Reaction dynamics I: Mechanisms and rates
17.1 Linking empirical kinetics to reaction dynamics
In Chapter 16 we investigated empirical kinetics. We defined and developed the mathematical treatment necessary to handle the description of reaction rates and the dependence of concentrations as a function of time. The purpose of this chapter is to get to the heart of chemistry. We aim to understand how a chemical reaction occurs and how the manner of making and breaking bonds affects reaction rates. To do so, we begin by taking a closer look at how molecular collisions and energy exchange actually lead to the making and breaking of bonds. Beginning with unimolecular and bimolecular elementary reactions, we will develop the foundation of a predictive theory to describe reaction rates, rather than just relying on empirical fits to data that have no predictive power. We will then show how a series of unimolecular and bimolecular processes can be combined to generate an overall reaction mechanism. At that point, we can understand more fully the effects of reaction conditions such as concentrations and temperature on reaction mechanisms. We will learn to understand when changes in reaction kinetics are produced by changes in mechanism – that is, a change in the steps involved in the reaction – as opposed to a change in which steps are determining the reaction rate. From a fundamental understanding of the relationship of unimolecular and bimolecular processes to reaction mechanisms, we will then be in a position to address complex reaction dynamics of the type described in Chapter 18, which are the reaction dynamics of useful reactions rather than model systems.
Figure 17.1 One of the most fundamental tools for the study of chemical dynamics is the supersonic molecular beam apparatus. The gas source is at the far left. The upper inset shows an expanded view of a pulsed nozzle in front of a skimmer. The skimmer has a tiny orifice through which the molecules expand into the next vacuum chamber. The chamber on the left is called the source chamber. The middle chamber (mostly obscured by the gate valve above the diffusion pump) contains a partition and a collimating orifice that allow it to provide two stages of differential pumping. The main chamber on the right is an ultrahigh vacuum chamber suitable for performing laser-induced fluorescence (LIF) and resonance-enhanced multiphoton ionization (REMPI) spectroscopy on either the beam directly or on molecules that are scattered, thermally desorbed or laser-desorbed from the surface of a sample. The lower inset is a schematic diagram of the molecular beam apparatus. Photograph reproduced with permission from Eckart Hasselbrink.
17.2 Hard-sphere collision theory
17.2.1 Collision density
Consider a bimolecular elementary reaction between reactants A and B leading to the product P,
which follows the rate law
For simplicity we take A, B and P as gas-phase species. We know that a reaction can only occur if A and B collide with the right amount of energy in the right orientation. Therefore, the reaction rate is proportional to three factors: the collision rate; the collision energy; and the orientation of the collision, which is known as the steric factor. Let us calculate estimates of these factors based on a model in which molecules are represented by hard spheres.
In Chapter 3 we derived that the collision density Z12 is the number of AB collisions per unit time and volume
where ⟨v12⟩ is the relative mean speed of A and B
where μ is the reduced mass, the number densities N1 and N2, and σ is the collision cross-section
defined in terms of the collision diameter d
for molecules with hard sphere diameters d1 and d2. The collision density Z12 is an upper limit to the rate of any reaction that requires the collision of A with B.
17.2.2 The energy requirement
The probability that a collision leads to a reaction depends on the energy of the collision. In Chapter 16 we used this premise to derive the Arrhenius equation. By assuming that the cross-section is zero if the collision energy is below the activation energy ϵa (on a per molecule basis, or Ea on a per mole basis), we arrive at an expression for the rate of formation of products by multiplication of the collision density by the fraction of collisions with an energy above the activation energy
We are assuming that the products are equilibrated to a Boltzmann distribution described by the absolute temperature T.
Equation (17.7) is in units of molecules per unit volume per unit time. To convert this equation to molar units, we multiply by Avogadro’s constant NA,
The rate is now expressed in the form
in which the rate constant
is written the form of the Arrhenius equation with the collision frequency factor (pre-exponential factor)
This has a weak temperature dependence compared to the exponential term, consistent with our previous discussion of the Arrhenius equation.
17.2.3 The steric requirement
Calculations of the pre-exponential factor A according to Eq. (17.11) are often good for simple reactions, but become increasingly less accurate for reactions involving more complex molecules. This suggests that the geometry of the collision – and not just its energy – is important for reactivity. We introduce the steric factor, P, to account for the difference,
where P is usually (much) smaller than 1. This is to be expected for hard spheres for which a fraction less than one of all possible orientations of the spheres during collision leads to reaction.
An exception here is a reaction that occurs via what is known as a harpooning mechanism. Two classes of reactive system in which this occurs are collisions between an alkali metal and a halogen, such as K + I2 → KI + I, or the reaction of an electronically excited rare gas with a halogen to form an exciplex, such as Ar* + F2 → ArF + F. An exciplex is a molecule that only exists in an excited electronic state, that is, its ground state is dissociative. In these reactions the cross-section is considerably larger than the collision cross-section, P > 1. The reason is that electron transfer between the collision partners with vastly different electronegativities occurs prior to collision at the hard sphere impact parameter. Electron transfer leads to the formation of an ion and a counterion that attract each other at long range. The K atom in the first reaction throws out an electron, much like a harpoon. The electrostatic attraction then acts like the rope that is used to reel in the K+ ion toward the I2− ion, causing the steric factor to be much larger than 1. Exciplexes such as ArF, KrF and XeCl are important for the creation of excimer lasers – the dissociative nature of the ground state ensuring a population inversion since the population cannot build up in the ground state.
A simple and intuitive model of how to estimate the steric factor P and improve upon simple collision theory was proposed by Ian W.M. Smith.1 We introduce two improvements to simple collision theory: (i) taking account of the geometry of the collision; and (ii) allowing the criterion for the minimum energy for reaction to change with this collision geometry. The factors needed to apply the corrections are defined in Fig. 17.2, including the impact parameter b. The impact parameter is the shortest distance that would separate the collision partners along the line of the collision if there were no interactions. The relative velocity v of the collision is defined in terms of the velocities of the collision partners v1 and v2 as 
. Note that v must be positive in order for a collision to occur.
Figure 17.2 The reaction of A + BC. (a) Defining the parameters that characterize a hard-sphere collision. The impact parameter b is the separation between two parallel lines drawn through the centers of mass of the two spheres (the distance of closest approach along the line-of-centers that would result if the particle trajectories were undeflected by the collision), d is the minimum distance attained when one hard sphere meets the other, and v is the relative velocity directed along the line of centers. (b) To calculate an improved value of the steric parameter for reaction the spheres must reach the point at which chemical interactions are significant. This distance is D, which is at a distance of closer approach than the hard-sphere collision diameter d; that is, D < d. The angle φ is the angle between the velocity vector and the internuclear axis of BC, which has length RBC. The circle with a dashed boundary corresponds to the hard-sphere radius of BC.
ϵa is identified as the minimum energy for reaction in a collinear geometry. This corresponds to the minimum activation energy from the PES. The B−C bond has length RBC. The angle φ between the B−C bond that has to be broken and the line joining the centers of mass of the reactants at impact is assumed to be the geometric property that determines the critical energy for reaction ϵc. The bond that forms is the A−B bond, which has an equilibrium bond length RAB.
Molecular interactions strong enough to initiate chemical reactivity inevitably start at distances below the hard-sphere collision diameter d. Thus, we introduce a critical separation D < d, which is the distance where chemical interactions are significant. This distance can be found from the position of the transition state on the PES. For the H + H2 reaction, d = 3.5 Å but D is only 1.35 Å. The critical separation is taken as the distance between the centers of mass at the transition state in the collinear geometry
where a = mC/(mB + mC) and has a value of 0.5 when BC is a homonuclear diatomic. The * in the distances indicates that they are the values in the activated complex rather than the equilibrium values.
The most important aspect of the model is that the criterion for reaction to occur is not when the energy of the collision exceeds the minimum barrier height ϵa. Instead, the criterion for reaction is that the component of energy transferred along the line of centers of the collision complex must exceed the minimum barrier height. The critical energy for reaction ϵc is defined in terms of the collision parameters according to
The energy parameter ϵ′ is related to characteristics of the potential energy surface,
where kbend is the force constant of bending in the activated complex. The force constant tells us whether the bending motion is floppy (small kbend) or rigid (large kbend). The more rigid the transition state species, the greater the critical energy for reaction. The distances are in the activated complex. The rate constant is then
The exact values of the distances and the force constant will not be known unless we have available a high-quality potential energy surface (PES) from quantum mechanical calculations. Nonetheless, the distances and the force constant can easily be estimated for approximate values. This provides not only a rational estimate of the steric factor, but also a refined conception of the critical energy of reaction.
The critical energy of reaction is not simply the energy of collision; rather, it is the energy transmitted along the line of centers when the molecules begin to interact chemically that is determinative for reaction.
17.3 Activation energy and the transition state
We turn now to a more quantitative definition of the activation energy in terms of the potential energy surface that governs the interaction between atoms in the reactive system. We have seen in Chapter 16 that a single PES describes the reaction, and that microscopic reversibility and detailed balancing can be used to connect the reactants and products through the activated complex in the transition state. The concept of a PES was developed by Henry Eyring and Michael Polanyi.2 Consider the reversible bimolecular elementary reaction
The reaction proceeds from the reactants through a transition state (TS) region of the potential energy surface to the products. The transition state is well defined for a reaction with a positive activation energy. It is at a col in the potential energy surface, which is located at a maximum in the lowest energy path that links the reactants to the products. A look at Figs 17.3 and 17.4 allows one to quickly identify where the transition state is located. The chemical species in the transition state is the activated complex.
Figure 17.3 The classical barrier height, Eca, 0, is the energy difference from the bottom of the well to the top of the activation barrier. The quantum mechanical barrier, Eqma, 0, is a similar difference defined between reactant and transition state zero-point energy levels ZPER and ZPE‡, respectively. Ea is defined as the difference between the mean energy of the reactants ⟨E⟩R and the mean energy of the molecules in the transition state ⟨E⟩‡. The standard enthalpy of activation Δ‡H° is also shown. Reproduced with permission from Kolasinski, K.W. (2012) Surface Science: Foundations of Catalysis and Nanoscience, 3rd edition. John Wiley & Sons, Chichester.
Figure 17.4 A multidimensional potential energy surface (PES, middle panel) is often represented by a simple one-dimensional profile along the path of lowest energy (upper panel). More information is provided by a contour plot (bottom panel), which includes information about the energy away from the lowest energy path along the reaction coordinate. Provided by Fleming Crim and reproduced from Crim, F.F. (2007) Science, 317, 1707, with permission from AAAS.
When the energy of a molecule is equal to the energy of the potential energy surface of Fig. 17.3, all of the molecule’s energy is equal to potential energy. This can only happen in a classical system at T = 0 K. At any finite temperature, a molecule in the system has a kinetic energy that corresponds on average to the mean kinetic energy of the ensemble of molecules at that temperature. The mean energy of the system lies slightly above the PES. At T = 0 K, the difference between the minimum energy of the reactants and the maximum at the transition state, which corresponds to the minimum energy of the activated complex, is equal to the classical barrier height Eca, 0. Quantum mechanically, the molecules possess zero-point energy (ZPE) and the minimum energy that they can possess lies slightly above the PES, even at T = 0 K. If the reactant molecules and the activated complex had exactly the same degrees of freedom, the thermal behavior of the reactants and activated complex would be exactly the same. The activation energy would be independent of temperature and equal to the classical barrier height. This is the 90% correct answer for the activation energy: it is equal to the minimum energy required for a reaction. This answer is 100% correct at T = 0 K, which unfortunately is a temperature at which no reactivity occurs. The 100% correct answer requires a little more subtlety.
As shown by Fowler and Guggenheim,3 the activation energy is given by the difference between the mean energy of the reactants ⟨E⟩R and the mean energy of the molecules in the transition state ⟨E⟩‡
Since both ⟨E⟩R and ⟨E⟩‡ are temperature-dependent, Ea is, in principle, also temperature-dependent. As can be seen in Fig. 17.3, the classical and actual activation energies are identical at T = 0 K. At any other temperature, Ea and Eca, 0 are slightly different.
To account for the expected temperature dependence, it is useful to introduce a more general mathematical definition of the activation energy
Frequently, it is found experimentally that Eq. (17.19) obeys the form
which is exactly what we derived in Eq. (16.48). The zero point energy of a molecule depends on its vibrational structure. The slight difference of zero-point energies between the reactants and products causes a slight difference between Eqma, 0 and Eca, 0.
Most reactions exhibit a positive activation energy. Some reactions – that is, radical recombination and non-activated adsorption – do not exhibit an activation barrier, so that Ea = 0. Some reactions, particularly if they proceed through a stable intermediate, have Ea < 0. For a simple reaction such as the dissociative adsorption of H2 on a metal surface, or the combination of atoms to form a diatomic molecule,4 the reaction coordinate is simply the distance between the two atoms. For a more complex reaction system, the PES corresponds to a higher dimensional surface (or hypersurface). A diagram of the type shown in Fig. 17.3 is a representation of the minimum energy path through this PES leading from reactants through the activated complex to the products. Such a one-dimensional curve is often called a Lennard–Jones diagram.5 The relationship of this minimum energy pathway to a contour plot of a PES for a bimolecular reaction is shown in Fig. 17.4.
The trajectory of a reactive collision can be thought of in terms of ‘bobsledding’ along this PES. If the collision partners could maintain the lowest energy configuration at zero kinetic energy, but still somehow manage to move forward, they would trace out the minimum energy pathway represented in the Lennard–Jones diagram at the top of the figure. Putting enough energy into the system allows the collision partners not only to overcome the activation barrier but also to ride up the walls and round the corner into the product valley. The shape of the PES causes different forms of energy (rotational, vibration and translational) to have different efficacies for promoting reaction. This point is discussed in Section 17.11.
17.4 Transition-state theory (TST)
A general theoretical foundation for the calculation of rate constants was developed by Eyring, Evans and Polanyi6 and is called transition-state theory. Assume that a rapid equilibrium exists between the reactants A and B and the activated complex AB‡ before it decays into products P. Rapid equilibrium demands that the first step of the reaction is reversible. The second step is irreversible. This leads to the following set of equations, rate constants and equilibrium constant that describe the system,
c0 is a standard concentration used to keep the equilibrium constant dimensionless. The rate of product formation is found from the unimolecular decay of AB‡
This expression is not experimentally accessible since we cannot measure the concentration of the activated complex [AB‡]. We solve the equilibrium constant expression in Eq. (17.22) for [AB‡]
substitute this into Eq. (17.23), and the rate express becomes
To derive a value for the rate constant k2, consider the following. Think of the activated complex as an unstable complex that can dissociate into either the product P or back into the reactants A and B. The dissociation occurs by breaking a bond that has a vibrational frequency ν. Each time the bond extends it has some probability to form a product. The probability of the activated complex forming the product is called the transmission coefficient κ. Therefore, the rate constant and rate law are
The transmission coefficient is often close to unity, and κ = 1 is generally assumed as a first approximation. There are exceptions. We will see below than an important exception is adsorption – particularly dissociative adsorption – at surfaces.
The change in Gibbs energy for entering the transition state is related to the equilibrium constant by
We can relate the rate constant to the Gibbs energy of activation Δ‡G°m. With a bit of statistical mechanics (which we do not need to go into here, see Laidler or Houston in Further Reading) we can show that
Further analysis using Δ‡G = Δ‡H − TΔ‡S shows us that the pre-exponential factor is related to the entropy of activation and the activation energy is related to the enthalpy of activation
This is the Eyring equation, from which we see that the pre-exponential factor is related to the entropy of activation A∝exp (Δ‡S°m) and the activation energy is related to the enthalpy of activation Ea∝Δ‡H°m. The exact dependence of A and Ea on Δ‡S°m and Δ‡H°m depends on the molecularity and the phase of the reaction, as stated explicitly in Table 17.1.
Table 17.1 Dependence of the expressions for activation energy and pre-exponential factor on molecularity and phase for elementary reactions.
Solutions | Gas phase | |||
|---|---|---|---|---|
| Molecularity | Ea | A | Ea | A |
| Uni | Δ‡H + RT |  | Δ‡H + RT |  |
| Bi | Δ‡H + RT |  | Δ‡H + 2RT |  |
| Ter | Δ‡H + 3RT |  | ||
The activation parameters Δ‡S°m, Δ‡H°m and Δ‡G°m are given in Table 17.2 for selected first-order reactions, and in Table 17.3 for selected second-order reactions. Reviewing these results we see that the behavior of first-order reactions is decidedly different than that of second-order reactions. For first-order reactions, the activation enthalpy is always positive and is often the dominant factor in determining Δ‡G°m, though the effects of entropy cannot be ignored. Most unimolecular elementary reactions do not exhibit the expected behavior that Δ‡S°m is unfavorable (negative). For second-order reactions, on the other hand, Δ‡S°m is in the range of −110 to −70 J K−1 mol−1 in the cases given. The entropy factor is unfavorable as expected, and plays a much more important role in determining the value of Δ‡G°m. The enthalpy of activation ranges widely from negative to positive values (−6 to 116 kJ mol−1).
Table 17.2 Activation parameters derived from the Arrhenius parameters for selected first-order reactions in the gas phase. Values calculated from the values in Table 16.2. No entry is made if the temperature lies outside the valid range of the Arrhenius parameters.
Δ‡S°m / J K−1 mol−1 | Δ‡H°m / kJ mol−1 | Δ‡G°m / kJ mol−1 | Δ‡S°m / J K−1 mol−1 | Δ‡H°m / kJ mol−1 | Δ‡G°m / kJ mol−1 | |
|---|---|---|---|---|---|---|
| Reaction | T = 380 K | T = 1200 K | ||||
| CH3CHO → CH3 + HCO | 33.45 | 318.4 | 278.3 | |||
| C2H6 → 2CH3 | 58.67 | 364.4 | 342.1 | 49.11 | 357.6 | 298.6 |
| O3 → O2 + O | −8.66 | 99.3 | 102.6 | |||
| HNO3 → OH + NO2 | 32.70 | 196.4 | 183.9 | 23.14 | 189.6 | 161.8 |
| CH3OH → CH3 + OH | 66.62 | 383.6 | 358.3 | 57.06 | 376.8 | 308.3 |
| N2O → N2 + O | 18.24 | 224.6 | 202.7 | |||
| N2O5 → NO3 + NO2 | 35.82 | 231.4 | 217.8 | |||
| C2H5 → C2H4 + H | 2.19 | 167.1 | 166.3 | −7.38 | 160.3 | 169.1 |
| CH3NC → CH3CN | 4.72 | 157.9 | 156.1 | |||
| c-propane → propene | 12.06 | 251.2 | 236.7 | |||
| c-butane → 2C2H4 | 29.28 | 243.5 | 208.4 | |||
Table 17.3 Activation parameters derived from the Arrhenius parameters for selected second-order reactions in the gas phase at T = 300 K. Values for A and Ea adapted from Houston, P.L. (2001) Chemical Kinetics and Reaction Dynamics. Dover Publications, Mineola, NY.
| Reaction | A / m3 mol−1 s−1 | Ea / kJ mol−1 | Δ‡S°m / J K−1 mol−1 | Δ‡H°m / kJ mol−1 | Δ‡G°m / kJ mol−1 |
|---|---|---|---|---|---|
| CH3 + H2 → CH4 + H | 6.7 × 106 | 52 | −100.23 | 47.0 | 77.0 |
| CH3 + O2 → CH3O + O | 2.4 × 107 | 121 | −89.62 | 115.7 | 142.6 |
| F + H2 → HF + H | 8.4 × 107 | 4.16 | −79.20 | −0.8 | 22.9 |
| H + H2O → OH + H2 | 9.7 × 107 | 86 | −78.01 | 81.0 | 104.4 |
| H + CH4 → CH3 + H2 | 1.9 × 108 | 50.8 | −72.42 | 45.8 | 67.5 |
| H + O2 → OH + O | 1.4 × 108 | 66.5 | −74.96 | 61.5 | 84.0 |
| H + CO2 → CO + OH | 1.5 × 108 | 111 | −74.38 | 105.6 | 127.9 |
| H + O3 → O2 + OH | 8.4 × 107 | 3.99 | −79.20 | −1.0 | 22.8 |
| O + H2 → OH + H | 5.2 × 107 | 41.6 | −83.19 | 36.6 | 61.5 |
| O + CH4 → OH + CH3 | 1.0 × 108 | 42.3 | −77.75 | 37.3 | 60.7 |
| O + NO2 → NO + O2 | 3.9 × 106 | −0.998 | −104.73 | −6.0 | 25.4 |
| O + C2H2 → products | 3.4 × 107 | 14.6 | −86.72 | 9.6 | 35.7 |
| O + O3 → 2O2 | 7.5 × 106 | 17.8 | −99.29 | 12.8 | 42.6 |
| O + C2H4 → products | 9.5 × 106 | 7.87 | −97.32 | 2.9 | 32.1 |
| OH + H2 → H + H2O | 4.6 × 106 | 17.5 | −103.35 | 12.5 | 43.5 |
| OH + CH4 → H2O + CH3 | 2.2 × 106 | 15.1 | −109.49 | 10.1 | 43.0 |
| OH + O → O2 + H | 1.4 × 107 | −0.915 | −94.10 | −5.9 | 22.3 |
Note further an interesting aspect of the data in Table 17.2. Entries in the table are only made when constant values of A and Ea found in Table 16.6 are valid at the chosen temperature. Constant values of A and Ea are able to model the reaction rate of a given reaction over an extended temperature range, even though neither Δ‡S°m nor Δ‡H°m is constant. The reason for this is that Δ‡S°m and Δ‡H°m change sympathetically with temperature. They both decrease with increasing temperature. Whereas, a decrease in Δ‡S°m decreases the rate, a decrease in Δ‡H°m increases the rate. The effects counterbalance one another and allow constant values of A and Ea to approximate the rate quite accurately over extended ranges.
17.4.1 Directed practice
Confirm the values of Δ‡S°m, Δ‡H°m and Δ‡G°m in Table 17.2 for the unimolecular dissociation of HNO3.
17.5 Composite reactions and mechanisms
Many reactions exhibit first-order or second-order kinetics. More often than not, this is not an indication that the reaction is a unimolecular or bimolecular reaction; rather, it is more likely a reflection of a well-defined rate-determining step that is unimolecular or bimolecular. Most reactions follow complicated, multistep mechanisms.
There are a number of clues that a reaction mechanism is composite rather than elementary, and these clues also help us to decipher the mechanism. A clear indication that a reaction is complex is the presence of an intermediate. An intermediate does not appear in the balanced stoichiometric equation for the reaction. The intermediate usually forms after a short induction period, with its concentration going through a maximum before disappearing completely as the reaction goes to completion. The presence of an intermediate indicates that there must be at least two steps in the reaction, because one step is required to generate the intermediate and a separate step is required to consume it.
If a reaction has an order other than zero, one, two, or three, it must be complex. In particular, noninteger orders indicate a complex, multistep reaction mechanism. A reaction order that changes as a function of temperature also indicates a multistep reaction. A noninteger order indicates that more than one elementary step is contributing to the reaction rate. If the rate-determining step were first-order at low temperature and second-order at high temperature, then in the intermediate temperature regime the order would be intermediate between first- and second-order. Curvature would also be noted in an Arrhenius plot of ln k versus T. A linear plot would be observed at low T, corresponding to the activation energy of the first rate-determining step, followed by a curved intermediate region, and ending in a linear region with a slope corresponding to the high-T rate-determining activation energy.
Recall the definition of a reaction mechanism.
A reaction mechanism is a complete sequence of elementary reactions (steps) that add together to give the stoichiometric equation of a reaction.
An individual step may have a stoichiometric number that is different from 1. For example, a step may have to occur more than once in a reaction to achieve the proper stoichiometry. The correct mechanism must give the correct mathematic form of the concentration versus time curves for every reactant, intermediate and product in the reaction. Always remember the asymmetry in the burden of proof: If a mechanism makes a prediction about the kinetics of a reaction and an accurate experiment disproves the prediction, the mechanism does not describe the reaction. It is the wrong mechanism. However, if a mechanism makes a prediction that is substantiated by an accurate experiment, this only proves that the mechanism is contained in the set of all possible mechanism that might describe the reaction. Proof requires many positive reinforcements. This is the difference between necessary and sufficient conditions.
17.5.1 Types of steps
There are a number of different ways elementary reactions can be combined into a reaction mechanism and we define them here. A mechanism may consist of a series of consecutive steps that occur one after the other. A simple two-step example is as follows.
The Arrhenius parameters of the rate constant are the pre-exponential factor and the activation energy. In any reaction composed of a series of consecutive steps, the kinetics of one step may be of singular importance and is known as the rate-determining step (RDS). We defined the RDS in Chapter 16. The concentration versus time curves of A, I and P for a reaction with the simple two-step mechanism defined above is shown in Fig. 17.5. Notice how the concentration of the intermediate progresses through a maximum.
Figure 17.5 Concentration versus time curves for a reaction mechanism composed of two consecutive unimolecular elementary steps in series. In panels (a) through (d) the rate constant of the first step remains constant, but the rate constant of the second step increases by one order of magnitude in each step. The concentration profile of the reactant A is unaffected by this change, but those of the intermediate I and the product P change dramatically in response. All concentrations are normalized to the initial concentration of the reactant a0.
The kinetics of this mechanism lead to the following rate equations. The reactant A is involved in only one step. Therefore, the rate of disappearance of A is related to the rate of the first step alone
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