3.Introduction to reactive molecular collisions We examine in this chapter how the motion of the reactants as they approach each other governs chemical reactivity.This allows us to use a two-body point of view where the internal structure of the colliding specie is not explicitly recognized.All that we can do therefore is lead the reactants up to a reaction.But we will not be able to describe the chemical rearrangement itself nor to address such questions as energy disposal in the products.Chapter 5 takes up these themes.On the other hand,without the approach of the reactants there cannot be a bimolecular reaction.The tools already at our disposal are sufficient to discuss this approach motion.As expected,the striking distance that we have called the impact parameter will be a key player.We do all of this in section 3.2.What we will obtain is information about the dependence of the reaction cross section on the collision energy In chemical kinetics one characterizes the role of energy in chemical reactivity by the temperature dependence of the reaction rate constant.In section 3.1 we review the input from chemical kinetics-the Arrhenius representation of the rate constant-then go from the rate constant to the reaction cross section.Next we go in the opposite direction,from the microscopic reaction cross section to the macroscopic rate constant.What we obtain thereby is the Tolman interpretation of the activation energy as the(mean)excess energy of those collisions that lead to reaction. MRD Chapter 3 page 1 ©RD Levine(2003)
3. Introduction to reactive molecular collisions We examine in this chapter how the motion of the reactants as they approach each other governs chemical reactivity. This allows us to use a two-body point of view where the internal structure of the colliding specie is not explicitly recognized. All that we can do therefore is lead the reactants up to a reaction. But we will not be able to describe the chemical rearrangement itself nor to address such questions as energy disposal in the products. Chapter 5 takes up these themes. On the other hand, without the approach of the reactants there cannot be a bimolecular reaction. The tools already at our disposal are sufficient to discuss this approach motion. As expected, the striking distance that we have called the impact parameter will be a key player. We do all of this in section 3.2. What we will obtain is information about the dependence of the reaction cross section on the collision energy. In chemical kinetics one characterizes the role of energy in chemical reactivity by the temperature dependence of the reaction rate constant. In section 3.1 we review the input from chemical kinetics-the Arrhenius representation of the rate constant-then go from the rate constant to the reaction cross section. Next we go in the opposite direction, from the microscopic reaction cross section to the macroscopic rate constant. What we obtain thereby is the Tolman interpretation of the activation energy as the (mean) excess energy of those collisions that lead to reaction. MRD Chapter 3 page 1 © R D Levine (2003)
3.1.The rate and cross section of chemical reactions This section is a review of the macroscopic notions that you may well be familiar with from chemical kinetics'.Our final purpose is to build rate constants from the bottom up and therefore to describe the rate of chemical reactions in systems that are not in thermal equilibrium.Thermally equilibrated reactants are more typical of the laboratory than of the real world and,even in the laboratory,it takes care and attention to insure that the reactants are indeed thermally equilibrated.Outside of the lab,whether in the internal combustion engine (which fires many thousands of times per minute),in the atmosphere and in outer regions of space,this is not the case. The second theme that we begin to explore is the role of energy in promoting chemical reactions.Macroscopically this is characterized by the temperature dependence of the reaction rate as summarized by the activation energy.We can already guess that the underlying root is the collision energy dependence of the reaction cross section.The new feature is that certain reactions have cross sections that decrease with increasing energy.In fact,this is rather typical for reactions with large cross sections2. MRD Chapter 3 page 2 ©RD Levine(2003)
3.1. The rate and cross section of chemical reactions This section is a review of the macroscopic notions that you may well be familiar with from chemical kinetics1 . Our final purpose is to build rate constants from the bottom up and therefore to describe the rate of chemical reactions in systems that are not in thermal equilibrium. Thermally equilibrated reactants are more typical of the laboratory than of the real world and, even in the laboratory, it takes care and attention to insure that the reactants are indeed thermally equilibrated. Outside of the lab, whether in the internal combustion engine (which fires many thousands of times per minute), in the atmosphere and in outer regions of space, this is not the case. The second theme that we begin to explore is the role of energy in promoting chemical reactions. Macroscopically this is characterized by the temperature dependence of the reaction rate as summarized by the activation energy. We can already guess that the underlying root is the collision energy dependence of the reaction cross section. The new feature is that certain reactions have cross sections that decrease with increasing energy. In fact, this is rather typical for reactions with large cross sections 2 . MRD Chapter 3 page 2 © R D Levine (2003)
3.1.1.The thermal reaction rate constant The rate of an elementary gas phase bimolecular reaction,say CI+CH4→HCI+CH3 O+CS→S+CO F+HCI→CI+HF is characterized by thermal reaction rate constant k(T)which is a function of the temperature only.This rate constant is 'constant'meaning that it is not a function of time and it is a measure of the rate of depletion of the reactants (that are kept in a thermal bath)or the rate of appearance of the products.Because the reaction is bimolecular,it is a second order rate constant,e.g. d四_dC=(LF]HC] (3.1) For our purpose it is essential to emphasize that the thermal reaction rate is defined only when the experiment does maintain a thermal equilibrium for the reactants.If necessary,the reaction needs to be slowed down,say by the addition of a buffer gas,so that non-reactive collisions rapidly restore the reactants to thermal equilibrium.If this is not possible,appendix 3.A introduces the reaction rate constant under more general conditions.Under non- equilibrium conditions the rate constant defined through equation(3.1))may however depend on other variables such as the pressure and even on time. The experimental temperature dependence of the thermal reaction rate constant is often represented in an Arrhenius form k(T)=Aexp(-Ea/kBT) (3.2) We write the Boltzmann constant k with a subscript B to avoid confusion with the reaction rate constant.If you prefer to measure the activation energy per mole,the exponent needs to be written as Ea/RT where R=NAkB is the gas constant and NA is Avogadro's number. MRD Chapter 3 page 3 ©R D Levine(2003)
3.1.1. The thermal reaction rate constant The rate of an elementary gas phase bimolecular reaction, say Cl + CH4 → HCl + CH3 O + CS → S+ CO F + HCl → Cl + HF is characterized by thermal reaction rate constant k(T) which is a function of the temperature only. This rate constant is ‘constant’ meaning that it is not a function of time and it is a measure of the rate of depletion of the reactants (that are kept in a thermal bath) or the rate of appearance of the products. Because the reaction is bimolecular, it is a second order rate constant, e.g. − d[F] dt = d[Cl] dt = k(T)[F][HCl] (3.1) For our purpose it is essential to emphasize that the thermal reaction rate is defined only when the experiment does maintain a thermal equilibrium for the reactants. If necessary, the reaction needs to be slowed down, say by the addition of a buffer gas, so that non-reactive collisions rapidly restore the reactants to thermal equilibrium. If this is not possible, appendix 3.A introduces the reaction rate constant under more general conditions. Under nonequilibrium conditions the rate constant defined through equation (3.1)) may however depend on other variables such as the pressure and even on time. The experimental temperature dependence of the thermal reaction rate constant is often represented in an Arrhenius form k(T) = Aexp(−Ea / kBT) (3.2) We write the Boltzmann constant k with a subscript B to avoid confusion with the reaction rate constant. If you prefer to measure the activation energy per mole, the exponent needs to be written as Ea / RT where R = NAkB is the gas constant and NA is Avogadro's number. MRD Chapter 3 page 3 © R D Levine (2003)
If measurement are carried out over a wide range in 1/T there is no real reason to expect that either the Arrhenius 'A factor'or the activation energy E is independent of temperature. Therefore a strict definition of the activation energy is as the local(meaning,possibly T dependent)slope of the Arrhenius plot of k(T)vs.1/T Ea=-kB- 0-知2D (3.3) d dT This is the definition that we shall use. The chemical change as observed in a macroscopic experiment is the result of many molecular collisions.In section 3.1.2 we define the reaction cross section from the macroscopic observable rate of reactive collisions.This is sufficient to discuss the dependence of the cross section on the collision energy.Then we discuss the temperature dependence of the reaction rate as arising from this energy dependence.To understand the origin of the energy needs of chemical reactions,we provide,in section 3.2,a microscopic interpretation of the reaction cross section.Finally,a caveat.In this chapter we are still not taking account of the internal structure of the reactants.This key topic has to wait until chapter 5.The only concession is that an appendix provides an extension of chemical kinetics to the important special case where we do resolve internal states 3.1.2 The reaction cross-section-a macroscopic view We define the reaction cross-section,R,in a way suggested by the definition of the total collision cross-section (section 2.1.5).For molecules colliding with a well-defined relative velocity v,the reaction cross-section is defined such that the chemical reaction rate constant k(v)is given by k(v)=VOR (3.4) MRD Chapter 3 page 4 ©R D Levine(2003)
If measurement are carried out over a wide range in 1/T there is no real reason to expect that either the Arrhenius 'A factor' or the activation energy Ea is independent of temperature. Therefore a strict definition of the activation energy is as the local (meaning, possibly T dependent) slope of the Arrhenius plot of k(T) vs. 1 / T Ea ≡ −kB d ln k(T) d 1 T = kBT2 d ln k(T) dT (3.3) This is the definition that we shall use. The chemical change as observed in a macroscopic experiment is the result of many molecular collisions. In section 3.1.2 we define the reaction cross section from the macroscopic observable rate of reactive collisions. This is sufficient to discuss the dependence of the cross section on the collision energy. Then we discuss the temperature dependence of the reaction rate as arising from this energy dependence. To understand the origin of the energy needs of chemical reactions, we provide, in section 3.2, a microscopic interpretation of the reaction cross section. Finally, a caveat. In this chapter we are still not taking account of the internal structure of the reactants. This key topic has to wait until chapter 5. The only concession is that an appendix provides an extension of chemical kinetics to the important special case where we do resolve internal states. 3.1.2 The reaction cross-section- a macroscopic view We define the reaction cross-section, σ R, in a way suggested by the definition of the total collision cross-section (section 2.1.5). For molecules colliding with a well-defined relative velocity v, the reaction cross-section is defined such that the chemical reaction rate constant k(v) is given by k(v) = vσ R (3.4) MRD Chapter 3 page 4 © R D Levine (2003)
Here k(v)is the reaction rate constant for a specific,well defined,relative velocity.It is not the same quantity as the thermal rate constant.We can imagine measuring it by passing a beam of reactant A molecules through a scattering cell as in figure 2.1.Then,the loss of flux due to reactive collisions is given by" -(dI dx)R=k(v)nAnB =InBOR 3.5) Here,as in eqns(2.1-2.4),Ix)is the flux of beam molecules A at position x and nB is the number density of the target B molecules.However,not all collisions need to lead to reaction. While we can conclude that oRso,it is not enough to measure the attenuation of the parent beam,because an appreciable loss of intensity can result from non-reactive scattering.We must specifically determine the loss of flux due to reactive collisions.Experimentally,this is easy to do for reactions producing ions:the ions are simply collected by the application of an electric field.One interesting class of reactions of this sort is that of the endothermic collisional ionization type;here two neutral molecules collide to form ions e.g., K+B→K++B5 without atom exchange,or: N2+CO→NO+CN Another type of reaction involving both reactant and product ions is the so-called ion- molecule class e.g., H+He→HeH+H Later on we shall generalize equation(3.5),for example looking at those reactive collisions where the products scatter into a particular direction. MRD Chapter 3 page 5 ©R D Levine(2003)
Here is the reaction rate constant for a specific, well defined, relative velocity. It is not the same quantity as the thermal rate constant. We can imagine measuring it by passing a beam of reactant A molecules through a scattering cell as in figure 2.1. Then, the loss of flux due to reactive collisions is given by k(v) ∗ −(dI / dx)R = k(v)nAnB = InBσ R (3.5) Here, as in eqns (2.1-2.4), I(x) is the flux of beam molecules A at position x and nB is the number density of the target B molecules. However, not all collisions need to lead to reaction. While we can conclude that σ R ≤σ , it is not enough to measure the attenuation of the parent beam, because an appreciable loss of intensity can result from non-reactive scattering. We must specifically determine the loss of flux due to reactive collisions. Experimentally, this is easy to do for reactions producing ions: the ions are simply collected by the application of an electric field. One interesting class of reactions of this sort is that of the endothermic collisional ionization type; here two neutral molecules collide to form ions e.g., K + Br 2 → K+ +Br2 - without atom exchange, or: N2 + CO→NO+ +CN- Another type of reaction involving both reactant and product ions is the so-called ionmolecule class e.g., H2 + +He→HeH+ +H ∗ Later on we shall generalize equation (3.5), for example looking at those reactive collisions where the products scatter into a particular direction. MRD Chapter 3 page 5 © R D Levine (2003)
Of course it is not enough just to collect the ionic or neutral products:it is also necessary to identify their chemical nature.This is often achieved by mass specrometric methods.Such identification is essential when several different reaction paths are possible e.g., KBr+Br K+Br2 K++B2 K*+Br+Br and one needs to determine the branching ratio or the relative contribution of each process to the total reaction cross-section. 3.1.2.1 The energy threshold of reaction We embark on our study of the role of energy in chemical dynamics by examining the dependence of the reaction cross section on the translational energy'of the colliding partners. Our first consideration is the operational concept of the threshold energy,E0,as the minimum energy needed for the reaction to take place.The reaction cross section vanishes for energies below this threshold value.For endothermic reactions,the conservation of energy implies that there is a minimal energy for reaction to take place.For example,for the ion molecule reaction H2+He>HeH+H,the minimal energy expected on thermochemical grounds is the difference between the binding energies of the reactants and products:Eo=Do(H2)-Do(HeH")=2.65-1-84=0.81 eV.The experimental results, shown in figure 3.1 are that this minimal energy is indeed the threshold.It is further seen that the reaction cross section increases rapidly as the translational energy increases above Eo. This behavior is typical for reactions with an energy threshold. As before,only the relative translational energy is of importance.For a binary collision,the motion of the center of mass cannot affect the outcome of the collision. MRD Chapter 3 page 6 ©R D Levine(2003)
Of course it is not enough just to collect the ionic or neutral products: it is also necessary to identify their chemical nature. This is often achieved by mass specrometric methods. Such identification is essential when several different reaction paths are possible e.g., K +Br2 → KBr + Br K+ + Br2 - K+ + Br- + Br and one needs to determine the branching ratio or the relative contribution of each process to the total reaction cross-section. 3.1.2.1 The energy threshold of reaction We embark on our study of the role of energy in chemical dynamics by examining the dependence of the reaction cross section on the translational energy* of the colliding partners. Our first consideration is the operational concept of the threshold energy, E0, as the minimum energy needed for the reaction to take place. The reaction cross section vanishes for energies below this threshold value. For endothermic reactions, the conservation of energy implies that there is a minimal energy for reaction to take place. For example, for the ion molecule reaction , the minimal energy expected on thermochemical grounds is the difference between the binding energies of the reactants and products: . The experimental results, shown in figure 3.1 are that this minimal energy is indeed the threshold. It is further seen that the reaction cross section increases rapidly as the translational energy increases above E H2 + + He→HeH+ + H E0 = D0(H2 +) − D0(HeH+) =2 ⋅ 65−1⋅84 =0 ⋅ 81 eV 0 . This behavior is typical for reactions with an energy threshold. * As before, only the relative translational energy is of importance. For a binary collision, the motion of the center of mass cannot affect the outcome of the collision. MRD Chapter 3 page 6 © R D Levine (2003)
H2(v =0)He-HeH++H 0.15 0.10 0.05 0 4 6 8 E州 ErleV Figure 3.1.Translational energy dependence of the reaction cross-section,oR(ET)for the H(v=0)+He>HeH+H reaction.[Adapted from T.Turner,O.Dutuit and Y.T.Lee,J. Chem Phys.81,3475(1984)].For this ion-molecule reaction the observed threshold energy is equal to the minimal possible value,the endoergicity of the reaction.Exoergic ion-molecule reactions often have no threshold3.By exciting the vibrations of the H2 reactant the cross section for the reaction above can be considerably enhanced. Reactions can have a finite energy threshold that is higher than the thermochemical threshold, meaning that oR is effectively zero below some threshold energy even though the reaction is thermodynamically 'allowed'.One then speaks of an activation barrier for reaction to take place.A particulalrly clear example are thermoneutral exchange reactions.The (actually,a shade endoergic,Problem A)reaction H+D2(v=O)→D+HD has a threshold energy of about 30 kJmol(0.3eV).Thus,while all endoergic reactions necessarily have an energy threshold,many exoergic reactions also have an effective energy MRD Chapter 3 page 7 ©R D Levine(2003)
Figure 3.1. Translational energy dependence of the reaction cross-section, σ R(ET ) for the reaction. [Adapted from T. Turner, O. Dutuit and Y. T. Lee, J. Chem Phys. 81, 3475 (1984)]. For this ion-molecule reaction the observed threshold energy is equal to the minimal possible value, the endoergicity of the reaction. Exoergic ion-molecule reactions often have no threshold H2 +(v = 0) +He→ HeH+ +H 3 . By exciting the vibrations of the H reactant the cross section for the reaction above can be considerably enhanced. 2 + Reactions can have a finite energy threshold that is higher than the thermochemical threshold, meaning that σ R is effectively zero below some threshold energy even though the reaction is thermodynamically ‘allowed’. One then speaks of an activation barrier for reaction to take place. A particulalrly clear example are thermoneutral exchange reactions. The (actually, a shade endoergic, Problem A) reaction H + D2(v = 0) → D + HD MRD Chapter 3 page 7 © R D Levine (2003) has a threshold energy of about 30 kJmol -1 ( ≈ 0.3eV). Thus, while all endoergic reactions necessarily have an energy threshold, many exoergic reactions also have an effective energy
threshold.The reaction energy threshold Eo can be no lower than the minimum energy AEo thermochemically required for the reaction,but may be higher or even significantly higher. An important class of reactions,of particular interest in atmospheric chemistry (aeronomy) and,in general,for interstellar chemistry,is that of the exoergic ion-molecule reactions e.g., N++O2 NO+O N+O Such reactions often show no threshold energy*and the reaction cross-sections are found to be a decreasing function of the translational energy,roughly as OR(ET)=AET12 (3.6) as shown in figure 3.2.It is the preference for low collision energies that makes ion-molecule reactions so important for the synthesis of molecules in the interstellar medium. MRD Chapter 3 page 8 ©R D Levine(2003)
threshold. The reaction energy threshold E0 can be no lower than the minimum energy ∆E0 thermochemically required for the reaction, but may be higher or even significantly higher. An important class of reactions, of particular interest in atmospheric chemistry (aeronomy) and, in general, for interstellar chemistry, is that of the exoergic ion-molecule reactions e.g., N+ + O2 → NO+ + O N + O2 + Such reactions often show no threshold energy4 and the reaction cross-sections are found to be a decreasing function of the translational energy, roughly as σ R(ET ) = AET −1/2 (3.6) as shown in figure 3.2. It is the preference for low collision energies that makes ion-molecule reactions so important for the synthesis of molecules in the interstellar medium. MRD Chapter 3 page 8 © R D Levine (2003)
Art+H2→ArH++H TTTTTTTT 100 Exptl. 10 ⊥LL4L 0.01 0.1 1.0 ETleV Figure 3.2.Log-log plot of the translational energy dependence of a reaction with no threshold energy.The solid curve is an experimental result for the system Ar+D>ArD"+D;the dashed curve has the slope of -1/2,cf.equation (3.28)and the potential shown in figure 3.8.[Adapted from K.M.Ervin and P.B.Armentrout,J.Chem. Phys.83,166 (1985)].For the drop in the reaction cross section at higher energies see problem I. 3.1.2.2 Translational energy requirements of chemical reactions On the basis of their translational energy requirements we can thus make the following rough correlation: (1)Reactions which have an energy threshold (this necessarily includes all endoergic reactions)have a reaction cross-section which is an increasing function of the translational MRD Chapter 3 page 9 ©R D Levine(2003)
Figure 3.2. Log-log plot of the translational energy dependence of a reaction with no threshold energy. The solid curve is an experimental result for the system ; the dashed curve has the slope of -1/2, cf. equation (3.28) and the potential shown in figure 3.8. [Adapted from K. M. Ervin and P. B. Armentrout, J. Chem. Phys. 83, 166 (1985)]. For the drop in the reaction cross section at higher energies see problem I. Ar+ + D2 → ArD+ + D 3.1.2.2 Translational energy requirements of chemical reactions On the basis of their translational energy requirements we can thus make the following rough correlation: (1) Reactions which have an energy threshold (this necessarily includes all endoergic reactions) have a reaction cross-section which is an increasing function of the translational MRD Chapter 3 page 9 © R D Levine (2003)
energy in the post-threshold region.This is the case chemists are more familiar with.It gives rise to a positive Arrhenius activation energy as discussed next. (2)Reactions which proceed without any apparent energy threshold(and this includes some, but not all,exoergic reactions)often have a reaction cross-section which is a decreasing function of the translational energy.However,as the translational energy is increased other, previously endoergic,reaction paths become allowed.These have a threshold and their cross- section will increase with energy,at the expense of the previously allowed reaction To rationalize these correlations we turn in section 3.2 to the microscopic interpretation of the reaction cross section and the concept of the reaction probability.Before that we reiterate that the energy requirements of chemical reactions appear,in the macro world,as the temperature dependence of the reaction rate constant. 3.1.2.3 The temperature dependence of the reaction rate constant The translational energy dependence of the reaction cross section translates into the temperature dependence of the rate constant.The rule is clear:take k(v)=voR,equation (3.4),and average it over a thermal distribution of velocities,k(T)=voR(v)).We wrote oR(v)to remind you that the reaction cross section itself depends on the collision velocity Sometimes the thermal averaging voR(v)required to compute k(T)is easy to implement. For example,for ion-molecule reactions for which,cf.equation(3.6),voR constant,k(T)is independent of temperature.How smart was nature to make ion-molecule reactions so that they can be operative in the cold regions of space!Othertimes,the averaging needs to be carried out.Explicitly,it means evaluating an integral over a Maxwell-Boltzmann velocity distribution fv)of the(collision energy dependent)reaction cross section kT)=∫vorf(v)d=(Uu/2πkI)3/2∫vGRexp((-212eI)4m2 (3.7) MRD Chapter 3 page 10 ©R D Levine(2003)
energy in the post-threshold region. This is the case chemists are more familiar with. It gives rise to a positive Arrhenius activation energy as discussed next. (2) Reactions which proceed without any apparent energy threshold (and this includes some, but not all, exoergic reactions) often have a reaction cross-section which is a decreasing function of the translational energy. However, as the translational energy is increased other, previously endoergic, reaction paths become allowed. These have a threshold and their crosssection will increase with energy, at the expense of the previously allowed reaction. To rationalize these correlations we turn in section 3.2 to the microscopic interpretation of the reaction cross section and the concept of the reaction probability. Before that we reiterate that the energy requirements of chemical reactions appear, in the macro world, as the temperature dependence of the reaction rate constant. 3.1.2.3 The temperature dependence of the reaction rate constant The translational energy dependence of the reaction cross section translates into the temperature dependence of the rate constant. The rule is clear: take k(v) = vσ R, equation (3.4), and average it over a thermal distribution of velocities, k(T) = vσ R(v) . We wrote σ R(v) to remind you that the reaction cross section itself depends on the collision velocity. Sometimes the thermal averaging vσ R(v) required to compute k(T) is easy to implement. For example, for ion-molecule reactions for which, cf. equation (3.6), vσ R ∝ constant , k(T) is independent of temperature. How smart was nature to make ion-molecule reactions so that they can be operative in the cold regions of space! Othertimes, the averaging needs to be carried out. Explicitly, it means evaluating an integral over a Maxwell-Boltzmann velocity distribution f(v) of the (collision energy dependent) reaction cross section k(T) = ∫ vσ R f (v)dv = (µ / 2πkBT) 3/2 ∫ vσ R exp(−µv 2 / 2kBT)4πv 2 dv (3.7) MRD Chapter 3 page 10 © R D Levine (2003)