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Boltzmann’s Concepts of Reaction Rates

Boltzmann’s Concepts of Reaction Rates. Distribution of Air Particles. Number. Height. Mathcad & EXCEL. P.S. 5. Perrin Experiment: Mathcad Calculation Picture File. Distribution of Molecular Energy Levels. Where: E = E i – E j & e -E/kT = Boltzman Factor.

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Boltzmann’s Concepts of Reaction Rates

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  1. Boltzmann’s Concepts of Reaction Rates

  2. Distribution of Air Particles Number Height

  3. Mathcad & EXCEL P.S. 5

  4. Perrin Experiment: Mathcad Calculation Picture File

  5. Distribution of Molecular Energy Levels Where: E = Ei – Ej & e-E/kT = Boltzman Factor

  6. (S14) The Barometric Formulation

  7. (S14) The Barometric Formulation E = Ei – Ej

  8. The Barometric Formulation – S11

  9. The Barometric Formulation • Calculate the pressure at mile high city (Denver, CO). [1 mile = 1610 m] Po = 101.325 kPa , T = 300. K . Assume 20.0 and 80.0 mole % of oxygen gas and nitrogen gas, respectively.

  10. Molecular Temperature

  11. The Kinetic Molecular Model for Gases ( Postulates ) • Gas consists of large number of small individual particles with negligible size • Particles in constant random motion and collisions • No forces exerted among each other • Kinetic energy directly proportional to temperature in Kelvin

  12. K-M Model: Root-Mean-Square Speed

  13. Maxwell-Boltzmann Distribution M-B Equation gives distribution of molecules in terms of: • Speed/Velocity, and • Energy One-dimensional Velocity Distribution in the x-direction: [ 1Du-x ]

  14. Mcad

  15. MB Distribution: Normalization Mcad Integral Tables

  16. 1D-x Maxwell-Boltzmann Distribution One-dimensional Velocity Distribution in the x-direction: [ 1Du-x ] One-dimensional Energy Distribution in the x-direction: [ 1DE-x ]

  17. 3D Maxwell-Boltzmann Distribution 3D Velocity Distribution: [ 3Du ] , Let: a = m/2kT Cartesian Coordinates:

  18. 3D Maxwell-Boltzmann Distribution Re-shape box into sphere of same volume with radius u . V = (4/3)  u3 with u2 = ux2 + uy2 + uz2 dV = dux duy duz = 4  u2 du

  19. 3D Maxwell-Boltzmann Distribution Mcad Low T High T

  20. 3D Maxwell-Boltzmann Distribution Conversion of Velocity-distribution to Energy-distribution:  = ½ m u2 ; d  = mu du

  21. Velocity Values from M-B Distribution • urms = root mean square velocity • uavg = average velocity • ump = most probable velocity Integral Tables

  22. Velocity Value from M-B Distribution – S14 Integral Tables

  23. Velocity Value from M-B Distribution – S14 • urms = root mean square velocity Integral Tables

  24. uavg = average velocity Velocity Value from M-B Distribution S14 Integral Tables

  25. Velocity Value from M-B Distribution S14 • ump = most probable velocity

  26. Comparison of Velocity Values

  27. Application to other Distribution Functions

  28. Collision Properties ( Ref: Barrow ) • ZI = collision frequency = number of collisions per molecule •  = mean free path = distance traveled between collisions • ZII = collision rate = total number of collisions • Main Concept => Treat molecules as hard-spheres

  29. Collision Frequency ( ZI ) Interaction Volume ( VI ): ( d = interaction diameter ) Define: N* = N/V = molecules per unit volume

  30. Mean Free Path (  )

  31. Collision Rate ( ZII ) Double Counting Factor

  32. Viscosity (  ) from Drag Effects

  33. Kinetic-Molecular-Theory Gas Properties - Collision Parameters @ 25oC and 1 atm Species Collision diameter Mean free path Collision Frequency Collision Rate d / 10-10 m d / Å l / 10-8 m ZI / 109 s-1 ZII / 1034 m-3 s-1 H2 2.73 2.73 12.4 14.3 17.6 He 2.18 2.18 19.1 6.6 8.1 N2 3.74 3.74 6.56 7.2 8.9 O2 3.57 3.57 7.16 6.2 7.6 Ar 3.62 3.62 6.99 5.7 7.0 CO2 4.56 4.56 4.41 8.6 10.6 HI 5.56 5.56 2.96 7.5 10.6

  34. Boltzmann’s Concepts of Reaction Rates

  35. Theories of Reaction Rates

  36. Arrhenius Concept The Arrhenius Equation • Arrhenius discovered most reaction-rate data obeyed the Arrhenius equation: • Including natural phenomena such as: • Chirp rates of crickets • Creeping rates of ants

  37. Extended Arrhenius Equation Experimentally, m cannot be determined easily! Implication: both A & Ea vary quite slowly with temperature. On the other hand, rate constants vary quite dramatically with temperature.

  38. Extended Arrhenius Equation

  39. Reaction Progress

  40. Collision Theory Main Concept: Rate Determining Step requires Bimolecular Encounter (i.e. collision) Rxn Rate = (Collision Rate Factor) x (Activation Energy) ZII (from simple hard sphere collision properties) Fraction of molecules with E > Ea : e-Ea/RT (Maxwell-Boltzmann Distribution)

  41. Fraction of molecules with E > Ea : e-Ea/RT (Maxwell-Boltzmann Distribution)

  42. Collision Theory: collision rate ( ZII ) For A-B collisions: AB , vAB

  43. Collision Diameter Number per Unit Volume

  44. Collision Theory: collision rate ( ZII )

  45. Collision Theory: Rate Constant Calculations Collision Theory: Kinetics: Combining Collision Theory with Kinetics:

  46. Collision Theory: Rate Constant Calculations A-A Collisions m2 per molecule m s-1 Units of k: dm3 mol-1 s-1 M-1 s-1

  47. Collision Theory: Rate Constant Calculations A-B Collisions Units of k: dm3 mol-1 s-1 M-1 s-1

  48. Collision Theory: Rate Constant Calculations Consider: 2 NOCl(g)  2NO(g) + Cl2(g) T = 600. K Ea = 103 kJ/mol dNOCl = 283 pm (hard-sphere diameter) Calculate the second order rate constant.

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