Integrated rate expressions

Differential equations can be solved by integration, and this results in linear equations relating [reactant] and time, so that the order and rate constant can be found directly from linear plots. 

In scientific work it is always much easier and more accurate to work with linear relationships. These have an easily defined gradient, and the intercept is accurately determinable. Also, it is much simpler to discriminate between systematic scatter and random scatter using linear plots rather than curves. 

Integrated rate expressions can also be used to demonstrate that the half-life of a reaction varies systematically with [reactant], and so is diagnostic of the reaction order. 

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Our initial experimental results indicated that the kinetic model— first order in liquid phase CO concentration—was the leading candidate. We designed an experimental program specifically for this reaction model. The integrated rate expression (see Appendix for nomenclature) can be written as ... 

First-order kinetics. Show that the first-order integrated rate expression can be written as... 

Equation (8) is the differential rate expression for a first-order reaction. The value of the rate constant, k, could be calculated by determining the slope of the concentration versus time curve at any point and dividing by the concentration at that point. However, the slope of a curved line is difficult to measure accurately, and k can be determined much more easily using integrated rate expressions. 

Equation (11) is the integrated rate expression for a first-order process and can serve as a working equation for solving problems. It is also in the form of the equation of a straight line ... 

In this section we discuss the mathematical forms of the integrated rate expression for a few simple combinations of the component rate expressions. The discussion is limited to reactions that occur isothermally in constant density systems, because this simplifies the mathematics and permits one to focus on the basic principles involved. We will again place a V to the right of certain equation numbers to emphasize that such equations are not general but are restricted to constant volume batch reactors. The use of the extent per unit volume in a constant volume system ( ) will also serve to emphasize this restriction. For constant volume systems,... 

Equations 5.1.5, 5.1.6, and 5.1.8 are alternative methods of characterizing the progress of the reaction in time. However, for use in the analysis of kinetic data, they require an a priori knowledge of the ratio of kx to k x. To determine the individual rate constants, one must either carry out initial rate studies on both the forward and reverse reactions or know the equilibrium constant for the reaction. In the latter connection it is useful to indicate some alternative forms in which the integrated rate expressions may be rewritten using the equilibrium constant, the equilibrium extent of reaction, or equilibrium species concentrations. 

Writing an integrated rate expression for a complicated reaction is difficult because we don t readily know the time-dependent concentration of the intermediate, [B]t. 

The schemes considered are only a few of the variety of combinations of consecutive first-order and second-order reactions possible including reversible and irreversible steps. Exact integrated rate expressions for systems of linked equilibria may be solved with computer programs. Examples other than those we have considered are rarely encountered however except in specific areas such as oscillating reactions or enzyme chemistry, and such complexity is to be avoided if at all possible. 

Caution 1. In the special case where reactants are introduced in their stoichiometric ratio, the integrated rate expression becomes indeterminate and this requires taking limits of quotients for evaluation. This difficulty is avoided if we go back to the original differential rate expression and solve it for this particular reactant ratio. Thus, for the second-order reaction with equal initial concentrations of A and B, or for the reaction... 

Thus the general integrated rate expression of Eq. 2 becomes, with Eq. 3,... 

For second-order kinetics, the integrated rate expression is... 

A] = 0.5[A]0. Substituting into the integrated rate expression, one obtains... 

The experimental data is concentration-time data, and integrated rate expressions give the observed rate constants. The technique can be extended to multi-step equilibria and to equilibrium reactions with consecutive steps. For multi-step equilibria, analysis results in a sum of exponentials, which may or may not reduce to first order. 

But note the graph for the integrated rate expression is the experimental graph of [reactant] versus time. This only happens for a zero order reaction, and it is immediately obvious from the experimental plot that reaction is zero order. 

One problem for kineticists is that only a relatively slight increase in complexity of the kinetic scheme results in differential equations which cannot be integrated in a straightforward manner to give a manageable analytical expression. When this happens the differential equations have to be solved by either numerical integration or computer simulation. This is a mathematical limitation of the use of integrated rate expressions which is not apparent in the kinetic scheme. Typical schemes which are mathematically complex are... 

The reaction is predicted to be first order in RX. Nonetheless, the observed rate cannot be written as rate = hs[RX] because, if the mechanism is correct, the rate at any time will also depend on [X ] and [OH ] in both cases. A plot of loge[RX] would therefore be non-linear, and the first order integrated rate expression in [RX] cannot be used. 

Total pressure/time data given use the integrated rate expression method. But first have to convert to partial pressure of reactant remaining. 

Since the mechanistic rate = k2[NH3]actual, reaction is second order. The data quoted is concentration/time data, and so the integrated rate expression is appropriate, and a plot of l/[NH4]totai versus time should be linear with slope kQbs-... 

The Use of Integral Rate Expressions We have already demonstrated this method in solving problems for reactions of various orders. This method can be used either analytically or graphically. In the analytical method, we assume a certain order for the reaction and calculate the rate constants from the given data. The constancy of the ic-values obtained suggests that the assumed order is correct. If the ic-values obtained are not constant, we assume a different order for the reaction and again calculate the ic-values using the new rate expression and see if ic is constant. 

Integrate rate expression for plug flow analytically (not graphically or by... 

Since we may consider k[03] to be constant after initial saturation is achieved, the integrated rate expression is that in Equation 15. 

Kinetic theory tells us that the extent of completeness of a reaction is expressed by the integrated rate expression ... 

Reaction (32) is probably the most well studied radical recombination reaction with data existing from 200-1700 K and 30-2 x 10 Pa. In contrast to the experiments described in previous sections, second order reactions, such as radical recombinations require a knowledge of absolute concentrations in order to determine rate coefficients. The integrated rate expression for a second order decay is ... 

Note General reaction types or conditions that correspond to the differential rate equations are given parenthetically. Some reactions are irreversible (denoted by — ) and others reversible (denoted by double arrows). Note that the rate constant, k is always positive. In the integrated rate expressions the concentration of A = Ao, at r = 0, and A = AJ2 at half-time (ti/i). A denotes the equilibrium, mineral saturation or steady state concentration of species A. 

Table 2.2 is a summary of the differential and integrated forms of some simple rate laws. Also listed are expressions for the reaction half-times corresponding to several of the integrated rate expressions. These are the times at which the concentration of a reactant is half its initial value. 

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