heptane systems

CHEMICAL PHYSICS LETTERS

Volume 140, number 2

25 September 1987

CRITICAL NUMBER DENSITY IN POLAR LIQUIDS: ORIENTATIONAL POLARIZABILITY OF POLAR MOLECULES IN n-DECANOLIn-HEPTANOL AND n-HEPTANOL/HEPTANE SYSTEMS H. MECHETTI, A. CATENACCIO and C. MAGALLANES L&oratorio de Dielkctricos,UniversidadNationalde San Luis, Chacabucoy Pedernera, 5700 San Luis, Argentina Received 16 October 1986; in final form 6 July 1987

The orientational polarizability of polar molecules in n-decanolln-heptanol and n-heptanollheptane systems has been determined. Experimental permittivities show that both systems (together with the family of aliphatic normal alcohols) are described by a unique function of the number density p and the absolute temperature T. The experimental data show that the polarizability increases rapidly as a function of p from a critical number density pc.

1. Introduction A recent paper [ I] on induced birefringence in binary systems (n-decanol/decane and n-octanol/ octane) showed that the electric-field-induced changes in the refractive index are described by a Kerr constant B=P& =@o(l -P&)

>

(1)

in which B, is the specific Kerr constant and B. a constant evaluated at a number density (the number of polar molecules per unit volume) p >p, where pc is a critical number density. Eq. (1) is important because: (i) it describes two systems with the same values of B. and p. (ii) it associates the collective molecular interactions with B,, (iii) it shows that for p=pc the system experiences something like a phase transition. Certainly, for p
systems studied in ref. [ 11. From now on these will be called systems S, and SZ respectively. These systems are interesting because it is possible to obtain the same value of p in both of them with appropriate proportions of each component and so it will be possible to check if eq. (1)) as a universal function, describes their behaviour. However it must be pointed out that equations such as (1) are likely to be approximate rather than exact. The family of aliphatic normal alcohols, from now on called system S, is of interest because: (i) they are associated liquids; (ii) the value of ~~~2.8 is the same for them all, which means that all contributions to the polarization - excluding molecular dipole orientation - are constant. em is the value of the permittivity for a frequency so high that the molecular orientation contributions, designated as contributions of the first zone [ 21, disappear. E, may be assumed constant over the range of temperatures studied in this paper (253-333 K) since it is only slightly affected by temperature.

2. Experimental The value for the complex permittivity was determined by means of a Boonton RX-Meter Bridge type 250 A in the frequency range 10 to 250 MHz using a cell described in ref. [ 31.

0 009-2614/87/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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The density of the mixtures was determined with a pyknometer calibrated with doubly distilled water. The concentrations of the mixtures and the purity of the materials were checked by means of an HewlettPackard 5711 A analytic gas chromatograph with flame detector. The normal errors in t’ and e” were estimated following ref. [ 31. These were found to be smaller than 2% for both c’ and E” and < 1% for the density. The values of e. and E, were calculated by multiparametric fitting based on Marquard’s algorithm [ 41 and assuming that the values measured for e’ and e” - the real and imaginary components of the dielectric permittivity, respectively - follow the Debye model [5]. The values of the relaxation times rb for the first zone were calculated by means of zD=t”/W(E’--E,)=X”fOX’

,

(2)

25 September

1987

are the real and imaginary components of the orientational susceptibility, w the frequency of the perturbing field, t (0, T) = e. and x( 0, T) the permittivity and orientational susceptibility respectively for o = 0 and absolute temperature T.The equation

[c(O,T)-~,]/4n=~(O,T)=pa(0,T)

(3)

allows the calculation of the susceptibilityx(O,T) and the polarizability Q( 0,T)at zero frequency. The values of (Yobtained for systems S, and S2 at 293 K and the values of cx obtained for system S at 253 and 273 K are shown in tables 1 and 2. The values of CYcorresponding to 293, 313 and 333 K were taken from ref. [2].

3. Discussion The Cole-Cole plot [ 61 of e Uversus E’ presented in fig. 1 for systems S, and S2 shows that both systems behave as polar components for p>pc (semicircumferences C, and Co). E, has on average the same value for both systems over the entire range of mixtures studied, with the exception of mixtures S2

in which X’(o,T)=X(O,T)(l+w*z~)-’

LETTERS

)

Table 1 Parameters corresponding to systems S, and S2 at 293 K. g corresponds to experimental values. The a! values were obtained by means of eq. (2) using t0 and e,=2.8. CY*corresponds to the values obtained by means of eq. (4) using the values given by eqs. (5) and (6). e$ co~espondstothevaluesobtainedusingeq.(3)withcy*ande,=2.8.ForsystemS,andsystemS,:p,=1.365~10~’cm-~;ru,=2.337x10-~~ cm3; T= 293 K lo-*‘p, (cm-‘)

lo-2’p (cm-‘)

10’61 SI (g cm*)

10%

102*a*

(s)

(cm’)

(cm3)

3.157 2.915 2.397 1.817 1.158 0.412

3.157 3.239 3.425 3.634 3.861 4.123 4.269

2.21 2.05 1.82 1.66 1.51 1.41 1.38

1.55 1.45 1.26 1.07 0.88 0.69 0.60

1.286 1.315 1.383 1.454 1.525 1.593 1.633

1.326 1.352 1.405 1.459 1.510 1.563 1.589

lo-*IN 7 (cme3)

lo-2’(p,

(cm-)) 4.269 3.827 3.397 2.967 2.560 2.126

0.425 0.849 1.271 1.706 2.126

lo-2’p, (cm-‘)

0.324 1.027 1.817 2.702 3.710 4.269

IO-Z’p, s2

190

lO%p

+A$)

lO%o

lo-=Isz

10%

1o*%Y*

(cm-3)

(s)

(g cm*)

(cm3)

(cm’)

4.269 4.252 4.246 4.238 4.266 4.252

1.38 1.25 1.20 1.13 0.89 0.81

0.60 0.54 0.48 0.42 0.36 0.30

1.633 1.508 1.378 1.196 0.926 0.840

1.589 1.503 1.397 1.261 1.090 0.836

eo

es

7.90 8.15 8.75 9.44 10.20 11.05 11.56

8.05 8.30 8.84 9.45 10.13 10.89 11.32

11.56 10.05 8.68 7.26 5.78 5.04

11.32 10.02 8.76 7.50 6.10 4.89

Volume140,number 2

CHEMICAL PHYSICS LETTERS

25 September 1987

Table 2 Parameters corresponding to system S at temperatures of 253 and 273 K. t,, corresponds to the experimental values. The (Yvalues were obtained by means of eq. (2) using e0and em= 2.8. (Y*corresponds to the values obtained by means of eq. (4) using the values given by eqs. (5) and (6). es corresponds to the values obtained using eq. (3) with LY*and e-=2.8 (4-01, 5-ol,... correspond to normal butanol, pentanol, hexanol,...). (I) ForsystemsS:p,=l.018~10*’ cm-3;(r,,=3.135~10-22cm3; T=253K. (II) ForsysternS:p,=l.l85~10~’ cm- ; (~~~2.692~ 10-zzcm3; T=273 K

I

%I

63 IO*%Y(cm3) ff5 10-21p (cm-3) II

63

4 10% (cm’) 1O%Y*( cm3) 10-21p (cm-))

4-01

5-01

6-01

7-01

22.38 25.60 2.289 2.666 6.81

20.39 21.41 2.43 1 2.581 5.76

18.06 18.36 2.444 2.493 4.97

15.60 16.07 2.321 2.408 4.39

20.30 21.43 2.079 2.215 6.70

17.75 17.92 2.102 2.128 5.66

15.61 15.35 2.08 1 2.040 4.90

13.60 13.43 1.985 1.955 4.33

with p near pC.Therefore we conclude that both systems behave in accordance with the Debye model. The values of rr, that correspond to systems S, S, and S2 are plotted in fig. 2 as a function of p. Fig. 2 shows that the values of rD for systems S and Sr are described by a unique function of p. As a consequence, there are mixtures S, with appropriate proportions of the components for which a pair of values (rb, p) similar to either n-nonanol or n-octanol can be assigned.

6E’ 5-

T=293

4

5

6

7

8

9

11.56 11.87 1.802 1.867 3.87

10.75 10.62 1.808 1.780 3.50

K

25 3

9-01

Fig. 2 shows that rb as a function of p has a different behaviour for system Sz than for systems S and S,. Using appropriate component proportions in system Sz, we obtain number densities in this system which are equal to those of n-octanol, n-nonanol and n-decanol but the values of rb that correspond to those mixtures by no means correspond to these alcohols. Nevertheless the values of e. which belong to the

001 2

8-01

10

11

12

E'

Fig. 1. Plots of e” versus c’ for mixtures of the systems S,, S2 and the components n-heptanol and ndecanol of the systems S. I-J system S,, A system S2, 0 n-heptanol C,, 0 n-decanol C,,,. The experimental points correspond to different frequencies between 10 and 250 MHz. -fitting using Marquard’s algorithm.

I

30

I

35 _

I

40

45 p 110-2'(mi3)

Fig. 2. relaxation time ~b determined with eq. (2) and the experimental values of c’ and e”. -.-.specific moment of inertia of the systems S, (0 ), S2 (A ) and S (0) versus number density p. The numbers indicate the alcohol (6 for n-hexanol, 7 for n-heptanol, ...). The concentration x, of n-heptanol in heptane was between 1 and 0.5.

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CHEMICAL PHYSICS LETTERS

4

lS-

6-

/’

30

3

2

I 4

I 5

-21 klli31 p x 10

I 6

Fig. 3. Dielectric permittivity e(O,T) =cO of the systems S, (Ii), Sz ( A ) and S (0 ) versus the number density p. The numbers indicate the alcohols: 4 for n-butanol, for n-pentanol, ... . dielectric permittivity calculated with eq. (3), eq. (4) and e,= 2.8.

components and mixtures of the systems S, S, and Sz plotted in fig. 3 as a function of p show that all of them can be described by a unique function. This result is interesting to analyze. As an example assume ~~3.15~ lO*l cm-3: in system S at 293 K this corresponds to n-decanol and to a certain mixture of the system S2 which will be called S:O, both giving the same value of e. but, according to the experimental data given in fig. 2, the relaxation time of n-decanol is 2.21 x 10e9 s and that of S:” is 1.17~ 10d9 s. As a consequence of eqs. (2) and ( 3)) the real and imaginary components of ndecanol and Si” will be equal only if (wz,,),,_~~~.=(w~T,,~). In other words, the points of the respective Cole-Cole plots are displaced in frequency, where o is the frequency of the external perturbing field. For a static applied field the response of n-decanol must be equal to the response of S:” (identical susceptibilities at zero frequency) which means that the average torque applied to each molecule and the average intermolecular interactions are the same for both systems. The dipole is attached to its own carbon chain and therefore determines a particular moment of inertia. It is interesting to compare the relaxation times for the systems S, and S2 with the moments of inertia defined as follows: System SI: System

192

S2:

IS, = (PIZI

+p212Yh

Zs2=p,Z,I(p, +N) ,

+P2)

,

25 September 1987

in which Z,=O.601 x 1O-36 g cm* and Z2= 1.55~ lo-36 g cm*; N is the number of non-polar molecules per unit volume. The moments of inertia I, and Z2(corresponding to n-heptanol and n-decanol respectively) were calculated with reference to the permanent dipole position because we suppose that among the possible torques acting on the molecule, the most important is the one acting on the permanent dipole which compels the molecule to rotate. The components of the rotations are in the X, Y and 2 directions where the OY direction is parallel to the dipole. Rotations around the OY axis do not contribute to the polarization. Rotations around the 02 axis can be neglected because the corresponding moment of inertia is very small (two orders of magnitude less than the remaining) which implies very high frequencies, greater than the critical frequency of the first zone. In consequence I, and I2 represent the moments of inertia of the carbon chain rotating around the OX axis, perpendicular to the permanent dipole and to the carbon chain. The geometry used for the moment of inertia calculations is [ 71: rc_c = 1.54 A )

rc_,=1.09A,

rc_o = 1.43 A,

rO_H= 0.97 A ,

LHCH= 109.1’ ,

LCOH= 107.3” .

Is, and Is1 can be interpreted as “specific moments of inertia”. Obviously non-polar molecules do not contribute to Is*. From fig. 2 it is evident that a relationship exists between rD and Is which implies that theories of polar dielectrics must include in calculations of the orientational susceptibility and polarizability the “specific moment of inertia” of the system. The polarizability (Y=x( O,T)/p is an average property assigned to each molecule by the whole, so it represents a collective characteristic of the system. Values of (Ywere calculated from eq. (3) and plotted as a function of p in tig. 4 at different temperatures. Fig. 4 shows that, as a function of p, (Ybehaves like B, of eq. (1). So rX=cGJ(l-Z&/p) .

(4)

CHEMICAL PHYSICS LETTERS

Volume 140, number 2

25 September 1987

ceptibility is therefore

3.0

X(W) =pa=p,ao(l

-PJPI)

.

The values of a0 and pC were calculated by multiparametric fitting based on Marquard’s algorithm [ 41 assuming that a! obeys eq. (4) at different temperatures. The values obtained for a0 and pCwere

Eq. (4) shows that forp=p,, cr=O. The first relaxzone disappears with eo=eoo. This condition could be attained in two ways: (i) In system SZ, with polar-non-polar components, by simply varying the numbers x, and x2 (x, +x2 = 1); (ii) raising the temperature, because according to eq. (6), pc increases with T*, until it reaches a value such that pc=p. In both ways the system experiences something like a phase transition in which cr plays the role of an order parameter. The behaviour of rD, similar to that of the specific moment of inertia I,, implies that molecular rotations are coupled. For p
cro=.4T-*=2.007x10-“2’-*,

(5)

~=~o[l-P(T-Tndl,

p,=AT*

(6)

ation

0

12

3

4

5 p"

6 l$'

7

0

9

ki33)

Fig. 4. Orientational polarizability (Yversus number density p at different temperatures T. 0 system S, 0 system S,. A system S2, The experimental data of the system S for the temperatures 333, 3 13 and 293 were taken from ref. [ 21. -values of (Ycalculated with eq. (4) using the values given by eqs. ( 5 ) and (6).

= 1.59x 1016T2 .

pC is the same critical number density used in both eq. (1) and eq. (4). The numbers A and A emphasize the universal character of eq. (4) which, together with T and the number density p, is able to describe the systems S, S , and S2 over a wide range of temperatures. Thus, the specific Kerr constant B, and the orientational polarizability (Yare related through a constant y such that B,= ya. In system S, the number density p=pI +p2 is the sum of the number densities corresponding to n-heptanol and n-decanol with pI=xIp, and p2=x2pIo where X, +x2= 1. The susceptibility for the system S, according to eqs. (3) and (4) is x(OJ) =~a=(/+

+PZ~O[

1 -PJ(PI

+p2)1

;

in system S2 we have p=x,p,=p, because the nonpolar component does not contribute and the sus-

(7)

where p. is the number density at the melting temperature T,,,(K), /I( K- ’ ) is a constant which is a characteristic number for each alcohol. Using eq. (7) in eq. (4) and eq. (6) for pc we obtain for the susceptibility x=aop-cu,p,=AT-*po[l-8(T-T,)]-Al,

which can be written as x=X,T-*-X~T-~

-C,

(8)

where X,=Ap0(l+/3Tm), X2=&Q and C=aOpc =0.319. Eq. (8) shows that there is a temperature T, for which x = 0. This implies that the cooperative molecular association disappears at such a temperature. Takingpo=3.2x lo*’ cm-3,j?=9.64x10-4 K-l and T,,, = 280 K as estimated values for n-decanol, eq. (8) gives T, of the order of 400 K which is less than the boiling point T, = 502 K. The more important temperature dependence is given by the T -* term in eq. (8). This represents a discrepancy with the Debye [ 51 model which is 193

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CHEMICAL PHYSICS LETTERS

explicitly T -’ dependent. A comparison with the theories of Onsager [8] and Kirkwood [9] is not ’ possible because their expressions (28 and 2 1 in the respective references) do not give an explicit dependence of x on temperature. Buckingham [ lo] derived from a statisticalmechanical theory an expression for the molecular Kerr constant which is applicable, at any density, to polar molecules. The terms in T -’ are dominant and depend on the molecular dipole moment; the remaining terms that depend on T - ’ are usually negligible. These are results which must be considered for a better understanding of CL

Acknowledgement We thank Professor A.D. Buckingham for comments and suggestions.

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References [ 1] H. Mechetti and E. Zakowicz, Chem. Phys. Letters 128 (1986) 563. [2] SK. Garg and C.P. Smyth, J. Phys. Chem. 69 (1965) 1294. [ 31 N.L. Federigi and G.H. Barbenza, J. Phys. E7 (1974) 719. [4] D.J. Marquard, Sot. Ind. Appl. Math. 11 (1963) 431. [5] P. Debye, Polar molecules (Chem. Cat. Co., New York, 1929). [ 61 K.S. Cole and R.H. Cole, J. Chem. Phys. 20 (1952) 1839. [ 71 Tables of interatomic distances and configuration in molecules and ions, special publication No. 11 (Chem. Sot., London, 1958) M 114 and M 138. [S] L. Onsager, J. Am. Chem. Sot. 58 (1936) 1486. [ 91 J.G. Kirkwood, J. Chem. Phys. 7 (1939) 9 11. [lo] A.D. Buckingham, Proc. Phys. Sot. A68 (1955) 910.