Chemical reaction kinetics

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Author: Dr. Richard J. Field, Department of Chemistry, The University of Montana, Missoula, MT 59812

Chemical kinetics describes the relationship between measured rates of chemical reactions, as expressed in sometimes nonlinear differential equations, and the detailed atomic and molecular mechanisms via which the observed chemical change occurs.

Contents 1 Introduction 2 Elementary chemical reactions 2.1 Collision Theory 2.2 Stoichiometry and reaction rates 3 Determination of Rate Expression 3.1 Flooding 3.2 Initial-rate methods 4 Complex mechanisms and approximate methods in chemical kinetics 4.1 Kinetic distinguishability 4.2 Reduction of very large mechanisms and numerical methods 5 References 6 Recommended reading 7 External links 8 See also

Introduction

Chemical kinetics (Laidler, 1987; Houston, 2001; Atkins and de Paula, 2006) is a branch of dynamics, the science of motion. It is that body of concepts and methods used to investigate and understand the rates and mechanisms of chemical reactions (rxns). For example, the rxn

(1)

is found (Nicovich and Wine, 1990) to be very fast and to occur via a simple mechanism consisting of a single bimolecular collision. Its rate is given by

,

where the proportionality (rate) constant, , may be either determined experimentally or estimated from theory.

Chemical kinetics is used for largely empirical analysis of rates of rxn for applied purposes and in fundamental research work associated with the elucidation of chemical and biochemical rxn mechanisms. A number of complex chemical rxns, e.g., the Belousov-Zhabotinsky (BZ) Reaction (Zhabotinsky, 1991; Scott,1994 ), have autocatalytic/feedback dynamics (Epstein and Pojman, 1998) leading to nonlinear behavior of chemical concentrations, including oscillation, Hopf bifurcation, excitability, multistability, traveling waves, Turing patterns, and deterministic chaos. The Oregonator model (Field and Noyes, 1974) of the BZ Reaction is obtained by reduction of a detailed chemical mechanism (Field, Kőrös, and Noyes, 1972) by use of approximations, e.g., rate-determining step, equilibrium, and steady state, described below.

Elementary chemical reactions

A bimolecular, elementary rxn is one in which a pair of atomic or molecular reactants [ e.g., and in Rxn 1] is converted to one or more products ( and ) via a single reactive (-) collision. Such a rxn may be generalized as (Fig. 1), where is a metastable, high-energy, activated complex (transition state) (Eyring, 1935; Atkins and de Paula, 2006). Fig. 1 shows this transition from a locale in phase space in which the atoms of the system are arranged in configurations we identify as reactants, through high-energy configurations identified as an activated complex, and ending in a locale of phase space we identify as products. The horizontal lines are reactant and product energy levels.

The rate of a bimolecular, elementary rxn is proportional to the product of the reactant concentrations, , which defines the frequency of collisions (Atkins and Depaula, 2006). Thus the so-called rate expression of such a rxn is , a form referred to as first-order in , first-order in , and second-order overall. The rate expression for a complex chemical transformation (see below) not occurring in a single reactive collision may be of a complex form not necessarily related to the rxn stoichiometry; it normally must be determined empirically.

Fig. 2 shows simulated (complete with typical 5% experimental noise) , , , and during Rxn 1. Simulation parameters are , , and . Setting , the amount of reacted at time , the second-order equation above becomes , and its integration for yields . The linear plot of the left- hand- side of this equation vs. (Fig. 3) verifies the data in Fig. 2 is indeed second-order. The least-squares line in Fig. 3 yields (compare to used in the simulation), suggesting the level of experimental error often present in chemical-kinetics parameters. Trial and error fitting of experimental data to various integrated rate expressions is a common way of determining the kinetics (rate expression) of a chemical transformation. However, rxns often must be followed through several half-lives (depending on the noise level) for this method to distinguish effectively among several possible rate expressions.

Collision Theory

An effective-AB-collision-frequency estimate of is given by . The quantity empirically adjusts for energetic collisions that are not oriented properly for rxn to occur, is the cross-section for an collision (), is the temperature-dependent mean relative velocity of and (), is the fraction of collisions having kinetic energy (the energy difference between separated and species and in Fig. 1) along the line of centers, and is the gas constant. For Rxn 1 at the experimental value of is . The experimental value of is (Nicovich and Wine, 1990). Thus using the quite good estimate and from kinetic theory yields a collision-theory rate constant of , perhaps slightly larger than the experimental value, suggesting is a bit less than one.

Other treatments (Eyring, 1935; Atkins and de Paula, 2006) of bimolecular chemical rxns involve a statistical calculation of for a particular and multiplied by the rate of passage of across the barrier to products (Fig. 1), as well as the calculation of classical or quantum-mechanical trajectories across a surface of total potential energy for various A, B, C, and D configurations (Houston, 2001).

An elementary unimolecular rxn (Atkins and de Paula, 2006) may occur when a reasonably complex, excited molecule with energy eventually finds and passes to products. There seems to be no unambiguous example of an elementary termolecular rxn resulting from the simultaneous collision of three reactant species (Laidler, 1987).

Stoichiometry and reaction rates

The stoichiometry of a non-elementary chemical transformation may be quite complex. For example, consider the non-elementary rxn below.

(2)

The various stoichiometric coefficients, i.e., 5, 1, 6 and 3, require the rxn rate to be carefully defined; i.e., it must be accounted for that disappears five times faster than . Thus we define the of this rxn as

(3)

The signs occur because the rates of change of reactants, e.g., , are . The necessarily positive term on the far right is the empirical rate expression of this rxn for a particular temperature and reactant concentration ranges. Its form may be different under other experimental conditions. The observed rate expressions of complex, non-elementary reactions are normally not directly deducible from the reaction stoichiometry.

Determination of Rate Expression

Flooding

Consider Rxn 4,

(4)

which is not an elementary bimolecular rxn, despite its simple stoichiometry, and its kinetics is complex. We expect its rate expression to be of the form :

.

The form of must be determined empirically. It is often convenient to flood such a system with one reactant whose concentration does not change more than 10% (preferably less) in the course of the rxn. The reaction order in the other reactant, present at much lower concentration, is then often simple and readily determined from its significant rate of change. Thus a series of experiments were carried out (Kice and Legan, 1973) in which with six values ranging from to , all at least 100 times greater than and thus essentially unchanging as is consumed. The flooded rxn was found to be first-order in , which follows a logarithmic integrated form, with the six measurements yielding six values of . The quantity of course normally depends on , and a plot (Fig. 4) of vs (form found by trial and error) yields a straight line with slope and intercept . This result indicates

(5)

which suggests the rxn takes place through two parallel channels, the first transition state containing one molecule and the second containing two molecules.

Initial-rate methods

Initial-rate methods are often used to infer a rate expression from experimental data (Espenson, 1995). Initial rates are readily measured graphically, may be assumed to be at the initial concentrations of reactants, and avoid problems with secondary rxns and instrumental instability. However, especially for fast reactions, initial rates may be significantly perturbed by mixing effects. Fig. 5 shows vs. time curves simulated for Rxn 4 on the basis of the empirical Eq. 5. Each curve has a straight line tangent drawn to it at defining the initial rate of the rxn at with various much higher values of . Eq. 5 suggests that a plot of vs. should be linear with slope = and intercept = , as is verified in Fig. 6.

Complex mechanisms and approximate methods in chemical kinetics

Rxn 2, , is suggested (Field et al., 1972; Pelle et al., 2004) to occur by the following collection of bimolecular elementary rxns, its mechanism.

The symbol states that there may be significant amounts (depending upon ) of both reactants and products present in each elementary Rxn () and the overall Rxn () when the reaction reaches thermodynamic equilibrium. Microscopic reversibility (Mahan, 1975) requires , etc. Note the elementary Rxns composing the mechanism sum to the overall stoichiometry, .

Analysis of the above mechanism so as to compare its dynamic behavior to that of the experimental system and for extraction of parameters (e.g., rate constants) from experimental data is difficult. Complete description of the model (assuming all reactions are elementary) requires a differential equation for each of the ten species involved, and each differential equation contains a term from each member of the subset of the fourteen rxns in which that particular species appears as reactant or product. See Eq. 7.

Fortunately, this system may be very greatly simplified by taking advantage of the variation in time scales of the elementary rxns involved. For example, the four proton-exchange rxns, , , , and , may be assumed to be (at reasonably high ) so rapid in both the forward and reverse directions that they are always at equilibrium on the time scale of the overall rxn. Thus the instantaneous, very small concentrations of , , , and may be defined by equilibrium expressions such as , thus eliminating four species and four differential equations. Furthermore, this approximation allows Rxns to be combined to yield the net stoichiometries below, which may be treated as elementary (single collision) rxns.

At very high values of the acidity, e.g. , it might be guessed that the reverse rates of Rxns and are unimportant compared to the rates of the forward rxns. This assumption eliminates two more rxns and makes the forward of Rxn rate determining for the appearance of the final product, . That is, every formed in Rxn must eventually continue on to . Thus if is small compared to the concentrations of the principal reactants and products, , , , and , Rxns and may be intuitively combined (not simply added) to yield , thus eliminating from the mechanism (Györgyi and Field, 1991).

Furthermore, if , as well as , , , , and , are always small in the same sense, then Rxn determines the rate of formation of according to Eq. 6.

(6)

The remaining rxns constitute a pre-equilibrium () followed by a rate-determining step () (Atkins and de Paula, 2006).

In order to evaluate from Eq. 6 in terms of only principal reactants, the quantity (recall that is an intermediate species) must be evaluated in terms of the concentrations of principal reactants. This may be done using the so-called pseudo-steady-state approximation (Atkins and de Paula, 2006), which assumes that is an unstable species involved in very fast removal rxns, i.e., and . Thus its concentration is always very small, and must then remain very near to zero. Eq. 7 is the differential equation for instantaneous on the basis of and .

(7)

Setting yields Eq. 8.

(8)

Substitution of Eq. 8 into Eq. 6 yields Eq. 9, the predicted rate expression for the overall stoichiometry, .

(9)

This still somewhat complicated approximate expression involving can be made even simpler by assuming the reverse of Rxn () is very slow compared to Rxn . Thus , and Eq. 9 becomes Eq. 10.

(10)

For this set of assumptions we conclude that is rate determining for under certain conditions, e.g., high . Eq. 10 is of the same form as the experimentally observed form given in Eq. 3, suggesting that this treatment of the kinetics of mechanism is reasonable for the prevailing conditions of Eq. 3 and supports the suggested mechanism. Suggested chemical mechanisms can be disproved but not proved. This treatment also suggests .

The equality of stoichiometrically adjusted rates given in Eq. 3 and the above analysis is true only if the concentrations of all intermediate species are small compared to the concentrations of the principal reactants and products. In general, the equilibrium, pseudo-steady-state and rate-determining step approximations are only applicable to species that are present in very small concentrations (Turányi, et al., 1993). The mathematical basis of these methods is the reduction of sets of differential equations using time scales (Tikonov, 1952, Vasil'eva, Butuzov and Kalachev, 1995), which control the concentrations of intermediate species.

Kinetic distinguishability

Often the details of a chemical mechanism are indistinguishable to kinetic analysis. For example, it would be a very difficult problem to determine in mechanism () whether the protonated species, e.g., , actually exist as separated species or the whole process () occurs nearly simultaneously in a solvent cloud containing , , and . However, there is an alternate mechanism to () that is in principle kinetically distinguishable. It is given by followed by and .

A similar mechanism has been strongly supported in the analogous rxn (Schmitz, 1999; Field and Agreda, 2000) in which the kinetics is commensurate with a substantial fraction of being tied up as , violating the requirement that the equilibrium and steady-state approximations may only be applied to species present at much smaller concentration than the concentrations of principal reactants. Thus in the bromine-atom mass balance must be maintained by requiring Eq. 11 to be met.

(11)

The rxn is normally followed by monitoring instantaneous , i.e., . The stoichiometric quantity is the that would be present if no were tied up as . The quantities and are the instantaneous concentrations of these species at time . The system is flooded with and , whose nearly constant concentrations are then and . The assumption that Rxns , and are at equilibrium yields , and substitution of this result into the mass balance (Eq. 11) yields Eqs. 12 and 13.

(12)

(13)

Assuming Rxn to be rate-determining for the overall process leads to Eq. 14.

(14)

Substituting Eqs. 12 and 13 into 14 leads to the predicted rate expression, Eq. 15.

(15)

A more usual steady-state treatment of , , , and assuming that only very little is tied up as yields Eq. 16.

(16)

Eqs. 15 and 16 suggest a second-order () contribution to , for which experimental evidence has recently been reported (Schmitz, 2006) when . However, considerable effort is required to confidently distinguish among the various detailed mechanistic alternatives presented above. The goal of chemical kinetics is to establish the dynamic structure of a complex chemical reaction at the level of resolution necessary to extract the information desired.

Reduction of very large mechanisms and numerical methods

The above section contains the basis of qualitative methods for treatment of relatively small but still complex mechanisms. These methods are useful to initiate analysis of a very large mechanism, e.g., a tropospheric chemistry system, but usually their application is limited, and very large mechanisms remain even after their use. These residual mechanisms are usually treated by numerical simulation (Szopa et al. 2005; Gillespie, 2007), and reduction is attempted using Tikhonov's theorem (Tikhonov, 1952; Tomlin et al. 1992) and sensitivity methods (Turányi, 1990) to separate time scales.

References

• Agreda B.; J. A.; Field, R.J. and Lyons, N. J. (2000) Kinetic evidence for accumulation of stoichiometrically significant amounts of H₂I₂O₃ during the reaction of I⁻ with IO₃⁻, J. Phys. Chem., 104(22), 5269-5274.
• Atkins, P. and de Paula, J. (2006) Physical Chemistry, 8th Edition, W. H. Freeman and Company, New York, N.Y. Chapters 21 - 25.
• Epstein, I. R. and Pojman, J. A. (1998) An Introduction to Nonlinear Chemical Dynamics, Oxford University Press, New York, N.Y.
• Espenson, J. A. (1995) Chemical Kinetics and Reaction Mechanisms, 2nd Edition, McGraw-Hill, Inc., New York, N.Y.
• Eyring, H. (1935) Activated complex in chemical reactions, J. Chem. Phys. 3, 107-15.
• Field, R. J.; Kőrös, E. and Noyes, R. M. (1972) Oscillations in chemical systems. II. Thorough analysis of temporal oscillation in the bromate-cerium-malonic acid system, J. Amer. Chem. Soc., 94(25), 8649 -8664.
• Field, R. J. and Noyes, R. M. (1974) Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction, J. Chem. Phys. 60(5), 1877-1884.
• Gillespie, D. T. (2007) Stochastic simulation of chemical kinetics, Ann. Rev. Phys. Chem., 58, 35-55.
• Györgyi, L. and Field, R. J. (1991) Simple models of deterministic chaos in the Belousov-Zhabotinsky Reaction, J. Phys. Chem. 95(17), 6594-6602.
• Houston, P. L. (2001) Chemical Kinetics and Reaction Dynamics, McGraw-Hill Higher Education, New York, N.Y.
• Kice, J. L. and Legan, E. (1973) Relative nucleophilicity of common nucleophiles toward sulfonyl sufur. II. Comparison of the relative reactivity of twenty different nucleophiles toward sulfonyl sulfur vs. carbonyl carbon. J. Amer. Chem. Soc. 95(12) 3912-17.
• Laidler, K. J. (1987) Chemical Kinetics, 3rd Edition, Harper Collins Publishers, Inc., New York, NY.
• Mahan, B, H. (1975) Microscopic reversibility and detailed balance. Analysis, J. Chem. Ed. 52(5), 299-302.
• Nicovich, J. M. and Wine, P. H. (1990). Kinetics of the reactions of atomic oxygen [O(3p)] and atomic chlorine [Cl(2P)] with hydrobromic acid and molecular bromine, Intl J. Chem. Kinetics, 22(4), 379-97.
• Pelle, K.; Wittman, M.; Lovrics, K. and Noszticzius, Z. (2004) Mechanistic investigations of the Belousov-Zhabotinsky Reaction with oxalic acid substrate. 2. Measuring and modeling of the oxalic acid-bromine chain reaction and simulating the complete oscillatory system. J. Phys. Chem. 108(37), 7554-7562.
• Schmitz, G. (1999) Kinetics and Mechanism of the iodate-iodide reaction and other related reactions, Phys. Chem. Chem. Phys. 1(8), 1909-1914.
• Schmitz, G. (2006) Kinetics of the bromate-bromide reaction at high bromide concentration, Intl. J. Chem. Kinetics, 39(1), 17-21.
• Scott, S. K. (1994) Oscillations, Waves, and Chaos in Chemical Kinetics, Oxford University Press, New York, N.Y.
• Szopa, A.; Aumont, B. and Madronich, S. (2005) Assessment of reduction methods used to develop chemical schemes: Building of a new chemical scheme for VOC oxidation suited to three-dimensional, multi-scale HOx-NOy-VOC chemistry simulations, Atmos. Chem. and Phys. 5(9), 2519-2538.
• Tikonov, A. N. (1952) Systems of differential equations containing small parameters in the derivatives, Mat. Sb., 31, 575- 586. In Russian.
• Tomlin, A. S.; Pilling, M. J.; Turányi, T.; Merkin, J. H. and Brindley, J. (1992) Mechanism reduction for the oscillatory oxidation of hydrogen: sensitivity analysis and quasi-steady-state analysis, Combustion and Flame, 91(2), 107-30.
• Turányi, T. (1990) Sensitivity analysis of complex chemical systems. Tools and Applications, J. Math. Chem., 5(3), 203-48.
• Turányi, T.; Tomlin, A.S. and Pilling, M. J. (1993) On the error of the quasi-steady-state approximation, J. Phys. Chem., 97(1), 163-72.
• Vasil'eva, A.B.; Butuzov, V. F. and Kalachev, L. V. (1995) The Boundary Function Method for Singular Perturbation Problems, Society for Industrial and Applied Mathematics. Philadelphia, Pa.
• Zhabotinsky, A. M. (1991) A history of chemical oscillations and waves, Chaos, 1(4), 379-86.

Recommended reading

• "Principles of Chemical Kinetics", 2nd Ed., James E. House, Academic Press, NY (2007).
• "Chemical kinetics and Reaction Dynamics", Paul L. Houston, Dover Press, NY (2006).

External links

• Richard Field's website
• Chemical Kinetics simulation at UC Irvine

See also

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