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 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) = 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)