1. ChE 452 Lecture 15 Transition State Theory 1

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

3. Motion Over PE Surface 3 Figure 7.6 A potential energy surface for the reaction H + CH 3 OH  H 2 + CH 2 OH 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 4

5. Derivation Continued Assume reaction rate=K 0 [ABC † ] Where K 0 is the rate constant for reaction of the hot molecules From equilibrium Combining reaction rate = K 0 K EQU [A][BC] or rate constant=K 0 K EQU 5

6. From Statistical Mechanics 6

7. Combining 7 Equation (7.38) is exact but we will need an expression for K 0. We can get it from collision theory. (7.38) here k A  BC is the rate constant for reaction (7.38); k B is Boltzman’s constant; T is the absolute temperature; K 0 is the rate constant; q A is the microcanonical partition function per unit volume of the reactant A; q BC 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.

8. Next: Estimate K 0 From Collision Theory 8 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: 9 (7.40)

10. Key Approximation 10

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

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

13. Data 13

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

15. Solution Continued 15

16. Next: Substitute Expressions From Tables 6.5 According to Table 6.5: 7.C.4 where q t is the translational partition function for a single translational mode of a molecule, m is the mass of the molecule, k B is Boltzmann’s constant, T is temperature, and h P is Plank’s constant. For our particular reaction, the fluorine can translate in three directions; the H 2 can translate in three directions; the transition state can translate in three directions. 16

17. Consequently 17

18. Performing The Algebra 18 (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: 19 (7.C.8) (7.C.9) (7.C.7)

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

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

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

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

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

25. Substitution In Values At h p And k B From The Appendix Yields 25 Note that we actually used h p c/N a and k B /N a in equation 7.C.16, and not hp where N a 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 7.C.18

26. Substituting 26 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: 27 (7.C.21)

28. Finally: Calculate k B T/h P 28 (7.C.22)

29. Putting This All Together, Allows One To Calculate A Pre-exponential 29 (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: 30 (7.C.26)

31. Comparison To Collision Theory 31

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

33. Comparison Of Results 33

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

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