Multiple second-order reaction

The examples in this section have treated a single, second-order reaction, although the approach can be generalized to multiple reactions with arbitrary... 

Heeb, T. G. and R. S. Brodkey (1990). Turbulent mixing with multiple second-order chemical reactions. AlChE Journal 36, 1457-1470. 

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. 

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 phase plane of C vs. T shows a separairix. [2nd Ed.. P8-22) A second-order reaction with multiple steady stales is carried out different solvents. 

Figure 13.23 Multiple steady states of two-reactants, second-order reaction in isothermal CSTR-Separator-Recycle system...

In Chapter 7 we discussed the basics of the theory concerned with the influence of diffusion on gas-liquid reactions via the Hatta theory for flrst-order irreversible reactions, the case for rapid second-order reactions, and the generalization of the second-order theory by Van Krevelen and Hofitjzer. Those results were presented in terms of classical two-film theory, employing an enhancement factor to account for reaction effects on diffusion via a simple multiple of the mass-transfer coefficient in the absence of reaction. By and large this approach will be continued here however, alternative and more descriptive mass transfer theories such as the penetration model of Higbie and the surface-renewal theory of Danckwerts merit some attention as was done in Chapter 7. 

Fig. 6 Second-order reaction in CSTR/Separator/Recycle a) Control structure relying on selfregulation b) bifurcation diagram showing multiple steady-states...

The derivation of another equation for the analysis of reversible second order reactions, which was obtained by Erickson et al. (1987), can be useful if the choice of reactant concentrations is limited. The treatment was specifically developed for the investigation of the rates of ligand binding to cell surfaces with multiple receptor sites. In this connection a relation is derived between k 2 and the density of receptor sites on cell surfaces. The same group (Goldstein et al., 1989)... 

Presto, a third-order rate law This multiplication should not be taken as representing a chemical event or as carrying such implications it is only a valid mathematical manipulation. Other similar transformations can be given,2 as when one multiplies by another factor of unity derived from the acid ionization equilibrium of HOC1. (The reader may show that this gives a second-order rate law.) These considerations illustrate that it is the rate law and not the reaction itself that has associated with it a unique order. 

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. 

Reaction of 3 with Ph3C+PF6" resulted in the formation of methylidene complex [(n-C5H5)Re(N0)(PPh3)(CH2)]+ PF6 (8) in 88-100% spectroscopic yields, as shown in Figure 11. Although 8 decomposes in solution slowly at -10 °C and rapidly at 25 °C (She decomposition is second order in 8), it can be isolated as an off-white powder (pure by H NMR) when the reaction is worked up at -23 °C. The methylidene H and 13C NMR chemical shifts are similar to those observed previously for carbene complexes [28]. However, the multiplicity of the H NMR spectrum indicates the two methylidene protons to be non-equivalent (Figure 11). Since no coalescence is.observed below the decomposition point of 8, a lower limit of AG >15 kcal/mol can be set for the rotational barrier about the rhenium-methylidene bond. 

We took the 4- sign on the square root term for second-order kinetics because the other root would give a negative concentration, which is physically unreasonable. This is true for any reaction with nth-order kinetics in an isothermal reactor There is only one real root of the isothermal CSTR mass-balance polynomial in the physically reasonable range of compositions. We will later find solutions of similar equations where multiple roots are found in physically possible compositions. These are true multiple steady states that have important consequences, especially for stirred reactors. However, for the nth-order reaction in an isothermal CSTR there is only one physically significant root (0 < Ca < Cao) to the CSTR equation for a given T. ... 

The high element effects for the reactions of amines with (11) and (12) suggest multiplicity of mechanistic routes. The second-order kinetics and the very slow exchange of cis-bromo-(ll-a-D) in isopropanol fit addition-elimination (Ghersetti et al., 1965). In methanol, exch/ nub values for reaction of cyclohexylamine with cis and iraws-chloro-(ll) and cis- and Jrans-bromo-(ll) are 13, 11, 20 and 23, respectively, and 0-9, 0-6, 1 4 and 0-8 for the corresponding reactions of di-n-butylamine. While (rans-bromo-(ll) and cis-chloro-(ll) showed normal kinetics in methanol and in ethanol, the rate constants with cis-bromo- (11) and (12) decreased with time, but steady second-order behaviour could be achieved by addition of the perchlorate of the amine used. While this fits an amine-promoted elimination-addition, where the ammonium salt formed shifts the equilibrium to the left (equation 15), the slow... 

In this section we analyze processes involving the second-order A + B —> 2P reaction. Such processes have been studied, among others, by Luyben andTyreus [10]. It has been noticed [11] that certain control structures lead to state multiplicity and instability. Here, we use dimensionless models to derive general feasibility and stability conditions. The reader is encouraged to check carefully the balance equations, writing them first in the dimensional form, and then deriving the dimensionless versions. To solve these equations, software such as Maple or the symbolic toolbox of Matlab can be used. 

SW1 substitution involves an intermediate carbonium ion prior to its reaction with DNA, and proceeds with a first-order kinetics. The rate limiting step is the formation of the carbonium ion. SN2 substitution depends on both the alkylating agent and DNA. Thus, the reaction proceeds with a second-order kinetics. Multiple sites on... 

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