Rate constant determination, integral method

A reading of Section 2.2 shows that all of the methods for determining reaction order can lead also to estimates of the rate constant, and very commonly the order and rate constant are determined concurrently. However, the integrated rate equations are the most widely used means for rate constant determination. These equations can be solved analytically, graphically, or by least-squares regression analysis. 

Ea s were also determined by the integral conversion method (17). This method does not require assumption of order or determination of rate constants. The integral conversion method may have limited usefulness since the values obtained did not always agree with the Efl values obtained by the Arrhenius equation of the 0—, 1st- or 2nd-order constants. 

Indices 1 and 2 denote the catalyzed and non-catalyzed reactions, respectively. If rate constant k2 and orders m2 and n2 have been determined Eq. (3) can be solved by an iterative integration method. 

Values for kj and kjj are assumed and the above equations are integrated subject to the initial conditions that a = 2, b = 0 at t = 0. The integration gives the model predictions amodel(j) and bmodel(j). The random search technique is used to determine optimal values for the rate constants based on minimization of and S. The following program fragment shows the method used to adjust kj and kjj during the random search. The specific version shown is used to adjust kj based on the minimization of S, and those instructions concerned with the minimization of S appear as comments. 

ILLUSTRATION 3.2 USE OF A GRAPHICAL INTEGRAL METHOD FOR DETERMINING THE RATE CONSTANT FOR A CLASS II SECOND-ORDER REACTION... 

Use a graphical integral method to determine the order of the reaction and the reaction rate constant. 

Integral Methods for the Analysis of Kinetic Data—Numerical Procedures. While the graphical procedures discussed in the previous section are perhaps the most practical and useful of the simple methods for determining rate constants, a number of simple numerical procedures exist for accomplishing this task. 

ILLUSTRATION 3.4 USE OF GUGGENHEIM S METHOD AND A NUMERICAL INTEGRAL PROCEDURE TO DETERMINE THE RATE CONSTANT FOR THE HYDRATION OF ISOBUTENE IN HYDROCHLORIC ACID SOLUTION... 

Determine the reaction order and rate constant for the reaction by both differential and integral methods of analysis. For orders other than one, C0 will be needed. If so, incorporate this term into the rate constant. 

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. 

It is also possible to use integral methods to determine the concentration dependence of the reaction rate expression and the kinetic parameters involved. In using such approaches one again requires a knowledge of the equilibrium constant for use with one of the integrated forms developed in Section 5.1.1. 

The first point to be established in any experimental study is that one is dealing with parallel reactions and not with reactions between the products and the original reactants or with one another. One then uses data on the product distribution to determine relative values of the rate constants, employing the relations developed in Section 5.2.1. For simple parallel reactions one then uses either the differential or integral methods developed in Section 3.3 in analysis of the data. 

If one monitors the rate of disappearance of the original reactant species the general differential and integral approaches outlined in Section 3.3 may be used to determine the rate expression for the initial reaction in the sequence. Once this expression is known one of several other methods for determining either absolute or relative values of the rate constants for subsequent reactions may be used. 

Use (a) the differential method and (b) the integral method to determine the reaction order, and the value of the rate constant. Comment on the results obtained by the two methods. 

However, the average rates calculated by concentration versus time plots are not accurate. Even the values obtained as instantaneous rates by drawing tangents are subject to much error. Therefore, this method is not suitable for the determination of order of a reaction as well as the value of the rate constant. It is best to find a method where concentration and time can be substituted directly to determine the reaction orders. This could be achieved by integrating the differential rate equation. 

In this method, the different rate equations in their integrated forms (given in Table 1) are used. The amount of reactant a - x or product x at different time intervals t is first experimentally determined. Then the values of x, a-x and time are introduced into the different rate equations and the value of rate constant k is calculated at different time intervals. The equation which gives the constant value of rate constant indicates the order of reaction. For example, the values of rate constants at different time intervals are same in equation... 

We see that the half-life is always inversely proportional to k and that its dependence on [A]o depends on the reaction order. Thereby the method can be used to determine both the rate constant and the reaction order, even for reactions with noninteger reaction order. Similar to the integral method, the half-life method can be used if concentration data for the reactant are available as a function of time, preferably over several half-lives. Alternatively the half-life can be determined for different initial concentrations in several subsequent experiments. 

Initial Rate Method. Using integrated equations like Eqs. (2.5), (2.6), or (2.7) to directly determine a rate law and rate constants is risky. This is particularly true if secondary or reverse reactions are important in equations like (2.5) and (2.6). One sound option is to establish these equations directly using initial rates (Skopp, 1986). 

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