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)