Ion Binding to Poly(oxyethy1ene) in Methanol
latter for the p-nitroperbenzoic acid results could be appreciable.
Effect of Brownian Motion of the Ligand When the target area is smaller than the mean free path of the ligand, the diffusion-controlledrate constant cannot be deduced from a simple diffusion equation.36 However, the mean free path, L, calculated from the following equation%is only 0.07 8,for p-nitroperbenzoic acid, where it4 is molecular weight and R is gas constant: l n/f 1112 L = De\%) Therefore, the effect of Brownian motion is almost negligible for the horseradish peroxidase-ligand system.
References and Notes
.,
(1) G. G. Hammes and P. R. Schimmel. Enzymes, 2, 67 (1970); A. Fersht, "Enzyme Structure and Mechanism", W.H. Freeman, San Francisco, 1977, pp 129-133. (2) S. H. Koenig and R. D. Brown, 111, froc. Natl. Acad. Sci. U .S.A ., 69, 2422 (i972). 13) . , S. Lindskoa and J. P. Coleman, froc. Nafl. Acad. Scl. U.S.A., 70, 2505 (1973). (4) B. Jonsson and H. Wennerstrom, Biophys. Chem., 7, 285 (1978). (5) E, M. Fielden, P. B. Roberts, R. C. Bray, D. J. Lowe, G. N. Mautner, G. Rotllio, and L. Calabrese, Biochem. J., 139, 49 (1974). (6) H. B. Dunford and W. D. Hewson, Biochemistry, 16, 2949 (1977). (7) P.4 Ohlsson and K . 4 . Paul, Acta Chem. Sand.. Sect. 6 , 30, 373 (1976).
The Journal of Physical Chemistry, Vol. 83, No. 20, 1979 2805
(8) M. L. Cotton and H. B. Dunford, Can. J . Chem., 38, 582 (1973). (9) D. M. Davies. P. Jones. and D. Mantle. Biochem. J.. 157, 247 (1976). ( i O j S. Marklund, P.4 Oh&, A. Opara, and K.G. Paul, Bkchim. Biophys. Acta, 350, 304 11974). (11) M. Eigen, Z. Elektrochem., 64, 115 (1960). (12) M. Elgen, 2. Phys. Chem. (Frankfut? am Main), 1, 176 (1954). (13) S. Loo and J. E. Erman. Blochim. BioDhvs. Acta. 481, 279 (1977). (14j M. W. Smoluchowski, Z. fhys. Chem:, 92, 124 (1917). (15) "Handbook of Chemisby and physics", 57th ed.,CRC Res,Cleveland, 1976, p F62. (16) T. L. Hill, Proc. Nati. Acad. Sci. U.S.A.,72, 4918 (1975); 73, 679 (1976). (17) A. Szabo, Proc. Natl. Acad. Scl. U . S . A . , 75, 2108 (1978). (18) R. W. Noble, Q. H. Gibson, M. Brunori, E. Antonlni, and J. Wyman, J . Biol. Chem., 244, 3905 (1969). (19) R. H. Austin, K. W. Beeson, L. Eisensteln,H.Frauenfelder and I. C. Gunsalus, Biochemistry, 14, 5355 (1975). (20) J. C. Kendrew, Brookhaven Symp. Biol., 15, 216 (1962). (21) S. E. V. Phillips, Nature (London), 273, 247 (1978). (22) W. J. Albery and J. R. Knowles, Biochemistry, 15, 5631 (1976). (23) J. R. Knowles and W. J. Albery, Acc. Chem. Res., 10, 105 (1977). (24) K. S. Schmitz and J. M. Schurr, J. fhys. Chem., 76, 534 (1972). (25) R. A. Alberty and 0.0. Hammes, J. fhys. Chem., 62, 154 (1958). (26) P. Rlchter and M. Eigen, Biophys. Chem., 2, 255 (1974). (27) F. C. Collins and G. E. Kimball, J . Colloid Sci., 4, 425 (1949). (28) R. M. Noyes, frog. React. Kinet., 1, 129 (1961). (29) A. S. Brill, Comp. Biochem., 14, 447 (1966). (30) J. Ricard, M. Santimone, and 0. Vogt, C . R. Hebd. Seances Acad. Sci., Ser. D ,267, 1414 (1968). (31) J. E. Critchlow and H. B. Dunford, J . Biol. Chem., 247, 3714 (1972). (32) 0. R. Schonbaum, J . Biol. Chem., 248, 502 (1973). (33) 6. H. J. Bielski and S. Freed, Biochim. Bbphys. Acta, 8% 314 (1964). (34) P. Douzou, R. Sireix, and F. Travers, frm. Natl. Acad. Sci. U.S.A., 66, 787 (1970). (35) A. L. Fink, Acc. Chem. Res., 10, 233 (1977). (36) M. Doi, Chem. fhys., 11, 107, 115 (1975).
Conductometric Studies of Ion Binding to Poly(oxyethy1ene) in Methanol Katsumlchi Ono, Hideo Konami, and Kenkichi Murakami Chemical Research Institute of Non-aqueous Solutions, Tohoku University, Sendai, Japan (Received Februav 2, 1979) Publication costs assisted by Tohoku University
The binding of alkali metal cation with poly(oxyethy1ene)was studied by means of conductometry in methanol. Apparent binding constants, KA, were determined by assuming the existence of discrete binding sites distributed every a monomer units along the polymer chain. KA increased in the order Li < Na < K < Cs < Rb. For large anions, such as thiocyanate or perchlorate, the formation of an ion pair complex was indicated. KA and a increased rapidly with decreasing salt concentration. This result was interpreted by the increased electrostatic repulsion between bound ions. The linear relationship between all2 and C;lI2 conforms to the Gaussian distribution of the segment between charges as well as to the partial retention of crystalline conformation of the poly(oxyethy1ene)-ion complex.
Introduction Recently, there has been a growing interest in investigating the interaction between metal ions and neutral macromolecules which have no electric charges on the molecular chain. Such interaction plays a fundamental role not only in the selective transport of metal ions through membranes, and in the solubilization of inorganic salts in organic solvents or phase transfer catalysis by polymeric additives, but also in biological enzyme systems. Linear poly(oxyethy1ene) (POE) seems to be an excellent model for the investigation of this type of interaction, since it has been reported that this polymer forms complexes with a variety of metal salts in organic solvents in a manner similar to macrocyclic polyethers.'" In cyclic polyethers, the strength of the complex formation is primarily de0022-3654/79/2063-2665$01 .OO/O
termined by the intrinsic properties of the given pair of metal ion and cyclic ligand, and does not depend on the concentration of metal ~ a l t s . ~On J the other hand, the electrostatic interaction between ions which are complexed with the different sites on the same polymer chain must be taken into account in the case of linear POE. One of the objects of this paper is to clarify the above difference between the mechanism of complexation of linear and cyclic polyethers. The abnormal properties of POE-salt solutions was first observed by Lundberg and others' in methanol. They observed an increase of the reduced viscosity of methanolic POE-potassium halide solutions with increasing dilution as well as the solubilization of POE in this solution. Similar polyelectrolyte behavior was also observed for ionic 0 1979 American Chemical Society
2666
The Journal of Physical Chemistty, Vol. 83, No. 20, 7979 10
L
K. Ono, H. Konami, and K. Murakami
I l12 3
t
/
-
II
'
/
3:
/
A-
{\I O -
d
0 98-
4L 0
I
0 05
0 10 Cp/mol d i 3
I
0 15
0 20
7 2
3
4
5
6
Log M
Figure 1. Apparent equivalent conductance, A, for methanolic POE-KSCN solutions plotted against POE concentration, C, (monomer mol). The average molecular weight of POE is (0)200, (0)1 X IO3, (0)3 X lo3, ( 0 )7.5 X IO3, and ( 0 )2 X lo4: [KSCN] = 5 X mol dmT3;temperature, 25 OC.
Flgure 2. Relationships between apparent equivalent conductance or Walden product and logarithmic molecular weight of POE for methanolic POE-KI solutions; qr is the relative viscosity. The concentrations of POE and K I were 5 X IO-' monomer mol dm3 and 5 X IO3 mol dm3; respectively. The temperature was 25 OC.
solutions of poly(crown ethers) and their copolymers in acetone or methyl ethyl ketone,8 and this may be attributable to the electrostatic repulsion between the complexed ions. The strong interaction between POE and alkali metal salts was also demonstrated by NMR2p9and ~alorimetry.~ In order to determine the stoichiometry of the reaction, attempts were made to separate the crystalline complex from the s01ution.lO-l~ Although these investigations have revealed some important features of this specific interaction in the solution, it seems that no studies on the equilibrium properties of the reaction have been made on the quantitative basis. I t is essential to determine the number of oxyethylene unit which take part in the complex formation for the proper estimation of binding constants of ions to POE chain. Besides, information concerning the density and the distribution of bound ions is necessary for the elucidation of the magnitude of the intramolecular electrostatic repulsion. In order to obtain this information, a precise determination of the degree of complex formation is required. In this respect, conductometry seems the most simple and effective tool, if the appropriate attention is given to data analysis. In this paper, the chemical equilibrium of POE-alkali metal salts in methanol was investigated by means of conductometry. The conductance data were analyzed on the assumption of the existence of discrete binding sites on the POE chain.
POE-alkali metal salts in methanol was measured for various concentrations of POE under a constant salt concentration.
Experimental Section Poly(oxyethy1ene) samples, with average molecular weight ranging from 200 to 2 X lo4, were obtained from Wako pure chemicals and used without further purification. The absence of ionic impurities in these samples was verified by conductance measurement of methanolic polymer solution. High molecular weight POE samples (Polyox, Union Carbide Co.) with mol wt 2 X lo5and >5 X lo6 were used after purification by dialysis or gel filtration. Monodisperse POE samples, which were kindly supplied by Toyo Soda Manufacturing Co., were also used after dialysis. Except where indicated, the measurements were carried out for a sample with mol wt 2 X lo4. All inorganic salts were reagent grade and dried under vacuum before use. The conductance measurements were performed with a transformer bridge at 3000 Hz in a liquid paraffin bath thermostated at 25 f 0.02 "C. The electrodes were the conventional two parallel platinum plates which were lightly platinized with platinum black. The cell constant was determined with standard methanolic KC1 solution^'^ and calculated to be 0.08487 cm-l. The conductance of
Results and Discussion The apparent equivalent conductance A of POE-KSCN solutions in methanol is shown in Figure 1 as a function of POE concentration, C,. As shown in the figure, a remarkable decrease in A was observed. This decrease was larger for high molecular weight POE, but A exhibited an almost constant value above mol wt 2 X lo4. For 0.2 M POE in 5 mM methanolic potassium thiocyanate, the decrease amounted to about 40%, as compared with POE-free solution. Two sources of decrease in conductivity should be taken into consideration. One is of course ion binding to POE, and the other is the viscosity effect. In electrolyte solutions, the conductivity-viscosity products show approximately a constant value, which is well known as Walden product.15 In order to estimate the viscosity effect, Walden products for 0.05 M POE solutions were plotted against the logarithmic molecular weight (Figure 2). It is evident that the products are far from constant and markedly increase in the high molecular weight region. This result, and the above described fact that A vs. C curves coincide with each other above mol wt 2 X 1 08, indicates that the mobility of ions is governed primarily by local viscosity rather than bulk viscosity, and that the effect of viscosity on the observed conductivity decrease is very small. This observation is consistent with the results obtained for the ionic solutions of other neutral macromolecules.16~17 The observed decrease in A, therefore, may be regarded as caused by ion binding alone. Molecular weight distribution did not exert any effect on the conductivity above mol wt 2 X lo4. The following assumptions were made in order to analyze the conductivity curves. (1)Ions are presumed to bind to discrete sites of the POE chain. If the sites are distributed every a monomer units along the chain, its concentration is given by C,/a. (2) Under a given concentration of salts, the apparent binding constant, KA,of free binding sites is virtually constant, regardless of the occupancy of the other sites along the same polymer chain. This means that the interaction between each pair of sites is negligible. (3) The electrostatic potential set up by bound ions does not exert a serious effect on the mobility of free ions. As has been pointed out by Manning,l* this is not true for ordinary polyelectrolytes, since the charge which was not compensated by counterion binding is capable of interacting with the free ions. In the present case, however, the
The Journal of Physical Chemistry, Vol. 83, No. 20, 1979 2667
Ion Binding to Poly(oxyethy1ene) in Methanol
TABLE I: Conductance Parameters for POE-Alkali Metal Iodide Solutions cation ionic diam, A aa log K*/dm3 mol-' NaI KI RbI CSI
2.08 .i: 0.14 2.67 i. 0.07 2.82 i 0 . 0 1 2.77 0.03
28.3 i. 6.1 12.3 ? 0.7 13.2 i 0.1 14.5 i. 0.4
1.94 2.66 2.94 3.34
*
a
103hps,aS m a mol-'
SEb
5.18 i. 0.11 5.29 f 0.07 5.38 i 0.01 5.36 i. 0.05
0.148 0.162 0.035 0.103
The error limits shown in this table are the standard errors of the conductance parameters which were determined by Standard error of estimate. nonlinear regression analysis. a
charge density on the POE chain is far less than that for typical polyelectrolyte as will be shown later. Suppose the complexation reaction is represented by the following equilibrium: binding site (Cpla) C,(1 - a )
KA + cation ? binding site-cation
C,(l
C,a
complex
- a)
(1) where C, is the salt concentration in molarity and a is the fraction of free salt. The concentration equilibrium constant, KA, for the complexation is given by KA = (1- a)/((Cp/a) (2) Here, the formation of ion pairs was neglected. On the basis of the assumptions mentioned above, a is given by A - A,, a=(3) A, - A,, where A, and Ap are the equivalent conductances for free salt and complex, respectively. Combining eq 2 and 3, we obtain the following expression for A:
In this equation, A is represented as a function of both C, and C,, and analysis of the conductivity curves may be performed at a constant C, as well as constant C,. In the former case, however, the apparent binding constant, KA, is presumably dependent on salt concentrations and cannot be regarded as a constant. In addition, the determination of parameters by this method is sensitive to experimental errors, since the calculation is based on small conductivity difference~3.l~ Accordingly, we analyzed A vs. C, curves at constant salt concentrations. The curves were fitted by a nonlinear multiple regression computer analysis, based on eq 4, with KA, a, and Aps as adjustable parameters. Standard errors, SE,given in the last column of the tables are those between the experimental conductance and the estimated values. The agreement between experimental and estimated values was satisfactory and S E was less than 0.2% in most cases. This value is within the experimental error. These results indicate the applicability of eq 4 to analysis of conductivity curves as well as the validity of the original assumptions. Another expression for eq 4 is given as follows:20 - =CP -
A,- A
aC,
A s - A,,
a 1 + KAA-___ A,,
(5)
This equation is a modified form of the well-known Scatchard plot,21applied to conductivity. If there are no interactions between sites, a plot of C,(A, - A)-I vs. (A A ,)-l should yield a straight line. This was actually ogserved in the present case (Figure 3). It must be noted that a linear relationship does not necessarily indicate the
"
0
5
10
15
20
I~ - ~ ~ - ~ ~ , ) - ' / n m ~ ~ m ~ t
Figure 3. Modified Scatchard plots for POE-KSCN solutions. The symbols are the same as in Figure 1.
absence of an electrostatic interaction between sites but it depicts a similar electrostatic environment for each binding site. Conductance parameters and their standard errors for POE-alkali metal iodide solutions are given in Table I. Data for lithium iodide were not presented, since this salt did not bind with POE in methanol. The observed difference in the binding ability is clear evidence for the formation of a cation complex, because the difference on the basis of anionic species with a common cation was not large as will be shown later. The binding constant for the sodium ion is an order of magnitude smaller than that for other cations. The order of the binding ability for series of cation is similar to the one reported for crown ethers. However, there seems to be a difference with regard to the relative magnitude of the binding constant. The binding constant for typical crown ethers with five or six oxygen atoms shows the largest value for the potassium ion and decreases for rubidium and cesium ions.22 The binding constant difference between potassium and cesium is about an order of magnitude in methanol. On the other hand, as shown in Table I, maximum binding was observed for the rubidium ion and the binding strengths for potassium and cesium are comparable for POE complexes. The relatively strong interaction with POE observed for large cations is probably due to the flexible structure of the POE chain which enables the chain conformation to change with the ionic diameter of the cation. The parameter a increases with an increase in ion size except for the sodium ion. This value may be regarded as the minimum number of oxyethylene unit between adjacent charges, and may be dependent upon the relative strength of the chemical binding and the electrostatic interaction. The large a value observed for the sodium ion can be interpreted by the weak binding force as compared with electrostatic one for this ion. As described above, ion size is a significant factor in determining the binding constant. It should be noted, however, that solvation to cations also plays an important role especially for lithium and sodium ions. The low binding constant for these ions may be explained by the strong solvation of methanol molecules, which in turn makes it difficult to interact with POE.
2668
K. Ono, H. Konami, and K. Murakami
The Journal of Physical Chemistty, VoL 83, No. 20, 7979
TABLE 11: Conductance Parameters for POE-Potassium Salt Solutions a log K ~ / d m mol-' ~ 103~,,, S m2 mol-' KCl KBr KI KSCN KCIO,
13.7 t 13.1 f 12.3 t 12.1 f 12.2 f
0.5 1.0 0.7 0.7 0.4
2.58 2.60 2.67 2.70 2.88
f
t i. jl
f
0.04 0.08 0.07 0.07 0.05
The dependence of conductance parameters on the types of anion for POE-potassium salt solutions is summarized in Table 11. A significant difference, although small in comparison with that for cation types, was observed. The hypothetical equivalent conductance of complex, Aps, may be expressed as Aps = ApMt AA(6)
+
where ApMt and AA- are the ionic equivalent conductances of the POE-cation complex and anion. From this equation, we can estimate APMt from the difference between Aps and the equivalent conductance of the anion. This value is also shown in Table 11. The ionic equivalent conductance for anions at a given concentration was estimated from the reported conductivity values for alkali metal halide in m e t h a n ~ l . It ~ ~should ? ~ ~ be noted that (A, - AA-) values for thiocyanate and perchlorate are negative. This result strongly suggests ion-pair formation between the cation complex PX- and the anion A- in these solutions which is represented by the equilibrium PX- + A- e PX-A+ (ion-pair complex) (7) The apparent increase of KA for thiocyanate and perchlorate can be also explained by ion-pair formation. Determined values for K Ainvolve the contribution not only from the cation binding but also from ion association, although the latter effect is very small. Since ion association of investigated ions in pure methanol can be regarded as negligible,24the difference between anion types may be solely explained by ion-pair formation of the cation complex represented by eq 7 . The order of the binding constants, KA,for various salts with a common cation is similar to that previously reported for the ion association ,~~ constant of tetrabutylammonium salts in l - p r o p a n ~ land may be regarded as the general pattern for the ion association in hydroxylic solventsaZ6 This is additional evidence for the existence of an ion-pair complex. The most important difference between the ion binding of POE and macrocyclic polyethers is the dependence of the binding constant on the ionic strength of the solution. The conductance parameters for potassium thiocyanate solutions are given in Figure 4 as a function of salt concentration, C,. The parameter a increased sharply with a decrease of C,. This behavior can be interpreted by the increase of electrostatic repulsion among bound ions, since screening by counteranions becomes ineffective at low salt concentrations. As a result, the charge density of binding ions on the polymer chain diminishes markedly in this region. In order to confirm the above argument, the a value for a 1 mM KSCN solution which contains 4 mM lithium thiocyanate was determined (Figure 4). As mentioned previously, lithium salts have little ability to form complexes with POE. The depression of a observed for the solution with LiSCN clearly shows that the decrease of a in the low C, region originates from the reduction of electrostatic screening. The observed behavior, therefore, resembles the expansion of salt-free polyelectrolytes. It is instructive to consider the relationship between a values and Debye's effective screening length K - ~ . In the first approximation, we assume that the separation between any given pair of binding cations is represented by
4.60 4.87 5.29 5.17 5.54
SE
0.05 0.09 0.07 0.08 0.04
f
t f
k f
103(Aps- A A - ) ,
0.069 0.164 0.162 0.196 0.153
s m2 mol-'
2.9 1.5 0.6 -0.6 - 3.5
40 CJ 30-
20
-
IO-
4 6 10%s/mol d m - 3
2
0
8
IO
-.-
Flgure 4. Parameter a and logarlthmic blnding constant plotted as a function of sat concentration for POE-KSCN solutlons. A crossed circle denotes the a value calculated for a mixed salt solution of 1 mM KSCN and 4 mM LiSCN. The temperature was 25 OC.
15
10 &2
5
0
I
I
IO
20
I
I
30 4 0 (Cs/mol dm-3?'2
I
50
Figure 5. Square roots of a and (N,/v) plotted against the Inverse square roots of KSCN concentration. A point on the vertical axis is calculated from the stoichiometric value of the crystalline POE-KSCN complex, a = 4. N,/v was calculated at a constant C, of 0.05 mol/dm3.
Gaussian distribution. In this case, the minimum charge separation should be proportional to the square root of a. This value is expected to have some correlation with the screening length K - ~ , Le., with the reci rocal of the square root of C,. The correlation between alE and C1;2/ is shown in Figure 5. The average charge separation length may also be correlated with C;1/2. If v cations bind on a POE molecule with N monomer units, the average number of monomer units getween any given pair of charges is denoted by ( N p / v ) . This value is given by (NP/v)= C,(1 - a)/Cp (8) where a is determined by eq 3, using the calculated value of Ap,. The average charge separation length may be proportional to the square root of (Np/v), which is also correlated with Ci1f2in Figure 5. In fact, good correlations exist between the square root of a or ( N p / v )and C;1f2. A plot of u1I2 vs. was approximated by a linear
The Journal of Physical Chemistry, Vol. 83, No. 20, 7979 2669
Ionic Mobility
least-squares fit with a correlation coefficient 0.995:
d 2= 2.06
+ 0.101C;1/2
(9)
The square of the intercept, ao, has the meaning of the minimum number of monomer units between charges at the high concentration limit and is calculated to be a. = 4.28. It must be emphasized that the stoichiometry of a crystalline POE-KSCN complex obtained from methanol solution was reported to be 4:l.l' The close agreement between the stoichiometry of the crystalline complex and the one calculated from the limiting behavior in solution strongly suggests that the crystalline structure is partially retained in the solution, and that the binding sites consist of four monomer units of POE. Furthermore, the above results conform to a Gaussian distribution of POE segments over a wide range of salt concentrations. The persistence of the Gaussian distribution may be attributed to the relatively high electrostatic screening and to the low charge density in the polymer coil in this peculiar system. There is also a good correlation between ( N and CL'/~. The slight deviation from the straight fine at high concentrations may be ascribed to the onset of electrostatic interaction between binding sites. A detailed analysis of electrostatic interaction between bound charges will be given in a following paper.27
Acknowledgment. The authors express their gratitude to Dr. M. Fukuda, Toyo Soda Manufacturing Co., for providing monodisperse poly(oxyethy1ene). Valuable comments of Professor N. Imai, Nagoya University, and Professor Y. Wada, The University of Tokyo, are also appreciated. References and Notes (1) R. D. Lundberg, F. E. Bailey, and R. W. Caliard, J . Polym. Sci., A 1 , 4, 1563 (1966).
K. Liu, Macromolecules, 1, 308 (1968). S. Yanagita, K. Takahashi, and M. Okahara, Bull. Chem. SOC.Jpn., 50, 1386 (1977). Z. N. Medved and A. K. Zbitinkina, Vys. Soed., B, 20, 475 (1978). K. Ono, H. Konaml, and K. Murakami, Rep. Rag. Polym. Phys. Jpn., 21, 17 (1978). J. J. Christensen, D. J. Eatough, and R. M. Izatt, Chem. Rev., 74, 351 (1974). D. F. Evans, S. L. Wellington, J. A. Nadis, and E. L. Cwsler, J. Solutlon Chem., 1, 499 (1972). S. Kopolow, T. E. Hogen Esch, and J. SmM, Macromolecules, 6, 133 (1973). S. Yanagita, K. Takahashi, and M. Okahara, Bull. Chem. SOC.Jpn., 51, 1294 (1978). A. A. Blumberg, S. S. Pollack, and C. A. J. Hoeve, J . Polym. Scl., A2. 2. 2499 119641. R. Icmoto, $. Saito, H. Ishihara, and H. Tadokoro, J. Polym. Scl., A2, 6, 1509 (1968). D. E. Fenton. J. M. Parker, and P. V. Wriaht, Pokmr, 14, 589 (1973). S. Yanagita, K. Takahashi, and M. Okakra, 5bll. Chem. Soc. Jpn:, 51. 3111 (1978). R. E. Jervis, D. d. Nuir, J. P. Butler, and A. R. Gordon, J. Am. Chem. Soc., 75, 2855 (1953). R. A. Roblnson and R. H. Stokes, "Electrolyte Solutions", 2nd ed, Butterworths, London, 1959. J. Komiyama and R. M. Fuoss, Proc. Nafl. Acad. Scl. U.S.A., 69, 829 (1972). H. 0. Phillips, A. E. Marclnkowsky, and K. A. Kraus, J . Phys. Chem., 81, 679 (1977). G. S. Manning, J. Phys. Chem., 79, 262 (1975). J. Jagur-Grodzinskl, Bull. Chem. Soc. Jpn., 50, 3077 (1977). I n our preliminary report (ref 5),a rough estlmation of conductance parameters was carried out by eq 5,in which a correction for the conductivity was made by subtracting the conductivlty decrease of a standard sample. This correction procedure was found to be unnecessary and was omitted in the present work. G. Scatchard, Annu. N . Y . Acad. Sci., 51, 660 (1949). H. K. Frensdorff, J . Am. Chem. Soc., 93, 600 (1971). J. P. Butler, H. I. Schiff, and A. R. Gordon, J . Chem. Phys., 19, 752 (1951). C. W. Davies, "Ion Assoclatlon", Butterworths, London, 1962. D. F. Evans and P. Gardam, J. Phys. Chem., 72, 3281 (1968). R. L. Kay, D. F. Evans, and M. A. Matesich in "Solute-Solvent Interactions", Vol. 2, J. F. Coetzee and C. D, Ritchie, Ed., Marcel Dekker, New York, 1976. K. On0 and K. Murakaml, J . Phys. Chem., to be submitted for publication.
Ionic Mobility. Theory Meets Experiment D. Fennel1 Evans,* T. Tomlnaga, Department of Chemical fnglneering, Carnegie-Mellon University, Pittsburgh, Pennsylvanla 152 I3
John B. Hubbard, and P. 0. Wolynes Depatfment of Chemistry, Harvard University, Cambridge, Massachusetts 02 138 (Received January 18, 1979) Publication costs assisted by the Natlonal Science Foundation and the Petroleum Research Fund
l'wo recent theories for ionic mobility, the Hubbard-Onsager theory which is based on a continuum model and the Wolynes theory which is based on a stochastic model, are critically tested by comparison with conductance data. Both theories predict finite mobilities as the ionic size decreases and thus can successfully account for many of the observed features of conductance data. The models upon which the theories are based are described in detail and the current state of ionic mobility theory is discussed. Introduction What determines the mobility of an ion in a liquid? This is one of the oldest unresolved questions in physical chemistry. The most thoroughly explored approach to the problem of ionic mobility at infinite dilution is based on continuum models. In 1906 Waldenl observed that the conductance-viscosity product was often constant for a salt in several solvents. Einstein2 and Nernst3 had established the formal basis for the relationship between ionic velocity 0022-365417912083-2669$0 1.OO/O
and solvent viscosity, using Stokes' law with the implicit assumption that a continuum model applied, even when solute and solvent were of comparable size. This initial continuum model was unsuccessful, as were numerous attempts at ad hoc correction to it. As early as 1920, Born4 pointed out one reason for the failure of Stokes' law, that the interaction between a moving ion and the solvent dipoles in its vicinity gives rise to additional friction, beyond the hydrodynamic friction considered in the 0 1979 American Chemical Society
