Ch. 24 Molecular Reaction Dynamics 1. Collision Theory 2....

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 16-1

• Ch. 24 Molecular Reaction Dynamics

1. Collision Theory

2. Diffusion-Controlled Reaction

3. The Material Balance Equation

4. Transition State Theory: The Eyring Equation

5. Transition State Theory: Thermodynamic Aspects

6. Reactive Collisions: will be skipped.

7. Potential Energy Surfaces

8. Some Results from Experiments and Calculations

9~12. Others: will be skipped.

Lecture 16

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013

• Now we are at the heart of chemistry.

• Here we examine the details of what happens to molecules at

the climax of reactions.

• The simplest quantitative account of reaction rates is in terms of

collision theory, which can be used only for the reactions in the

gas phase.

• Consider the bimolecular elementary reaction,

Lecture 16-2

]B][A[ P B A 2kv

]B][A[

collision of rate

M

Tσ

M

Tσ

cσ

v

BA

BA

NN

NN

where is the collision cross-section.


• However, a collision will be successful only if the kinetic energy

exceeds the activation energy (Ea) of the reaction.

• Therefore the rate constant should also be proportional to a

Boltzmann factor of the exponential form.

Lecture 16-3

]B][A[2kv ]B][A[M

Tσv

M

Tσk 2Comparing,

RT

Ea

eM

Tσk

2


• Not every collision will lead to reaction, even if the energy

requirement is satisfied, because the reaction may need to collide

in a certain relative orientation.

• This steric requirement suggests that a further factor, P should

be introduced.

Lecture 16-4

RT

Ea

eM

TσPk

2

trequiremenenergy rateencouter t requiremen steric2 k


• Recall that the collision frequency (z) for like molecules (A) is:

Arelcσz N

where NA is the number density of A molecules and is their

relative mean speed. relc

m

kTccrel

8 and c2

Therefore,

kT

m

kT

m

kTcrel

8288 2

where is the reduced mass.

• For two identical molecules, = m/2.

• For two dissimilar molecules, BA mm

111


• This expression also applies to the mean relative speed of

dissimilar molecules.

• The total collision density is the collision frequency multiplied by the number density (NA) of A molecules:

kTcrel

8

2

2

1

2

1ArelAAA cσzZ NN

where the factor of ½ has been introduced to avoid double

counting of the collisions.

Arelcσz N


BArelAB cσZ NN

• For collisions of A and B molecules, the collision density is:

• Note that the factor of ½ has been discarded, because we are

considering an A molecule colliding with any of the B molecules

as a collision.

• Since the number density of a species J is NJ = NA[J],

]B][A[8 2

AAB NkT

σZ

where is the collision cross-section,

2 and 2 BA dd

ddσ

Note that d is the radius of .


• Similarly, the collision density for like molecules can be

expressed with [A],

2222 4

4

288

2

1

2

1A

A

A

A

AArelAAm

kTσ

m

kTσ

kTσcσZ NNNN

22 ][4

ANm

kTσZ A

A

AA

Since NJ = NA[J],

• For example, in N2 gas (d = 280 pm) at room T and p,

Z = 51034 /m3·s

Very large number !!!


• According to collision theory, the rate of change in the number

of A molecules per unit volume per unit time can be written as the

product of this collision density and the probability that the

collision occurs with sufficient energy:

fZdt

dAB

A N

where f is the fraction of collisions that occur with a kinetic

energy along the line of approach of the atoms in excess of

some threshold value Ea.

By the Boltzmann distribution, RT

Ea

ef

RT

E

A

ABa

eN

Z

dt

d

]A[Since NJ = NA[J],


• Therefore,

RT

E

A

ABa

eN

Z

dt

d

]A[

]B][A[8 2

AAB NkT

σZ

]B][A[8

]B][A[8

]A[

2

RT

E

ART

E

A

A aa

eNkT

σeN

NkT

σ

dt

d

• It follows that: RT

E

A

a

eNkT

σk

82

•Therefore, the pre-exponential factor is a measure of the rate at

which collisions occur in the gas. called frequency factor.

RT

E

relA

a

ecNk

2or

T

• This equation has the Arrhenius form, provided the exponential

T dependence dominates the dependence of the pre-

exponential factor.


• For gas-phase reactions, the simplest procedure for calculating

A (i.e., k) is to use for the values obtained for non-reactive

collisions or from tables of molecular radii.

RT

E

A

a

eNkT

σk

82 AN

kTσA

8

>

<< <<

Good agreement

Large discrepancy

Large discrepancy

Reaction occurs more

quickly than the collision??

2 and 2 BA dd

ddσ


• This large discrepancy suggests that the collision energy is not

the only criterion for reaction and that some other feature, such

as the relative orientation of the colliding species, is important.

• The disagreement can be accommodated by introducing a

steric factor (P).

P*

RT

E

A

a

eNkT

Pσk

82

• The reactive cross-section (*) is

expressed as:

• Then the rate constant becomes:


• Estimate the steric factor for the reaction H2 + C2H4 C2H6 at

628 K given that the pre-exponential factor is 1.24106 dm3/mol·s.

RT

E

A

a

eNkT

Pσk

82 s/moldm 1024.1

8 36

exp ANkT

PσA

kg 1012.3

K 628

J/K 10380.1

/mol10022.6

27

23

23

422

422

mm

mm

T

k

N

HCH

HCH

A

Lecture 16-13

• As a result, the steric factor P is the ratio of Aexp/Acalc.


2

22

422

HCH dd

• However, because the molecules are not spherical, more

approximate collision cross-section is the average of them.

222

nm 46.02

nm 64.0nm 27.0

2

422

HCH

(From Table 21.1)

• It follows that P = 1.7 10-6.

• As a general guide, the more complex the molecules, the

smaller the value of P.

Lecture 16-14


>

• In this case, P = 4.8, indicating an anomalously large reaction

cross-section. * = 4.8

• It has been proposed that the reaction proceeds by a harpoon

mechanism. • For the reaction, the K atom approaches

the Br2 molecules, and when the two are

close enough an electron (the harpoon) flips

across to the Br2 molecule.

• Now they are ions, and so there is a

Coulombic attraction (the line on the

harpoon). Lecture 16-15

http://en.wikipedia.org/wiki/Image:Whaling_harpoon.jpg


• Under the Coulombic attraction, the ions move together and the

reaction take place.

• Consequently, the harpoon extends the cross-section for the

reactive encounter.

• The value of P can be estimated by calculating the distance at

which it is energetically favorable for the electron to leap from K

to Br2.

Lecture 16-16


• Estimate the value of P for the harpoon mechanism by

calculating the distance at which it is energetically favorable for

the electron to leap from K to Br2.

• There are three contributions to energy of the process K + Br2

K+ and Br2-.

1. Ionization (I) of K

2. Electron affinity (Eea) of Br2

3. Coulombic interaction. When the separation is R, R

e

0

2

4

• The electron flips across when the sum of three contributions

changes from positive to negative (i.e., when the sum is zero).

Lecture 16-17


• The net change in energy when the transfer occurs at a

separation R is:

R

eEIE ea

0

2

4

• Since the ionization energy I is larger than Eea, only when R has

decreased to less than some critical value of R*, E becomes

negative.

*4 0

2

R

eEI ea

• At this separation R*, the reactive cross-section is given by:

2** R


• Thus, the steric factor is:

2** R

2

0

2

2

2

)(4

**

eaEId

e

d

RP

2BrK RRd

*4 0

2

R

eEI ea

where

• With I = 420 kJ/mol, Eea ~ 250 kJ/mol, and d = 400 pm, P = 4.2

Good agreement with the experimental value of 4.8!


• Next Reading:

8th Ed: p.876 ~ 880

9th Ed: p.839 ~ 843

Lecture 16-20

Ch. 24 Molecular Reaction Dynamics 1. Collision Theory 2....

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