Series of first-order reactions

Schematic representation of the time dependence of the concentration of the first intermediate in a series of first-order reactions. Initial intermediate concentration is nonzero.

If each type of sulfur compound is removed by a reaction that was first-order with respect to sulfur concentration, the first-order reaction rate would gradually, and continually, decrease as the more reactive sulfur compounds in the mix became depleted. The more stable sulfur species would remain and the residuum would contain the more difficult-to-remove sulfur compounds. This sequence of events will, presumably lead to an apparent second-order rate equation which is, in fact, a compilation of many consecutive first-order reactions of continually decreasing rate constant. Indeed, the desulfurization of model sulfur-containing compounds exhibits first-order kinetics, and the concept that the residuum consists of a series of first-order reactions of decreasing rate constant leading to an overall second-order effect has been found to be acceptable. 

Let us begin simply Consider a series of first-order reactions as in fig. 5.1, which shows an unbranched chain of reversible reactions. We shall not be restricted to first-order reactions but can learn a lot from this example. Let there be an influx of ko molecules of Xi and an outflow of molecules of Xs per unit time. We assume that the reaction proceeds from left to right and hence the Gibbs free energy change for each step and for the overall reaction in that direction is negative. The mass action law for the kinetic equations, say that of X2, is... 

The disappearance of A is the result of two reactions that represent a series of first-order reactions that have rate constants ki and k. Therefore, the rate law can be written as... 

Series of First-Order Reections The phenyl rearrangement (reaction 4, Scheme 7.2) proceeds through an intermediate rather than a transition state. In this case, the unimolecular reaction becomes a series of first-order reactions ... 

The phenyl rearrangement is a series of first-order reactions described by Eq. (7.31), for which the steady-state rate constant was derived in Section 13.2.2. 

Displacement of oxalate from [Pt(ox)2] by chloride takes place by a series of first-order reactions in the pH range 5—8. In the proposed reaction sequence (6)... 

A similar series of first order reactions with a cooperative initiation is... 

Another very common kinetic scheme involves a series of first-order reactions, leading first to the formation of an intermediate B, which subsequently reacts to give the final product... 

Numerical calculations are the easiest way to determine the performance of CSTRs in series. Simply analyze them one at a time, beginning at the inlet. However, there is a neat analytical solution for the special case of first-order reactions. The outlet concentration from the nth reactor in the series of CSTRs is... 

It is readily apparent that equation 8.3.21 reduces to the basic design equation (equation 8.3.7) when steady-state conditions prevail. Under the presumptions that CA in undergoes a step change at time zero and that the system is isothermal, equation 8.3.21 has been solved for various reaction rate expressions. In the case of first-order reactions, solutions are available for both multiple identical CSTR s in series and individual CSTR s (12). In the case of a first-order irreversible reaction in a single CSTR, equation 8.3.21 becomes... 

The optimum size ratio for two mixed flow reactors in series is found in general to be dependent on the kinetics of the reaction and on the conversion level. For the special case of first-order reactions equal-size reactors are best for reaction orders n > 1 the smaller reactor should come first for n < 1 the larger should come first (see Problem 6.3). However, Szepe and Levenspiel (1964) show that the advantage of the minimum size system over the equal-size system is quite small, only a few percent at most. Hence, overall economic consideration would nearly always recommend using equal-size units. 

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 ... 

A series of first order irreversible reactions is one in which an intermediate is formed that can then further react. A generalized series reaction is... 

With two electrocatalytic steps in series, the concentration of the intermediate B (Eq. 70) goes through a maximum with time (or space-time for a flow reactor). Solution of the kinetic equations for each species (60) yields for the simple case of first-order reactions... 

A plot of the conversion as a function of the number of reactors in series for first-order reaction is shown in Figure 4-4 for various values of the Damkohlt... 

The conventional data analysis involves the fitting of data to an equation describing the time dependence of the reaction, leading to the best estimates for the constants defining the equations. Analytical solutions to most simple reaction sequences can be obtained (7, 5, 63). Solutions of differential equations describing the series of first-order (or pseudo-first-order) reactions will always be a sum of exponential terms [Eq. (22)]. Thus for a single exponential, the fitting process provides the amplitude (A), the rate of reaction (X), and the end point (C)... 

Such a sequence is known as series or consecutive reactions. In this case, B is known as an intermediate because it is not the final product. A similar situation is very common in nuclear chemistry where a nuclide decays to a daughter nuclide that is also radioactive and undergoes decay (see Chapter 9). For simpficity, only the case of first-order reactions will be discussed. 

Program to plot dimensionless E-curve for tanks in series model Fvmction subroutine to calculate factorial of n Program to plot dimensionless E-curve for laminar flow reactor Program to calculate conversion of first-order reaction in non-ideal reactor using various models... 

More terms of the series are usually not justifiable because the higher moments cannot be evaluated with sufficient accuracy from e)meri-mental data. A comparison of the fourth-order GC with other distributions is shown in Fig. 23-12, along with calculated segregated conversions of a first-order reaction. In this case, the GC is the best fit to the original. At large variances the finite value of the ordinate at... 

Concentration-time curves. Much of Sections 3.1 and 3.2 was devoted to mathematical techniques for describing or simulating concentration as a function of time. Experimental concentration-time curves for reactants, intermediates, and products can be compared with computed curves for reasonable kinetic schemes. Absolute concentrations are most useful, but even instrument responses (such as absorbances) are very helpful. One hopes to identify characteristic features such as the formation and decay of intermediates, approach to an equilibrium state, induction periods, an autocatalytic growth phase, or simple kinetic behavior of certain phases of the reaction. Recall, for example, that for a series first-order reaction scheme, the loss of the initial reactant is simple first-order. Approximations to simple behavior may suggest justifiable mathematical assumptions that can simplify the quantitative description. 

First-order kinetics. Consider a first-order reaction studied with a series of initial concentrations, as depicted here. Show that the initial rate tangents come to a common intercept t" on the x axis, and find an expression for what time this is. 

A solution to Equation (8.12) together with its boundary conditions gives a r, z) at every point in the reactor. An analytical solution is possible for the special case of a first-order reaction, but the resulting infinite series is cumbersome to evaluate. In practice, numerical methods are necessary. 

For a first-order reaction n=l, and the dimensionless time is given by k t. Make a series of runs with different initial concentrations and compare the results, plotting the variables in both dimensional and dimensionless terms. 

Tanks-in-series reactor configurations provide a means of approaching the conversion of a tubular reactor. In modelling, they are employed for describing axial mixing in non-ideal tubular reactors. Residence time distributions, as measured by tracers, can be used to characterise reactors, to establish models and to calculate conversions for first-order reactions. 

Referring back to the rate equation for a first-order reaction (Equation A1.2), we have a differential equation for which the derivative of the variable ([S]) is proportional to the variable itself. Such a system can be described by an infinite series with respect to time ... 

For first-order and pseudo first-order reactions of the series type several methods exist for determining ratios of rate constants. We will consider a quick estimation technique and then describe a more accurate method for handling systems whose kinetics are represented by equation 5.3.2. 

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