Chemical reaction kinetics

Figure 1: Energy profile for an elementary bimolecular chemical reaction such as Reaction (1).

Chemical kinetics describes the relationship between measured rates of chemical reactions and the detailed atomic and molecular mechanisms via which the observed chemical change occurs.

[edit] 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, typically occurring either in a well-mixed, homogeneous gaseous or liquid system or on a catalytic surface (Freund and Knozinger, 2004).

For example, Reaction 1 \[\tag{1} Cl(g) + Br_2(g) \rightarrow BrCl(g) + Br(g) \]

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 \[ \begin{array}{lcl}- d[Cl]/dt = - d[Br_2]dt = d[BrCl]/dt = d[Br]/dt = k[Cl][Br_2]\end{array} \ ,\] where the proportionality (rate) constant (coefficient), \(k\ ,\) may be either determined experimentally or estimated from theory.

Chemical kinetics is used for largely empirical analysis of rates of reaction for applied purposes and in fundamental research work associated with the elucidation of chemical and biochemical reaction mechanisms. A number of complex chemical reactions, e.g., the Belousov-Zhabotinsky (BZ) Reaction (Zsabotyinszkij, 1964; Zhabotinsky, 1991; Scott,1994; Epstein and Pojman, 1998), have autocatalytic/feedback dynamics (Epstein and Pojman, 1998) leading to nonlinear behavior of chemical concentrations in a well-mixed medium, including oscillation, Hopf bifurcation, excitability, multistability, deterministic chaos, and when diffusion is important in an unstirred medium, traveling waves and Turing patterns.

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 standard approximations of chemical kinetics, e.g., the rate-determining step, fast-equilibrium, and pseudo-steady-state methods described below.

[edit] Elementary Chemical Reactions

A bimolecular, elementary reaction is one in which a pair of atomic or molecular reactants [ e.g., \(Cl\) and \(Br_2\) in Reaction (1)] is converted to one or more products (\(BrCl\) and \(Br\)) via a single reactive (\(Cl\)-\(Br_2\)) collision. Such a reaction may be generalized as \(A + B {\rightleftharpoons}X^{\mp} \rightarrow C + D\ ,\) where \(X^{\mp}\) is a metastable, high-energy, activated complex (transition state) (Eyring, 1935; Atkins and de Paula, 2006). Figure 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 rotational and vibrational energy levels. Only activated complexes traveling from reactants to products are counted in a reaction-rate calculation.

The rate of a bimolecular, elementary reaction is proportional to the product of the reactant concentrations, \([A][B]\ ,\) which partially defines the frequency of \(A-B\) collisions (Atkins and de Paula, 2006). Thus the so-called rate expression of a bimolecular reaction is \(-d[A]/dt = k_{bimolecular} [A][B]\ ,\) a form referred to as first-order in \([A]\) (which appears to the exponent one), first-order in \([B]\ ,\) and second-order (1+1) overall. The rate expression for a complex transformation (See below) not occurring in a single reactive collision may be of a complex form not necessarily related to the reaction stoichiometry; it must be determined empirically. The empirical order of a chemical reaction is to be distinguished from molecularity, which is the number of reactant molecules incorporated into the activated complex in a reaction occurring in a single step. Reaction 1 is then referred to as bimolecular.

Figure 2: Simulated time profiles of \([Cl]\ ,\) \([Br_2]\ ,\) \([BrCl]\ ,\) and \([Br]\) during Reaction (1) on the basis of a second-order rate expression and parameters given in the text. Simulated concentrations are perturbed by random Gaussian noise scaled to have a standard deviation of about 1.5%. This is done to illustrate the effect of experimental noise on kinetic parameters

Figure 3: Second-order plot of simulated Reaction (1) data from Figure 2 according to the integrated form given in text.

Figure 2 shows simulated time profiles of \([Cl]\ ,\) \([Br_2]\ ,\) \([BrCl]\ ,\) and \([Br]\) during Reaction (1) assuming a typical \(\pm\) 1.5% experimental noise.

Simulation parameters are \(k_{bimolecular}= 9.00 \times 10^{10} M^{-1}s^{-1}\ ,\) \([Br_2]_0 = 1.00 \times 10^{-9} M\ ,\) and \([Cl]_0 = 5.00 \times 10^{-10} M\ .\) Setting \(x = [Cl]_0 - [Cl]\ ,\) the amount of \(Cl\) reacted at time \(t\ ,\) the second-order equation above becomes \(dx/dt = k_{bimolecular} ([Cl]_0 - x)([Br_2]_0 - x)\ ,\) and its integration for \([Br_2]_0 \neq [Cl]_0\) yields \(ln\left(\frac{[Br_2][Cl]_0}{[Br_2]_0[Cl]}\right) = k_{bimolecular}([Br_2]_0 - [Cl]_0)t\ .\) The linear plot of the left-hand-side of this equation vs. \(t\) ( Figure 3) verifies that the data in Figure 2 are indeed of second-order. The least-squares line in Figure 3 yields \(k_{bimolecular} = 8.0 \pm .3 \times 10^{10} M^{-1}s^{-1}\) (compare to \(9.00 \times 10^{10} M^{-1}s^{-1}\) 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, reactions often must be followed through several half-lives (depending on the noise level) for this method to distinguish effectively among several possible rate expressions.

[edit] Collision Theory

A reasonably accurate AB-collision-frequency estimate of \(k_{bimolecular}\) is given by \(k_{bimolecular} = P_{steric}N_{A}{\sigma_{A-B} c^{rel}_{A-B}e^{-E_0^{\mp}/RT}}\ .\) The quantity \(P_{steric} <1\) empirically adjusts for energetic collisions that are not oriented properly for reaction to occur, \(N_{A}\) is Avogadro's number, \(\sigma_{A-B}\) is the cross-section for an \(A-B\) collision (\(nm^2/molecule\)), \(c^{rel}_{A-B}\) is the temperature-dependent mean relative velocity of \(A\) and \(B\) (\(m/s\)), \(e^{-E_0^\mp/RT}\) is the fraction of collisions having kinetic energy \({\geq}E_0^\mp\) (the energy difference between separated \(A\) and \(B\) species and the complex \(X^\mp\) in Figure 1) along the line of \(A-B\) centers, \(T\) is the Kelvin temperature, and \(R\) is the gas constant. For Reaction (1) at \(350 K\) the experimental value of \(k_{bimolecular}\) is \(\left(9.6 \pm 5\right) \times 10^{10} M^{-1}s^{-1}\ .\) The experimental value of \(E_0^\mp\) is \(\approx 0\) (Nicovich and Wine, 1990). Thus using the quite good estimate \(\sigma_{A-B} = 0.55nm^2\) and from kinetic theory \(c^{rel}_{A-B} = 500 m/s\) yields a collision-theory rate constant of \(1.67 \times 10^{11} M^{-1}s^{-1}\ ,\) perhaps slightly larger than the experimental value, suggesting \(P_{steric}\) is a bit less than one.

Other theoretical treatments (Eyring, 1935; Atkins and de Paula, 2006) of bimolecular chemical reactions involve a statistical calculation of \([X^\mp]\) for a particular \([A]\) and \([B]\) multiplied by the rate of passage of \(X^\mp\) across the barrier to products ( Figure 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 reaction (Atkins and de Paula, 2006) may occur when a reasonably complex, excited molecule with energy \(> E_o^\mp\) eventually finds \(X^\mp\) and passes to products. There seems to be no unambiguous example of an elementary termolecular reaction resulting from the simultaneous collision of three reactant species (Laidler, 1987).

[edit] Variability of Reaction Rates

Reaction rates are measured by observation of changes in reactant or product concentrations with time - a moving target. This is often a difficult experimental challenge, especially because the half-lives of chemical reactions vary from a few femto(\(10^{-15}\))seconds (Baskin and Zewail, 2001) to many years. Reaction rates depend exponentially on temperature and typically follow the empirical Arrhenius law, \(k_{experimental} = Ae^{-E^{\mp}/RT}\ ,\) over relatively small (~\(50^{o}\)C) temperature ranges. The quantities \(A\) and \(E^{\mp}\) are experimental parameters related to a particular chemical reaction. Comparison of the Arrhenius relationship with the theoretical \(k_{bimolecular}\) given above suggests that for a bimolecular reaction \(A \)(closely related to the collision frequency) ~ \(P_{steric}N_{A} \sigma_{A-B}c^{rel}_{A-B}\) and \(E^{\mp}\) (the empirical activation energy) ~ \(E_{0}^{\mp}\ .\)

Reaction rates then vary over very wide ranges with the values of \(A\ ,\) \(E^{\mp}\ ,\) \(T\ ,\) and reactant concentrations. This variation requires use of a very large stable of experimental methods (Atkins and dePaula, 2006) for reactions of vastly different rates. Indeed, several methods often are necessary only to measure the rate of a particular chemical reaction over a few hundred K.

[edit] Stoichiometry and Reaction Rates

The stoichiometry of a non-elementary chemical transformation may be quite complex. For example, consider the non-elementary reaction below. \[\tag{2} 5Br^-(aq)+BrO_3^-(aq)+6H^+(aq) {\rightleftharpoons} 3Br_2(aq)+3H_2O(l) \]

The various stoichiometric coefficients in Reaction 2, i.e., 5, 1, 6 and 3, require the reaction rate to be carefully defined; i.e., it must be accounted for that \(Br^-\) disappears five times faster than \(BrO_3^-\ .\) Thus we define the \(Rate\) of this reaction as

\[\tag{3} Rate = -\frac{1}{5}\frac{d[Br^-]}{dt} = -\frac{1}{1}\frac{d[BrO_3^-]}{dt} = -\frac{1}{6}\frac{d[H^+]}{dt} = \frac{1}{3}\frac{d[Br_2]}{dt} = \frac{1}{3}\frac{d[H_2O]}{dt} = k_{experimental}[Br^-][BrO_3^-][H^+]^2 \]

The \(-\) signs occur because the rates of change of reactants, e.g., \(d[Br^-]/dt\ ,\) are \(< 0\ .\) The rate-of-change of each individual species is divided by its stoichiometric coefficient in Reaction 2 in order to maintain the extended equality. The necessarily positive term on the far right is the empirical rate expression of this reaction 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.

[edit] Determination of Rate Expression

[edit] Isolation or Pseudo-Zero-Order Method

Figure 4: Plot suggesting the form of Eq. (5).

Consider Reaction (4), \[\tag{4} PhSO_2SO_2Ph + N_2H_2 \rightarrow PhSO_2NHNH_2 + PhSO_2H \ ,\]

which is not a simple elementary bimolecular reaction, despite its stoichiometry, and its kinetics is complex. We expect its rate expression to be of the form : \[ \begin{array}{lcl} -d\frac{[PhSO_2SO_2Ph]}{dt} &=& \frac{-d[N_2H_4]}{dt} = \frac{d[PhSO_2NHNH_2 ]}{dt} = \frac{d[PhSO_2H]}{dt} = f([PhSO_2SO_2Ph][N_2H_2], T) \end{array} \ .\]

The form of function \(f\) must be determined empirically. It is often convenient to flood such a system with one reactant whose resulting high concentration then does not change more than 10% (preferably less) in the course of the reaction. The reaction may then be said to be pseudo-zero-order in the concentration of this species. The reaction order in the other reactant, present at much lower concentration and said to be isolated, 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) at 25 \(^o\)C in which \([PhSO_2SO_2Ph]_0 = 5.0 \times 10^{-5} M\) with six \([NH_2NH_2]_0\) values ranging from \(5.0 \times10^{-3}\) to \(4.0 \times10^{-2} M\ ,\) all at least 100 times greater than \([PhSO_2SO_2Ph]_0\) and thus essentially unchanging as \([PhSO_2SO_2Ph]\) is consumed. That is, \([N_2H_2]\) = \([N_2H_2]_0\) may be assumed throughout the reaction. The flooded reaction was found to be first-order in \([PhSO_2SO_2Ph]\ ,\) which follows a logarithmic integrated form, with the six measurements yielding six values of \(k_{flood}\ .\) The quantity \(k_{flood}\) of course depends on \([NH_2NH_2]_0\ ,\) and a plot ( Figure 4) of \(k_{flood} / [NH_2NH_2]_0\) vs \([NH_2NH_2]_0\) (form found by trial and error) yields a straight line with slope \(k^{''}\) and intercept \(k^{'}\ .\) This result indicates

\[\tag{5} \begin{array}{rcl} -d[PhSO_2SO_2Ph]/dt = f([PhSO_2SO_2Ph],[NH_2NH_2],T) = (k^'[NH_2NH_2] + k^{''}[NH_2NH_2]^2)[PhSO_2SO_2Ph] \end{array} \ ,\]

which suggests that the reaction takes place through two parallel channels, with the second channel being assisted by a second \(NH_2NH_2\) molecule.

[edit] Initial-Rate Method

Figure 5: Simulation of Reaction (4) [sulfone] vs. time profiles and initial-rate tangents for three values of \([NH_2NH_2]_0\)

Figure 6: Plot of Reaction (4) initial rates from Figure 5 according to Eq. (5).

The initial-rate method is 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 reactions and instrumental instability. However, especially for fast reactions, initial rates may be significantly perturbed by mixing effects. Figure 5 shows \([PhSO_2SO_2Ph]\) vs. time curves simulated for Reaction (4) on the basis of the empirical Eq. (5). Each curve has a straight line tangent drawn to it at \(t = 0\) defining the initial rate of the reaction at \([PhSO_2SO_2Ph]_0 = 5 \times 10^{-5} M\) with various much higher values of \([NH_2NH_2]_0\ .\) Equation (5) suggests that a plot of \((d[PhSO_2SO_2Ph]_0/dt)_0/ ([PhSO_2SO_2Ph]_0 [NH_2NH_2]_0)\) vs. \([NH_2NH_2]_0\) should be linear with slope = \(k^{''}\) and intercept = \(k^'\ ,\) as is verified in Figure 6.

[edit] Complex Mechanisms and Approximate Methods

[edit] Analysis of A Complex Mechanism

Reaction (2), \(5Br^{-}(aq) + BrO_3^{-}(aq) + 6H^{+}(aq) \rightleftharpoons 3Br_2(aq) + 3H_2O(l)\ ,\) is suggested (Field et al., 1972; Pelle et al., 2004) to occur by the following collection of bimolecular elementary reactions.

\( \begin{array}{rclcl} BrO_3^{-} + H^{+} &\rightleftharpoons& HBrO_3 & & (M1) \\ HBrO_3 + H^{+} &\rightleftharpoons& H_2BrO_3^{+} & & (M2) \\ Br^{-} + H_2BrO_3^{+} &\rightleftharpoons& HBrO_2 + HOBr & & (M3) \\ HBrO_2 + H^{+} &\rightleftharpoons& H_2BrO_2^{+} & & (M4) \\ H_2BrO_2^{+} + Br^{-} &\rightleftharpoons& HOBr + HOBr & & (M5) \\ 3(HOBr + H^{+} &\rightleftharpoons& H_2OBr^{+}) & & (M6) \\ 3(H_2OBr^{+} + Br^{-} &\rightleftharpoons& Br_2 + H_2O) & & (M7) \\ ----------&-&--------&-&---\\ 5Br^{-} + BrO_3^{-} + 6H^{+} &\rightleftharpoons& 3Br_2 + 3H_2O & & (M8) \end{array} \)

The equilibrium symbol (\(\rightleftharpoons\)) states that under some conditions the forward and reverse rates of reaction may be competitive and that there may be significant amounts (depending upon \([H^{+}]\)) of both reactants and products present in each elementary Reaction (\(M1 - M7\)) and the overall Reaction (\(M8\)) as the system reaches thermodynamic equilibrium. Microscopic reversibility (Mahan, 1975) requires that at equilibrium \(k_{M1}/k_{M-1} = K_{M1} = [HBrO_3]_{eq}/[H^{+}]_{eq}[BrO_3^{-}]_{eq}\ ,\) etc. Note that sum of elementary Reactions \(M1 + M2 + M3 + M4 + M5 + 3M6 + 3M7\) yields the overall stoichiometry, \(M8\ .\)

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 that 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 reactions 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 reactions involved. For example, the four proton-exchange reactions, \(M1\ ,\) \(M2\ ,\) \(M4\ ,\) and \(M6\ ,\) may be assumed to be (at reasonably high \([H^+]\)) so rapid in both the forward and reverse directions that they are always near to equilibrium on the time scale of the overall reaction. Thus the instantaneous, very small concentrations of \(HBrO_3\ ,\) \(H_2BrO_3^+\ ,\) \(H_2BrO_2^+\ ,\) and \(H_2OBr^+\) may be defined by equilibrium expressions such as \([H_2BrO_3^+] = K_{M1}K_{M2}[BrO_3^-][H^+]^2\ ,\) thus eliminating four species and four differential equations. This is referred to as the fast-equilibrium approximation. Furthermore, this approximation allows Reactions \(M1-M7\) to be combined to yield the net stoichiometries below, which it is readily shown may be treated as having the dynamic form (rate expression) of elementary (single collision) reactions (See Eq.7).

\[ \begin{array}{rclcl} Br^- + BrO_3^- + 2H^+ &\rightleftharpoons& HBrO_2 + HOBr &~& (M3^') \\ k_{M3^'} = K_{M1}K_{M2}k_{M3} ~&and&~ k_{-M3^'} = k_{-M3} \\~\\ Br^- + HBrO_2 + H^+ &\rightleftharpoons& HOBr + HOBr &~& (M5^') \\ k_{M5^'} = K_{M4}k_{M5} ~&and&~ k_{-M5^'} = k_{-M5} \\~\\ Br^{-} + HOBr + H^{+} &\rightleftharpoons& Br_2 + H_2O &~& (M7') \\ k_{M7^'} = K_{M6}k_{M7} ~&and&~ k_{-M7^'} = k_{-M7} \end{array} \]

At very high values of the acidity, e.g. \([H^+] \sim 1M\ ,\) it might be guessed that the reverse rates of Reactions \(M5^'\) and \(M7^'\) are unimportant compared to the forward rates when the overall reaction is not near to equilibrium. This assumption eliminates two more reactions and makes the forward of Reaction \(M5^'\) rate-determining for the appearance of the final product, \(Br_2\ .\) That is, every \(HOBr\) formed in Reaction \(M5^'\) must quickly continue on to \(Br_2\ .\) Thus if \([HOBr]\) is small compared to the concentrations of the principal reactants and products, \(H^+\ ,\) \(BrO_3^-\ ,\) \(Br^-\ ,\) and \(Br_2\ ,\) Reactions \(M5^'\) and \(M7^'\) may be intuitively combined (not simply added) to yield the non-stoichiometric \(M5^{''}\ ,\) thus eliminating \(M7^'\) from the mechanism (Györgyi and Field, 1991). \[ \begin{array}{rcl} Br^- + HBrO_2 + H^+ &\rightarrow& Br_2 + H_2O ~~~~~~ (M5^{''}) \\ & k_{M5^{''}} = k_{M5^'} \end{array} \]

Reaction \(M5''\) is the combination of the left side of Reaction \(M5'\) with the right hand side of Reaction \(M7'\ ,\) indicating that Reaction \(M5'\) is \(rate-determining\) for Reaction \(M7'\)

Thus if \([HOBr]\ ,\) as well as \([HBrO_3]\ ,\) \([H_2BrO_3^+]\ ,\) \([HBrO_2]\ ,\) \([H_2BrO_2^+]\ ,\) and \(H_2OBr^+\ ,\) are always small in the same sense as HOBr above, then Reaction \(M5^{''}\) determines the rate of formation of \(Br_2\) according to Eq. (6). \[\tag{6} \left(\frac{1}{3}\right)\frac{d[Br_2]}{dt} = -\left(\frac{1}{5}\right)\frac{d[Br^-]}{dt} = k_{M5^'} [Br^-][HBrO_2][H^+] \]

The remaining reactions constitute a fast pre-equilibrium (\(M3^'\)) followed by a rate-determining step (\(M5^{''}\)) (Atkins and de Paula, 2006; Turányi et al, 1993).

\[ \begin{array}{rclcl} Br^- + BrO_3^- + 2H^+ &\rightleftharpoons& HBrO_2 + HOBr & & (M3^') \\ Br^- + HBrO_2 + H^+ &\rightarrow& Br_2 + H_2O & & (M5^{''}) \end{array} \]

In order to evaluate \(d[Br_2]/dt\) from Eq. (6) in terms of only principal reactants, the quantity \([HBrO_2]\) (recall that \(HBrO_2\) 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 \(HBrO_2\) is an unstable species involved in very fast removal reactions, i.e., \(-M3^'\) and \(M5^{''}\ .\) Thus its concentration is always very small, and \(d[HBrO_2]/dt\) must then remain very near to zero. Equation (7) is the differential equation for instantaneous \([HBrO_2]\) on the basis of \(M3^'\) and \(M5^{''}\ .\)

\[\tag{7} \frac{d[HBrO_2]}{dt} = k_{M3^'}[Br^-][BrO_3^-][H^+]^2 - k_{-M3^'}[HOBr][HBrO_2] - k_{M5^'} [Br^-][HBrO_2][H^+] \]

Setting \(d[HBrO_2]/dt = 0\) yields Eq. (8).

\[\tag{8} [HBrO_2] = k_{M3^'}[Br^-][BrO_3^-][H^+]^2/(k_{-M3^'}[HOBr] + k_{M5^'} [Br^-][H^+]) \]

Substitution of Eq. (8) into Eq. (6) yields Eq. (9), the predicted rate expression for the overall stoichiometry, \(M8\ .\)

\[\tag{9} \frac{d[Br_2]}{dt} = 3k_{M3^'}k_{M5^'}[Br^-]^2[BrO_3^-][H^+]^3/(k_{-M3}[HOBr] + k_{M6^'} [Br^-][H^+]) \]

This still somewhat complicated approximate expression involving \(HOBr\) can be made even simpler by assuming the reverse of Reaction \(M3^'\) (\(-M3\)) is very slow compared to Reaction \(M5^'\ .\) Thus \(k_{-M3^'}[HOBr] << k_{M5^'} [Br^-][H^+]\ ,\) and Eq. (9) becomes Eq. (10).

\[\tag{10} \frac{d[Br_2]}{dt} = 3k_{M3^'}[Br^-][BrO_3^-][H^+]^2 \]

For this set of assumptions we conclude that \(M3^'\) is rate-determining for \(M8\) under certain conditions, e.g., high \([H^+]\ .\) Equation (10) is of the same form as experimentally observed (See Eq. (3), suggesting that this treatment of the kinetics of mechanism \(M1 - M7\) is reasonable for the prevailing conditions of Eq. (3) and supports the suggested mechanism. Chemical mechanisms can be disproved but not proved. This treatment also suggests \(k_{experimental} = 3k_{M3^'} = 3K_{M1}K_{M2} k_{M3}\ .\)

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. The \(fast-equilibrium\) approximation is valid only if the opposing reactions of the equilibrium are much faster than their competitors and the rate-determining reactions in the scheme. The \(rate-determining-step\) approximation is applicable when the occurrence of one reaction effectively determines the rate of one or more following reactions. Finally, the \(pseudo-steady-state\) approximation (PSSA) is valid only if the PSSA species is present at a very small concentration and has a very short lifetime. For a more complete discussion see (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, Lam and Goussis, 1994; Maas and Pope, 1992).

[edit] 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 (\(M1 - M7\)) whether the protonated species, e.g., \(H_2BrO_3^+\ ,\) actually exist as separated species or the whole process (\(M3^'\)) occurs nearly simultaneously in a solvent cloud containing \(2H^+\ ,\) \(BrO_3^-\ ,\) and \(Br^-\ .\) However, there is an alternate mechanism to (\(M1 - M7\)) that is in principle kinetically distinguishable. It is given by \(M1 + M2\) followed by \(M9 -M11\) and \(M5^{''}\ .\) \[ \begin{array}{rclcl} H_2BrO_3^+ + Br^- &\rightleftharpoons & H_2Br_2O_3 & & (M9)\\ Br^- + H_2Br_2O_3 + H^+ &\rightarrow& Br_2 + Br(OH)_3 & & (M10)\\ Br(OH)_3 &\rightarrow& HBrO_2 + H_2O & & (M11) \end{array} \]

A similar mechanism has been strongly supported in the analogous reaction \(5I^-+ IO_3^- + 6H^+ \rightleftharpoons I_2 + 3H_2O\) (Schmitz, 1999; Field and Agreda, 2000) whose kinetics is commensurate with a substantial fraction of \(I^-\) being tied up as \(H_2I_2O_3\ ,\) violating the requirement that the fast-equilibrium and pseudo-steady-state approximations may only be applied to intermediate species present at much smaller concentration than the concentrations of principal reactants. Thus in \(M9 - M11\) the bromine-atom mass balance must be maintained by requiring Eq. (11) to be met. \[\tag{11} \frac{5}{3}([Br_2]_{\infty} - [Br_2]_t) = [Br^-]_s = [Br^-]_t + [H_2Br_2O_3]_t \]

Reaction \(2\) is normally followed by monitoring instantaneous \([Br_2]\ ,\) i.e., \([Br_2]_t\ .\) The stoichiometric quantity \([Br^-]_s\) is the \([Br^-]_t\) that would be present if no \(Br^-\) were tied up as \(H_2Br_2O_3\ .\) The quantities \([Br^-]_t\) and \([H_2Br_2O_3]_t\) are the instantaneous concentrations of these species at time \(t\ .\) The system is flooded with \(BrO_3^-\) and \(H^+\ ,\) whose nearly constant concentrations are then very close to \([BrO_3^-]_0\) and \([H^+]_0\ .\) The assumption that Reactions \(M1\ ,\) \(M2\ ,\) and \(M9\) are near to equilibrium yields \([H_2Br_2O_3]_t = K_{M1}K_{M2}K_{M9}[BrO_3^-]_0[H^+]_0^2[Br^-]_t\ ,\) and substitution of this result into the mass balance (Eq. (11) yields Eqs. (12) and (13). \[\tag{12} [Br^-]_t = \frac{[Br^-]_s}{(1 + K_{M1}K_{M2}K_{M9} [BrO_3^-]_0[H^+]_0^2)} \]

\[\tag{13} [H_2Br_2O_3]_t = \frac{K_{M1}K_{M2}K_{M9}[BrO_3^-]_0[H^+]_0^2[Br^-]_s}{(1 + K_{M1}K_{M2}K_{M9} [BrO_3^-]_0[H^+]_0^2)} \]

Assuming Reaction \(M10\) to be rate-determining for the overall process \((M8)\) leads to Eq. (14). \[\tag{14} - \frac{1}{5}\frac{d[Br^-]_s}{dt} = \frac{1}{5}(\frac{d[Br^-]_t} {dt} + \frac{d[H_2Br_2O_3]_t}{dt}) = k_{M10}[Br^-]_t[H_2Br_2O_3]_t \]

Substituting Eqs. (12) and (13) into (14) leads to the predicted rate expression, Eq. (15). \[\tag{15} - \frac{1}{5}\frac{d[Br^-]_s}{dt} = \frac{k_{M10}K_{M1}K_{M2}K_{M9}[BrO_3^-]_0[H^+]_0^2[Br^-]_s^2}{(1 + K_{M1}K_{M2}K_{M9}[BrO_3^-]_0[H^+]_0^2)} \]

A more usual steady-state treatment of the transient species \(H_2BrO_3^+\) and \(H_2Br_2O_3\) in \(M1\ ,\) \(M2\ ,\) \(M9\ ,\) and \(M10\) assuming that very little \(Br^-\) is tied up as \(H_2Br_2O_3\) yields Eq. (16).

\[\tag{16} - \frac{d[Br^-]}{dt} = \frac{k_{M9}k_{M10}K_{M1}K_{M2}[BrO_3^-]_0[H^+]_0^2[Br^-]^2}{(k_{-M9} + k_{M10}[Br^-])} \]

Eqs. (15) and (16) suggest a second-order (\([Br^-]_s^2\)) contribution to \(d[Br^-]/dt\ ,\) for which experimental evidence has recently been reported (Schmitz, 2006) when \([Br^-] >> [BrO_3^-]\ .\) However, considerable effort would be required to confidently distinguish among the various detailed mechanistic alternatives under various experimental conditions presented above. It is likely that each mechanism is dominant under suitable experimental conditions. 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. In this case it might be sufficient to know only that Reactions \(M9 - M11\) are important under some conditions.

[edit] 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 further reduction is attempted using the methods of time-scale separation (Tikhonov, 1952; Tomlin et al. 1992; Lam and Goussis, 1994; Maas and Pope, 1992) and sensitivity methods (Turányi, 1990; Tomlin et al, 1997; Zsély and Turányi, 2003).

[edit] References

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Atkins, P. and de Paula, J. (2006) Physical Chemistry, 8th Edition, W. H. Freeman and Company, New York, N.Y. Chapters 21 - 25.
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Internal references

Yuri A. Kuznetsov (2006) Andronov-Hopf bifurcation. Scholarpedia, 1(10):1858.

Anatol M. Zhabotinsky (2007) Belousov-Zhabotinsky reaction. Scholarpedia, 2(9):1435.

James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.

Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.

Richard J. Field (2007) Oregonator. Scholarpedia, 2(5):1386.

Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.

Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

David H. Terman and Eugene M. Izhikevich (2008) State space. Scholarpedia, 3(3):1924.

[edit] 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).