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J . Phys. Chem. 1987, 91, 4414-4416
liquid structure neglected in the HO theory; it has been concluded above that the shortcomings of the continuum theory come out distinct in the liquid structure formed by hydrogen bonds. To make this important conclusion decisive, however, more systematic conductance and dielectric studies are needed in a variety of aprotic
solvents over a wide range of temperature. Acknowledgment. We are grateful for the financial support by the Research-Grant-in-Aid from the Ministry of Education, Science and Culture (No. 601 2903 1 and 61 134043).
Test of the Hubbard-Onsager Dielectric Friction Theory of Ion Mobility in Nonaqueous Solvents. 3. Pressure Eftect 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 against the effect of pressure on the mobility of the alkali metal and tetraalkylammonium ions in acetone, methanol, and water as far as the relevant data were available. 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 predicts that the pressure dependence of A[ is governed mainly by that of the solvent viscosity. This agrees well with experimental results for the small ions like Li' and Na' in all the solvents. In acetone, the HO theory can explain the pressure dependence of A( even for the tetraalkylammonium ions; however, this is not the case in such hydrogen-bonding solvents as methanol and water.
Introduction
This is the third paper in a series of devoted to the comprehensive study of the reliability and limitations of the Hubbardansager (HO)3*4 dielectric friction theory of ion mobility in solution at infinite dilution. Although a decade has passed since the appearance of the HO theory, it is not properly recognized as yet that at least within the level of the continuum model, the new framework developed by the H O electrohydrodynamic equation for ion dynamics is to replace the classical one made up by Stokes, Einstein, and Walden on the basis of ordinary hydrodynamic~.~-*In the present series, we have attempted to exemplify the application of the epoch-making HO theory and to bridge a gap between the theory and experiments on ion dynamics. We show here how efficient high pressure is when used to couple theory and experiment. The HO theory presents the slow translational motion of a charged hard sphere in viscous dielectric continuum in terms of solvent properties in the bulk. In view of the structure of this theory, it is of great interest to test it against pressure because pressure can vary solvent properties over a wide range; most organic Iiquids are more compressible than water. Such pressure as the few kilobars considered here is high enough to vary liquid density to a much larger extent than temperature does at normal pressure with the phase preserved. The examination of the HO theory in water at high pressure has been accomplished in the previous ~ o r k , ~and - ' ~the effects of pressure in organic solvents (1) Part 1 of this series: Ibuki, K.; Nakahara, M. J . Phys. Chem. 1987, 91, 1864. (2) Part 2 of this series: Ibuki, K.; Nakahara, M. J . Phys. Chem., pre-
ceding paper in this issue. (3) Hubbard, J.; Onsager, L. J . Chem. Phys. 1977, 67, 4850. (4) Hubbard, J. J . Chem. Phys. 1978, 68, 1649. (5) Ibuki, K.; Nakahara, M. J . Chem. Phys. 1986, 84, 2776. (6) Ibuki, K.; Nakahara, M. J . Chem. Phys. 1986,84, 6979. (7) Nakahara, M.; Ibuki, K. J . Phys. Chem. 1986, 90, 3026. (8) Ibuki, K.; Nakahara, M. J . Phys. Chem. 1986, 90, 6362. (9) Nakahara, M.; Torak, T.; Takisawa, N.; Osugi, J. J. Chem. Phys. 1982, 76, 5145. (10) Takisawa, N.; Osugi, J.; Nakahara, M. J . Chem. Phys. 1982, 77, 4717. (1 1) (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
and Technology; Homan, C., MacCrown, R. K., Whalley, E., Eds.; NorthHolland: New York, 1984; Part 11, p 169.
0022-365418712091-44 14$01SO10
are investigated here extensively as far as the relevant data are available in the literature. To our surprise, the HO theory was successful for small ions like Li' even in water that has a viscosity minimum against pressure as a well-known dynamic anomaly. l 3 A comparative study of the pressure effect in various solvents leads to a better understanding of general features of the dielectric friction theory having predictive power. Ion transport process in water plays an important role in retaining our life in various ways, and it is important to elucidate to what extent it is affected by the structure neglected in the continuum theory; nowadays, not only physical chemists but also biochemists and biophysicists are interested in the question how the anomalous behavior of water is related to the structure and its fluctuation. The self-consistent HO theory provides a firm base for elucidating the effect of the liquid structure on the ion dynamics; deviations from this continuum theory may be used as a measure of the structural effect. In addition to the reliable theoretical model, we want a reference solvent which the model fits fairly well. A point to be attacked is measured relative to well-defined reference points in triangulation; the former and the latter are water and reference solvents, respectively, in our study. Although the structure of simple liquids like argon is fairly well understood at present, these liquids lack polarity of our interest as well as ion solubility and are too far away from water to meaningfully compare theory and experiment on the ion dynamics. Parts 1 and 2 of this series tell us that aprotic solvents polar enough to dissolve various ions can be used as such a practical reference; it has been found that the hydrogen-bonding ability of the solvent is the key factor that restricts the applicability of the continuum theory. This important criterion is examined further here by applying pressure to protic and aprotic organic solvent systems. Theoretical Section
When an ion is introduced into a flowing liquid, the pattern of the solvent flow is perturbed by the presence of the ion due to the size and charge effects. As a result, two kinds of frictional forces are exerted on the ion by the solvent which may be modeled (12) Nakahara, M.; Zenke, M.; Ueno, M.: Shimizu, K . J . Chem. Phys. 1985, 83, 280. (13) Nakahara, M.; Osugi, J. Reti. Phys. Chem. Jpn. 1980, 50, 66.
0 1987 American Chemical Society
The Journal of Physical Chemistry, Vol. 91, No. 16, 1987 4415
Ion Mobility in Nonaqueous Solvents as viscous dielectric continuum. To take account of both frictions at the same time, H ~ b b a r d - O n s a g e r ~have , ~ developed the electrohydrodynamic equation of motion. We have scrutinized the solution of the HO equation for translation of a spherical ion and fit the result to the following polynomial e x p r e s ~ i o nfor ~ ~the ~ residual friction coefficient A{
where 7 is the solvent viscosity, RH0 the H O radius, R the crystallographic radius of the i ~ n , ' ~and . ' ~a, the universal coefficients which depend only on the boundary condition (the slip one is considered in our study) and the numerical values of which are summarized elsewhere.' The residual friction coefficient A{ is suitable for our purposes mentioned above and superior to the conventional Walden product based on ordinary hydrodynami c ~ . ' ~ , ' 'The quantity A{ is defined as follows
l-
{s
(2)
{ = lelF/Xo
(3)
{s = 4 ~ 7 R
(4)
A{=
where {, lS, e, F , and Xo denote the total ionic friction coefficient, the Stokes friction coefficient for slip, the ionic charge, the Faraday constant, and the limiting ionic conductance, respectively. Since the purely viscous friction is taken as a reference in eq 2, A{ reflects mainly the dielectric friction and all the other factors neglected in the simple hydrodynamic continuum model for lS. The key parameter RHOis given by
r
.
1114
where 7 , to, and t, are the dielectric relaxation time, the static, and high-frequency dielectric constants of the solvent, respectively. As seen in eq 5 , the parameter RHO,which determines the coupling of dielectric and viscous frictions, depends on the ionic charge and the bulk properties of solvent; the charge is fixed here for the monovalent ion. The larger RH0 is, the larger the ratio of dielectric friction to viscous friction; see Figure 1 in ref 5 . Equation 1 expresses the residual friction coefficient as an explicit function of the solvent viscosity, ion radius, and H O radius; of these variables the ion radius is assumed to be independent of pressure. As shown below, the variation of RHOwith pressure is relatively small in most solvents. The neglect of the pressure dependence of RH0 leads to the simplification of eq 5 as
where the superscript indicates the pressure. Thus the H O theory predicts that the effect of pressure on A{ is determined primarily by the solvent viscosity and that the pressure dependence of A{ becomes larger as the ion size decreases. These predictions will be compared with experimental results below.
Results and Discussion Data Sources. We first show the data sources used in the computation of RH0 and the theoretical and experimental values of A{ defined by eq 5 , 1, and 2 , respectively; we need solvent properties and conductance data measured at high pressures. Solvents so far investigated to a satisfactory extent are just acetone, methanol, and water; the first is aprotic and the last two are protic (hydrogen bonding). Although, in general, the high-pressure data are still limited, the study of aqueous systems has so far been made most extensively because water is the most popular and the most (14) Pauling, L. The Nature ofrhe Chemical Bond; Cornel1 University: New York, 1960. (15) Robinson, R. A.; Stokes, R. H. Electrofyte Solutions; Butterworths: London, 1968. (16) Takisawa, N.; Osugi, J.; Nakahara, M. J . Phys. Chem. 1981, 85, 3582. (17) Nakahara, M.; Ibuki, K. J . Phys. Chem. 1986, 90, 3026
TABLE I: Solvent Viscosities and RHOfor Monovalent Ions in a Pressure Rangeo solvent TIOC Plkbar nlcP R d A 30
acetone methanol water
0-2 0-2
25 25
0.295-0.713 0.544-1.037 0.890-0.886
0-1
2.14-2.06 3.16-3.09 1.51-1.44
"The numerical values before and after the dash for P , 7, and RHO correspond to each other.
4
I
I
I
I
p
I
I
I
I
Li'
I
1000 2000 Press. ( bar) Figure 1. Variation of Arwith pressure in acetone at 30 OC. The broken and full lines indicate the theoretical and experimental results, respectively.
interesting liquid. Water and methanol systems are studied at 25 O C and acetone systems at 30 OC. We pay attention to the alkali metal and tetraalkylammonium ions to make a systematic test of the HO theory in these solvents as a function of pressure. Values of 7 and Q for acetone are tabulated by Inada.Is Values of T and e, for this solvent are not measured at high pressure and are evaluated on the assumption that 717 and t, do not vary with pressure; their atmospheric values are given by Evans et al.19 Limiting molar conductances A' for the alkali metal and tetraalkylammonium salts are reported by Inada'* and Adams and Laidler,20 respectively. The Ao values are split into the ionic components at 25 O C by using a reference electrolyte,2' and the resultant limiting cation transference numbers are assumed to be independent of temperature and pressure in the evaluation of limiting ionic conductances at each pressure at 30 O C . The Xo values thus obtained for the I- ion from different salts agree within an error of 2%; an uncertainty of this magnitude causes no problem in the present comparison of theory and experiment. Values of 9 and eo for methanol are provided by Woolf and co-workers22and Srinivasan and Kay,23respectively. Values of T and e, for methanol at high pressure are obtained in the same manner as in the case of acetone. The conductance data are due to the work by Watson and Kay.24 The physical properties of water and the relevant conductance data for the alkali metal ions are taken from ref 9. The limiting conductances for the tetraalkylammonium ions in water are read from the graphical presentation by Kay.2s It is important to see to what extent the key parameter RH0 varies with increasing pressure and how its dependence on pressure (18) Inada, E. Rev. Phys. Chem. Jpn. 1978, 48, 72. (19) Evans, D. F.; Tominaga, T.; Hubbard, J. B.; Wolynes, P. G. J . Phys. Chem. 1979, 83, 2669. (20) Adams, W. A.; Laidler, K. J. Can. J . Chem. 1968, 46, 1977. (21) Evans, D. F.; Thomas, J.; Nodas, J . A.; Matesich, M. A. J . Phys. Chem. 1971, 75, 1714. (22) Isdale, J. D.; Easteal, A. J.; Woolf, A. In?. J . Thermophys. 1985, 6, 439. (23) Srinivasan, K. R.; Kay, R. L. J . Solution Chem. 1975, 4, 299. (24) Watson, B.; Kay, R. L., cited in ref 25. (25) Kay, R. L. In Wafer,Franks, F., Ed.; Plenum: New York, 1973; Vol. 3, Chapter 4.
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The Journal of Physical Chemistry, Vol. 91, No. 16, 1987 I
I
I
I
Bu4N'
Ibuki and Nakahara
1
1
- /
Ln
a
0
1000
2000
Press. ( bar )
Figure 2. Variation of A{ with pressure in methanol at 25 "C. The broken and full lines have the same meaning as in Figure 1.
differs from one solvent to another. Variations of R H 0 together with the related solvent properties are listed in Table I. When the solvents are compressed to 1 kbar, the R H 0 values decrease only by 1-5%. The decrease in R H 0 is the largest in water, which is caused by the anomalous "decrease" in the dielectric relaxation timez6 involved in eq 5 ; T / T is not constant but decreases in this case. Acetone. Theoretical and experimental A{ for the alkali metal and tetraalkylammonium ions in acetone are plotted against pressure in Figure 1 . All the cations have a positive pressure coefficient and the pressure dependence is almost linear. These features of the observation are correctly predicted by the HO theory. The conformity indicates that the increase in A{ is caused by the increase in 7;see the negligible "decrease" in RHOin Table I. For the case of the alkali metal ions, the observed dependence of A{ on the ion size is also in agreement with the theoretical prediction at each pressure. In this way the HO theory is successful in predicting the pressure effect on A{ in this aprotic solvent . One should not place any strong accent on differences between theory and experiment at 1 bar in Figure 1 because many assumptions have been used in deriving the final form of the H O theory and because no adjustable parameters are involved. For the HO theory we are not allowed to expect such a quantitative success as that of the Debye-Hiickel-Onsager-Fuoss theory for the concentration dependence of ion mobility which is determined mainly by the long-range interactions; the complicated nature of the dynamic ionsolvent interactions in the short range is simplified in the continuum model on which the H O theory is based. For the case of the tetraalkylammonium ions, the largest ion has the largest A{ in contrast to the theoretical prediction. Such an irregular size dependence for the tetraalkylammonium ions is also observed in acetonitrile at 1 bar.lS2 This limitation of the HO theory with the slip boundary condition has been discussed in part 1. Methanol. Theoretical and experimental A{ for the alkali metal and tetraalkylammonium ions in methanol are plotted against pressure in Figure 2. For the alkali metal ions, theory agrees with experiment in the protic solvent as in the aprotic solvent referred to above. However, the tetraalkylammonium ion shows a negative pressure coefficient of A( contrary to the H O prediction and the tetrabutylammonium ion has a positive pressure coefficient much larger than that the theory predicts. Similar discrepancies ( 2 6 ) Pottel, R.;Asselborn, E. Ber. Bunsen-Ges. Phys. Chem. 1979, 83, 29.
1000
2000
Pressure ( bar )
Figure 3. Variation of A{ with pressure in water at 25 "C The broken and full lines have the same meaning as in Figure 1.
have been observed in the previous study on the temperature effect; see Figure 1 in ref 2. These limitations of the HO theory would be ascribed to the structural effect of methanol. An increase in pressure corresponds to a decrease in temperature with respect to the structural effect; both tend to solidify this solvent. Water. Theoretical and experimental A{ for the alkali metal and tetraalkylammonium ions in water are plotted against pressure in Figure 3. The HO theory predicts that A{ "decreases" slightly with increasing pressure as a result of the decrease in R H O ; as seen in Table I, the viscosity of water is virtually invariant in the pressure range of our interest in sharp contrast to organic liquids. This negative pressure coefficient is observed only for the small Li' ion having a large surface charge density. It is surprising that the continuum theory is successful for the small ion in water despite the anomalous pressure dependence of the solvent viscosity. The positive pressure coefficients observed for the larger alkali metal ions are not explained by the continuum theory and are discussed in detail The A( values for the tetraalkylammonium ions all increase with increasing pressure. The observed trend is not understandable in terms of the continuum theory combined with the bulk solvent properties; this would be related somehow to the structural effect. Conclusions The experimental values of A{ (defined by eq 2) for the small ions like Li' increase and decrease with increasing pressure in organic solvents (acetone and methanol) and water, respectively. These trends are successfully explained by taking account of the response of the solvent properties to the applied pressure in the HO theory; while the increase is controlled by the pressure dependence of the solvent viscosity, the anomalous decrease is brought about by the decrease in the HO radius defined by eq 5. In fact, the dielectric friction plays an important role in determining ion mobility in solution. When we estimate ion mobility in solution at high pressure, the dielectric friction theory is to be taken into account instead of the Stokes law. Although the HO theory works relatively well in the aprotic solvent, it shows limitations for the larger ions in the protic solvents. We believe that whether or not these limitations are explained is a touchstone of the adequacy of a molecular theory. Acknowledgment. We are grateful for the financial support by the Research-Grant-in-Aid from the Ministry of Education, Science and Culture (No. 601 2903 1 and 61 134043).
