J . Phys. Chem. 1986, 90, 3026-3030
3026 34 \
1
I
0
100% c02
c
9.5% MeOH
I
the energy levels of the absorbance spectrum. Therefore, the absorbance spectrum shifts from a "gas-phase-like" region to a regime where the probe molecule becomes effectively solvated by C 0 2 . The precise density where the qualitative change observed in Figure 5 takes place is less readily explained but may correspond to the beginning of some ordering of adjacent molecules in the cybotactic region. Mixed Fluids. Methanol was added to C 0 2at 5.6 and 9.5 wt %, and the spectral shift of 2-nitroanisole was determined. Figure 6 shows the effect of methanol on the absorbance maxima with pressure. At low percentages of methanol in COz the spectra are red-shifted toward the value seen for pure methanol (see Table I). At 9.5% methanol i n , C 0 2 , pressure has no effect on the absorbance maximum. One can hypothesize that the cybotactic region of 2-nitroanisole experiences an effective enrichment in methanol at small percentages of methanol in C 0 2 that leads to an environment similar to that in pure methanol. Once this methanol cybotactic environment is established, then pressure would have a small effect on this region and hence on the energy of the P A * transition for 2-nitroanisole. Further studies are in
progress to better understand the effects in mixed fluid systems. Conclusion The A* solvent polarity-polarizability scale has been applied to four supercritical fluids based on the solvatochromic shift of the absorption peak maximum. Measurement of the solvatochromic shift in position of the absorption maximum directly probes the cybotactic region of the solute, providing information on the solute-solvent intermolecular interactions in the solvation
reaction field function is effective in relating the state of the cybotactic region of the solute as one progresses from gas-phase densities to liquidlike densities for the supercritical fluid. A significant effect of a solvent modifier in the fluid on the cybotactic region was observed. The cybotactic region can apparently become enriched with the solvent modifier under conditions of relatively small modifier concentrations. Extension of these studies is expected to lead to a more fundamental understanding of the intermolecular interactions occurring during supercritical fluid extraction or supercritical fluid chromatography and solutesolvent interactions relevant to reaction processes in supercritical fluid media. In the broader sense, these studies allow an enhanced understanding of solvation phenomena by probing the continuous range of interactions between the gas- and liquid-phase limits.
Acknowledgment. This work has been supported by the U S . Department of Energy, Office of Basic Energy Sciences, under Contract DE-AC06-76RLO 1830. Registry NO. CClF,, 75-72-9; N20, 10024-97-2;CO,, 124-38-9;"3, 7664-41-7;2-nitroanisole, 91-23-6;pentane, 109-66-0;tetrahydrofuran, 109-99-9;methanol, 67-56-1; acetonitrile, 75-05-8.
Is the Walden Product Useful? M. Nakahara* and K. Ibuki Department of Chemistry, Faculty of Science, Kyoto University, Kyoto 606, Japan (Received: November 19, 1985)
The invalidity of the use of the Walden (conductance-viscosity) product in the interpretation of the limiting ionic conductance Xo has been elucidated by comparing the experimental result and the prediction made for the relatively small alkali metal ions as a function of temperature, solvent, and pressure by the Hubbardansager dielectric friction theory. It is recommended to transform Xo into a more meaningful quantity such as the residual friction coefficient A{ which is defined as the overall friction coefficient subtracted by the Stokes friction coefficient for slip. Taking the residual friction coefficient comes from the dielectric friction theory, while taking the Walden product is based on the Stokes law where the effect of the charge on ion is not taken into account. It turns out that, contrary to what has so far been believed, the contribution of the viscosity qo to ion mobility is not eliminated by the multiplication of Xo by 7' but remains in the "corrected" quantity.
Introduction The limiting ionic conductance Xo has been determined in an exhaustive manner for a variety of ions in aqueous and nonaqueous solvents and used for a long time as a probe for getting insight into the dynamic aspects of ionsolvent interactions. Interpretation of this useful quantity is, of course, influenced by the form into which it is transformed for the discussion; any undesirable transformation obscures or distorts the original physical meaning. Unfortunately, however, it continues to take the Walden product, where Xo is multiplied by the solvent viscosity qo, in order to shed light on factors determining ion migration in solution even after the Walden rule1 was recognized to be invalid in the rigorous 0022-3654/86/2090-3026$01.50/0
sense;2 the afterglow of the Walden rule is too strong in view of the fact that the contribution of the dielectric f r i ~ t i o nas ~ .well ~ as the hydrodynamic one to the overall friction coefficient for moving ion was noticed a long time ago. This kind of transformation is seen in almost every paper on ion mobility whether it is theoretical or experimental. This prolonged convention will not (1) Walden, P. 2.Phys. Chem. 1906, 55, 207. (2) Fuoss, R. M.; Accascina, F. Electrolytic Conductance; Interscience: New York, 1959. (3) Wolynes, P. G. Annu. Rev. Phys. Chem. 1980, 31, 345. (4) (a) Hubbard, J.; Onsager, L. J . Chem. Phys. 1977, 67, 4850. (b) Hubbard, J. J . Chem. Phys. 1978, 68, 1649.
0 1986 American Chemical Society
Is the Walden Product Useful?
The Journal of Physical Chemistry, Vol. 90, No. 13, 1986 3021
be abandoned unless it is clearly shown how illegitimate and dangerous it is to take the Walden product. It is emphasized here that a preferable transformation of Xo is the residual friction coefficient which is defined as the overall friction coefficient subtracted by the Stokes friction coefficient for slip. One of the conceivable motivations for taking the Walden product would be the feeling that the viscous drag acting on a moving ion through the ion size is properly or almost completely corrected in the transformed form from which the other factors controlling Xo are expected to be picked up directly; the solvent (liquid) structure is often considered as such a remaining factor. This intuition is by no means self-consistent because of the presence of the charge on the ion, and may be rationalized only with tremendous approximations according to the Stokes-Einstein law widely used for estimating transport properties of ions and molecules. This law comes from the stationary solution to the Navier-Stokes hydrodynamic equation where electrostatic interactions between an ion and solvent dipoles are completely neglected. Now the time matures when the Navier-StokesEinstein-Walden framework for the interpretation of ion mobility in terms of solvent viscosity is replaced by a new one that is based on the more developed Hubbard-Onsager4 electrohydrodynamic equation where the viscous and dielectric frictions on moving ion are taken into account at the same time. Taking the residual friction coefficient is compatible with the new framework provided by the dielectric friction theory and matches well with the formalism of the Wolynes molecular theory for ion t r a n s p ~ r t . ~ It has been shown in a series of previous that the dielectric friction theory refined to an almost final form by Hubbard and Onsager (HO) is successful in explaining the effects of p r e s ~ u r e , ~solvent - l ~ isotope,1° and mixed solvent]’ on the mobility of relatively small ions even in hydrogenbonded solvents; only systematic deviations of experimental results from the HO prediction are attributed to the effect of the liquid structure of the solvent neglected in the continuum theory. In these studies, Xo has been transformed into the residual friction coefficient Acdefined in the following way instead of the Walden product A{= .f-
b
{ = (zelF/Xo
(2)
{s = 4rq0R
(3)
where [ and lS are the overall ionic friction coefficient and the Stokes friction coefficient for perfect slip, respectively, and I , e, F, Xo, vo, and R denote the ionic valence, the protonic charge, the Faraday constant, the limiting ionic condctance, the solvent viscosity, and the ionic radius,12 respectively. On the basis of the confirmed reliability of the H O theory for the relatively small ions, we have concluded that the residual friction coefficient is more useful than the conventional Walden product.
Historical Section Since the Walden product has played a historically indispensable role in the study of the relationship between ionic conductance and solvent viscosity, it is worthwhile to review in brief the historical background. In order to find an empirical relation between ionic or electrolytic conductance and solvent viscosity, it is required to vary ( 5 ) (a) Wolynes, P. G . J . Chem. Phys. 1978,68,473. (b) Wolynes, P. G.; Colonomos, P. J. Chem. Phys. 1979, 71, 2644. (6) Takisawa, N.; Osugi, J.; Nakahara, M. J. Phys. Chem. 1981,85,3582. (7) Nakahara, N.; Torok, T.; Takisawa, N.; Osugi, J. J. Chem. Phys. 1982, 76, 5145.
(8) Takisawa, N.; Osugi, J.; Nakahara, N. J . Chem. Phys. 1982,77,4717. (9) Takisawa, N.; Osugi, J.; Nakahara, M. J. Chem. Phys. 1983,78, 2591. (10) Nakahara, M.; Zenke, M.; Ueno, M.; Shimizu, K. J. Chem. Phys. 1985, 83, 280. (1.1) (a) Ibuki, K.; Nakahara, M. J. Chem. Phys. 1986, 84, 2276. (b) Ibuki, K.; Nakahara, M. J. Chem. Phys., in press. (12) Pauling, L. The Nature of the Chemical Bond; Cornell University: New York, 1960.
the solvent viscosity in some way. It was the pressure that was used first for the purpose. In the 1890s, ROntgen,l3 Fanjung,I4 and Tammann” first found a parallelism between the pressure dependence of the electrical conductivities of aqueous solutions and that of the solvent fluidity (1/qo)l6 which exhibited a maximum against pressure. However, the early phenomenological studies did not lead to any simple quantitative relation between the two types of transport properties. The parallelism between the two quantities with respect to temperature variation was noticed by Kohlrausch before Walden.” Walden found in 1906 that the product of the limiting electrolytic conductance Ao and the solvent viscosity qo was constant within some uncertainty when a variety of organic solvents were used for an electrolyte composed of large organic ions.] The empirical rule was utilized to estimate conductance of an ion in a solvent in which neither conductance nor transference number but viscosity was measured. But at present it is well-known from more precise experiment that the Walden rule does not hold; the AOqois not exactly constant when qo is changed by solvent, temperature, or pressure. Let us see the theoretical background of the Walden product. A simple relation between the stationary velocity v of a macroscopic sphere flowing in a fluid and the fluid viscosity qo was hydrodynamically derived by Stokes.l* His result is given by
X = yqoRv
(4)
where X and R denote the force applied to the sphere and the radius of the sphere, respectively, and y is equal to 6?r and 47 depending on the stick and slip boundary conditions, respectively. The factor yqoR is the Stokes friction coefficient corresponding to eq 3. Equation 4 tells us that the velocity-viscosity product is invariant for the constant external force. Although as referred to above, there were a few experimental attempts to find such a simple relation for ion transport process by the application of pressure, just after the time (in 1887) when Arrhenius established the complete ionization theory for electrolyte solution, it was Einstein who first applied the hydrodynamic relation eq 4 to the Brownian motion.19 Lorentz followed the extended application by Einstein to derive ion size from ion mobility through the solvent viscosity.20 These theoretical applications of the hydrodynamic relation to the Brownian and microscopic particles influenced the foundation of the empirical Walden It is to be noted that at that age people were anxious to obtain the firm evidence for the atomic theory. In this respect it was very important to determine the atomic or ion size even in an approximate manner in advance of the X-ray diffraction technique; to us the reliable ion size seems to have been established by Pauling on the basis of the quantum mechanical interpretation of bond lengths determined by the diffraction technique by the end of the 1930s.’* Thus the hydrodynamic approach has made an important contribution to our estimation of dimensions of ions or atoms in solution. In the reverse way, once the ion size was established, it became actually possible to make a stringent test of any hydrodynamic expressions for microscopic transport processes. Thus several fortunate factors underlay the finding of Walden’s rule. The most important background was formed by the preceding hydrodynamic relation between velocity and viscosity as expressed by eq 4. Since we understand that in the treatment of ion transport, the Navier-Stokes hydrodynamic equation is to be replaced by the more developed Hubbard-Onsager electrohydrodynamic equation within the framework of the sphere-incontinuum model, there remain no reasons why we adhere to the (13) (14) (15) (16) (17) (18) (19) (20) (21)
Rontgen, W. C. Nach. K . Ges. Wiss. Gottingen 1893, 509. Fanjung, I. Z . Phys. Chem. 1894, 14, 673. Tarnmann, G. Z. Phys. Chem. 1895, 17, 725. Cohen, R. Wied. Ann. 1892, 45, 666. Walden, P. Samml. chem. chem.-tech. Vortrage 1910, 15, 277. Stokes, G. G. Trans. Cambridge Philos. SOC.1850, 9, 8. Einstein, A. Ann. Phys. (Leipzig) 1905, 17, 549. Lorentz, R. Z . Phys. Chem. 1910, 73, 252. Walden, P. 2. Elektrochem. 1910, 16, 1003.
3028 The Journal of Physical Chemistry, Vol. 90, No. 13, 1986
Nakahara and Ibuki
Walden product based on the primitive hydrodynamic equation.
Theoretical Section According to the Hubbard-Onsager t h e ~ r ythe , ~ equation of motion for the dielectric fluid is given by 1
q0V2v = V p - -[Eo X (V X P * ) 2
+ Eo(V*P*)]
ta
(5)
where qo, p, Eo, and P* denote the viscosity, the pressure, the electrostatic field due to the presence of ion, and the polarization deficiency relevant to the dipole relaxation, respectively. The second term in eq 5 represents the contribution of the dielectric friction to the total force. When the charge on ion is neglected (E, = 0), eq 5 reduces to the linearized form of the Navier-Stokes equation leading to the Stokes law. In order to obtain the friction coefficient { for a moving ion, Hubbard numerically solved the ordinary differential equation for the velocity field which is derived after the linearization of P*. The numerical results of the friction coefficients are given in terms of the dimensionless quantity of the form
x = {/?'RHO
4
(6)
where x depends only on R and RHO, and RH0 is the HubbardOnsager radius expressing the coupling between the viscous and dielectric frictions. The HO theory has been extended to solvent mixtures by us.ll In this case, RH0 is given as follows
1
i 0
-3.0
40
20 t
In the case of the slip boundary conditions considered here, y = 4 a and the coefficients are given as follows: a , = -2.786 64, a2 = 8.621 63, a3 = -3.342 52, and a4 = 0.395 501 for RIRHo > 0.3. When the bulk properties of solvent involved in eq 7 are all available, the H O value of the ionic friction coefficient 1:can be calculated according to eq 6-8 under any conditions and then converted into ho through eq 2. The theoretical value of the residual friction coefficient is obtained by substituting eq 6 into eq 1 as where y = 47r for the slip case considered here. Equation 9 indicates that not the total friction coefficient but the residual friction coefficient can be expanded to the fourth power of RHoIR to a good approximation. Since the hydrodynamic friction is taken as a reference in eq 1 or 9, A{ reflects mainly the dielectric friction and is more sensitive to it than the Walden product. Equation 9 involves both dielectric and viscosity properties of solvent, reflecting the coupling between the viscous and dielectric frictions. If the earlier dielectric friction theory by ZwanzigU is used instead, the resultant form of A( involves only the solvent dielectric properties as a result of the separate treatment of the viscous and dielectric frictions. Recently Wolynes5 used a stochastic model to develop a molecular theory for ion migration in solution. In his molecular theory the friction coefficient is divided in the following way where S and H denote the soft and hard forces between an ion (22) Zwanzig, R. J . Chem. Phys. 1970, 52, 3625.
40
20 t ('C)
Figure 1. Comparison of plots of A{ and ?IO$ for the alkali metal ions in water against temperature. The broken and full lines indicate the HO
theoretical and experimental results, respectively. and solvent molecules; the soft force (S) is regarded as the slowly varying part of the interactions between the ion and solvent molecules. In his treatment of the force-force correlation functions, the cross terms P Hand psin eq 10 are neglected in view of the large difference between the relaxation times of the two kinds of forces and p His assumed to be given by eq 3. Therefore, we have A{=
where T~ and r2 denote the two relaxation times, and eo, ec, and t, are the static, intermediate-frequency, and high-frequency dielectric constants, respectively. In the case of a single relaxation time T , RH0 is given by taking r = T ] and ec = e, in eq 7 . In order to facilitate a systematic application of the HO theory to a variety to conditions, Hubbard's numerical solution has been improved and fitted to the following polynomial equation instead of the tabulation:"
0
('C)
ps
(11)
Thus taking the residual friction coefficient is in harmony with the formalism of the Wolynes molecular theory. In the Walden product, the contribution of the viscous drag is considered to be corrected simply by the multiplication of solvent viscosity. This might have no problem if the force acting on the ion were only the viscous force as expressed by eq 4; in this case the Walden product is equal to IzelF/yR according to eq 4. On the contrary, the number of the forces is not limited to one in the actual system; they are at least the viscous and dielectric frictional forces. When ion mobility is multiplied by the solvent viscosity, any factors other than the viscous friction are unduly masked by the viscosity as suggested by eq 8 and IO. The correction of the hydrodynamic factor in the additive manner as in eq 1 is more natural in view of the additive nature of the forces acting on moving ion. This problem of the Walden product will be discussed in the following section by using experimental results so far obtained under various conditions.
Discussion It is interesting to compare (HO) theoretical and experimental values of the residual friction coefficients A{ and the Walden products XOqofor the alkali metal ions under various conditions in order to decide which transformation of Xo is more meaningful and useful. Temperature, pressure, and solvent are taken here as the variable conditions because attention is paid to the effect of these variables on the Walden p r o d ~ c t . ~ ~ , * ~ Temperature Effect. The limiting conductance for the alkali metal ions in have been transformed in the two ways described above by using relevant properties of the solvent: and the transformed quantities are plotted against temperature in Figure 1 for comparison. Figure la shows that the theoretical value of the residual friction coefficient decreases with increasing temperature and increasing ion size (decreasing surface charge density); according to eq 9, the decrease in A{ with temperature is caused by the decrease (23) Kay, R. L.; Cunningham, G. P.; Evans, D. F. In Hydrogen-Bonded Soluent Systems; Covington, A. K., Jones, P., Eds.; Taylor & Francis: London, 1968; p 249. (24) Kay, R. L. In Water; Franks, F., Ed.; Plenum: New York, 1973; Vol. 3, Chapter 4. (25) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworths: London, 1968.
The Journal of Physical Chemistry, Vol. 90, No. 13, 1986 3029
Is the Walden Product Useful?
I
(a)
6-
Theor. -4
-
-
0.8(b)
-
m
l
i
0.6
E
-
F
k
0.4
9
LT
a
0.2
0
u 10
20
10
t
(‘C)
20
O
50
loo
EtOH ( mol ’la )
.
0.8
I
I
0.7-
-
$ -
I
~ 10
L 20
t
d d 10 20
(‘C)
Figure 2. Comparison of plots of A{ and AD$ for the alkali metal ions in H20and D20against temperature. The broken and full lines indicate the results in D20and H20, respectively.
in q0 because RHOfor water changes little with temperature. The HO theory correctly predicts the temperature dependence of A{ for the relatively small ions like Li+ and K+. As expected, the agreement between theory and experiment gets better a t higher temperatures. The anomaly observed for the Cs+ ion that A{ is negative with its positive temperature coeffkient has been ascribed to the special mechanism of the ion transport in the open structure of water which is called the passing-through-cavities mechani~m.~,’ Thus the residual friction coefficient is of great use to clarify the reliability and limitations of the continuum theory. The Walden products are plotted against temperature in Figure lb. Despite the use of the same data as those in Figure l a , comparison of the theory and experiment is rather complicated by this transformation. The problems with the Walden product are summarized as follows: (1) the size dependence is not correctly predicted; (2) although the negative temperature coefficient is predicted to be the largest for the smallest ion Li+, the contrary trend is observed; (3) the predicted temperature dependence for the K+ ion is quite different in magnitude from the observed one in sharp contrast to the excellent agreement between theory and experiment in the case of AC (4) it is unlikely that the continuum theory tends to meet experiment at the higher temperatures where the open structure of water is destroyed; and ( 5 ) the theoretical and experimental curves for the K+ and Cs+ ions crms with a large angle at quite different temperatures. Thus the theoretical value of the Walden product cannot serve as a reliable reference from which any remaining factors controlling ion mobility are abstracted. These confusions which are brought about by the improper transformation indicate the inferiority of the conventional Walden product to the residual friction coefficient based on the new theoretical framework. Solvent Isotope Effect. A study of solvent isotope effect on ion mobility plays an important role in the elucidation of fundamental aspects of the solvent effect on ion dynamics.1° When the solvent is changed from light water H 2 0 to heavy water D20, such dynamic properties as viscosity and dielectric relaxation time markedly (by about 20%) vary with most static properties unvaried. In consequence, the solvent isotope effect is hoped to serve as another criterion for judging which transformation of Xo is more meaningful. The limiting ionic conductances in both isotopic solventsz6have been transformed into the two quantities in question by using the solvent proper tie^,^^.^' and the results are plotted against temperature in Figure 2. As seen in Figure 2a, the direction of the solvent isotope effect is correctly predicted for the smaller ions by the HO theory at both temperatures when the solvent isotope effect is inspected by means of Afi the behavior of the Cs+ ion is again peculiar, and (26) Kay, R. L.; Broadwater, T. L. J . Solution Chem. 1975, 4, 745. (27) Hasted, J. B. In Water; Franks, F., Ed.; Plenum: New York, 1972; Vol. 1, Chapter 7.
0
50 EtOH ( mol ‘le )
100
Figure 3. Comparison of plots of A{ and AD$ for the alkali metal ions in ethanol-water mixtures against the ethanol composition (mol %) at 25 “ C . The broken and full lines indicate the HO theoretical and ex-
perimental results, respectively. the reverse solvent isotope effect is discussed elsewhere.1° On the other hand, such a good correspondence between theory and experiment is lost in Figure 2b. Although the HO theory predicts a considerable difference in the temperature coefficient of XOqO between the two isotopes, such a difference is not clear in the experimental result transformed in this way. Thus as in the case of the temperature coefficient of XOqoin water, the solvent isotope effect on the temperature coefficient of the Walden product makes no sense. Mixed-Solvent Effect. Since the solvent viscosity which plays a central role in A{ and XOqoas explained above can be varied systematically over a wide range by mixing two kinds of liquids, the dependence of the two quantities on the solvent composition is expected to serve as a key criterion for comparing the meaningfulness of the two quantities of our interest. Ethanol-water mixtures are taken here because the full knowledge of the solvent properties required for the application of the dielectric friction theory is available so far only in this case. The theoretical and experimental values of A{ and XOqOwhich are calculated for the alkali metal ions in ethanol-water mixtures from the relevant data compiled elsewhere]’ are plotted against the solvent composition in Figure 3. In the case of the residual friction coefficient, the HO theory correctly predicts the size and composition dependencies. Although a minimum is noticed for the Cs+ ion around 10 mol % of ethanol, the general trends of the observed A{ are surprisingly well reproduced by the continuum theory in spite of the neglect of the molecular nature of the solvents and complicated factors in the vicinity of ion. Furthermore, the development of the minimum with increasing ion size makes sense because it can be ascribed to the effect of the solvent structure ignored in the continuum model. As shown in Figure 3b, correspondence between theory and experiment is again lost by the transformation of Xo into the Walden product. Neither the size dependence nor the composition dependence is comparable with the corresponding theoretical prediction. There appears a maximum in the plot of the experimental value of Xoqo against composition in sharp contrast to the theoretical prediction. The maximum is brought about by the multiplication of Xo by qo which shows a sharp maximum around 20 mol % of ethanol. This is a typical example where the contribution of the viscosity to the ion mobility is neither corrected nor eliminated by taking the Walden product; that is to say, the influence of the solvent viscosity still remains in the “corrected” quantity. Pressure Effect. Pressure is a remaining variable that can vary in a unique manner such dynamic properties of solvent as viscosity and dielectric relaxation time. The theoretical and experimental values of A{ and XOqoin water are calculated as a function of pressure by using the relevant data’ and plotted in Figure 4. As shown in Figure 4a, the HO theory predicts that A{ decreases with pressure as a result of the decrease in RHOin eq 9; note that qo of water is constant within 2% in the range of pressure. In the case of the smallest ion Li+, the pressure effect is correctly
3030
Nakahara and Ibuki
The Journal of Physical Chemistry, Vol. 90, No. 13, 1986 1
I
I
I
I
I
I
I
1000 P ( kg cm-2 )
I
2000
Figure 4. Comparison of plots of A{ and XovOfor the alkali metal ions in water against pressure at 25 OC. The broken and full lines indicate the HO theoretical and experimental results, respectively. 1 kg cm-* = 0.9807 bar.
0lo&5
I
1
a 1
500 P
( 1000 kg
) 1500
0.6
0 1
500
1000
1500
P ( kg cm-’ )
predicted by the dielectric friction theory. However, the observed pressure coefficient tends to change its sign with increasing ion size and this tendency has been ascribed to the pressure effect on the open structure of water.’ As shown in Figure 4b, the pressure effect of the Walden product for the smallest ion Li+ in water is soundly predicted by the dielectric friction theory; this is in sharp contrast to the troubles of the Walden product pointed out above. With respect to the pressure effect in water, A{ and XOqOare almost equally useful fortuitously. The exceptional success of the Walden product for the pressure effect cannot be taken as a justification of this transformation. The viscosity of water is almost constant in the pressure range as an indication of its anomaly, so that the problem caused by taking the multiplication of Xo by 8’’ does not take place; in other words, the pressure effect on both A{ and Xoqa is dominated by the “decrease” in the dielectric relaxation time. In order to confirm this insight, the pressure effect in mixed solvents will be discussed below. Pressure Effect in Mixed Solvent. As referred to above, the viscosity of water can be varied to a large extent by adding ethanol, and the viscosity of the mixture can be increased by the application of pressure. We use here the conductance data and related properties of the solvent reported in a recent paper2*where the effect of pressure on the Walden product is discussed in terms of the structural effect. Figure 5 shows the pressure dependence of the residual friction coefficient and Walden product for the K+ ion in ethanol-water mixtures for which the dielectric friction theory works well as exhibited by Figure 3a. The viscosity of the solvent mixtures increases with pressure in contrast with that of water, the rate becoming larger with the ethanol ccomposition. Hence we predict
an increase in the residual friction coefficient expressed by eq 9 with increasing pressure and an increase in the pressure coefficient with the ethanol composition. It is very clear in Figure 5a that what we have predicted is really observed in the proper transformation. In Figure Sb, on the other hand, the curves for different solvent compositions cross in a very complicated manner. Therfore, it is dangerous to try to interpret the pressure coefficients of the Walden products in spite of the wide popularity. If people who do not understand the significance of the dielectric friction theory dare to interpret the temperature, pressure, or solvent-composition dependence of the Walden product, the following two cases are conceivable: (a) one may place an unreasonable accent on the effect of the solvent liquid structure neglecting the effect of temperature, solvent, and pressure on the dynamic electric interactions between the charge on ion and solvent and (b) the other may attribute the variation of the Walden product wholly to that of the “Stokes radius”. The latter attitude which may lead to the solvent-berg model is based on the old Stokes-Einstein-Walden framework and should be replaced by the new dynamic view explained in the present paper. The extent to which the structural effect contributes to the mechanism of ion migration has to be judged on the basis of some reliable reference. In this respect it is desired that the Hubbard-Onsager theory which works better than the Zwanzig theory is applied to the interpretation of experimental results which are analyzed in accordance with the procedure described here.
(28) Ueno, M.; Tsuchihashi, N.; Shirnizu, K. Bull. Chem. SOC.Jpn. 1985, 58, 2929.
Acknowledgment. This work is supported in part by the Research Grant-in-Aid from the Ministry of Education, Science and Culture (No. 60129031).
Figure 5. Comparison of plots of A( and Xoqo for the potassium ion in ethanol-water mixtures against pressure at 25 O C . Each numerical value near the line indicates the solvent composition (mol % of ethanol). 1 kg cm-2 = 0.9807 bar.
