ch. 24 molecular reaction dynamics 1. collision theory 2...

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 18-1

• 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 18


• This reaction profile shows how the potential

energy of the reactants A and B change in the

course of a simple bimolecular reaction.

• The horizontal axis (called the reaction

coordinate) represents the course of the

individual reaction event.

SN2 Reaction

‡

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

• Transition state theory (also called activated complex theory)

pictures a reaction between A and B as proceeding through the

formation of an activated complex (C‡), in a rapid pre-equilibrium:

A + B C‡

BA

o

C

oB

oA

o

Cp

pp

pp

pppp

ppK

‡‡

)/)(/(

/‡

Using pJ = RT[J], ]B][A[

]C[

]B[]A[

]C[ ‡‡‡ ‡

RT

p

RTRT

pRT

pp

ppK

oo

BA

o

Cp

]B][A[]C[ ‡‡po

Kp

RTTherefore,

• The activated complex falls apart by unimolecular decay into

products.

• The rate of product formation: ]C[ P C ‡‡‡ kv

‡k

Lecture 18-3


]B][A[]C[ ‡‡po

Kp

RT ]C[ P C ‡‡‡ kv

‡k

• It follows that ‡‡22 e wher]][[ po

Kkp

RTkBAkv

• Now, our task is to calculate the unimolecular rate constant

and the equilibrium constant, . ‡pK

‡k

Lecture 18-4


• The criterion for the activated complex turning into products is

that it pass through the transition state.

• If its vibration-like motion along the reaction coordinate occurs

with a frequency ‡, then the frequency with which the cluster of

atoms forming the complex approaches the transition state is

also ‡.

• However, not every oscillation along the reaction coordinate

takes the complex through the transition state.

• Therefore, we suppose that the rate of passage of the complex

through the transition state is proportional to the vibrational

frequency along the reaction coordinate:

‡‡ k

where is the transmission coefficient (dimensionless).

In many cases = 1


where rE0 is the difference in molar energies of the ground

states of the products and reactants. is the standard molar

partition function of species J.

Lecture 18-6

A + B C‡

RT

E

oB

oA

o

CART

E

AoBA

oA

Ao

Cp e

qq

qNe

NqNq

NqK

0‡

0‡

)/)(/(

/‡

• For the pre-equilibrium,

dD cC bB aA RT

E

b

A

o

mB

a

A

o

mA

d

A

o

mD

c

A

o

mCr

eNqNq

NqNqK

0

)/()/(

)/()/(

,,

,,

• According to the statistical thermodynamics, the equilibrium

constant for the reaction,

o

mJq ,

(B)(A))(C where 00

‡

00 EEEΔE

Note that since NA and qJ have the units of mol-1, K‡p is

dimensionless.


• Assuming that a vibration of the activated complex (C‡) tips it

through the transition state, the partition function for this vibration

is:

kT

h

e

q ‡

1

1

where ‡ is its frequency (the same frequency that determine k‡).

• This frequency is much lower than for an

ordinary molecular vibration and so the force

constant is very low.

‡‡‡

h

kT

kT

he

q

kT

h

11

1

1

1

• Provided h‡/kT << 1,


‡

‡

‡‡‡

popoK

h

kT

p

RTKk

p

RTk

2

Lecture 18-8

• Therefore, we can write ‡‡ ‡ CC

qh

kTq

where denotes the partition function for all the other modes

of the complex.

‡Cq

RT

E

oB

oA

o

CA

p eqq

qNK

0‡‡

‡‡22 e wher]][[ po

Kkp

RTkBAkv

• The constant is therefore, ‡pK ‡

‡

‡

pp Kh

kTK

• is a kind of equilibrium constant, but with one vibrational

mode of C‡ discarded.

• Now combining all the parts of the calculation into:

‡pK

‡

2 Kh

kTk called the Eyring Equation

‡‡

poK

p

RTK Set


• If we accept that is an equilibrium constant (despite one

mode of C‡ discarded), we can express it in terms of a Gibbs

energy of activation (‡G ) through the definition,

‡‡ ln pKRTG

• All the ‡X in this section are standard thermodynamic

quantities ( ), but we shall omit the standard state sign to

avoid overburdening the notation.

• Then, the rate constant becomes:

‡pK

oX‡

RT

G

opoe

p

RT

h

kTK

p

RT

h

kTK

h

kTk

‡

‡‡Δ

2


• By the definition, G = H – TS, the Gibbs energy of activation can

be divided into an entropy of activation and an enthalpy of

activation.

RT

G

oe

p

RT

h

kTk

‡Δ

2

STHG ‡‡‡ ΔΔΔ

R

S

RT

H

oRT

STH

oRT

G

oee

p

RT

h

kTe

p

RT

h

kTe

p

RT

h

kTk

‡‡‡‡‡ ΔΔΔΔΔ

2

• When ~ 1, we obtain:

oR

S

RT

H

p

RT

h

kTBeBek

ere wh

‡‡ ΔΔ

2


• Using the formal definition of activation energy, after some

calculation we find that:

oR

S

RT

H

p

RT

h

kTBeBek

ere wh

‡‡ ΔΔ

2

Td

kdR

dT

kdRTEa

/1

lnln2

• For reactions of P B A

In the gas phase,

In solution,

RTHEa 2Δ‡

RTHEa ‡Δ


• For a gas phase reaction,

oR

S

RT

H

p

RT

h

kTBeBek

ere wh

‡‡ ΔΔ

2

RTEH a 2Δ‡

RT

RT

RT

E

R

S

RT

RTE

R

S

eeBeeBekaa 2Δ2Δ

2

‡‡

RT

E

R

S a

eBeek

‡Δ

22

• Therefore, the pre-exponential factor of the Arrhenius equation

can be identified as:

R

S

BeeA

‡Δ

2


• The entropy of activation is negative because two reactant

species come together to form one species (the activated

complex).

• Hence, the negative value of ‡S reflects the occurrence of the

collisions.

• Furthermore, collisions with well-defined relative orientations

correspond to an even greater reduction of entropy.

• Indeed, we can identify that additional reduction in entropy

(‡Ssteric), as the origin of the steric factor of collision theory,

R

S

BeeA

‡Δ

2

R

Ssteric

eP

‡Δ


• The more complex the steric requirements of the encounter, the

more negative the value of ‡Ssteric. the smaller value of P

R

Ssteric

eP

‡Δ


RT

G

oe

p

RT

h

kTk

‡Δ

2

RT

GK

orln

KRTGor ln

RT

Gk

‡

2

Δln

• Using correlation analysis in which ln K is

plotted against ln k, in many cases the

correlation is linear.

• This signifies that as the reaction

becomes thermodynamically more

favorable, its rate constant increases.


• Next Reading:

8th Ed: p.884 ~ 892

9th Ed: p.849 ~ 856

Lecture 18-16

ch. 24 molecular reaction dynamics 1. collision theory 2...

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