Zero-order integrated rate equation

The reactant concentration-time curve for a typical zero-order reaction, A products, is shown in Fig. 1.1(a). The rate equation for a zero-order reaction can be expressed as 

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The zero-order integrated rate law in Equation 13.17 is also in the form of an equation for a straight line. A plot of the concentration of the reactant as a function of time yields a straight line with a slope of -k and an intercept of [A]q, as shown in Figure 13.10 . 

Finally, although rare, we mention the occurrence of zero-order reactions. The special case of a pseudo-zero order reaction arises if a reactant is present in large excess, and the reaction does not noticeably change the concentration of the reactant. The differential and integral rate equations for a zero-order reaction R —> P are... 

The differential rate equations, corresponding integral rate equations and rate constants for various reactions (having order zero to three) under different sets of conditions are summarized in Table 1.1. 

You must choose the form of the rate-law expression or the integrated rate equation —zero, first, or second order—that is appropriate to the order of the reaction. These are summarized in Table 16-2. One of the following usually helps you decide. 

For a zero-order reaction, we can rearrange the integrated rate equation... 

Integration between the limits of c = c° when r = 0 and c = c when t = t gives the integrated zero-order rate equation. 

We can reach two useful conclusions from the forms of these equations First, the plots of these integrated equations can be made with data on concentration ratios rather than absolute concentrations second, a first-order (or pseudo-first-order) rate constant can be evaluated without knowing any absolute concentration, whereas zero-order and second-order rate constants require for their evaluation knowledge of an absolute concentration at some point in the data treatment process. This second conclusion is obviously related to the units of the rate constants of the several orders. 

Initially, it could be postulated that the reaction could be zero order, first order or second order in the concentration of A and B. However, given that all the reaction stoichiometric coefficients are unity, and the initial reaction mixture has equimolar amounts of A and B, it seems sensible to first try to model the kinetics in terms of the concentration of A. This is because, in this case, the reaction proceeds with the same rate of change of moles for the two reactants. Thus, it could be postulated that the reaction could be zero order, first order or second order in the concentration of A. In principle, there are many other possibilities. Substituting the appropriate kinetic expression into Equation 5.47 and integrating gives the expressions in Table 5.5 ... 

Integration of Equation 3 results in a first order rate decay equation. Equation 4 describes the pesticide concentration in the wastewater which will decrease exponentially, but never reach zero. 

Differential rate equations like these are not much use to the practising chemist, so it is usual to integrate the differential form of the rate equation, shown above, to obtain more useful expressions. This can be carried out as follows for a first-order reaction. In this reaction, compound A reacts to form products. At the start of the reaction (time 0) the concentration of A is equal to a mol IT1, while the concentration of products will be zero (since the reaction has not started). At some later time, t, the concentration of products has increased to % mol IT1 and as a result the concentration of A has fallen to (a — x) mol IT1. This can be represented mathematically as... 

Assume concurrent zero-order and first-order reactions (as the authors did), integrate the rate equation, and determine the best values of the rate constants. 

Many authors propose alternative mathematical treatments for kinetics equations. Some examples are a general approach based on a matrix formulation of the differential kinetic equations (Berberan-Santos Martinho, 1990) spreadsheets in which rate equations are integrated using the simple Euler approximation (Blickensderfer, 1990) a method for the accurate determination of the first-order rate constant (Borderie, Lavabre, Levy Micheau, 1990) a simplification of half-life methods that provides a fast way of determining reaction orders and rate constants (Eberhart Levin, 1991) a general approach to reversible processes, the special cases of which are shown to be equivalent to basic kinetic equations (Simonyi Mayer, 1985) an equation from which zero-, first- and higher order equations can be derived (Tan, Lindenbaum Meltzer, 1994). 

For the inner zone, in which both transport and reaction occurs, the differential equations are those of the first stage, but the boundary conditions arc dC Jdr = 0 at r = 0 and Eq. 4.2-11 at the boundary with the outer zone. This mt el corresponds to that set up by Ausman and Watson, to describe the rate of burning of carbon deposited inside a catalyst particle [8]. Analytical integration of this fairly general two-stage model is only possible for a zero-order, first-order or pseudo-first-order rate law, whereby Eq. 42-8 reduces to... 

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