KAY~JIOKD AI. Fuoss
3304 [COUTRIBUTION No. 1431 FROM
THE
AND
CHARLES X.KR.IUS
T*(Il,79
STERLING CHEMISTRY LABORATORY OF YALEUNIVERSITY A N D THE METCALF RESEARCH LABORATORY O F BROWNUNIVERSITY]
Ionic Association. 11. Several Salts in Dioxane-Water Mixtures BY RAYMOND hf. FuosS AND CHARLES KRAUS RECEIVED MARCH1, 1957 new method of mathematical analysis is applied t o conductance data for several salts in dioxane-water mixtures. It is shown that the association constant .4 is a simple function A Oexp(e2/aDkT) of the dielectric constant of the solvent mixture, unless a dipole is present in one of the ions (e.g., the bromate ion), when an additional ion-dipole free energy appears in the exponent. Data for quaternary ammonium salts show the presence of a previously unsuspected linear term in conductance, which is due to the viscosity increase produced by bulky ions. Consequently solution viscosity rather than solvent viscosity should be used in treating conductance. Finally, i t is concluded that the Bjerrum theory of ion association and the Fuoss-Shedlovsky extrapolations should both be replaced by the methods used in the present analysis.
I n the preceding paper,’ it was shown that association constants may be derived from conductance data, even when the fraction of ions associated to pairs is small. A fundamental prerequisite is that the data be of high precision (=I=0.02% a t least); a t a concentration of 0.005 N , for example, an association constant A = 5 corresponds to only about 2.57, association. A further prerequisite is that data be available over a range of values of dielectric constant; the range must be wide enough to include solvent mixtures in which the association constant has a value of a t least 10. The reason for the latter requirement is simply that association gives a leading term of order concentration in the conductance equation, and there are also present linear terms from long range ionic interaction. IYhen A is small, the difference of the interaction term Jl(u)and -lLiocan be determined but the quantity cannot be resolved into its components. When one (or both) of the ions is bulky compared to the solvent molecules, an additional linear term (5,i&’2) also appears, which can only be separated from the coefficient of the linear term if .1 and d are independently known. From the data in solvents of low dielectric constant, the ion size a can be determined ; given the ion size, .4 can be calculated for the solvents of higher dielectric constant, within the validity of the assumption that the parameter a is independent of composition. Recent work by Kraus2 and co-workers satisfy the criteria described above. (There are many data of high precision in the literature for aqueous solutions, but they cannot be examined for ionic association because, if association occurs for say the alkali halides in water, it is so slight that it is undetectable by our present analysis; a small 31.53. %‘hen (1)
R 32
(-7)
R W Martel
(19ii)
FuoSS THISJ O U R N A L , 79, 3301 (1957)
and C A
Kraus, Proc N o t Acad S a , 41, 9
treated by the method appropriate for unassociated electrolyte^,^ the data give linear A”’ DS. c plots and extrapolate to an unambiguous value of 110, but the value of a required $0 make the A”’ vs. c plot horizontal is only 3.00 A., which seems unrealistically small. Hence, despite the fact that the conductance curve lies above the Onsager tangent, we are led to suspect ion association. In 557, dioxane, the phoreogram is catabatic, clearly indicating association. On the basis of the Bjerrum theory, it would be possible to have ion association a t low dielectric constants, with an abrupt cessation of pairing above a critical value of the dielectric ~ o n s t a n t . ~ But this critical value is merely the consequence of a device used by Bjerrum to avoid divergence of an integral and corresponds to no physical reality; it seems more reasonable to assume that, as the dielectric constant is raised, the fraction of solute present as ion pairs gradually decreases a t a given stoichiometric concentration, but never completely vanishes. Indeed, as already has been pointed out,’ just as soon as a non-zero ion size is introduced into the model, the existence of ion pairs is tacitly admitted, because a non-zero size can only make itself visible experimentally by the effects of actual contacts of ions, and for the duration of the contact, neither ion can be considered to be “free” in any of the senses implied by that adjective. Availability of data for a salt over a range of dielectric constant in which the phoreogram starts below the Onsager tangent and crosses it as dielectric constant increases provides an ideal means of testing the above hypotheses. For the four mixtures of lower dielectric constant, the functions y and Y. defined by equations 21 and 22 of the preceding paper were computed; the corresponding plots are shown in Fig. 1. For the 35 and 5OY0 mixtures, the calculation was made to second approximation, as described in the previous paper. It should be emphasized that y is extremely sensitive to experimental error a t low concentrations, because it contains in the numerator a diference which approaches zero as c approaches zero, while the denominator is proportional to c itself. For example, a t 5 X 11’ in the 557, mixture, an error of only 0.020 A-unit in A makes an error of 10% in y. Hence relatively more weight is given to the points at higher concentrations; the latter must, of course, satisfy the condition AU < 0.20, in order to permit application of the model (3) R.M . Fuoss and L Onsager, J P h y s Chem , 61, 668 (1957) (1) R RZ Fuoss a n d C A Kraus. THISJ O U R N A L , 66, 1019 (1933)
600,
3 305
CONDUCTANCE OF SALTS IT\‘DIOX AXE-WATER XIXTURES
July 3 , 19.57
-
I
400 ~-
3001 -
-
20
40
30
50
6.
60
Fig. 1.-Extrapolation plots for sodium bromate in low range of dielectric constants.
and method.) Within experimental error, the plots are linear; the slopes determine the association constants A immediately. Extrapolation to x = A0 gives the ordinate ~ ( 0= ) JI(u)- AAo
from which Jl(a)is evaluated, because the product is now known. Then from a plot of Jl(a) against a,the value of a is determined. The results of these calculations are shown in the last four lines of Table I ; it will be noted that the parameter a remains substantially constant.
Ah0
TABLE I SODIUM BROMATE IN DIOXANE-WATER MIXTURESAT 25’ %
Dioxane
D
A
J I (a)
Li
0 10 20 30 35 40 50 55
78.48 70.33 61.86 53.28 48.91 44.54 35.8c5 31.53
0.50 0.68 0.90 1.33 2.10 2.73 6.87 11.8
191 202 225 2 72 307 360 570 790
(4.00) (4.00) (4.00) (4.00) 3.96 3.94 4.03 4.17
xo 105.755 90.415 77.315 66.47 61.785 57.66 50.74 47.92
At dioxane contents of less than 35% (D> 50), the y VS. x plots are so nearly horizontal that no reliable value of the slope (and hence of A ) could be obtained; that is, the (nearly) constant ordinate evaluates the quantity (-71 - A&) but separation of the terms is no longer possible. We therefore assumed that a remained constant a t 4.00, and computed A t ‘ / , using a = 4.00 to evaluate J1 and J2,and plotted A”‘ against z = cAf2/(1 - CYC’/~); according to equation 30, the plot should be A”’ = ho
00
4.
02
04
06
Fig. 2.-Extrapolation plots for sodium bromate in high range of dielectric constants.
merical results are summarized in the upper half of Table I. The A column of Table I thus contains two sets of values: four values directly determined as slopes of the y-x plots where a was simultaneously evaluated with A and i l o , and four values from the slopes of the At‘‘ us. z plots, whose construction required an a priori value of a. For the latter, we used the average of the first four values; the self-consistency of the ensemble of values of the constants is tested in Fig. 3, where log A is plotted against the reciprocal of the dielectric constant. Both sets of points line on the same graph, which moreover is linear. If a were not a constant, the lowest points in Fig. 3 obviously could not lie on a prolongation of the line through the top four, except by an amazing compensation.
X I
k
/
i I -1
- AZ
linear with slope -4and intercept Ao. The graphs are shown in Fig. 2 , where for compactness in presentation, the ordinate scales are shifted vertically by arbitrary amounts. The values of A0 are indicated by the arrows. The unit of the vertical scale is shown by indicating 0.10 A-unit between the vertical lines a t the lower left; this distance corresponds to about O.lO’% for the aqueous system and 0.15% for the 30% mixture. The points lie on the lines within 0.01%. The nu-
0
, -I 0 10
I
20
100ID
30
40
of association on dielectric constant : 6, sodium bromate, from y ZJS. x plots; 8 , sodium bromate, from A”’ DS. z plots; coordinates left and below. 0 , tetraisoamylammonium nitrate, from y us. x plots; coordinates, right and above. Fig. 3.-Dependence
0, tetrabutylammonium iodide, from y Z J ~ x .plots;
The linearity of the log A that A has the simple form
D-l plot suggests
From the slope of the log A zls. D-l line, we obtain center-to-center distance in the ion paris. It will be noted that the .-I= .lo exp(ii/kT) B u N I line is less steep in Fig. 3 than the one for where b is an electrostatic free energy. But if we NaBrOa; the difference is due first to the smaller assume that u is simply a charge-charge energy size of the sodium ion and second to the reinforcee2 aD, the slope of the line in Fig. 3 gives if = 3.30 ment of charge-charge attraction by the chargeinstead of 4.00, the electrostatic center-to-center dipole interaction already discussed. distance found from the y us. x analysis. The disIf now values of Jl(a) are calculated by adding crepancy can be removed, quickly however, if we Ah0 to the ordinate y(0) obtained by extrapolating recall that the bromate ion contains a dipole, be- the y vs. x plots of Fig. 4 to x = bo,absurdly small cause it has a pyrirnidal structure with Br+f a t the (1.5-3.5) and variable values of the parameter & apex and three 0 - ions a t the base in one of its result from the values of J l ( a ) so obtained. This canonical forms.5 The energy ZL therefore is the is in marked contrast to the case of sodium brosum mate, where it was shown that the d value from the u = (e2/aD)+ (ge/dzD) slope of the log A vs. D-l plot was consistent with where is the dipole strength and d is the distance the values obtained from the y E’S. 3c intercepts, from the center of the cation to the electrostatic i.e., remained constant over the entire range of solcenter of the dipole. For 10% = 2.00, 1018u vents investigated. Clearly, an effect is appearing in BuoNBr which is absent (or a t least iiegligiblej = 1.03 and for 10Yd= 4.00, loi5 p = 4.10, if we set 8. = 4.00 above. These values lie in an entirely in the case of KaBrOs. The outstanding physical difference between the two salts is the disparity in reasonable range. This result naturally suggests a conductimetric size of the cations; the tetrabutylammonium ion, study of a variety of other ions; nitrates, for ex- according to molecular models, should exclude a cm. in radius, while ample, should give the same value of n from the spherical volume 7-S X y vs. x plot and the log A v s . D-’ plot, because the the sodium ion is about the same size as n water nitrate ion is electrically ~ymmetrical.~Chlorates, molecule. Using equation 34 of the preceding on the other hand, should have a steeper log .4 n ~ . paper in the form D-’ plot than corresponds to the a determined 5Ad/2 = .Ti(.) - ~ ( 0-) AAIo from the y us. x plot. Conversely, conductance and using d = 0.55 to evaluate J l ( u ) ,the values of data should be helpful in determining previously 6 obtained are shown in the fifth column of Table unknown structures. 11. The next column gives the corresponding Tetrabutylammonium Iodide.”-The conduct- values of the hydrodynamic radius R , calculated as ance was measured in water and in four dinxaneR = 7.326”a X lo-* water mixtures, covering the range 33.83 < D < 78.48. As is shown in Fig. 4, the y ns. x plots are The resulting quantity is reasonably constant and linear; the slopes give the association constants. agrees remarkably well with the value expected on The latter are plotted as the open circles on Fig. .3. the basis of the model, assuming the Einstein formula to give the viscosity contribution due to the +203 I large cations. It should be mentioned that earlier Bu‘p I work on the viscosity of bolaform electrolytesG 050 e 3 0 shows that the Einstein limit of 5 ;2 for the ratio of 045 0 I5 specific viscosity to volume fraction is approached as these ions approach spherical shape. IntroducY \ tion of the (56c/2) term into the conductance equation is thus not completely an ad hoc hypothesis. The fact that the viscosity term leads to reasonable R values argues for its reality; granting this, the long debated question of solution viscosity versus solvent viscosity as the pertinent variable in conductance theory appears to have found its answer. BS.
d = 5.55 as the nitrogen-iodine
TABLE I1 3 83,
TETRABUTYI-AMMO~IUM IODIDE I N
75
“I-
IO
70
30
40
Fig. 4.-Extrapolation plots for tetrabutylammonium iodide. Abscissa scales: 50%. as shown; 45%, ( x - 10); 30%, ( X - 20) ; 157,, ( X - 30) ; 0%, ( X - 60). ( 5 ) J.
C .Slater, P h y s . Rev., 38, 325 (1931)
D J O X A R E - ~ ~ A LIrXTER
TURESAT25’
~
Dioxane
D
A
Ji(5.5)
I
K
‘12
0 15 30 45 50
78.48 66.10 53.28 40.20 35.85
2.73 3.65 5.1 9.5 13.0
245 253 303 4’76 604
0.98 1.03 1.30 1.46 1.56
7.3 7.3 8.0 8.3 8.5
96.30 70.74 d3.835 43.805 41.69
Tetraisoamylammonium Nitrate.2-This salt was measured in 0, 10, 20, 30 and 50y0 dioxane, but (6) P . Goldberg and R. M. Fuoss, Pvoc. N a l . A c a d . Sci , 38, 758 (1952).
July 3 , 19.57
CONDUCTANCE OF SALTS I N DIOXANE-LTATER
only a t 20 and 5Oy0 was the concentration range wide enough to permit a reliable determination of slope (.-I) and intercept (J1 - 5,106,'2 - -4b0). The span from D = 61.86 (20%) to D = 35.55 (50yo)is rather narrow to use to evaluate d from the slope of the log A vs. D - l plot, but fortunately some older data on this salt in 79.8y0 dioxane are available'; the y vs. x plot (second approximation) is shown in Fig. 5 , From the slope, we obtain A
330'7
Lf IXTURES AM, N NO,
_i
0. IO
-75
Fig. 6.-Extrapolation plots for tetraisoamylammonium nitrate in 0, 10 and 30% dioxane.
satisfactory. Other details of the calculation are summarized in Table 111. Again, an entirely reasonable value of the hydrodynamic radius is found; since this constant in effect collects all of the residual uncertainties of the calculation in this particular example, we are inclined to place considerable confidence in both the data and the theoretical treatment, because such agreement among six sets of data has practically zero probability of being fortuitous. TABLE 111 TETRAISOAMYLAMMOSKIM
Fig. 5.-Extrapolation plot for tetraisoamylammonium nitrate in 79.8y0 dioxane.
1030. This point, together with the points for the 20 and .!joy0mixtures, establish a satisfactory straight line (solid points, Fig. 3 ) : A = 0.354 exp(96.1/1)). Equating the exponent to e2/aDkT, we find d = 5.83. This equation was then used to evaluate A for the 0, 10 and 30y0 mixtures, and the corresponding values of J1 and J;r were computed, using 8 = 3.53. Using these constants, the =
XJTRATE
IS
DIOXANE-WATER
MIXTURES AT 25' %
Dioxane
0 10 20 30 50 80
D A 78.48 (1.20) 70.33 (1.38) 61 86 1.67 53.28 (2.14) 35.85 5 . 2 12.01 1050
Ji(5.83)
6
i?
Ao
248 252 274 320 643 1 . 2 X 10'
1.02 1.04 1.17 1.13 1.53
7.3 7.4 7.7 7.6 8.4
89.275 74.845 63.13 54.04 42 575 34.8
..
..
We finally remark that, since the a-value obtained from the slope of the line in Fig. 3 leads to a quantity was then computed; if the conductance satisfactory value of R , the charge-charge energy is obeys the equations derived in the previous paper, all that is required to account for the slope of the log A vs. D-' plot; in other words, as expected, the AIv includes the effect of both ion association and ion atmosphere, leaving only the viscosity term t o be nitrate ion shows no dipole. Tetramethylammonium Picrate.*-This salt preevaluated. From equation 33, we have sents an unusually interesting case: i t was measAIv = A 0 - 5AoSc/2 ured in a series of mixtures including 7oy0dioxane. The corresponding plots are shown in Fig. 6; where the dielectric constant is only 19.07. The they are linear, well within the limit of experi- conductance curve lies a little below the Onsager mental error, and 6 is determined from the slopes. tangent in the dioxane-rich systems, and one might For the 20 and 5070 mixtures, 6 was determined be tempted to assume a small amount of ion from y ( 0 ) as for tetrabutylammonium iodide, us- association. When, however, the y v s . x plots are ing d = 5-83to evaluate J l ( a ) ; for the 79.8y0 mix- constructed, they are seen to be practically horizonture, the (J1 - A&) term is so large that it com- tal; in other words, within the experimental error, From Fig. 5 , A is zero. The fact that no association due t o pletely masks the 5A06/2 term. Jl(a) = 1.47 X loJ; calculated for it = 5.83, J1 = Coulomb forces appears for this salt confirms the 1.20 X lo4. Considering the sensitivity of J ( a ) generally accepted picture of the picrate ion: the to a when b is large (low dielectric constant) and single negative charge is not localized a t the phenthe uncertainty in determining J1 from the graph olic oxygen but is distributed over the peripheral when A is so large, we consider the agreement a-electron system of the whole molecule. ConseAI\' =
4-A z
(7) R. M.Fuoss, Thesis, Brown University, 1932; C . A. Kraus a n d
R. h l . Fuoss, THISJ O U R X A L , 66, 21 ( 1 9 3 3 ) .
( 8 ) P.L. Mercier a n d C. A. Kraus. Proc. Not. Acad Sci., 41, 1033 (1955).
quently the electrostatic force field is so weak that thermally stable pairs with the cation cannot form. The only pairing possible is that due to chance collisions, which evidently do not reduce conductance appreciably.
plots in the mixtures containing 50% or more dioxane. These values gave a straight line on a log A us. 1/D plot, which can be represented by the equation A = 0.11 exp(133/D); the corresponding d-value is 4.21. On the other hand, if we calculate d from Jl(d) obtained from the intercepts y(O), usTABLE IV ing 6 = 0.535 (see later), the values shown in Table TETRAMETHYLAMMOUIUM PICRATE IN DIOXANE-WATER V result. Unlike tetrabutylammonium iodide and MIXTURESAT 25' tetraisoamylammonium nitrate, a different ion size To Dioxane D Ji(a) a 10 is required to reproduce the experimental associa
