Kinetics Review

Adapted from an Exercise developed by the Physical Chemists at the University of Kansas by J. M. McCormick

Last Update: October 15, 2008

The general expression of the rate law for the reaction aA + bB + gC → products can be written as Eqn. 1, where k is the reaction’s rate constant and a, b and c are the orders of the reaction with respect to the reactants A, B and C, respectively.  The order of reaction for each reactant is not necessarily equal to its stoichiometric coefficient,¹ and in general will not be.  The order for each reactant derives from the mechanism by which the reaction occurs, which can not be determined by knowing only the identities of the reactants and products; it must be determined experimentally.

The general approach to the determination of a rate law is to use the isolation method in which the initial concentration of one reactant, let’s say B, whose initial concentration is [B]₀, is much less than the initial concentrations of A, or C ([A]₀ and [C]₀, respectively). In this scenario, [A] and [C] will change only very little as the reaction proceeds, while B will be completely used up. It can then be assumed that the concentrations of A and C at any time, t, are equal to their initial values, or [A]_t = [A]₀ and [C]_t = [C]₀. Since only [B] varies appreciably with time during the course of the reaction, Eqn. 1 can be rewritten as Eqn. 2, where the constant k_AC equals k [A]₀^a[C]₀^c. The order with respect to B is then determined graphically using the integrated rate laws. Once the order with respect to B is known, the process would be repeated by making the [B]₀ large and the concentration of one of the other reactants small until the order of each reactant is found. The rate constant for the reaction is then found from any of the rate data.¹

In some cases, it is not possible to change the concentrations to satisfy the conditions set by the isolation method.  For example, if we were following the reaction by monitoring the absorbance of species B, we would not be able to increase [B] without Beer's Law being no longer applicable.  Since the absorbance would be no longer proportional to the concentration, we would not have an accurate measure of the [B] and therefore an accurate measure of the rate.  In this case, we leave [B] at whatever concentration is needed for us to measure it and make the [A] and the [C] much larger than [B].  We then make systematic changes in the concentration of one reactant (let's say A), while holding the concentration of the other constant (C, in this case).  When we do this the rate will change, if a reactant has a non-zero order, because the constant k_AC changes.  Since k_AC = k [A]₀^a[C]₀^c, we can obtain an equation that we can graph and fit to extract the order with A by taking the log of both sides to give Eqn. 3, where [A]_0i is the initial concentration of A for a particular kinetics run and k_ACi is the measured rate constant for that run (obtained using the integrated rate law that was used to find the order with respect to B).  As long as we only change [A]₀ (leave [C]₀ constant), a graph of logk_ACi as a function of [A]_0i will be a straight line. If the order with respect to A is 0, then this graph is a horizontal line; otherwise it has slope of a (the order with respect to A) and an intercept of k+clog[C]_0i.   If the order with respect to C is 0 (or there is no third reactant), then we would have been able to obtain the overall rate constant, k, from the graph of logk_ACi as a function of [A]_0i. Once the order with respect to A is found, the process is repeated, but now varying C, to determine the order with respect to C.  The value of the overall rate constant, k, would then be found from the intercepts and the known initial concentrations of A and C.

There is a special case of this rate law when the rate of reaction is independent of the concentration of B (the order of the reaction with respect to B is 0; i. e., b = 0).² In this case Eqn. 2 simplifies to Eqn. 4. The [B] at any time, [B]_t, can be easily obtained by the integration of Eqn. 4 to give Eqn. 5. So a graph of [B]_t as a function of time will yield a straight line with an intercept of [B]₀ and a slope of -k_AC. As a check of whether the reaction is truly independent of B, the experiment can be repeated with different [B]₀, keeping [A] and [C] constant, which should give graphs with the same slope, but with different intercepts.

In this case, where the order with respect to B is zero, one determines the order with respect to A and C by simply keeping [A]₀ and [C]₀ much greater than [B]₀, and then systematically varying [A]₀ and [C]₀ in turn. For example, if we will hold [C]₀ constant and vary [A]₀, we will have i kinetics runs with initial concentrations of A in each run equal to [A]_0i. Remembering that b equals 0 allows us to rewrite Eqn. 1 as Eqn. 6, where R_i is the rate of the i^th run and k_C is equal to k[C]₀^c. Thus, a graph of [B]_t as a function of time for each run should give a straight line with a slope of -R_i, and Eqn. 6 predicts that the slopes of these lines (values of R_i) are proportional to [A]₀^a. To obtain a form of Eqn. 6 from which a can be extracted, one simply takes the logarithm of both sides of Eqn. 6 to give Eqn. 7. A graph of log R_i as a function of log[A]_0i will give a straight line with a slope equal to a. This procedure is then repeated for reactant C ([B]₀ and [A]₀ are kept constant while varying [C]_0i) gives c.

There are, of course, some cases that can't be treated using integrated rate laws or the simple methods given here.  There are a number of more advanced methods to treat kinetics data, such as the Guggenheim method and Job's method, and the reader is referred to any physical chemistry textbook or books on kinetics for more information.

References

1. Atkins, P. Physical Chemistry, 6^th Ed.; W. H. Freeman and Co.: New York, 1998; p. 765ff.

 

2. Daniels, F.; Williams, J. W.; Bender, P.; Alberty, R. A. and Cornwell, C. D. Experimental Physical Chemistry, 6^th Ed.; McGraw-Hill: New York, 1962; pp. 143-146.