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

Distribution of Air Particles NumberNumber Height

P.S. 5MathcadMathcad & EXCELEXCEL

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

(S14) The Barometric Formulation

The Barometric Formulation – S11

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

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

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

K-M Model: Root-Mean-Square Speed

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 ]

Mcad

MB Distribution: Normalization Integral Tables

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 ]

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

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

3D Maxwell-Boltzmann Distribution Low T High T Mcad

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

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

Velocity Value from M-B Distribution – S14 Integral Tables

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

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

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

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

Application to other Distribution Functions

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

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

Mean Free Path ( )

Collision Rate ( Z II ) Double Counting Factor

Viscosity (  ) from Drag Effects

Kinetic-Molecular-Theory Gas Properties - Collision 25 o C and 1 atm Species Collision diameter Mean free path Collision Frequency Collision Rate d / md / Å / m Z I / 10 9 s -1 Z II / m -3 s -1 H2H He N2N O2O Ar CO HI

Boltzmann’s Concepts of Reaction Rates

Theories of Reaction Rates

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

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,

Extended Arrhenius Equation

Reaction Progress

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)

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

Collision Diameter Number per Unit Volume

Collision Theory: collision rate ( Z II )

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

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

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

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.

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 ( ‡ )

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.

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.

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

Diff-paper

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

Boltzmann’s Concepts of Reaction Rates

Theories of Reaction Rates