BERTR. STAPLESAND GORDON ATXINSON
520 order-of-magnitude agreement with the measured value (3 X 10ls ern+), This description is, however, t o a large extent unsatisfactory; the mobilities seem to be low enough as to be representative of mean free paths shorter than
intermolecular spacings.lg It is hoped that further measurements will pinpoint in greater detail the mechanism of carrier transport. (19)A. F. Joffe, solid State m y s . (USSR), I , 1 (1963).
Structure and Electrolyte Properties in Bolaform Electrolytes. 111. The Hydrodynamics of Potassium Salts of Several Rigid Bolaform Disulfonic Acids in Dioxane-Water Mixtures at 25” by Bert R. Stapled and Gordon Atkinson Department of Chemistry, University of Maryland, College Park, Maryland
60740
(Received June 1 7 , 1 9 6 8 )
The conductances of potassium benzenesulfonate, potassium p-benzenedisulfonate, potassium 4,4’-biphenyldisulfonate, and potassium 4,4”-terphenyldisulfonate in dioxane-water mixtures of 0 to 70% dioxane content were measured a t 25’. The hydrodynamic properties of these elongated ions agree with the calculated properties based on a rigid ellipsoid model. The distance of closest approach in solution between the cation and anion is compared as calculated by five distinct methods. This distance was determined from thermodynamics of association, hydrodynamics, conductaiice J parameter, and the dielectric relaxation drag effects. This distance of closest approach agrees fairly well between all methods for each salt. The trend observed was generally in increasing distance of closest approach with increasing charge separation.
Introduction The effects of ion structure on the conductance parameters of salts of rigid bolaform electrolytes have been investigatedS2 The present authors have presented the basic conductance behavior of the potassium salts of benzenesulfonic acid (KBS) , p-benzenedisulfonic acid ( K2BDS), 4 ,4’-biphenyldisulfonic acid (KZBPDS), and 4 ,4”-terphenyldisulfonic acid (K2TPDS) in dioxane-water mixtures at 25’. This series of rigid bolaform electrolytes represents a unique, systematic increase in size and charge separation of the anion. Rice8 has examined models for a theoretical treatment of bolaforni salts to obtain transport properties, and Atkinson and coworkers have applied these models to calculate the hydrodynamic parameters of bolaforni salts in water4 and dioxane-water2 mixtures. Assuming that the usual equations of viscous fluid motion describe the hydrodynamics of the model, Perrinj gives the expression for the frictional coefficient of a rigid ellipsoid as
s
=
R 5
6~ro(b’’ - a”) l’’ In [(b’la’) d(b’/a’)2 -
+
6~7ob‘
In (2b’la’)
(b‘
11
>> a’)
All terms used are defined in the Appendix. The Journal of Physical Chemistry
For the diquaternary ammonium bolaform ions Rice found that the Peterlin? investigated by FUOSS,~ model of beads separated by massless rods was superior t o the rigid ellipsoid model. Using the rigid bolaform ions such as the 4,4’-biphenyldisulfonate ion, BPDSZ-, Atkinson4 found that the rigid ellipsoid model enabled one to accurately calculate the frictional coefficient of BPDS2- from the parameters of the benzenesulfonate ion, BS-. Both of these results seem valid since, in the case of the diquaternary ammonium salts a nonrigid polymethylene chain separates the charge sites, but a rigid aryl framework lies between the charge sites of the BPDS2- ion. Also, Rice, Thompson, and Nagasawa*have measured the diffusion coefficients of IGBDS, IGBPDS, and K2TPDS and found that the Perrin rigid ellipsoid gave a very accurate description of that property. I n 1959, Fuossg had proposed a method of getting a (1) Taken in part from an M.S. thesis submitted t o the Graduate School of the University of Maryland; National Bureau of Standards, Washington, D . C. 20234. (2) B. R. Staples and G. Atkinson, J . Phys. Chem. 71, 667 (1967). (3) S. A. Rice, J. Amer. Chem. Soc., 80, 3207 (1958). (4)G.Atkinson and S.Petrucci, J . Phys. Chem., 67, 1880 (1963). (5) F. Perdn, J. Phys. Rad., 7, 1 (1936). (6) Q. V. Brody and R. M. Fuoss, J. Phys. Chena., 60, 156 (1956). (7)A Peterlin, J. Chem. Phys., 47, 6 and 669 (1950). (8) G. Thomson, 9. A. Rice, and 41. Nagasawa, J . Amcr. Chem. Soc., 85, 2537 (1963). (9)R. M . Fuoss, Proc. N a t . A c a d . Sci. U.S., 45, 807 (1959).
STRUCTURE AND ELECTROLYTE PROPERTIES IN BOLAFORM ELECTROLYTES distance of closest approach from a semiempirical method utilizing the effect of die1ect)ricrelaxation drag on ions in polar solvents. This was demonstrated theoretically on a macroscopic basis consistent with Stokes’ law by Boyd’O in 1961. A year later Boyd’s derivation was improved upon and refined by Zwan.zig.ll This dielectric effect was described by Fuoss as the resultant effect of the motions of ions creating an electrostatic field in the surrounding polar medium which opposes that motion. More simply, it is an electrostatic coupling of ion with solvent causing an effective increase in viscosity, since there is extra work to orient the solvent dipoles as the ion passes among them. The Fuossg empirical equation Xi0
52 1
Table I: Diffusion Coefficients
Salt
Kz (P) BDs KzBPDS KzTPDS
%3Q X 10e (Rice6 exptl,) cm2 sec-1
9 0 x 100 (calcd from conductance), cmz sec-1
12.0 10.25 9.50
13.2 11.74 10.79
APO/PO (cond.)
10.0 14. 5 13. 6
measurements. The method for calculating the mutual diffusion coefficients from conductance data may be found in Robinson and Stokesls and is briefly outlined below. The Nernst-Hartley relation is
5 2
+ S/D)
677-N7(ri
(2)
can be rearranged and then upon multiplying by D one obtains
Then the limiting value at infinite dilution where d In y,./ci In c 4 0 is given by
(3) Thus a plot of (S2D/6nN7Xi0) vs. D should yield a straight line with a slope of (an)i. However, one may also use the Zwanzigl’ equation as tested by Atkinson and Mori12 xio =
Frictional CoefJicients. Based on a rigid ellipsoid model the authors have calculated the ionic frictional coefficients (Table 11) where a’ has been taken t o equal
52
(4) Table 11:
Frictional Coefficients of Salts
7 - r % Dioxane
rearranged to KBS
so that a plot of L* vs. T / O [ ( e O - E , ) /eo2] or R* will give ~ r i from another distance of closest approach ( u z )E the slope. Thus a distance of closest approach determined by five methods may be compared: thermodynamics of association, conductance J parameter, dielectric relaxation [Fuoss empirical equation (2) and Zwanzig equation ( 5 ) 1, and hydrodynamics.
0.0 35.94 49.68 56.36 66.52
KzBDS
0.0
KiBPDS
35 I94 47 76 54.79 59.14 65 59 0.0 24.17 46 83 52.27 59.14 68.69 I
I
I
Results Difusion Coeficients. Reasonable agreement was found between the experimentally determined diffusion coefficients in water, made by RiceJ5 who based his measurements on the Perrin rigid ellipsoid model, and those calculated from conductance data. The calculated mutual djffusion coefficients mere consistently about 10% higher than experiment, as shown in Table I. This seems satisfactory, since there is probably a few per cent experimental error in the determination of these mutual diffusion coefficients and these measurements are not in as dilute a range as the conductance
KzTPDS
0.0 41.94 46.83 52.27 58.32
X 108 sec cm-1Calcd from Perrin model Exptl
... ... # . .
< . a
... 0.607 1.11 1.27 1.33 1.34 1.33 0.797 1.23 1.66 1.73 1.77 1.72 0 928 1.83 1.95 2.01 2.05 I
0.442 0.819 0.956 0.945 1.13 0.529 1.22 1.42 1.74 1.86 2.67 0.631 1.11 1.47 1.61 1.84 3.11 0 727 1.67 2.00 1.83 2.38 I
90
x
102
0.895 1.636 1.908 1.971 1.952 0.895 1.636 1.876 1.962 1.981 1.958 0.895 1.380 1.861 1.943 1.981 1.931 0 895 1.766 1.876 1.943 1.979 I
(10)R. H.Boyd, J. Chem. Phys., 3.5, 1281 (1961). (11)R. Zwanzig, ibid., 38, 1603 (1963). (12) G.Atkinson and Y. Mori, J. Phys. Chem., 71, 3523 (1967). (13) R. A. Robinson and R . H. Stokes, “Electrolyte Solutions,” Butterworth and Co. Ltd., London, 1955.
Volume 19, Number 3 March 1969
BERTR. STAPLES AND GORDON ATKINSON
522 0.5aJ value for KBS and b' has been measured from molecular models (Figure 1).
Figure 1. Dimensions of the molecular models.
creases its length and thereby appears more like an ellipsoid. This also indicates good agreement between Stokes' hydrodynamic radii from both t,he theoretical and experimental lines, as pictured in Figure 2. The random scattering above 50% dioxane can be explained by the failure of Stokes' law in this region where the local viscosity is no longer described by the bulk viscosity. The scattering is probably due to nonideality of the solvent mixture as shown by the parabolic curve of the 70 us. per cent dioxane in regions of high dioxane content as well as other properties. Dielectric Relaxation. Based on eq 3, a plot of (52D/6aNqXi0)vs. D is shown in Figure 3 and another
Frictional coefficients calculated using the Perrin rigid ellipsoid model (eq 8) agree well with those calculated from experiment (Table 11) thus demonstrating the probable validity of the rigid ellipsoid model proposed by Perrin.
40 ,-I-T--r--J
/
TPDS'-
)
I
When one plot's { us. 70)Figure 2 graphically illustrates the agreement between the theoretical slope using geometric dimensions of the Perrin model compared 2.5 20
:EXPERIMENT/ PERRIN MODEL
1.5
'5
6
K,TPDS
---
1.0 0.5
0 L-1.IO
U
f
20
40
30
50
70
60
8(
D
1.5
0
Figure 3. The ionic radius as a function of the dielectric constant.
X
L 0.5
15 1.0
I
, ~ , l
plot, based on eq 5 , is illustrated in Figure 4. The ionic radii in solution, derived from these two interpretations of dielectric friction are compared in Table 111. These calculations demonstrate an increasing ai distance with an increasing charge separation: TPDS2Table 111: Comparison of Ionic Distances Using the Dielectric Friction Approach
Figure 2. The frictional coefficient as a function of viscosity.
with the experimental slope. One can observe that the theoretical slope approaches that of the experimental slope as the size of the anion increases. That is, the description of the data by the model chosen closely approaches the experimental results as the anion inThe Journal of Physical Chemistry
Ion
(aD)i, a
K+ BSBDS2BPDS2TPDW Fuoss interpretation.
1.10 2.53 2.55 3.33 4.18 b
A
(az)i.
A
1.9
2.9 1.3 1.6 2.4
Zwanaig derivation (from slope)
STRUCTURE AKD ELECTROLYTE PROPERTIES
IX
BOLAPORM ELECTROLYTES
523
Table IV: Correlation of the Distance of Closest Approach
-
Thermodynamics Salt
KBS KpBDS KzBPDS KiTPDS
[FUOSS]~K
3.4 5.0 4.6 3.0
Distance of closest approach Hydrodynamics Conductance2 Dielectric ralaxation[Fuoss-Onsagerla~ [FUOSS~~D [Zwanziglaz (intercept) (slope)
---
3.216 3.6 4.2 4.5
> BPDS2- > BDS2-. It appears that the a, obtained from the intercept is without solvation whereas the a, derived from the slope includes solvation. The Distance of Closest Approach. The comparison of the distances of closest approach, ai, of the cation (K+) and anion (Table IV) determined by five different approaches-thermodynamically, hydrodynamically, by conductance and through the effect of the dielectric relaxation drag-resulted in the following. a. Thermodynamics. FUOSS' method of obtaining aK from a plot of log K,, vs. 1/D yielded a trend opposite to all the other resulk2 The UK decreased in size as the charge separation increased. This decrease might be attributed to the attempted interprettation of a spherical model, embodied in the theory, with an ellipsoidal anion, to the failure of the Fuoss-Ede1son2J4 technique for such a large ion, or to any combination of each. The possibility also exists that the cation might prefer to occupy an end position on the large terphenyldisulfonate anion. There probably is a particular distance of the separation of charges beyond which the cation will favor an ion pair where it is
6 1
!
3.6 3.7 4.4 5.3
3.7 2.2 2.6
3.5
4.7 3.1 3.3 4.2
.
Viscosity [8tOkeS]Ug
4.0 6.0 6.2 7.2
directly in contact with a single sulfonate group. This distance may have been reached with three benzene rings between the negative charges of the anion, but the following results seem to indicate that this is not the case. There appears to be no way of deciding which of these factors is or is not operating in this particular instance. b. Conductance. A consistent increase in the U J values was observedJ2 as the separation of charges increased. c. Hydrodynamics. The Perrin rigid ellipsoid model seemed to give the best description and agreed with experimental results when diffusion coefficients (Table I) and hydrodynamic properties (Table 11) were investigated. Some of the small differences (about 10% high) jn the diffusion coefficients calculated from conductance parameters and those determined by Rice and coworkers* may .i?lell be due t o slight association, that is, the presence of KA- ion pairs in addition to single Kf and A2- ions, or to the dimensions of the ellipsoid assumed by Rice in his calculations. The a' value for the rigid ellipsoid was evaluated by Yokoi and AtkinsonI6who determined the U J for KBS in water. The b' value for the rigid ellipsoid model was measured from end to end on molecular models to give a comparison between calculated and experimental frictional coefficients illustrated in Figure 2. The US values increased with increasing charge separation and it can be seen that this Stokes radius is generally larger than other ai's since more solvation is probably included in this model. d . Dielectric Relaxation. Once again an increasing a D , determined by eq 3, was noted as charge separation increased, as was also indicated by the Zwanzig equation, eq 5 (uz). Position of the Cation in an I o n Pair. Probably the most pertinent question to be decided is: "Can one determine what site the cation prefers to occupy in an ion pair with a charge-separated anion?" These investigators feel that a good educated guess can be advanced based on the determinations of the distance of closest approach, ai. If a contact ion pair of KBS,
R'x IO"
Figure 4. L* vs. R*. (Points marked "?" were not included in least-squares calculations. Omission of those points values of less than 0.1 d.) amounted to a change in ~i
(14) R. M. Fuoss and D. Edelson, J. Amer. Chem. SOC.,73, 269 (1951). (15) M.Yokoi and G. Atkinson, i b i d . , 83, 4367 (1961).
Volume 78, Number 9 March 1960
BERTR. STAPLESAND GORDONATKINSON
524
CaH,SO*-K+ has an aJ of about 3.5 A in dioxane-water mixtures,2 then an increase in ai with an increase in charge separation would indicate that the cation occupies an intermediate position between the two negatively charged sulfonate groups, as pictured in Figure 5. This is indicated since, if the cation was
/--. \
\
Figure 5. Probable position of the cation in an ion pair.
intimately associated with a single sulfonate group, a ((contact” ion pair, the ai should show no tendency to change with an increase in the separation of sulfonate charges. It is important to realize that all the distances of closest approach that have been determined are based on a distance, ai, from charge center to charge center. Thus, if the cation was situated at one end of the charge separated anion, the ai determined would be constant,
.i-
a
(SLOPE)
Figure 6. The experimental distance of closest approach as a function of the radius of the anion model. (Straight lines are used to show trends.)
The Journal of Physical Chemistry
within the experimental error, for all salts investigated. This is not the case as demonstrated by the results of the conductance aJ, the Stokes’ hydrodynamic radius, as, and the and az determined from the dielectric relaxation friction (Table IV) , where a definite change occurs in the distance of closest approach, beyond the limits of experimental error. The distances of closest approach (average values) experimentally determined by conductance (uJ), hydrodynamics (as) and the dielectric relaxation (uD and az) are shown in Figure 6 as a function of the anion radius measured from molecular models. It is therefore concluded that the cation assumes an intermediate position approximately equidistant between the two charges on the anion.
Acknowledgments. The authors wish to express their gratitude to the National Institutes of Health for their support under Grant GM 9232. The computer time used for this research project was supported by National Aeronautics and Space Administration Grant NsG-398 to the Computer Science Center of the University of Maryland. Appendix. Symbols
CC(Q)++ (ao1-1
+ (ad-1 + (as)-] Avogadro’s number CC(as)+ CC(as)+
Bulk viscosity of solvent “Stokes” radius of ith ion Empirical constant of the Fuoss equation Static (low frequency) dielectric constant of solvent Optical (infinite frequency) dielectric constant of solvent Dielectric relaxation time of solvent Electronic charge Mutual diffusion coefficient Faraday Frictional coefficient Number of ions which the cation produces Number of ions which the anion produces Equivalent ionic conductance of the cation a t infinite dilution Equivalent ionic conductance of the anion at infinite dilution Equivalent ionic conductance of the ith ion a t infinite dilution Mean molar activity coefficient Gas constant (8.314 J deg-1 mol-’) Concentration of salt in moles/liter
