First-order reactions mathematics

Linear differential equations with constant coefficients can be solved by a mathematical technique called the Laplace transformation . Systems of zero-order or first-order reactions give rise to differential rate equations of this type, and the Laplaee transformation often provides a simple solution. 

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. 

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. 

The first-order reaction is a special case mathematically. For n --has the exponential form of Equation (1.24) ... 

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 ... 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

A reagent in solution can enhance a mass transfer coefficient in comparison with that of purely physical absorption. The data of Tables 8.1 and 8.2 have been cited. One of the simpler cases that can be analyzed mathematically is that of a pseudo-first order reaction that goes to completion in a liquid film, problem P8.02.01. It appears that the enhancement depends on the specific rate of reaction, the diffusivity, the concentration of the reagent and physical mass transfer coefficient (MTC). These quantities occur in a group called the Hatta number,... 

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. 

At the start of a reaction, the reactants are present in large excess pseudo-first order), and this simplifies the mathematical description of rate to that of a first order reaction, where rate is proportional to the concentration of a reactant or... 

In a first order reaction, the change in concentration of the reactants or products with time is exponential (see Figure 2.8), and this can be illustrated mathematically by integrating and rearranging Equation 2.6 to give ... 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

Mathematical Analysis. Reactions 3 to 5 are first-order or pseudo-first-order reactions. Thus, the pseudo-first-order constant for Reaction 3 is ksE. For brevity we rewrite the two intermediates (E2+ 02H A )2 and (+E—E+ 02H A )g as Ci and C2, respectively. We assume that the light intensity is proportional to C2 during any one run. Thus, dl/dt = dC2/dt and the calculated decay of C2 can be related to k0. 

The model developed here assumes that only unimolecular reactions are involved, but that the first-order reaction velocity constant varies with the fraction extracted. To derive a suitable mathematical relationship between the first-order reaction velocity constant, k> and the fraction extracted, x, one proceeds as follows ... 

Here c(x, t)dx is the concentration of material with index in the slice (x, x + dx) whose rate constant is k(x) K(x, z) describes the interaction of the species. The authors obtain some striking results for uniform systems, as they call those for which K is independent of x (Astarita and Ocone, 1988 Astarita, 1989). Their second-order reaction would imply that each slice reacted with every other, K being a stoichiometric coefficient function. Only if K = S(z -x) would we have a continuum of independent parallel second-order reactions. In spite of the physical objections, the mathematical challenge of setting this up properly remains. Ho and Aris (1987) have shown how not to do it. Astarita and Ocone have shown how to do something a little different and probably more sensible physically. We shall see that it can be done quite generally by having a double-indexed mixture with parallel first-order reactions. The first-order kinetics ensures the individuality of the reactions and the distribution... 

It is useful to examine the consequences of a closed ion source on kinetics measurements. We approach this with a simple mathematical model from which it is possible to make quantitative estimates of the distortion of concentration-time curves due to the ion source residence time. The ion source pressure is normally low enough that flow through it is in the Knudsen regime where all collisions are with the walls, backmixing is complete, and the source can be treated as a continuous stirred tank reactor (CSTR). The isothermal mole balance with a first-order reaction occurring in the source can be written as... 

Remark 1 The mathematical model is an MINLP problem since it has both continuous and binary variables and nonlinear objective function and constraints. The binary variables participate linearly in the objective and logical constraints. Constraints (i), (iv), (vii), and (viii) are linear while the remaining constraints are nonlinear. The nonlinearities in (ii), (iii), and (vi) are of the bilinear type and so are the nonlinearities in (v) due to having first-order reactions. The objective function also features bilinear and trilinear terms. As a result of these nonlinearities, the model is nonconvex and hence its solution will be regarded as a local optimum unless a global optimization algorithm is utilized. 

Consider the exothermic first-order reaction A —> B taking place batchwise at reactor temperature Tr and coolant temperature 7j. The mathematical model describing the system is given by the mass balance on reactant A and the energy balance in the reactor ... 

The mathematical model of the reactor consists of the mass and energy balances written for all the compartments and an energy balance written for the jacket. The mass balance written for the reactant and for a first-order reaction in a generic compartment on the central level holds ... 

PROBLEM 6.4.3. The first-order reaction rate law assumes mathematically that you only exhaust [A] at infinite time is this reasonable ... 

Random thermal degradation can usually be described as a first-order reaction (loss of weight as a parameter) if the decomposition products are volatile. For the mathematical treatment we refer to Van Krevelen et al. (1951), Reich (1963,1967) and Broido (1969). 

For a first-order reaction taking place in a ring-shaped catalyst pellet, the effectiveness factor was already calculated by Gunn [1], That solution, however, is very complex because the mathematical techniques used are not the most suitable with which to solve the equations that arise. Therefore, the following solution derives another expression for the effectiveness factor. 

A. First-order Reactions General Treatment. There has been a considerable amount of work done on the solution of particular and general systems of first-order reactions. All such systems are capable of exact, explicit mathematical solutions. If we consider the most general case of a system of s components Ci, C2,. . . , C in which first-order reactions of the following type may take place between any two components... 

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