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). In 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 subtractingthe 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. Einstein2and Nernst3had 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
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The Journal of Physical Chemistry, Vol. 83, No. 20, 1979
Stokes’ law model. The importance of Born’s idea was not appreciated until the late 1 9 5 0 ’ ~when , ~ attempts to incorporate the dielectric friction effect into the continuum model began. Other approaches to the problem have also been tried. Ionic mobilities have been treated in terms of the Eyring transition state theory6 and the Hildebrand free volume t h e ~ r ybut , ~ neither approach has been pursued very far because of the large number of parameters associated with both methods. In 1977 and 1978 there have been two major theoretical attempts to predict ionic mobilities. The first, by Hubbard and Onsager,8p9is based on the continuum model originally proposed by Born. The second, by Wolynes,lo is based on a stochastic model employing a correlation function. Neither theory has been rigorously tested against experimental data. In this paper we describe the premises on which these theories are based, test them against experiment, and discuss future directions in the field.
Evans et al.
static) drag force on a moving ion which is proportional to the product of TD and ion velocity, and this effect is added to the usual viscous or hydrodynamic drag which, in a first approximation, is given by p = 6 ~ q R i i (stick)
p = 4nqRii (slip) (3) with 7 as the solvent viscosity and R as the ionic radius. A linearized set of hydrodynamic equations, of the form qV2v’ = grad p
+ PDF
av’ = 0
(4)
yas derived and solved for fluid velocity v’, where RDF(9, Po, Q) is the contribution of dielectric friction to the total force density in the fluid. A friction coefficient was extracted from v’ by standard hydrodynamic procedures and it was ascertained that dielectric friction is a significant factor in ion mobility, provided that
R4/a I1 Continuum Theories of Dielectric Friction where In this section we shall discuss the relevance of continuum models to the problem of ionic mobility, where we (5) define a continuum model as any idealization based on a discrete ion immersed in a solvent whose properties (density, velocity, viscosity) vary continuously from point Thus, dielectric friction (DF) is important for small ions, to point in space. We wish to make a clear distinction multivalent ions, and ions in a solvent having a low dibetween continuum models and stochastic models based electric constant or characterized by a long relaxation time. on the theory of Brownian motion, which will be discussed For the condition R4/a >> 1, Hubbard and Onsager (HO) later. Continuum models are characterized by linear obtained for the friction coefficient, 6 integral or differential equations whose solutions are completely determined once the boundary conditions are 6 = 67rqR + €0 - E , (stick) specified. The stochastic theory, which is based on the computation of a force autocorrelation function, allows for (61 much more flexibility in both formulation and numerical calculation. At this time the only continuum models of 6 = 47r7R + 15 R3 (slip) ionic mobility are those based on an extension of Stokes’ law so as to incorporate the so-called “dielectric friction For solvents with a high dielectric constant, the earlier effect”. This model was proposed as far back as 1920 by Born and has been recently reviewed by Boyd,ll F U O S S , ~ ~analysis by Zwanzig yielded a factor of 3/16 instead of 17/280 for stick, and 3/8 instead of 1/15 for slip. The Zwanzig,13and Hubbard and O n ~ a g e r Inasmuch .~~~ as the discrepancy is in part due to Zwanzig’s neglect of the F D F physical basis underlying all the proposed continuum term appearing in eq 4. models is the same, disparities in their predictions should An important result of the HO analysis is that the be attributed to an inconsistent or incomplete analysis; we functional form of eq 6 no longer holds when DF is imare dealing with only one continuum theory of dielectric portant and, in fact, an infinite series in inverse powers friction and not a collection of theories. of R is required for R4/a > 1, while an infinite series in The most thorough analysis of a continuum model is due positive powers of R is generated for R4/a < 1. If R is to Hubbard and Onsager, who pointed out inconsistencies taken to be the crystallographic radius of the ion, it was in the earlier publications on this subject. The ion is noted that condition 5 applies for the alkali metal ions and treated as an impenetrable sphere with a symmetric charge nearly all multivalent ions in water or methanol at room distribution, while the surrounding solvent is regarded as temperature. For example, R 4 / a = 0.17 for Na+ and 0.63 an incompressible fluid having a uniform viscosity and for K+ in water a t 25 “C. This indicates that within the dielectric constant. Electrostriction and dielectric satucontext of the model, dielectric friction is the dominant ration effects are ignored. Each infinitesimal volume in source of drag for many common ion-solvent systems. the fluid supports a static polarization In the HO treatment, a curious “hydrodynamic saturation effect” becomes apparent for R 4 / a 0; the friction coefficient tends to a finite limit independent of the boundary conditions (stick or slip); that is where gois taken as the Coulomb field of an ion, e , in a G(point) = (15.624)~Al/~ (7) medium of dielectric constant €6
(%)$( 7) (L)@(
-)
I^\
-
-
In addition, each fluid element is endowed with a single Debye dielectric relaxation time, TD,which is taken to be uniform throughout the medium. Since dielectric relaxation in the medium does not occur instantaneously, the local polarization in the medium exerts an (electro-
Recall that Stokes’ law gives 6 = 0, while the first-order perturbation theory (eq 6) indicates 6 m in this limit. Also note that it is not necessary to set R = 0 to arrive at eq 7, a sufficiently large ionic charge with finite R would serve just as well. Of course, a small ionic radius or large value of 6 also implies that dielectric saturation effects (a local depression in the static dielectric constant, local increase in viscosity, etc.) may play an important role.
Ionic Mobility
The Journal of Physical Chemistry, Vol. 83, No.
TABLE I: Hubbard-Onsager Correction Factor A Pstick = Ast,ick6Tr, pslip = Aslip4nr for a / R 4 < 1. Pstick = A a k k p ( p o i n t ) , pslip = Aslipp(point) for R 4 / a < 1. a/R4
Astick
A slip
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.0000
1.0000
1.0157 1.0305 1.0445 1.0579 0.0710 1.0806 1.0960 1.1081 1.1252
1.0255 1.0510 1.0752 1.0983 1.1205 1.1438 1.1712 1.2225 1.3395
R4/a
Actirk
0 0.1
1.0000
1.0000
1.0779 1.1262 1.1661 1.2013 1.2331 1.2609 1.2880 1.3082 1.3354
0.93644 0.92460 0.92384 0.92834 0.90207 0.94490 0.95494 0.96928 1.0003
0.2 0.3 0.4
0.5 0.6 0.7 0.8 0.9
A rlia
That is to say, it may be that one cannot have a large dielectric frictional effect without a corresponding saturation effect, so that it becomes virtually impossible to disentangle the two processes. We shall have more to say on the matter at the end of this section. As shown in Table I, several interesting consequences emerge from a numerical calculation of ionic friction coefficients: (1) The most obvious result is that 6 is completely determined once the boundary conditions at the ion surface are specified and a value of R4/a is chosen. (2) 6(stick) is always greater than G(s1ip) at any given value R4/a,and &(stick)increases monotonically with increasing R4/a;there is no local minimum in 6 for the stick boundary condition. (3) G(s1ip) varies only slightly over the entire range 0 C R4/a C 1.5, with a very shallow minimum located a t about R4/a = 0.5. This implies that 6(point) (eq 7 ) is a very useful scaling parameter to consider when discussing dielectric friction, and, in fact, for R4/a C 1,the friction coefficients for both stick and slip are displayed as multiples of G(point). (4) The friction coefficients computed via a thorough hydrodynamic treatment are significantly smaller than those given by crude estimates (eq 61, except, of course, in the large R case where Stokes’ law is recovered. This behavior is closely related to the principle of least dissipation; its application to the dielectric friction problem is disclosed in HO. We now turn to the obvious question: “When should the Hubbard-Onsager model work and when should it fail?” A better question would be: “When should local solvent “geometry” be important, and when should a continuum model be appropriate?” We believe that the HO theory provides us with some useful answers to this question. First we consider the physical principles that lead to the hydrodynamic saturation effect (eq 7 ) . The case of Couette (constant velocity gradient) flow in a uniform, transverse (with respect to flow) electric field was treated explicitly by Hubbard, with the result that the viscosity q is enhanced by the presence of the field: r
If we now take E to be the Coulomb field of an ion,
20, 7979 2671
evaluated at a distance RH from the ion’s center, we see that, for (Y = 1,we must have RH = all4,which is the same as our earlier criterion (eq 5) for dielectric friction to be important. Thus, in a heuristic sense, dielectric friction can be regarded as an effective increase in the local viscosity around an ion; this is precisely Fuoss’ original interpretation of the effect. Now since momentum diffusion is very fast for r C R H ,becoming “normal” for r > RHm we may infer that the frictional drag on an ion of radius R should be independent of R if R C C RH. Since it makes no sense to speak of momentum diffusing through a single solvent molecule, we must also have the condition that RH be significantly larger than the solvent molecular size. Continuum processes should then outweigh solvent “size” effects only if the length RH = a1/4is significantly larger than the solvent diameter. For water and methanol at room temperature, we have RH = 1.5 and 3.2 A, respectively, and solvent size should therefore be important in both cases, probably less so for methanol than for water. Since RH is proportional to the square root of the ionic charge, we see that a continuum model might work better for multivalent ions. This brings to mind a question posited earlier in this section: whether or not dielectric friction and dielectric saturation effects can ever be considered as mutually independent. Recall that the HO condition for dielectric friction to be important is simply
If we substitute for
TD
where y I 1and a is the radius of a solvent molecule, then eq 9 becomes
We now use the Onsager formula for eo - E,: Eo
- E, =
3P2 -Eo[ a3KBT
2€0
+
Em
]
( 2 +~ 1)2
(12)
where p is the dipole moment of a solvent molecule in the medium, so that eq 11 becomes for large to
where the quantity in brackets is just the potential energy of a solvent dipole at the ion surface divided by the thermal energy. Note that dielectric saturation is important if this ratio is comparable to unity. We therefore conclude that the HO theory, which does not include saturation, is not entirely consistent in a physical sense, and we must infer that a rigid solvent-berg, held together by ion-dipole forces and obeying Stokes’ law, is also physically unrealistic. The above argument does not rule out Stokes’ law solvent-bergs stabilized by other forces, but it does place an important restriction on the nature of those forces; namely, that they be strong and short-ranged.
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The Journal of Physical Chemistry, Vol. 83, No. 20, 1979
Evans et al.
TABLE 11: Solvent Properties solvent
temp, C 0
water
10
25 45 25 25 25 25 25 25 25 25
methanol 1-butanol ethylene glycol formamide acetone acetonitrile dimethyl sulfoxide hexamethylphosphorous triamide
lO"T,/S
€0
Em
87.7' 84.0' 78.3a 71.5' 32.62d 17.1" 37.70f 109.5h 20.56' 35.95m 47.3O 29.6O
5.0a 5.0' 5.2" 5.7" 5.6" 2.95a 2,04f 3' 1.91 2" 5.7O 3.3O
1.77b 1.26b 0.82b 0.53b 4.770 47.70 10.5f 3.9' 0.31' 0.39" 1.96' 8.0°
rl, c p 1.787' 1.306' 0.8903' 0.5963' 0.55 13e 2.571e 16.1gg 3.313' 0.303k 0.340grn 1.998" 3.25p
F. Buckley and A. A. Maryyott, Natl. Bur. Stand. Circ., No. 589 (1958). R. A. Robinson and R. H. Stokes, "Electrolyte Solutions", 2nd ed, Butterworths, London, 1970, p 12. J. F. Swindells, J. R. Coe, and T. B. Godfrey, J. Res. Natl. Bur. Stand., 48, 1, 457 (1952). G. P. Cunningham, G. A. Vidulich, and R. L. Kay, J. Chem. Eng. Data, 12, 336 (1967). e "Selected Values of Properties of Chemical Compounds", Thermodynamics Research Center Data project. A. R. Tourky, H. A. Rizk, and I. M. Elanwar, 2.Phys. Chem. (Frankfurt and Main), 31, 161 (1962). R. P. DeSieno, P. W. Greco, and R. C. Mamajek, J. Phys. Chem., 75, 1722 (1971). h G. P. Johari and P. H. Tewari, J. Phys. Chem., 69, 2862 (1965). S. J. Bass, W. I. Nathan, R. M. Meigham, and R. H. Cole, J. Phys. Chem., 68, 509 (1964). L. R. Dawson, E. D. Wilhoit, and P. G. Sears, J. Am. Chem. SOC.,79, 5906 (1957). D. F. Evans, J. Thomas, J. A. Nadas, and M. A. Matesich, J. Phys. Chem., 75, 1714 (1971). J. Calderwood and C. P. Smyth, J. Am. Chem. SOC.,78, 1295 (1956). rn C. P. Cunningham, G. A. Vidulich, and R. L. Kay, J. Chem. Eng. Data, 12, 336 (1967). " R. Krishmaji and A. Marsingh, J. Chem. Phys., 41, 827 (1964). O H. Behret, F. Schmithals, and J. Barthel, 2. Phys. Chem. (Frankfurt and Main), 96, 73 (1975). p N. P. Yao and D. N. Bennion, J. Electrochem. SOC.,118, 1097 (1971). * E. M. Hanna, A. D. Pethybridge, J. E. Prue, and D. J. Spiers, J. Solution Chem., 3, 563 (1974). a
Acetone /!/4~ / l / 6 ~ 0.5
051-
-
,
l&--
Li+
I
04
..oA0 7
0.2i
0.3
0.2
02
0.4
0.6
l/R (L-'l
0.8
1 .o
0 , 2 y
1.2
Flgure 1. Experimental and predicted values of limiting conductance-viscosity products of various ions as a function of crystallographic size in acetone at 25 O C . Experimental values are taken from D. F. Evans, J. Thomas, J. A. Nadas, and Sr. M. A. Matesich, J. phys. Chem., 75, 1714 (1971).
:::wzo *---
-______ -------------_----
B--
//-----
The limiting ionic conductances are related to the HO frictional coefficient by lzle96500 =p 300
7 '
0.2
,lo:
'
0.2 1/R t i - ' )
~
&-
(15)
Flgure 2. Experimental and Hubbard-Onsager theoretical values of limiting conductance-viscosity products in acetonitrile (AN), dimethyl sulfoxide (DMSO), and hexamethylphosphorous triamide (HMPT) at 25 OC. Experimental values are taken from the followlng: C. H. Springer, J. F. Coetzee, and R. L. Kay, J. Phys. Chem., 73, 471 (1969) for AN; R. Gopal and J. S.Jha, ibid., 78, 2405 (1974), for DMSO; M. I. McElroy, Thesis, Case Western Reserve University, 1973; E. M. Hanna, A. D. Pethybridge, J. E. Prue, and D. J. Spiers, J. Solution Chem., 3 , 563 (1974); C. Atlani and J . 4 . Justice, ibid., 4, 955 (1975), for HMPT.
for R4/a € 1 and p = f r r A for a / R 4 > k. We will compare the predictions of continuum models to experiment. In particular, we will examine how ionic mobilities determined from conductance and transference
measurements vary with ion size, charge, and valency in several solvents that differ markedly in their physical properties. The required solvent properties are given in Table 11.
where IzI is the ionic charge (in esu) and p is expressed in g s-l. Therefore X = lzl(1.544 X lO-'/p). The frictional coefficient, 6, is evaluated by using eq 5 to calculate a, eq 7 to calculate p(point), and Tables I and I1 for a given value of R4/a to obtain the correction factor A for p = p(point)A
The Journal of Pbysical Chemistty, Vol. 83, No. 20, 1979 2673
Ionic Mobility
Water Cs'
t
Rb' K' 4 1
ClO; I- BiCI' i 4 4 4
0.8
0.6
MeOH 1
?
04
06
I
08 1/R(A-'l
*
1'0
-*
Figure 4. Experimental and Hubbard-Onsager theoretical values of limiting condutance-viscosity products in water at 0 (0),10 (O),25 (A),and 40 O C (v).Experimental values are taken from R. L. Kay and D. F. Evans, J. Pbys. Cbem., 70, 2325 (1966); R. A. Robinson and R. H. Stokes, "Electrolyte Solutions", Butterworths, London, 1970, p 465.
in all solvent systems and is a consequence of the very large value of B. In the limiting forms of the HO equation (eq 6 ) , the maximum in Xoq occurs at a larger value of R and Xoq Figure 3. Experimental and Hubbard-Onsager theoretical Values of decreases more gradually as R becomes small. However, limiting conductance-viscosity products in ethylene glycol (EG), forneither the slip nor stick condition describes the change mamide (FA), methanol (MEOH), and 1-butanol (1-BuOH) at 25 OC. Xoq with size in a satisfactory way, and the largest in Experimental values are taken from the following: R. P. Desieno, P. deviations are observed for the smallest ions where the W. Greco, and R. C. Mamajik, J . Pbys. Cbem., 75, 1722 (1971); P. Kirby and 0. Maass, Can. J . Cbem., 36, 456 (1958); F. Accascina dielectric frictional contribution is most important. The and M. Goffredi, Ric. Sci., 37, 1126 (1967), for EG; J. Thomas and conclusions given for both the limiting HO equation and D. F. Evans, J. Pbys. Cbem., 74, 3812 (1970), for FA; R. L. Kay and the Zwanzig equation apply to all solvent systems. D. F. Evans, ibid., 70,2325 (1966); R. L. Kay, J. Am. Chem. SOC., Consequently, we will not repeat this discussion for the 62, 2099 (1960); R. L. Kay, D. F. Evans, and G. P. Cunningham, J. Pbys. other solvents. Cbem., 73, 3322 (1969); R. W. Kunze and R. M. Fuoss, ibid., 67, 385 In the extended HO theory, the predicted values of Xoq (1963), for MeOH; D. F. Evans and P. Gardam, ibid., 73, 158 (1969); Hanns-Ulrich Fusban, Thesis, Freien Universitat Berlln, 1974, for I-BuOH. in acetone for both stick and slip approaches a common finite limit as R approacheszero. There are no pronounced In Figures 1-4, the Walden product, io?, is plotted vs. maxima in the curve. The data for all ions lie below the the reciprocal of ionic radius l/R,14 for monovalent ions values for perfect slip. In fact, except for the alkali metal in aprotic, hydrogen-bonding, and aqueous solutions. ions, the experimental data points lie in the region between There are a number of features of Walden products that slip and stick where hydrodynamic theories are applicable. are common to all of the solvent systems. First, for any At its present level of development, the HO theory cannot given solvent the change of Xoq with solute size is very account for the alkali metal values. The features observed different for cations and anions and gives rise to two in acetone appear to be general for other aprotic solvents. distinct curves. Second, the Xoq curves exhibit maxima. Figure 2 shows the predictions of the HO theory and the These are most clearly seen for the cations where R, data for ions in acetonitrile, dimethyl sulfoxide, and occurs at an ionic size corresponding to the larger alkali hexamethylphosphorous triamide. metal or smaller tetraalkylammonium ions. Third, except The values for Xoq for ions in the hydrogen-bonding for aqueous solutions, the temperature dependence of hOq solvents ethylene glycol, formamide,methanol, and butanol is small. are shown in Figure 3. There are a number of features of Clearly, the different behaviors of cations and anions in the data in this figure that should be pointed out. First, a given solvent cannot be satisfactorily explained by a dein the alcohols, methanol through butanol, (XOq),, continuum theory since there is no direct way to account creases. In methanol R, x R(Me4Nt) while in butanol for cation-solvent or anion-solvent interactions. R,, = R(Et4N+)for the cations. Although the Zwanzig The values of X0v for ions in acetone, a typical aprotic model predicts (XOV)~, which are too low, one-half of the solvent, are compared to the predictions of the Zwanzig experimentally observed values in the alcohols, it does and the HO equations in Figure 1. The predictions for predict a shift of the maximum to larger values of R as one both perfect slip and perfect stick are shown. The Zwanzig goes from MeOH to BuOH. Secondly, for most solvent equation (f = 4)gives a maximum that for cations shows systems, smooth continuous curves connecting the data for the correct positions (Rma = R(Me4N+))and magnitude, cations or for anions can be shown. However, in butanol however, for anions the data are clearly above the predicted Xo(Cs+) is larger than Xo(Me4N+)and X0(C104-)is larger curve. In addition, the predicted curve falls rapidly and than Xo(Me4B-). Thus there are clearly defined disconapproaches zero for the smaller ions. This behavior for tinuities in both the cation and anion curves. Similar smaller ionic sizes is characteristic of the Zwanzig equation trends are apparent but less pronounced in propanol and
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The Journal of fbysical Chemistry, Vol. 83, No. 20, 1979
possibly in ethanol. The most likely explanation is that the transport mechanisms for alkali metal and tetraalkylammonium cations, or for the corresponding groups of anions, are markedly different. Whether such behavior is limited to the higher alcohols is not known. No such discontinuities have been observed in aprotic solvents, perhaps because they have not been sought. A comparison with the extended HO theory shows that the values of the Walden products can be either larger or smaller than the predicted limits. Although one can argue that data points below the predicted range might be accounted for by further refinements of the theory, it is difficult to account for Walden products larger than the predicted values by a similar approach. The temperature dependence of Aoq observed in MeOH at 10 and 25 " C is small and similar to that predicted by the HO theory. The values of AOv for ions in water at several temperatures are shown in Figure 4. It has been argued that the mobility of many ions in aqueous solution is greater than one would expect and that for the larger alkali metal ions and halide ions that excess mobility can be attributed to the unique structure properties of water.15 For the slip condition, the HO theory predicts a maximum Walden product in water at 25 " C of 0.734 (Ao = 82.4) at R,,, = 1.26 A. Thus the predicted maximum conductance is approximately correct, but the size dependence is not. A larger and more complex temperature dependence is observed in water than in most other solvents. For the larger tetraalkylammonium ions, the Walden product increases with temperature, while the opposite behavior is observed for the larger alkali metal and halide ions. The Walden product for E,+N+ and Li+ is almost independent of temperature. The predicted temperature dependence of the HO theory (Figure 4) is considerably smaller and always decreases. We will now examine the behavior of the Walden product for multivalent ions. Figure 5 shows the Walden products for such ions in water, methanol, and formamide as well as the predictions of the Zwanzig and HO theories. The dependence upon charge of the limiting forms of the equation (eq 6) gives Z / ( l + 9). Thus, the predicted Inspection curves decrease rapidly with 1/R after A(o&.,. of eq 5 and 14 shows that the extended HO equation goes as (z)lI2,so that predicted maxima for z = 1 , 2 , and 3 are in the ratio 1:1.4:1.7. In water this corresponds to maximum conductances for perfect slip of 82.4,116, and 142 and for perfect stick of 74.4, 105, and 129, respectively. Thus, the HO theory can account for the magnitude but not the size dependence of ionic conductance of multivalent ions.
A Molecular Approach The ideal theory of ionic motions in dilute electrolyte solutions would explain the mechanism of ionic mobility in terms of the ion-solvent interaction and the molecular motion of the solvent molecules. The development of such a theory is most challenging since the ion-solvent interaction is not perfectly understood, and the theory of molecular motion in pure, simple molecular fluids has only recently begun to be quantitatively useful.16 A beginning in this direction has been made by Wolynes and gives an interesting perspective on the issues involved in the theory of ionic conduction. We shall present in this section a short summary of this approach. The standard language for describingtransport problems from a microscopic, molecular standpoint is that of equilibrium time correlation f ~ n c t i 0 n s . l ~Transport coefficients are related to the dynamical correlation of fluctuations in the system. In particular, the ionic drag
Evans et al.
.--
Lo 9
0
04
08 1/R
12
16
(I-')
Flgure 5. Experimental and Hubbard-Onsager theoretical values of limiting conductance-viscosity products of multivalent ions in water, formamide, and methanol at 25 OC. DMD++ = Me3N(CH,CH )NMe,'+, DiEt" = Et3N(CHz)4N(Et),2+,DiPr" = (n-Pr),N(CH2),N(n-Pr),2', DiBu" = (~-Bu),N(CH,),N(~-BU)~'+,BDS;+= m-C6H,(S03)$-, 1 = [Fe (2,2'-bpy),]'+, 2 = [Fe(l,lO-phen),] , en = enthylenediamine, pn = propylenediamine. Experimental values are taken from the following: R. A. Robinson and R. H. Stokes, "Electrolyte Solutions", Butterworths, London, 1970, p 463; T. L. Broadwater and D. F. Evans, J. fbys. Chem., 73, 3985 (1969); Y. Yamamoto, E. Sumimura, K. Miyoshi, and T. Tominaga, Anal. Cbim. Acta, 64, 225 (1973); S. Katayama and R. Tamamushi, Bull. Cbem. SOC.Jpn., 41, 606 (1968); I. L. Jenkins and C. B. Monk, J. Cbem. Soc., 68 (1951), for water; J. Lange, Z.Pbys. Cbem., A188, 284 (1941); G. P. Johari and P. M. Tewari, J. fbys. Cbem., 69, 2862 (1965), for formamde; T. L. Broadwater, T. J. Murphy, and D. F. Evans, J. fbys. Cbem., 80, 753 (1976); E. Kubota and M. Yokoi, Bull. Cbem. SOC. Jpn., 49, 2674 (1976), for methanol.
coefficient is related to the fluctuations in the random forces exerted on the ion by the solvent molecules: 1 { = - l"(FR(0).FR(+)) dt KBT o The random force contains a rapidly varying part due to the hard collisions of solvent molecules with the ion and a more slowly varying part which comes from the softer, attractive forces operating between the ion and the solvent molecules. The slowly varying part of the fluctuations can be analyzed approximately in terms of the molecular motions of the solvent molecules. The result of such an approximate analysis gives an expression for the drag coefficient:
where ( ( p ) 2 is)the static mean square fluctuation in the attractive forces acting on the ion and T F is their characteristic decay time. Thus we see that, if the attractive forces have very large fluctuations or persist for long times, they will lead to a substantial increase in the drag on the ion, over what it would be if no attractive forces were present. The soft force correlation time, Tp, is determined by the way in which the motion of the solvent molecules changes the force on the ion. The force exerted by a
Ionic Mobility
w Figure 6. I f we imagine cutting off the electrostatic potential inside a sphere of radius R , hydrodynamic effects become small and Hubbard-Onsager theory reduces to Zwanzig’s original result.
solvent molecule on the ion depends on both the relative position and orientation of the solvent molecules and the ion; therefore, both rotational and translational diffusive motions of the solvent molecules are involved. We must realize that motions of different solvent molecules may be correlated and this must be taken into account. The simple molecular theory gives an expression for 733
The Journal of Physical Chemistty, Vol. 83, No. 20, 1979 2675
STOKES LAW
0.E
0.5
0.4 07)
O.?
0.2
0.1
I
0.5
where DiiWand DipRare the translational and rotational diffusion coefficients of the ith solvent molecule, which are experimentally accessible and Dijn’, DiFT,and DijRRare the cross-diffusion coefficients which take into account correlated solvent motions. This expression leads to an estimate of the characteristic decay time of the soft force in terms of the equilibrium solvent structure of the fluid around the ion and the transport properties of the pure solvent. The first important feature of the molecular theory is that it makes clear the microscopic, dynamical underpinnings of the two classic pictures of ionic motion. Both the continuum dielectric friction model and a solvent-berg picture, where the solvated ion acts as a dynamically large composite body, emerge as two different limiting cases of the molecular theory. The dielectric friction picture correspondswith the limit of a very weak, but very long-range potential. One could imagine forcing the ion-dipole potential to be of this form by smoothly cutting it off within a sphere of large radius. In that case both continuum dielectric friction models, Zwanzig and Hubbard-Onsager, give the same result. If the potential is infinitely long range, translational motions of the solvent molecules change the force they exert on the ion much more slowly than the rotational motions of solvent molecules. Thus T F depends only on the rotational diffusion constants of the solvent. Since the potential is assumed to be long range, only the solvent structure far from the ion is needed to evaluate the equilibrium averages in the molecular theory. This structure far from the ion is given, however, by the results of the continuum theory. One finds then that the molecular expression for {reduces to that of the Zwanzig model. This strongly argues that Zwanzig’s original result is exact for such hypothetical, infinitely weak, long-range potentials. In fact, this statement is buttressed by the HO theory. The HO theory reduces to Zwanzig’s result if we consider a situation where the ion-dipole force is made weak and long range by artificially cutting off the potential inside a radius R, >> R (see Figure 6). In this case a moment’s meditation (and calculation) is sufficient to convince one that purely hydrodynamic effects become small and the extra drag
I
1.0 1/R
I
I
1.5
2.0
[PI
Figure 7. Limiting conductance-viscosity products from the molecular theory for water (curve 2) and methanol (curve 3) obtained by using hard sphere radial distribution functions and molecular dynamics radial distribution functions for water (curve 1).
computed by HO theory is the same as Zwanzig’s first calculation, i.e.
The molecular theory also gives sensible results in the opposite limit of strong but short-range ion-solvent interactions. Here we find the result of the rigid solvation picture; no matter how strong the interaction, the drag is essentially that of a rigid shell of solvent molecules surrounding the ion. This saturation to a finite drag comes about in an interesting way. As the interaction between an ion and a molecule in the first solvation shell becomes stronger, the fluctuations in the ion-solvent distance become smaller. Therefore the solvent molecules move in general a shorter distance and the force fluctuations decay faster. The increase in the initial value of the force correlation function is precisely balanced by this decrease in the fluctuation decay time, which then leads to a finite drag. In fact, the drag calculated by the molecular theory in this limit is very close to that one would get by considering an ion with its first coordination shell as a rigid body moving hydrodynamically through the solvent. The molecular theory can be used also in situations which do not correspond to either limiting case. To do this, one must know the actual ion-solvent interaction potential and a considerable amount of information about the structure of the solvent around the ion. Colonomos and Wolynes (to be published) have obtained results using only the ion-dipole interaction and an approximate evaluation of the needed equilibrium averages. A complete description of this work is forthcoming. The preliminary results of this study for water and methanol are enormously revealing (see Figure 7). The maximum conductances in water and methanol are in reasonable
2676
The Journal of Physical Chemistry, Vol. 83, No. 20, 7979
1/ R
(i-’ 1
Flgure 8. Comparison of limiting conductance-viscosity products and diffusion coefficient-viscosity products in I-butanol and acetone at 25 OC. Diffusion coefficients are taken from ref 18 and 19.
agreement with experiment. Also the theory gives the mobilities of the smallest ion Li+ to within a factor of 2. This type of agreement is the best that can be expected on the basis of our present knowledge of the potential function and the equilibrium distribution functions. While the numerical results are not enormously different from the Hubbard-Onsager continuum theory, the physics of the problem seems considerably different. First we note that the finite drag on point ions arises from the finite size of the solvent molecules. In a continuum theory, the dielectric fluid penetrates to the surface of the ion, but in a molecular theory no solvent molecule approaches any closer than one solvent radius. Second, the main con- ~ from translational diffusive motions tribution to T ~ comes of the solvent molecules, not from rotational motion. Thus we might call the mechanism of the ionic motion “dipole diffusion friction”. Dipole diffusion is left out of continuum theories. The molecular theory shows that the finite size and geometry of solvent molecules lead to a qualitatively different picture of the participation of solvent molecules in ionic motion. Discussion The main differences between the two theories discussed here and those previously proposed concerns the predicted behavior of small ions. Both the HO continuum theory and the Wolynes molecular theory give finite single ion conductance in this limit, although the physical reasons for this “saturation” behavior are very different. These results are in sharp contrast to other approaches which predict either zero or infinite ionic mobilities under similar conditions. In particular it is clear that Stokes’ law should not be applied unless R 4 / a >> 1 and that ionic radii calculated from this equation under any other circumstances are meaningless. That the HO model tends to work better in aprotic solvents than in hydrogen-bonded solvents is not a t all surprising. The predicted conductances (for various ion sizes) tend to be larger than the observed A’s, but this does not imply that dielectric saturation should be invoked to account for the discrepancy. It goes without saying that
Evans et al.
any continuum model of ion mobility neglects solvent structural effects, and these are especially important in the case of hydrogen-bonded solvents or situations in which RH is comparable to the solvent molecular size. On the other hand, certain general trends and even novel physical effects emerge from the continuum HO treatment which are not present in static structural theories of solvation. Another useful guidance is the development of the more rigorous molecular theories of ionic migration. However, the limitations cited above mean that the theory is unlikely to be useful in making quantitative predictions. The molecular approach to ion mobility, though it represents a break with established hydrodynamic tradition, is firmly grounded in linear response theory and should be regarded as less heuristic and more systematic than a phenomenological continuum model. Perhaps its most attractive asset is that physical intuition may be applied directly at the molecular level, rather than having to synthesize continuum analogies to molecular processes. The latter task has proven itself to be surprisingly difficult and deceptive. The input requirements for the microscopic approach consist of equilibrium distribution functions and interaction potentials for ion-solvent systems, both of which are the subjects of intensive research in modern chemical physics. A judiciously applied molecular theory will therefore inevitably result in a significantly more realistic mosaic than we have at present, though the nature of the input information should receive careful scrutiny. An additional criterion that can be employed in testing any theory of ionic mobility involves the transport behavior as the ionic charge is diminished. When the charge of an ion goes to zero, the diffusion coefficient for the corresponding neutral molecule should be obtained. Recent measurements of the diffusion coefficients of the tetraalkyltins and rare gases in a number of solvents by Evans and c ~ - w o r k e r s provide ~ ~ J ~ the data required for such a test. Shown in Figure 8 is a plot of D q / k T vs. 1/R for ions and neutral symmetrical solutes in acetone and butanol. The deviations from Stokes’ law for the neutral compounds are surprisingly large, particularly for the smaller solutes. The difference between the diffusion coefficient for an ion and that for its corresponding neutral solute in a given solvent provides a direct measure of the effect of charge upon mobility. The diffusion results also give a possible reason for the discrepancies observed in the HO theory for smaller ions, particularly in the hydrogen-bonding solvents. I t is for these systems that the basic assumptions regarding Stokesian behavior are in most error. Similar tests of the molecular theory are a t present more difficult to make. With regard to possible future studies related to ion migration, we should like to stress the importance of (1) an investigation of the role of dielectric friction in motion of highly charged, well-defined complexes, (2) the possible significance of quadrupole moments in accounting for the observed discrepancies between cation and anion mobilities, and (3) the discontinuities observed in the higher alcohols. Acknowledgment. J.B.H. and P.G.W. acknowledge the National Science Foundation and the donors of Petroleum Research Fund administered by the American Chemical Society for their support of this work. D.F.E. and T.T. acknowledge the National Science Foundation for their support through Grant No. ENG 77-02129. References and Notes (1) P. Waiden, Z. fhys. Chem., 55, 207-246 (1906). (2) A. Einstein, Ann. Phys., 17, 549 (1905). (3) W. 2. Nernst, 2. fhys. Chem., 2, 613 (1888).
The Journal of Physical Chemistry, Vol. 83, No. 20, 1979 2677
Bond Length-Force Constant Relationship
12) R. M. Fuoss, Proc. Natl. Acad. Sci. U.S.A., 45, 807 (1959). 13) R. ZwBnzig, J. Chem. Phys., 36, 1603 (1963); 52, 3625 (1970). 14) J. T. Edwards, J. Chem. Educ., 47, 261 (1970); Chem. Id.(London), 774 (1956); A. Bondi, J. Phys. Chem., 66, 441 (1964). 15) R. L. Kay and D. F. Evans, J. Phys. Chem., 70, 2325 (1966). 16) J. T. Hynes, Annu. Rev. Phys. Chem., 26, 301 (1977). 17) R. Zwanzig, Annu. Rev. Phys. Chem., 16, 67 (1965). 18) D. F. Evans, C. Chan, and B. C. Lamartine, J. Am. Chem. SOC., 99. 6492 - - - (1977). (19) D. F. Evan‘s,-T. Tominaga, and C. Chan, J. Solution Chem., 9, 461 (1979).
(4) M. Born, Z. Phys., 1, 221 (1920). (5) H. S. Frank, “Chemlcal Physics of Ionic Solutions“, 8. E. Conway and R. G. Barradis, Ed., Wiley, New York, 1966, p 60. (6) S. Glasstone, J. J. Laider, and H. Erging, “The Theory of Rate Processes”, McGraw-Hill, New York, 1941. (7) J. H. Hildebrand. Science. 174. 490 (1971): J. H. Hildebrand and R. H. Lamoreaux, Proc. Natl. Acad. Sci. U:S.A., 71, 3321 (1974). (8) J. Hubbard and L. Onsager, J. Chem. Phys., 67, 4650 (1977). (9) J. 6.Hubbard, J . Chem. Phys., 66, 1649 (1978). (10) P. G. Wolynes, J . Chem. Phys., 68, 473 (1978). (11) R. H. Boyd, J. Chem. Phys., 35, 1261 (1961).
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A Simple Relation between the Internuclear Distances and Force Constants of Diatomic Molecules’ Roman F. Nalewajski” Department of Chemistry, University of North Carolina, Chapel Hill, North Carollna 275 14, and Department of Theoretical Chemistry, Institute of Chemistry, Jaglellonian University, M. Karasia 3, 30-060 Cracow, Poland (Received Aprll 9, 1979) Publication costs assisted by the National Institutes of Health
It is shown empirically that for diatomic molecules consisting of nontransition elements the relation between the harmonic force constant, k,, and the equilibrium internuclear distance, Re,is quite accurately given by the ~ ) / 2 ~ ~ + ’ ~ -where ~ R e 3for , an atom X from groups IA-VA of simple universal expression k , = (2, + 6 ~ ) (+26 ~ the periodic table 6x = 0, while for an atom X from groups VIA and VIIA 6x = 5 - ux;2, is the atomic number and nx and ux denote respectively the period and group numbers. The expression when combined with Pearson’s empirical relation, k , = 2zAzB/Re3,gives the following expression for the effective charge, zx, in terms of the “coordinates”of atom X in the periodic table: zx = (2, + 6~)/2’x-~’~. A chart of such effective charges is given. The proposed relation is applied both in its k , = k,(R,) and R, = R,(k,) forms to the ground states of typical diatomic molecules and predictions are tested against those resulting from Pearson’s empirical charges, empirical Badger’s rules, and known spectroscopic data. The proposed R, = R,(k,) relation gives an average error of less than 5% for the 97 molecules compared.
Introduction One would expect that there should be a relation between the equilibrium internuclear distance, Re, and the harmonic force constant, k,, of a diatomic molecule. Various simple relationships of this kind have been discovered both by analyzing trends exhibited by known experimental data2s3and by developing theoretical descriptions of molecular force constant^.^-^ In particular, Badger2 was able to formulate a well-known set of empirical rules relating Re and k , for a large number of diatomic molecules. Similar rules have been proposed by Frost and Musulin3 who examined the possible existence of an universal reduced potential energy function for diatomic molecules. Recently, using the Poisson equation approach,’ Anderson and Parr7b derived an empirical relation involving a linear dependence of k , on the product of exact nuclear charges, and an exponential dependence of k , on Re. Among the various simple theoretical models of a vibrating diatomic molecule, the Platt4 model, the simple bond-charge model of Parr and Borkman: and the Poisson equation approach of Anderson and Parr7 have proven very successful in providing a theoretical basis for interpreting the observed trends in the k,-R, c o r r e l a t i ~ n s . ~ ~ J ~ When formulating relations between parameters of molecular potential energy functions it is desirable to make maximum use of the information known a priori such as the exact nuclear charged1 and the “coordinates” of atoms in the periodic table. In the present paper an empirical *Address correspondence to this author at Jagiellonian University. 0022-365417912083-2677$0 1.OO/O
relation between k, and Re is proposed which satisfies this postulate. The proposed expression leads to a simple formula for calculating universal effective charges of atoms-in-molecule.
Bond Length-Force Constant Relationship It follows from the perturbation theory12and the simple bond-charge model of a diatomic molecule A-B that k , should be proportional to R L ~ . Moreover, both these treatments suggest a proportionality between k , and the product of some effective charges of constituent atoms in a molecule, zA and Z B (eq 1). The proportionality constant, a, is 714 in the simple bond-charge model and 2 in Pearson’slO empirical simplification of the corresponding expression derived from the perturbation theory. It has been observed for experimentally determined values of such effective nuclear charges in homonuclear diatomic molecules5J0(see also Figure 1)that, as one moves horizontally across the periodic table, the trend in zx parallels that observed for the number of actively bonding valence electrons of atom X. For example, the maximum values of zx for each row of the nontransition elements in the periodic table are those of atoms in group VA, which is consistent with the maximum bond order of the corresponding homonuclear diatomics. This parallel is also consistent with a dual interpretation of zx in the simple bond-charge model, i.e., (1) effective, shielded nuclear charge, or, (2) contribution of atom X to the bond-charge shared by two atoms in a molecule, and accumulated in 0 1979 American Chemical Society
