Associating behaviour of pure polar liquids: dielectric properties of lauric acid

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Associating behaviour of pure polar liquids: dielectric properties of lauric acid

E. Mognaschi, L. Laboranti

To cite this version:

E. Mognaschi, L. Laboranti. Associating behaviour of pure polar liquids: dielectric properties of lauric acid. Journal de Physique II, EDP Sciences, 1994, 4 (9), pp.1469-1475. �10.1051/jp2:1994212�.

�jpa-00248056�

Classification Physics Abstracts 61.20

Associating behaviour of _pure polar liquids: dielectric

properties of lauric acid

E-R-

Mognaschi

and L.M. Laboranti

Dipartimento di Fisica "A. Volta" dell'Universith, Pavia, Italy

(Received

15 April 1994, accepted ^{14 June}

1994)

Abstract. In this paper measurements of the static dielectric perrnittivity of lauric acid at different temperatures from about 10 K above the MP to 352 K _are reported. These data, together with the dependence of the refractive index and density _on temperature, _are used in order to investigate the associative behaviour of lauric acid. The Kirkwood correlation factor

calculated both with the classical Kirkwood-Fr6hlich equation and with that corrected for ellip- soidal shaped molecules _are reported and discussed. Both correlation factors obtained indicate the existence of _a prevailing antiparallel order of dipole moments. The static perrnittivity and the correlation factors increase with increasing temperature and this suggests ^{that the number} of apolar dimers decreases _on going from low to high temperature.

1 Introduction.

It is well-known

that,

in the

liquid phase

^of

monocarboxylic fatty acids, hydrogen

bonds be- tween

carboxylic

_{groups are}

responsible

^{for the formation of dimers in} statistical

equilibrium

between pairs ^of molecules and that

they

_are also the main _source of short range interactions between molecules. The

associating

behaviour of _pure

polar liquids

_can be studied

directly

from their dielectric

properties, particularly by

_means of the static and

high frequency

dielec- tric

permittivity.

To this aim the statistical mechanical

theory developed by

Kirkwood [I] ^and extended

by

Fr6hlich [2, 3] ⁱⁿ the _case of

polarizable

molecules with permanent

dipole

_mo- ment, is _a useful tool for

obtaining

information about molecular stucture and intermolecular

interactions.

The main feature of the model

proposed by

Kirkwood and Fr6hlich with respect to the

widely

used

Onsager theory

_[4], is the introduction of _a correlation factor _g which takes into account the correlations between molecular and

dipolar

orientations due to

short-range ordering

interactions.

On the basis of the

analogy occurring

between the

Onsager

^and Kirkwood-Fr6hlich _equa- tions, the Abbott-Bolton [5] ^correction for

ellipsoidal

molecules to the

Onsager

formula has

been extended to the Kirkwood-1i6hlich equation, ⁱⁿ order to take into account the

elongated

1470 JOURNAL DE PHYSIQUE II N°9

molecular

shape

of

fatty

acids.

Thus,

the correlation factor

g'

evaluated in this _{way assumes} values different from those of _g, calculated via the usual Kirkwood-Fr6hlich

equation,

but _more

realistic.

In the present work measurements of the static

permittivity

of lauric acid _{as a} function of temperature _are

reported

and the results will be discussed in view of the above considerations.

Moreover _an attempt is done of

interpreting

the associative behaviour of lauric acid

by

means of _a different model [6]

developed

within the frame of the Kirkwood-Fr6hlich

theory

for solutions. In this model

associating compounds

_are treated _{as a} mixture of dimers and

monomers which behave _as

apolar

^solvent and

polar soluted, respectively.

A _new correlation

factor _{gm is}

introduced, taking

into account the correlation _among the orientations of the

dipole

moments carried

by

_monomers and it _can be put ^{in relation} with the

degree

of association

calculated via the association factor obtained

by

adiabatic

compressibility

data.

2.

Properties

of Iauric acid.

2, I GENERAL. Lauric acid

CH3(CH2 )IOCOOH,

is _a saturated

monocarboxylic fatty acid,

with molecular

weight

M

m 200.32 g mol~~. The total chain

length

is about 13.5

I

[7]

and,

_as

occurs for all saturated

fatty acids,

the molecule is rod-like and the molecular

dipole

_moment is localized in the acid _group [8]. ^The

melting point

is about 44 ° C. Lauric acid of

purity

greater than 99.5% _was obtained from Fluka AG and

employed

without further treatments.

2.2 DENSITY. The temperature

dependence

of the

density

^oflauric acid in the

liquid phase

was obtained from literature data

[9-12].

The inverse of the

density

_was found _to

depend linearly

_on the absolute temperature:

d~~

= A ₊ BT

(1)

with A

= 8.14 x 10~~

g~~

^cm~ and B

= 1.03 _x 10~~

g~~

^{cm~ K~~}

The number N of molecules _per unit volume is calculated

by

N =

NAd/M, (2)

where NA is the

Avogadro

number.

2. 3 REFRACTIVE _{INDEX AND HIGH} FREQUENCY PERMITTIVITY. The refractive _{index nD}

of lauric

acid,

for the sodium D-

line,

_was taken from literature

[9-12].

The

plot

of refractive index _versus absolute temperature follows _a linear law for 318<

T/K<353

nD = C + DT

(3)

where C

= 1.54 and D

= ~3.4651 _x 10~~ K~~ _are obtained from _a best fit of literature data.

For the

high frequency permittivity

_e~, I.e. the

permittivity

due to induced

polarization,

we used [13]

e~ =

1.05n[. ₍₄₎

3.

Experimental

details.

STATIC PERMITTIVITY. The

permittivity

of lauric acid _was

measured,

in the

liquid phase, by

_a General Radio

bridge,

mod. 1616. A three-terminal cell for measurements _on

liquids,

with

platinum electrodes,

_was used.

Permittivity

_was measured in _a field of 50 V m~~ and the results _were found to be

independent

of field

intensity

and

frequency.

In table I the values of

permittivity

of lauric acid from about 10 K above the MP to 352 K _are

reported.

All the values

reported

were obtained from measurements made at I kHz. At this

frequency

the

permittivity

values _can be considered static, because the relaxation times _are of the order of10~~° _s [8].

Table I. Static

permittivity

_e of lauric

acid,

values of (~~~)~phere ^and (~~~)~ph~rojd ^obtained

by

_means of

(6)

and

(9)

and correlation factors _g and

g'

evaluated for _~o

= 1.5 D.

T _f (jLefl)sphere g

(/lefl)spheroid g'

lK) ID) ID)

318.44 2.295 0.539 0.129 1.065 0.504

321.84 2.298 0.556 0.138 1.080 0.519

328.03 2.290 0.560 0.139 1.090 0.528

338.48 2.281 0.577 0.148 1.112 0.549

344.33 2.280 0.594 0.257 1.130 0.568

349.82 2.281 0.614 0.167 1.151 0.589

351.39 2.281 0.619 0.171 1.157 0.594

353.19 2.280 0.623 0.172 1.161 0.599

357.27 2.291 0.655 0.191 1.193 0.633

365.44 2.295 0.688 0.211 1.229 0.671

368.69 2.300 0.708 0.223 1.248 0.693

372.88 2.306 0.732 0.238 1.273 0.720

379.23 2.310 0.759 0.256 1.302 0.754

382.77 2.311 0.773 0.265 1.317 0.771

393.05 2.321 1.823 0.301 1.370 0.835

4. Discussion and conclusions.

The correlation factor _g, which characterizes the Kirkwood-Fr6hlich statistical mechanical the- ory of dielectrics

taking

into account the correlations between the orientations of

dipole

_mo-

ments due _to

short-range ordering interactions,

is defined _as:

fit

g #

£(COBRq) ⁽⁵⁾

j=I

where,

_Rq is the

angle

between the orientations of the I-th and

j-th dipole

moment, fit is the number of molecules which enter the Kirkwood

sphere

and the

angle

brackets denote the

1472 JOURNAL DE PHYSIQUE II N°9

statistical mechanical _average. from the above

expression

for _g, it _can be _seen that _{g c~} 0 _or g - cc,

respectively,

when most of the molecules of the

liquid

lies with

antiparallel

_or

parallel alignment

of their

dipole

moments. When there is _no correlation between

dipoles

then _{g c~} 1.

According

to this model the

Onsager equation

derived in the continuum

approach

must be rewritten in this _way]

~~~ _{~~~ ~ ~ ~~~ ~}

~~~

47rN

e(ef

+

2)2

~ ~~~

where _e and _{e~ are} the static and

high frequency permittivity respectively,

T is the absolute temperature, ^{N the number of molecules in the}

liquid

_per unit volume and ~o the permanent

dipole

moment of the molecule in _vacua. One _may observe that

(6)

is very similar to the

Onsager equation

obtained

by neglecting short-range interactions,

apart from the left hand side term. To extend the

validity

of the

Onsager equation including short-range correlations,

one

usually

introduces _an effective

dipole

moment for

spherical

molecules

(~~~)~phere

related to the true

dipole

moment in _{vacuo ~o} of the molecule

by

(/lefl)~phere "

g/l~ (~)

and whose

meaning

is that of the average

dipole

moment to be attributed _to each molecule when

short-range

interactions _occur.

Equation (6)

has been derived for

polar liquids

with

spherical

molecular

shape

and this is not the _case for

monocarboxylic fatty

acids whose molecules _are indeed rod like. Nevertheless this

equation

also holds for

liquids

with molecules of _a

shape

different from

spherical

since it has been observed [14] ^{that the dielectric}

properties

of _a considerable number of

liquids

can

satisfactorily

be described

by

_a hard

sphere

model. This is

probably

due to the

partial

compensation of different factors

including

the presence of

hydrogen bonds,

the existence of real

(not ideal) multipoles,

the _non

spherical shape

of molecules and their

flexibility.

On the other

hand,

_{one may} take the molecular

shape

of the molecule into account and use, in the _case of

fatty acids,

the

equation

derived

by

Abbot and Bolton _[5] which extends the

Onsager equation

_to

liquids

with molecules whose

shape

is

approximately

that of _a

prolate spheroid. Moreover,

in the _same way as one

generalizes

the

Onsager equation introducing

the effective

dipole

moment

(~~~)sphere

in order _to consider the

short-range

correlation between

molecules,

_one

might

introduce _an effective

dipole

_moment for

spheroids

defined

by

(/lefl)~pheroid " g'/l~>

(8)

where

g'

is a correlation factor whose values

might ^obviously

^{differ from those of} g obtained in the _case of

spherical

molecules. This

generalizing equation

_may be

presented

in the

following

form:

~~~~

~~~ ^~~~

~~ ~~~~ °~~~

~~~

^~

^{~ ~}

+ ~~ _°~~ ~~~

where

y m

3~~

^~~

^~~~~~~~

^~~,

(lo)

n + 2

e(Ai

1) Ai

Kn

_m i

jje i)le)An (o

_{m 1,}

2) (ii)

and the

An

_are the

depolarizing

factors of the

prolate spheroid

whose numerical values _were tabulated

by

^Osbom [15].

The purpose of this work is to determine the correlation factor from

experimental

data

by

means of

(6)

or

(9), and, possibly,

to find _a link between _g and the

degree

of association

fl

defined _as the ratio between the number d of dimers and the maximum attainable number

N/2

of dimers, I-e-

fl

=

~. ₍₁₂₎

In table I the measured values of the static

permittivity

_e, at different temperatures, for lauric acid in the

liquid phase

_are

reported, together

with the values _g and

g'

of the correlation factors calculated

by (6)

and

(9)

and those of the effective

dipole

moments,

(~~~)sphere

^and

(~~~)spheroid

^evaluated

by (7)

and

(8) assuming

_{~o =} 1.5 D. This

assumption

has been made _on the basis of _a

previously reported

discussion [16], even

though

the value of lauric acid molecular

dipole

_moment, at 20

°C,

in dioxane

solution,

_was found to be _~o

= 1.64 D [8].

The discussion of the results _may be

approached

in two different _ways, based _{on two}

possible interpretations

of the correlation factor. In the first

approach,

since g ^accounts for

short-range

interactions, _{we may suppose} that these correlation effects

play

_a relevant role

only

_among

nearest

neighbours

and _use the

following approximate expression

for the Kirkwood correlation factor [13]:

g = I +

z(cos Rq). (13)

where _z is the number of nearest

neighbours. Moreover,

in the

associating compounds,

_we

mainly

deal with dimers and _monomers and this would induce _to consider

couples

^of

interacting

molecules and take _z

= I. In this way, if the

degree

of dimerization is

high,

that is, a great number of dimers is present in the

liquid,

_{we may} expect that

(cos

_{R,~ ci} -I since the molecules

constituting

each dimer

couple

with

antiparallel dipole

moments. Thus _we may conclude

that,

if _{g -} 0, then _a strong dimerization _occurs, while when _{g -} I _no association is present.

The

reliability

of this

approach

is indeed denied

by

the

experimental

evidence. In

fact,

measurements of the

degree

of dimerization evaluated

by

_means of acoustical measurements

[17], ^{for the first terms of the series of}

monocarboxylic fatty acids,

showed that the

degree

of association increases with

decreasing

molecular

weight.

from _our dielectric measurements [6] we observed that the Kirkwood correlation factor _g increases from 0 to I with molecular

weight

and this reveals _a

tendency

in the associative behaviour of

fatty

acids which is in clear contrast with

experimental

data deduced from acoustic measurements.

The

inadequacy

of this

interpretation

of the values obtained for _g must

undoubtly

be at- tributed to the fact that the

assumption

_{z =} I is too

rigid.

On the other

hand, considering

a nearest

neighbours sphere

with _{z >} I, the presence of _{one or more} dimers in the

sphere

does not

significantly

influence the values of _g since in

expression (13)

_one must _use the value of

R,j averaged

for all the

dipole

moments

entering

the

sphere.

In the second

interpretative approach

_we consider the

associating liquid according

to _a pre-

viously presented

model [6] in which the fluid is considered _{as a} system of

polar

molecules

(monomers)

in _an

apolar

medium

(dimers).

In this framework

equation (6)

must be revised

according

to the Kirkwood-libhlich

theory

for mixtures and _{a new} correlation factor _{gm ac-}

counting

for the short _range interactions among monomers must be introduced. This correla- tion factor _can be related to the Kirkwood correlation factor _g which _appears in

(6)

and the

degree

of association

fl by

the

following

relation

gm =

g(I fl)~~ (14)

We may ^note that gm represents ^the

degree

of orientational correlation among monomers.

As _one

might

expect, ^if

fl

_- 0, I.e. _no association occurs, gm = g, while if

fl

_- I, I.e.

strong

association occurs, gm becomes very great

suggesting

^{that the}

remaining

monomers

tend to

align

themselves with

parallel dipole

moments. To evaluate _{gm one} should know g

1474 JOURNAL DE PHYSIQUE II N°9

and

fl independently.

For

example

_we could get

fl starting

from values of association factor _x obtained from adiabatic

compressibility

measurements. The association factor _x is defined _as the factor

by

^{which the} _average ^molecular

weight ^R

increases with respect to the molecular

weight

when association _occurs. In

fact,

if association is assumed _to be the

grouping

of _some of the molecules to form dimers _or

multimers,

the _average molecular

weight

increases to xM.

According

to

this,

_{x can vary} from _one, when _no association _occurs, to two, ^{when all the} molecules dimerize; x can also be greater ^than two if multimers _are also present. It is easy to show that the association factor _x is linked to the

degree

of association

fl by

the

following

relation:

p

=

2ji 1lx). jis)

For lauric acid the

only

value

reported

in the literature for the association factor

[17],

obtained

by

_means of adiabatic

compressibility

measurements

is, surprisingly,

less than _one, which should _mean that the _average molecular

weight

is decreased with respect to the molecular

weight

and this is

clearly impossible.

We may ^note that the model used

by

^the authors to calculate the association factor is based _on the assumpion that the molecules of the

associating compounds

_are

spherical

and this is

nearly

true for the low molecular

weight

terms of the series of

fatty acids,

but it does _not hold for the

higher

terms.

So,

for lauric

acid,

_an evaluation of

the correlation factor _{among monomers} is not, ^{at this}

point, possible.

from the results

reported

in table I _we may

observe,

at

first,

that the values of

g'

_are about three times greater ^{than the}

corresponding

values of _{g, at} the _same temperature. Even

though

we may suppose that the values of

g'

_are more realistic than those of _g, since

they

take into account the molecular

shape

of the

molecules,

nevertheless the

general interpretation

does not

change

since both

g'

and _{g are} less than 1.

So, finally,

_{we may} conclude

according

to the

general meaning

^of the Kirkwood correlation factor that the correlation _among the

dipole

moments of each molecule _occurs

mainly

with

antiparallel orientation,

whose

intensity

decreases with

increasing

temperature, as

expected

since thermal

agitation

contrasts

dipolar

orientation.

From the

experimental

results

reported

in table I _{one can} observe that the

permittivity

values of Iauric

acid,

in the range of temperatures ^of our

investigation,

lies between those of

undecylic

acid

in

=

11)

_[18] and of

margaric

acid

in #17)

_[19] at the _same temperature.

Here below, we also report the value of molar refraction R for lauric acid calculated

using

the

extrapolated

values of refractive index _nD and

density

d

by

_means ^of

(3)

and

(1)

at 20 °C and

by

the

following expression:

~

~)

₊

^j~'

^~~~~

In this way we get ^R = 58.733 cm~ which is in

good

_agreement with the value of R

= 59.411 cm~ obtained from the

atomic,

_group and structural contributions [20].

Acknowledgements.

The collection of

physical

data _on lauric acid from literature and dielectric measurements made

by

A. Chierico are

gratefully acknowledged.

Thanks _are also due to L. Cattaneo for the

construction of the cell for measurements. The Consorzio Interuniversitario Nazionale _per la Fisica della Materia

(C-INFM)

of the Ministero dell'Universith _e della llicerca Scientifica _e

Tecnologica supported

this work.

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