Electrolyte Diffusion irt Acetonitrile 1667

Electrolyte Dlflfusion in Acetonitrile. Harned Conductometric Technique

Terrence A. Renner' and Philip A. Lyons*

Yale University, New Haven, Connecticut 06520 (Received July 73, 7973; Revised Manuscript Received April 76, 7974)

Publicabon costs assisted by Yale University

Diffusion coefficients for solutions of sodium tetraphenylboride in acetonitrile and rz-tetrabutylammo- niuni tetraphenylboride in acetonitrile have been determined at 25" using the Harned conductometric technique. In the concentration range below 0.01 M, the data are compared with values predicted by the Debye-Huckel-Onsager-Fuoss theory as well as by the extended theory of Gronwall, La Mer, and Sandved. The electrophoretic effect in acetonitrile is much larger than in aqueous solutions. As a result, the new data provide the most reliable test of the Onsager-Fuoss theory yet possible. Each term in the theory has been separately validated. Precise, internally self-consistent activity coefficients were com- puted from the diffusion data.

Introduction Harned's conductometric method has been used success-

fully to determine diffusion coefficients in many aqueous electrolyte trolutions.2 We propose in this work to extend this study to nonaqueous solutions. The systems NaBPhl- acetonitrile and (n-C4H~)~NBPh4--acetonitrile were used since mobility data iire available for these system^^-^ and also since, below 0.0 1 M , the electrolytes above display no measurable association to neutral pairs.

The purposes of the work were several. First, it was necessary to discover whether or not precise measure- ments could be made in the very dilute solutions of theo- retical interest. Secondly, it was important to determine whether or not a self-consistent set of activity coefficients could be evaluated from the diffusion data since, if this were possible, such a procedure would often be the meth- od of choice for experimental convenience. Finally, the study should provide a decisive test for the Onsager-Fuoss theory of the electrophoretic e f f e ~ t , ~ J ? since the magni- tude of the electrophoretic contribution is very large for acetonitrile solutions.

Experimental !section The Harned method for the determination of electrolytes

diffusion coefficients IS based on an estimate of the flow of electrolyte iJn a rectangular parallelepiped by measure- ments of the electrical resistance of the solution at differ- ent levels in the cell. Except for the use of all-glass cells, the procedure followed was substantially that which has al- ready been well de~c r i lbed .~ ,~ - l l

Fisher ACS Certified acetonitrile was refluxed over cal- cium hydride in a dr:y nitrogen atmosphere for 1 day and then distilled. In the middle fraction of the distillate which was retained, there was no chromatographic evi- dence of an37 water. Fisher ACS Certified sodium tetra- phenylboride was recrystallized once from a mixture of acetone and toluene.-'2 The product needles decomposed a t 280". Tetrabutylarrtmonium tetraphenylboride (hereaft- er, TBA-TPB) was prepared metathetically from reagent grade tetrabutylammonium bromide and sodium tetra- phenylboride. This product was twice recrystallized from a 2: l acetone-water mixture.6 The purified crystals melt- ed at 232.5" and began to decompose at about 236".

All of the diddng surfaces of the diffusion cells were lu- bricated with Apiezon M grease which had been subjected to acetonitrile (extraction for 1 week to remove soluble

components. At no time during or after the diffusion ex- periments was there any evidence of solution contamina- tion by Apiezon M residue.

In the process of restricted diffusion, the solution of Fick's second law

ac/at = D ( E + ~ C / ~ X ~ ) (1) leads, for times 2 48 hr, to the expression

In ( K ~ * -kKt*) = -rr2D2/A2 -C constant (2)

In (a ) , A is the cell height, K b * and &* are the recipro- cals of the resistances measured at the bottom and top electrode pairs (at A / 6 and 5A/6 , respectively), and k is the ratio, &/kb, of the cell constants of the electrode pairs. Measurement of the conductance difference be- tween the electrode pairs as a function of time permits a determination of the diffusion coefficient, D, from a plot of In (Kb* - kKt*) against time.

Results and Discussion Experimental data were fit to eq 2 by least squares to

give the slope ( -n2D/A2) and hence L). Typical plots for each system are shown in Figure 1.

Measured diffusion coefficients are summarized in Table I. They may be compared with values calculated from the Onsager-Fuoss expression

D = ( D o + A, + A , ) ( l + d In y+/d In c ) (3) Do is the Nernst limiting value for the diffusion coefficient as c approaches zero, c is the molar concentration, yi is the mean ionic activity coefficient on the molar scale, and Ax and A 2 are the Onsager-Fuoss electrophoretic terms for symmetrical electrolytess

where

Q z ( ~ a ) = (KU)' [exp(m)/(l + K L ~ ) ~ ] ' E ~ ( ~ K C ~ ) (6) and

E i ( 2 ~ a ) = E i ( x ) = j m exp(-2) d f / t = -0.5772 - X

X 2 I n % + % - - + . . . ( 7 )

22

The Journalof Physical Chemistry, Vol. 78, No. 16, 1974

1668 Terrence A. Renner and Philip A. Lyons

Figure 1. Typical plots of In (Kb* - k K t * ) vs. time for Harned diffusion runs.

TABLE I: Determination of the Experimental Diffusion Coefficients by the Conductometric Methoda

NaBPh4 TEA-TPB ______I__

I O ~ C , 1 0 5 ~ ~ ~ ~ ~ . Y * loa,, lO~D,,,t, Y * M cmz/nec (eq 11) M cm2/sec (eq 11)

1.886 1 6 7 1 0 2.281 I 6621 2.417 16625 3.667 1 6 4 2 6 4.119 1 6 4 3 3 4.354 1 6388 4.871 1 6356 5.377 1 6360 5.706 1 6260 8.534 I 6182

0.861 1 .901 1.5037 0.849 2.467 1,5042 0.845 3.125 1.4844 0.817 4.017 1.4809 0.809 5.032 1.4800 0.805 6.004 1.4742 0.796 6.911 1.4710 0.789 0.784 0.752

a NaBPhn in acetonitrile at 25'; DO = 1.7662 X 10 -6 cmZ/sec. in acetonitrile at 25': DO = 1.5894 X 10-5 cm*/sec.

0,860 0.844 0.828 0.811 0.794 0.780 0.768

TBA-TPB

TABLE PI: Auxiliary Constants and Parameters for Theoretical Computation of Diffusion and Activity Coefficients

De.rcription of constant or warameter NaBPh4 TBA-TPB

E

Sf

die1ec:tris: constant of 36.0

viccoi;ity of ,acetonitrile, P 0.003412 liniiting conductance of 77.3 cation, ohm!-1, cm2

limiting conductance of 5 8 . 1

acetonitrile

equiv -1

anion, ohm-' cm2 Iequiv - 1

dope for activity coefficient

M-.l

to 'zero

108, cm

~ u n i of individual :ionic radii 108, cm

DEE theoretical limiting 1.6409

activity B coefficient, - 0.1095

limiting value a8 c goes - 1.8892

D1'I imon-sizo parameter 5.15

GLS ion-size parameter 5.15

36.0

0.003412 61.75

58 .1

1.6409

-0.3130

- 1.8892

5.18

8.94

Other terms art? definled and listed in Table 11. The thermodynamic correction, (1 + d In y,/d In c),

was computed from an extended form of the Debye-

I I l.528!000 ,." JjO &p.L Inale)i-lrte;.&y 0.125 0.150 0.175

Figure 2. Diffusion coefficients for NaBPh4 in acetonitrile at 25"

1 . 5 5 0

- i 1 . 5 0 0

1.4oc

1.330

Figure 3. Diffusion coefficients for TBA-TPB in acetonitrile at 25".

Huckel theoryl3,

logy, = -sfC*'z/(l + KCL)

where M I and Mz are the molecular weights of solvent and solute, respectively. The same thermodynamic term was also calculated from a different extended form due to Gronwall, La Mer, and Sandved.14 Diffusion coefficients computed using the different thermodynamic corrections are included in Figures 2 and 3 as are values obtained by omitting the electrophoretic terms from eq 3.

The (extended) Debye-Huckel theory incorporates an em- pirical correction for short-range ion-solvent interaction in the form of the Bc term of (9). The ion-size parameter, a, used in computing activity coefficients from the DH ex- pression is derived from the conductance data of Kay, Hales, and Cunningham3 for NaBPh4; for TBA-TPB, the value of "a" is taken from the conductance data of Fuoss and Brown.6 In the GLS theory, no provision is made for interactions other than coulombic attraction and repul- sion. The only adjustable parameter is the ion size, a'. They recommended using the sum of the ionic radii in the activity coefficient expression which they derived.14 For the sodium r+ = 0.95 A. For the tetraphenylboride ion, the ionic radius, r- , was determined from a molecular model according to the method suggested by Grunwa1d;lG thus, r- = 4.20 A. There seems to be some disagreement in the literature in assigning ionic radii to tetraalkylam-

The Journaiof Phy:xca/ Chemistry, Vol. 78. No. 76, 1974

Electrolyte Diffusion in Acetonitrile

TABLE III: Electrophoretic Terms for Diffusion Equation

1669

0.0 O.”Z 0.04 0 . 0 6 0.08 0.10

c+ imole*-l,rer-k)

Figure 4. D‘ lc”2 vs. c1/* for evaluation of activity coefficients from Harned diffusion data.

monium ions, which may coil when the hydrocarbon chains become long. We have selected a value of rt = 4.74 6, for the tetrabutylammonium ion, which is taken from data of Fuoss and Tuan5 Hence, for all GLS theory calcu- lations, the thermodynamic correction terms were evalu- ated with a’(NaBPbi4) = 5.15 6, and a’(TBA-TPB) = 8.94 A. Regardless of the exact choice of a’ for either salt in the GLS development, the importance of including the electrophoretic effect in computations of the diffusion coefficient is still evident from Figures 2 and 3.

The empirical constant, B, in eq 9 was determined from the diffusion (lata using a method proposed by Harned.a Rearranging eq 3 to the form

D’ = ( D / ( D o A, + Az) - 1.0) = C ( d In y,/dc) (IO) and integrating after a change of variable gives the ex- pression

(2D’/ci’‘) dc”‘ 0

Plots of D’/ella against e l l 2 are given in Figure 4. Linear least-squares analyses of these plots, subject to the con- straint

Xim D ’ / C * ’ ~ = 02.3038,/2 (12) c - 0

yield analytical expressions for D’/e1/2, which can be nu- merically integrated to determine In y* from eq ll. See Table I.

To obtain a value of the empirical constant, B, an aver- aging procedure was employed. Equations 9 and 10 were combined to give

D’+ 1 . 1 5 1 4 S , ~ ~ ’ ~ / ( 1 + K U ) ‘ + c\k(d) = 2 . 3 0 3 B ~ (13) \k (d) is the derivative with respect to concentration of the final density term in eq 9. Individual values of B were cal- culated from eq 13 for pairs of experimental points (D’,e). The average of these values was used in eq 9 to yield a working exprespion for In yi .

It is evident from Table 11 that the value of B in each system studied was quite small and negative. Averaged B values resulted in a fit of activity coefficients to about 10.15%. At the lowest concentrations, the Bc term in eq 9 is obviously negligible; but, even at 0.01 M, Bc contrib- utes only -0.001 to log yf for NaBPh4 and -0.003 for

While the wlork of Harned and coworkers had estab- lished without reasonable doubt that the electrophoretic

TBA-TPB.

x 103, M 1 0 7 ~ ~ lO6Az

NaBPhr in Acetonitrile at 25 O

1.886 -0.2448 0.2588 2.281 -0.2665 0.2931 2.417 - 0.2735 0.3042 3.667 - 0.3286 0.3951 4.119 -0.3455 0.4238 4.354 -0.3538 0.4380 4.871 -0.3712 0.4679 5.377 -0.3871 0.4955 5.706 -0.3969 0.5125 8.534 -0.4688 0.6388 9.999 - 0.4998 0.6933 x 103. M 108Az 1 0 6 A ~

TBA-TPB in Acetonitrile at 25” 1.901 - 0.0920 0.2591 2.467 -0,1034 0.3070 3.125 -0.1147 0.3564 4.017 -0.1280 0.4155 5.032 -0,1409 0 I4745 6.004 - 0.1518 0.5248 6.911 -0.1610 0.5674 9.999 -0,1871 0.6a94

theory of Onsager and Fuoss was correct both in sign and in order of magnitude, it must be conceded that the in- trinsic error in the experiments (about *0.2%) was sizable compared with the electrophoretic contribution. As an ex- ample, one finds a variation of only 0.8% in the quantity D/(1 + d In yJd In c) over the concentration range 0- 0.01 M for aqueous KC1, the result quoted most often.& Guggenheiml? had also commented QTB the fact that the variation in the same quantity for NaCI. solutions is of the same magnitude as the scatter in the data,18 iniplying that the experimental data did not constitute compelling evidence for the validity of the theory.

Happily, the electrophoretic contributions are much larger for acetonitrile solutions than for aqueous systems. Thus the residual variation in D/(1 + d In y,/d In e) from 0-0.01 M was 3.6% for NaBPh4 and 4.2% for TBA- TPB. The striking deviations from experiments which are observed from omission of the electrophoretic terms in theoretical computations of diffusion coefficients are dis- played in Figures 2 and 3. These figures include some ad- ditional diffusion coefficients measured by the Gouy opti- cal method. All results lie on the same curve for a given salt. The internal consistency lends some confidence to the procedure employed for the calculation of the values of B in each system.

For the case of TBA-TPB, the first term, A I , in the On- sager-Fuoss theory is negligible compared with A2. This is due to the fact that the tetrabutylammonium ion and the tetraphenylboride ion have nearly the same limiting ionic mobilities, Xlo , in acetonitrile; and, since i s propor- tional to (t2O - t1°)2 where tl0 = X,O/A0, this first electro- phoretic term is a very small fraction of A2. For TBA- TPB the electrophoretic effect is due then essentially to theA2 term.

For solutions of NaBPhd in acetonitrile, A 1 makes a larger relative contribution. Table I11 lists A 1 and A2 for both systems. In the range of expected theoretical appli- cability, i.e., less than 0.01 M, the Be term, although in- cluded for completeness, could well be eliminated from (9) without significant effect on diffusion coefficients cal- culated using the Debye-Huckel activity coefficient ex-

The Journalof Physical Chemistry, Vol. 78, No. 76, 1974

1670 M. E. BaLr and D. A. Horsma

pression. It should be pointed out that the activity coeffi- cients evaluated from diffusion data were only used to ob- tain the empirical parameter, B. One would furthermore expect that the simple DH activity coefficient treatment, with the effect of ion size included, would adequately de- scribe the thermodynamic behavior of these systems in the dilute solution region which is of interest here; and in- deed, in the absence of any activity data for the systems studied, such ai, assumption must be made.

The agreement of theory with experiment is excellent, ltO.1% for NaRPh4 and 10.2% for TBA-TPB; a t the same time, the estimated error in our experimental data is 10.15%. It would appear that the validity of the Onsager- Fuoss electrophoretic theory has been established term by term, and only an the very unlikely event that the Debye- Huckel theory For activity coefficients of dilute electrolyte solutions is invdia would such a conclusion be unduly op- timist] c.

Acknowletlgnaents. We thank Mr. E. Ford for aid in the diffusion cell design and for the fabrication of the cells. One of us, T. 11. R., expresses his appreciation to the Na-

tional Science Foundation for support in the form of a Graduate Student Fellowship.

References and Notes Ph.D. Thesis, Yale University, 1973. (a) H. S. Harned and D. M. French Ann. N. Y . Acad. Sci., 46, 267 (1945); (b) H. S. Harned, Discuss. FaradayScic., 24, 7 (1957). R. L. Kay, B. J. Hales, and G. P. Cunningharn, J. Phys. Chem., 71, 3925 (1 967), R. L. Kay and D. F. Evans, J. Phys. Chem.. 70, 2325 (1966). R. M. Fuoss and D. F. Tuan, J. Phys. Chem., 6'7, 1343 (1963). R. M. Fuossand A. M. Brown, J. Phys. Chem., 64, 1341 (1960). L. Onsager and R. M. Fuoss, J. Phys. Chem.. 26,2689 (1932). R. A. Robinson and R. H. Stokes "Electrolyte Solutions," 2nd ed, Butterworths, London, 1959, Chapter I 1 M. Blander. Dissertation, Yale University, 1953. R . M. Hudson, Dissertation, Yale University, 1350. C. A. Blake, Dissertation, Yale University, 1950. J. F. Skinner and R. M. Fuoss, J. Phys. Chem.. 68, 1682 (1964) P. Debyeand E. Huckel, Phys. Z., 24, 185 (1923). T. H. Gronwall, V. K. La Mer, and K. Sandved. Phys. 2.. 29, 35%

L. Pauling, "The Nature of the Chemical Bond," Cornel1 University Press, Ithaca, N. Y., 1940, Chapter X. E. Grunwald, "Electrolytes," B. Pesce, Ed., Pergamon Press, New York, N. Y., 1962, p 74. E. A. Guggenheim, Trans. FaradaySoc., 50, 1048 (1954). H. S. Harned and C. L. Hildreth, J. Amer. Chem. SOC., 73, 650

(192%).

(1951).

Dielectric: Properties of Liquid Sulfur in the Ring-Chain Transition Regionla

M. E. Baur* and D. A. Horsma

Department of Chemistry, University of California, Los Angeies, California 900241b Manuscript Received March 1, 7974)

(Received September 20 '973, Revmed

The dielectric constant of liquid sulfur has been determined to within a precision of 10.0001 over the temperature range 134-206" a t frequencies up to 10 kHz. The data reveal some interesting new features not noted in previous less precise investigations. The liquid below the ring-chain transition at 159", as- sumed to consist entirely of s g rings, exhibits a temperature-dependent molar polarization [PI. This be- havior is explained by postulating the existence of both crown and chair conformers of the Sg ring, with the Iaitter having either a dipole moment or enhanced polarizability relative to the crown. The parame- ters which must be associated with the chair conformer appear reasonable. Above the transition. [PI in- creases proportionally to W, the total concentration of S S units incorporated in chains. This is taken to imply a similarity in conformational flexibility and symmetry between the high-temperature s g rings and SI8 units in the (infinite) chain. The increase of [PI with W is interpreted in terms of an increment in the isotropic temperature-independent polarizability of Ss units when converted from ring to chain. This increment is attributed to a slight increase in r-electron bonding in the chain. Extrapolation of [PI to the transition temperature indicates a finite discontinuity at that point. This anomaly is accounted for qual- itatively by treating the ring-chain transition from the standpoint of fluctuation theory. No significant dependence of dielectric constant on frequency was found.

1. htroducti.on

The liquid sulfur system possesses the unusual feature of a thermodynamic transition between two distinct fluid modifications; below T , = 159", a liquid of relatively low viscosity consisting in the main of SS rings, and above this temperature a highly viscous material in which conversion of a significant fraction of the material to polymeric chains (SS), hais occurred.2a The onset of the transition is

marked by sharp changes in both equilibrium and non- equilibrium proper tie^.^^,^ Analysis of the transition in terms of a simple chemical equilibrium mode1435 leads to excellent agreement of predicted degree of polymerization with experiment, leaving no doubt as to the central role of the ring-chain conversion in producing the transition.

Although this equilibrium polymerization theory is un- doubtedly correct .and self-consistent as far as it goes, it represents an analysis at a thermodynamic level and

The Journal of Physical Chenlistry, Vol. 78, No. 16, 1974