1930
The Journal of Physical Chemistry, Vol. 82,
No. 17,
M.-K. Ahn and 2. J. Deriacki
1978
anions rather than MA cations takes place below the low-temperature phase transition. (MA)2Te16yielded the usual curve for the temperature dependence of v 1 frequencies. Although various hexaiodotellurates(1V) studied are known to give rise to intense NQR signals, particularly in the K2PtC16 type cubic phase,17J8only a weak signal was observed for the present complex. With decreasing temperature the v 1 signal gradually weaken below about 180 K and faded out in the noise level at 156 K. Below this temperature, no resonance was detected down to 77 K. On heating, the reverse process could be followed exactly. These facts indicate that (MA),TeI6 maintains its cubic structure down to 156 K. To detect possible phase transition, we carried out DTA measurements and found a heat anomaly ascribable to a first-order phase transition at 119 K as shown in Figure 5. Since the second moment of lH NMR absorptions observed a t 77 K was 5.0 G2,the motion of MA cations in the low-temperature phase is thought to be somewhat restricted as compared with that of the overall rotations in the high-temperature phase. The restricted motion of MA cations may be responsible for the absence of NQR signals below the Ttr. N
Acknowledgment. The authors express their thanks to Professor J. Tanaka and Dr. C. Katayama for their help in carrying out a preliminary X-ray analysis. This work
was partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education.
References and Notes (1) D. Nakamura, R. Ikeda, and M. Kubo, Coord. Chem. Rev., 17, 281 (1975). (2) R. L. Armstrong and H. M. van Driel, Adv. Nucl. QuadrupoleReson., 2, 179 (1975). (3) R. L. Armstrong, J . Magn. Reson., 20, 214 (1975). (4) L. Ramakrishnan, S. Soundararajan, V. S. S. Sastry, and J. Ramakrishna, Coord. Chem. Rev., 22, 123 (1977). (5) Y. Kume, R. Ikeda, and D. Nakamura, J. Magn. Reson., 20, 276 (1975). (6) Y. Kume, R. Ikeda, and D. Nakamura, J . Magn. Reson., submitted for publication. (7) R. W. G. Wyckoff, "Crystal Structures", 2nd ed, Vol. 5,Interscience, New York, N.Y., 1966, p 159. (8) R. Ikeda, Y. Kume, D. Nakamura, Y. Furukawa, and H. Kiriyama, J. Magn. Reson., 24, 9 (1976). (9) D. Nakamura, Y. Kurita, K. Ito, and M. Kubo, J . Am. Chem. SOC., 82, 5783 (1960). (10) W. C. Fernelius, Inorg. Syn., 2, 188 (1946). (1 1) T. 6. Brill and W. A. Welsh, J. Chem. Soc., Dalton Trans.,357 (1973). (12) I.D. Brown, Can. J . Chem., 42, 2758 (1964). (13) R. W. G. Wyckoff, "Crystal Structures", 2nd ed, Vol. 3, Interscience, New York, N.Y., 1965, p 349. (14) Y. Furukawa, H. Kiriyama, R. Ikeda, and D. Nakamura, 4th International Symposium on Nuclear Quadrupole Resonance Spectroscopy, Sept 13-16, Osaka, Japan. (15) E. R. Andrew and C. P. Canepa, J . Magn. Reson., 7 , 429 (1972). (16) G. P. O'Learv and R. G. Wheeler. Phvs. Rev.. 61. 4409 (1970). (17) D. Nakamura, K. Ito, and M. Kubo, J.-Am. Chem.'Soc., 84, 163 (1962). (18) D. Nakamura and M. Kubo, J . Phys. Chem., 68, 2986 (1964).
Translational Diffusion Coefficients of Vanadyl Acetylacetonate in Organic Solvents Myong-Ku Ahn" Department of Chemistry, Indiana State University, Terre Haute, Indiana 47809
and 2. J. Derlacki Diffusion Research Unit, Research School of Physical Sclences, Australian Natlonal University, Canberra, ACT 2600, Australia (Received April 3, 1978) Publication costs assisted by ISU Research Committee
Translational diffuson coefficients D were measured at 25.00 "C for vanadyl acetylacetonate (VOAA) in toluene, chloroform, methanol, ethanol, 1-propanol,and 2-propanol using a constant volume tracer diaphragm technique. Because VOAA is unstable in organic solvents in the presence of dissolved oxygen a method was developed to rigorously exclude air from the system. Kivelson's anisotropic interaction parameter K was reevaluated using the hydrodynamic radii resulting from the D values. For the two propanols K > 1 were observed.
1. Introduction Recently there has been a renewed interest in measuring diffusion coefficients in 1iquids.l This is because the theory of nonequilibrium statistical mechanics can relate the fluid transport properties to molecular motions through a correlation function approachS2 The simplicity in the resulting expression for the particle (self-) diffusion coefficient D not only facilitates the theoretical interpretations of the liquid state but also provides a means of comparing the results of molecular dynamics calculations with experimental measurements of fluid system^.^ Experimental values of D are known to obey the functional dependency on the shear viscosity 7 and temperature T according to the Stokes-Einstein equation4 D = kT/6.rrqro (1) for a probe molecule that is larger than surrounding solvent 0022-3654/78/2082-1930$01 .OO/O
molecules. Thus eq 1 can be used to estimate the hydrodynamic radius, ro, of the probe molecule. Hwang, Kivelson, and Plachy5 reported the first experimental measurement of D for vanadyl acetylacetonate (VOAA) in benzene by McClung.6 They used the resulting value of ro in their estimate of the anisotropic interaction parameter, K , which is defined by the modified Debye equation for the reorientational correlation time, T~~ 72
= 4.rrrO3(7~/3kT)
(2)
Theoretical considerations of Kivelson, Kivelson, and Oppenheim7show that K is related to the ratio of the mean square torque to the mean square force. There is a large number of T~ reported for VOAA in various solvents5~&10because the paramagnetic VOAA molecule gives electron spin resonance spectra with eight hyperfine lines due to the 51Vnuclear spin. The alternating 0 1978 American Chemical Society
Diffusion Coefficients of VOAA in Organic Solvents
The Journal of Physical Chemistty, Vol. 82, No. 17, 1978
2. Experimental Section All solvents were analytic reagent grade material, used directly without further purification, and stored over molecular sieves. Diffusion coefficients are known to be insensitive to small amounts of impurities. The y-active VOAA tracer was prepared from the vanadyl (&V) chloride supplied by the Radiochemical Centre Ltd. of Amersham, England as follows. The carrier-free material was added to 100 mg of nonradioactive vanadyl sulfate and converted to sulfate by evaporating to dryness with dilute sulfuric acid. The residue was dissolved in 2.5 mL of water and 10% excess of acetylacetone was added. A few drops of sodium carbonate solution (3 g in 25 mL of water) were added until effervescence stopped. VOAA precipitated out of solution at this stage. The solid product was washed repeatedly with cold water and with acetone. The wash solution was also saved and VOAA was extracted from it by adding 2 mL of chloroform. The chloroform solution was combined with the solid product and dried with a small stream of nitrogen gas. The blue-green product was recrystallized from acetone. This procedure is similar to the one already described12except that care was taken to keep VOAA under a nitrogen atmosphere at all stages. The values of D for nonradioactive VOAA in tetrahydrofuran, toluene, and 2-propanol were measured by a combined ESR-capillary diffusion method for which details were published e1~ewhere.l~ The radioactive tracer diaphragm technique14was used to measure D in toluene, chloroform, methanol, ethanol, 1-propanol, and 2-propanol as described below. Solid VOAA is stable indefinitely. However, it decomposes slowly when dissolved in liquids in the presence of oxygen. Even when the oxygen is carefully excluded from the system, trace amounts of VOAA were found to decompose on the diaphragm (sintered glass). For this reason diffusion runs were carried out with a uniform concentration of nonradioactive VOAA ranging between and M. In order to exclude oxygen the system shown in Figure 1was used as follows. A weighed amount of VOAA and 170 mL of the solvent were put in the 500 mL flask, F, and degassed on a vacuum line by repeated freeze-thaw cycles. Solid tracer was dissolved in 2 mL of CHCl,; 100 WLwas taken into the centrifugal tube, B, diluted with 2 mL of the solvent, and degassed in a similar manner. The system was assembled in the arrangement shown in Figure 1 and evacuated. The diffusion cell, D, was filled with the nonradioactive solution by raising the vapor pressure in F with a hot-air gun. When a portion of A was filled the Teflon stopcock E closed, F was detached, and the cell was brought to 25 "C in a water bath. A portion of the solution in the top compartment of D was poured into reservoir A. Sometimes it was necessary to lower the vapor pressure in A by pouring a small amount of liquid nitrogen over it. After closing off A the tracer in B was introduced into the top compartment, and the cell was filled again with the solution in A. The diffusion process was considered to start when the system was again put into the constant temperature water bath and the soluton was magnetically stirred. There was a time lag of approximately 2 min between the introduction of the tracer
x n
line widths and line shapes can be readily analyzed to yield T ~ The , ~previous ~ calculations of K have been based on the assumption that ro obtained from D in eq 2 is independent of the ~ o l v e n t .We ~ ~have ~ measured D of VOAA in six organic solvents using a diaphragm technique in order to examine this assumption and in an attempt to understand the transport properties of the probe molecule in these solvents.
A
1931
n
i \ B
U
n
Figure 1. Vacuum filling apparatus for a diaphragm cell. All Jointsare 3 10/18 and stopcocks are greaseless type. Teflon stopcocks C and E fitted with Delrin stems were machined in our lab (DRU): A, 50-mL reservoir; B, 15-mL calibrated tube for the tracer solution: D, 100-mL diaphragm cell with a slntered glass plate; F, 500-mL flask.
TABLE I: Vanadylacetylacetonate
D,b solvent
THF
q,a
CP
toluene
0.460 0.552
chloroform methanol ethanol 1-propanol 2-propanol
0.536 0.545 0.585 1.995 2.081
ro, A (0.860) 3.81 1.010 3.92 (1.01) 1.074 3.79 0.992 4.02 0.532 3.82 0.354 3.09 0.334 3.14 (0.32) cm2/s
A
K
3 . 0 1 ~ 0.50 3.17 0.53 3.66 3.58 3.57 3.76 3.37
0.90 0.84 0.83 1.80 1.24
a Reference 15. Values in parentheses are measured b y the capillary diffusion method and others b y the tracer diaphragm technique. Calculated f r o m the data in r e f 8 except for THF. M. K. Ahn and D. E. Ormond,
J. Phys. Chem., 82, 1635 (1978). and the start of stirring. Counting of the radioactivity of the tracer was carried out by a standard procedure.14 Once the solutions were taken out at the end of the diffusion run no precautions were taken to exclude oxygen. The decomposed product was, however, redissolved by adding 20% alcohol by weight when necessary. The experimental results were reproducible within &570. The temperature of the water bath was maintained at 25.00 f 0.01 "C. The pressure inside the diaphragm cell during a diffusion run was close to the solvent vapor pressure and the completely filled and closed system provided a constant volume diffusion condition. 3. Results and Discussion The experimental values of D are listed in Table I along with 015 and other parameters that are discussed later. The diaphragm diffusion technique requires the cell constant which is obtained by calibrating against a system with a known D value.14 For this purpose we used the capillary diffusion value of D for VOAA in toluene as the standard
1932
M.-K. Ahn and 2. J. Derlacki
The Journal of Physical Chemistry, Vol. 82, No. 17, 1978
reference. The consistency of the cell constant is demonstrated by the fact that the diaphragm technique reproduces the result of the capillary method within 3% for VOAA in 2-propanol as shown in Table I. The errors are estimated to be 3% for the capillary method and 5% for the diaphragm system. The latter value is an order of magnitude larger than the optimum 0.5% error limit of diaphragm t e c h n i q u e ~ ' ~because of the difficulty in handling the unstable free-radical system. The hydrodynamic derivation of the Stokes-Einstein relationship in eq 1 is based on the usual hydrodynamic assumptions of low Reynolds number, continuity, and incompressibility of the fluid.lG The stick-limit boundary condition for a spherical probe particle gives the numerical factor of 6s. The same assumptions and boundary condition yield K = 1 for the hydrodynamic expression of T~ in eq 2." We define the reorientational hydrodynamic radius r' as r f 3 = Kr? (3) Comparison with eq 2 shows that r' corresponds to the hydrodynamic radius of a rotating sphere with stick boundary conditions. The values of ro from D in eq 1are given in Table I along with r'and K obtained from the literature values8 of T~ and using eq 2 and 3. For the nonalcoholic solvents tetrahydrofuran, toluene, and chloroform the values of ro agree well within 3.8 f 0.1 8, while r' and the resulting K vary to a greater extent. An opposite trend is observed for the alcohols, Le., variations of ro are larger than those of r' for the four alchols examined. Therefore the variations in K for these solvents are primarily due to the variations in ro values. Note that T~ determined by ESR is the correlation time corresponding to the reorientational motion of the symmetry axis of a VOAA m ~ l e c u l e . ~ The values of the hydrodynamic parameters for the alcohols suggest the effect of hydrogen bonding ability on the Brownian motion of VOAA molecules, which contain an axial oxygen atom and four equitorially positioned oxygen atoms derived from the acetylacetones. Both methanol and ethanol with the possibility of a higher degree of hydrogen bonded association'* exhibit larger ro and consequent slower translational diffusion rates than the propanols. The vanadyl acetylacetonate is nearly a symmetric top molecule with the X-ray structure of the axial radius rii= 1.95 8, and the perpendicular radius r L = 4.2 A,'' If we assume that the molecular shape may be approximated by the spheroidal shape for which analytic hydrodynamic solutions are available with the stick boundary condition,I6 it is possible to estimate the directionally dependent, spherically equivalent hydrodynamic radii.20 For a spheroid moving along the symmetry axis the equivalent spherical radius, rllo,is given byzo 4 (R2 - 1) (4) rllo = ~r11~---(R - 2)s+ 1 where R = rL/rll (5)
S = (R2 - 1)-ll2tan-l (R2- 1) The above expression for S is valid for an oblate spheroid. When the spheroid is moving along one of the perpendicular axes, the equivalent radius, rLo,is given byzo 8 (R2- 1) r L o= - r (6) 3 11(3R2- 2)s- 1
14
1
c
10
50
100
150
200
Figure 2. Equivalent spherical hydrodynamic radii and their cubic ratios, K , of obbte spheroids in units of one half of the short axis, rl,,as functions of the axial ratios, R. Po is the directionally averaged equivalent spherical radius for transbtonal diffusion and rlr is the equivalent spherical radius for the reorientational Brownian motion of the symmetry axis. The Kivelson parameter is given by K = (rLr/?$.
where R and S are given in eq 5. If there is no preferential direction for the translational motion of a spheroid, the directionally averaged equivalent spherical radius Po can be calculated fromz1 (see Appendix) p0-' = (rllo-l 2r10-')/3 (7)
+
For the VOAA molecule the X-ray dimensions give rilo= 3.78 A, rLo= 3.26 A, and t o= 3.42 8,. The experimental values of ro are larger than tofor the solvents listed in Table I except for the propanols. For propanols ro values are close to the rCp= 2.97 8, obtained from the density 1.49 g/cm3 of the VOAA crystal assuming a close packing arrangement. Values of K greater than 1are observed for the propanols. In order to see that K > 1 is possible we examine the hydrodynamic solution of a rotating spheroid with the stick boundary conditions.20 If the rotation axis coincides with one of the perpendicular axes, the spherically equivalent hydrodynamic radius, rl', is expressed aszo (8)
where the symbols are defined before. We may estimate K by combining eq 3-8 as 2 S3(R4- 1) K = rl'3/po3 = -(9) 3(R2- 2)s+ 1 The X-ray dimensions of VOAA in eq 9 give K = 1.01. Equation 9 suggests that K increases as R becomes larger reaching the limiting value, limR,, K = 1.64. Figure 2 shows to,rlf, and K as functions of R. As R increases all of these functions also increase. For the propanols the experimental values of K are larger than 1while both ro and r' are smaller than the corresponding values for other solvents. This suggests that the stick boundary conditions might be not be adequate to described the motions of VOAA molecules in these solvents. Dependency of K on R has been observed previously for pure liquids.22 Stokes4and Zwanzig and B i ~ o suggested n~~ that the slip boundary condition is more appropriate for molecular systems than the stick boundary condition. Molecular dynamic calculations of a hard sphere fluid confirmed the
Diffusion Coefficients of VOAA in Organic Solvents
The Journal of Physical Chemistry, Vol. 82, No. 77, 7978
validity of the slip boundary ~ o n d i t i o n .For ~ a spherical probe the expression for D with slip boundary condition is obtained by merely replacing the numerical factor of 6 in eq 1 by 4. This process increases the values of the hydrodynamic radius since ro(slip) = 1.5ro(stick).For most pure liquids ro(slip) N rCpand the slip boundary condition is j u ~ t i f i e d . However ~ VOAA molecules are larger than those solvent molecules that have been studied and we find robtick) > rcp,which suggests that complexing with the solvent molecules could be important. An additional difficulty is encountered by assuming a spherical shape with the slip boundary condition since there is no torque acting on a perfectly slipping sphere and slip) = 0. The hydrodynamic problem of a rotating spheroid with a slip boundary condition has been solved numerically by Hu and Z ~ a n z i g .Hoe1 ~ ~ and Kivelson* examined their values of K with the Hu and Zwanzig resultz4and found that the slip boundary condition alone is not sufficient to account for the deviation of K from 1. We have been unable to find the slip boundary condition solution of a translating spheroid that is needed to carry out a similar analysis.
Acknowledgment. One of us (M.K.A.) thanks Dr. R. Mills for the hospitality of the Australian National University and Dr. D. J. Reuland and D. E. Ormond for their comments. He also acknowledges partial support of the NSF under US-Australia Cooperative Program, ISU research committee, and the donors of the Petroleum Research Fund, administered by the American Chemical Society. Appendix Equation 7 can be obtained from the average hydrodynamic force expression in ref 20. Here we show a simple derivation from the definition of D2 6tD
(r-r) = ( x 2 ) + ( y 2 )+ (9) E 2t(D, + D,
+ D,) (AI)
1933
where r is the probe particle position vector whose coordinates are x , y, and z. D is the average diffusion coefficient. D,, D,, and D, are the respective directional diffusion coefficients and t is time. We assume that all the diffusion coefficients in (Al) obey the Stokes-Einstein relationship in eq 1 with ro replaced by the appropriate rollfor D , D,, respectively, equivalent spherical radii, and r I for D, and D,. When the resulting S-E equations are combined with eq A1 we obtain the desired eq 7 .
References and Notes (1) H. J. V. Tyrrell and P. J. Watkiss, Cbem. SOC.(London),Annu. Rep. A . 35 (19761. (2) M.’ K. Ahn, 6. J. K. Jensen, and D. Kivelson, J . Cbem. Phys., 57, 2940 (1972). (3) B. J. Alder, D. M. Gass, and T. E. Wainwright, J . Cbem. Pbys., 53, 3813 (1970). (4) R. H. Stokes, Aust. J . Sci., 19, 35 (1957). (5) J. Hwang, D. Kivelson, and W. Plachy, J . Chem. Pbys., 58, 1753 (1973). (6) R. McClung, dissertation, UCLA, Los Angeles, Calif., 1967. (7) D. Kivelson, M. G. Kivelson, and I. Oppenheim, J. Chem. phys., 52, 1810 (1970). (8) D. Hoe1 and D. Kivelson, J . Chem. Pbys., 62, 1323 (1975). (9) N. S.Angerman and R. B. Jordan, J. Chem. Pbys., 58, 837 (1971). (10) B. Kowert and D. Kivelson, J . Cbem. Pbys., 64, 5206 (1976). (11) R. Wilson and D. Kivelson, J . Cbem. Pbys., 44, 154 (1966). (12) B. E. Bryant and W. C. Fernelius, Inorg. Syn., 5, 115 (1957). (13) M. K. Ahn, J. Magn. Reson., 22, 289 (1976). (14) R. Mills and L. A. Woolf, “The Diaphragm Cell”, Australian National University Press, Canberra, 1968. (15) J. A. Riddick and W. B. Bunger, “Organic Solvents”, 3rd ed, Wiley-Interscience, New York, N.Y., 1970. (16) H. Lamb, “Hydrodynamics”, 6th ed, Dover Publicatlons, New York, N.Y., 1945. ( 17) A. Abragam, “The Principlesof Nuclear Magnetism”, Oxfwd University Press, London, 1961. (18) W. Dannhauser and L. W. Bahe, J. Cbem. Pbys., 40, 3058 (1964). (19) R. P. Dodge, D. H. Templeton, and A. Zalkin, J. Cbem. Phys., 35, 55 (1961). (20) H. Shimizu, J. Cbem. Pbys., 37, 765 (1962). (21) J. Happel and H. Brenner, “Low Reynolds Number Hydrodynamics”, Prentice-Hall, Englewood Cliffs, N.J., 1965. (22) M. Fury and J. Jonas, J . Cbem. Pbys., 65, 2206 (1976). (23) R. Zwanzig and M. Bixon, Pbys. Rev., A2, 2005 (1970). (24) C. M. Hu and R. Zwanzig, J . Chem. Pbys., 80, 4354 (1974).
