Bolzmann factor for the state i and the averaging in (A.3.3)defines the familiar thermal reaction rate constant.The point is that(A.3.3)is valid whether the internal states of the HCI molecules are or are not in thermal equilbrium. We have already seen,Fig.1.4,that the reaction rate does depend on the internal excitation of the reactants.Yet the measurement of k(T)for reactions in thermal equilibrium can give no indication of such an effect.(For a gas in thermal equilibrium,we are unable to vary the pi's independently).In other words,at thermal equilibrium we are unable to state-select the reactants.It is only by imposing non-equilibrium reactant distributions that we can characterize the role of reactant excitation.Otherwise,when we vary T we vary both the occupations pi of the different internal states of HCI and the kinetic energy of the collision (which is why the state selected rates ki(T)are T-dependent).The measurement of k(T)only cannot tell the two apart,without making assumptions. So far we have shown that the observed reaction rate constant is an average over the rate constants for the selected state reactants.If we do state resolve the products then =-∑TFHC(】 (A.3.5) dt and proceeding as before, -amn-na网-ΣAmn (A.3.6) i,l In summary:the rule is asymmetric.To define the overall rate constant we are to sum over the states of the products but to average over the internal states of the reactants. The experimental evidence,as reiterated in figure 3.A.1,is that the detailed reaction rate constants,k(T)do in general depend upon both the initial and final states (i and )These state-to-state reaction rate constants also depend on the collision energy and hence on the translational temperature T.Our purpose is to reduce this extensive averaging.Among the MRD Chapter 3 page 16 ©R D Levine(2003)
Bolzmann factor for the state i and the averaging in (A.3.3) defines the familiar thermal reaction rate constant. The point is that (A.3.3) is valid whether the internal states of the HCl molecules are or are not in thermal equilbrium. We have already seen, Fig. 1.4, that the reaction rate does depend on the internal excitation of the reactants. Yet the measurement of k(T) for reactions in thermal equilibrium can give no indication of such an effect. (For a gas in thermal equilibrium, we are unable to vary the pi' s independently). In other words, at thermal equilibrium we are unable to state-select the reactants. It is only by imposing non-equilibrium reactant distributions that we can characterize the role of reactant excitation. Otherwise, when we vary T we vary both the occupations pi of the different internal states of HCl and the kinetic energy of the collision (which is why the state selected rates ki(T) are T-dependent). The measurement of k(T) only cannot tell the two apart, without making assumptions. So far we have shown that the observed reaction rate constant is an average over the rate constants for the selected state reactants. If we do state resolve the products then d[HF(j)] dt = − kij(T)[F][HCl(i)] i ∑ (A.3.5) and proceeding as before, k(T) = pi i, j ∑ (T)kij(T) = pi i ∑ (T) kij j ∑ (T) = pi i ∑ (T)ki(T) (A.3.6) In summary: the rule is asymmetric. To define the overall rate constant we are to sum over the states of the products but to average over the internal states of the reactants. The experimental evidence, as reiterated in figure 3.A.1, is that the detailed reaction rate constants, kij(T) do in general depend upon both the initial and final states (i and j). These state-to-state reaction rate constants also depend on the collision energy and hence on the translational temperature T. Our purpose is to reduce this extensive averaging. Among the MRD Chapter 3 page 16 © R D Levine (2003)
features that we would want to explore would be:(a)the 'energy requirements'of chemical reaction.In particular,the threshold energy or the minimum energy required for the reaction to occur'and the variation of the reactivity with the reactants'translational (and internal) energy.(b)The steric effect or the variation of the reactivity with the relative orientation of the reactants.(c)The energy disposal into the products.(d)The angular distribution of the products after they have separated from the region of interaction.We have made a start on(c) and (d)already in Chapter 1.But before we continue on this road we need to know how to characterize quantitatively the reaction rate under non-equilibrium conditions. F+HCI(W)→CI+HF(N) Fig.3.A.1.Influence of the vibrational state of the reactant molecule upon the distribution of vibrational states of the product molecule,for the reaction F+HCI(v)>Cl+HF().Plotted is k vs.v'for v=0 vs.v=1.Note the strong effect of reagent vibration upon the overall reaction rate also.(The reaction is about five-fold more efficient from the v=1 state).[Adapted from J.L.Kirsch and J.C.Polanyi,J.Chem.Phys.57,4498(1972).] The unraveling of the averaging that goes into the definition of a thermal reaction rate constant shows that its temperature dependence is not quite a simple matter.The state-to-state reaction rate constants vary with temperature because the state-to-state cross section depends on the collision energy.In addition,for a system in thermal equilibrium,the populations of the different initial states are themselves temperature dependent.Problem F shows that if an increment in the collision energy is as effective in promoting reaction as an increment in the MRD Chapter 3 page 17 ©R D Levine(2003)
features that we would want to explore would be: (a) the ‘energy requirements’ of chemical reaction. In particular, the threshold energy or the minimum energy required for the reaction to occur5 and the variation of the reactivity with the reactants’ translational (and internal) energy. (b) The steric effect or the variation of the reactivity with the relative orientation of the reactants. (c) The energy disposal into the products. (d) The angular distribution of the products after they have separated from the region of interaction. We have made a start on (c) and (d) already in Chapter 1. But before we continue on this road we need to know how to characterize quantitatively the reaction rate under non-equilibrium conditions. Fig. 3.A.1. Influence of the vibrational state of the reactant molecule upon the distribution of vibrational states of the product molecule, for the reaction F . Plotted is k + HCl(v) → Cl + HF(v' ) vv’ vs. v’ for v = 0 vs. v =1. Note the strong effect of reagent vibration upon the overall reaction rate also. (The reaction is about five-fold more efficient from the v=1 state). [Adapted from J.L. Kirsch and J.C. Polanyi, J. Chem. Phys. 57, 4498 (1972).] The unraveling of the averaging that goes into the definition of a thermal reaction rate constant shows that its temperature dependence is not quite a simple matter. The state-to-state reaction rate constants vary with temperature because the state-to-state cross section depends on the collision energy. In addition, for a system in thermal equilibrium, the populations of the different initial states are themselves temperature dependent. Problem F shows that if an increment in the collision energy is as effective in promoting reaction as an increment in the MRD Chapter 3 page 17 © R D Levine (2003)
internal energy of the reactants,then the two sources of T dependence will be equivalent.As we have noted,such equivalence is not necessarily the case. MRD Chapter 3 page 18 ©R D Levine(2003)
internal energy of the reactants, then the two sources of T dependence will be equivalent. As we have noted, such equivalence is not necessarily the case. MRD Chapter 3 page 18 © R D Levine (2003)
3.2 Two-body microscopic dynamics of reactive collisions This section is a simple microscopic approach to reactive collisions.What we want is to determine the rate of those collisions that result in a chemical reaction.The reaction cross- section oR is a measure of the effective size of the molecules as determined by their propensity to react,at a given collision energy.We will not explicitly indicate that the reaction cross-section is velocity dependent but please bear this in mind.In order to keep the view simple in this section we overlook the role of the internal energy states of molecules. That is,we use the discussion of the approach motion of structureless particles,as in section 2.2,augmented by the notion of the reaction probability.We will be able to make quite a bit of headway but much will be left for chapter 5 where the polyatomic nature of the collision is explicitly recognized. 3.2.1 The opacity function As the reactants collide (at a given energy)we characterize their initial approach in terms of the impact parameter.We define the reaction probability or opacity function,P(b),as the fraction of collisions with impact parameter b that lead to reaction.Two properties of the opacity function are obvious.Since at most all collisions can lead to reaction,0<P(b)s1. Moreover,for a chemical reaction to take place,it is necessary for the reactant molecules to get close to each other so that 'chemical forces'will operate and the atomic rearrangements that constitute the chemical change,can take place.For high impact parameter collisions the centrifugal barrier(section 3.2.6)acts to keep the molecules apart(recall that the distance of closest approach Ro->b for large b).We therefore expect that reaction will take place only when b is 'small',i.e.of the order of the range of the chemical force,and that the reaction will fail to occur,i.e.P(b)-0,for higher values of b. MRD Chapter 3 page 19 ©R D Levine(2003)
3.2 Two-body microscopic dynamics of reactive collisions This section is a simple microscopic approach to reactive collisions. What we want is to determine the rate of those collisions that result in a chemical reaction. The reaction crosssection σ R is a measure of the effective size of the molecules as determined by their propensity to react, at a given collision energy. We will not explicitly indicate that the reaction cross-section is velocity dependent but please bear this in mind. In order to keep the view simple in this section we overlook the role of the internal energy states of molecules. That is, we use the discussion of the approach motion of structureless particles, as in section 2.2, augmented by the notion of the reaction probability. We will be able to make quite a bit of headway but much will be left for chapter 5 where the polyatomic nature of the collision is explicitly recognized. 3.2.1 The opacity function As the reactants collide (at a given energy) we characterize their initial approach in terms of the impact parameter. We define the reaction probability or opacity function, P(b), as the fraction of collisions with impact parameter b that lead to reaction. Two properties of the opacity function are obvious. Since at most all collisions can lead to reaction, 0 ≤ P(b) ≤1. Moreover, for a chemical reaction to take place, it is necessary for the reactant molecules to get close to each other so that ‘chemical forces’ will operate and the atomic rearrangements that constitute the chemical change, can take place. For high impact parameter collisions the centrifugal barrier (section 3.2.6) acts to keep the molecules apart (recall that the distance of closest approach R0 →b for large b). We therefore expect that reaction will take place only when b is ‘small’, i.e. of the order of the range of the chemical force, and that the reaction will fail to occur, i.e. P(b)=0, for higher values of b. MRD Chapter 3 page 19 © R D Levine (2003)