10-15, with the result that the influence of HC12- (20) and AgC12- (16) formation was insignificant in these calculations. The mean pKa value obtained for hydrogen chloride is 14.5 f 0.1, but in view of the indirect method of calculation, its uncertainty must be considerably larger than fO.l unit. In Table I11 dissociation constants of several Brfinsted acids in sulfolane are compared with those in acetonitrile. For those dissociation constants that are based on calibration of an acidity scale, our provisional calibration gives 6 to 7 units, while that of Benoit pKa(SL) - pKa(AN) and Pichet (8) gives 3 to 4 units, which is close to the estimated difference in p K a ( S H + ) values of the two solvents. Unfortunately, these comparisons can only be inexact at present, for a t least two major reasons. First, it is not known how the (somewhat uncertain) difference in pKa(SH+) values measured in sulfuric acid is related to the relative basicities of the two bulk solvents. Second, in comparing the position of the equilibrium
-
HA(so1vated)
+ S z=SH+(solvated) + A-(solvated)
(15) in two solvents, it is necessary to consider not only the relative proton basicities of the two solvents, but also their relative abilities to stabilize the species HA, SH+ and A - . For the two ions SH+ and A - , such a comparison would require extrathermodynamic assumptions which introduce uncertainties of generally unknown magnitude.
ACKNOWLEDGMENT We thank R. L. Benoit and P. Pichet for information prior to the paper's publication. We also acknowledge the help of J. Bykowski, E. Devitt, and J. Meyn in carrying out some of the conductimetric titrations. Received for review October 18, 1972. Accepted December 20, 1972. We thank the Kational Science Foundation for financial support under Grant GP-16342.
Single-Ion Activity of Fluoride in Mixed Alkali Halide Solutions John Bagg' and G. A. Rechnitz Department of Chemistry, State University of New Y o r k . Buffalo, N. Y . 14274
The activity coefficients of F - in mixtures of trace concentrations of NaF, and KF in NaCI, KCI, KBr, and K I , at concentrations up to 4 molal have been measured in a cell with a F--selective electrode. The single-ion activities of F- in NaF-NaCI mixtures up to 1 molal, and KF-KX mixtures up to 4 molal, were, within experimental error, equal to the values in pure N a F or KF of the same ionic strength. The values in pure fluoride solutions were assigned using the convention based upon hydration theory as proposed by Robinson, Duer, and Bates. The experimental and theoretical difficulties in assigning single-ion activities in mixed solutions are relevant to the determination of selectivity parameters for ion-selective electrodes. In the light of the results reported in this paper, a procedure is described which maintains a simple ion-activity convention and absorbs the uncertainty in activity coefficients in the selectivity parameter.
The primary advantage of an ion-selective electrode is its ability to measure the activity of one particular ion in the presence of several others. T o make full use of this ability in analytical applications, it is necessary to devise a single-ion activity convention for mixed solutions in order to convert activities into concentrations. Although there has been considerable research into the measurement of mean molal activity coefficients in mixed electrolytes ( I ) , there has been little corresponding work for single-ion activities ( 2 ) . 'On study leave from the Department of Industrial Science. University of Melbourne, Parkvilie, Victoria 3052, Australia H. S. Harned and R. A. Robinson, "Multicomponent Electrolyte Solutions, International Encyclopedia of Physicai Chemistry and Chemical Physics," Topic 15, Voi. 2, Pergamon Press, Elmsford, N.Y., 1968. J. V. Leyendekkers, Anal. Chem., 43, 1835 (1971).
I t is desirable to choose a cell for the estimation of single-ion activities in which a minimum of assumptions is required in calculations derived from the measured emf. Several workers have used combinations of two electrodes in order to remove junction potentials (2-4); in particular, Leyendekkers ( 2 ) has used a F--selective electrode in combination with a liquid-membrane C1- -electrode to determine the trace activity of F - in NaF-XaC1 and NaFKC1 mixed solutions. Another type of cell, consisting of a double-junction reference electrode and an ion-selective electrode was applied recently to single electrolyte solutions (5) and offers some advantages for determination of trace activities. Consider the following solution chain for a binary electrolyte mixture, NY-MX, with ml > K12mzy2, it will be shown later that, from Equation l, the single-ion activity coefficient of MX can be related to the conventional assignment of single-ion activity coefficient of NY in pure solution. A further advantage of this solution chain is that only a single ion-selective electrode is required, thus, avoiding problems of interference with the second electrode, and allowing measurements in solutions for which no suitable second electrode exists. Because of its nearly ideal selectivity and Nernstian response, the F--selective electrode is, a t present, the electrode which comes closest to meeting the requirements for determining trace activities a t high concentrations of a second electrolyte. Measurements of the F - activity coefficient a t trace concentrations were, therefore, made for NaF and K F in NaC1, KC1, KBr, and KI, up to 4 molal using the above solution chain.
EXPERIMENTAL Reagents a n d Apparatus. All solutions were prepared from analytical grade chemicals dissolved in deionized water. Solutions were contained in poly(ethy1ene) beakers maintained at 25 f 0.1 "C and stirred by a Teflon (Du Pontbcoated magnetic stirrer. Electrodes were Orion Research Inc. (94-90) F - -selective electrode and Orion Research Inc. double-junction reference (90-02-00) with the outer filling solution (m,) 4m KCl, 4m NaC1, 4m KBr, or 4m KI, to match the electrolyte NY in the test solution. Procedure. The procedure used by Leyendekkers (2), in which K F or NaF was added to a known weight of 0.01m NY to make a concentration approximately 10-4m in fluoride, was adopted in this work. The molality of NY was increased by adding solid NY to the original solution in order that the fluoride concentration should remain unchanged. Stable emf readings were obtained (0.1 mV drift/l5 min) within 1 min up to 0.5, NY but at higher concentrations up to 10 min was required to achieve such stability. The assumption of a constant F- concentration requires a very low F - impurity in NY.Potentiometric measurements in pure iXY at the highest concentrations used showed that the impurity concentration was < 5 x 10-'jrn F - . Careful measurements have shown negligible interference by C1- on the F--selective electrode a t a composition of 10-4m NaF3m KCl (Si, and the assumption was made in this work that m l y >> K12m2y2 for all the solutions. i . e . , that
E
=
E,"
-
R T / F log m,y,
represents the half-cell potential of the F - -electrode. On the basis of the Harned coefficients for the mean molal activities of alkali halides f7), it is reasonable to equate the value of y p in a mixed solution where mz = 0.01m to the value in pure NaF of the same concentration. Remembering that both ml and m , remain fixed in these experiments, the difference between the emf of a cell measured, first. at concentration m2, and, second, with a reference solution, m2 = 0.01m, is given from Equation 1,
(3)
In the calculation of yF m 2 r from Equation 3 the integral was evaluated graphically using values of t,, and u N y from the literature 18) Where data were not available, e g , for KBr, KI, and to extrapolate to higher concentrations of NaC1, the following formula for the transport number was used (9):
t,
=
A, -30.325fi/(l 4-0.329afi) AKy- 60.65fi / (1 0.329afi)
+
RESULTS AND DISCUSSION For all the halide solutions, yy = ycl was employed, where ycl is the single-ion activity coefficient from the convention proposed by Bates (11) for I < 0.1. At higher ionic strengths, the values of ~ ~ ( were ~ calculated 2 ) from the following formula (12): with ( h N - hy), the difference between the cation and anion hydration numbers set equal to 1.9, 2.2, and 2.5, for KC1, KBr, and KI, respectively (10); Y N Y the mean molal activity coefficient of NY, and cp the osmotic coefficient (12). On the basis of conductivity (23) and mean molal activity data (14), it appears that K F can be treated as completely dissociated in solution. NaF, however, does appear to be appreciably associated (14) and so corection was made for NaF-NaC1 mixtures. The degree of dissociation, a d , of trace NaF in NaCl is given by,
where y N a F is the mean molal activity coefficient of the dissociated species and K = 1.88 is the thermodynamic dissociation constant (13, 14). As an approximation, YNaF was equated to the value calculated for pure NaF of the same ionic strength as the mixture (14). The observed stoichiometric activity coefficient is then related to the ideal activity coefficient by YF(ideal)
= YF(rn2)Iad
(7)
The plot of log yF(ldea1) us. m2 for NaF-NaC1 is shown in Figure 1, and log y F ( m 2 ) us. m2 for KF-KY and NaF~ pure 2 ~ KF KC1 in Figures 2-4. Also shown is log y ~ ( for of the same ionic strength as the mixture using the values of Robinson, Duer, and Bates (14). The value of log YFildeal) for pure NaF shown in Figure 1 was calculated by the formula of the above workers modified to account for association as follows:
with the observed mean molal activity coefficient of NaF, h N a = 3.5, hF = 1.8. The degree of dissociation is, in this case, given by (14), Bock and Strecker have also made measurements in mixed NaF-NaC1 using the F--selective electrode and SCE reference with liquid-junction (15). In order to compare their results with the present work, some method of estimating junction potentials was required. We have successfully used the Henderson equation for this purpose and obtained results consistent with the single-ion activity scale for halides ( 5 ) . The Henderson equation was applied to Bock and Strecker's results and the comparison is
(4)
with AN, A N Y , the limiting equivalent conductances for N + and NY, respectively, e. the molar concentration, and, a , an ionic (6) J. N. Butler, Nat. Bur. Stand. I U . S . JSpec. Pub/.. 314, 162 (1969). (7) J. V . Leyendekkers, J , Phys. Chem., 75,946 (1971). (8) R. A . Robinson and R. H . Stokes, "Electrolyte Solutions,'' Butterworth, London, 1959. p 492. (9) R. A . Robinson and R. H . Stokes, Ref. 8. p 171
1070
radius parameter. In those cases where, a, had not been determined from transport data, the values KBr 3.9, KI 4.2, taken from the two-parameter equation for mean molal activity coefficients, were used (10).
A N A L Y T I C A L C H E M I S T R Y , VOL. 45, NO. 7, J U N E 1973
R. A . Robinson and R. H. Stokes, Ref. 8, p 246. R. G . Bates and M. Alfenaar, Nat. Bur. Stand. I U S . ) Spec. Pub/., 314, 191 (1969). R. G. Bates, B. R. Staples, and R . A. Robinson, Anal. Chem.. 42, 667 (1970). W. C. Duer. R. A . Robinson, and R. G . Bates, J . Chem. S O C . . Faraday Trans.. 1 , 6 8 , 716 (1972). R. A . Robinson, W. C. Duer, and R. G . Bates, Ana/. Chem.. 43, 1862 (1971). R. Bock and S. Strecker. Z. Anal. Chem., 235, 322 (1968).
O.O
t
O.Oi
V
0
t 1 1.0
0
1
Figure 1 . Log Y F ( i d e a l , vs. I (m2) for NaF-NaCI. ( i ) - for pure NaF; ( i i ) experimental values, 0 , this work; X, ,Bock and Strecker (75); V , Leyendekkers (2). Error bar shows estimated 95% confidence limits
1.o
2 .@
I
3 .O
4.O
Figure 3. Log Y F , ~ vs. , ~ I ( m 2 ) for KF-KI. ( i ) -for ( i i ) experimental values, 0, this work
pure KF;
O O t
V
1 IO
0
0
IO
2 0
3.0
4 0
I Figure 2. Log Y F ( ~ vs. ~ ) I ( m 2 ) for KF-KCI and NaF-KCI. (i)for pure K F , (ii) experimental values, 0, KF-KCI this work; X, NaF-KCI this work: V , NaF-KCI Le-yendekkers ( 2 )
shown in Figure 1. The third set of results displayed in Figure 1 are those of Leyendekkers (2) who used a liquidmembrane C1- -electrode (Orion Research Inc. 92-17) in combination with the F--selective electrode. His results have been adjusted to the same scale for ycl and the same association constant [log K = -0.28 compared to log K = -1.2 (2)] used in this paper in order to eliminate the effects of the different assumptions upon the comparison. Bock and Strecker’s results are in good agreement with the present work over the whole concentration range but Leyendekker’s results deviate above 0.2m. One possible explanation for this deviation is that the liquid-membrane electrode has been shown to be non-Nernstian compared to the Ag, AgCl electrode at NaCl concentrations above 0.2m (11). The deviations observed under these circumstances were of the same sign and magnitude as those shown in Figure 1. It seems possible that, at high concentrations, NaCl is appreciably soluble in the membrane, as has been proposed for the Ca-selective liquid-membrane (16, 17). The results in Figure 1 agree, within experimental error with the values calculated for pure NaF up to l m . Measurements in NaF-NaCl were extended up to 6 m and the (16) R. Huston and J. N. Butler. Anal. C h e m . , 41, 200 (1969) (17) J. Bagg and R . Vinen, Anal. Chem., 44, 1773 (1972)
2 0
I
30
4 0
Figure 4. Log Y F ( m Z ) vs. I ( m 2 ) for KF-KI. ( i ) -for ( i i ) experimental values, 0,this work
pure KF;
results for log ~ ~ ( given ~ 2 in) Table I. Due to the uncertainty in extrapolating y N a F at such high concentrations no attempt was made to calculate Y F ( i d e a ] ) . The results in Figures 2-4 for KF-KY and NaF-KC1 also showed reasonable agreement with the values for the pure salt, KF. Leyendekker’s data for NaF-KC1 as shown in Figure 2., again adjusted to the same ycl, and from the assumption of partial association [log K = - 1.5 estimated in (2)] to our assumption of no association. The adjusted data are in agreement with this work up to 0.5m but show significant deviations at higher concentrations in the same direction as in NaF-NaCl. For the system we have examined, all the conventions adopted and assumptions made concering association seem, therefore, to be internally consistent with the assumption that the single-ion activity coefficient of F- depends only upon the total ionic strength of the mixture. Such a conclusion must, of course, be an oversimplification, and the following approach is suggested for application as more data from a variety of systems become available. Taking the present system as an example; in KY solutions, with only trace amounts of KF, Y K ( m 2 )can reasonably be set equal to the value in pure KY, and then Y F ( m 2 , is given by log
YF(rn2) =
log
YKF(trace)
- log
YK(m2)
(lo)
For a large number of electrolytes, Harned’s rule is applicable, and assuming this rule holds for KF-KY mixtures (I), A N A L Y T I C A L C H E M I S T R Y , VOL. 45, NO. 7, J U N E 1973
1071
Table I. Log YF for NaF-NaCI at High Concentrations NaCl (m2)
-log Y F
1
0.255
2
0.300 0.331
4 6
log YF(m2)
=
0.352
log YKF(m2) -
2aKF(KY)m2
- log YK(m2) (11)
In this way, YF could be derived from mean molal and single-ion activity coefficients in single electrolyte solutions provided an estimate of ~ K F ( K Y )is available. At present ( Y K F ( K Y ) has not been measured, but, when the experimental values of 7F were substituted into Equation 11, values for a of 0.006-0.010 were calculated, these were of the same order of magnitude as those observed in other mixed halide systems (1, 7). Several approaches have been adopted toward the theoretical estimation of a (18, 19). One which has been successful in fitting activity coefficients in single electrolyte solutions ( I , 5, lo), is based upon solvent-ion interaction and is capable of extension to mixed solutions (20). For KF-KY the theory predicts,
with q K F , q K Y , the osmotic coefficients of pure KF and KY a t molality m2. Clearly, accurate values of hydration numbers as functions of concentration are required for reliable estimation of a. Such data are becoming available (21) and will be applied in future studies. Using the hydration numbers obtained by fitting mean molal activity coefficients, and assuming an uncertainty of f l in these numbers (12), gives CYKF(KCI,= 0.015 f 0.015. The difficulties of assigning single-ion activity coefficients in mixed solutions has a direct bearing upon the determination of electrode selectivity. Moody and Thomas (22, 23) have stressed that the most informative means of expressing selectivity is to determine the selectivity parameter, K l 2 , defined in the following equation (24):
for a binary mixture containing A-2(m1), B-Y(m2) with E, the half-cell potential for an A-2-selective electrode. In certain cases, the A-selective electrode can be combined with a suitable second selective reference electrode to form a cell without liquid-junction. In practice, however, the calibration and use of electrodes is carried out in solutions of arbitrary composition and reference electrodes with liquid junctions are employed. Attempts to determine Kl2 from potentiometric measurements in cells with liquid-junction, lead, without making some assumptions, to a circular line of reasoning. The determination of K12 requires a knowledge of 71 and 7 2 in the mixed solution; the estimation of 71 and 7 2 from potentiometric results requires a knowledge of Kl2. Under certain circumstances, as shown in this work with a nearly ideal electrode, the uncertainty can be minimized. However, with many other electrodes which are less selective, this E. A . Guggenheim, Phil. Mag., 19, 588 (1935). G. Scatchard, Chem. Rev., 19, 309 (1936). R. A. Robinson and R. H. Stokes, Ref. 8, p 452. J. O'M. Bockris and P. P. S. Saluja, J. Phys. Chem., 76, 2140 (1972). (22) G. J. Moodyand J. D. R. Thomas, Talanta, 18, 1251 (1971) (23) G. J. Moody and J. D. R. Thomas, Talanta, 19, 623 (1972). (24) G. Eisenman, Nat. Bur. Stand. (U.S.) Spec. Pub/., 314, 1 (1969).
(18) (19) (20) (21)
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A N A L Y T I C A L CHEMISTRY, VOL. 45, NO. 7, JUNE
1973
approach is not possible. To overcome this problem, a t the present time, it is suggested that the convention be adopted in the reporting of selectivity parameters of equating the single-ion activity coefficient with the value for a pure salt solution of the same ionic strength as the mixture. This assumption has already been examined by some workers (25, 26) and received support from the results presented in this paper. Several experimental procedures (23, 27) have been suggested for determining K l 2 , but the following procedure used by earlier workers (28, 29), in conjunction with the single-ion activity coefficient, seems to offer some advantages over other methods. As an illustration, consider the following cell: Reference 1 1 MX ( m l ) , NY ( m z )1 X- selective electrode Measurements can be made with pure MX ( m l = I) and in mixtures of the same constant ionic strength ( m l - m2 = I). The difference in emf between pure solution and one of the mixtures is given by,
E,,, - E,,,t,
RT =
Iog(1 + K , Z m ~ Y n / m l Y J+
with 4 E J the residual junction potential. One advantage of the constant ionic strength is that the residual junction potential is likely to be small, in contrast to other methods, where, because the ionic strength varies, AEJ is likely to be several millivolts in magnitude (30). K12 can be determined from the slope of the graph { F ( E M-~E,,,,)/ RT - log(ml/IJ us. m ~ y 2 l m l y 1(29). If Kl2 is determined a t several different total ionic strengths, then Kl2 given as a function of total ionic strength is a very concise way of reporting a large amount of information about the electrode, suitable for practical application, without resorting to extensive graphical display. The constant ionic strength technique may be especially used to advantage in those cases where a suitable ion activity coefficient has not yet been formulated. At constant ionic strength the ratio, 72/71, is constant, using the convention proposed, and this ratio may be included into the selectivity parameter to give a concentration parameter, K12', rather than activity parameter. When the electrode is being used in an analytical application, the appropriate value of Kl2' or K l 2 for the total ionic strength of the sample solution may be inserted into Equation 14 to estimate errors due to foreign ion interference. In summary, we believe that the application of the single-ion activity convention of Bates ( I I ) , Bates, Staples, and Robinson (12), extended to mixtures by the further convention of dependence only upon total ionic strength and coupled with the method of constant ionic strength mixtures, would lead to considerable standardization and rationalization in the reporting and usage of selectivity parameters. Received for review August 23, 1972. Accepted December 18, 1972. We gratefully acknowledge support of this work by the National Science Foundation. (25) R. M. Garrels. "Glass Electrodes for Hydrogen and other Cations," G. Eisenman, Ed., Marcel Dekker, New York, N.Y.. 1967, p 344. (26) E. W. Moore, Naf. Bur. Stand, (U.S.) Spec. Publ., 314,215 (1969). (27) K. Srinivasan and G. A . Rechnitz, Anal. Chem., 41, 1203 (1969). (28) H. P. Gregor and H . Schonhorn. J . Amer. Chem. SOC., 81, 3911 (1959). (29) J. Bagg and H . P. Gregor, J. Amer. Chem. SOC.,86,3626 (1964). (30) R . G. Bates, Nat. Bur. Stand. IU.S.J Spec. Publ. 314,416 (1969).
