6142
J . Phys. Chem. 1991, 95, 6142-6148
Dielectric Relaxation and Libration Spectroscopy of Some Aliphatic Ketones and Their Molecular Behavior J. K. Vij* and F. Hufnagelt Department of Microelectronics and Electrical Engineering, Trinity College, Dublin 2, Ireland, and Institut f i r Physik, Johannes Gutenberg Universitdt, 6500 Mainz, Federal Republic of Germany (Received: February 27, 1990; In Final Form: April 23, 1991)
The dielectric loss in the frequency range 2-150 GHz and the power absorption coefficient from 300 GHz to 6 THz have been measured in dilute solutions of cyclohexane at 20 OC for acetone, hexanone-2, heptanone-2, heptanone-4, nonanone-5, and undecanoned. Power absorption coefficient for neat acetone and undecanone-6 have also been measured. The results are analyzed in terms of the relaxation and libration processes of molecules. The integrated intensities for diluted and undiluted acetone are higher than calculated by using Gordon's formula by 57% and 90%. respectively. These are interpreted in terms of an enhancement of the effective dipole moment of acetone in its dynamic liquid structure. From extreme dilution of acetone to its neat liquid, the libration frequency increases from 24 to 66 cm-I, whereas the half-power points bandwidth increases similarly from 86 to 115 cm-I. A similar though not so dramatic effect is observed for undecanone-6; the libration frequency increases slightly from 75 to 80 cm-' for diluted to undiluted solutions. The results show that acetone behaves as a rigid molecule and relaxes predominantly through a tumbling motion of the asymmetric top. The same holds for undecanone-6 but to a lesser degree, whereas for the intermediate members of the family, there is an increased contribution from the intramolecular rotations to the relaxation and libration processes.
1. Introduction The zero to tetrahertz frequency spectra of a number of rigid polar molecules in neat liquids and dilute solutions have been studied by a number of investigator^.'-^ Several interesting aspects of the dynamics of the molecules have been revealed from the instant of time of a fraction of a picosecond to several minutes. This has given rise to a considerable impetus toward the development of computer simulation of liquids. The prediction of an orientational correlation among the dipoles of the symmetric top moleculesc6 such as methyl chloride and methyl fluoride in the condensed phase by computer simulation has been verified experimentally.' Klages&'O has recently reported dielectric spectroscopic studies on a number of nonrigid aromatic polar molecules in dilute solutions of inert solvents. The results obtained for the molecular, intramolecular relaxation, and resonance processes have not only given a valuable insight into the dynamics of nonrigid polar molecules but have also helped in clarifying the nature of the relaxation and the resonance phenomena in polar liquids. The far-infrared spectrum of diphenyl ether," a nonrigid polar molecule, has provided us with an answer to a longstanding problem for a mechanism that is responsible for an anomalously small dielectric relaxation time. A comprehensive dielectric spectroscopic study is lacking for aliphatic polar nonrigid molecules and especially in providing a comparison between the extremely dilute solution and the neat liquid. The intensity and characteristics of intermolecular dipole-dipole interaction can be studied effectively when spectral comparisons are effected between these two. Crossley'2 has reporteddielectric spectra for some aliphatic ketones for a few spot frequencies up to 145 GHz. The availability of an extensive range of microwave, millimeter, and submillimeter wave equipment13has prompted us to undertake a systematic study of aliphatic ketones in dilute solutions of cyclohexane and two of their representatives: acetone and undecanone-6 in their neat liquids. Owing to their large dipole moments (a2.7 D), ketones are also ideal liquids for investigating the effect of the dipole-dipole coupling on the far-infrared spectral band shape. Acetone is somewhat bigger than the globular molecules such as CH3CI and CH3F but nevertheless is also nearly a symmetric top molecule. The moments of inertia of acetone calculated from first principles are found to be I , = 77.5 X lo4, I2 = 77.8 X IO4, and I3 = 155.3 X IO4 g cm2; II is about the long axis, taken to be x whom correspondence should be directed at Trinity College. '*To Johannes Gutenberg UnivcrsitBt.
direction, and I, is about the direction of the dipole moment, taken to be y direction. A comparison of the dilute solution spectra with that of its neat liquid may also throw some light onto whether highly polar molecules of acetone tend to aggregate into microclusters of various sizes. In the wavenumber range 220-340 cm-', a broadband far-infrared absorption has already been shown for these ketones,14 except for acetone, which is found to be almost transparent in this range. In this paper we therefore limit ourselves to the results obtained for a study of the absorption spectra up to 200 cm-I. 2. Experimental Section
The dielectric loss c"(w) for acetone, hexanone-2, heptanone-2, heptanone-4, nonanone-5, and undecanoned in dilute solutions of cyclohexane was measured in the frequency range 2-1 50 GHz with a combination of techniques.I3 For each polar sample, five dilute solutions with varying in the mole fractionf2 from 0.2 to 1.5% were prepared. For frequencies above 150 GHz, power absorption coefficient A(27rcs) was measured with a molecular laser'* as a source and employing a two-chamber differential ceI1.l6 The spot wavelengths given in micrometers for which the measurements were made are 1258.3 (238.4 GHz), 447.1 (671.0 GHz), 250.8 (1.196 THz), 191.5 (1.567 THz), 118.8 (2.525 THz), ( I ) Evans, M. W.; Evans, G.J.; Coffey, W. T.; Grigolini, P. Molecular dynamics and theory of broadband spectroscopy; Wiley Interscience: New York, 1982. (2) Reid, C. J.; Evans, M. W. J . Chem. Phys. 1982, 76, 2576. (3) Jain, S. R.; Walker, S. J . Phys. Chem. 1971, 75, 2949. (4) Neumann, M.; Steinhauser, 0.;Pawley, G. S. Mol. Phys. 1984,52,
97. (5) Hesset-Bczot, C.; Bossis, G.; Brot, C. J . Chem. Phys. 1984,80, 3389. (6) Brot, C.; Gerschel, A. Z . Phys. D 1987, 5 , 367. (7) Gerschel, A.; Grffihulski, T.; Kisiel, Z.; Pszczolkowski, A,; Leibler, K. Mol. Phys. 1985, 54, 97. (8) Klages, G.;Wieczorek, E. Z . Naturforsch. 1982, 390. 101, 113. (9) Klages, G. Z . Naturforsch. 1985, 40a, 1206. (10) Klages, G. Z . Narurforsch. 1988, 430, 1. (1 I ) Reid, C. J.; Vij, J . K. J . Chem. SOC.,Faraday Tram. 2 1982, 78, 1649. (12) Crwley, J. J . Chcm. Phys. 1972.56.2549. (1 3) Vij, J. K.; Hufnagel, F. In Dynamical processes in condensed matter, Evans, M . W., Ed.;Wiley: New York, 1984. Also Adu. Chem. Phys. 1985, 63. 751. (14) V$ J. K.; Hufnagel, F. Chcm. Phys. ktf.1987, 139, 77. (15) VI], J . K.; Hufnagel. F.; Helker, M.; Reid, C. J. IEEE J . Quanfum Elecfron 1986, QE-22- - -, 1121 .(16) Vij, J. K.; Hufnagel, F. J . Phys. E Sci. Instrum. 1989, 22, 749.
__.
~
Q 1991 American Chemical Society
The Journal of Physical Chemistry, Vol. 95, No. 16, 1991 6143
Aliphatic Ketones and Their Molecular Behavior 96.52 (3.108 THz), and 70.5 pm (4.255 THz). These measurements were combined with those made by using a FTS l 13C Bruker spectrometer in the spectral range 80-250 cm-I. Static dielectric permittivity and refractive index for the sodium D line was also measured for these solutions. A ( o ) measurements in the FIR band of frequencieswere also made for neat acetone and neat undecanone-6. Measurements for acetone were combined with the static permittivity and dielectric loss in the radio and microwave band of frequencies from the literature.” Static dielectric permittivity, refractive index at the sodium D line, dielectric loss, and the power absorption coefficients of solutions were related to the mole fraction, f2, by the following equations, which are valid for dilute solutions only:
+ (1
-f2)em1vcnt(O)
(1)
( l -f2)(nD2)soIm1
(2)
+ (1 -/,)e”so~vcnt
(3)
Asode) = /,Asolutc(p) + (1 -/,)AsoIvcnt(P)
(4)
emdo)
/,~so~utc(O)
f22(nD2)solu~e +
(nD2)dn
€”sodo)= /,~”solutc(o)
-
For/, say up to 0.015, 1 -f2= 1 , and hence eqs 1-4 in the limit
h
0 simplify to
where [ = 8ra3q,a is the radius of the molecule and q is the viscosity. The time TJ
where UF is the volume fraction. In the limit offi to/, by the following linear equation:
-.
0, OF is related
a(r) is the angular velocity vector at time r and the brackets denote the ensemble average at equilibrium. We note that TJ is inversely proportional to [,wherease T D is directly proportional to 4. For the reference dipole, the rotational kinetic energy can be written by using the Boltzmann expression 72kT On combining (1 1) and (1 2), we get TJ = I/2r~kT On use of (14), T J becomes %I( Q2(Q )o
a,
-
3. Tbeoreticrl Description Following Rocard,I* Powled9 incorporated inertial effects into the Langevin equation for a spherical molecule and has derived the following equation for the complex polarizability a ( w ) : 1
ab)
-3
4s)
1
+ ~ U T D-
T,T&
= ~/(~TD(Q’(O))O)
(14) (15) (15 4
Equations 10-15 were obtained on the basis of the rotational diffusion model in the limit of large [(TJ 1. These processes may be considered to correspond to the molecular and intramolecular relaxations of nonrigid molecules or as discussed in section 5.1. The third process is resonance or librational in character since here T D , / T J ~is of the order of unity or less. For i = 3, the denominator of eq 18 is a minimum for f = fo = w0/27r such that
5i c m l 1
18
1
188
18
-
7D37J3
= 1/w02
FIHz Figure 3. Same as Figure 1 for heptanone-2.
(19)
and the power absorption coefficient A(w) =
nc W€”(U)
is a maximum; n is the refractive index. The librational frequency is interpreted in terms offo = w o / 2 r for which A(@)is a maximum since it is a resonance phenomenon. Equation 18 indicates that forf>>fo, c”,,,,,(w) falls off as l/f (provided the first term is neglected). If we needed to increase the rate with which c”(w) falls off with frequency, the third term in (1 8) is replaced by the following: c3wTD~
(1 - w2rD17J,)2 + w2(7Dl+ 7M[1 - drD17J3])2
(20)
In (20), T~ is the torque correlation time. Since the accuracy of the fit with an additional time constant is poor, this is not being attempted here. Equation 20 predicts that forf >> fo = (7D37J3)’/2,e”(@) falls off as 1/y. We can also write TD3
= rIw0
TJ1 =
1/w0r
where I’ is the dimensionless damping constant such that
r = (TD~/TJ,)’/~
(21) One can show that r = half-width/fo of the librational band, from which T~~ and 7 J 3can initially be estimated. The ratios of TDJTJ,and the amplitudes of the molecular processes, Ci, for i = 2 and 3 can, in principle, yield information about the molecular behavior. This approach though semiempirical is very useful in understanding the relative behavior of the ketone type molecules where the length of the chain and the position of the polar group relative to the chain are being altered. For rigid and semispherical molecules, the collisions are discrete, T J is well defined and can be expected to have a longer correlation time than for the semiflexible rod-shaped molecules. The collisions between the latter type. may be too frequent and be rather soft.
f1Hz Figure 4. Same as Figure 1 for heptanone-4.
4. Results
Experimental At”(w)/fi values at the microwave and far-infrared band of frequencies for the five ketones are shown in Figures 1-5. These are marked by full circles. Each figure shows three other curves: the Debye curve and curves for i = 2 and i = 3 given by eq 18. An addition of these curves with their appropriate weights is shown to provide a reasonable fit to the experimental data for each case. The various parameters of the fit are shown in Table Ia and their ratios of the characteristic times are given in Table Ib. Parameters US and uD,defined by ( 5 ) and (6), and the measured refractive indices for the six ketones are listed in Table 11. The experimental dipole moments for these ketones are calculated by using the Cohen Henriquez equationUand given in Table 11:
(23) Cohen Henriqucz, P.Thesis, Delft, 1935.
The Journal of Physical Chemistry, Vol. 95, No. 16, 1991 6145
Aliphatic Ketones and Their Molecular Behavior TABLE I
material acetone hexanone-2 heptanone-2 heptanone-4 undecanone-6
(a) Relaxation Times and Their Amplitudes or Relaxation Strengths CAos - am) amp, % ~ q / p WS - 0-1 amp, % r D 3 / p 0.72 8.9 1.09 0.101 7.2 89.3 0.042 2.6, 0.047 5.72 23.5 71.6 0.146 1.88 i 0.03 3.62 0.132 5.29 70.7 0.161 1.468 19.6 0.115 4.87 68.0 22.7 3.72 0.264 1.628 5.05 0.046 3.97 52.2 0.142 3.243 42.6
TD~/PS
* **
3.83 0.19 7.28 & 0.27 11.36 0.51 8.51 0.42 19.78 0.60
-
CAUS 0.) 0.141 0.391 0.725 0.665 0.400
TJ,/P
0.167 0.075 0.066 0.033 0.046
amp, % 1.8
4.9 9.7 9.3 5.3
(b) Ratios of T n , / T j , and T n , / T I , and the Experimental and Calculated Values of the Dispersion Step material acetone hexanone-2 heptanone-2 heptanone-4 undecanone-6
~DI/~JI
TDJTJ~
10.4 52.8 27.4 32.0 101.0
(0s
-4
(0s
x p
7.28 7.78 7.62 7.35 7.43
0.2 2.0 2.4 8.0 3.0
- om)ak' 8.06 1.99 7.48 7.16 7.61
+ + C3 = 1
#(o, - a,) is calculated on assuming C, C2
TABLE 11: Parameters as, a h and nom and Dipole Moments for Ketoms in Dilute Solutions of Cyclohexane
dipolar molecule acetone hexanone-2 heptanone-2 heptanone-4 nonanone-5 undecanone-6
os
OD
n~~
7.10 7.65 7.46 7.21 7.35 7.22
-0,180
-0.125 -0,160 -0.135 -0.103 -0.205
1.3595 1.4012 1.4088 1.4071 1.4198 1.4283
~lcxplD 2.65 2.73 2.71 2.66 2.68 2.67
20 -
EO,'
F1it.P
2.75 2.62 2.6 2.7 2.63
16/
I
'
/
12-
/
1
$84- A /
- 00
2 4 6 NUMBER OF (CHJ GROUPS
F i 6. TD, in picoseconds vs maximum number of methyl group from ketone group. Points indicate as follows: A, acetone; B, hexanone-2; C, heptanone-2; D, heptanone-4; E, undecanone-6.
E o
I /
a'
m
u+ss
m10
lltll
lSCl2
C
1KC13
FIHz Figure 5. Same as Figure 1 for undecanone-6.
The physical constants, el, p l , and M Irelate to those for the solvent-cyclohexane in this case. In the literature, (22) is also known as Higasi's equation.24
5. Discussion 5.1. Dielectric Relaxation and Molecular Behavior. Dielectric relaxation T D ~for the systems studied is the longest time for the dipolar orientation. For a given value of I, the corresponding angular velocity correlation time is negligible since is the largest for this process. From Table Ia we find that the amplitude or the relative strength of this process varies from 9% for acetone to 43% for undecanone-6. For all cases this is lower than for T D . We observe that the ratio T D J T J ~ (Table Ib) lies in the range l b I T D J T J ~I101 for the five liquids analyzed. This shows that the process involving T D like that for 7~~ can be interpreted to be a Debye orientationaf one. The dipole moment, which arises mainly from the c-0double bond, acts with a greater probability at an angle of an almost 90° 25 to the axis of the zigzag carbon chain of the molecule when extended. For acetone, the dipole moment direction bisects the C-C-C angle. The rotation around the dipolar axis evidently does not produce any change in the ~~~
(24) Higari. K. Denki Kcnkyusho Iho 1952.4, 231. ( 2 5 ) Smyth, C. P. Dielccrrlc Behaulour ond Structure; McGraw-Hill: New York, 1955.
/
e*'
2 4 6 NUMBER OF METHYL GROUPS Figure 7. Same as Figure 6 for
7D2.
direction of the polarization. The other two possible degrees of freedom, one around the molecular axis and the other normal to the plane containing the molecular axis and the dipole moment, can also be represented by T D and ~ T D , , respectively. The former motion is faster as it is relatively less restricted and involves a sweeping of lower volume of the liquid than the latter. In addition the former is supplemented by a small contribution from the internal rotations of the parts of the molecule involving the dipole moment in cases where the steric hindrance is not too large. This would be supported by the observed higher relaxation strength C2 for T D than ~ C1 for T D . The former relaxation strength for acetone is the highest 89.3k (Table Ia), whereas for undecanoned it is the lowest, being 52.5% where the internal rotations are relatively severely sterically hindered. The Debye time T D , of 7.28 ps for hexanone-2 is approximately the same as found by Crossley,12 7.2 ps. However, our proposed model for the relaxation mechanism differs somewhat from him. He suggests that for the molecules such as hexanone-2, the main relaxation mechanism is through the intramolecular rotation of the terminal acetyl group. We feel that such a rotation will contribute only partly to 7 ~ In~ addition, . T D corresponds ~ to an overall molecular rotation about an axis at right angles to the plane containing the molecule. We find that T D , (Figure 6) lengthens
6146 The Journal of Physical Chemistry, Vol. 95, No. 16, 1991
Vij and Hufnagel
TABLE III: Fnqwmics of Maximum Dielectric Loas, Power Absorption, Maximum Absorption Coefficient, and tbe Effective Radius (r,,,)
dipolar molecule acetone hexanone-2 heptanone-2 heptanone-4 nonanone-5
fmx/GHz 1 IO
33.5
3.7 4.7s
36 24 16
6.63 9.9s
43
undecanone-6
%O/P 1.45
dilute solution s,/cm-l 24 f 2
(AA,/uF)/Neper 220 f 5
cm-l
so/cm-l
neat liquid A,,/Nepcr cm-l
68 f 4
207 f 10
(ren)/A 2.22
55 f 2 60 f 2 52 f 2
4.42
7s dz 2
Figure 9. Same as Figure 8 for hexanone-2 in cyclohexane. 0,spot frequency measurement; 0 , FTS measurement. iicm-1
Figure 8. Power absorption coefficient A(s)/Neper cm-l vs s/cm-I for /U u for Facetone in cyclohexane,dotted line shows acetone. (1) U ( P )vs the Debye plateau. 0,spot frequency measurements completed by FTS measurement. (2) A(P)vs B for neat acetone,dotted line shows the Debye
plateau. 0,spot frequency measurement.
0 , FTS
measurement.
considerably, whereas TD also does increase somewhat (Figure 7) with an increase in the length of the chain. The slope for these curves are found to depend on the position of the polar group in the chain. For this to happen, the molecule must rotate about the major and minor axes of the ellipsoid in addition to the intramolecular rotations envisaged above. On the basii of our results, of the relaxation strengths (Table IA), we p r o p that a relaxation mechanism in smaller ketone molecules is predominantly via the tumbling motion of the ketone group, i.e., rotation around the major or molecular axis. The relaxation spectrum (Figure 1) reflects an almost rigid structure for the smallest aliphatic ketone molecule-acetone. A phenomenological relation% between the dielectric relaxation time and the effective molecular radius is used to determine the effective radius (rcff)of the molecules: TDo
= 70 exp(o(rcff))
Figure 10. Same as Figure 8 for heptanone-2 in cyclohexane. 0,spot frequency measurement; 0 , FTS measurement. Dotted line provides an extension of the experimental data.
t
(23)
is a prefactor time constant and u is a structural factor de~ determined pendent on the solvent and the temperature. T D is from the relation W , T ~ = 1, where w, is the angular frequency for which e'' is maximum. Based on the rDolisted in Table 111, the effective radii of the molecules are calculated and given in Table 111. For cyclohexane solvent, T~ and uZ6are taken to be 0.087 ps and 1.27 A, respectively. On the basis of the space-filling models for these molecules, the results suggest that for smaller ketones the various segments of the molecules coil up together to form a ball-like structure. However, with an increase in the number of the methyl groups on each side of the ketone group, the flexibility of the molecular segments increases, giving rise to an increase in the sponginess of the molecular shape. Table Ib shows that 7Da/7JI lies in the range 0.2 IT? /iJ3 I 8. Such an observation is interpreted to imply that the libration time ID is affected strongly by the kind of molecules involved in molecular collisions. This is reflected in the angular velocity , strongly on the correlation time T j 1 . We find that I ~ depends T~
(26) Hufnagel, F. Z . Narurforsch. 1970, 25A, 1143.
.'. 40
80
120
160
i 0
Figure 11. Same as Figure 8 for heptanone-4 in cyclohexane. 0,spot frequency; 0 , FTS measurements. Dotted line is drawn after having subtracted for the higher frequency mode.
molecules under study. For acetone T J , is the highest, and for heptanone-4 it is the lowest. 7J, for undecanone-6 is slightly higher than for heptanone-4. For the reasons stated in section 3, 7J is interpreted in terms of the type of molecule. The more rigid the molecule, the longer the time for which the angular velocity remains preserved. This results in accordance with that expected from the molecular considerations. We find that the dipole moments (Table 11) for the six ketones agree with the l i t e r a t ~ r ewhere * ~ ~ ~known. ~ These lie within 2% (27) McClellan, A. Tables of Experimental Dipole Moments; W. H. Freeman: San Francisco, 1963.
The Journal of Physical Chemistry, Vol. 95, No. 16, 1991 6147
Aliphatic Ketones and Their Molecular Behavior TABLE I V obscned Integrated Absorption Intensities liquid JA(v)/iJ dv acetone (neat) acetone (cyclohexane) hexanone-2 (cyclohexane) heptanone-2 (cyclohexane) heptanone-4 (cyclohexane) undccanone-6 (cyclohexane) undecanone (neat)
(IA)ap = SA@) ds/103 cm-2 . ~ A ( u ) $dP/10’ cm-‘ 24.28 19.66 20.21 11.01 11.18 9.18 8.30 4.73 7.85 6.97 5.42 5.26 7.00 7.60
36.70 54.0 30.48 21.19 19.93 9.22 6.40
-...\\
.\ 1.
40
80
120
160
-‘*.
200
ilo”
Figure 12. Same as Figure 8 for undecanone-6. 0,spot frequency and
FTS measurements for its dilute solution in cyclohexane. A, FTS measurements for neat undecanone-6. of the average value, 2.68 D (Table 11). The percentage change is approximately twice the experimental error. 5.2 Power AbsorptiOn specba The power absorption spectrum shows the details of the high-frequency loss for ketones except for nonanone-5, which is not discussed here. These spectra are shown in Figures 8-12. Figure 8 shows the spectrum of neat acetone and the normalized dilute solution spectrum in cyclohexane extrapolated to full concentration. We note two important features of the results: first the frequency of maximum absorption for acetone rises from 24 cm-’ for its dilute solution to 66 cm-’ for neat liquid; second, the plot of A(r) for neat acetone is broader than its dilute solution counterpart. The librational frequency is associated with the frequency of maximum power absorption. This follows from the nature of the absorption arising from librations: it is resonance in character, as opposed to the Debye relaxation being Lorentzian. The torsional frequency of librations in harmonic approximationBis given by Qo = (VO/Zr)’/2
cm-2
12.87 12.87 5.36
further dramatic increase in the libration frequency of undecanone-6 from hexanone-2 is observed. These observations are interpreted in terms of an increase in the steric hindrance to rotations and an increase in the molecular flexibility. The bandwidtbJAB) for the dilute solution spectrum of heptanone4 (140 cm-l): is much greater than for any other member of the ketone family studied here (AB for acetone = 86 cm-I). 5.3. Debye Plateau and the Poley Absorption. The magnitude of the absorption coefficient for the Debye plateau of the polar liquid (DPL) is given by the formula
I
’
I&/103
(24)
where 4V0 is the height of the potential barrier in which a dipole is situated and I, is the reduced moment of inertia of a molecule, I;’ = (II + 13)/I,13. The dipole moment is along the direction labeled as 2. An observed increase in the frequency of maximum absorption predicts an almost 8-fold increase in the height of the potential barrier in going from the dilute solution to the neat liquid state of acetone. The dipoltdipole coupling between the molecules in the polar liquid and the intermolecular steric effects seem to be the dominant contributing factors giving rise to an increase in the height of the potential barrier. Vois higher in the neat liquid due to the larger dipolar torques than for dilute solutions. The increase in the bandwidth of the absorption curve for neat acetone from its dilute solution is reflected in an increase of the damping coefficient r (since r = half-widthlf,); this in turn affects the ratio TD,/TJ, using eq 21. The spectra in Figures 9-12 show that the resonant behavior of the larger ketones is more complicated than for acetone. A (28) Marcheroni, F.; Vij, J. K.; Coffey, W. T. Z . Phys. B 1985,6/,357. (29) Coffey, W. T.;Corcoran, P.M.; Vij, J. K. Chem. f h y s . Lrr.1986, 129,375. (30) Vij, J. K.; Grochulski. T.; Kocot, A,; Hufnagel, F. Mol. fhys. 1991, 2, 353.
For acetone, es = 21.20, e, = 2.765, n = 1.3600, TDO= 3.22 ps, and (A)DPL= 140 neper cm-’, shown in Figure 8. The corresponding formula for dipolar solution (DPS) extrapolated to full volume fraction can be given as
where a, zz aDand n is the refractive index of cyclohexane for the sodium D line. For as = 7.10, OD = 4.180, nD = 1.4259 (for cyclohexane), ( ~= 1.45 4 ps, ~( A ) Dfor~ acetone solution = 172.3 neper cm-I; this IS also shown in Figure 8. We recall that the parameters as and a, are determined with respect to the mole fraction. (If these were determined with respect to volume fraction, then the factor M 1 p 2 / M f l Iwould have been replaced by unity). For dilute solutions, we observe that (i) the effect of the resonance absorption (Poley absorption) over and above the Debye plateau appears to be relatively small compared to that for the neat liquid. (ii) The half-power points bandwidth for dilute solution (86 an-’) is lower than for undiluted acetone (1 16 cm-’) by 34%. We suggest that both the magnitude and the bandwidth of the absorption spectrum increase with the dipole-dipole coupling and the intermolecular steric hindrance effected by the neighboring molecules. 5.4. Spectral Moments. Areas under the absorption curves SA(r)/r2 dv, SA@) dr, and SA(s)s2dr, which are measures for the zero, second, and fourth spectral moments, respectively, are listed in Table IV. 5.4.1. Second Spectral Moment. The second moment is expressed in terms of the molecular quantities by the Gordon sum rule,” which leads to the integrated intensity called the Gordon integrated intensity IAo:
In cgs units to is replaced by 1 / 4 ~ ,N is the number of molecules per unit volume; N = NAp2/M2.p2 is the density of the solute, and NA is Avogadro’s number. For the liquid phase, the expression on the right-hand side is usually multiplied by the Polo and Wilson3*internal field correction factor, Apw:
=
? a(,
+ 2)2
(28) 9nOpt where noptis the refractive index at a frequency above the farinfrared band. The correction factor arises from the polarizability APW
(31) Gordon, R. G. J. Chem. fhys. 1963, 38, 1724. (32) Polo, S. R.; Wilson, M. K. J . Chem. fhys. 1955,23,2376.
Vij and Hufnagel
6148 The Journal of Physical Chemistry, Vol. 95, No. 16, 1991
TABLE V: Comprisoo of Correction Factors exutl corriction liquid factor A,, acetone (cyclohexane) 1.57 acetone (neat) 1.90
Polo-Wilson correction factor Apw 1.27 1.21
Kirkwood correlation factor g
eq 28 and (d2(0)) is the mean-square angular acceleration such that Z:(02(0)) is the mean-square torque. G is given by
G=
1.o
1.38
(31)
of the molecules in the liquid phase. For a dilute solution, the normalized experimental integrated intensity (IA)cxpis defined as follows:
U ( s ) is the increment in the absorption coefficient of the solution over the solvent and uF is the volume fraction of the solute. We find (IA)c,p for a diluted acetone is 57% greater than that calculated by using Gordon's formula (Table IV). Polo and Wilson32 internal field correction factor listed in Table V is insufficient to account for the observed discrepancy between the measured and calculated values. Furthermore the measured integrated intensity for the undiluted acetone is almost twice that of the calculated value. To quantify these results, we introduce the experimental correction factor, Acx , with which the calculated value needs to be multiplied to mat& it with the experimental value. kXp and Apw for acetone are listed in Table V. The discrepancy between Polo-Wilson and the experimental correction factors suggests that previously the internal field theory needs to be modified. discovered a similar discrepancy for acetonitrile, and he accounted for it by assuming an anisotropy in the polarizability of the molecule. The dipoldipole interactions however are suggested to account mainly for a difference in the correction factors observed between the diluted and undiluted acetone. This is also reflected in Kirkwood correlation factor, g. We find that (IA), for diluted hexanone-2 (Table IV)is almost twice that of the calculated value. The moments of inertia of hexanone-2 are calculated on the assumption of it being a rigid molecule: they are II = 90.0 X lo4, I2 = 827.0 X lo4, and I3 = 2034 X 10" g cm2, with I, = 78.9 X lO*g cm-2 (I;! = ZI-l + Iyl 1 ~ ~Since ) . the molecule is nonrigid, the effective reduced moment of inertia could be a great deal lower than the above value. This should increase l A o with a closer agreement between the measured and calculated values. Figure 12 shows the plot of the absorption coefficient versus v for diluted and undiluted (neat) undecanone-6. The results of frequency of maximum absorption coefficient vo and the value of this absorption coefficient A,, are listed in Table 111. Table IV lists the integrated absorption intensities. We find that unlike acetone, so does not change very much with concentration. We can explain it by assuming that steric hindrance to the rotations of the dipole changes only a little with dilution. The difference in the integrated intensities between diluted and undiluted undecanone-6 is found to be similar to acetone. 5.4.2. Higher Spectral Moments. The higher spectral moments are related to the molecular quantities by Davies et
+
JmA(v)v2 dv =
Nr:
A . P w [ ( W ) ~+) GI
487r2kT8to
(30)
where Apw is the Polo-Wilson internal field correction given by
11,Z2, and I3 are the moments of inertia along the three principle directions of the molecule. The constants are calculated as follows: G = 1.56 X 1050s-" for undiluted and diluted acetone, (G2(0)) = 2.29 X lO5I s4 (undiluted acetone) and (G2(0)) = 7.01 X l P s4 (diluted acetone). These show that the mean-square angular acceleration for neat acetone is 3 times that of its dilute solution value. The effect of dilution on acetone as reflected in (G2(0)). This is found to be similar to that observed in methylene chl0ride.9~ These were calculated as 5.87 X lo5' s-" for neat methylene chloride an4 3.44 X lo5' s4 for its 10% solution in CCl,; G was found to be 8.58 X los0 s-". We find that the mean-square angular acceleration of acetone is lower than dichloromethane by a factor of 2.4. This is possibly due to its higher reduced moment of inertia by a factor of 2.2 and due to a difference in the shape of the molecule compared to that for methylene chloride. 6. Conclusions
The dielectric relaxation studies suggest that smaller ketone molecules (particularly acetone) relax predominantly via the tumbling motion of the symmetric top or the rotation around the molecular axis. For intermediate ketones such as hexanones and heptanones, the results indicate that internal rotations of the parts of the molecule involving the dipole moment contribute increasingly toward the relaxation process. The steric hindrance to internal rotations is higher when the dipole is in the middle of the chain than toward the end. For larger ketones such as undecanone-6, the steric hindrance to the internal rotation increases such that the relaxation mechanism is again mainly via the tumbling motion involving the ketone group. From a comparison of the dilute solution and the neat acetone spectra, we find that the librational frequency increases from 24 to 66 cm-', which is explained by an increase in the barrier height to the librations of the dipole. The increased barrier height may arise from the dipole-dipole coupling and from the intermolecular steric factors. (IA)e, for neat acetone is greater than its dilute solution counterpart %y 20% which seems to arise from the dipole-dipole coupling. This may mean that the effective dipole moment of a dipolar molecule in neat polar liquid is enhanced by the reaction field from the neighboring molecules. For hexanone-2, the experimental integrated intensity is almost twice that of the calculated value; this shows the internal rotations contribute significantly to the absorption due to its nonrigid molecular structure. For undecanone-6, we find that unlike acetone, the frequency of maximum absorption is largely unaffected by dilution.
Acknowledgment. We are grateful to Professor B.K.P. Scaife of Dublin and Dr. Y.P. Kalmykov of the Academy of Sciences, Moscow, for useful discussions. J.K.V. thanks DAAD Germany and Trinity Trust Dublin for financial assistance. Registry No. Acetone, 67-64-1; 2-hexanone, 591-78-6; 2-heptanone, 1 10-43-0; 4-heptanone, 123-19-3;5-monanone, 502-56-7;6-undecanone,
927-49-1. (33) Bossis, G. Physica 1982, llOA, 408. (34) Davies, G. J.; Veerappa, M.; Evans, M.W. Chem. Phys. 1981, 61, 13.
(35) Reid, C. J. Spcctrochim. Acta 1982, 38A, 691.
