Integrated rate, zeroth order

Thirdly one needs a drastic step to turn this integral equation into a differential equation. This is the Markov approximation , which comes in two varieties. The first variety consists in replacing ps(t — t) with ps(t). The error is of relative order rc/rm (where l/rm is the unperturbed rate of change due to S s) and of absolute order a2T2/rm. In this approximation one may as well omit the S s in the exponent of (4.13) and the result is the same as (3.19). The second variety takes the zeroth order variation of ps into account by setting ps(t — r) = e T sps(t). The result is the same as was obtained in (3.14) by means of the interaction representation, and the only requirement is arc <[Pg.444]

Beginning with the integrated rate law, derive a general equation for the half-life of a zeroth-order reaction of the type A — Products. How does the length of each half-life compare with the length of the previous one Make the same comparison for first-order and second-order reactions. 

The kinetic equation for homogeneous systems is given by Eq. (7.47). The evolution equation for the zeroth-order moment of the NDF is null, which is due to the fact that the collision integral does not change the number of particles, or, more explicitly, f Cdf = 0. If the rate of change of the particle velocity (i.e. particle acceleration) is a linear function of the particle velocity (i.e. f = a + b ), then the evolution equation for the first-order moments are... 

For a zeroth order reaction the rate of reaction is constant and does not depend of the concentration of reactant. By integration one achieves simply the following expression for the concentration of A as function of time, the parameter k as well as the initial concentration [A]o -... 

This second-order ordinary differential equation given by (16-4), which represents the mass balance for one-dimensional diffusion and chemical reaction, is very simple to integrate. The reactant molar density is a quadratic function of the spatial coordinate rj. Conceptual difficulty arises for zeroth-order kinetics because it is necessary to introduce a critical dimensionless spatial coordinate, ilcriticai. which has the following physically realistic definition. When jcriticai which is a function of the intrapellet Damkohler number, takes on values between 0 and 1, regions within the central core of the catalyst are inaccessible to reactants because the rate of chemical reaction is much faster than the rate of intrapellet diffusion. The thickness of the dimensionless mass transfer boundary layer for reactant A, measured inward from the external surface of the catalyst,... 

If the kinetics are not zeroth-order, then these integral expressions are more tedions to nse than the ones developed earlier in this chapter based on mass transfer across the external snrface of the catalyst. The preferred expressions for the effectiveness factor are summarized below for nth-order irreversible chemical kinetics when the rate law is only a function of the molar density of one reactant ... 

There are a few other simple rate laws, and the integrations of those rate laws follow the same type of steps we used to find integrated rate laws for first- and second-order reactions. Rather than repeat such derivations, they will be left to the student. The discussion will be confined to some more interesting attributes of other rate laws. For example, a reaction following zeroth-order kinetics has a rate law of... 

We can consider equation 20.23 in several equivalent ways. Because the rate of disappearance of reactant A is a constant, a plot of [A] versus time is a straight line, as shown in Figure 20.4. Also, we can integrate equation 20.23 to get the integrated zeroth-order rate law... 

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