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Unformatted text preview: CE 561 Lecture Notes Fall 2009 p. 1 of 7 Day 8: Collision theory of bimolecular reactions 1 st change in direction for the course: From a phenomenological point of view to a molecular point of view Up to this point, we have considered macroscopic, or phenomenological, chemical kinetics. Rate parameters were numbers to be obtained from experiment, and there was no molecular basis for interpreting them. Nearly everything we have done so far could be done without any reference to the discrete atomic/molecular nature of matter. For the next few lectures, we will take a more microscopic point of view, in which we consider the description of chemical reactions in terms of ensembles of molecular collisions. We will develop this approach for gas phase reactions, because they usually occur as isolated events involving one or two molecules. Eventually, we will extend this to reactions in liquids (where the effects of the surrounding molecules cannot be neglected) and gas-solid reactions, where one of the molecules is effectively infinitely large. The hard-sphere collision rate and simple collision theory The simplest model of a bimolecular reaction is one in which the two molecules are pictured as hard spheres that do not interact at all until they collide. When they collide, if they have sufficient energy (either kinetic energy along the line of centers, or total energy, or energy in certain degrees of freedom, depending on the exact flavor of theory being used) they are assumed to react. If they have less energy, they do not react. So, we will calculate the reaction rate as Rate = ( collision rate )*( fraction of collisions with sufficient energy to react ) The non-reactive collisions are assumed to be elastic that is, both the total momentum and total kinetic energy are unchanged in the collision. An elastic collision is what we observe when macroscopic hard spheres (billiard balls or bocce balls or bowling balls or marbles) collide. In an inelastic collision (which could be reactive or non-reactive) the total kinetic energy changes during the collision. Momentum is conserved in any isolated collision. We will first use the kinetic theory of gases to find the number of collisions per unit time between two different types of hard-sphere molecules (say A and B) in an ideal gas. We also want to know the fraction of these collisions in which the kinetic energy along the line of approach is greater than some value. This will be taken to be the fraction of the collisions that leads to reaction. Simple, not-so-rigorous derivation of hard sphere collision rate Molecules A and B have diameters and corresponding collision cross-section d A , d B , A = d A 2 , and B = d B 2 . Note that the collision cross-section is not the cross-sectional area of the molecule itself, but the cross-sectional area of a cylinder whose radius is equal to the diameter of the molecule (see below). An A and B molecule collide whenever their centers of mass come within a distance d AB = ( d A + d B )/2 of each other. CE 561 Lecture Notes...
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- Fall '09