Two reactions in series

One of the simplest cases of two consequtive reation steps is the Michaelis-Menten kinetics  

Assuming that the second step is irreversible and using the steady-state approximation, we can determine the coverage 

From the rate we can calculate the activation enthalpy for the reaction 

Going back through the calculation, we see that the first term comes from k2 and is caused by the rate limiting step speeding up when the temperature is increased. The second term comes from the differentiation of 0. When the temperature is increased, the rate will change partially due to changes in 0. The second term describes this effect. 

For two consequtive reactions the reaction enthalpy for the overall reaction is the sum of the reaction enthalpies for each of the steps 

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In this section, some analytical solutions of fluidized-bed models are presented. Specifically, model solutions will be given for the case of a gas-phase reactant and a single solid-catalyzed reaction of the form A —> products and bubbling fluidized bed (Type B fluidization). The same analysis holds for a reaction of the form A + B —> products, if the reaction depends only on the concentration of A. Some solutions for the cases of a single reversible reaction, for two reactions in parallel, and two reactions in series will be given as well. 

Species-concentration profiles, obtained by integrating Equation (13.20), are plotted in Figure 13.7 for the case of two reactions in series (A —> B —> C) occurring in a catalytic hollow-fiber membrane. The reactant A contained in the bulk phase on shell-side flow through the membrane where reacting produces the intermediate product B then, B is converted in the final product C. Variations on concentration profiles are present only inside the fiber, outside the fiber there is no variation due to the reaction. No diffusion limitation in the films were considered in the present model in order to focus on transformation inside the membrane. 

If therefore experiments have shown the existence of a distinct bend in the Arrhenius line, the curvature shows whether the reaction is composed of two parallel reactions (case 1) or of two reactions in series (case 2). Of course the sequences may be more complicated, but in principle the result is always the same. If the reactions are in parallel the curve is concave upwards, and if they are in series it is concave downwards. 

Another technique is often used to follow the progress for two reactions in series. The concentrations of A, B and C are plotted as a singular point at different space times (e.g., t/, ) on a triangular diagram (see Figure 6-4). 

Fig. 4. List of kinetic parameters which determine the electrical response of model electrochemical reaction schemes. The single electrode reaction (E), two reactions in series (E.E), and two reactions in parallel (E + E) are common types of electrode reactions, discussed in the text, which every electrochemical investigation has to be capable of assessing as a matter of course.

The initiation step consists of two reactions in series. The first is the production of free radicals, which can be accomplished in many ways. The most common method, however, involves the use of a thermolabile compound, called an initiator (or catalyst), which decomposes to yield fre Radicals when heated. Thus, the homolytic dissociation of an iiiitiator I yields a pair of radicals R, as shown by Eq. (6.3), where kd is the rate constant for initiator dissociation at the particular temperature. Its magnitude is usually of the order of 10 -10 s (Being derived from the initiator, R is referred to as an initiator radical and often as a primary radical.) The second step of the initiation process is the addition of the radical R to a monomer molecule as shown in Eq. (6.4), where RM is the monomer-ended radical containing one monomer unit and an end group R. The rate constant for the reaction is ki. For a vinyl monomer, this second step involves opening the r-bond to form a new radical ... 

Two reactions in series. Consider the following two irreversible reactions,... 

Both upper and lower cycles normally proceed at different rates. But since they represent two reactions in series, the maximum amount of intermediate product is reached when the rates of both successive reactions become equal. Thus, the optimum selectivity, i.e. the maximum amount of formaldehyde will be reached when ... 

For these reasons, societal needs are increasingly being formulated that call for new processes to avoid or sharply reduce the handling of hazardous chemicals. As an example, consider the manufacture of ethylene glycol, the principal ingredient of antifreeze. Ethylene glycol is produced commonly by two reactions in series ... 

Modeling of process equipment often requires a careful inspection of start-up problems that lead to differential equations of initial value type, for example, modeling of chemical kinetics in a batch reactor and modeling of a plug flow reactor. For the first example with two reactions in series... 

The plug flow reactor also gives rise to an IVP. If we write down a mass balance equation for the thin element at the position z with a thickness of Az (see the control volume of Fig. 1.1b) and then allow the element thickness to approach zero, we obtain the following equations for the two reactions in series ... 

A sequence of chemical reactions with a first-order kinetics is the archetype of chain processes in which the concepts of intermediate species, transition state, and activation energy barrier find a thorough application. This is for saying that the study of such a system with this case study of two reactions in series is of paramount importance. 

In many practical cases, several reaction pathways are possible, and we may even have a complicated network of parallel and serial reactions. Simple cases are two parallel reactions A —> B and A —> C, or two reactions in series (A B C). For consecutive reactions, the selectivity depends on the ratio of the rate constants as well as on the reaction progress and conversion of the reactant. 

Simple combinations of reactor elements can be solved direc tly. Figure 23-8, for instance, shows two CSTRs in series and with recycle through a PFR. The material balances with an /i-order reaction / = /cC are... 

Part (c) in Example 15.15 illustrates an interesting point. It may not be possible to achieve maximum mixedness in a particular physical system. Two tanks in series—even though they are perfectly mixed individually—cannot achieve the maximum mixedness limit that is possible with the residence time distribution of two tanks in series. There exists a reactor (albeit semi-hypothetical) that has the same residence time distribution but that gives lower conversion for a second-order reaction than two perfectly mixed CSTRs in series. The next section describes such a reactor. When the physical configuration is known, as in part (c) above, it may provide a closer bound on conversion than provided by the maximum mixed reactor described in the next section. 

The kinetics observed are typical of two first-order reactions in series. 

A reaction network could then consist of two steps in series in which CH4 forms HCHO, which subsequently oxidizes to CO, together with a third step in parallel in which CH4... 

Although this example comprises two reactions in a sequence, many reactions involve a vast series of steps. Some radioactive decay routes, for example, have as many as a dozen species involved in a sequence before terminating with an eventual product. 

If the two steps of first-order reactions in series have very different values for their rate constants, we can approximate the overall behavior as follows ... 

Consider a two-step first-order irreversible reactions in series... 

From Fig. E17.1 we see that two steps in series are involved—mass transfer of oxygen to the surface followed by reaction at the surface of the particle. 

While reactions in series influence the rate of conversion of reactants and intermediates, parallel reactions may affect the selectivity towards different possible products of the overall process. This would correspond to the situation of having two (or more) drains in the bathtub. Each drain symbolizes a different final product of the chemical reaction,... 

The continuous-flow set-up can use either a reaction, similar to semi-batch, to remove the oxygen transferred or two reactors in series. The ozone or oxygen is removed from the liquid in the first reactor by stripping or vacuum degassing and then it flows into the absorber. After having passed through the absorber, the liquid can be recycled or discharged. More information on full-scale application of this method in municipal waste water treatment plants can be found in Redmon (1983) and ASCE (1991). 

When reactions in series are considered, it is not possible to draw any very satisfactory conclusions without working out the product distribution completely for each of the basic reactor types. The general case in which the reactions are of arbitrary order is more complex than for parallel reactions. Only the case of two first-order reactions will therefore be considered ... 

Reaction mixtures encountered in industrial practice often show complex behavior and the overall reaction rate comprises several individual reactions, forming a multiple reaction scheme. There are two basic reaction schemes allowing the construction of more complex ones [5]. The consecutive reactions are also called reaction in series ... 

Wu and Gschwend (1986) reviewed and evaluated several kinetic models to investigate sorption kinetics of hydrophobic organic substances on sediments and soils. They evaluated a first-order model (one-box) where the reaction is evaluated with one rate coefficient (k) as well as a two-site model (two-box) whereby there are two classes of sorbing sites, two chemical reactions in series, or a sorbent with easily accessible sites and difficultly accessible sites. Unfortunately, the latter model has three independent fitting parameters kx, the exchange rate from the solution to the first (accessible sites) box k2, the exchange rate from the first box to the... 

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