Boltzmann Concept

822019 Boltzmann Concept

httpslidepdfcomreaderfullboltzmann-concept 154

Boltzmannrsquos Concepts of Reaction Rates


V e l o c

C o l l i s i

M e a n

V i s c o

D e r i v

E n e r

M a x w e l l - B

B a r o m e t r

B o l t z m a n

022712



0 5000 1 104

15 104

2 104

25 104

0

24 104

48 104

72 104

96 104

12 105

P2 h1( )

Pa

P3 h1( )

Pa

P h1( )

Pa

h1

m

Distribution of Air Particles


Num

ber

Height



PS 5



Distribution of Molecular Energy Levels


Equation Boltzmanne g

g

N

N kT E

j

i

j

i ∆minus

sdot=

Where ∆ E = Ei ndash E j amp e-∆ EkT = Boltzman Factor

If Boltz Factor Comment

∆ E ltlt kT Close to 1 Ratio of population is equal

∆ E ~ kT 1e = 0368 Upper level drops suddenly

∆ E gtgt kT About 0 Zero upper level population



The Barometric Formulation






bull Calculate the pressure at mile high city (Denver CO) [1 mile= 1610 m] Po = 101325 kPa T = 300 K Assume 200 and800 mole of oxygen gas and nitrogen gas respectively



Molecular TemperatureMolecular Temperature

Distribution Measurement of

Vibrational Temp in Hot GasesPlasmas Explosions

Rotational Low Temp in Interstellar Gases

Electronic High Stellar Temp of Atoms and Ions



The Kinetic Molecular Model for

Gases ( Postulates )The Kinetic Molecular Model for

Gases ( Postulates )

bull Gas consists of large number of smallindividual particles with negligible size

bull Particles in constant random motion andcollisions

bull No forces exerted among each other

bull Kinetic energy directly proportional totemperature in Kelvin

T R KE sdotsdot=

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

bullSpeedVelocity and

bullEnergy

One-dimensional Velocity Distribution in the x-direction

[ 1Du-x ]

x

T k um

due A N

dN x

sdotsdot=sdotsdotsdotminus

2

1 2



1500 1000 500 0 500 1000 15000

5 104

0001

00015

0002

00025

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m 1minus ssdot

15001500minus u

m s1minussdot

x

T k um

due A

N

dN x


2

1 2

Mcad



MB Distribution NormalizationMB Distribution Normalization



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

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

x

T k um

u D

dueT k

m

N

dN x

x

sdotsdot

sdotsdotsdot

=

sdotsdotsdotminus

minus

2

1

1

2

2 π

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

xT k

x

E D

d eT k N

dN x

x

ε ε

π

ε sdotsdotsdotsdotsdotsdot

=

sdotminusminus

minus

2

1

1 4

1



x

T k um

u D

dueT k

m

N

dN x

x

sdotsdotsdotsdotsdot

=

sdotsdotsdotminus

minus

2

1

1

2

2 π

x

T k

x E D

d eT k N

dN x

x

ε ε

π


=

sdotminusminus

minus

2

1

1 4

1



3D Maxwell-Boltzmann Distribution3D Maxwell-Boltzmann Distribution

3D Velocity Distribution [ 3Du ] Let a = m2kT

xau

u D

duea

N

dN x

x

sdotsdot=

minus

minus

2

1 π

Cartesian Coordinates

z y x

uuua

D

dududuea

N

dN z y x sdotsdotsdotsdot

=

++minus ][23

3

222

π

y

au

u D due

a

N

dN y

y sdotsdot=

minus

minus

2

1 π z

au

u D due

a

N

dN z

z sdotsdot=

minus

minus

2

1 π




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

V = (43) π u3 with u2 = ux2 + uy

2 + uz2

dV = dux duy duz = 4 π u2 du

2223

3

4 ua

D

euadu

N dN sdotminussdotsdotsdot=

π

z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π



0 500 1000 1500 2000 25000

0001

0002

0003

00035

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m1minus

ssdot

25000 u

m s

1minus

sdot


Low T

High T

Mcad




Conversion of Velocity-distribution to Energy-distribution

ε = frac12 m u2 d ε = mu du

2223

3

4 ua

u D

euadu

N dN sdotminus

minussdotsdotsdot=

π

kT

D

ekT d

N dN ε

ε

ε π ε

minus

minus

sdotsdot

sdot=

21

23

3

12



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

bull urms = root mean square velocity

bull uavg = average velocity

bull ump = most probable velocity

int

sdot=

x

naverage

n

N dN x x )(



Velocity Value from M-B DistributionVelocity Value from M-B Distribution bull urms = root mean square velocity

2223

3

4 ua

D

euadu


π



Velocity Value from M-B DistributionVelocity Value from M-B Distribution bull uavg = average velocity

2223

3

4 ua

D

euadu


π



Velocity Value from M-B DistributionVelocity Value from M-B Distribution bull ump = most probable velocity

2223

3

4 ua

D

euadu


π



Comparison of Velocity ValuesComparison of Velocity Values

Ratio in terms of

urms uavg ump

173 160 141

m

kT

m

kT sdot3

m

kT sdot

π

8

m

kT sdot2



Application to other Distribution FunctionsApplication to other Distribution Functions



Collision Properties ( Ref Barrow )Collision Properties ( Ref Barrow )

bull ZI = collision frequency = number of collisions per molecule

bull λ = mean free path = distance traveled between collisions

bullZ

II= collision rate = total number of collisions

bull Main Concept =gt Treat molecules as hard-spheres



Collision Frequency ( ZI )Collision Frequency ( ZI )

Interaction Volume ( VI ) ( d = interaction diameter )

avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π

Define N = NV = molecules per unit volume

2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88



Mean Free Path ( λ )Mean Free Path ( λ )

I

avg

Z

u==

timeunitinwithcollidesitmolecules

timeunitpertraveleddistanceλ

2

12

N d sdotsdotsdot= π λ



Collision Rate ( ZII )Collision Rate ( ZII )

sdotsdot=

2

1 N Z Z I II

Double Counting Factor Double Counting Factor

22 )(2

1 N ud Z avg II sdotsdotsdotsdot= π



Viscosity ( η ) from Drag EffectsViscosity ( η ) from Drag Effects

λ η sdotsdotsdotsdot= m N uavg 2

1

2

12 N d sdotsdotsdot= π

λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where



Kinetic-Molecular-Theory Gas Properties - Collision Parameters

25oC and 1 atm

Species

Collision diameter Meanfree path CollisionFrequency CollisionRate

d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106



Boltzmannrsquos Concepts of Reaction RatesBoltzmannrsquos Concepts of Reaction Rates

V e l o c

C o l l i s i o M e a n V i s c o

D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(



Theories of Reaction RatesTheories of Reaction Rates

C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p

R e a c t i o D i f f u s i o

P a r t i t i o

T r a n s i t i T D T r e a

C o l l i s i o H a r d S p

A r r h e n i



The Arrhenius Equation

bull Arrhenius discovered most reaction-rate data obeyed theArrhenius equation

bull Including natural phenomena such asbull Chirp rates of crickets

bull Creeping rates of ants

Arrhenius ConceptArrhenius Concept

T R Ea

e Ak sdotminus

sdot=

A i i



Extended Arrhenius EquationExtended Arrhenius Equation

2-32-121m where plusmn=sdotsdot= sdotminus T R

E meT ak

Experimentally m cannot be determined easily

T Rm E EaT ea A mm sdotsdot+=sdotsdot=

Implication both A amp Ea vary quite slowly with temperatureOn the other hand rate constants vary quite dramatically withtemperature

E d d A h i E i







Reaction ProgressReaction Progress

C lli i Th



Collision TheoryCollision Theory

Main Concept Rate Determining Step requires Bimolecular Encounter (ie collision)

Rxn Rate = (Collision Rate Factor) x (Activation Energy)

ZII (from simple

hard sphere collision properties)

ZII (from simple


Fraction of molecules with E gt Ea e-EaRT (Maxwell-BoltzmannDistribution)




Fraction of molecules with E gt Ea e-EaRT (Maxwell-Boltzmann

Distribution)


Distribution)

C lli i Th lli i t ( Z )

C i i i i ( )



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

][)(2

1 22

avg II vvvd N Z equivsdotsdotsdotsdot= π

M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88

For A-B collisions micro AB vAB

AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced



Collision DiameterCollision Diameter

2

B A AB

d d d

+=

Number per Unit VolumeNumber per Unit Volume

V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=

Collision Theory collision rate ( Z )

C lli i Th lli i ( Z )




21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8

sdotsdotsdotsdotsdot=sdotsdotsdotsdot=

AB

AB B A AB AB B A AB II

kT d N N vd N N Z

micro π

π π

C lli i Th R t C t t C l l ti



Collision Theory Rate Constant CalculationsCollision Theory Rate Constant Calculations

Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate

Kinetics ][][)( 2 B A N N k sdotsdot=Rate

Combining Collision Theory with Kinetics

T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π

m2 m s-1 per molecule

sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1

Collision Theor Rate Constant Calc lations

C lli i Th R C C l l i




A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π

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

2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro

Collision Theory Rate Constant Calculations





Consider 2 NOCl(g) 2NO(g) + Cl2(g) T = 600 K

Ea = 103 kJmol d NOCl = 283 pm (hard-sphere diameter)

Calculate the second order rate constant

httpwwwubccaindexhtml



mass ratio k2 M s-1

Reaction Ea kJ mol-1

atom-1atom-2 atom-1 + atom-2-CO2----gt atom-1-atom-2 + CO2

-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39

Simple Collision Theory Comparison of MuoniumHydrogenDeuterium Abstractions

00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1

Transition State Theory

T iti St t Th



Transition State TheoryTransition State Theory

Concept Activated Complex or Transition State( Dagger )

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 ( Dagger )

Potential Energy Surfaces

P t ti l E S f



Potential Energy SurfacesPotential Energy Surfaces

Consider D + H2 DH + H

D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H

Most favorable at θ = 0o 180o

Calculate energy of interaction at different r 1 r 2 and θ Get

3D Energy Map

Reaction coordinate = path of minimum energy leading from

reactants to products

Reactions in Solutions

R ti i S l ti



Reactions in SolutionsReactions in Solutions

Compared to gaseous reactions reactions insolutions require diffusion through the solventmolecules

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 gettrapped in ldquosolvent cagesrdquo and could have agreat number of collisions before escaping thesolvent cage

Diffusion Controlled Solutions

Diff i C t ll d S l ti



Diffusion Controlled SolutionsDiffusion Controlled Solutions

Smoluchowski (1917) D = diffusion coefficient

)(4 B A ABdiff D D Ld k +sdotsdotsdotsdot= π

a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity

Diffusion Controlled (Aqueous) Reactions




viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot

3 ηsdot R 83145 Jsdot mol1minus

sdot K 1minus

sdot=T2 95 27315+( ) K sdot=

k 25C8 R sdot T1sdot

3 η25Csdot

= k 95C8 R sdot T2sdot

3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=

Arrhenius Equation k A e

Eaminus

R Tsdotsdot kJ 10

3Jsdot=

EaR minus T1sdot T2sdot

T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus

sdot= Ea 17kJ mol1minus

sdot=

Therefore all aqueous solutions whose rate is determined by the

diffusion of species should have an Activation Energy of about 17kJmol

Diff-paper

Quantum Mechanical Tunneling

Q t M h i l T li



Quantum Mechanical TunnelingQuantum Mechanical Tunneling

)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot

sdot=Tunnelingof Prob

bull curvature in Arrhenius plots

bull abnormal A-factors

bull relative isotope effects

bull low Ea


Bolt mannrsquos Concepts of Reaction Rates



Boltzmann s Concepts of Reaction RatesBoltzmannrsquos Concepts of Reaction Rates

V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(

Theories of Reaction Rates

Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o

T r a n s i t i o T D T r e a


A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdotsdotsdotsdot=

21

23

)(2

810 π

η sdot sdotsdot= 38 T Rk diff



0 5000 1 104

15 104

2 104

25 104

0

24 104

48 104

72 104

96 104

12 105

P2 h1( )

Pa

P3 h1( )

Pa

P h1( )

Pa

h1

m



Num

ber

Height



PS 5






g

N

N kT E

j

i

j

i ∆minus

sdot=































T R KE sdotsdot=

2

3









bullEnergy


[ 1Du-x ]

x

T k um

due A N

dN x


2

1 2



1500 1000 500 0 500 1000 15000

5 104

0001

00015

0002

00025

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m 1minus ssdot

15001500minus u

m s1minussdot

x

T k um

due A

N

dN x


2

1 2

Mcad








x

T k um

u D

dueT k

m

N

dN x

x

sdotsdot

sdotsdotsdot

=

sdotsdotsdotminus

minus

2

1

1

2

2 π


xT k

x

E D

d eT k N

dN x

x

ε ε

π


=

sdotminusminus

minus

2

1

1 4

1



x

T k um

u D

dueT k

m

N

dN x

x


=

sdotsdotsdotminus

minus

2

1

1

2

2 π

x

T k

x E D

d eT k N

dN x

x

ε ε

π


=

sdotminusminus

minus

2

1

1 4

1





xau

u D

duea

N

dN x

x

sdotsdot=

minus

minus

2

1 π


z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π

y

au

u D due

a

N

dN y

y sdotsdot=

minus

minus

2

1 π z

au

u D due

a

N

dN z

z sdotsdot=

minus

minus

2

1 π





V = (43) π u3 with u2 = ux2 + uy

2 + uz2


2223

3

4 ua

D

euadu


π

z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π



0 500 1000 1500 2000 25000

0001

0002

0003

00035

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m1minus

ssdot

25000 u

m s

1minus

sdot


Low T

High T

Mcad






2223

3

4 ua

u D

euadu

N dN sdotminus

minussdotsdotsdot=

π

kT

D

ekT d

N dN ε

ε

ε π ε

minus

minus

sdotsdot

sdot=

21

23

3

12







int

sdot=

x

naverage

n

N dN x x )(




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




Ratio in terms of

urms uavg ump

173 160 141

m

kT

m

kT sdot3

m

kT sdot

π

8

m

kT sdot2









bullZ







avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π




PS 5






g

N

N kT E

j

i

j

i ∆minus

sdot=































T R KE sdotsdot=

2

3









bullEnergy


[ 1Du-x ]

x

T k um

due A N

dN x


2

1 2



1500 1000 500 0 500 1000 15000

5 104

0001

00015

0002

00025

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m 1minus ssdot

15001500minus u

m s1minussdot

x

T k um

due A

N

dN x


2

1 2

Mcad








x

T k um

u D

dueT k

m

N

dN x

x

sdotsdot

sdotsdotsdot

=

sdotsdotsdotminus

minus

2

1

1

2

2 π


xT k

x

E D

d eT k N

dN x

x

ε ε

π


=

sdotminusminus

minus

2

1

1 4

1



x

T k um

u D

dueT k

m

N

dN x

x


=

sdotsdotsdotminus

minus

2

1

1

2

2 π

x

T k

x E D

d eT k N

dN x

x

ε ε

π


=

sdotminusminus

minus

2

1

1 4

1





xau

u D

duea

N

dN x

x

sdotsdot=

minus

minus

2

1 π


z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π

y

au

u D due

a

N

dN y

y sdotsdot=

minus

minus

2

1 π z

au

u D due

a

N

dN z

z sdotsdot=

minus

minus

2

1 π





V = (43) π u3 with u2 = ux2 + uy

2 + uz2


2223

3

4 ua

D

euadu


π

z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π



0 500 1000 1500 2000 25000

0001

0002

0003

00035

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m1minus

ssdot

25000 u

m s

1minus

sdot


Low T

High T

Mcad






2223

3

4 ua

u D

euadu

N dN sdotminus

minussdotsdotsdot=

π

kT

D

ekT d

N dN ε

ε

ε π ε

minus

minus

sdotsdot

sdot=

21

23

3

12







int

sdot=

x

naverage

n

N dN x x )(




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




Ratio in terms of

urms uavg ump

173 160 141

m

kT

m

kT sdot3

m

kT sdot

π

8

m

kT sdot2









bullZ







avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π







g

N

N kT E

j

i

j

i ∆minus

sdot=































T R KE sdotsdot=

2

3









bullEnergy


[ 1Du-x ]

x

T k um

due A N

dN x


2

1 2



1500 1000 500 0 500 1000 15000

5 104

0001

00015

0002

00025

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m 1minus ssdot

15001500minus u

m s1minussdot

x

T k um

due A

N

dN x


2

1 2

Mcad








x

T k um

u D

dueT k

m

N

dN x

x

sdotsdot

sdotsdotsdot

=

sdotsdotsdotminus

minus

2

1

1

2

2 π


xT k

x

E D

d eT k N

dN x

x

ε ε

π


=

sdotminusminus

minus

2

1

1 4

1



x

T k um

u D

dueT k

m

N

dN x

x


=

sdotsdotsdotminus

minus

2

1

1

2

2 π

x

T k

x E D

d eT k N

dN x

x

ε ε

π


=

sdotminusminus

minus

2

1

1 4

1





xau

u D

duea

N

dN x

x

sdotsdot=

minus

minus

2

1 π


z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π

y

au

u D due

a

N

dN y

y sdotsdot=

minus

minus

2

1 π z

au

u D due

a

N

dN z

z sdotsdot=

minus

minus

2

1 π





V = (43) π u3 with u2 = ux2 + uy

2 + uz2


2223

3

4 ua

D

euadu


π

z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π



0 500 1000 1500 2000 25000

0001

0002

0003

00035

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m1minus

ssdot

25000 u

m s

1minus

sdot


Low T

High T

Mcad






2223

3

4 ua

u D

euadu

N dN sdotminus

minussdotsdotsdot=

π

kT

D

ekT d

N dN ε

ε

ε π ε

minus

minus

sdotsdot

sdot=

21

23

3

12







int

sdot=

x

naverage

n

N dN x x )(




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




Ratio in terms of

urms uavg ump

173 160 141

m

kT

m

kT sdot3

m

kT sdot

π

8

m

kT sdot2









bullZ







avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π



























T R KE sdotsdot=

2

3









bullEnergy


[ 1Du-x ]

x

T k um

due A N

dN x


2

1 2



1500 1000 500 0 500 1000 15000

5 104

0001

00015

0002

00025

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m 1minus ssdot

15001500minus u

m s1minussdot

x

T k um

due A

N

dN x


2

1 2

Mcad








x

T k um

u D

dueT k

m

N

dN x

x

sdotsdot

sdotsdotsdot

=

sdotsdotsdotminus

minus

2

1

1

2

2 π


xT k

x

E D

d eT k N

dN x

x

ε ε

π


=

sdotminusminus

minus

2

1

1 4

1



x

T k um

u D

dueT k

m

N

dN x

x


=

sdotsdotsdotminus

minus

2

1

1

2

2 π

x

T k

x E D

d eT k N

dN x

x

ε ε

π


=

sdotminusminus

minus

2

1

1 4

1





xau

u D

duea

N

dN x

x

sdotsdot=

minus

minus

2

1 π


z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π

y

au

u D due

a

N

dN y

y sdotsdot=

minus

minus

2

1 π z

au

u D due

a

N

dN z

z sdotsdot=

minus

minus

2

1 π





V = (43) π u3 with u2 = ux2 + uy

2 + uz2


2223

3

4 ua

D

euadu


π

z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π



0 500 1000 1500 2000 25000

0001

0002

0003

00035

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m1minus

ssdot

25000 u

m s

1minus

sdot


Low T

High T

Mcad






2223

3

4 ua

u D

euadu

N dN sdotminus

minussdotsdotsdot=

π

kT

D

ekT d

N dN ε

ε

ε π ε

minus

minus

sdotsdot

sdot=

21

23

3

12







int

sdot=

x

naverage

n

N dN x x )(




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




Ratio in terms of

urms uavg ump

173 160 141

m

kT

m

kT sdot3

m

kT sdot

π

8

m

kT sdot2









bullZ







avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π










bullEnergy


[ 1Du-x ]

x

T k um

due A N

dN x


2

1 2



1500 1000 500 0 500 1000 15000

5 104

0001

00015

0002

00025

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m 1minus ssdot

15001500minus u

m s1minussdot

x

T k um

due A

N

dN x


2

1 2

Mcad








x

T k um

u D

dueT k

m

N

dN x

x

sdotsdot

sdotsdotsdot

=

sdotsdotsdotminus

minus

2

1

1

2

2 π


xT k

x

E D

d eT k N

dN x

x

ε ε

π


=

sdotminusminus

minus

2

1

1 4

1



x

T k um

u D

dueT k

m

N

dN x

x


=

sdotsdotsdotminus

minus

2

1

1

2

2 π

x

T k

x E D

d eT k N

dN x

x

ε ε

π


=

sdotminusminus

minus

2

1

1 4

1





xau

u D

duea

N

dN x

x

sdotsdot=

minus

minus

2

1 π


z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π

y

au

u D due

a

N

dN y

y sdotsdot=

minus

minus

2

1 π z

au

u D due

a

N

dN z

z sdotsdot=

minus

minus

2

1 π





V = (43) π u3 with u2 = ux2 + uy

2 + uz2


2223

3

4 ua

D

euadu


π

z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π



0 500 1000 1500 2000 25000

0001

0002

0003

00035

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m1minus

ssdot

25000 u

m s

1minus

sdot


Low T

High T

Mcad






2223

3

4 ua

u D

euadu

N dN sdotminus

minussdotsdotsdot=

π

kT

D

ekT d

N dN ε

ε

ε π ε

minus

minus

sdotsdot

sdot=

21

23

3

12







int

sdot=

x

naverage

n

N dN x x )(




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




Ratio in terms of

urms uavg ump

173 160 141

m

kT

m

kT sdot3

m

kT sdot

π

8

m

kT sdot2









bullZ







avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π




1500 1000 500 0 500 1000 15000

5 104

0001

00015

0002

00025

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m 1minus ssdot

15001500minus u

m s1minussdot

x

T k um

due A

N

dN x


2

1 2

Mcad








x

T k um

u D

dueT k

m

N

dN x

x

sdotsdot

sdotsdotsdot

=

sdotsdotsdotminus

minus

2

1

1

2

2 π


xT k

x

E D

d eT k N

dN x

x

ε ε

π


=

sdotminusminus

minus

2

1

1 4

1



x

T k um

u D

dueT k

m

N

dN x

x


=

sdotsdotsdotminus

minus

2

1

1

2

2 π

x

T k

x E D

d eT k N

dN x

x

ε ε

π


=

sdotminusminus

minus

2

1

1 4

1





xau

u D

duea

N

dN x

x

sdotsdot=

minus

minus

2

1 π


z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π

y

au

u D due

a

N

dN y

y sdotsdot=

minus

minus

2

1 π z

au

u D due

a

N

dN z

z sdotsdot=

minus

minus

2

1 π





V = (43) π u3 with u2 = ux2 + uy

2 + uz2


2223

3

4 ua

D

euadu


π

z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π



0 500 1000 1500 2000 25000

0001

0002

0003

00035

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m1minus

ssdot

25000 u

m s

1minus

sdot


Low T

High T

Mcad






2223

3

4 ua

u D

euadu

N dN sdotminus

minussdotsdotsdot=

π

kT

D

ekT d

N dN ε

ε

ε π ε

minus

minus

sdotsdot

sdot=

21

23

3

12







int

sdot=

x

naverage

n

N dN x x )(




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




Ratio in terms of

urms uavg ump

173 160 141

m

kT

m

kT sdot3

m

kT sdot

π

8

m

kT sdot2









bullZ







avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π









x

T k um

u D

dueT k

m

N

dN x

x

sdotsdot

sdotsdotsdot

=

sdotsdotsdotminus

minus

2

1

1

2

2 π


xT k

x

E D

d eT k N

dN x

x

ε ε

π


=

sdotminusminus

minus

2

1

1 4

1



x

T k um

u D

dueT k

m

N

dN x

x


=

sdotsdotsdotminus

minus

2

1

1

2

2 π

x

T k

x E D

d eT k N

dN x

x

ε ε

π


=

sdotminusminus

minus

2

1

1 4

1





xau

u D

duea

N

dN x

x

sdotsdot=

minus

minus

2

1 π


z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π

y

au

u D due

a

N

dN y

y sdotsdot=

minus

minus

2

1 π z

au

u D due

a

N

dN z

z sdotsdot=

minus

minus

2

1 π





V = (43) π u3 with u2 = ux2 + uy

2 + uz2


2223

3

4 ua

D

euadu


π

z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π



0 500 1000 1500 2000 25000

0001

0002

0003

00035

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m1minus

ssdot

25000 u

m s

1minus

sdot


Low T

High T

Mcad






2223

3

4 ua

u D

euadu

N dN sdotminus

minussdotsdotsdot=

π

kT

D

ekT d

N dN ε

ε

ε π ε

minus

minus

sdotsdot

sdot=

21

23

3

12







int

sdot=

x

naverage

n

N dN x x )(




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




Ratio in terms of

urms uavg ump

173 160 141

m

kT

m

kT sdot3

m

kT sdot

π

8

m

kT sdot2









bullZ







avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π




x

T k um

u D

dueT k

m

N

dN x

x


=

sdotsdotsdotminus

minus

2

1

1

2

2 π

x

T k

x E D

d eT k N

dN x

x

ε ε

π


=

sdotminusminus

minus

2

1

1 4

1





xau

u D

duea

N

dN x

x

sdotsdot=

minus

minus

2

1 π


z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π

y

au

u D due

a

N

dN y

y sdotsdot=

minus

minus

2

1 π z

au

u D due

a

N

dN z

z sdotsdot=

minus

minus

2

1 π





V = (43) π u3 with u2 = ux2 + uy

2 + uz2


2223

3

4 ua

D

euadu


π

z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π



0 500 1000 1500 2000 25000

0001

0002

0003

00035

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m1minus

ssdot

25000 u

m s

1minus

sdot


Low T

High T

Mcad






2223

3

4 ua

u D

euadu

N dN sdotminus

minussdotsdotsdot=

π

kT

D

ekT d

N dN ε

ε

ε π ε

minus

minus

sdotsdot

sdot=

21

23

3

12







int

sdot=

x

naverage

n

N dN x x )(




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




Ratio in terms of

urms uavg ump

173 160 141

m

kT

m

kT sdot3

m

kT sdot

π

8

m

kT sdot2









bullZ







avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π






xau

u D

duea

N

dN x

x

sdotsdot=

minus

minus

2

1 π


z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π

y

au

u D due

a

N

dN y

y sdotsdot=

minus

minus

2

1 π z

au

u D due

a

N

dN z

z sdotsdot=

minus

minus

2

1 π





V = (43) π u3 with u2 = ux2 + uy

2 + uz2


2223

3

4 ua

D

euadu


π

z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π



0 500 1000 1500 2000 25000

0001

0002

0003

00035

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m1minus

ssdot

25000 u

m s

1minus

sdot


Low T

High T

Mcad






2223

3

4 ua

u D

euadu

N dN sdotminus

minussdotsdotsdot=

π

kT

D

ekT d

N dN ε

ε

ε π ε

minus

minus

sdotsdot

sdot=

21

23

3

12







int

sdot=

x

naverage

n

N dN x x )(




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




Ratio in terms of

urms uavg ump

173 160 141

m

kT

m

kT sdot3

m

kT sdot

π

8

m

kT sdot2









bullZ







avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π






V = (43) π u3 with u2 = ux2 + uy

2 + uz2


2223

3

4 ua

D

euadu


π

z y x

uuua

D

dududuea

N


=

++minus ][23

3

222

π



0 500 1000 1500 2000 25000

0001

0002

0003

00035

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m1minus

ssdot

25000 u

m s

1minus

sdot


Low T

High T

Mcad






2223

3

4 ua

u D

euadu

N dN sdotminus

minussdotsdotsdot=

π

kT

D

ekT d

N dN ε

ε

ε π ε

minus

minus

sdotsdot

sdot=

21

23

3

12







int

sdot=

x

naverage

n

N dN x x )(




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




Ratio in terms of

urms uavg ump

173 160 141

m

kT

m

kT sdot3

m

kT sdot

π

8

m

kT sdot2









bullZ







avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π




0 500 1000 1500 2000 25000

0001

0002

0003

00035

0

F1 u( )

m1minus

ssdot

F2 u( )

m1minus

ssdot

F3 u( )

m1minus

ssdot

25000 u

m s

1minus

sdot


Low T

High T

Mcad






2223

3

4 ua

u D

euadu

N dN sdotminus

minussdotsdotsdot=

π

kT

D

ekT d

N dN ε

ε

ε π ε

minus

minus

sdotsdot

sdot=

21

23

3

12







int

sdot=

x

naverage

n

N dN x x )(




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




Ratio in terms of

urms uavg ump

173 160 141

m

kT

m

kT sdot3

m

kT sdot

π

8

m

kT sdot2









bullZ







avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π







2223

3

4 ua

u D

euadu

N dN sdotminus

minussdotsdotsdot=

π

kT

D

ekT d

N dN ε

ε

ε π ε

minus

minus

sdotsdot

sdot=

21

23

3

12







int

sdot=

x

naverage

n

N dN x x )(




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




Ratio in terms of

urms uavg ump

173 160 141

m

kT

m

kT sdot3

m

kT sdot

π

8

m

kT sdot2









bullZ







avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π








int

sdot=

x

naverage

n

N dN x x )(




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




Ratio in terms of

urms uavg ump

173 160 141

m

kT

m

kT sdot3

m

kT sdot

π

8

m

kT sdot2









bullZ







avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




Ratio in terms of

urms uavg ump

173 160 141

m

kT

m

kT sdot3

m

kT sdot

π

8

m

kT sdot2









bullZ







avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





2223

3

4 ua

D

euadu


π




2223

3

4 ua

D

euadu


π




Ratio in terms of

urms uavg ump

173 160 141

m

kT

m

kT sdot3

m

kT sdot

π

8

m

kT sdot2









bullZ







avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





2223

3

4 ua

D

euadu


π




Ratio in terms of

urms uavg ump

173 160 141

m

kT

m

kT sdot3

m

kT sdot

π

8

m

kT sdot2









bullZ







avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





Ratio in terms of

urms uavg ump

173 160 141

m

kT

m

kT sdot3

m

kT sdot

π

8

m

kT sdot2









bullZ







avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π










bullZ







avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π






avg relative

avg I

uuwhere

ud V

sdot=

sdotsdotsdot=

2

2 2π


2

)()(

2 N ud Z

N V Z

avg I

I I

sdotsdotsdotsdot=

sdot=

π M

T R

m

T k uavg sdot

sdotsdot=

sdotsdotsdot

=π π

88




I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





I

avg

Z

u==



2

12





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





sdotsdot=

2

1 N Z Z I II


22 )(2






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π






1

2


λ

222 d

muavg

sdotsdot

sdot=

π η

123

100226

minus

sdot=

sdotsdot

=

mol L

T R

L P N

where




25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





25oC and 1 atm

Species


d 10-10 m d Aringλ 10-8

m

ZI 109 s-1 ZII 1034 m-3 s-1

H2 273 273 124 143 176

He 218 218 191 66 81 N2 374 374 656 72 89

O2 357 357 716 62 76Ar 362 362 699 57 70CO2 456 456 441 86 106

HI 556 556 296 75 106




V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

eua

du


π

int

sdot= x

naverage

n

N

dN x x )(




C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





C o l l i s i o

P o t e n t i

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i








T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π









T R Ea

e Ak sdotminus

sdot=

A i i





E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π






E meT ak




E d d A h i E i








C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π









C lli i Th






ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π







ZII (from simple


ZII (from simple







Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





Distribution)


Distribution)


C i i i i ( )




][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





][)(2

1 22


M

T R

m

T k v

sdotsdotsdot

=sdotsdotsdot

=π π

88


AB

AB

B A

B A

AB

T k v

mm

mm

micro π

micro

sdot

sdotsdot=equiv gt

+

sdot

=equiv gt

8VelocityRelative

MassReduced




2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





2

B A AB

d d d

+=


V

N

V

Ln N A A

A =sdot

=V

N

V

Ln N B B

B =sdot

=






21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





21

22)(

8)(

2

1

sdot

sdotsdotsdotsdot= A

A A AA II m

kT d N Z

π π

21

22)(

8


AB


kT d N N vd N N Z

micro π

π π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





Collision Theory

sdot= sdot

minusT R

Ea

II e Z )(Rate



T R

Ea

B A

II e N N

Z k sdot

minussdot

sdot=

)()(2


C i i C C i




A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





A-A Collisions

T R

Ea

A

Ae

m

T k d k sdot

minussdot

sdotsdotsdot

sdotsdotsdot=21

22

8

2

1

π π


sdotsdot

mol

molecule

m

dm

1

100226

1

1023

3

33

T R

Ea

A

A AAe

m

T k d

Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot

=21

23

)(2

8

2

10

π π

Units of k dm

3

mol

-1

s

-1

equiv M

-1

s

-1






A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





A-B Collisions

T R

Ea

AB

AB ABe

T k d Lk sdot

minussdot

sdotsdotsdot

sdotsdotsdotsdot=21

23)(2

810

micro π π


2

B A AB

d d d +=

B A

B A AB

mm

mm

+sdot= micro












mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π











mass ratio k2 M s-1



-

1 120E+08 H + HCO2----gt H2 + CO2

-

05 230E+07 H + DCO2----gt HD + CO2

-

011 340E+06 Mu + HCO2----gt MuH + CO2

-33

0056 990E+05 Mu + DCO2----gt MuD + CO2

-39


00E+00

20E+07

40E+07

60E+07

80E+07

10E+08

12E+08

14E+08

000 020 040 060 080 100

mass ratio

k M -

1 s

- 1


T iti St t Th






Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π







Saddle point

H H

DD

H H

DD

H H

DD

H2 + D2 2 HD

H2 + D2 2 HD



P t ti l E S f





D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π






D

HA HBr 2

r 1 θ r 1= dH-D

r 2 = dH-H



3D Energy Map




R ti i S l ti
















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π

















a

T k D

sdotsdotsdotsdot

=η π 6

η sdotsdotsdot

=3

8 T Rk diff

a = radius

η =viscosity





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





viscosity η

η25C

08904103minus

sdot kg m1minus

sdot s1minus

sdot= η95C

02975 103minus

sdot kg m1minus

sdot s1minus

sdot=

T1 25 27315+( ) K sdot=k

8R Tsdot


sdot K 1minus

sdot=T2 95 27315+( ) K sdot=


3 η25Csdot


3 η95Csdot

=

k 25C 742 109

times L mol1minus

sdot s1minus

sdot= k 95C 274 1010

times L mol1minus

sdot s1minus

sdot=


Eaminus

R Tsdotsdot kJ 10

3Jsdot=


T2 T1minusln

k 25C

k 95C

sdot=

Ea 17 104

times J mol1minus


sdot=



Diff-paper


Q t M h i l T li




)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





)(22

14

E Eam L

e E

Ea

E

Eaminussdotsdot

minussdot

minussdot





bull low Ea






V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





V e l o c


D e r i v E n e r g

M a x w e l l - B o

B a r o m e t r i

B o l t z m a n

2223

3

4 ua

D

euadu


π

int

sdot= x

naverage

n

N

dN x x )(


Th i f R ti R t




C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π





C o l l i s i o

P o t e n t i a

U n i m o l e

M u o n i

Q M T

I s o t o p


P a r t i t i o



A r r h e n i

T R

Ea

AB ABe

T k d Lk sdot

minussdot


21

23

)(2

810 π


Boltzmann Concept

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Transcript of Boltzmann Concept