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Boltzmann’s Concepts of Reaction Rates 5/4/2015

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Distribution of Air Particles NumberNumber Height

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P.S. 5MathcadMathcad & EXCELEXCEL

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Distribution of Molecular Energy Levels Where: E = E i – E j & e - E/kT = Boltzman Factor If Boltz. FactorComment E << kT Close to 1Ratio of population is equal E ~ kT 1/e = 0.368Upper level drops suddenly E >> kT About 0Zero upper level population

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(S14) The Barometric Formulation

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The Barometric Formulation – S11

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

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Molecular Temperature DistributionMeasurement of Vibrational Temp. in Hot Gases, Plasmas, Explosions Rotational Low Temp. in Interstellar Gases Electronic High Stellar Temp. of Atoms and Ions

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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

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K-M Model: Root-Mean-Square Speed

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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: [ 1D u-x ]

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Mcad

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MB Distribution: Normalization Integral Tables

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1D-x Maxwell-Boltzmann Distribution One-dimensional Velocity Distribution in the x-direction: [ 1D u-x ] One-dimensional Energy Distribution in the x-direction: [ 1D E-x ]

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

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3D Maxwell-Boltzmann Distribution Re-shape box into sphere of same volume with radius u. V = (4/3) u 3 with u 2 = u x 2 + u y 2 + u z 2 dV = du x du y du z = 4 u 2 du

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3D Maxwell-Boltzmann Distribution Low T High T Mcad

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3D Maxwell-Boltzmann Distribution Conversion of Velocity-distribution to Energy-distribution: = ½ m u 2 ;d = mu du

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Velocity Values from M-B Distribution u rms = root mean square velocity u avg = average velocity u mp = most probable velocity Integral Tables

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Velocity Value from M-B Distribution – S14 Integral Tables

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Velocity Value from M-B Distribution – S14 u rms = root mean square velocity Integral Tables

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Velocity Value from M-B Distribution S14 u avg = average velocity Integral Tables

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Velocity Value from M-B Distribution S14 u mp = most probable velocity

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Comparison of Velocity Values Ratio in terms of : u rms u avg u mp 1.731.601.41

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Application to other Distribution Functions

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Collision Properties ( Ref: Barrow ) Z I = collision frequency = number of collisions per molecule = mean free path = distance traveled between collisions Z II = collision rate = total number of collisions Main Concept => Treat molecules as hard-spheres

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Collision Frequency ( Z I ) Interaction Volume ( V I ): ( d = interaction diameter ) Define: N* = N/V = molecules per unit volume

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Mean Free Path ( )

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Collision Rate ( Z II ) Double Counting Factor

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Viscosity ( ) from Drag Effects

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Kinetic-Molecular-Theory Gas Properties - Collision Parameters @ 25 o C and 1 atm Species Collision diameter Mean free path Collision Frequency Collision Rate d / 10 -10 md / Å / 10 -8 m Z I / 10 9 s -1 Z II / 10 34 m -3 s -1 H2H2 2.73 12.414.317.6 He2.18 19.16.68.1 N2N2 3.74 6.567.28.9 O2O2 3.57 7.166.27.6 Ar3.62 6.995.77.0 CO 2 4.56 4.418.610.6 HI5.56 2.967.510.6

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Boltzmann’s Concepts of Reaction Rates

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Theories of Reaction Rates

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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 Arrhenius Concept

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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,

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Extended Arrhenius Equation

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Reaction Progress

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

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Collision Theory: collision rate ( Z II ) For A-B collisions: AB, v AB

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Collision Diameter Number per Unit Volume

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Collision Theory: collision rate ( Z II )

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

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Collision Theory: Rate Constant Calculations A-A Collisions m2m2 m s -1 per molecule Units of k:dm 3 mol -1 s -1 M -1 s -1

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Collision Theory: Rate Constant Calculations A-B Collisions Units of k:dm 3 mol -1 s -1 M -1 s -1

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Collision Theory: Rate Constant Calculations Consider:2 NOCl(g) 2NO(g) + Cl 2 (g)T = 600. K Ea = 103 kJ/mold NOCl = 283 pm (hard-sphere diameter) CalculateCalculate the second order rate constant.

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http://www.ubc.ca/index.html

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Transition State Theory Concept: Activated Complex or Transition State ( ‡ ) 3D Potential Energy Surface Saddle point HH DD HH DD HH DD H 2 + D 2 2 HD H 2 + D 2 2 HD Activated Complex or Transition State ( ‡ )

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Potential Energy Surfaces Consider:D + H 2 DH + H D HAHA HBHB r2r2 r1r1 r 1 = d H-D r 2 = d H-H Most favorable at: = 0 o, 180 o Calculate energy of interaction at different r 1, r 2 and . Get 3D Energy Map.3D Energy Map Reaction coordinate = path of minimum energy leading from reactants to products.

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Reactions in Solutions Compared to gaseous reactions, reactions in solutions require diffusion through the solvent molecules. The initial encounter frequencies should be substantially higher for gas collisions. However, in solutions, though initial encounters are lower, but once the reactants meet, they get trapped in “solvent cages”, and could have a great number of collisions before escaping the solvent cage.

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Diffusion Controlled Solutions Smoluchowski (1917): D = diffusion coefficient a = radius; = viscosity

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Diff-paper

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Quantum Mechanical Tunneling curvature in Arrhenius plots abnormal A-factors relative isotope effects low Ea

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Boltzmann’s Concepts of Reaction Rates

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Theories of Reaction Rates

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