Dielectric relaxation of mixtures of non-rigid polar molecules in benzene

JournalofMolecularLiquids,38(1988)23-33 Elsevier Science PublishersB.V., Amsterdam-

DIELECTRIC

RELAXATION

J. M. GANDHI

Department

OF MIXTURES

OF NON-RIGID

POLAR MOLECULES

IN BENZENE

and GOPAL LAL SHARMA

of Physics,

(Received

23 PrintedinThe Netherlands

14 April

University

of Rajasthan,

Jaipur-302004,

India

1987)

ABSTRACT

The dielectric

absorption

and Allyl-methacrylate, studied

of non-rigid

at 9.92 GHz over a range of temperatures

benzene.

The various

parameter,

and the viscous

dielectric

single component rotations, component

relaxation

data.

system.

relaxation

process

in the single

relaxation

flow processes

such as existence

The values indicate

of molar

A comparison

has been solutions

for the activated

of

of intramolecular

phase,

in the two

of activation orientation

into non-cooperative

time rc,with those calculated

values

for the of the molecules

orientation

can successfully

by M.P. Madan

predict

in the

of the most probable

on the basis of simple

suggested

of the

and overall

are retained

entropy

both

using the

that the properties

of the experimental

rules and also by a procedure

states,

have been determined

that the cooperative

system changes

none of these procedures

in the dilute

indicate

of solid rotator

component

binary system.

parameters

The results

systems,

and absence

of Isobutyl-methacrylate molecules,

times ro, r(l) and r(2), the distribution

and the thermodynamic

a

for relaxation measured

mixing

of three compositions

both consisting

and reciprocal

[5] reveals

the experimental

that

data.

INTRODUCTION

Dielectric solutions Firstly

because

literature. validity

frequencies

and Smyth Schallamach

and concluded

only one relaxation

0167-7322/88/$03.50

time.

of polar liquids in dilute

gain considerable

importance

very few such measurements

because

of the conclusions

[71.

of mixtures

solvents

at present

Secondly

Bos [2], Forest Vashisth

measurements

of non-polar

these measurements

arrived

131, Garg and Kadaba 111 made measurements that a mixture His conclusions

are available

provide

at by various

these days.

workers

[4], Madan

like Schallamach

[ll,

[5,61 and Vyas and

on dipolar

mixtures

of two polar components were at variance

0 1988ElsevierSciencePublishersB.V.

in

a good check for the

at radio

should have

with the theories

24

of Debye

[al, Kauzmann

experimental solvents

solutions

[lo] has established the molecules

relaxation, relaxation

time

associative

for each molecular

identity

species

of two highly non-rigid

are on three different relaxation

30°C, as reported temperatures

by Khatry

is a sensitive

means

these investigations of relaxation

will be helpful

behaviour

these

of the mixture.

The

temperatures.

are in the ratio of lo:17 at

The measurements

over a range of

on the interaction interaction.

and temperature

to the mixtures

energy which

It is hoped that

in better understanding

will show how differently

behave with respect

investigations

of Isobutyl-

at four different

molecular

on the composition

investigations

polar molecules

[ll].

information

of investigating

The present

molecules

component

compositions

and Gandhi

useful

of dielectric

(AMA), one of them being

times of these esters

will provide

of the

in nonpolar

and so give a separate

present.

and the other as non-associative

The individual

Further,

that in the phenomenon

(IMA) and Allyl-methacrylate

measurements

the large amount

of single polar liquids

retain their

are made on binary mixture methacrylate

Further,

[9] and others.

work on dilute

the dependence of the mixture.

mixtures

of non-rigid

of rigid polar molecules.

EXPERIMENTAL

The dielectric

constant

9.92 GHz using the apparatus maximum

possible

percent

respectively.

distribution described

errors

composition

of benzene.

of the mixture fractions

from Merck,

(India).

by volume

of the mixture

were obtained

from M/s.

[ill, the

time ro, the as

Fluka, A.G. Benzene

(AR Grade)

It was first dried over sodium wire and

Mixtures

of three different

was treated

concentrations

were prepared.

as a single solute

had solute concentrations

f O.l'C.

varying

The dilute

from 0.0032

All these measurements The temperature

Each

in the dilute

The values of c' and E" for each composition as for a single polar component.

for each system.

within

relaxation

Germany.

of one component)

ures of 30°C, 40°C, 50°C and 60°C. constant

before

at

at f 1.0 and + 5.0

times r(l) and r(2) were computed

(purum) was obtained

twice before use.

(25, 50 and 75 percent

solutions

of most probable

o, the relaxation

by M/s. B.D.H.

then distilled

as described

and E" were estimated

and Allyl-methacrylate

was supplied

mixture

and procedure

loss E" were determined

[ill.

Isobutyl-methacrylate Switzerland

E’

The values

parameter

previously

in

E' and the dielectric

of the solutions

to 0.0243 weight

were made at four temperatof measurement

was kept

25

RESULTS

AND DISCUSSIONS

The values distribution relaxation mixture

of the dielectric

parameter

1.

Table

most probable

CL, the most probable

relaxation

field

be different

E

temperatures

calculated

of IMA and AMA are nearly

from the internal

for the systems

=

time ro, the

and three different

at four different

in the pure component

are reported

values

of the

of the same order, the

of the mixture

would not be expected

field of the mixture. constant

investigated

by a simple mixing

0

loss s", the

times for comparison.

fact that the static dielectric obtained

relaxation

1 also lists some theoretically

The dipole moments internal

E', the dielectric

times r(l) and r(2) for pure components

ratios of these components

in Table

errors

constant

relation

E

0

This is evident

and the dielectric

can be expressed

within

to

from the

constant

E'

experimental

of the type

xlcol + x2 co2

where x

and x are the mole fractions of the components, E is the static 1 2 0 dielectric constant of the mixture and co1 and co2 are the static dielectric constants

of the pure components

The values each temperature component

of measurement,

in the mixture.

of the rotating molecules,

results

E" increases

more difficult

in higher

of the mixture

fields

in the mixture

1 show that at

with the increase

value of E".

the variation

in Table

because

of the heavier

the overall

with the increase

rotation

of the heavier

For a particular

of E" with temperature

same order as that for the pure components. that the internal

of measurement.

loss E", reported

This is to be expected

unit becomes

which

concentration

at the temperature

of the dielectric

This again supports

are nearly

is of the the assumption

the same as those in the

pure components. The values

of the distribution

for the pure components AMA) mixture

of more than one relaxation

four temperatures

it has the highest

relaxation

of IMA.

the contribution is highest.

two other mixture

value

indicating

both in the pure as well

of a at a given temperature trend.

However,

for the composition

for all the

of the mixture

This shows that for this particular

of the processes

The a values

concentrations

1 show.that of the (IMA-

these are finite,

processes[l2]

The variation

does not follow any particular

0.47 mole fraction

composition,

in Table

as well as for all the three compositions

of these molecules.

with composition

having

a, recorded

and for all the four temperatures,

the existence as mixtures

parameter

mixture

other than the molecular

for the pure components

as well as for

viz. 0.30 and 0.81 mole fractions

of IMA in

1.642 4.008 9.507 14.968 23.002

1.326 4.125 8.834 13.317 20.046

1.182 3.495 7.881 11.841 18.057

0.819 3.037 8.463 10.445 15.916

2.272 2.279 2.289 2.296 2.309

2.251 2.258 2.269 2.271 2.289

2.232 2.239 2.249 2.260 2.271

2.214 2.223 2.229 2.241 2.256

0.00 0.30 0.47 0.81 1.00

0.00 0.30 0.47 0.81 1.00

0.00 0.30 0.47 0.81 1.00

in sets

0.233 0.175 0.456 0.170 0.188

0.256 0.234 0.449 0.193 0.202

0.280 0.288 0.428 0.216 0.227

4.4 4.1 3.9 6.4 6.5

4.9 4.7 5.0 7.4 7.5

5.6 5.4 7.3 9.3 9.4

0.320 6.8 0.313 6.7 0.401 9.1 0.221 11.3 0.235 11.7

E" x103 a

0.00 0.30 0.47 0.81 1.00

Mole frac. of E' IMA in (AMAt1MA)

4.9 4.7 4.3 6.4 6.5

5.3 5.2 4.8 7.1 7.1

5.7 5.5 5.7 8.1 8.1

6.2 6.1 6.5 9.1 9.2

in sets

12.5 10.2 23.9 12.2 13.0

14.0 13.2 24.9 14.1 14.5

15.9 16.1 25.8 16.8 17.1

19.0 18.6 25.9 18.8 19.7

in sets

7rzB,

7-91

9 nf

I nf

9 nf

in in sets tion of tion of tion of sets cal (Madan) cal (Madan) 3 (S.M:) (f;aIk) (R.M.) Temperature=30"C 8.3 7.8 7.7 23.9 16.4 14.9 9.1 8.5 8.7 0 -6.6 -4.4 10.8 10.3 10.0 -4.4 -8.8 -8.5 Temperature=40'C 6.7 6.4 6.3 24.0 18.5 '6.7 7.4 6.9 7.1 1.4 -5.5 -2.7 8.7 8.3 8.1 -6.5 -10.8 -12.9 Temperature=50°C 5.7 5.4 5.4 21.2 14.9 14.8 6.1 5.8 6.0 22.0 16.0 20.0 7.0 6.8 6.7 -5.4 -8.1 -9.5 Temperature=GO'C 5.0 4.9 4.8 21.9 19.5 17.0 5.4 5.2 5.2 38.5 33.3 33.3 6.1 6.0 5.8 -4.7 -6.3 -9.4 -

in sets (S.M.)

X1O12 r(2) T(l) xlO12xlO12 7ca11

Table 1. Values of the Permittivity E', the Loss Factor E", the Distribution Parameter a, the Experimental values of Relaxation Times TV, ~(1) , ~(2) and the Theoretically Calculated Values of ~~ for the (IMA-AMA) mixture at Different Temperatures.

%

21

the (IMA-AMA)

mixture,

decrease

of 0.47 mole fraction

with temperature

of IMA in the (IMA-AMA)

This shows that at this particular somewhat

different

orientation uniform

fashion.

throughout

at higher

temperatures,

rise of the temperature

hindering

other than the molecular

systems

systematically are distinctly

orientation

with temperature. different

characteristics

mixtures.

1 show that these

Here again the mixture

ratio, the contribution group rotation

at this concentration, increase

giving

domination

When measurements probable

relaxation

to the relaxation

the solute-solvent

behaviour

these values

of course the T

of the mixture.

appear.

Davidson

investigations,

values

plot

separate

ment.

It

regions

results

for the mixture

appear

seems that the increased in a relaxing

0

at of

vary with the may

Debye regions.

ratio of the two components different,

are not likely to of two distinct

time of individual

component

three

at all the four temperatures,

to provide

overlap

mixture

in

Such a behaviour

In the present

mixture,

to have merged

the

listed

values

times are not widely

loss maxima

[14], the relaxation

ratios of the (IMA-AMA)

the two polar media

the r

of two individual

should be about 5-8 times that of the other. concentration

r. values

polar components.

relaxation

of the most

regarding

of the (MA-AMA)

[13] has shown that for the resolution

arcs in a Cole-Cole

in the

the values

lie between

varies with the concentration

If the individual

as in the present

0

always

of the simple overlap

The degree of overlap

that

interactions

information

of two polar components.

ratio of the constituent because

higher

It appears

resulting

frequency,

very useful

1 show that for all the three compositions

be expected

is relatively

rotation.

and the solute-solute

are made at a single

all the four temperatures,

concentration

is highest

group rotations.

of mixture

the pure components;

of IMA is typical

process

of local environments

time r. can provide

relaxation

in their

the r(1) and r(2) values

than by the overall

rise to a variety

of internal

that the

are retained

with 0.47 mole fraction

between

for all the five

indicating

This again shows that for this particular

at all the four temperatures.

by the internal

compositions

Also r(2) values

of the pure components

in the fact that the difference

of the composition,

are more dominant. in Table

from r(1) values

individual

mixture

with the

composition

both for the pure as well as for the three mixture

decrease

to become more

due to the fact that for this mixture

of ro, r(l) and r(2) reported

in a

the molecular

is expected

in the case of this particular

processes

Table

barrier

in general

increase.

the system behaves

becomes more and more non-uniform

This is probably

values

for the composition these values

composition

The potential

the liquid, which

mixture.

The values

mixture

while

mixture,

a single

of two nearly

unit with very little change

local environ-

equal Deybe in the shape of the

28 For the concentration

dispersion.

r o values of the mixture value computed

ratio of 0.47 mole fraction

at 30°C and 40°C are nearly

from the single component

concentration

ratio,

from the corresponding

temperatures

giving

rise to different

behaviour

molecules

in identical

relaxations

with the relaxation other workers Table

behaviour

concentration

for the mixture

and then increase

At lower concentrations

interactions,

heavier

region

of the

are consistent reported

by

first decrease

with the rise of the heavier

as mole fractions

time.

showing

contribute

by the rate process

more resulting

concentrations

orientation

equations

[211,

processes

as well as for the viscous

dynamical

parameters

i.e., the molar

molar enthalpy

of activation

the dielectric

relaxation

using rate process

therefore

flow processes.

relaxation

The various

free energy of activation

1211.

times

can be represented

both for the dielectric

and for the viscous

equations

of the mixture

to that of the individual

AH and the molar entropy

process

of the relaxing

that the decay of relaxation

i.e. similar

thermo-

AF, the

of activation flow process

These thermodynamic

AS for can be

parameters

in Table 2.

The molar

free energies

of activation

in the

of the

of ro.

These relationships

of the mixture.

increase.

the overall

much but due to solute-solvent

plots for the three composition

is exponential

component

in the mixture,

At higher

values

of the mixture.

of heavier

component

rotations

into higher

linear relationships

variation

component

relaxation

resulting

with temperature

are given

of the dispersion

These results

unit is not affected

The log (roT) vs $

obtained

when dissolved

is the resultant

the ease of the rotational

component,

components

similar

of the heavier

the intra-molecular

of the average

unit is reduced

exhibit

which

of some other polar mixtures

of the heavier

of the relaxing

lowering

is somewhat

investigation,

yield a slight widening

polar

rigid components

1 shows that To values do not have any systematic

The r. values

rotation

under

components.

that the

14, 18, 19, 201.

with increasing

component

characters

of two non-associating

of the mixture

of the individual

and the different

it appears

two highly non-rigid

and non-associative

environment

that the

at different

local environments

Thus, binary mixtures

to a relaxation

This means

values.

containing

for the mixture

16, 15, 15, 171.

leading

of a mixture

for the same

at 50°C and 60°C

is different

From these results

units.

of associative

to the behaviour

average

of the two Debye regions

sizes of the relaxing relaxation

0

equal to the average

However,

- values of the mixture

are different

degree of overlap

the T

times.

the

of IM,

for the two processes

i.e.

303 313 323 333

303 313 323 333

303 313 323 333

303 313 323 333

303 313 323 333

0.00

0.30

0.47

0.81

1.00

Mole fraction Temperaof IMA in ture in (AMA+1MA) OK

11.7 9.4 7.5 6.5

11.3 9.3 7.4 6.4

9.1 7.3 5.0 3.9

6.7 5.4 4.7 4.1

6.8 5.6 4.9 4.4

Experimental T x 1012 (Pn sets)

2.22

3.15

5.06

2.63

2.15

2.50

2.47

2.47

2.47

2.50

AHE in -1 AHn in -1 kcalmole kcalmole

AH

2.90 2.94 2.94 2.98

2.57 2.56 2.52 2.51

2.57 2.56 2.53 2.51

2.25 2.24 2.24 2.26

2.88 2.89 2.92 2.95

2.91 2.93 2.95 2.98

2.90 2.92 2.95 2.97

2.90 2.92 2.95 2.97

2.90 2.92 2.95 2.97

2.91 2.93 2.95 2.98

AF, in -1 AFq in -1 kcalmole kcalmole

AF

-2.24 -2.30 -2.23 -2.28

1.91 1.88 1.95 1.92

8.2 8.0 7.8 7.7

1.25 1.25 1.21 1.11

-2.41 -2.36 -2.38 -2.40

ASE in _, calm0 e -1 deg

-1.35 -1.37 -1.39 -1.44

-1.42 -1.44 -1.49 -1.50

-1.42 -1.43 -1.48 -1.50

-1.42 -1.43 -1.48 -1.50

-1.35 -1.37 -1.39 -1.44

AS,.in -1 calm01 -'i deg

AS

Molar Entropy of Activation Table 2. Values of Relaxation time T Molar Free Energy of Activation (AF ,AF ) (ASE,AS~) and Molar Enthal& of Activation (AH AH f for Dielecfric'Reiaxationand Viscous Flow for the E' rl (IMA-AMA) Mixture.

30 increase with temperature

AFc and AF

in the case of mixtures

in similar

n fashion

as in the case of pure components.

behaves

like a single system, therefore,

results

in the fall of the viscosity

thermal

agitation,

the molecules

to come to the activated compositions corresponding Perhaps the lowering values

listed

values

medium

which

and rise in the

of AFc and AFn for the three are always

less than the

for their pure components. between

of the activated

the two dissimilar

state.

It is obvious

in Table 2 for the three mixture state is reduced

ratio of the mixture. for the three mixture

6

The values

at any given temperature

the interaction

to which the activated

AF

of the surrounding

of the system as a whole require more energy

states.

of the mixture

This means that the mixture

with the rise of temperature

Further,

AF

are always

results

in

rl that the extent

compositions,

depends

as expected

compositions

molecules

from the AFc and AF

on the particular

concentration

values for the pure as well as rl greater than the corresponding

values. The values

processes

are negative, and Smyth positive

of molar entropy

present

an interesting

the explanation

[221.

of activation case.

of such negative

For the three mixture

at all the temperatures.

composition

the AS values are E that these molecules which

of cooperative

orientation

of the existence

component

become non-cooperative

into the activated ASc values

state which

are highest

of the two components for this particular

in a two component

is more disordered

for the mixture

composition

(0.47 mole fraction

concentration

the molecules

of the two components,

is enhanced.

This also indicates results

ASn values

for the three compositions are negative

the viscous

formations

for the larger relaxing The knowledge under varying because

it helps

in obtaining

prediction

adequate

data available

and calculation

on mixtures

gain sufficient

with The

of activation

Hennelly

are positive

ones.

of polar components is of great

of liquid relaxation

processes

in mixtures.

is very limited, methods importance.

This is

to the complex

and temperature models

[22].

interactions.

for the smaller

about the relaxation

with the

of IMA molecules

-1.4 to -1.5.

times of a mixture

of composition

in formulating

information

the experimental

units and negative

of the relaxation

conditions

of entropies

that

the interaction

as well as for the pure

vary between

or solute-solvent

et al. [23] also find that the values

The

equal moles

of steric forces

is not much sensitive

due to the solute-solute

state.

and their interaction

of the mixture

and their values

flow process

having nearly

that presence

in the reduction

components because

than the normal

ratio of the mixture,

solvent

in benzene

in the single system resulting

of IMA), once again indicating

between

AMA molecules

for the pure components

values has been given by Branin

This indicates

give the evidence system,

ASc for the relaxation

The ASc values

importance and also As of

31 The simplest

and the most straight

itself for the estimation linear molar mixing of mole fraction

forward

of the relaxation

rule.

procedure

time of a mixture

If ri is the relaxation

x, in the mixture,

which

may be the

time of the i thcomponent

then the relaxation

time T

1

is

suggests

a

of the mixture

given by n T

=X

a

Another

(2)

Ti

1

simple procedure

According

=x

is the receprocal

mixing

1

rule.

(3)

i=l T

a

[5]

i

proposed

an ingenious

rigid polar molecules decays exponentially

l,cn T a

for estimating

ra for mixtures

of polarization

order

just as it does for single component of a number

with a non-polar

represented

method

in which the degree

For a system consisting solution

~~ values

X.

n

1

T

for computing

to this rule

-

Madan

X.

i=l

by a relation

solvent,

liquid systems.

of rigid polar molecules the relaxation

time r

a

in dilute

of the mixture

is

of the form

A. 1 i=l

of

in the mixture

(4)

T. 1

where A

are the parameters representing the effect of molecular environment, i shape and size, viscosity, fractional volumes, solute-solvent interactions,

and other factors

affecting

the molecular

on a particular

relaxation

process

dipole moments,

the parameter

depends

reorientation.

As the weight

upon the square of the associated

A of the i th component

may be written

as:

C.p2 11

Ai =

(5) ;

C&

i=l where Ci are parameters moments

pi.

For a

involving

binary mixture, C u:

L= T

a

effects

other than those due to dipole

equation

(4) becomes

uf

(f-5)

+ (c P: + lla,r,

(C P: + lG)r,

32

c1 where C = -

.

The value of C in dilute solutions

c2 considering it to be roughly be estimated

listed in Table

values

of ra employing

1 in the columns

by the three different

when compared

a mixture deviations absolute

methods

of the same order,

compare

deviations

The computed

for the values

However,

are quite large These

At 30°C and 40°C, for

of the constituent

are least for all the three computed

percentage

Ci can thus

well among themselves.

at 50°C and 6O'C.

equal mole fractions

by

(Z), (3) and (6) are

the deviations

values,

"1 -. v2

for all the three methods.

become more pronounced of nearly

equations

8,9 and 10 respectively.

with the experimental

and are nearly deviations

fractions

from molar volumes.

The calculated

values

equal to the volume

can be estimated

values.

computed

components

The overall by equations

It is

4O'C; 16.0, 13.0 and 14.8 at 50°C and 21.7, 19.7 and 19.9 at 60°C. from these results

successful present eqns.

in predicting Guided

study.

that none of the three computational

the ra values

for the system

by the percentage

deviations, Equation

(3) and (6) are of the same order.

better fit with the experimental

results,

be assigned

to C to include the effects

encountered

in the rotations

the relaxation different

behaviour

becomes

interactions

development

of

relations,

times of various

mixture

ratios and over a range of temperatures

values

Further,

in solutions.

solutions

large number

could

of conditions

(6).

solvents,

For the

of more elaborate

systems with different

as

among the

of non-polar

term should be added to equation

of more suitable

for relaxation

the predictions

(6) may give relatively

upon the interactions

in dilute

are

in the

if more appropriate

dependent

methods

investigated

of the variety

of polar molecules

types of polar molecules

a perturbing

average (21, (3)

9.4, 10.6 and 10.3 at 3O'C; 10.6, 11.6 and 10.8 at

and (6) are respectively

evident

the

data

concentration

are needed.

ACKNOWLEDGEMENTS

The authors University

are grateful express

for providing

the junior

of Scientific research

of Physics, facilities,

discussions.

One of us (Gopal La1

and Industrial

fellowship.

and

The authors

Gupta and Miss Chhavi Maithel

of experimentation.

to the Council

for awarding

the laboratory

for some useful

thanks to Miss Rashmi

in the process

is thankful

New Delhi,

to the Head of the Department

Jaipur

to Dr. M.L. Sisodia

their sincere

their assistance Sharma)

are thankful

of Rajasthan,

Research,

for

33

REFERENCES

1

A. Schallamach,

2

N.E. Hill, W.E. Waughan, and Molecular (E.E. BOS,

3

E. Forest

Trans.

Faraday.

'Dielectric

(Van Nostrand-Reinhold,

Behaviour'.

Ph.D.

Sot. A (GB) 42 (1946) 180.

A.H. Price and M. Davies,

Dissertation,

Leiden,

J. Phys. Chem.

and C.P. Smyth,

4

S.K. Garg and P.K. Kadoba,

5

M.P. Madan,

Can. J. Phys. 58 (1980) 20.

6

M.P. Madan,

J. Molec.

7

A.D. Vyas and V.M. Vashisth,

8

P. Debye,

9

W. Kauzmann,

10

J.G. Powles

(U.S.A.)

J. Phys. Chem.

Liquids

London)

The Netherlands

1302.

(USA), 69 (1965) 674.

(Netherlands),

Chemical

1969.

1958).

69 (1965)

25 (1983) 25.

Ind. J. Pure and Appl.

'Polar Molecules'

properties

Catalog

Phys. 23 (1985) 527.

Co., Inc., New York, N.Y.

1929. Rev. Mod. Phys.,

14 (1942)

Vol. I, 2nd Ed., Part III, Chapter New York, N.Y.

1.

'Physical Methods

and C.P. Smyth,

KKKV,

of Organic

Interscience

Chemistry',

Publishers,

Inc.,

1954.

11

Madhulika

12

J. Crossley

Khatry

and J.M. Gandhi,

13

D.W. Davidson,

14

K.S. Cole and R.H. Cole, J. Chem. Phys. 9 (1941) 341.

15

M.P. Madan,

M. Shelfoon

16

M.P. Madan,

Can. J. Phys. 51, (1973)

17

M.P. Madan,

Z. Phys. Chem. Neue Folge 86 (1973) 274.

18

R.L. Dhar and M.C. Saxena,

Z. Phys. Chem. Neue Folge 8 (1972)

19

R.L. Dhar and J.P. Shukla,

2. Phys. Chem. Neue Folge 84 (1973) 25

20

A. Mathur,

and S. Walker,

Can. J. Chem.

S.N. Sharma

J. Molec.

J. Chem.

Liquids

30 (1985) 63.

Phys. 45 (1966) 4733.

39, (1961) 571.

and I. Cameron,

Can. J. Phys. 55 (1977) 878.

1815.

and M.C. Saxena,

Ind. J. Pure and Appl.

189.

Phys.

12 (1974) 370 21

S. Glasstone,

K. Laidler

and H. Eyring,

(McGraw Hill Co., New York) 22

F.H. Branin,

23

E.J. Hennelly, (1948) 4102.

'The Theory

of Rate Processes'

1941.

Jr. and Smith, C.P. 1952, J. Chem. Phys. 20 1121. W.M. Heston

Jr and C.P. Smyth,

J. Am. Chem. Sot. 70