Rate equations, chemical integrals

A general method is presented for the development of chemical reaction rate equations from integral reactor and single-sample batch reactor data such as are obtained in process development studies. Following the scope of the method, three earlier foundation stones upon which the method rests, the method itself, and a simple illustration are presented. 

For fast irreversible chemical reactions, therefore, the principles of rigorous absorber design can be applied by first estabhshing the effects of the chemical reaction on /cl and then employing the appropriate material-balance and rate equations in Eq. (14-71) to perform the integration to compute the required height of packing. 

A final comment on the interpretation of stochastic simulations We are so accustomed to writing continuous functions—differential and integrated rate equations, commonly called deterministic rate equations—that our first impulse on viewing these stochastic calculations is to interpret them as approximations to the familiar continuous functions. However, we have got this the wrong way around. On a molecular level, events are discrete, not continuous. The continuous functions work so well for us only because we do experiments on veiy large numbers of molecules (typically 10 -10 ). If we could experiment with very much smaller numbers of molecules, we would find that it is the continuous functions that are approximations to the stochastic results. Gillespie has developed the stochastic theory of chemical kinetics without dependence on the deterministic rate equations. 

Sections 3.1 and 3.2 considered this problem Given a complex kinetic scheme, write the differential rate equations find the integrated rate equations or the concentration-time dependence of reactants, intermediates, and products and obtain estimates of the rate constants from experimental data. Little was said, however, about how the kinetic scheme is to be selected. This subject might be dismissed by stating that one makes use of experimental observations combined with chemical intuition to postulate a reasonable kinetic scheme but this is not veiy helpful, so some amplification is provided here. 

Throughout Section 11.3, balanced chemical equations are written in such a way that the coefficient of die reactant is 1. In general, if the coefficient of the reactant is a, where a may be 2 or 3 or..., then k in each integrated rate equation must be replaced by the product ak. (See Problem 101.)... 

Models of population growth are analogous to chemical reaction rate equations. In the model developed by Malthus in 1798, the rate of change of the population N of Earth is dN/dt = births — deaths. The numbers of births and deaths are proportional to the population, with proportionality constants b and d. Derive the integrated rate law for population change. How well does it fit the approximate data for the population of Earth over time given below ... 

The concentrations of each reactant and product will vary during the course of a chemical reaction. The so-called integrated rate equation relates the amounts of reactant remaining in solution during a reaction with the time elapsing since the reaction started. The integrated rate equation has a different form according to the order of reaction. 

The reaction set was numerically modeled using the computer program CHEMK (9) written by G. Z. Whitten and J. P. Meyer and modified by A. Baldwin of SRI to run on a MINC laboratory computer. CHEMK numerically Integrates a defined set of chemical rate equations to reproduce chemical concentration as a function of time. Equilibria can be modeled by Including forward and reverse reaction steps. Forward and reverse reaction rate... 

Differential rate equations such as Eq. 2.38 are often used in integrated form, so that the rate constant can be evaluated simply by measuring concentrations rather than rates at selected times t. Integration of the rate equation, however, may be difficult or may give an unwieldy result. In this book, we are concerned primarily with reaction rates as such, and the reader is therefore referred to texts devoted to chemical kinetics30 for information on integrated rate equations. 

An overall conversion rate may depend on rates of mass transfer between phases as well as chemical rates. In the simplest case, mass transfer and chemical transformation occur in series advantage is taken of the equality of these two rates at steady state conditions to eliminate interfacial concentrations from the rate equations and thus to permit integration. Item 8 of Table 17.2 is an example. 

However, the reverse process, in going from speed to distance, involves integration of the rate equation (6.2). In chemistry, the concept of rate is central to an understanding of chemical kinetics, in which we have to deal with analogous rate equations which typically involve the rate of change of concentration, rather than the rate of change of distance. For example, in a first-order chemical reaction, where the rate of loss of the reactant is proportional to the concentration of the reactant, the rate equation takes the form ... 

The Integration of Chemical Rate Equations on a Vector Computer... 

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