3922
RICHARD M. WALLACE
Determination of the Second Dissociation Constant of Sulfuric Acid by Donnan Membrane Equilibrium1
by Richard M. Wallace Savannah River Laboratorg, E . I . du Pont de Nemours and Co., Aiken, South Carolina
(Received June 21, 1966)
A method was developed for determining the degree of dissociation of sulfuric acid by measuring the distribution of 22Natracer between solutions of sulfuric acid and perchloric acid separated by a cation-permselective membrane. The degree of dissociation and the stoichiometric activity for sulfuric acid were combined to obtain the second dissociation equilibrium constant. Equilibrium constants at 25, 35, and 50" were 0.0131,0.0094,and 0.0062 mole kg-l, respectively. An average AH was calculated to be -5.6 kcal/mole.
Donnan membrane equilibrium across permselective membranes is used for the determination of charges on ions in solution.2 This paper describes the extension of this technique to measure dissociation constants of acids. The method was tested by determining the second dissociation constant of sulfuric acid because previous measurements of this constant by a variety of methods3 are available for comparison. Also, knowledge of the free sulfate ion concentration in sulfuric acid solutions of varying concentration is necessary for further studies of complex formation between sulfate and metal ions in progress in this laboratory.
Basis of Method The distribution of 22Na+tracer between a solution of perchloric acid on one side (p) of a cation-permselective membrane and a solution of sulfuric acid on the other side (s) is measured. From t.his distribution and knowledge of the concentrations of the solutions, the concentration of free hydrogen ion in the sulfuric acid solution can be calculated along with other quantities necessary for determining the degree of dissociation. The use of sodium tracer 22Na+permitted convenient and accurate measurement of the relative concentration of sodium ion in the two solutions at an absolute concentration too low to affect the concentrations of other ions in the solutions. The following assumptions were made in developing equations for calculating the equilibrium constant for the bisulfate dissociation (HS0,H + Sod2-) :
+
The Journal of Physical Chemistry
(1) perchloric acid is completely dissociated; (2) the first stage of the dissociation of sulfuric acid is complete; (3) sodium ion is not complexed by any of the anions present. Conditions for Donnan membrane equilibrium across cation-permselective membranes require that
-
(Hs+>- _ (Nas+) ___ _ _ -YNas - YH, (Hp+>
(Nap+> ?"ap
(1)
YHB
where (H+) and (Na+) are the respective concentrations of hydrogen and sodium ions, while the subscripts s and p refer to the sulfuric acid and perchloric acid solutions, respectively. The symbol y with the appropriate subscript is the activity coefficient of the designated ion. The concentration of free hydrogen ion in the sulfuric acid solution can then be calculated from the equation
where /3 is the collection of activity coefficients in eq 1, which is assumed to be unity when the ionic strength of the solutions on opposite sides of the (1) The information contained in this article was developed during the course of work under Contract AT(07-2)-1 with the U. S. Atomic Energy Commission. (2) R. M. Wallace, J . Phys. Chem., 68, 2418 (1964). (3) L. G. Sillen and A. E. Martell, "Stability Constants of MetalIon Complexes," The Chemical Society, London, 1964, p 232.
3923
SECOND DISSOCIATION CONSTANT OF SULFURIC ACID
membranes are the same. This assumption will be discussed in detail later. The total analytical sulfate concentration, St, and the total analytical acidity, Ht, can be represented in terms of the concentrations of free hydrogen ion (Ha+), bisulfate ion (HS04-), and sulfate ion (s04’-)
+ (HSOa-) Ht = (Ha+>+ (HSOr-)
St = (s04’-)
(3)
(4)
Since the sulfuric acid solution is essentially pure
(5)
Ht = 2St
Finally, if the degree of dissociation, a, of bisulfate is defined as a=-
(so42-) St
then a combination of eq 2-6 yields
(7) from which a can be determined if 6 is known. In solutions of moderate concentration (up to a few tenths molar), the activity of an ion can be represented quite well by the Debye-Huckel equation
Where A and B are known constants, Zi is the charge on the ion, p is the ionic strength, and it is the ionsize parameter. d need not be the same for two different cations in the same solution. Differences in d between the hydrogen ion and sodium ion in the two solutions will cause p to differ significantly from unity, particularly in the more concentrated solutions. Differences in d are awkward to handle because of the form of eq 8. However, Robinson and Stokes4 have shown that in dilute solutions, variations in it produce approximately the same effect as an additional term, linear in I.(. The activity coefficients can therefore be most simply represented by
where 3 is an average ion-size parameter, assumed to be constant for a given solution, which may however vary from one solution to another, while ci will depend on the particular ion and the solution. With the above assumption log 6 =
(CNas
-
CHhPs
-
(CNap
CHp)Pp
(9) and 6 is the difference between the actual value and the average, then eq 9 can be written
log P =
&3
- PPI +
6(,,8
+
PPI
(10)
The first term on the right of eq 10 becomes zero when p s = pp, while the second term represents an uncertainty in log 6 that cannot be evaluated exactly. A comparison of the activity coefficients of the sodium salts and acids of various univalent anions indicates that the absolute value of S is probably no greater than 0.03. This value would cause an uncertainty of only 0.006 in log (1 a) at 1.1 = 0.1; and the uncertainty decreases with decreasing ionic strength, approaching zero at infinite dilution. For the present purposes 6 can therefore be assumed to be zero. The validity of this assumption is derived from the internal consistency of the results and their agreement with other observations, as will be apparent later in this paper. The best value of a should be obtained by equilibrating a solution of sulfuric acid with one of perchloric acid of the same ionic strength. Unfortunately, the ionic strength of sulfuric acid depends on a, and the correct concentration of perchloric acid to use is not known a priori. The problem may be solved, however, by equilibrating a given concentration of sulfuric acid with several different concentrations of perchloric acid that bracket the ionic strength of the sulfuric acid solution. The correct value of a can then be obtained by interpolation, as shown below. From eq 7 and 10, assuming 6 = 0
+
(Ep+)(Na,+) log __ ___ = log (1 St (Nap+)
+ a) - a,,, + App
(11)
with distribution measurements at constant sulfuric acid concentration but variable perchloric acid concentrations. The value of the left side of eq 11 can be determined as a function of perchloric acid concentration. The graph of this relationship should be a straight line with a slope equal to the constant 5. When the value of A has been established, a can be determined from eq 12, which is derived from (11) by substituting the equivalent St(l 2a) for ps.
+
log (1
+ a) - ZSt(1 + 2a)
(12)
With a constant sulfuric acid concentration, the left side of eq 12 must remain constant (except for small variations caused by scatter in the data). Equation 12 can then be graphically solved for a.
(9)
If A is the average of the two Values Of C N ~ CH in
(4) R. A. Robinson and R. H. Stokes, “Electrolytic Solutions,” Butterworth and CO. Ltd., London, 1955, p 234.
Volume 70,Number 18 December 1066
RICHARD M. WALLACE
3924
Calculation of Equilibrium Constants The usual procedure for determining equilibrium constants when the degrees of dissociation are known at various concentrations is to calculate the equilibrium quotient, Q, in terms of concentrations or molalities, and then by trial and error to determine a value of the ion-size parameter, d, that will yield a constant value for the equilibrium constant, K , in terms of activities. Since stoichiometric activity data for sulfuric acid are a ~ a i l a b l e equilibrium ,~ constants can be calculated in a more convenient manner and simultaneously tested for the consistency of the two types of measurements. The equation for this calculation is derived as follows. The equilibrium constant,, K , to be determined is defined by eq 13, where a is the activity of the designated ion.
K =
a H taSOd2-
(13)
aHSOi-
The stoichiometric activity of sulfuric acid, a2, in (14) can be represented in terms either of (1) the activities of the hydrogen ions and sulfate ions or (2) the molality, m, of sulfuric acid, the degree of dissociation of bisulfate, a, and the activity coefficients, y, of the individual ions. a2 = aH+2aso4*= m3(1
+ a)2ayH+2yso4z- (14)
The equilibrium constant can also be expressed in terms of the stoichiometric activity and the activities of the hydrogen and bisulfate ions or in tzrms of m, a, and the activity coefficients of these ions. Thus
K =
az
~H+~HSO,-
m2(1
+ a)(l -
~YH+YHSO,-
(15)
Equations 14 and 15 can each be solved for the product of activity coefficients to give
YH+YHSO,- =
a2
Kmy1
+ a)(l - a)
(17)
If eq 17 is divided by the cube root of eq 16 and az is expressed in terms of the molality and the stoichiometric mean activity coeffcient, y+, of sulfuric acid (a2 4may*3), then
If a single ion-size parameter d in the DebyeHuckel equation (eq 8) exists that applies to all of The Journal of Physical Chemistry
the ions involved, then the left side of eq 18 will be unity, and K can be expressed in terms of experimentally determined quantities
Experimental Section Membrane equilibrations were run in an apparatus described previously1 that consists of two Teflon (Du Pont trademark for its fluorocarbon plastic) blocks each containing a 20-ml cylindrical cavity and a filling passage. The equilibration cell was assembled by placing a circular membrane between the two blocks and bolting them together. Solutions were added to the cavities on each side of the membrane through the filling passages, which were subsequently sealed. The solutions were agitated during equilibration by rotating the cells at 60-120 rpm in a constanttemperature bath. The membranes were AAlFion C103 (Trademark of American Machine and Foundry Co.) cation-permeable membranes, converted from the sodium form to the hydrogen form. About 15 ml of water containing a small amount of 22Na tracer was introduced into the cell on each side of the membrane. The cells were rotated for about 1 hr to absorb the tracer on the membranes; then the water was removed and the cells were disassembled, carefully dried, and reassembled. Accurately analyzed solutions of sulfuric and perchloric acids were placed in each cell on opposite sides of the membrane and equilibrated in a constant-temperature bath, after which the solutions were removed for analysis. The length of the equilibrations varied with the temperature: 6 hr at 25 i 0.lo, 4 hr at 35 f 0.lo, and 2 hr at 50 i 0.1'. Preliminary kinetic studies indicated that equilibrium with respect to the distribution of sodium ions was virtually complete after 1 hr at 25O, but longer periods were desirable to assure that equilibrium was more nearly attained. Attempts to equilibrate for 20 hr or more resulted in a significant leakage of sulfate ion; e.g., after 24 hr at 25' almost 1% of the sulfate was transferred into the perchloric acid solutions and even larger fractions were transferred at the higher temperatures. At the equilibration times used, however, less than 0.3% of the sulfate was transferred in the worst case. No direct analysis of the amount of perchlorate transferred into the sulfate solutions was made, but a material balance based on acid and sulfate analyses indicated that it was no greater than the sulfate transport.
SECOND DISSOCIATION CONSTANT OF SULFURIC ACID
The solutions were counted for the 22Nay activity after removal from the cells. Because of osmosis during the equilibrations, the solutions in which the acid concentrations were 20.05 M were reanalyzed for total acidity. 22Sa was counted in a well-type y-scintillation counter with an RIDL solid-state scaler and timer. All solutions were counted for a sufficient time to obtain at least lo5 counts, which limited statistical error to about 0.3%. Perchloric and sulfuric acids were determined by titrating (to pH 7.0 on a pH meter) with carbonatefree sodium hydroxide using a 1-ml micrometer buret. Duplicate results always agreed within 0.4%. SuIfate leakage across the membrane was determined independently by titration with BaC12. Titrations were performed in 80% ethanol-20% water mixture at an apparent pH between 2.5 and 4 with Thoron as an indicator. This titration was not nearly so precise as the acid-base titration. Consequently, the pure sulfuric acid was titrated with base.
Results and Discussion Detailed results of equilibrium measurements at 25' are shown in Table 1. The distribution of 22Na was determined among three different concentrations of perchloric acid for each of six concentrations of sulfuric acid.
Table I: Dissociation of HSOa- at 25O HC104
&SO4 molarity (St)
molarity
0.001010 0.001010 0.001010
0.001013 0.001520 0.002018
1.894 1.253 0.951
1.899 1.886 1,900
0.005026 0.005026 0.005026
0.005056 0.007613 0.01018
1.688 1,126 0.836
I . 698
0.01004 0.01004 0.01004
0.01018 0.01529 0.02021
I . 573 1.048 0.783
1.596 1.597 1.577
0.590
0.02515 0.02515 0.02515
0,02522 0.03751 0.05030
1.433 0.954 0.7091
1.437 1.424 1.418
0.420
0.05050 0.05lOl 0.05123
0.05065 0,07491 0.09985
1.3414 0.8998 0.6669
1.345 1.321 1,300
0.314
0.1015 0.1022 0.1033
0.1008 0.1502 0.1984
1.2556 0.8374 0.6268
1.248 1.231 1.204
0.227
(HP)
(Nan+)/ (Nap+)
(Hp+) (Nan+) -__ St
(Nap+)
1.706 1.693
a
0.895
0.699
3925
Fractional dissociations, a, for the three higher concentrations of sulfuric acid for which (H,) (Na,)/ &(Nap) varied significantly with the concentration of perchloric acid were calculated by interpolation to the conditions at which the ionic strength was the same on both sides of the membrane using eq 12. This procedure was developed assuming constant sulfuric acid concentration for a set of measurements; however, this condition was difficult to maintain exactly at the higher concentrations because of differences in the amount of osmosis, and consequently the sulfuric acid concentration varied slightly. Hence, the average value of the sulfuric acid concentration was used in the calculations. Since a changes slowly with concentration in the more concentrated solutions, these small variations in the concentration did not contribute significantly to the error. At concentrations below 0.025 M , where differences between the ionic strength on opposite sides of the membranes were small, the quantity (HP+)(Nal+)/ &(Nap+) was virtually independent of the perchloric acid concentration. a was calculated from eq 7 with 0 = 1 in these cases, and an average of the three measurements was taken. Table I1 summarizes the dissociation values calculated from membrane equilibrium measurements at 25, 35, and 50'. The values of a which were measured as a function of molarity have been interpolated to the nearest round value of the molality for easy comparison with activity data.5 The values of a at 35 and 50' were determined in the same manner except that distribution measurements were made at only two different perchloric acid concentrations for each sulfuric acid concentration. The values of the stoichiometric activity coefficients of sulfuric acid -yk are those of Harned and Hamer5 determined potentiometrically with a hydrogen and a lead peroxidelead sulfate electrode. Quadratic interpolation was used where necessary to obtain values not given explicitly in their tables. The values of K shown were calculated from the other data in the table using eq 19. Average values of K at each temperature together with their standard deviations are given at the bottom of each column. The average value of AH between 25 and 50' for the reaction was determined to be -5.6 0.5 kcal/ mole from a least-squares fit of log K as a linear function of the reciprocal of the absolute temperature. The constancy of K at 25' over a 100-fold variation in the concentration demonstrates consistency between the values of a determined by membrane equilibrium ( 5 ) H. S. Harned and W. J. Hamer, J . Am. Chem. SOC.,57, 27
(1935).
V o l u m e 70, Number 12 December 1966
RICHARD M. WALLACE
3926
~~
~
Table 11: Dissociation Data for HSOdat Various Temperatures &SO4
molality
a
0.001 0.005 0.010 0.025 0.050 0.100
0.896 0.700 0.591 0.421 0.316 0.229
K
Y I
25 0.830 0.639 0.544 0,423 0.340 0.265
0.0130 0.0128 0.0131 0.0130 0.0133 0.0131 Av 0.0131 0.0002
*
0.857 0.635 0.500 0.361 0.268 0.182
0.001 0.005 0.010 0.025 0.050 0.100
0.784 0.534 0.409 0.270 0.191 0.139
0.814 0.608 0.511 0.392 0.313 0.240
Table 111: Calculated Values of the Ion-Size Parameter d Ha804
35 O 0.001 0.005 0.010 0,025 0.050 0.100
values are those obtained at the highest concentrations, where deviations from the limiting law are the greatest. Virtually the same value of d was obtained at the highest concentration at all three temperatures, and very good agreement was obtained in all cases between the values at 25 and 35'. The most reliable values of d are -6.0 A, which is a reasonable value for the ion-size parameter of acid solutions.
0.0090 0.0093 0.0091 0.0097 0.0100 0.0095 Av 0.0094 =k O.ooo4
molality
25O
6, A350
50'
0.005 0.010 0,025 0.050 0.100
4.5 4.7 6.5 6.4 5.9
5.7 7.7 6.5 6.3 6.0
10.7 9.9 9.0 8.1 6.1
50 O 0.790 0.566 0.467 0.352 0.279 0.214
0.0055 0.0061 0,0062 0.0064 0.0066 0.0067 Av 0.0062 & 0.0004
measurements and the stoichiometric activity coefficients determined potentiometrically. K varied more at the higher temperatures, although the variations were still small. The larger variations at the higher temperatures may derive from the experimental technique in the present work. Although the equilibrations were run in a constant-temperature bath, the cells had to be removed for sampling. The cells were sampled as rapidly as possible, but some reequilibration may have occurred. The changes in K with molality at 50' which appear to be systematic rather than random may be a result of this reequilibration. The ion-size parameters d (eq 8) were calculated as follows. The product of activity coefficients, YH + 2 ~ ~ ~ , nat - , each molality was first calculated with eq 16. These values were then substituted into the Debye-Hiickel equation together with the appropriate 2a) and values value of the ionic strength p = m(1 of A and B at the appropriate temperatures tabulated by Robinson and Stokes.6 The equation was then solved ford. Values of d calculated for the five highest concentrations are shown in Table 111. The most reliable
+
The Journal of Physical Chemiatry
Membrane equilibrium measurements are consistent with activity data, and the AH of reaction, -5.6 f 0.5 kcal mole-', obtained is in substantial agreement with the calorimetric values of Pitzer' (-5.2 f 0.5 kcal mole-') and of Austin and Mair* (-5.74 f 0.20 kcal mole-'). K values did not agree very well with results of other methods. Davies, Jones, and Monkg obtained a value of 0.0103 for K at 25' by potentiometric measurements on mixtures of hydrochloric acid and sulfuric acid using hydrogen and silver-silver chloride electrodes. Nair and Nancollaslo obtained a value of 0.0110 by the same method. Young, Klotz, and Singleterry" obtained a value for K of 0.01015 by a spectrophotometric method, while Kerker12 derived a value of 0.0102 from conductance and transport 'data. All these values are considerably lower than 0.0131 found in the present study. Similar discrepancies also occur at other temperatures. The potentiometric method with mixtures of sulfuric and hydrochloric acids or mixtures of sodium sulfate, sodium bisulfate, and sodium chloride has
(6) See ref 4, p 491. (7) K.S. Pitaer, J. Am. Chem. Soc., 59, 2365 (1937). (8) J. M. Austin and A. D. Mair, J . Phya. Chem., 66, 519 (1962). (9) C. W.Davies, H. W. Jones, and C. B. Monk, Trans. Faraday Soc., 48, 921 (1952). (10) V. 5. K.Nair and G. H. Nancollas, J. Chem. SOC.,4144 (1958). (11)I. M. Klotz and C. R. Singleterry, Thesis, University of Chicago, 1940. (12) M.Kerker, J . Am. Chem. SOC.,79, 3664 (1957).
SECOND DISSOCIATION CONSTANT OF SULFURIC ACID
been criticized by Hamerla because the extrapolated value of the equilibrium constant depends upon the ion-size parameter chosen for the calculations. He showed that the choice of d = 4 A gave agreement between the two potentiometric methods and the spectrophotometric method but stated that the agreement might be fortuitous. The agreement between the values obtained by the potentiometric methods and these of the present study would, however, not be improved by using tt = 6 A in the potentiometric calculation, since an increase in the ion-size parameter causes K to change in the wrong direction. The discrepancy between the results of the present study and those obtained by conductivity measurements is probably not serious. Kerker’s12 values for a are only 3-5% lower than those of the present study in the region between 0.001 and 0.005 M . This difference decreases with increasing concentration until both methods give essentially the same value at 0.025 &I. At still higher concentrations the conductance method gives higher values for a than the membrane method (0.266 for the conductance method vs. 0.229 for membrane method at 0.1 M ) . The exact method of extrapolation used with the conductance data was not specified. It appears, however, that Q, the equilibrium quotient in terms of concentrations, was calculated, and a plot of log Q os. some function of the concentration was extrapolated to zero concentration by drawing a line through the last few points. This procedure places greatest weight on the points that are most sensitive to slight errors in a; for example, at 0.00025 M where a was found to be 0.959 a 1% error in a would cause a 24% error in either Q or K . A recalculation of K from the values of a obtained from conductivity data using eq 19 gave values that varied between 0.0107 at 0.001 M and 0.0143 at 0.1 M ; all values of K obtained at concentrations above 0.01 M were in excess of 0.0120.
3927
It therefore appears that the results from the conductance method agree reasonably well with those from the membrane method. Although the values of a obtained by the membrane method are consistent with the activity data for sulfuric acid, the lower values of K obtained by other methods need explanation. This question can be answered by assigning any desired value of K in eq 19 and calculating values of a consistent with it and the stoichiometric activity coefficients of sulfuric acid. These values of a may then be substituted into eq 16 - , product of activity coto determine Y H + ~ Y B O ~ ~the efficients of the free ions. This product can then be substituted into the Debye-HiickeI equation, and the ion-size parameter, d, determined. This procedure was carried out for K = 0.0110 and resulted in values of 11 that varied between 11 and 16 A (values of it for completely dissociated acids fell between 5.6 and 8.2 A when calculated with eq 8 at 1.1 = 0.1 from data in ref 4, p 476). Since these values are unreasonably high, it appears that the lower values of K are not consistent with activity data. The purpose of the present paper was to demonstrate the utility of the Donnan membrane equilibrium method in determining dissociation constants of acids and not to challenge previously determined values of the dissociation constant of bisulfate ion. The consistency of the values of a obtained by the membrane method with the activity data for sulfuric acid over a large range of concentrations, together with fair agreement with conductivity measurements at the higher concentrations, indicates that the equilibrium constants found in the present study may be more reliable than those reported previously.
(13) W. J. Hamer, “The Structure of Electrolytic Solutions,” John Wiley and Sons, Inc., New York, N. Y . , 1959, Chapter 15.
Volume 70, Number 18 December 1966
