Boltzmann’s Concepts of Reaction Rates 5/4/2015. Distribution of Air Particles NumberNumber Height.

Boltzmann’s Concepts of Reaction RatesBoltzmann’s Concepts of Reaction Rates

Velocities

Collision Num bers M ean Free Path Viscosities

Derivations Energies

M axw ell-Boltzm ann Distribution

Barom etric Form ulation

Boltzm ann Distribution

04/18/23

0 5000 1 104

1.5 104

2 104

2.5 104

0

2.4 104

4.8 104

7.2 104

9.6 104

1.2 105

P2 h1( )

Pa

P3 h1( )

Pa

P h1( )

Pa

h1

m

Distribution of Air ParticlesDistribution of Air Particles

Number

Height

P.S. 5Mathcad & EXCEL

http://course1.winona.edu/wng/Assign/c414assignmentss01.htm

Distribution of Molecular Energy LevelsDistribution of Molecular Energy Levels

EquationBoltzmanneg

g

N

N kTE

j

i

j

i /

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

If Boltz. Factor Comment

E << kT Close to 1 Ratio of population is equal

E ~ kT 1/e = 0.368 Upper level drops suddenly

E >> kT About 0 Zero upper level population

(S14) The Barometric Formulation(S14) The Barometric Formulation

(S14) The Barometric Formulation(S14) The Barometric Formulation

The Barometric Formulation – S11The Barometric Formulation – S11

The Barometric FormulationThe 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.

Molecular TemperatureMolecular Temperature

Distribution Measurement of

VibrationalTemp. in Hot Gases, Plasmas, Explosions

RotationalLow Temp. in Interstellar Gases

ElectronicHigh Stellar Temp. of Atoms and Ions

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

TRKE 2

3

K-M Model: Root-Mean-Square SpeedK-M Model: Root-Mean-Square Speed

Maxwell-Boltzmann DistributionMaxwell-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 ]

x

TkumdueA

N

dN x

/2

1 2

1500 1000 500 0 500 1000 15000

5 104

0.001

0.0015

0.002

0.0025

0

F1 u( )

m1

s

F2 u( )

m1

s

F3 u( )

m1

s

15001500 u

m s1

x

TkumdueA

N

dN x

/2

1 2

Mcad

MB Distribution: NormalizationMB Distribution: Normalization

Integral Tables

1D-x Maxwell-Boltzmann Distribution1D-x Maxwell-Boltzmann Distribution

One-dimensional Velocity Distribution in the x-direction: [ 1Du-x ]

x

Tkum

uD

dueTk

m

N

dN x

x

/2

1

1

2

2

One-dimensional Energy Distribution in the x-direction: [ 1DE-x ]

xTk

xED

deTkN

dNx

x

/2

1

1 4

1

x

Tkum

uD

dueTk

m

N

dN x

x

/2

1

1

2

2

xTk

xED

deTkN

dNx

x

/2

1

1 4

1

3D Maxwell-Boltzmann Distribution3D Maxwell-Boltzmann Distribution

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

xau

uD

duea

N

dNx

x

2

1

Cartesian Coordinates:

zyxuuua

D

dududuea

N

dN zyx

][

2/3

3

222

yau

uD

duea

N

dN y

y

2

1 zau

uD

duea

N

dNz

z

2

1


Re-shape box into sphere of same volume with radius u .

V = (4/3) u3 with u2 = ux2 + uy

2 + uz2

dV = dux duy duz = 4 u2 du

222/3

3

4/ ua

D

euadu

NdN

zyxuuua

D

dududuea

N

dN zyx

][

2/3

3

222

0 500 1000 1500 2000 25000

0.001

0.002

0.003

0.0035

0

F1 u( )

m1

s

F2 u( )

m1

s

F3 u( )

m1

s

25000 u

m s1


Low T

High T

Mcad


Conversion of Velocity-distribution to Energy-distribution:

= ½ m u2 ; d = mu du

222/3

3

4/ ua

uD

euadu

NdN

kT

D

ekTd

NdN

2/1

2/3

3

12/

Velocity Values from M-B DistributionVelocity Values from M-B Distribution

• urms = root mean square velocity

• uavg = average velocity

• ump = most probable velocity

x

naverage

n

N

dNxx )(

222/3

3

4/ ua

uD

euadu

NdN

Integral Tables

Velocity Value from M-B Distribution – S14Velocity Value from M-B Distribution – S14

Integral Tables

Velocity Value from M-B Distribution – S14Velocity Value from M-B Distribution – S14• urms = root mean square velocity

222/3

3

4/ ua

D

euadu

NdN

Integral Tables

Velocity Value from M-B Distribution S14Velocity Value from M-B Distribution S14 • uavg = average velocity

222/3

3

4/ ua

D

euadu

NdN

Integral Tables

Velocity Value from M-B Distribution S14Velocity Value from M-B Distribution S14 • ump = most probable velocity

222/3

3

4/ ua

D

euadu

NdN

Comparison of Velocity ValuesComparison of Velocity Values

Ratio in terms of :

urms uavg ump

1.73 1.60 1.41

m

kT

m

kT3

m

kT

8

m

kT2

Application to other Distribution FunctionsApplication to other Distribution Functions

de

kTN

dN kT

D

2/12/3

3

12

x

naverage

n

N

dNxx )(

Collision Properties ( Ref: Barrow )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 )Collision Frequency ( ZI )

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

avgrelative

avgI

uuwhere

udV

2:

2 2

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

*2

*)()(

2 NudZ

NVZ

avgI

II

M

TR

m

Tkuavg

88

Mean Free Path ( )Mean Free Path ( )

I

avg

Z

u

timeunit in with collidesit molecules #

timeunit per traveled distance

*2

12 Nd

Collision Rate ( ZII )Collision Rate ( ZII )

2

1*NZZ III

Double Counting FactorDouble Counting Factor

22 *)(2

1NudZ avgII

Viscosity ( ) from Drag EffectsViscosity ( ) from Drag Effects

mNuavg *2

1

*2

12 Nd

222 d

muavg

12310022.6

*

molL

TR

LPN

:where

Kinetic-Molecular-Theory Gas Properties - Collision Parameters @ 25oC and 1 atm

Species

Collision diameterMean free path

Collision Frequency

Collision Rate

d / 10-10 m d / Å / 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 8.1N2 3.74 3.74 6.56 7.2 8.9O2 3.57 3.57 7.16 6.2 7.6

Ar 3.62 3.62 6.99 5.7 7.0CO2 4.56 4.56 4.41 8.6 10.6

HI 5.56 5.56 2.96 7.5 10.6


Velocities






222/3

3

4/ ua

D

euadu

NdN

x

naverage

n

N

dNxx )(

Theories of Reaction RatesTheories of Reaction Rates

Collision Param eters

Potential E -surfaces

Unim olecular Reactions

M uonium Kinetics

QM T unnelling

Isotope Effects

Reactions in Solutions Diffusion Controlled

Partition Functions

T ransition State T heory T D T reatm ent of T ST

Collision T heory Hard Sphere Diam eters

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

Arrhenius ConceptArrhenius Concept

TR

Ea

eAk

Extended Arrhenius EquationExtended Arrhenius Equation

2- 3/2,- 1/2, 1,m :where

TR

Em eTak

'

Experimentally, m cannot be determined easily!

TRmEEaTeaA mm '

Implication: both A & Ea vary quite slowly with temperature. On the other hand, rate constants vary quite dramatically with temperature,

Extended Arrhenius EquationExtended Arrhenius Equation

TRmEEaTeaA mm '

Reaction ProgressReaction Progress

Collision TheoryCollision 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)

ZII (from simple hard sphere collision properties)

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




Collision Theory: collision rate ( ZII )Collision Theory: collision rate ( ZII )

][*)(2

1 22avgII vvvdNZ

M

TR

m

Tkv

88

For A-B collisions: AB , vAB

ABAB

BA

BAAB

Tkv

mm

mm

8Velocity Relative

Mass Reduced

Collision DiameterCollision Diameter

2BA

ABdd

d

Number per Unit VolumeNumber per Unit Volume

V

N

V

LnN AA

A

*V

N

V

LnN BBB

*

Collision Theory: collision rate ( ZII )Collision Theory: collision rate ( ZII )

2/122

)(8

*)(2

1

A

AAAAII m

kTdNZ

2/122

)(8

****

AB

ABBAABABBAABIIkT

dNNvdNNZ

Collision Theory: Rate Constant CalculationsCollision Theory: Rate Constant Calculations

Collision Theory:

TR

Ea

II eZ )(Rate

Kinetics: *][*][)'( 2 BA NNk Rate

Combining Collision Theory with Kinetics:

TR

Ea

BA

II eNN

Zk

*)(*)('2


A-A Collisions

TR

Ea

AA e

m

Tkdk

2/12

28

2

1'

m2 m s-1 per molecule

mol

molecule

m

dm

1

10022.6

1

10 23

3

33

TR

Ea

AAAA e

m

Tkd

Lk

2/12

3

)(28

2

10

Units of k: dm3 mol-1 s-1 M-1 s-1


A-B Collisions

TR

Ea

ABABAB e

TkdLk

2/123

)(28

10

Units of k: dm3 mol-1 s-1 M-1 s-1

2BA

ABdd

d

BA

BAAB mm

mm


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.

mass ratio k2 / M s-1 Reaction Ea / kJ mol-1

atom-1/atom-2 atom-1 + atom-2-CO2- ---> atom-1-atom-2 + CO2

-

1 1.20E+08 H + HCO2- ---> H2 + CO2

-

0.5 2.30E+07 H + DCO2- ---> HD + CO2

-

0.11 3.40E+06 Mu + HCO2- ---> MuH + CO2

- 33

0.056 9.90E+05 Mu + DCO2- ---> MuD + CO2

- 39

Simple Collision Theory: Comparison of Muonium/Hydrogen/Deuterium Abstractions

0.0E+00

2.0E+07

4.0E+07

6.0E+07

8.0E+07

1.0E+08

1.2E+08

1.4E+08

0.00 0.20 0.40 0.60 0.80 1.00

mass ratio

k /

M-1

s-1

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

Transition State TheoryTransition State Theory

Concept: Activated Complex or Transition State ( ‡ )

3D Potential Energy Surface

Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD

Activated Complex or Transition State ( ‡ )

Potential Energy SurfacesPotential Energy Surfaces

Consider: D + H2 DH + H

D

HA HB

r2

r1 r1= dH-D

r2 = dH-H

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 SolutionsReactions 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 SolutionsDiffusion Controlled Solutions

Smoluchowski (1917): D = diffusion coefficient

)(4 BAABdiff DDLdk

a

TkD

6

3

8 TRkdiff

a = radius;

= viscosity

Diffusion Controlled (Aqueous) Reactions

viscosity

25C 0.8904103 kg m

1 s1 95C 0.297510

3 kg m1 s

1

T1 25 273.15( ) Kk

8R T

3 R 8.3145J mol1 K

1T2 95 273.15( ) K

k25C8 R T1

3 25C k95C

8 R T2

3 95C

k25C 7.42 109 L mol

1 s1 k95C 2.74 10

10 L mol1 s

1

Arrhenius Equation: k A e

Ea

R T kJ 103

J

EaR T1 T2T2 T1

lnk25C

k95C

Ea 1.7 10

4 J mol1 Ea 17kJ mol

1

Therefore all aqueous solutions whose rate is determined by thediffusion of species should have an Activation Energy of about 17kJ/mol.

Diff-paper

Quantum Mechanical TunnelingQuantum Mechanical Tunneling

)(22

14EEamL

eE

Ea

E

Ea

Tunneling of Prob.

• curvature in Arrhenius plots

• abnormal A-factors

• relative isotope effects

• low Ea


Velocities






222/3

3

4/ ua

D

euadu

NdN

x

naverage

n

N

dNxx )(

Theories of Reaction RatesTheories of Reaction Rates

Collision Param eters

Potential E-surfaces

Unim olecular Reactions

M uonium Kinetics

QM Tunnelling

Isotope Effects

Reactions in Solutions Diffusion Controlled

Partition Functions

Transition State T heory TD T reatm ent of TST

Collision Theory Hard Sphere Diam eters

Arrhenius Concept

TR

Ea

ABABAB e

TkdLk

2/1

23)(2

810

3

8 TRkdiff

Boltzmann’s Concepts of Reaction Rates 5/4/2015. Distribution of Air Particles NumberNumber Height.

Documents

distribution functions

d ux slide

kelvin slide

effects slide

probable velocity slide

d velocity distribution

d ex slide

energy distribution