Lecture 4

Lecture 4 - University of Southern California Version 1.3...

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University of Southern California ©Hai Wang Version 1.3 1 AME599 Combustion Chemistry and Physics Lecture 4 4. Bimolecular Reaction Rate Coefficients In the last lecture, we learned qualitatively the reaction mechanisms of hydrocarbon combustion. To make this description quantitative, we will need to have a basic knowledge of reaction rate theories. While the rate coefficients of a large number of combustion reactions are experimentally measured, reaction rate theories are often necessary to interpret the experimental data. We shall focus our discussion here to bimolecular reactions of the type A + B C + D and leave the discussion for unimolecular reactions to a later time. 4.1 Hard Sphere Collision We discussed earlier that an elementary chemical reaction requires molecular collision. The rate coefficient of a bimolecular reaction is essentially the product of reaction probability of two reactants upon collision at a given temperature γ ( T ) and the frequency of collision Z AB , k ( T ) = ( T ) Z AB (4.1) Here we assume that molecules may be described by rigid spheres. Figure 4.1 shows several scenarios of molecular encounter. Starting from the head-on collision, an offset of the two axes of molecular motion leads to off-center collision. Suppose the diameter of A and B are σ A and B , respectively. The limiting off-center collision would have a spacing equal to ( A + B )/2 between the two axes of molecular motions. It may be concluded from this simple analysis that two molecules with their axes of motions lie within a cylindrical volume of cross section equal to Figure 4.1 Various scenarios of molecule collision. AB Head-on collision Off-center collision Limiting off-center collision A B AB =( A + B )/2
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University of Southern California ©Hai Wang Version 1.3 2 2 2 2 AB AB S σσ πσ π ⎛⎞ + == ⎝⎠ (4.2) would collide with each other. Here AB σ is the collision diameter. Suppose an N B number of B molecules are at rest and confined to an arbitrary volume V . An A molecule travels through the volume with a mean velocity equal to A v . The cylindrical collision volume that A sweeps through per unit time is 2 AB A v πσ (see, Figure 4.2). The number density of B is N B / V . Therefore the number of collision per unit time is () 12 s B BA B A N Zv V πσ = . (4.3) If we have N A A molecules in the same volume, the collision rate is equal to 31 2 cm s AB AB A NN VV πσ −− = . (4.4) Figure 4.2 Schematic illustration of the cylindrical collision volume in a total volume V . The above derivation is simple, but it has one problem. That is, it treats the molecules like “ghost” particles since collision does not lead to changes in the direction of motion of the A molecule. This problem is easily mended by realigning the cylindrical volume with the velocity vector of the A molecule every time it undergo scattering with a B molecule. Since the number densities of A and B are independent of the orientation of the cylindrical volume, the result is not affected by the “ghost” particle assumption.
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Lecture 4 - University of Southern California Version 1.3...

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