Second order reaction - a special case

In the previous chapters we developed equations for zero and first order reactions. In this chapter we will look at a special case of a second order reaction, which involves dimerization reactions. We will define what a second order reaction is and how we can address it mathematically. The aim of this chapter is to give an introduction to second order reactions, in which both reactants are the same chemical. By the end of this chapter you should be able to  

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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. 

Caution 1. In the special case where reactants are introduced in their stoichiometric ratio, the integrated rate expression becomes indeterminate and this requires taking limits of quotients for evaluation. This difficulty is avoided if we go back to the original differential rate expression and solve it for this particular reactant ratio. Thus, for the second-order reaction with equal initial concentrations of A and B, or for the reaction... 

Most elementary reactions are second-order reactions. There are two t) es of second-order reactions 2A C and A + B C. The first type (special case) of second-order reactions is... 

In the special case of ion exchange and unfavorable equilibrium, i.e. aA B < 1, with A originally in the solution, under the condition of sufficiently long bed, Walter s solution could be used. Walter s equation is a special case of the Thomas model for arbitrary isotherm and the kinetic law equivalent to a reversible second-order chemical reaction (Helfferich, 1962) ... 

A special case of the second order reaction arises when there is only one substance reacting, i.e. A-j-A—>A2. When A reacts with A equation (5) becomes simplified, giving... 

Three examples are popular here. The first two start with flash photolysis, where an intense flash irradiates the whole cell at t = 0, instantly producing an electrochemically active species that decays chemically in time, either by a first-order reaction, or a second-order reaction. The labile substance is assumed to be formed uniformly in the cell space with a bulk concentration of c. These are cases where the concentration at the outer boundary is not constant, falling with time. The third case, the catalytic or EC7 system (see [73,74]), is of special interest because of the reaction layer it gives rise to. 

Substituting the rate law for the special case of a second-order reaction gives us... 

The kinetics for P700 and P430 recombination was apparently non-exponential, as the semi-logarith-mic plot (not shown) was non-linear. The kinetics of the recombination reaction could be fitted to a linear plot by the application of a special case of a second-order reaction where the two reactants P700 and P430 existed in equal amounts throughout the reaction course The second-order rate constant was calculated to be 3T0 and the ty, was approximately 45 ms. 

In order to simplify the interpretation of this somewhat complicated looking equation, the special but very common case of a second order reaction shall be considered first. This reduces Equ.(4-159) to ... 

The procedure as shown so far for the interpretation of the sensitivity for the special case of a second order reaction can be performed in an identical way for the complete validity range of the approximation. For crit and Bignit the following relationships are obtained, which are also graphically presented in Figure 4-44. 

In the previous chapter we investigated how the order of a reaction can be determined by changing the concentration of a reactant and measuring the resultant change in the rate of the reaction. In this chapter we will build up on this knowledge and investigate how we can analyse reactions with several reactants. In chapter 6 we already discussed a special case of this type of reactions, namely a second order reaction, in which the two reactants are not identical. By the end of this chapter you will be able to ... 

We assume that El can be represented by a second order reaction of the t) e A + A, in which the reactants are the same (we have discussed this special case in chapter 5). 

With Bernoulli mechanisms, the ultimate unit of the growing chain has no influence on the linkage formed by a newly polymerized unit. With first-order Markov mechanisms, the ultimate unit does exert an influence, and in second-order Markov mechanisms, the penultimate, or second last, unit exerts an influence. In third-order Markov mechanisms it is the third last unit that exerts the influence on the linkage of newly joined units. Thus, Bernoulli mechanisms are a special case of Markov mechanisms, and could also be called zero-order Markov mechanisms. Second- and higher-order Markov mechanisms cannot be stated with confidence to occur in polyreactions, and, so, will not be discussed further. In addition, the discussion will be confined to binary mechanisms, that is, polyreactions where the unit possesses only two reaction possibilities. 

An analytical method provides the solution in closed form , i.e. a formula can be given for the time evolution of the concentrations. This possibility only exists for a rather limited class of differential equations, even in the special subclass of kinetic differential equations. Methods of how to derive solutions can be found in the book by Kamke (1959) or in problems books like those by Filippov (1979), Matveev (1983), or Krasnov et al. (1978). The special case of kinetic differential equations is treated by Rodiguin Rodiguina (1964) who published the explicit solution for many first order reactions, and by Szabo (1969), who collected results for second order reactions too (together with realistic chemical examples). 

The pool chemical approximation (also called the pool component approximation) is applicable when the concentration of a reactant species is much higher than those of the other species, and therefore the concentration change of this species is considered to be negligible throughout the simulation period. For example, a second-order reaction step A -r B C can be converted to first-order, if concentration b of reactant B is almost constant dimng the simulations. In this way, the product = ft of concentration ft and rate coefficient k is practically constant therefore, the second-order expression can be converted to a first-order one Ac At = kab = k a.hitAAs special case, the pool chemical approximation is called... 

In the special case in which the concentrations of A and B at t = 0 are identical (i.e., [Ag] = [Bg] = [Rg]), the rate law is given by Equation 7.17, in which [R] is the remaining concentration of A or B at any time t. The integrated rate law is given by Equation 7.18, which is also valid for second-order reaction involving a single reactant such as A and, in this case, the elementary reaction is represented as 2A P with second-order rate constant k. 

This is the general result for a second-order batch reaction. The mathematical form of the equation presents a problem when the initial stoichiometry is perfect, ao =bo- Such problems are common with analytical solutions to ODEs. Special formulas are needed for special cases. 

At first glance the Diels-Alder reaction represents an organic transformation which is relatively insensitive to solvent effects (Table 9). For the dimerization of cyclopentadiene, the second-order rate constants in a broad range of organic solvents are quite similar5. The data of Table 9 refer to the special case of a Diels-Alder reaction between two pure hydrocarbons. Usually, Diels-Alder reactions only proceed at an appreciable rate when either the diene or the dienophile is activated by electron-donating or electron-withdrawing... 

Brilman et al. [42] and Lin et al. [44] using a numerical method, Nagy [48] by using an analytical method, investigated the effect of the second, third, etc. particles (perpendicular to the gas-liquid interface) on the absorption rate. They obtained that, in most cases, the first particle determines the absorption rate. However, in special cases, the effect of these particles can also be important. Nagy solved the mass transfer problem analytically for the number of particles in the diffusion path [48]. For the sake of completeness we will give the absorption rate for that case, as well (for details see [48] ). The mass transfer is accompanied here by a first-order chemical reaction. This situation is illustrated in Fig. 1 where three particles are located behind each other. The absorption rate... 

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