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ChE 452 Lecture 22

ChE 452 Lecture 22. Transition State Theory. Conventional Transition State Theory (CTST). TST Model motion over a barrier Use stat mech to estimate key terms. Motion Over PE Surface.

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ChE 452 Lecture 22

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  1. ChE 452 Lecture 22 Transition State Theory

  2. Conventional Transition State Theory (CTST) • TST • Model motion over a barrier • Use stat mech to estimate key terms

  3. Motion Over PE Surface Figure 7.6 A potential energy surface for the reaction H + CH3OH  H2 + CH2OH from the calculations of Blowers and Masel. The lines in the figure are contours of constant energy. The lines are spaced 5 kcal/mole apart.

  4. Approximate Derivation Of TST For A+BCAB+C Assume Arrhenius’ Model • Two populations of A-BC complexes • Cold A-BC complexes • Hot A-BC complexes that are in right configuration and have enough energy to react (A-BC could be far apart). • Equilibrium between the 2 populations

  5. Derivation Continued • Assume reaction rate=K0 [ABC†] Where K0 is the rate constant for reaction of the hot molecules • From equilibrium • Combining reaction rate = K0 KEQU[A][BC] or rate constant=K0 KEQU

  6. From Statistical Mechanics

  7. Combining (7.38) here kABC is the rate constant for reaction (7.38); kB is Boltzman’s constant; T is the absolute temperature; K0 is the rate constant; qA is the microcanonical partition function per unit volume of the reactant A; qBC is the microcanonical partition function per unit volume for the reactant BC; is the average energy of the hot molecules and, is the average partition function of the molecules which react. Equation (7.38) is exact but we will need an expression for K0. We can get it from collision theory.

  8. Next: Estimate K0 From Collision Theory First, let us define a new partition coefficient q+, by: (7.39) In equation (7.39) is the partition function for the translation of A toward BC and q+ is the partition function for all of the other modes of the reacting A-B-C complex.

  9. Combining Equation (7.38) And (7.39) Yields: (7.40)

  10. Key Approximation

  11. Derivation Continued We want TST to go to collision theory when qv’s are all one. After pages of algebra we obtain: (7.42) Substituting equation (7.42) into equation (7.41) yields: (7.43)

  12. Example 7.C A True Transition State Theory Calculation

  13. Data

  14. Solution According to transition state theory: (7.C.2)

  15. Solution Continued

  16. Next: Substitute Expressions From Tables 6.5 According to Table 6.5: 7.C.4 where qt is the translational partition function for a single translational mode of a molecule, m is the mass of the molecule, kB is Boltzmann’s constant, T is temperature, and hP is Plank’s constant. For our particular reaction, the fluorine can translate in three directions; the H2 can translate in three directions; the transition state can translate in three directions.

  17. Consequently

  18. Performing The Algebra (7.C.6)

  19. Next: Calculate The Last Term In Equation (7.C.6) Rearranging the last term shows: Plugging in the numbers yields: Doing the arithmetic yields: (7.C.7) (7.C.8) (7.C.9)

  20. Solution Continued Combining Equations (7.C.6) And (7.C.9) Yields:

  21. Next: Calculate The Ratio Of The Rotational Partition Functions

  22. Combining (7.C.12) And (7.C.13) Yields

  23. Next: Calculate The Vibrational Partition Functions. (7.C.16)

  24. First Get An Expression For The Term In Exponential In Equation (7.C.16) (7.C.17)

  25. Substitution In Values At hp And kB From The Appendix Yields 7.C.18 Note that we actually used hpc/Na and kB/Na in equation 7.C.16, and not hp where Na is Avogadro’s number and c is the speed of light in order to get the units right Doing the arithmetic in equation 7.C.18 yields: 7.C.19

  26. Substituting The vibrational partition function ratio equals: (7.C.20)

  27. Next: Calculate The Ratio Of The Partition Functions For The Electronic State Only consider the ground electronic state: (7.C.21)

  28. Finally: Calculate kBT/hP (7.C.22)

  29. Putting This All Together, Allows One To Calculate A Pre-exponential (7.C.23) Plugging in the numbers: (7.C.24)

  30. Note: Calculation Used A Fitted Geometry If one uses the actual transition state geometry, the only thing that changes significantly is the rotational term. One obtains: (7.C.26)

  31. Comparison To Collision Theory

  32. Collision Theory Continued (7.C.29)

  33. Comparison Of Results

  34. Summary: Transition State Theory Makes Two Corrections To Collision Theory

  35. Question • What did you learn new in this lecture?

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