Steric hindrance of diffusion controlled reactions

Journal of Molecular Liquids, 29 (1984) 243-261 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

STEKIC

M.A.

HINDRANCE

OF DIFFUSION

LOPEZ-QUINTELA*,

Fakulttit

fur

J.

Chemie,

CONTROLLED

SAMIOS

and

Universittit,

W.

243

REACTIONS

KNOCHE

II4800

Eielefeld

ABSTRACT

For

diffusion

proposed

controlled

which

includes

encounter

complex,

encounter

complex.

equations

of

this

reactions

the

and

the

For

some

scheme

in

diffusion rotation

of

types

are

solution

process,

of

solved

the

the

is

lifetime

of

an

in

the

reacting

reaction and

a theory

species

the

analytical

discussed.

INTRODUCTION

There lar

are

two

and

sion

processes

are

are

expressed

as

general

is

model

assume

cases cal

models to

have

the

The siders

view.

have

been

difficulty

a

large

approach,

In

its

motion the

term used,

0167-7322/84/$03.00

most

the

encounter

and

different

only

by

relative

reaction reaction

that

the

The

most

(ref.l), More

simple

quasichemi-

approach

they

encounter-pairs,

numerical

‘quasiphysical’,

from

a more this

Elsevier Science Publishers B.V.

con-

physical

model

point

applies

the

function for

(ref.6-11). the

and

methods.

call

accounted

medium

assumes

and

pair.

The

distribution is

rates

steady-state

pairs.

a realistic

for

of

diffu-

their

Stockmayer

that

formulation

the

approach 0 1984

Sole

of

model

constant the

encounter

by

we will

refined

chemical in

of

reactions

for

rate Then

bimolecu-

‘quasichemi-

and

recently.

soluble which

the

reactions

kind

of

quasichemical

analysed

number

is

theory

solution,

a diffusion

some

kinds

of and

a sink

frequently

proposed

diffusion-controlled

equation bodies

been

also

the

the

involved.

to

problem

other

of

reactants

applied

the

In

chemical

different

consider

therefore

product

the

has

four

(ref.Z-5)

need

the

as

to

in

model. treated

of

approximation

who

reactions

‘quasiphysical’

concentrations

of

approaches

diffusion-controlled

cal’

of

different

reaction

by

the

of

two

presence

Another, produces

more

244

microscopic is

gradients

applied

reaction

times the

background,

(ref.8)

rent

to

models

meters pect of

present

for

bimolecular

the

the

the

are

can

be

theoretical to

apply

it

Stockmayer

only

to

(ref.9) models

approximate

(ref.l,14).

Moreover,

these

theories

which

are

when

diffe-

involved

framework

of

the

kinetics

is

applied,

to

those

in

a

quasichemical

of

analytically the of

theory

the

para-

physical

the

presented,

a

encounter

and

underlying

is

and

kind

approximation

involved

kinetic

general

solved

A first

parmeters

some

law for

quasiphysical

cases

a more

related

phenomenon.

to

and

obtained

adjust

reactions

equations

various

applied

the

incorporates

involved of

Sole

incorporated

article,

which

Thus

be

a solid

attempts

developed

special to

Fick’s

(ref.13).

one

they

which

aproximation

(ref.4).

the

proposed

pair.

to

reaction

In

but

to

good

has

Thus

difficult

have

when

a few

for

very

parameters

particular

is

be

a

ps

have

symmetry,

solutions

seems

0.1

occur

situation.

non-spherical

be

approach

problems

Schurr

medium

to

about

and

analytical it

than

experimental

Schmitz

with

reaction

appears

quasiphysical

many

concrete

the

This

longer

Although

and

in

(ref.12).

and

as-

calculation the

theory

is

examples.

THEORY

Formulation

Let

of

us

the

problem

suppose

and

a general

definitions

bimolecular

reaction

A + B -Products

between

two

(1)

non-charged

reactants

a solvent

S.

Furthermore

molecules

is

small

compared

a dilute

solution

a system

which

is

B do

not

react.

with

each

we

discuss

start

with

molecules many

A and

collisions

chemical assume of

transformation that

A and

imagine

we

at

B in processes

time the

t=O solvent. to

take

A and

suppose

that

to

of

that

of

A and

other

and

into

other

there

is

Then, place

the the

to

S.

For

the

real

molecules also

substances

on

i.e.

simplicity one

we

except

take

place.

but

scale

no

We also

distribution

1.

that

B perform

solvent,

figure

B

molecules,

uniform

in

in

A and

the

a microscopic shown

‘immersed‘ of

A and

with

a spatially

as

are

number solvent

B in

identical I.e.

B which

we

can

a

245

b)

a) 1.

Fig.

Model

uniform

Figure

1 shows

motion

of

llisions in

(or

the

0.1

ps

cage

in

to

solvent

or

free

(ref.18). in

a cage.

This

velocity

(ref.17).

sufficient

can

cule,

i.e. its B. At

defined

there molecule

random this as

the

walk

moment the

19). is

the

it

figure

function can

has

no memory

a molecule A and

‘structure’

of

formed

an by

is

aproximately

molecule if

the

attractions (ii)

an empty

molecule

can

becomes

i.e.

the

motion

original the

encounter El and

cage pair,

solvent

sothe

by

times of

the

the

CC14 calculated for

jump

trapped

represents

that

a

escapes

(i)

the

which

its

A,

A in

numerous

2,

A may enter

B form

surrounding

through

in

them)

the

uncorrelated,

seen

no correlation

co-

call

diffuses

for be

to

and

the again

are

as

in

It

cage,

which

jumps

repeated

possible

molecule

walk’

seen

(ref.

in

the

the

solvent

liquids,

molecule

overcome

to

their

A makes

is

into

autocorrelation

2 ps the

of

be nicely

dynamics

about

it

jump,

a ‘random

liquids

therefore

available,

this

by

Spatially same cage.

of

of

by the

escape

holding

is

in

spent

energy

in

blocked

a)

the

prefer

Eventually,

This

jump.

directions

vi(t)

than

be

cage

molecules

The

molecular

cule

and

A performs

lution.

During

longer

volume

is

S.

theory largely

time

much

Following

molecule

exit time

A acquires

the

site

that

B ‘traped’

A consists

the

a diffusive

ET in

A and B in

statistical

collision

The

b)

A and

its

However, be

S.

as many authors

since

cage,

may

collisions molecule

like

A and

of

classical

a molecule

(ref.15).

and makes

molecules

the

quasi-vibrations

(ref.16).

single

a solution

A and B in

the

to

solvent

molecules

for

of

According

cages. the

assumed

distribution

longer

the

mole-

situation. of

a mole-

which

may

molecules

246 where the

the

motions

same

cage

of

(see

A and

B are

correlated,

i.e.

A and

B are

in

fig.lb).

100

c(r)

a50

0.00

I

-OK

Fig.2.

In

of

be

course,

place

to

only

function

generalize

molecules

may

the

when

our have

whole

model,

active

we

and

occurs

r/ps

for

CC14.

suppose,

centers

molecules)

collision

I

1.00

autocorrelation

order

the

I

a50

Velocity

that

L

I

000

OA,

that

between

furthermore, OB (which

the

these

can,

reaction

can

specific

take

reactive

centers. The will and

at

encounter

denote 48 For

the

calculations

initial

: the

average

time which B : the

enter

per

timi

enter

per

which

place
be

denoted

pairs,

>:

by

where

which

average enter

suppose then

number a cage

average

between the

we

positions;

per

takes

will

encounter

(A,B),

whereas

a collision

(A.B)

between

@A

occurs.

their

pairs

those

of of

number a cage their

molecules

a cage

of of

molecules

of

molecules

A and

respective number

the

A are

fixed

define B per

volume

and

A. of

of

that

we can

of

whose

first

reactive molecules

A and

for

B per

volume

collision centers,

of which

B per

OA, volume

a collision

and with OB. and takes

A

241 place

between :

per

titnz

does

@A and

the

which

not

involve

According

a cage

eA

they

place

these

of of

leave

the

molecules

of

A and

@B but

and

take to

before number

enter

@B does

OA and

4B

average

for

before

whose

which they

definitions

first

volume

and

collision

a collision

leave

we

cage. B per

the

with

A

between

cage.

have

= +
(2)

i

Furthermore, B


i

is

> /

the



molecule

B

/

the

the

-

) 1

probability

first

is

An

B to

which

neither

the

of not

time

eA and we

equation

can

can

fixed

at

A nor

eB.

B and

the

Also,

a cage

place

spends

of

between

in

this

A so

that

eA and

cage

so

that

OB.

write:

-

also

their

B are

B enters

take it

+ fB(,)[fB(@)-’

be

between

(4)

A does

(2)-(4)

analogous

molecules

eA and

a molecule

between

fB(@){l

collision

fB(T)

=

during

relations

=

first

between

with

rotates a collision

With

the

that

collision it

there

in

that A occurs

(

eB but



(3)

probability

fixed

is

:: f’(e)

131

be

obtained

initial

fixed

(5)

for

A assuming

positions.

we can

For

generalize

the

this

the case

equation

to:



f

=

=

f(o){l

f(e)

+ f(r)Cl

per

per

is

time

(A,B)

+

the

which

cages

(or

f(T)[f(O)-’

-

=

-111

f

(6a)

f(e)1

average

(6b)

number

enter

the

encounter

same

of

reacting

cage,

pairs)

molecules

i.e.

which

the are

per

average formed

volume

number per

volume

and

of and

time, CN~>

volume

is

and

the per

average time

which

number are

of

encounter

formed

and

in

pairs

(A.B)

per

which

A and

B perform

248 at least

collision

one

B before 0A and CD

between

they

leave

the

cage, f(0) El in

and

is

the probability

encounter

the

takes

aB, f(l) represents

that place

the

first

between

the probability

in the encounter taking place OS, given that the first

the reactive

of

between collision

and

collision

a collision

between centers between

A and

GA A

and B

the reactive centers 0 was not between aA atd

@f3* Relation

with

the kinetic

Now we will

relate

constants the above

the

‘classical’ phenomenological The formation of (A,B) reaction. ted

equations

and definitions

description encounter

of a bimolecular pairs can be represen-

with

by kcl

Scheme

with of

A + B =

1

an

equilibrium

(A.6)

encounter

k-O KD = k,,/k_D,

constant pairs

(A,B)

and

the

formation

as kl

Scheme

A + B m

2

(A.B) k-l

with

Kl If

= k,_/k_l. we

translate

equation

(6a)

into

‘kinetic’

terms,

we can

write



dCA,BI formation/dt

=



= dcA.81

(7a)

= kOCAlCB1

formation/dt

= kl[A][6]

=

(7b)

f

then

kl

= f

Assuming

CA.BI/CA,BI k-l

(7c)

kg

= k-$1

that

= f(s)

no

relative

orientation

is

preferential,

i.e.

we obtain

+ f(+f(@)-l

- 111

(8)

249

Eq.

(8)

Schemes

gives

1,

Reaction

2 and

far

we

equilibrium bility

Continuing be

have

probability

a system

encounter

our

take

place

by

may

3

leading

the

analyse

be

supposed

From

dCPl/dt

this

mass

always

we

the

possi-

centers products this

kinetic

in

eA

and

P. new

possibility

scheme:

kr (A.B)B

when

balance

the

B are

introduce

description,

k-l scheme

new

A and

reactive to

following

A + B =

To

in

f(r)..

which

Now we the

kl Scheme

constants

f(O),

in

pairs.

between

phenomenological

represented

rate

molecules

a collision can

the

parameters

considered the

at

QB a reaction

between

non-charged

with that

relation

the

between

So

can

the

we

A and

assume

that

B are

not

[A,Bl

<<[Al,[Bl(this

charged).

obtain

(9)

= kefCA1CB3

where

k

= kl/(l

ef

If

(10)

+ k-l/k,)

we

introduce

equations

(7)

and

(8)

in

equation

(10)

we

obtain

=

l’kef

l/fkB

This

but

inferred

(but

their

k

ef

now

a)Fast

reactions

this

neglected

= kl

should

not

form be

formally

Our

to

noted deduced)

derivation

may

that

of

that their be

equation

Schurr

and

(2)

equation

considered

in

Schmitz from as

a

a proof

equation.

We can

In be

a similar

it

model.

(11)

0)

has

(14),

quasiphysical of

l/KBkrf(

equation

reference have

+

analyse

case and

= fkB

two

(kr the

extreme

cases:

>>k_l)

second

term

on

the

rhs

of

equation

(11)

can

we obtain:

(12)

This

expresion

constant of

of

an

addit’ional

factor’ the

k

This an

(i.e.

of

(11) the

these

reaction

has

the

of

(f(r)

the

the

‘steric

reactants

-see

and

below-).

show

parameters

and

except

rate

constant

can

of

to

and the

be

theoretically

are

an

reaction

the the

” point

reaction

restrictive

of

there

is

in in

products

There-

a possible earlier. do

not

in-

autocatalysed.

we

assume

nearly

no

a

parameters

view.

of

not

because

therefore

the

defined

the

is

constant) observe

because,

introduction

when

kef

a case

and

parameters

only

the

for

advantage

‘physical

on

to

defined

is

in

rate

interesting

the

that

justified i.e.

not

B,

k,

forget

involved

obtained is

This

influence

This

dilute

solu-

interference

products.

various

can

the

parameters

It

kr.

a purely

is

E and are

the

place.

from

reaction,

A,

that

assumption

is

to

phenomenologically

little

A and

similar

reaction.

we

the

approximation

There

pair

i)

gives:

is

kef

k_B

approximation

of

account

introduction

k-1)

(11)

takes

calculated we make

between

<<

parameters

approximation,

the

(6)

k,,,

first

tions

into

rate

the

parameters

reaction

fluence

so-called with

(13)

the

no

This

the

diffusion

(kr

macroscopic

all

be

takes

encounter

for

and

f(O),

fore,

that

rotational

equation

which

may

to reaction,

the

controlled

f(r),

that

the

expression

Calculation

are

similar

= f(4)KBkr

activation

Eq.

(f)

of

case

is

controlled

reactions

this

= KIk,

ef

, ii)

lifetime

b)Slow In

kef

factor

f(@)

iii)

for

a diffusion

levels

of

We have

estimated. more

approximation

rigurous

and

at

chosen those

which

the

a compromise with

more

above

between

practical

applications.

Estimation kB cess,

of

represents namely,

kB the

the

rate

For

the

present

purposes

and

the

solvent

to

rate

of

the

constant

encounter

reaction

be

of we

of

the

assume

our This

a continuum. if

i)

kr

>>

a purely

reactant

k,,

and

diffusive

molecules

molecules rate ii)

to

constant the

entire

pro(A

be

and

6).

spherical would surface

be

the

251 was

reactive

(see

not

wish

the

parameters,

to

and

B enter

are

in

the

eq.

introduce

and

the

and

so

same

cage.

same

cage

they

the

concentration

ach

from

is

B (d

=

but

rate

become to

and

we mentioned

at

also

for

the

estimation

the

molecules

supposed

that

entity,

an

well

known

‘black

at

the

distance

we obtain

before,

which

a new

the

A vanishes

2)

as

concept

the

We have

according of

6),

reaction

kD

the

Therefore,

pair.

12

when

(A,B)

of A A and

B

model

closest

Smoluchowsky’s

do

encounter

trap’

of

we

appro-

equation

(ref.12,20,21):

kD= the

4 n NDAB

relative

diffusion the

DB (ref.22), It, and

r

A

for

usually for It

on

of

r i

is

n and

the

+ rB,

agrees

with

second

non-charged

coefficients

we k ‘10

order

influence

can

be

10

rate

substances

the

DA +

coefficients. -5 2-l cm s

=lO

= rA

for

discuss

hydrodynamic

for

Van

in

of

the

estimated

by

conssolution. temperature the

Stokes

the

der

Waals

radius,

non-ideal radii

reaction

n is

the

behaviour r i

for

solvents

of

bulk the

different

can

be

viscosity,

and

molecules. molecules

obtained

from

Values in

literature

.23).

= 4RT/

(14)

and

(15)

II [ l/nArA

We may

Q

to

value

limit

between

D A=DB

by

(15)

common

temperature the



the

aproximated

II n ri

accounts

From

kD

assume This

is

diffusion

that

. as

single

(14)

equation:

various (ref

taken

the

DAB

suppose

r&-1

diffusion

= kBT/n

where n/6

nm and

interesting

The

kD.

of we

reactions

is

-Einstein

Di

value

= 0.25 B kD = 8x10’

+ D,]

coefficient

example,

=r

obtain M-ls-l tants

= 4 II N[DA

equation

assume and

+

that that

obtain

1

l/nBrB

n,

ri

and

n varies

proposed

= AoexpC(Ev/RT)

we

by

+

(rVo/Vf)



with

Macedo

1

(16)

and

the

do

not

vary

temperature

Litovitz

much

with

according

to

(ref.18):

(17a)

(17b)

252 Ev brium

is

packed a

the

height

positions, molecular

factor

lume.

between Then

ko =

specific

but

kB

Vf

l/2

and

1 to

(16)

and

C-E,/RT

bonded

liquids

second

term

-

is

of

(the in

with most

the

the

between V.

average

correct

the

is

equili-

the

close-

free

volume

overlap

of

and the

Y

is

free

vo-

(17),

is

the

increase

barrier volume,

~V~/v,l

equation

values

always

potential

molecular

volume,

Arrhenius’

the

the

the

combining

const-T-exp

whether

of

V is

(18)

complied

with

parameters T.

For

common

esponential

or

which Van

der

solvents factor

will into

Waals

in can

not

enter

and

on

(18),

hydrogen-

reaction

be

depend eq.

kinetics)

disregarded

the

and

we

obtain

kD=

const.T+exp(-Ev/RT)

where

Ev

lies

between

Estimation =

suppose the i

of

that

that

is

then

molecule

Di

mol-‘.

to

the

time

lifetime

takes of

of

place.

A and

self-diffusion

related a

kJ

the

reaction

molecules

The

after

20

k_B

no

the

solution.

4 and

represents

l/k_,,

‘enc in case

A,B

(19)

the

To

B undergo

Di

of

displacement

pair

lenC

Brownian

coefficient

mean-square

encounter

estimate

motion

the cr.

we

1

in

molecule 2>

of

the

1 by

= /6r

for

(20)

This

relation

any

correlation

supposes in

that

the

the

motion

time

of

the

t

is

sufficiently

molecule

to

long

have

disap-

peared. Analogously

the

relative

diffusion

coefficient

DAB is

defi-

ned

(21)

= /6T

DAB

2>
where the

We can B are

is

the

A and

apply

together

mean-square

eq.(21) in

the

relative

displacement

between

B. to same

obtain cage,

~~~~~ if

we

the also

average suppose

time that

A and the

mole-

253 cules

undergo

ment

Brownian

of

the

molecules

2

for

them

l/k_D

=

If

T

14)

we

to

introduce

For

the

The

equilibrium

and

k_o

is

by

tions.

For

North

is

A and oA

the

B’

then

~~~~

estimated

= 40

from

the

kD

ps. (eq.

8,

of

of the

the

place

first

the

assump-

area

collision

between

the

we

a reactive of

molecule.

KD was

different

probability

carry

for

dm3mol-‘.

that

takes

and

applied

= 0.3

this

ratio

expresion

he

K.

encounter estimate

area

to

obtain

spherical

surface

patch

the

Let

reactive

assume

that

on

reactive

e be

the

their center

solid

eAeB/16n2

(25)

function

= 4n).

to

angle

then

a probability

f(r)

is

contact

pair

TenC. and

defined at

oB come

and

eA

obtain

As

an

ranging

example,

if

form eA

0

=

(eA

or

eB = 2n

eB

= 0)

then

= 0.25.

f(@)

in

as

To

total

= eB

obtain

(23)

be

although

f(z)

ace.

is

we

a similar

probability

f(0)

o

=

that

the

the

(e,

15)

(24)

and

surf

f(e)

and

(22)

we

KD can

of

are

=

eq.

we

molecules

1

by

earlier

(ref.21),

the

corresponding

to

+ l/nBrB)

note

B in

and

cp is

cage,

displaceequal

enc>2’30AB

example

defined

between

average

:

22)

to

our

centers

to

(given

The

approximately

4 II N3/3

Estimation f(o)

cage.

be

a shared

=
constant

(eq.

the to

of

discussed

interesting

which

out

DAB

example

=

obtained

f(0)

get

/3DAB

=

KD = kg/k-D

It

in has

(Il.*r enc>2/3kBT)(l/nArA

=

enc

lenC

motion

eB,

tion

of

time

needed

as

the

into

contact

Therefore, ii)

T

molecules for

the

probability

formation

an

during f(z)

enc

and

in the

the

of

that

the

depends iii) cage.

transition

trot

OA and

encounter

pair

lifetime upon

i)

which

of the

the

for

defined

as

is

(A,B)

-(A.B),

not

but

‘A

encounter

reaction

allows

trot

QB are

(A,B)

angles

rotathe under

average the

condi-

254 tion

that

the

same

are

not

A and solvent in

rotation

cage

contact,

of

tricted

B cannot

A and

B to

i)

If

iii)

then

These

TenCztrOt

as

the

are

a function

of

O<

an

time aA

trot

B stay oA

OS.

long

in

and

needed

and

infinitely

and

OB

for

The

the

res-

~~~~~

shows: f(r),

probability f(r)

A and

position,

between

becomes

f(T)

=

tends

to

1.

0.

~1.

summarized T

if

average

probability

the

then

conclusions

sketched

to lenC

I.e.

starting the

contact

then

Tenc<
is

between

Tenc>>trot If

get

apart. the

trot

corresponds

comparision

ii)

in

and, then

condition

The

diffuse

in

figure

3,

where

f(r)

is

enc’trot’

f(x) I_,______----

____--_

.

/

.

,

/-

/ / / / / / /

,I’

0

I

-1 Fig.3.

f(

To assume the

versus

7)

obta

i n an

that

the

trot

Tent/trot.

approximate molecules

angle

average

Zenc/

0

expression undergo

a molecule

for

the

rotational

rotates

function

Brownian

during

time

f(r) motion.

t

is

given

we Then by

(ref.24-26)

(t)>


Here

= exp(-2Drt)

Now we between cules

is

Dr

can

two at

time

the

(26)

rotational

write

a

molecules 0,

if

similar A and

we

diffusion

coefficient.

expression B at

introduce

time

for t

a relative

and

the between rotational

angle the

eAiB(t) same diffusion

mole-

255 coefficient

(DrA,B)

.sA,B(t)>

ccos

in

eq.

26.

For

which

we

The

value

‘enc >>’ rA,B Therefore lity

have

)

to

1


‘rot

A and

i.e.

=

make

This

rrot

as

t

>

on can

is

renc/lrA,B and

obtained on

the

two

molecules

the

case

in

which

as

as

the

an

approximate

probabiex-

cage

which

may

average

time

between

this

way,

is

not

with

f(e)

the

we

= kBT/mnn

size

be

of

rate

their

expressed

of

by

and

OB when

turns rrA,8

make

out is

= 0.5

the

a relative

different

tent

the

mole-

reactive

centers by

de-

the

to

to

be

aproximately

time

the

rotate

of

= 0.5.

for

of

about

average in

order

A and

B

equal

required

rotation from

have

molecules f(e)

70°,

angle to

that

make

a

(bB.

= 1/2Dr = ~~ A/8 A/B can apply Stokes’s

-

=

(30)

f(e))3

1/2(DrA

+ DrB).

law

r3i

for

for

explicitely

needed

expC-rencf(b)/Trot(l

T,.~~

corrects

eA

7rot

very

@ and A we can write

Di

and

expression

rotation

(29)

between

f(T) = 1 -

approximate

relative

f(o))/f(o)

the

Finally,

an

the

dependence

angle

collision

m/8

on

rent/trot,

have

in

this

i

(for

~e;st~~,~).

(28)

a collision

and

Dr

0

which

write

(defined earlier): to lrA,8 the molecules A and B to

For

in

depends

depends

-

rot(l

Defined

in

then

(27)

= 1/2DrA,B. A/B varies from

t:dBcase

we

B in

eB.

*rot

to

r

(for

f(r)

trot

@A fining

T 0

sA,B>

We already

and

have

1 - exp(-rent/trot)

(eq.23). cules

we

A/B)

ccos

depends

for

f(T) =

defined

of

function

pression

= ~~~~

= exp(-2DrAiBTenc) = exp(-renc/Tr

in

t

(31)

the

non-ideal

behaviour

of

the

molecules.

256 Then

As

an

example:

300K

and

f(@)

= 0.25

and

eq.

f

with k ef

is

6,

= 0.6.

equation

Estimation

(ref.

= rB

(32)

= 0.25

nm;

we obtain

= 40

ps

yields

That

is,

for

than

(eq.

14)

the

~~~~ f(r)

to

= 20

ps.

= 0.49

fast

predicted

the

value,

according

(kr>>

k_l),

the

Smolu-

by

restricted

T =

This

and,

reactions

value

due

mA = mB = 8;

active

site.

kr thz

transition

state

theory

kr

is

given

by

27):

(rQ, kBTs”/hqAqB)exp(

=

where

denotes

q”

for

the

the

motion

qB

are

E,

is

and

the

The

required

close

to

quantity

Qt for

temperature

racteristic

the

known,

level

is

in

to

pass

of

the

which

be

to

represents the

rate

high

freedom

unit

omitted);

volume

from

the

allows

for

the For

a tunnel

correction

of

the fact

reactants.

to and

reactants

levels

state; for

volume

qA

the

lowest

transition

unit

corresponding

being

constants,

enough of

the

transition

surfaces

obtained

and

(T>>hv#/kB,

partition

energy

may

of

per

A

factor that

xis

some

adiabatic

transi-

1.

obtaining

potential kr

per

revert

frequency

ficulties

degree

function

coordinate

coefficient

expression

the

(the

reaction

complexes

x is

partition

functions

lowest

transmission

sical

the

energy

(33)

incomplete

partition

the

-Eo/RT)

state

along

the

B to

tions

the

transition

transition

se

rA

less

of to

+ l/mgr3B)

poises,

rent

40%

According

a

for

n = 0.009

about

chowsky

kr

write

(h/2k,T)(l/mAr3A

=

‘rot

we can

can

where complex).

functions needed

in to

experimentally

of

the

clas-

bj;3 neglected v

is Due

the to

the

solution,

calculate

E.

(assuming

if

chadif-

and

becau-

are

not

x = Q,

= 1)

from

k,

= kBTK”/h

=

is

the

where sition tion.

Ki’

complex

(kBT/h)exp(-hG”/RT)

equilibrium and

AG”

(34)

constant the

activation

between free

reactants energy

for

and the

tranreac-

257 APPLICATIONS

As

are

application

an

chosen

a series

known.

transfer

These in

TABLE

of

of

reactions

various

obtained

for

enough

experimental

ref.4)

refer

which

(taken

solvents

of

the

theory

Reaction

2,4_dinitrophenol

at

from 25’6

and are

above,

we have

to

defined

data

proton

in

Table

1.

magenta

blue

acid

blue

picric

taken

K A6 /

a A exp ’ l kJmol-

b A cal’ k Jmol-’

kr’ lo-lls-l

0.4

0.2

0.53

12

0.7

0.1

1.6

9

1.4

0.2

0.65

11

1.3

0.1

0.33

13

kJmo1

-1

ref.4

from

calculated order

by eq.35 to

rate

we substitute

eq.

11 and obtain = n/A

l’kef

reactions.

+

acid

In

transfer

+

trichloracetic

aA exp

proton

E +

tri-n-butylamine

bA cal

to

+

tri-n-butylamine

nile

equations

1

Application

nile

the

reactions

get

the the

influence

of

expressions

the

viscosity

we have

derived

+ l/B

on the

reaction

in

rhs

the

of

(35)

where

A =

4RT
*cl B =

(4/3)

enc

-

>(l/nArA

+

l/nBrB)If(@)

exp(-rencf(@)/Trot[l

II N3f

(@)k,

+

-

f(@)l)ll

Cl

-

f(@)l.

(35a)

(SSb)

258 The the

dependende

of

experimental The

experimental

‘* Acal lated by

is

Edward’s ni

Wirtz

(ref.28),

is

same We

ni/6. in

the

from

discuss

the of

this

of

magnitude

state’ of

the

for

E,

AND

In

most

by

the

few

an

+

Caldin

estimate

et

mi

diffusion mi/8

that

al.

we

rotational

observed

the

it

=

discrepancies

with

is

purely

in

sion

as

North). are

Raoult’s

T

implies

law.

for

in

table

eq.

1, by

34

these is

are

tran-

we obtain

reactions,

within

normally

the

range

to

iden-

used

an

to The

the

the

that

that

that A/B

situation

the

linear

we

in

kef

is

found

k,,,

kr

fac-

because for

are,

arrive

of at

interactions

with

if

in

f(Q) 24) an

the

is ancounter

same

A/Solvent i.e.,

most

however,

involved or

KB (eq.

formation

different

determi-

exponentially

parameters f,

interactions, is

is

the

There

But

the

that

kef

increase

behaviour.

(so

of by

behaviour

of

likely

assumption

means

than

decreases

kef

definition

hardly

we obtain

(37)

more

This

which

statistical

equal

from

This

eq.35b.

predicted

(ref.4,29,30).

surprising

This

by

given

values

dependence much

positive.

the

the

k,

AG”/RT)

temperature

in

rather

also

the

obtain

discussed

reactions

is

are

a criterion

factors

always

From ll),

can

activation

+ II .T .exp(

increase

pair

B/Solvent

table

Spernol

by

so

kJmol-‘.

calculations

(ref.31)

this

estimated

obey

the

cases,

cases

cause

+ rB by

in calcu-

REMARKS

controlled

(eq.

ri

reactions.

exponential

temperature. k ef

To for

Therefore,

of 10

19>,

eq.

FINAL

to

&G# are

and

32.

the

than

= 0.

about

= C .T .exp(Ev/RT)

diffusion a

BG”

controlled

According

and

with

given

(i)

estimated

B we

which

energy

be

(see

of

smaller

for

to

Ev

CONCLUSIONS

tors,

A are

proposed

value

for

value

free

out

diffusion

ned

of

diffusion,

parameter,

theory

turns

l’kef

agrees

%rA

equation the

reasons

experimental

orders

expected

35

values:

(ii)

the f(@)

the

values

value

eq.

section.

two

tify

values

translational

The

the

by

following

(ref.23),

(iv)

for

will

From

which

the

method

as

next

sition

predicted

(v) 7 /T rot from eq. 23 and enc assumption that the non-ideality

the

the

n

calculated

with

obtained

(iii)

and

(ref.4),

on

and

calculated

rS,

make

kef

observations.

and

solutes these

expres-

A and

interac-

B

259

tions

are

tions

we vary

KD

not

the

same.

To

eq.

24

introducing

by

account

for

these

different

interac-

AGO:

= KDoexp(-AG’/RT)

tions

(38)

is given KDO between

interactions

and

possibility could

by eq.

A and

into

increase

34.

B are

is

AGo

positive

weaker

than

negative

in

opposite

account,

the

or

decrease

of

the

the

if

the

A/Solvent

second

case.

term

depending

on

of the

interac-

and If

B/Solvent we

the

rhs

values

take of

eq.

AG

of

this

#

37 and

AGO. The

origin

14

eqs. tional

and

in

the

motions

A/B

kr

the

and

are

only

for

the

by

coupling

molecular

By the

beams

‘ab

rot. The and ’ for macroscopic

diffusion

tained tional

and

enc

These A

in

may

be

may

pair

are

cage

(mofor

only

equal

interactions.

are

gas

medium

be

in

coeffitient

B/S

involved

phase

modes) either

methods)

or

in

eq.

reaction and

obtained

be

an

error

in

coeffitients but

motions

the

the

initio’

motions,

translational

table

and

seen

therefore theoreti-

experimentally

experiments). there

1

the

be

transla-

two

which

for

could

can

same

These

A/S

same

with

reason,

of

same

the

or

the

cage. to

38

relative

encounter

is

parameters

parameters

semiempirical

the

equal

other

eq. the

22

eq. pair

nearly

these

(by

in

are

to

in

outside

DAB

encounter

f(e)

principle

(e.g.

and

term

DAD represents

coeffitient

respect

(corrected

cally

14

interactions

With 11,

exponential

eq.

bulk)

inside

the

in

In

diffusion

tions

if

22.

reasons

the

calculation

involved

here

they

inside

for

the

are

are

used

a solvent

ob-

for

rota-

cage.

difference

between

Acal

and

1.

exp In

conclusion,

consider and

the

f(e))

cially

in

difficult

cules

a first

in

tively

the firstly

series

of

with

in

which

the

gas

how

the

theory

the

various

reactions small reaction

gas

These

to

a reaction

phase

in

rrot.

for

relatively

solution

molecules

measurements

controlled no or

the

in

reaction

and

a way

improve

precise

same rent

if

outlined

involved more

diffusion for

calculate

To

errors

(preferably

the kD,

approximation

solution.

quires

solvents)

to

a reaction

for

addition

We have

law.

in

interpret

parameters

and

Raoult’s ters

to

we

have

phase

(kr

are

espe-

do not obtain

we have

obey

these

between

to

parame-

rate

between dipole

know

for

neutral moments

quantitaThis

constants

parameters

mole-

neutral

approximations. of

to

rea

molecules in are

inert known;

260

and

secondly

a thorough ~~~~~

investigation

parameters

like

rro+,

reactions.

Research

in

and

Stockmayer,

for

these

of

the

the

various

‘physical’

involved

molecules

two directions

is

in

carried

the

out.

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W. Scheider, K.S. Schmitz K. Sole and

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