First Order Reversible Reactions

FIG. 23-17 Multiple steady states of CSTRs, stable and unstable, adiabatic except the last item, (a) First-order reaction, A and C stable, B unstable, A is no good for a reactor, the dashed line is of a reversible reaction, (h) One, two, or three steady states depending on the combination Cj, Ty). (c) The reactions A B C, with five steady states, points 1, 3, and 5 stable, (d) Isothermal operation with the rate equation = 0 /(1 -I- C y = (C o Cy/t. 

Consider the reversible first order reaction A R. It is possible to determine the minimum reactor volume at the optimum temperature Tgp( that is required to obtain a fractional conversion X, if the feed is pure A with a volumetric flowrate of u. A material balance for a CESTR is... 

To take the inverse Laplace transform means to reverse the process of taking the transform, and for this purpose a table of transforms is valuable. To illustrate, we consider a simple first-order reaction, whose differential rate equation is... 

One further system will be solved by the transform method. Scheme XV constitutes two consecutive reversible first-order reactions. 

Systems of reversible first-order reactions lead to sets of simultaneous linear differential equations with constant coefficients. A solution may be obtained by means of a matrix formulation that is widely used in quantum mechanics and vibrational... 

First, we consider Scheme I, a single reversible first-order reaction Eyring et al." treated this case. 

Any combination of first-order reactions can be simulated by extension of this procedure. Reversible reactions add only the feature that reacted species can be regenerated from their products. Second-order reactions introduce a new factor, for now two molecules must each be independently selected in order that reaction occur in the real situation the two molecules are in independent motion, and their collision must take place to cause reaction. We load the appropriate numbers of molecules into each of two grids. Now randomly select from the first grid, and then, separately, randomly select from the second grid. If in both selections a molecule exists at the respective selected sites, then reaction occurs and both are crossed out if only one of the two selections results in selection of a molecule, no reaction occurs. (Of course, if pseudo-first-order conditions apply, a second-order reaction can be handled just as is a first-order reaction.)... 

An interesting method, which also makes use of the concentration data of reaction components measured in the course of a complex reaction and which yields the values of relative rate constants, was worked out by Wei and Prater (28). It is an elegant procedure for solving the kinetics of systems with an arbitrary number of reversible first-order reactions the cases with some irreversible steps can be solved as well (28-30). Despite its sophisticated mathematical procedure, it does not require excessive experimental measurements. The use of this method in heterogeneous catalysis is restricted to the cases which can be transformed to a system of first-order reactions, e.g. when from the rate equations it is possible to factor out a function which is common to all the equations, so that first-order kinetics results. 

Reversible reactions. Consider an experiment on the reversible first-order reaction A =i P in which two identical solutions were prepared at different times. The age separation between the two is denoted as t. The older solution is placed in the reference beam of a... 

This equation is known as the rate law for the reaction. The concentration of a reactant is described by A cL4/df is the rate of change of A. The units of the rate constant, represented by k, depend on the units of the concentrations and on the values of m, n, and p. The parameters m, n, and p represent the order of the reaction with respect to A, B, and C, respectively. The exponents do not have to be integers in an empirical rate law. The order of the overall reaction is the sum of the exponents (m, n, and p) in the rate law. For non-reversible first-order reactions the scale time, tau, which was introduced in Chapter 4, is simply 1 /k. The scale time for second-and third-order reactions is a bit more difficult to assess in general terms because, among other reasons, it depends on what reactant is considered. 

Determine the maximum batch reactor yield of B for a reversible, first-order reaction ... 

The numerator of Equation (10.13) is the expected form for a reversible, first-order reaction. The denominator shows that the reaction rate is retarded by all species that are adsorbed. This reflects competition for sites. Inerts can also compete for sites. Thus, the version of Equation (10.13) that applies when adsorbable inerts are present is... 

The overall reaction between CO2 and GMA was assumed to consist of two elementary reactions such as a reversible reaction of GMA and catalyst to form an intermediate and an irreversible reaction of this intermediate and carbon dioxide to form five-membered cyclic carbonate. Absorption data for CO2 in the solution at 101.3 N/m were interpreted to obtain pseudo-first-order reaction rate constant, which was used to obtain the elementary reaction rate constants. The effects of the solubility parameter of solvent on lc2/k and IC3 were explained using the solvent polarity. 

It will be of interest to present mathematically the picture of the course of consecutive reactions. In the simplest case the substance A considered in the present example undergoes a first-order reaction to yield C the reverse reactions are neglected. The reaction occurring in two first-order steps can now be written as ... 

We have examined the effects of concentration, temperature, solvent and added electrolyte on the kinetics of this structural interconversion. In all instances, the kinetics are well described by the rate law for a reversible first-order reaction [Equation 1] ... 

This technique is readily adaptable for use with the generalized additive physical approach discussed in Section 3.3.3.2. It is applicable to systems that give apparent first-order rate constants. These include not only simple first-order irreversible reactions but also irreversible first-order reactions in parallel and reversible reactions that are first-order in both the forward and reverse directions. The technique provides an example of the advantages that can be obtained by careful planning of kinetics experiments instead of allowing the experimental design to be dictated entirely by laboratory convention and experimental convenience. 

For all reaction orders greater than unity, the appropriate order is plug flow, small CSTR, large CSTR. In the case of reaction orders less than unity, the reverse order should be employed. For a first-order reaction the conversion will be independent of the arrangement of the various reactors. 

For the reversible first order reaction followed by a first reaction... 

These consecutive reversible first order reactions, 1 3... 

The cis-trans isomerization of 1,2-dimethylcyclopropane at 453 C is a reversible first order reaction. The percentage of cis is shown as a function of t in sec in the table. Find the constants of the rate equation. 

The tabulated data of rate versus concentration refer to a reaction that is believed of the second order in the forward direction and first order in reverse. Initial concentrations of the two reactants were 1.2 mol/cuft each and there was no product to start with, [a) Find the specific rates (b) How long does it take to convert 60% of the reactants ... 

In an experiment at 25°C, starting with pure compound C at 0.02250 mols/liter, the concentration of benzaldehyde was found to be 0.01025 mol/liter after 53.8 hr. The equilibrium constant is 0.424. The reaction is believed second order in the forward direction and first order in reverse. Find the specific rate, x = change in concentration of C C = C0-x = 0.0225 - x A = B = x... 

Since less conversion is obtained than predicted by a first order reaction, it may be that the reaction is reversible and equilibrium is being approached. 

A reversible first order reaction, A < > B, is conducted in a CSTR. Initial concentration in the tank is Ca0 = 0, feed concentration is Caf = 1,... 

SAQ 8,22 A simple first-order reaction has a forward rate constant of 120 s 1 while the rate constant for the back reaction is 0.1 s F Calculate the equilibrium constant K of this reversible reaction by invoking the principle of microscopic reversibility. 

Figure 8.22 Kinetic graph for a reversible first-order reaction with the axes for an integrated rate equation ln([A], — A fcq ) (as 3/ ) against time (as V). The gradient is —5.26 x 10 3 min 1...

SAQ 8.23 Consider a reversible first-order reaction. Its integrated rate equation is given by Equation (8.50). People with poor mathematical skills often say (erroneously ) that taking away the infinity reading from both top and bottom is a waste of time because the two infinity concentration terms will cancel. Show that the infinity terms cannot be cancelled in this way take [A](eq) = 0.4 moldrrT3, [A]o = 1 moldrrT3 and [A]t = 0.7 mol dm 3. 

The reverse reaction, the retro-Diels-Alder reaction is a clean first order reaction, both in the gaseous and condensed phase. 

This equation resembles (1.26) but includes [A], the concentration of A at equilibrium, which is not now equal to zero. The ratio of rate constants, Atj/A , = K, the so-called equilibrium constant, can be determined independently from equilibrium constant measurements. The value of k, or the relaxation time or half-life for (1.47), will all be independent of the direction from which the equilibrium is approached, that is, of whether one starts with pure A or X or even a nonequilibrium mixture of the two. A first-order reaction that hides concurrent first-order reactions (Sec. 1.4.2) can apply to reversible reactions also. 

The reversible first-order reaction (1.47) can be converted into an irreversible A X process by scavenging X rapidly and preventing its return to A. Thus the intramolecular reversible electron transfer in modified myoglobin (Sec. 5.9)... 

Conditions OR = const. reverse rate zero [X] = [X]o at f = 0 Reversible first-order reactions ... 

Reversible first-order reactions are given by Eq. (5.15) of Table 5.1. Equation (5.15c) can be rewritten in a more elegant form by considering that at equilibrium t = °°) the net rate is equal to zero, that is... 

Since all of the above-mentioned interconversion reactions are reversible, any kinetic analysis is difficult. In particular, this holds for the reaction Sg - Sy since the backward reaction Sy -+ Sg is much faster and, therefore, cannot be neglected even in the early stages of the forward reaction. The observation that the equilibrium is reached by first order kinetics (the half-life is independent of the initial Sg concentration) does not necessarily indicate that the single steps Sg Sy and Sg Sg are first order reactions. In fact, no definite conclusions about the reaction order of these elementary steps are possible at the present time. The reaction order of 1.5 of the Sy decomposition supports this view. Furthermore, the measured overall activation energy of 95 kJ/mol, obtained with the assumption of first order kinetics, must be a function of the true activation energies of the forward and backward reactions. The value found should therefore be interpreted with caution. 

Fig. 10. Conversion in an exothermic reversible first-order reaction carried out in (a) a plug-flow reactor and (b) a continuous stirred tank reactor.

When reversible steps occur in a reaction scheme, distinctions between consecutive and parallel reactions cannot always be made. For example, the consecutive first-order reactions... 

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