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ChE 452 Lecture 26

ChE 452 Lecture 26 . Transition State Theory Revisited. Last Time We Discussed Advanced Collision Theory:. Method Use molecular dynamics to simulate the collisions Integrate using statistical mechanics. (8.20). Summary:.

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ChE 452 Lecture 26

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  1. ChE 452 Lecture 26 Transition State Theory Revisited

  2. Last Time We Discussed Advanced Collision Theory: Method • Use molecular dynamics to simulate the collisions • Integrate using statistical mechanics (8.20)

  3. Summary: Key finding – we need energy and momentum to be correct to get reaction to happen. • Many molecules which are hot enough do not make it. Today how does that affect transition state theory?

  4. Next: Review Arrhenius’ Model vs Transition State Theory Arrhenius’ model: • Consider two populations of molecules: • Hot molecules in the right configuration to react (i.e. molecules moving toward each other with the right velocity, impact parameter etc to react). • Molecules not hot enough or not moving together in the right configuration. • Assume concentration of hot reactive molecules in equilibrium with reactants. • Assume reaction occurs whenever hot molecules collide in right configuration to react.

  5. Can Derive Tolman’s Equation Where kABC is the rate constant for reaction (9.1), is Boltzmann’s constant; T is the absolute temperature; hP is Plank’s constant; qA is the microcanonical partition function per unit volume of the reactant A; qBC 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; E+ is the average energy of the molecules which react and, q+ is the average partition function of the molecules which react, divided by the partition function for the translation of A toward BC. Tolman’s Equation is almost exact.

  6. A Comparison of Tolman’s Equation and Transition State Theory

  7. Approximate Derivation Of TST: Let’s go back to the statistical mechanics definition of the partition functions. Need to assume Phot = PTST (Distribution of hot molecules poised to react equals the distribution of states at the transition state).

  8. Assumptions • Ignores: • Re-crossing trajectories • Changes in q’s • Dynamics Figure 9.7 A recrossing trajectory.

  9. Let Me Examine The Assumption That q’s Are Equal (9.38) Assume! Vibration q = translation q Figure 9.3 The minimum energy pathway for motion over the barrier as determined by the trajectory calculations in Chapter 8.

  10. Additional Assumptions In Conventional State Theory All the in equation (9.40) are 0 for states with energies below the barrier and unity for states with energy above the barrier so that . Motion along the Rx direction is pure translation. Motion perpendicular to the Rx direction is pure vibration. The are determined only by the properties of the transition state; and independent of shape at barrier. (9.46)

  11. Example HNCHCN TST assume that the bending mode is pure translation, the other modes are pure vibration. Cancellation of error.

  12. Successes Of Transition State Theory • Transition state theory generally gives preexponentials of the correct order of magnitude. • Transition state theory is able to relate barriers to the saddle point energy in the potential energy surface; • Transition state theory is able to consider isotope effects; • Transition state theory is able to make useful prediction in parallel reactions like reactions (7.27) and (7.29).

  13. Key Prediction Experimentally, if one lowers the energy of the transition state, one usually lowers the activation energy for the reaction.

  14. Example Of Isotope Effects

  15. Isotope Effects Continued

  16. Limitations Of TST Correction to collision theory.

  17. Figure 9.9 A diagram showing the extent of the wavefunction for a molecule. In A the molecule is by itself. In B the molecule is near a barrier. Notice that the wavefunction has a finite size (i.e. there is some uncertainty in the position of the molecule.) As a result, when a molecule approaches a barrier, there a component of the molecule on the other side of the barrier. TST ignores Tunneling

  18. Tunneling Data For H+H2H2+H

  19. Modern Versions Of Transition State Theory • Variational transition state theory. • Tunneling corrections.

  20. Variational Transition State Theory Figure 9.4 The activation barriers for the reactions in Table 9.6. Key idea: TST is saddle point in the free energy plot, not the energy plot

  21. Example: X+H+CH3HCH3+X Little improvement over TST except for methane.

  22. Tunneling Corrections: Figure 9.9 A diagram showing the extent of the wavefunction for a molecule. In A the molecule is by itself. In B the molecule is near a barrier. Notice that the wavefunction has a finite size (i.e. there is some uncertainty in the position of the molecule.) As a result, when a molecule approaches a barrier, there a component of the molecule on the other side of the barrier.

  23. Figure 9.8 Tunneling through a barrier. Tunneling through a barrier.

  24. Derive: An Equation For Tunneling Figure 9.11 A plot of the square-well barrier. Solve Shroedinger equation for Motion near a barrier. ħ (9.73)

  25. Solution: Figure 9.12 The real part of the incident (i) scattered (r) and total wavefunction (T) for the square-well barrier. ħ2 (9.81)

  26. Eckart Barriers Better Assumption Figure 9.14 The Eckart potential. (9.63) (9.89) Significant improvement over TST.

  27. Example 9.A Tunneling Corrections Using The Eckart Barrier In problem 7.C we calculated the preexponential for the reaction: F+H2 HF+H (9.A.1) How much will the pre-exponential change at 300 K if we consider tunneling?

  28. Solution

  29. Next Evaluate К(T) According to Equation (9.101) Where according to Table 7.C.1 =310 cm-1. (9.A.4)

  30. Solution Continued Equation (7.C.17) says: (9.A.5)

  31. Substituting Into Equation (9.A.4) Yields Therefore the rate will only go up by 10% at 300K. (9.A.7)

  32. At 100K Substituting into Equation (9.A.9) (9.A.8) (9.A.9)

  33. Note The tunneling correction in this example is smaller than normal, because the barrier has such a small curvature. A typical number would be 1000 cm-1. Still it illustrates the point that tunneling becomes more important as the temperature drops.

  34. If We Replace The Hydrogen With A Deuterium:

  35. Semiclassical Approximation For Tunneling (9.96) Figure 9.15 Some of the paths people assume to calculate tunneling rates. ħ2

  36. Exact Calculations Figure 9.18 Some trajectories from the reactants to products.

  37. Conclusions: Conclusions: Key assumptions in TST 1) no recrossing trajectories 2) qhot=qtst 3) no tunneling OK to factor of 20 for bimolecular reactions Hard to do better

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

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