441 1
J . Phys. Chem. 1987, 91, 441 1-4414
Test of the Hubbard-Onsager Dielectric Friction Theory of Ion Mobility in Nonaqueous Solvents. 2. Temperature Effect K. Ibuki and M. Nakahara* Department of Chemistry, Faculty of Science, Kyoto University, Kyoto 606, Japan (Received: February 17, 1987)
The Hubbard-Onsager (HO) dielectric friction theory was tested systematically against the effect of temperature on the mobility of the alkali metal, halide, and tetraalkylammonium ions in water, methanol, ethanol, 1-propanol, and acetonitrile as far as the relevant data were available in the literature. Instead of the conventional Walden product, we used the residual friction coefficient (A{) defined as the total friction coefficient subtracted by the Stokes friction coefficient for slip. The HO theory is successful in predicting the negative temperature coefficient of A{ for small ions like Li’ and Na’ in all the solvents; the continuum theory works better in the alcohol which has a longer alkyl chain and best in the aprotic solvent. However, the negative temperature coefficient of A{ observed for such larger ions as the tetramethylammonium ion in the hydrogen-bonding solvents is not explained by the continuum theory.
It is important to show that the widely spread classical Stokes-Einstein-Walden (SEW) framework based on the Navier-Stokes (NS) equation is to be replaced by the modern one provided by the Hubbard-Onsager (HO)]-*dielectric friction theory for a better understanding of ion mobility or diffusion in solution; the response of the solvent to the charge characteristic of the ion introduced into it can be taken into account in a self-consistent manner by the HO electrohydrodynamic equation of motion but not by the NS hydrodynamic equation. We attempt to promote the desirable transition to the new framework for the interpretation of the ion dynamics by making systematic and thorough tests of the HO theory in organic solvents in the present series; this paper discusses the temperature effect on ion mobility, while the first paper3 is concerned with the effect of ion size. Although water and water-organic solvent mixtures have been examined in the previous organic solvents with a lower dielectric constant are more interesting in view of the following feature of the solvent polarization: the latter are polarized to a longer range because of the less screening of the ionic field, so that the charge effect on the solvent velocity field, which is completely neglected in the SEW framework, plays a more important role in such organic solvents. By taking advantage of this merit, we discuss here the effect of temperature on ion mobility in organic solvents as far as the relevant data are available in the literature, and compare present results with previous ones for aqueous systems to get insight into a more general aspect of the HO dielectric friction theory which has wide implications and application~.~~J~ Although no continuum model itself can give any final answer to the updated question how the liquid structure of the solvent
affects ion mobility in solution, a primary step necessary for recognizing the structure effect is to choose a reliable continuum model as a reference where the effect of the charge is properly taken into account as in the HO theory. Any significant deviations from such a continuum reference may be attributed to the effect of the dynamic liquid structure the fluctuation of which depends on temperature. Along this line, the negative residual friction coefficient has been found as a challenging anomaly for medium-sized ions not only in water but also in other hydrogen-bonding solvent^,^ and it is also known that the anomaly is enhanced as water is ~ o o l e d . ~Thus ~ ~ Jthe ~ critical test of the HO theory with respect to temperature in a variety of solvents makes a key contribution to the confirmation of limitations of the continuum theory by which a more rational molecular theory may be induced. Since attention is now paid to the water structure which is controlled by the hydrogen bonds sensitive to temperature, a comparative study of the temperature effect on the ion dynamics in various solvents is worthwhile; it should be clarified whether water alone shows the unique temperature effect or not. Pressure is another important variable that can vary the liquid structure, and its effect on ionic residual friction coefficients in organic solvents will be discussed in the accompanying paper.16
(1) Hubbard, J.; Onsager, L. J . Chem. Phys. 1977, 67, 4850. (2) Hubbard, J. J . Chem. Phys. 1978, 68, 1649. (3) Part 1 of this series: Ibuki, K.; Nakahara, M. J . Phys. Chem. 1987, 91, 1864. (4) Takisawa, N.; Osugi, J.; Nakahara, M. J. Phys. Chem. 1981,85, 3582. (5) Nakahara, M.; Torok, T.; Takisawa, N.; Osugi, J. J . Chem. Phys. 1982, 76, 5145. (6) Takisawa, N.; Osugi, J.; Nakahara, M. J. Chem. Phys. 1982, 77,4717. (7) (a) Takisawa, N.; Osugi, J.; Nakahara, M. J . Chem. Phys. 1983, 78, 2591. (b) Nakahara, M.; Takisawa, N.; Osugi, J. In High Pressure in Science
where 7 is the solvent viscosity, v the velocity field in the fluid, p the pressure, and F D F the contribution of dielectric friction to the total force density. Here F D F is coupled to viscous friction. When the charge on an ion is neglected, eq 1 reduces to the linearized form of the N S equation given by
and Technology; Homan, C., MacCrown, R. K., Whalley, E., Eds.; NorthHolland: New York, 1984; Part 11, p 169. (8) Nakahara, M.; Zenke, M.; Ueno, M.; Shimizu, K. J . Chem. Phys. 1985, 83, 280. (9) Ibuki, K.; Nakahara, M. J. Chem. Phys. 1986, 84, 2776. (IO) Ibuki, K.; Nakahara, M. J. Chem. Phys. 1986, 84, 6979. (11) Nakahara, M.; Ibuki, K. J. Phys. Chem. 1986, 90, 3026. (12) Ibuki, K.; Nakahara, M. J . Phys. Chem. 1986, 90, 6362. (13) Nakahara, M.; Zenke, M. Bull. Chem. SOC.Jpn. 1987, 60, 493. (14) Nakahara, M.; Ibuki, K. J . Chem. Phys. 1986, 85, 4654. (15) Ibuki, K.; Nakahara, M. J. Chem. Phys. 1986, 85, 7312.
0022-3654/87/2091-4411$01.50/0
Theoretical Section Hubbard and Onsager1s2have extended the N S hydrodynamic equation of motion to ionic systems in a self-consistent manner by adapting the Debye dielectric relaxation theory to fluids flowing around a charged sphere. The following HO electrohydrodynamic equation is derived for slow motions of the viscous and dielectric fluid
vV% = v p which leads to the well-known Stokes law in the form {s = 4 ~ v R
(2) (3)
where Ts is the Stokes friction coefficient for the slip boundary Thus condition and R is the crystallographic radius of the ior~.~’
