1 Molecular collisions

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1 Advanced Kinetics Solution 9 April 29, Molecular collisions 1.1 The bimolecular rate constant for the reaction is defined as: dc A dt = k(t )C A C B. (1) The attenuation of the intensity of the beam can be cast into a Lambert-Beer-type form of expression: di dx = σc BI. (2) Using I = v A C A and v A = dx/dt, the left side of Eq (2) becomes: di dx = d(c Av A ) dc A = v A dx dx Using I = v A C A again, the right side of Eq (2) becomes: Thus, Eq (2) can also be expressed as: = dc A dt. (3) σc B I = σc B v A C A. (4) dc A dt = σc B v A C A. (5) Comparing the Eq (5) with Eq (1) yields: k(t ) = σv A, (6) which is an universal expression linking the scattering rate constant with the cross section. 1.2 This is an open question. Experimentally, it is easier to study 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 endothermic collision ionization type; here two neutral molecules collide to form ions, e.g. K + Br 2 K + + Br 2 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 spectrometric methods. Such identification is essential when several different reaction paths are possible, e.g. K + Br 2 KBr + Br K + Br 2 K + + Br 2 K + Br 2 K + + Br + Br One needs to determine the branching ratio or the relative contribution of each process to the total reaction cross-section. 1

2 Advanced Kinetics Solution 9 April 29, 216 Spectroscopic methods usually based on laser ionization are also possible which would be introduced in Chapter 5 of the lecture notes. 1.3 The rate constant for reactive scattering is a concept of the greatest significance in chemistry for a chemical reaction, for instance: A + BC AB + C, v = k(t )C A C BC The reactants will be characterized by various quantum numbers including the orbital angular momentum of A relative to the diatomic molecule BC, rotational and vibrational quantum numbers BC, and quantum numbers for the electronic states of A and of BC. The set of quantum numbers will specify a reactant channel, denoted λ. Similarly the products will be characterized by a set of quantum numbers that will specify the product channel, denoted γ. At a fixed total reaction energy E, we need to concern ourselves only with those channels λ and γ that are energetically accessible (that is, open channels). The probability that a transition (that is, a chemical reaction) occurs from reactant scattering (S) matrix, the latter being one of the fundamental concepts in scattering theory. For a given energy E, the transition probability is given by P γλ (E) = J (2J + 1) S J γλ (E) 2, (7) where J designates the total angular momentum of the system. At a reaction energy E, there are usually many reactant and product channels open and the sum over all possible channelto-channel reactive transition probabilities is called the cumulative reaction probability P (E): P (E) = γ,λ P γλ (E) (8) The temperature-dependent rate constant for the chemical reaction is given by P (E)e E/kt de k(t ) = hθ r (T ) where Θ r (T ) is the partition function density (the partition function divided by the volume) occupied by the reactants at the temperature T. The latter expression provides a critical link between an experimentally measurable quantity (a rate constant) and a theoretically calcuable quantity, P (E), and thus illustrates an example of the connection between bulk data and scattering theory. 1.4 The types of scattering processes are illustrated in the exercise sheet. (9) (i) (ii) (iii) (iv) (v) The process is inelastic because the electronic state of atomic oxygen changes. The process is elastic because the initial and final states are the same. The process is elastic because the initial and final states are the same. The process is reactive because a chemical reaction has occurred. The process is inelastic because the vibrational state of HF changes. 2

3 Advanced Kinetics Solution 9 April 29, Angular distributions in reactive molecular collision If the total energy in the reactants (the sum of collisional energy E coll, vibrational energy E v, rotational energy E r and electronic energy if applicable is higher than the barrier height, the reaction can proceed in principle. The available energy E avl after the collision is distributed among the products. 2.1 The total energy available to the products is the enthalpy of reaction is given by the collision energy can be calculated from E avl = E coll + E v + E r H r, (1) H r = kj mol 1 = 17 kj mol 1, (11) where the reduced mass is given by E coll = 1 2 µν2 rel = 1.81 kj mol 1, (12) µ = The rotational energy is given by and the vibrational energy by u = kg. (13) E r = RT = 2.49 kj mol 1, (14) E v = hcω e /(e θv/t 1) = kj mol 1. (15) Finally we find the total energy available E avl = kj mol The integral cross section can be expressed as σ = bmax P (b)2πbdb πb 2 max, (16) where P (b) is the probability of reaction at a given impact parameter b. We are assuming P (b) = 1 for b b max. We know the σ = 17 Å 2 and therefore b max = 7.4 Å. The maximum impact parameter and the orbital angular momentum quantum number, l max can be calculated as follows: L max = µν rel b max = = (17) and we get l max 312. L max = h l max (l max + 1) h(l max ) (18) 3

4 Advanced Kinetics Solution 9 April 29, The rotational energy is E r = B ej (j + 1) (19) and we assume j = l max = 312. Then E r = 5951 cm 1 = 71.2 kjmol 1. Therefore, the state j = 312 is energetically accessible since the rotational energy E r is smaller than the available energy E avl (184.7 kjmol 1 ). 2.4 From the momentum conservation after the reaction and the defnition of the relative velocity vector we obtain the center-of-mass velocity of KI products v KI = m I v I = m KI v KI (2) v rel = v I v KI, (21) v relm KI v rel + m KI = 453 m s 1 (22) 2.5 The Figure. 3 suggests that the reaction K+I 2 leads to a forward scattering, i.e. in the direction of the incident K atom, since most of the flux is in the area for the scattering angle θ 9 o. This forward scattering is typical of the stripping mode behavior. Since the integral cross section of this reaction is large, encounters at large impact parameters are also effective in yielding the reaction. The reaction occurs by the harpoon mechanism. The fact that the scattered KI products have smaller velocity than the maximum possible velocity implies that the KI products are internally excited. The K + CH 3 I reaction leads to the scattering into the backward hemisphere with respect to the incident K atom. This is called a rebound mode. The backward scattering implies that the collisions must occur at low-impact-parameters (so called head-on collisions). Since the reaction occurs only for small impact parameters, the integral cross section for the reaction is small - only 35 Å 2 compared to 17 Å 2 for K+I The fact that both reactions lead to a preferential scattering suggests that the reaction must be over quickly with respect to the rotational period of the collision complex. We talk about a direct reaction whenever the angular distribution is not symmetrical with respect to the forward vs. backward hemisphere. 2.7 The differential cross section for reaction S( 1 D) + H 2 is maximal near the poles θ or π and minimal near the equator π/2. The forward-backward symmetry of the differential cross section indicates the presence of the long-lived (in comparison to its rotational period) collision complex of H 2 S. The differential cross section is σ(θ, φ) = f(θ, φ) 2 = sin 2 (θ)cos 2 (φ) (23) 4

5 Advanced Kinetics Solution 9 April 29, 216 and the total integral cross section is given by σ = = π 2π π = σ(θ, φ)sin(θ)d(θ)d(φ) = 2π π 2π sin 3 (θ)d(θ) cos 2 (φ)d(φ) (24) (1 cos 2 (θ))sin(θ)d(θ) 1 cos(2φ + 1)d(φ) (25) 2 [ ] π [ ] cos(θ) cos3 (θ) 1 sin(2φ) 2π + φ = 4π (26) 2.8 The integral cross section is given by σ = π 2π σ(θ, φ)sin(θ)d(θ)d(φ) = π 2π Csin(θ)d(θ)d(φ) = 4πC. (27) 3 A case study: the S N 2 reaction Cl + CH 3 I I + CH 3 Cl 3.1 The images in Figure. 5 in the exercise sheet reveal many details of the reaction dynamics. For the lowest relative collision energy of.39 ev, there is an isotropic distribution of product velocities centered around zero with all scattering angles equally probable. This pattern points to the traditional reaction mechanism mediated by a collision complex whose lifetime is long compared to the time scale of its rotation. The complex-mediated mechanism is accompanied by a velocity distribution that drops to zero far before the kinematic cutoff is reached, as can be inferred from the position of the outermost ring in the image. Thus, the largest fraction of the available energy is partitioned to internal rovibrational energy of the CH 3 Cl product. Figure 1: Reaction profile for Cl + CH 3 I I + CH 3 Cl showing a characteristic double-well potential-energy profile along the reaction coordinate. 5

6 Advanced Kinetics Solution 9 April 29, 216 A distinctly different reaction mechanism becomes dominant at the higher relative collision energy of.76 ev. The I product is back-scattered into a small cone of scattering angles. This pattern indicates that direct nucleophilic displacement dominates. The Cl reactant attacks the methyl iodide molecule at the concave center of the CH 3 umbrella and thereby drives the I product away on the opposite side. The direct mechanism leads to product ion velocities close to the kinematic cutoff. In addition, part of the product flux is found at small product velocities with an almost isotropic angular distribution, indicating that for some of the collisions there is a significant probability of forming a long-lived complex. At a collision energy of 1.7 ev (Fig. 5C), the complex-mediated reaction channel is not observed any more. The reaction proceeds almost exclusively by the direct mechanism, with a similar velocity and a slightly narrower angular distribution relative to the.76-ev case. At an even higher collision energy of 1.9 ev, the dominating backward scattering pattern of the I ion spreads over an increased range of scattering angles. The inserted rings demonstrate a concomitant broadening of the velocity distribution. In addition, a new feature appears that consists of two distinct low-velocity peaks symmetric in the forward and the backward directions with respect to the center of mass. 3.2 Figure 1 presents the reaction profile for Cl + CH 3 I I + CH 3 Cl showing a characteristic double-well potential-energy profile along the reaction coordinate. In this crossed-molecular beam scattering experiment, three types of product angular distribution T (θ) are observed indicating three different reaction mechanisms: (i) Isotropic at low collision energies indicating the classic mechanism via a long-lived reactive complex. (ii) Forward-scattered scattered I indicating a fast, direct nucleophilic displacement of the I. (iii) Additional forward-backward-scattered I products at highest E c indicate a new indirect roundabout reaction mechanism. The exercise was created by Dr. Jun Ma and Dr. Tran Trung Luu. For the questions regarding this exercise, please contact jun.ma@phys.chem.ethz.ch. 6