq and... 

However, in the case of non-linear kinetics (e.g. a second-order reaction) the rate of reaction also depends upon the concentration of the reaction partners and thus on the earliness of the mixing . If the mixing vessel precedes the pipe reactor, the concentration of the reaction partners immediately drops to the level of the outlet concentration due to back-mixing. The reaction rate is therefore low in both reactors. If, however, the pipe reactor precedes the stirred vessel, the reaction can proceed at a high concentration, the conversion being higher for reaction orders greater than 1. 

By Jensen s inequality it is easy to show that (C)Da>i < (C)Dalower frequency of annihilation encounters and slower overall decay when the reactants are uniformly distributed, compared to the case of a distribution concentrated in some small regions. Therefore, the steady state average concentration, or total amount, is an increasing function of the stirring rate in this nonlinear annihilation process. Such change of the overall reactivity of a second order reaction due to chaotic mixing was observed experimentally by Paireau and Tabeling (1997). 

To quantify the effect of the incomplete mixing on reaction rates in the front of the reactor channel, this same simulation was repeated assuming second order kinetics (first order in each of the two components) and Cjj = C2j = 100 mol m. A rate constant of 1.0 X 10 m moh s was used to give an intermediate level of conversion (near 25%). This case can be compared with a simulation in which the inlet boimdary conditions were changed to assume complete mixing (50 mol m of each component across the entire inlet cross section). The axial fractional conversion profiles for these two cases (unmixed and premixed feeds) are shown in Fig. 13.4, where the unmixed feed curve is the average of the calculated values for the two components. The computed conversions for the two components were... 

If we were to change the kinetics so that the first reaction was second order in A and the second reaction was first order in B, then we would see largely the same picture emerging in the graphs of dimensionless concentration versus time. There would of course be differences, but not large departures in the trends from what we have observed for this all first-order case. But what if the reactions have rate expressions that are not so readily integrable What if we have widely differing, mixed-order concentration dependencies In some cases one can develop fully analytical (closed-form) solutions like the ones we have derived for the first-order case, but in other cases this is not possible. We must instead turn to numerical methods for efficient solution. 

Experimentally it has usually been found that anion-catalyzed reactions are first order with respect to iron(II) (8, 13, 15, 33, 34, 35). The reaction in very concentrated hydrochloric acid solutions (27) should be included in this class. Uncatalyzed reactions have usually been reported to be second order in iron(II) 14, 19, 24), although King and Davidson (15) observed mixed-order kinetics at high temperatures. In no case has the influence of iron (III) been carefully investigated, although Pound (29) reported that addition of iron (III) had no influence on the oxidation rate of iron (II) sulfate. 

The next example considers the slightly more complicated case of a second-order reaction in a perfectly mixed reactor, and also introduces a subtle assumption that has actually been made in the derivation of Eq. 12.4-1,2. 

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