1. ChE 452 Lecture 20 Collision Theory 1

2. So Far This Course Has Shown 2

3. Theory Of Reaction Rates Has Two Parts Theory of Preexponentials Collision Theory, Transition State Theory, RRKM, Molecular Dynamics Theory of Activation Barriers Polanyi Relationship, Marcus Equation, Blowers-Masel, Quantum Methods 3

4. Models For Preexponentials Collision theory (old collision theory) – simple model for preexponential - ~10 13 /sec, ~10 13 Å 3 /sec, ~10 13 A 6 /sec Transition state theory – slightly better model for preexponential – bimolecular (small correction to collision theory). RRKM – better model for preexponential – unimolecular-explains rate constraints at 10 18 /sec Molecular Dynamics & Tunneling – accurate method, but time consuming 4

5. Plan For Today Describe Arrhenius’ Model (1889) Describe Trautz and Lewis model (1918) Show limitations 5

6. Arrhenius Model For A  B Cold unreactive molecules Hot reactive molecules 6 Divides molecules into two populations

7. Next Derive Equation For Rate Equilibrium: Rate equ 7 (7.4) (7.2)

8. Derivation Continued 8 (7.6) (7.7) (7.8) (7.4) (7.5)

9. Result of Arrhenius’ Model Rate constant varies exponentially with T -1 No expression for K o 9 (7.8)

10. Collision Theory Assume K o equals the collision rate 10 (7.2)

11. Collision Theory 11 Figure 7.2 A collision between an A molecule and BC molecules. (7.12) (7.10) (7.11)

12. Next: Consider Billiard Ball Collisions 12 Collisions occur whenever molecules get close Figure 7.3 Some typical billiard ball collisions

13. Next: Calculate How Many Collisions Occur 13 Consider the volume swept out by a BC molecule in time to L ABC = v A  BC t c (7.13)

14. Next: Calculate How Many Collisions Occur 14

15. Derivation Continued 15

16. Derivation Continued 16

17. Derivation Continued 17 (7.19) The total number of collisions is (7.20) Combining equations (7.17), (7.18), (7.19), (7.20) yields. (7.21)

18. Deviation Continued 18 (7.21)

19. After Pages Of Algebra We Obtain 19 (7.24) (7.25)

20. Trautz & Lewis’ Approximation If ∆S † =0, one obtains the standard result 20 (7.26) Equation (7.26) is the key result for simple collision theory.

21. Pages Of Algebra Yields Trautz & Lewis’ Approximation 21 (7.26) Equation (7.26) is the key result for simple collision theory. Derivation

22. Additional Assumption Calculate the molecular velocity ignoring that molecules are hot. Where: 22 and m A, m B and m C are the masses of A, B and C in atomic mass units (1 AMU = 1.66  10 -24 g).

23. Simplified Equation In lecture 14 we showed 23 (7.29)

24. Example 7.A A Collision Theory Calculation 24

25. Solution: 25 According to collision theory: (7.A.2)

26. Step 1: Calculate V ABC According to equation (7.26): with 26 (7.A.3) (7.A.4)

27. Step 1 Continued For reaction (7.A.1) (7.A.5) Substituting the numbers shows that 500K: (7.A.6) 27

28. Step 2: Estimate d coll Trautz’s approximation Were d A and d B are the Van der Waals radii of A and B Therefore 28 (7.A.7)

29. Solution Continued Substituting (7.A.5) and (7.A.6) into equation (7.A.2) yields: 29 (7.A.8)

30. Discussion Problem Use collision theory to calculate the rate constant for the reaction F + H 2  H + HF Assume a collision diameter of 2.3Å 30

31. Solution: Step 1 Calculate  31

32. Step 2: Calculate v 32

33. Solution 33 k o =(4  10 12 Å/sec)  (  (3Å) 2 ) = 1.1 × 10 14 Å 3 /sec

34. Key Predictions Of Collision Theory Preexponentials always between 10 13 and 10 14 /sec for small molecules No special configurations effects Lighter species (i.e. H atoms tend to react faster). Larger molecules have larger cross sections than smaller molecules 34

35. Preexponentials Usually The Same Order As Collision Theory? 35

36. Comparisons Between Collision Theory And Experiments 36

37. Cases Where Collision Theory Fails 37

38. Why Does Collision Theory Fail For Reaction 7.30 38 Reaction 7.30 requires a special collision geometry: (7.33) (7.34)

39. Summary Collision theory: reaction occurs whenever reactants collide. Gives correct order of magnitude or slightly high pre-exponential Some spectacular failures TST theory after exam 39

40. Class Question What did you learn new today? 40