Second-order irreversible reaction

Section 5.1 shows how nonlinear regression analysis is used to model the temperature dependence of reaction rate constants. The functional form of the reaction rate was assumed e.g., St = kab for an irreversible, second-order reaction. The rate constant k was measured at several temperatures and was fit to an Arrhenius form, k = ko exp —Tact/T). This section expands the use of nonlinear regression to fit the compositional and temperature dependence of reaction rates. The general reaction is... 

Irreversible second-order reactions A + B C + D, small scale ... 

Irreversible second-order reaction A - - B large scale... 

There are two other limiting forms of these equations that are also of interest. If k 1 k2, the first step is very rapid compared to the second, so that it is essentially complete before the latter starts. The reaction may then be treated as a simple irreversible second-order reaction with the second step being rate limiting. On the other hand, if k2 ku the first step controls the reaction so the kinetics observed are those for a single second-order process. However, the analysis must take into account the fact that in this case 2 moles of species A will react for each mole of B that is consumed. 

Another class of irreversible second-order reactions obeys the rate law r = k[A][B]... 

For an irreversible second-order reaction to which eqn. (13) applies, integration of the design equation, again assuming isothermal conditions, yields... 

A number of unsteady state problems in diffusion have been considered, in which chemical reactions are occurring (F6, G8). Sherwood and Pigford (S9, pp. 332-337) have studied the unsteady state absorption of a substance A which diffuses into the solvent S and undergoes an infinitely fast, irreversible, second-order reaction with a solute B (that is A + B— AB). It is assumed that Fick s second law adequately describes the diffusion process and that because of the infinitely fast reaction of A and B there will be a plane parallel to the liquid surface at a distance a from it, which separates the region containing no A from that containing no B. The distance s is a function of t, inasmuch as the boundary between A and B retreats as B is used up in the chemical reactions. 

If the rate-limiting step in an irreversible second-order reaction to produce P from reactants A and B is the collision of single molecules of A and B, then the reaction rate should be proportional to the concentrations (C) of A and B that is, (kmolm 3) and Cg (kmolm ). The rate of reaction can be given as... 

Irreversible Second-Order Reaction In the case where the reactants A and B are converted to a product P by a bimolecular elementary reaction ... 

Derive an integrated rate equation similar to Equation 3.22 for the irreversible second-order reaction, when reactants A and B are introduced in the stoichiometric ratio ... 

An irreversible second-order reaction in the liquid phase 2A P... 

With single irreversible second-order reactions, the maximum of the heat release rate is reached at the beginning of the feed. At this stage, the heat exchange area may only be partially used, due to the increasing volume. This limits the effective available cooling capacity. Therefore, the knowledge of the maximum heat release... 

Another question is important for the safety assessment At which instant is the accumulation at maximum In semi-batch operations the degree of accumulation of reactants is determined by the reactant with the lowest concentration. For single irreversible second-order reactions, it is easy to determine directly the degree of accumulation by a simple material balance of the added reactant. For bimolecular elementary reactions, the maximum of accumulation is reached at the instant when the stoichiometric amount of the reactant has been added. The amount of reactant fed into the reactor (Xp) normalized to stoichiometry minus the converted fraction (A), obtained from the experimental conversion curve delivered by a reaction calorimeter (X = Xth) or by chemical analysis, gives the degree of accumulation as a function of time (Equation 7.18). Afterwards, it is easy to determine the maximum of accumulation XaCfmax and the MTSR can be obtained by Equation 7.21 calculated for the instant where the maximum accumulation occurs [7] ... 

This is the most common mode of addition. For safety or selectivity critical reactions, it is important to guarantee the feed rate by a control system. Here instruments such as orifice, volumetric pumps, control valves, and more sophisticated systems based on weight (of the reactor and/or of the feed tank) are commonly used. The feed rate is an essential parameter in the design of a semi-batch reactor. It may affect the chemical selectivity, and certainly affects the temperature control, the safety, and of course the economy of the process. The effect of feed rate on heat release rate and accumulation is shown in the example of an irreversible second-order reaction in Figure 7.8. The measurements made in a reaction calorimeter show the effect of three different feed rates on the heat release rate and on the accumulation of non-converted reactant computed on the basis of the thermal conversion. For such a case, the feed rate may be adapted to both safety constraints the maximum heat release rate must be lower than the cooling capacity of the industrial reactor and the maximum accumulation should remain below the maximum allowed accumulation with respect to MTSR. Thus, reaction calorimetry is a powerful tool for optimizing the feed rate for scale-up purposes [3, 11]. 

For an irreversible second-order reaction, the optimization of the reaction temperature and feed rate can be performed by using the following equation [14] ... 

Figure 5 shows concentration profiles that commonly occur when solute A undergoes an irreversible second-order reaction with component B, dissolved in the liquid, to give product C,... 

This discussion applies to an irreversible second-order reaction. For reversible reactions the relationships are more complex and are discussed in the texts by Sherwood et al. (1975) and by Danckwerts (1970). 

Figure 25 Irreversible second-order reaction A + B —> C + D, with Pr(A,B) = 0.1 arui initial conditions [A]0 = 100 cells, [B]o = 200 cells.

Some specific aspects in the modeling of gas-liquid continuous-stirred tank reactors are considered. The influence of volatility of the liquid reactant on the enhancement of gas absorption is analyzed for irreversible second-order reactions. The impact of liquid evaporation on the behavior of a nonadiabatic gas-liquid CSTR where steady-state multiplicity occurs is also examined. 

In a previous work ( 5), the film theory was used to analyze special cases of gas absorption with an irreversible second-order reaction for the case involving a volatile liquid reactant. Specifically, fast and instantaneous reactions were considered. Assessment of the relative importance of liquid reactant volatility from a local (i.e., enhancement) and a global (i.e., reactor behavior) viewpoint, however, necessitates consideration of this problem without limitation on the reaction regime. 

The occurrence of steady-state multiplicity in gas-liquid CSTRs has been demonstrated in experimental (9) and theoretical investigations (cf., 10). The irreversible second-order reaction system, in particular, has been treated extensively in several theoretical studies (10-15). These studies are however based on neglecting energy and material losses which result from evaporation of the liquid. 

The simplest case of parallel second-order steps is that of formation of two different dimers of a reactant A, corresponding to the network 5.23 and rate equations 5.24 (see next page). At all times, both products are formed in the same ratio rP rQ = kAP kAQ, so that the decay of A is an ordinary second-order reaction with rate coefficient k = kAP + kAQ. Likewise, the product formations are ordinary second-order reactions. (One could think of the initial amount of A as divided into two portions in the ratio kAV kAQ that react independently of one another and at the same rate, one to P and the other to Q.) All equations and plots for irreversible second-order reactions thus are valid (see Section 3.3.1). 

The type of chemical system that has received the most attention is the one in which the dissolved gas (component A) undergoes an irreversible second-order reaction with a reactant (component B) dissolved in the liquid. For the present, the gas will be taken as consisting of pure A, so that complications arising from gas film resistance can be avoided. The stoichiometry of the reaction is represented by... 

Transient absorption of gas, followed by irreversible second-order reaction, has been studied extensively by Brian et al. (B26) and Danckwerts (Dl). The average rate of absorption for a contact time d is also given in terms of the enhancement factor... 

T iming to Danckwerts model, we observe that irreversible second-order reactions are not conveniently dealt with except where the reaction is pseudo-first-order based on component A. In this case (D2), it is found for... 

Other Correlations for an Irreversible Second-Order Reaction... 

We again consider the irreversible second-order reaction... 

Solutions for two types of irreversible second-order reaction are presented in Table 9.5. The first (second-order reaction (a)) is a... 

For a graph of an irreversible second order reaction with a single reactant of the form 2A —> products rate = kJiA]2... 

The use of the PDF is best illustrated by use of a simple example for a single irreversible second order reaction at isotherm conditions, defined by ... 

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