Boltzmann’s Concepts of Reaction Rates
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Boltzmann’s Concepts of Reaction Rates. Distribution of Air Particles. Number. Height. Mathcad & EXCEL. P.S. 5. Distribution of Molecular Energy Levels. Where: E = E i – E j & e -E/kT = Boltzman Factor. (S14) The Barometric Formulation. (S14) The Barometric Formulation.

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Mathcad & EXCEL

P.S. 5


Distribution of Molecular Energy Levels

Where: E = Ei – Ej & e-E/kT = Boltzman Factor





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.



The kinetic molecular model for gases postulates
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



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 ]




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 ]


3D Maxwell-Boltzmann Distribution

3D Velocity Distribution: [ 3Du ] , Let: a = m/2kT

Cartesian Coordinates:


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



3D Maxwell-Boltzmann Distribution

Conversion of Velocity-distribution to Energy-distribution:

 = ½ m u2 ; d  = mu du


Velocity Values from M-B Distribution

  • urms = root mean square velocity

  • uavg = average velocity

  • ump = most probable velocity

Integral Tables



Velocity Value from M-B Distribution – S14

  • urms = root mean square velocity

Integral Tables


  • uavg = average velocity

Velocity Value from M-B Distribution S14

Integral Tables


Velocity Value from M-B Distribution S14

  • ump = most probable velocity




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


Collision Frequency ( ZI )

Interaction Volume ( VI ): ( d = interaction diameter )

Define: N* = N/V = molecules per unit volume



Collision Rate ( ZII )

Double Counting Factor


Viscosity (  ) from Drag Effects


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




Arrhenius Concept Parameters @ 25

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


Extended Arrhenius Equation Parameters @ 25

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,



Reaction Progress Parameters @ 25


Collision Theory Parameters @ 25

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)


Fraction of molecules with E > Ea : e Parameters @ 25-Ea/RT (Maxwell-Boltzmann Distribution)


Collision Theory: collision rate ( Z Parameters @ 25II )

For A-B collisions: AB , vAB


Collision Diameter Parameters @ 25

Number per Unit Volume



Collision Theory: Rate Constant Calculations Parameters @ 25

Collision Theory:

Kinetics:

Combining Collision Theory with Kinetics:


Collision Theory: Rate Constant Calculations Parameters @ 25

A-A Collisions

m2

per molecule

m s-1

Units of k: dm3 mol-1 s-1 M-1 s-1


Collision Theory: Rate Constant Calculations Parameters @ 25

A-B Collisions

Units of k: dm3 mol-1 s-1 M-1 s-1


Collision Theory: Rate Constant Calculations Parameters @ 25

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.



H Parameters @ 25

H

H

H

H

H

D

D

D

D

D

D

Transition State Theory

Concept: Activated Complex or Transition State ( ‡ )

3D Potential Energy Surface

Saddle point

H2 + D2 2 HD

H2 + D2

 2 HD

Activated Complex or Transition State ( ‡ )


Potential Energy Surfaces Parameters @ 25

Consider: D + H2 DH + H

D

r1= dH-D

r1

r2 = dH-H

r2

HA

HB

Most favorable at:  = 0o , 180o

Calculate energy of interaction at different r1, r2 and . Get 3D Energy Map.

Reaction coordinate = path of minimum energy leading from reactants to products.


Reactions in Solutions Parameters @ 25

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 Parameters @ 25

Smoluchowski (1917): D = diffusion coefficient

a = radius;

 = viscosity


Diff-paper Parameters @ 25


Quantum Mechanical Tunneling Parameters @ 25

  • curvature in Arrhenius plots

  • abnormal A-factors

  • relative isotope effects

  • low Ea




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