Kinetics and Dynamics of Elementary Gas Reactions: Butterworths Monographs in Chemistry and Chemical Engineering

By Ian W. M. Smith

Kinetics and Dynamics of Elementary Gas Reactions surveys the state of modern knowledge on elementary gas reactions to understand natural phenomena in terms of molecular behavior. Part 1 of this book describes the theoretical and conceptual background of elementary gas-phase reactions, emphasizing the assumptions and limitations of each theoretical approach, as well as its strengths. In Part 2, selected experimental results are considered to demonstrate the scope of present day techniques and illustrate the application of the theoretical ideas introduced in Part 1. This publication is intended primarily for working kineticists and chemists, but is also beneficial to graduate students.

• Language: English
• Publisher: Elsevier Science
• Release date: Oct 22, 2013
• ISBN: 9781483161990

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Part 1

The theoretical background

Outline

Chapter 1: Macroscopic and microscopic kinetics

Chapter 2: The electronic potential energy in molecular systems

Chapter 3: Molecular collision dynamics

Chapter 4: Statistical theories of reaction rates

Chapter 1

Macroscopic and microscopic kinetics

Publisher Summary

When chemical change occurs, it usually does so through a network of interacting steps known as the reaction mechanism. The identification of mechanism in this sense is one of two major problems in the field of chemical kinetics. The other is to investigate and attempt to understand the details of chemical reactions that cannot be further subdivided into the processes of lesser molecular complexity. These processes, which necessarily involve the participation of a small integral number of molecules, are termed elementary reactions and they can be studied in greatest detail in the gas phase. This chapter describes the results of microscopic and macroscopic experiments on elementary bimolecular reactions and on establishing the relationship between various quantities that have been defined during this process. In these experiments, four interrelated energy quantities are considered: the classical energy barrier to reaction, the difference between the zero-point quantum levels of reactants and the activated state, the threshold energy, and the activation energy. Despite the connection between them, these quantities are neither conceptually nor usually quantitatively the same and therefore, the terms are used with care.

1.1 Elementary reactions

When chemical change occurs it usually does so through a network of interacting steps which is known as the reaction mechanism. The identification of mechanism in this sense is one of two major problems in the field of chemical kinetics. The other is to investigate and attempt to understand the details of chemical reactions which cannot be further subdivided into processes of lesser molecular complexity. These processes, which necessarily involve the participation of a small integral number of molecules, are termed elementary reactions and they can be studied in greatest detail in the gas phase, since in solution the role played by nearby solvent molecules cannot be neglected.

Elementary chemical reactions can usually be classified as either collisional or decay processes. The former are generally bimolecular, that is two species collide in each microscopic event that leads to reaction. Reactions of this type in which a single atom is transferred have been studied in greatest detail and they are featured very prominently in this book. An example is

(1.1)

Decay processes are unimolecular; they involve a single species that changes to a different form. A familiar example is provided by the spontaneous radiative decay which follows photochemical excitation of an atom or molecule. In unimolecular chemical processes, a molecule with high internal energy may fragment, as in

(1.2)

in which case the reverse reaction is bimolecular, or it may isomerize, as in

(1.3)

when the reverse reaction, as well as the forward reaction, is unimolecular.

The separation of elementary reactions into these two categories is useful but must be carried out with care. Thus, neither the dissociation of ethane nor the isomerization of methyl isocyanide are as simple as one might be led to believe by equations (1.2) and (1.3) and the statement that these reactions are unimolecular. As well as the unimolecular chemical step represented by these equations, collisions in which energy is transferred but no chemical change takes place play a vital role in the kinetics. A somewhat similar situation arises when a bimolecular encounter is ‘sticky’, that is, it leads to the formation of a collision complex which survives for a time which is longer than the characteristic periods of its vibrations and rotations. In these circumstances, the molecular event may be thought of as a bimolecular collision followed by a unimolecular decay process. An example is provided by the reaction of methyl and trifluoromethyl radicals at low pressure. This proceeds via the formation of a 1,1,1-trifluoroethane molecule with considerable internal energy, i.e.,

(1.4)

In recent years, new techniques have enabled the experimentalist to study the molecular collisions that are so important in elementary chemical reactions in increasingly fine detail. Theoretical developments have kept pace–and occasionally outstripped–the experimental progress. The purpose of this book is to introduce its readers to some of this exciting research into reaction dynamics, and to encourage them to think about the molecular level events that underlie any kinetic observation.

1.2 Macroscopic kinetics¹

The past 20 years have seen a huge improvement in the quality and quantity of kinetic data. This situation has been brought about by the invention of a whole armoury of new laboratory techniques that have enabled the experimentalist both to study elementary reactions in isolation and also to observe the rapid changes which frequently accompany them. Although this advance has been spectacular, the aim of much of this work has, in one important sense, remained conventional: to measure the progress of reaction in terms of the rate of change in concentration of a particular chemical entity (e.g., the disappearance of a reagent) over as wide a range of temperature as possible. The definition of a temperature implies that the constituents of the reacting mixture are thermally equilibrated, and this is the usual situation in what are often called ‘bulk’ or ‘bulb’ experiments.

1.2.1 The rate constant and the Arrhenius equation

When a temperature is defined for a reacting system, the rate of elementary reactions–as well as those of more complex ones–can be expressed in terms of the concentrations of the chemical species present. To illustrate this, let us consider a generalized bimolecular reaction

(1.5)

where A, B, C and D all represent different atomic or molecular species. Allowing for the possibility of reaction in both directions, then the rate equation is given by

(1.6)

The square brackets denote concentrations, and kf(T) and kr(T) are coefficients which relate the rates of the bimolecular reactions in the forward and reverse directions to the concentrations of reactants (that is the species written on the left-hand side of the chemical equation) and of products (i.e., the species on the right-hand side). Clearly, kf(T) and kr(T) must have the dimensions of concentration−1 time−1. Because a major preoccupation in this book is to relate macroscopic observables, such as kf(T) and kr(T), to molecular properties, molecule cm−3 will be used as the unit of concentration, so bimolecular rate constants will be expressed in unitsa of cm³ molecule−1 s−1. Although kf(T) and kr(T) are temperature-dependent, emphasized here by the symbols used, we shall follow the customary practice and refer to them as rate constants.

In the majority of cases, the variation of an observed rate constant with temperature can be described, within the accuracy of the experimental measurements, by the Arrhenius relationship

(1.7)

Differentiating this equation with respect to (1/RT) leads to the following definition for the activation energy, Eact

(1.8)

These two equations–and these equations alone–should be used to define Eact and A; A possesses the same dimensions as k(T) and is termed the pre-exponential factor.

The purpose of conventional kinetic studies is to measure k(T) over a wide range of temperature and thus to determine A, Eact, and whether these parameters themselves vary with T. Partly to stress the macroscopic nature of the thermal rate constant, the relationship between the Arrhenius parameters and their thermodynamic counterparts is considered next.

1.2.2 Kinetics and thermodynamics

A system at chemical equilibrium is not microscopically static but no overall change is observed because each elementary reaction is proceeding at an equal rate in the forward and reverse directions. Therefore, if reaction (1.5) is again taken as an example, at equilibrium

(1.9)

where the subscripts are used to denote values at equilibrium, and kf and kr replace kf(T) and kr(T) for the sake of simplicity. Rearrangement of this equation yields

(1.10)

where the right-hand side is simply Kc, the equilibrium constant expressed in terms of concentrations. Hence

(1.11)

This equation, relating the rate constants of chemical kinetics to the equilibrium constant of chemical ‘statics’ or thermodynamics, gives algebraic form to the principle of detailed balancing which states that at equilibrium every molecular process proceeds on average equally fast in both directions. The microscopic basis for this law will be considered in Section 1.3.2.

The connection between kinetic and thermodynamic quantities can be carried further by differentiating the equation

(1.12)

with respect to (1/RT). Remembering equation (1.8), it is clear that the left-hand side of this equation is the difference in activation energies for the forward and reverse reactions, i.e., — (Eact, f – Eact, r). The right-hand side can be evaluated using the appropriate form of the van’t Hoff equation, i.e.,

(1.13)

Actually the standard state symbols are unnecessary for ideal gases, since E depends only on temperature and not on concentration or pressure. Therefore, one can write simply

(1.14)

A similar derivation can be used to relate the pre-exponential factors to the standard entropy change for the reaction. For a reaction such as (1.5), in which there are equal numbers of molecules on both sides of the chemical equation, this leads to

(1.15)

and once again the definition chosen for the standard state is unimportant. However, the general form of this equation is

(1.16)

It allows for a change, Δv, in the number of molecules on going from left to right in the chemical equation: thus, Δv = 1 in reaction (1.2). For reactions of order higher than one, the standard state must be specified in the same concentration units as are used in expressing the rate constants and pre-exponential factors. Further details of these relationships can be found elsewhere.¹

Since accurate thermodynamic data are available for most species, equations (1.14) to (1.16) can be useful in one of two related ways. When values of the Arrhenius parameters for a reaction in both directions have been determined independently, the equations provide an important check on the measurements. Alternatively, when only Af and Eact, f have been obtained directly, they can be used to calculate Ar and Eact, r. The equations are not, however, much help in predicting rate data in the absence of any kinetic measurements, although, as the activation energy is never both negative and large in magnitude, equation (1.14) does lead to the conclusion that where ΔE

These relationships between kinetic and thermodynamic quantities suggest that elementary reactions proceeding in opposite directions pass through a common critical state or configuration. Where Eact is positive for both reactions, then this activated state must lie at an energy higher than those of the separated reactants or separated products. Figure 1.1 illustrates this, and emphasizes diagrammatically that it is possible for the activated state to have any energy and still satisfy equation (1.14).

Figure 1.1 Relationship between ΔE and the activation energies for forward and reverse reactions. Possible activation energies are represented by the vertical arrows (Eact, f on the left; Eact, r on the right), and the diagram shows that the equation ΔE = (Eact, f – Eact, r) holds for each pair of values of Eact, f and Eact, r

Comparison of equations (1.14) and (1.16) suggests that, as well as energies of activation, it may be possible to consider entropies of activation, such that

(1.17)

and

(1.18)

Since the activated state is always a single molecular conglomerate,

(1.18a)

It then becomes possible to formulate general expressions for the rate constants which include the entropies of activation:

(1.19)

(1.20)

where ρ is a constant for all reactions.

These expressions, which relate rate constants to differences in thermodynamic quantities for the system (a) as separated reactants and (b) in the activated state, imply that in both configurations the species are in internal thermal equilibrium; that is, they are distributed among the available energy states according to the Boltzmann laws of statistical thermodynamics. This type of equilibrium is usually maintained by collisions in which energy is transferred. The assumption of thermal equilibration of the reactants will be valid as long as molecules in every state making a significant contribution to the reaction rate react over a time which is longer than the ‘relaxation’ time characterizing the energy transfer processes. For bimolecular reactions this is likely to be so under any but the most extreme conditions, but for unimolecular reactions, as was indicated in Section 1.1, the energy transferring processes must be included in any general discussion of the kinetics. The question of whether species in the activated state are in internal thermal equilibrium is much more difficult to answer. This problem will be considered at length in Section 4.2.2 where the fundamental assumptions of transition state theory are examined.

In the past few pages, some emphasis has been placed on the macroscopic nature of the thermal rate constant. Clearly, even for an elementary reaction, k(T) measures the average result of a multitude of individual molecular events which differ in their detailed characteristics–for example, the collision energy or the orientation of the species at ‘impact’. Any fully detailed theory of reaction rates should start at the molecular level, and derive an expression for k(T) by performing correctly weighted summations or integrations over the variables that characterize the molecular events. Since the statistical laws governing the distributions of these variables are well known, this averaging is not difficult. Unfortunately, the reverse process of deriving the microscopic probabilities of reaction from the measured rate constant and its variation with temperature is impossible. Consequently, any experiment which determines the probability of reaction in events which are, at least to some extent, selected provides a more useful test of a detailed theory than the rate data provided by conventional kinetic experiments. It is this kind of approach which is introduced next.

1.3 Microscopic kinetics

The realization that conventional rate data provide little information about elementary reactions at the molecular level has prompted a growing number of kineticists to devise experiments under non-equilibrium conditions. Those which are potentially most powerful, and which differ most dramatically from ‘bulb’ experiments, use crossed molecular beams.

In these experiments, each reactant is formed into a molecular beam characterized by a well-defined direction. The velocities of the species in each beam and their internal states can–at least, in principle–be selected before the beams intersect in a well-defined crossing zone. The spatial distribution of the products of the collisions that occur in the crossing region, and possibly their translational velocity and internal energy states, are measured. The method is powerful, because the results of single well-defined collisions are observed, and because the results are more closely related to the dynamics in the collisions and the forces that control them than the data from conventional ‘bulk’ experiments.

1.3.1 The total and differential reaction cross-sections²,³

In order to specify the rate or probability of reaction in the kind of experiment that has just been described, it is necessary to define the reaction cross-section, a quantity which occupies a position in ‘microscopic’ kinetics that is equivalent to that of the rate constant in ‘macroscopic’ kinetics. The meaning of the reaction cross-section can be made clear by reference to an idealized experiment in which a stream of species A, all with the same velocitya v, move through a collection of B molecules having a uniform concentration [B].

If the concentration in the beam is [A], its intensity is given by

(1.21)

As the beam passes through the volume containing B some A particles will be scattered and the beam intensity reduced. Assuming that only single two-body collisions are important, the change in intensity of the beam on passing through a thickness dl of the scattering medium is

(1.22)

This equation is exactly analogous to Beer’s law describing the attenuation of light (i.e., a ‘beam’ of photons) by an absorbing medium, and it defines S(v), the total scattering cross-section, which generally depends on v and corresponds to an effective area presented by a molecule of B to the approaching molecules of A (or vice versa).

When chemical reaction between A and B can occur, the total reduction in IA results from reactive and non-reactive collisions, and S(v) can be considered as the sum of two independent quantities: Sr(v), the reaction cross-section, defined by

(1.23)

where dIA, r is the change in beam intensity due only to reactive collisions, and S(v) – Sr(v), the cross-section for non-reactive scattering. Since IA = v[A] and v = dl/dt,

(1.24)

and this equation shows that vSr(v) defines a rate coefficient for reaction in collisions between A + B occurring at a specified relative velocitya, v.

Measuring the overall decrease in incident beam intensity does not distinguish reactive from non-reactive scattering and cannot by itself yield a value for Sr. To obtain Sr, it is necessary to detect one of the scattered reaction products. Further discussion requires the introduction of the differential reaction cross-section. This is related to the flux of product molecules reaching a detector that is positioned at an angle θ relative to v and that subtends a solid angle dω at the point where the beams intersect. Alternatively, the differential reaction cross-section, σr(v, θ), can be defined by

(1.25)

It is that part of the total reaction cross-section which results in product being scattered at an angle θ into an element of solid angle dω. Clearly

(1.26)

The relationships between θ and dθ, and ω and dω, are explained fully in Section 3.3.

Thus far, no mention has been made of the internal states of A, B, C and D. If it is supposed that n and n′ denote sets of quantum numbers specifying the states of A + B and of C + D, respectively, then σr(n′|n; v, θ) represents the differential cross-section for reactive scattering, it being assumed that both selection of the internal states of A and B and the specific detection of the states of C and D are possible. The final relative speed of the reaction products, i.e., v,′ need not be specified explicitly, as to conserve energy it is necessary that the difference between the relative translational energy before and after the collision, εT = p²/2/μ and εT′ = p′²/2μ′, is equal to the difference in energy of the states specified by n and n′, i.e.,

(1.27)

where Δεint = εn′–εn + Δε0, as shown later in Figure 1.3a.

Figure 1.3 Diagram showing the relationships between several of the energy quantities introduced in Section 1.3. The solid line represents the profile of electronic potential energy, and the dashed line the energy of the lowest quantum state, along the path of minimum energy leading from reactants to products; εn and εn′ are energies of a particular pair of reactant and product states

1.3.2 Microscopic reversibility

³

A stage has now been reached at which it is possible to consider the principle of microscopic reversibility. This principle arises from the invariance of the Schrödinger equation–and the classical equations of motion–under time reversal. Because of this, the probability of a transition per unit time between a fully specified state of the reactants and a fully specified state of the products is independent of the direction in which time is chosen to move, i.e.

(1.28)

Because of this relationship, the ratio of the differential detailed rate coefficients [see equation (1.24)] connecting any pair of fully specified reactant and product states must be equal to the ratio of the ‘phase space volumes’ associated with the relative momenta of the reactants and products which are connected through equation (1.27). Further discussion of the principle of statistical mechanics on which this assertion is founded is given in Section 4.1. It leads to the equation

(1.29)

which is the microscopic equivalent of equation (1.11). As vσr(n′|n; v, θ) and v′σr(n | n′; v′, θ) are ‘differential detailed rate coefficients’ for the reaction in the forward and reverse directions involving state-specified species, this equation may be thought of as defining a ‘differential detailed equilibrium coefficient’. Equation (1.29) can be simplified by noting that differentiation of (1.27) yields v′dp′ = vdp so that

(1.30)

These expressions require some modification when the species involved possess internal angular momentum. Equation (1.30) becomes

(1.31)

where gn and gn′ are the total degeneracies associated with the angular momentum quantum states of A + B and of C + D. The σr are cross-sections averaged over such states, and defined by

(1.32)

where mn and mn′ are quantum numbers or combination of quantum numbers which differ only in the projection of angular momentum vectors along a chosen axis. Now n and n′ indicate specification of all quantum numbers other than these m quantum numbers.

Equations (1.28) to (1.31) incapsulate the principle of microscopic reversibility. In addition, this section has introduced the parameters which could describe the results of experiments at the ultimate microscopic level. Next we consider how these parameters are related to data which it is possible to obtain from experiments of different types through to those in which only the thermal rate constant is determined.

1.3.3 The relationship between the rate constant and the reaction cross-section

The ultimate goal of experiments in reaction dynamics is to determine sets of differential cross-sections for processes connecting specified reactant and product states in collisions of well-defined relative velocity. At the present time, it remains a distant objective, since the present generation of experiments only yield quantities that are averaged over the parameters that define individual collisions to a greater extent than σr(n′|n; v, θ).

Only crossed molecular beam experiments yield differential reaction cross-sections. It is now fairly standard in such experiments (Section 7.2) to employ velocity selection of the reactants and velocity analysis of the products, but rarer to select the internal reactant states or to measure the distribution over product states directly. However, the distribution of products with respect to total internal energy can be obtained by making use of the energy conservation equation

(1.33)

where εT and εn are the relative translational and internal energies of the reactants (unprimed) and products (primed), and Δε0 is the energy difference between the zero-point levels of products and reactants, as shown in Figure 1.3a. Because, even in a thermal distribution, the spread of εn values is usually small compared with Δε0, this equation yields the distribution of εn′ values if εT is selected and the distribution of εT′ values is measured directly.

If differential reaction cross-sections are determined over a sufficiently wide range of scattering angles, the integration in equation (1.26) can be carried out, and a total reaction cross-section obtained. Where no selection or analysis of internal states is performed, the value of Sr(v) is one averaged over the distribution of reactant states and includes equally weighted contributions from all the processes in which products are formed independent of their states, i.e.,

(1.34)

Here fn is the fraction of the total number of collisions that occurs with A and B in the combination of states denoted by n. If the distributions over these internal states are equilibrated at temperature T,

(1.35)

where Qint, A and Qint, B are the partition functions associated with the internal states of A and B (see Appendix 3), and gn and εn are the degeneracies and energies for the specified combination of internal states, the latter being referred to the zero-point level of the reactants.

The form of the variation of Sr(v) with v is usually termed the (state averaged) excitation function. These have been determined for a few reactions in experiments using either velocity selected crossed molecular beams or the photolytic production of reactive ‘hot’ atoms with defined amounts of excess translational energy.

Next we can consider the results of experiments where either the internal states of the reactants are selected, or those of the products are observed, but collisions occur with the full thermal spread of relative velocities defined by a temperature T. Then the detailed rate constant for formation of products in states denoted by n′ from reactants in states n is given by

(1.36)

where f(v; T) dv is the fraction of collisions which, in a thermal distribution of relative velocities defined by the temperature T, occur with a relative translational velocity between v and v + dv. This distribution function can be derived by straightforward application of the methods of statistical mechanics (Appendix 3). Substitution of the result into equation (1.36) yields

(1.37)

The equation for the detailed rate constant for the reverse reaction connecting this combination of reactant and product states has an identical form

(1.38)

Now, equation (1.27) and its derivative form, μvdv = μ′v′dv′, together with the microscopic reversibility equation (1.29), can be used to derive the relationship between k(n′ | n; T) and k(n | n′; T):

(1.39)

This equation is, of course, analogous to equation (1.29) on the one hand and equation (1.11) on the other.

Some experiments, such as those where infrared chemiluminescence is observed (Section 6.2.1), yield detailed rate constants for the formation of products in specific states but from reactants that are totally equilibrated. Consequently, values of k(n′ |; T) are obtained where

(1.40)

and fn was defined by equation (1.35).

The final connection to the thermal rate constant of macroscopic kinetics then simply requires a further summation over all the states of the products, i.e.,

(1.41)

Similar equations to (1.40) and (1.41) exist for k(n |; T) and kr(T), that is for the reverse reaction, and combining all of these with equation (1.39) leads to

(1.42)

where ΠQint is the product of partition functions associated with internal states of the reactants, i.e., Qint, A Qint, B, and ΠQ′int is the equivalent quantity for the products. From statistical mechanics (see Section 4.1), the expression on the right-hand side of this equation is simply Kc, the equilibrium constant for the reaction, so this relationship consolidates the connection that has been established between the parameters of microscopic and macroscopic kinetics.

Finally, it should be noted that if the state-specified reaction cross-section is first averaged over states according to equation (1.34) and then the averaging over f(v; T) is carried out, one obtains

(1.43)

This equation shows clearly that k(T) is the average over the relative velocity distribution of the rate coefficient {vSr(v)} that was introduced in equation (1.24). Thus

(1.44)

Frequently, a (mean) cross-section is quoted following the measurement of a rate constant in a bulk experiment. This is calculated from an approximate form of this expression:

(1.45)

where < v > = (8kBT/πμ)½ is the average relative velocity.

1.4 Towards a detailed rate theory for collisional reactions²–⁴

The links between the microscopic and macroscopic descriptions of kinetics having been firmly established, it is now possible to consider how a fully detailed theory of molecular collision processes may be formulated. Such a theory should first involve the calculation of fully detailed differential cross-sections, i.e., values of σr(n′|n; v, θ), and then use the equations derived in Section 1.3 to determine values of the parameters–for example, excitation functions, detailed rate constants, total rate constants–that are required.

In calculating the molecular dynamics of fully specified collisions use is made of the famous Born–Oppenheimer principle. This states that the electrostatic forces acting in a molecular system can be assumed to be independent of the nuclear motions, and can therefore be calculated with the nuclear positions .

1.4.1 The potential hypersurface

for a system of N(>2) atoms depends on 3N − 6 coordinates defining the relative positions of the nuclei, the potential cannot be represented graphically, unless the number of independent variables is reduced. For an atom-transfer reaction that involves only three atoms, for example

(1.46)

where A, B and C now each represent a single atom, this might be done by just considering collinear configurations. Then can be plotted as a contour diagram of the type shown in Figure 1.2, and reference can be made to a potential energy surface (rAB, rBC, rCA) is termed a potential energy hypersurface.

Figure 1.2 Potential-energy surface for the collinear reaction HA + HBHC → HAHB + HC, as calculated by Porter and Karplus [Journal of Chemical Physics, 40, 1105 (1964)]. The contour lines join points of equal potential energy (1 eV ≡ 96.5 kJ mol−1). The dashed line indicates the reaction path of minimum energy and the cross the highest point on this path

In general, there are a number of such hypersurfaces, each corresponding to a different electronic state of the molecular system. Except where explicitly stated we shall be considering reactions which proceed ‘across’ a single hypersurface–usually the lowest. Such reactions are termed ‘electronically adiabatic’. Where hypersurfaces intersect, it is necessary to consider the influence of additional quantum mechanical effects (Sections 2.4 and 4.4).

For atom-transfer reactions involving neutral species, one frequently finds that any ‘route’ from reactants to products must pass through configurations of higher electronic potential energy. As an example of this, the dashed line in Figure 1.2 traces out the path of minimum energy between separated reactants and separated products and the cross marks the highest point on this curve. The difference in energy between this point and the separated reagents constitutes the classical energy barrier for collinear reaction. There are, of course, an infinite number of potential energy surfaces (each with a different restriction on the independent variables) on which barriers of varying height are located, but it is common practice to assume that the expression ‘energy barrier’ refers to the minimum potential barrier between reactants and products, and there are indications that for reactions like (1.46) this barrier is often associated with a collinear conformation.

The barrier is defined as ‘classical’, because no reference has yet been made to the quantized behaviour of nuclei moving under the influence of the potential. As rsolving the Schrödinger equation for the nuclei yields the vibrational – rotational states of BC, and the lowest of these lies some way above the minimum of the potential curve. Similar quantum effects are present at all points on the hypersurface, but the magnitude of the ‘zero-point energy’ varies. One result of this is illustrated in , the difference in the potential energies at these points.

1.4.2 The relationship between the threshold energy and the activation energy³

The discussion in the last two paragraphs suggests that, for an individual collision to have any chance of leading to reaction when there is an energy barrier separating reactants and products, the sum of the internal and relative translational energies, εn + εT, will need to exceed some critical value. This introduces the concept of a threshold energy, which is taken to be the value of the collision energy below which Sr(n′ | n; εT) is effectively zero, i.e., for

and for

(1.47)

Consequently, if equation (1.37) for the detailed rate constant is rewritten in terms of εT, making the substitutions v² = 2εT/μ and v dv = dεT/μ,

(1.48)

can be written as the lower limit of the integration. The threshold energy for collisions without state selection can clearly be defined in an exactly analogous way.

. A further complication arises from the fact that atoms obey quantum, rather than classical, mechanics. This means that the energy barrier, which cannot be surmounted by classical systems possessing kinetic energy less than the barrier height, can be penetrated with low probability by quantum mechanical systems (see Sections 3.6 and 4.4). As a result of this quantum mechanical ‘tunnelling’ effect, Sr(εT) will never be exactly zero for reactions where Δεint is negative.

Ignoring tunnelling, equation (1.48) or the corresponding equation for k(T) can be used to derive the relationship between the activation energy and the threshold energy. Remembering that the activation energy is defined by

(1.8)

a general expression for Eact for a state-specified reaction can be obtained by operating on equation (1.48) according to (1.8). This yields

(1.49)

where NA is Avogadro’s constant. The first term on the right-hand side of this equation is the average collision energy of those collisions that actually lead to reaction, whereas 3/2 RT is, of course, the mean collision energy of all collisions. Therefore,

(1.50)

The corresponding treatment of k(T), which is algebraically somewhat more complex, leads to the result that the activation energy is the difference between the mean collision energy plus internal energy for those collisions leading to reaction and the same quantity for all collisions, i.e.,

(1.51)

The nature of the relationship between the activation energy and the threshold energy for some different forms of the excitation function is discussed in Section 3.4.4.

1.5 Summary

), and the activation energy (Eact). Despite the connection between them, these quantities are neither conceptually nor usually quantitatively the same, and therefore the terms should be used with