1 Molecular collisions

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1 1 Molecular collisions The present exercise starts with the basics of molecular collisions as presented in Chapter 4 of the lecture notes. After that, particular attention is devoted to several specific chemical reactions in order to achieve a better understanding of the scattering process. Every chemical reaction of order 2 entails a collision, i.e. a scattering event. Consider a simple experiment in which a beam of molecules A with intensity I 0 enters a chamber filled with a gas of molecules B. A reacts with B, and after passing a distance l through the chamber, the intensity is reduced from I 0 to I 1 because of reactive collisions. Figure 1: A simple experiment setup for the study of molecular collisions. The rate for the bimolecular reation A+B Products is given by: v r = dc A dt = k(t )C A C B, (1) where k(t ) is the rate constant at a certain temperature T and C A and C B are the number densities of the molecules A and B, respectively. The intensity I of the beam of molecules A (molecules passing through a surface per second) is given by: I = v A C A, (2) where v A is the velocity of the beam of molecules A. If we assume that the B molecules are much slower than the A molecules (v B 0), the attenuation of the intensity of the beam can be cast into a Lambert-Beer-type form of expression: where σ represents the cross section. di dx = σc BI, (3) 1.1 Derive a universal expression for the rate constant k(t ) as a function of the scattering cross section σ. 1.2 In fact, it is not enough to just measure the attenuation of the parent beam, because an appreciable loss of intensity can result from non-reactive scattering (σ r < σ tot ). We must specifically determine the loss of flux due to reactive collisions. Can one distinguish between reactive and non-reactive collisions experimentally? 1

2 1.3 Provide the link between an experimentally measurable quantity (a rate constant) and a quantity that is related to the reactant scattering (S) matrix, that is theoretically accessible. Hint: See (Chapter 14, Page 509, Peter Atkins, Molecular Quantum Mechanics, Oxford University) or (Chapter 5, Gabriel et al. Theory of Molecular collision, Royal Society Of Chemistry). All you need to know is just the definition of S matrix, and then find the relation between transition probabilities and rate constant. Molecular collisions or scattering processes are usually classified into elastic, inelastic and reactive processes. During elastic collisions, the total kinetic energy of the incident particles A and B is conserved in the centre-of-mass frame, but it leads to a change of the velocity vectors of the particles. Inelastic collisions are characterized by a change in the internal state of one or several collision partners, but their chemical identities are unchanged. The total kinetic energy during inelastic collisions is not conserved because it involves a transfer of energy between translational motion and internal degrees of freedom. In the case of the scattering of atoms, the internal energy is the electronic excitation energy, while for a molecule the only internal energy can also be vibrational or rotational. During a reactive scattering event, one or more chemical bonds are broken, while new chemical bonds can be formed. 1.4 Characterize each of the following scattering processes as either elastic, inelastic, or reactive. (Note that v and j are the vibrational and rotational quantum numbers, respectively). (i) O( 3 P) + O( 3 P) O( 3 P) + O( 1 D) (ii) O( 3 P) + O( 3 P) O( 3 P) + O( 3 P) (iii) Cl( 2 P) + HF(v=0, j=0) Cl( 2 P) + HF(v =0, j =0) (iv) Cl( 2 P) + HF(v=0, j=0) F( 2 P) + HCl(v =0, j =0) (v) Cl( 2 P) + HF(v=0, j=0) Cl( 2 P) + HF(v =1, j =0) 2 Angular distributions in reactive molecular collisions Consider a reaction K + I 2 studied at a mean relative velocity of 800 ms 1, with I 2 in thermal equilibrium at 300 K. The reaction cross section was found to be 170 Å 2. Use the constants in the following table to calculate the quantities described under items 2.1 to 2.4 below: I 2 KI D 0 / kj mol ω e / cm B e / cm where D 0 is the dissociation energy of molecules relative to their zero point energy. ω e and B e are vibrational and rotational wavenumbers, respectively. 2.1 The total energy available to the products, Hint: The mean vibrational energy of the I 2 reactants may be calculated assuming E v = hcω e /(e θv/t 1) with θ v = hcω e /k B, where k B is the Boltzmann constant. The rotational energy is obtained from E r = RT, assuming the equipartition of rotational energy of I 2, where 2

3 Advanced kinetics Exercise 9 April 29, 2016 R is the gas constant. 2.2 The maximum impact parameter bmax and the maximum angular momentum quantum number, lmax. Hint: In scattering processes, the reaction cross section can generally be formulated R bclassical max P (b)2π b db. The impact parameter b is defined as the distance of closest approach as σ = 0 of the reactants in the absence of an interaction potential. P (b) is the probability for reaction at collision at a given value of b. Assume that P (b) = 1 for 0 b bmax and therefore the classical relation σ r = πb2max is applicable. 2.3 The rotational energy of the KI product if j 0 = lmax, where j 0 is the rotational angular momentum quantum number of KI. Is this rotational state energetically accessible? 2.4 The maximum center-of-mass velocity of the KI products. The angular distribution of scattering products reflecting the differential scattering cross section can be measured in crossed molecular beam experiments. The distribution of scattering angles θ and product velocities vab in the centre-of-mass (CM) frame can be inferred from a Newton diagram (velocity diagram) as shown in Figure 2. Figure 2: Newton diagram for the reaction A + BC AB + C (left), and reconstruction of the CM angular distribution from a crossed molecular beam experiment (right). The reconstructed CM product flux distribution ICM (θ,v) can be decomposed into two different components: ICM (θ, v) = T (θ) P (Et0 ), (4) where T (θ) is the product angular distribution and P (Et0 ) is the product translational energy distribution (kinetic energy release). The CM product flux distributions are usually represented in a polar plot. The contour lines (Figure 2) indicate the product flux scattered into a certain angle θ with a given velocity v or kinetic energy Et0. 3

4 Figure 3: KI product flux contour plots in the center-of-mass (CM) frame as a function of KI velocity (i.e. flux as a function of scattering angle and KI velocity) for the K+I 2 (left) and K + CH 3 I (right) reactions. The reaction mechanism manifests itself directly in the angular distribution of the reaction products. Two important types of mechanisms can be distinguished: (I) Direct mechanisms entail a direct scattering event. (II) Indirect (or complex-forming) mechanisms entail the formation of an intermediary reaction complex. Let us look at two reactions involving the molecular collision of K + I 2 and K + CH 3 I. The KI product flux contour plots in the center-of-mass (CM) frame as a function of KI velocity from these two reactions are displayed in Figure What may be learnt about the dynamics of the two reactions from these figures? The outer rings of dots in Fig. 3 show the maximum CM velocities of KI in the two reactions. We calculated this velocity for the K + I 2 reaction in task 2.4. For discussing the K+ I 2 reaction, keep in mind what we learned in the previous section, namely the large integral cross section and large impact parameter. 2.6 What does the fact that both reactions lead to a preferential scattering (forward or backward) suggest about the duration of the reaction (lifetime of the intermediate complex)? Contrast the scattering behavior observed for the above two reactions with that found for the reaction S( 1 D) + H 2, the differential cross section for which is shown in Fig Assuming that the scattering amplitude has the simple analytical form f(θ, φ)= sin(θ)cos(φ), find an expression for the differential cross section, σ(θ, φ) and evaluate also the integral cross section σ. 4

5 Figure 4: Quantum mechanical and experimental total differential cross section versus centerof-mass scattering angle for S( 1 D) + H 2 SH + H 2. The collsion energy is 2.24 kcal/mol. QM contribution of each vibrational state is also show as thin solid lines. (Honvault et al. Chem. Phys. Letts. 370, 2003, 371.) 2.8 Evaluate the integral scattering cross section for a case in which the differential cross section σ(θ, φ) is a constant C independent of the angles θ and φ. 3 A case study: the S N 2 reaction Cl + CH 3 I I + CH 3 Cl Figure 5: Center-of-mass images of the I reaction product velocity from the reaction Cl + CH 3 I I + CH 3 Cl with CH 3 I at four different relative collision energies. Anion-molecule nucleophilic substitution S N 2 reactions X + R-Y Y + R-X reactions are known for their rich reaction dynamics, which caused by a complex potential energy sur- 5

6 face with a submerged barrier and by a weak coupling of the relevant rotational-vibrational quantum states. According to the conventional picture, the reaction proceeds via a back-side attack on the R-Y bond leading to an inversion of the molecular configuration. For the model reaction Cl + CH 3 I I + CH 3 Cl, the angular distribution T (θ) of the product I as a function of relative collision energies are measured in a crossed molecular beam experiment (see Fig. 5). (J. Mikosch et al., Science 319, 2008, 184). 3.1 Interpret why at lower energy E rel = 0.39 ev the angular product distribution is isotropic. How many different reaction mechanisms would you expect? 3.2 Try to draw a reaction profile for Cl + CH 3 I I + CH 3 Cl according to the Figure. 5 and give your interprations. It is not necessary to present it in strict mathematical form. 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