1271
DETERMINATION OF STABILITY CONSTANTS BY DONNAN MEMBRANE EQUILIBRIUM
Determination of Stability Constants by Donnan Membrane Equilibrium: the Uranyl Sulfate Complexes'
by Richard M. Wallace Savannah River Laboratory, E. I . du Pont de Nemoura and Company, Aikcn, South Carolina (Received August 8,1966)
,80801
~~
A new method for the determination of stability constants of inorganic complexes was developed. The distribution of the cation of interest between complexing and noncomplexing solutions that are separated by a permselective membrane is measured simultaneously with that for an unreactive univalent cation. Stability constants are calculated from this distribution and from the conditions necessary for Donnan equilibrium across permselective membranes. This method was applied to the reactions between uranyl and sulfate ions. The equilibrium constants for the reactions U0z2+
+
504'-
u02'+ f 2s04'-
uo&o4
(1)
U02(S04)22-
(11)
were determined at 25, 35, and 50'. The measurements were made at ionic strengths between 0.15 and 0.01 M , and the results were extrapolated to zero ionic strength. The stability constants (in terms of activity) were KI = 1390, 1830, and 2700 at 25, 35, and 50', respectively, for reaction I and KII = 1.61 X lo4, 2.31 X lo4, and 3.97 X lo4 a t the same respective temperatures for reaction 11. AH1 and AH11 were calculated to be 5.0 and 6.98 kcal/mole, respectively.
Donnan membrane equilibrium across permselective membranes has been used to measure charges of ions in solution2 and dissociation constants of acids.a This paper describes the use of this method to determine the stability constants of inorganic complexes in aqueous solutions. The reaction of uranyl ion with sulfate ion was chosen as a model system.
where ME+is the concentration of a cation of charge 2, and N + is the concentration of a univalent cation. The subscripts s and p refer to the particular solutions on opposite sides of the membrane, and p is a quotient of activity coefficientsdefined as
Basis of Method
The value of ,f3 will normally depend on the composition of the two solutions. If, however, the ionic strengths of both are the same, P may be assumed to be unity. This condition certainly applies with the same electrolyte in both solutions. However, p might differ slightly from unity if different electrolytes are placed on opposite sides of the membrane, even if the
If two solutions of electrolytes are separated by a membrane ideally permeable to cations but impermeable to anions and solvent, the anion concentrations on both sides of the membrane must remain constant. The cations, however, will be redistributed between the two solutions until Donnan equilibrium is established. The condition for equilibrium in such a system is
(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) R. M. Wallace, ibid., 70, 3922 (1966).
Volume 71, Nu-
6
ApnZ 1967
1272
RICHARD M. WALLACE
ionic strengths of the solutions are the same. This change from unity could result from differences in the ion-size parameters a of the various ions in the DebyeHuckel equation
1% Yr
=
-ZI2Adi 1 4- a B d i
(3)
A more complete equation for activity coefficients contains terms linear and quadratic in 1; however, for the moderately dilute solutions employed in the present study, eq 3 is assumed to give an adequate descrip tion. In the absence of detailed information concerning activity coefficients, @ can reasonably be assumed to be unity. With this assumption, equilibrium constants can be calculated from measurements of the distribution of ions between the two solutions by more or less conventional methods for determining equilibrium constants from equilibrium data, as discussed below. Consider the case in which a solution containing an anion Y'- that complexes the metal ion MZ+(but not the univalent cation N+) is placed on side s of the membrane, while the solution on side p contains anions that complex neither cation. Let the anion Y form a series of complexes with the metal ion represented by
hlz+ + y z -
MY[(Z+)-(Z-)l
Mz+ + 2yz-
MY2[(Z+)-(2z-)1
(4)
The total concentration of M,, the metal in the solution on side s, will be the sum of the concentrations of the various species
+ (MY). + ( M Y & + .. .
M E = (Mz+)s
(5)
If eq 5 is combined with the expressions for the equilibrium quotients for the reactions, one obtains
M, = (M"),(l
+ QIY + QpY2 + .-..)
(6)
where Q1, Q 2 ,etc. are the equilibrium quotients in terms of concentrations, which are assumed to be constant at constant ionic strength, and Y is the concentration of the complexing anion. The concentration of free Mz+ in solution s can be determined from its concentration in solution p and the ratio of the concentrations of the univalent cation N between the two solutions, since eq 1 applies to the free, uncomplexed cations in solutions only. When eq 1 and 6 are combined, one obtains
R -1 -Y
- &I
+ QaY + .. . + QnY"-l
where R is defined as Ths Journal of Physical Chemistry
(7)
(8)
All of the quantities on the right side of (8) are total concentrations and therefore measurable. The equilibrium quotients Q ican then be found by measuring R as a function of Y at constant ionic strength. The stability constants Kt in terms of activities can be calculated by measuring the Qi values at various ionic strengths and extrapolating to infinite dilution. Since R is determined by ratios of concentrations, only a relative measure of the concentrations is needed. This method is therefore well suited for radioactive tracers, whose concentrations can be kept so small that they do not affect the gross composition of the solutions. Measurements can thus be made in dilute solutions without the need for complicated corrections. In the preceding discussion, the membranes were considered to be ideally selective. The properties of real membranes, however, impose certain limitations on the method. These membranes are made of ion-exchange resine and acquire permselectivity by Donnan excl~sion.~They therefore lose their selectivity in contact with highly concentrated solutions. On the other hand, in very dilute solutions, highly charged ions are absorbed by the membrane so strongly that the remaining solutions are too dilute to analyze, unless the membrane is presaturated with the cation. Another limitation is the incompatibility between the requirement that the two solutions have the same ionic strength and the necessity for isoosmotic conditions on both sides of the membrane. The change in concentration caused by osmosis changes the ionic strength; therefore, equilibration times must be short enough to prevent large changes in solution concentrations, yet long enough to reach equilibrium with respect to the cations. This change in concentration from osmosis also requires that either the solutions be analyzed for all pertinent components after equilibration or some correction be made. In spite of certain limitations, the membrane method has several potential advantages over other methods for determining stability constants. The membrane method can be applied to any cation that will pass through the membrane and thus can be applied to ions that cannot be studied by spectrophotometric or potentiometric methods because of the lack of an absorbing species or a reversible electrode. The conventional ion-exchange equilibrium method can be applied to all ions to which the membrane method is applicable; however, the latter has two advantages. First, the (4) F. Helfferich, "Ion Exchange," McGraw-Hill Book Co., Inc., New York, N . Y., 1962, Chapter 8.
DETERMINATION OF STABILITY CONSTANTS BY DONNAN MEMBRANE EQUILIBRIUM
ion-exchange method requires measurement of the concentration of the cation in both the resin phase and the solution phase, while the membrane method requires the measurement of its concentration in two solution phases; analysis of solutions is generally more convenient than analysis of resins. Second, calculation of stability constants from equilibrium data with ionexchange resins requires (ref 4, p 212) the evaluation (or elimination) of the selectivity coefficients of each of the pertinent species involved in complex formation and the assumption that they remain constant during a series of measurements. In contrast,, the membrane method requires no assumptions concerning the behavior of the species in the ion-exchange membrane so long as the property of permselectivity is maintained. A 22Natracer can be used conveniently as the reference univalent cation even in systems involving salts of other univalent cations. Its distribution between the two solutions after equilibration can be measured easily and used directly for calculating R. In addition, the concentrations after equilibration can be calculated from the easily measured distribution of 22Naand the initial composition of the solutions. Thus, if the two solutions contain salts of a single univalent cation N whose initial concentrations are and N,,o in solutions s and p, respectively, if the same volumes of solutions s and p are used, and if changes in concentration are caused only by the osmotic transport of water, the concentration N, after equilibration is given by the equation (9)
where (22Na,)/(22Nap) is the ratio of sodium concentrations measured after equilibration. A similar equation can also be derived for N,. The concentrations of the anions in the solutions can then be calculated from their initial concentrations and N, and N,.
Uranyl Sulfate Complexes The stability constants of the uranyl sulfate complexes were calculated from measurements of the distribution of 233U022+ and 22Na+between solutions of ammonium perchlorate on side p and mixtures of ammonium perchlorate and ammonium sulfate on side s. It was assumed that the perchlorate ion complexes neither uranyl nor sodium ion, and that sulfate complexes only the uranyl ion. The reactions involved are
+
U0z2+ 504’uozso4 u022+ 2s0d2- J_ u 0 ~ ( S 0 4 ) 2 ~ -
+
(1) (11)
1273
The existence of a trisulfate complex is doubtful6 and will not be considered in this study. When applied to the uranyl sulfate system, eq 7 becomes
where R is now defined as
In determining &I and &2 in solutions of various ionic strength, a series of measurements should be made with varying sulfate concentration at each ionic strength. In very dilute solutions, the effect of the linear term in eq 10 is too small to be accurately determined but too large to be ignored completely. Allen6 has pointed out that &2/&1 should be independent of the ionic strength in the uranyl sulfate system if the Debye-Huckel equation is obeyed. If this condition prevails, &I and Q2 need only be determined in some reasonably concentrated solution in which Q2 can be measured accurately. In less concentrated solutions, can be calculated from the equation R-1 = (S042-)( [1 [(&2/&1)(504~-)1)
+
(12)
The equilibrium constants K1 in terms of activities can be calculated from the values of &I at various ionic strengths if the activity coefficients of all ions involved are assumed to obey the Debye-Huckel equation with a single ion-size parameter.
The constants A and B in (13) are tabulated by Robinson and Stokes’ at various temperatures. Since 0 is determined experimentally as a function of the ionic strength, p, it is necessary to find a value of d that will make log K independent of 1.1. Different values of 12 are chosen and a series of tentative values of log K are calculated for each d. These tentative values were than assumed to be linear functions of p ; i.e., log K = a bp. A least-squares determination of the constants a and b is then made, and the correct value of d is assumed to be the one that causes b to vanish. In our study, K1 was defined as the average of the values calculated with that particular value
+
(5) P.E. Stein, U. S. Atomic Energy Report Y-1398 (1962). (6) K.A. Allen, J. Am. Chem. Soc., 80,4133 (1958). (7) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions,” Butterworth and Co.Ltd., London, 1955, p 491.
Volume 71,Number 6 April 1967
RICHARD M.WALLACE
1274
of d. After K1 has been established, K2 can be determined from K I and &?/&I.
Experimental Section Membrane equilibrations were run in an apparatus described p r e v i o ~ s l y . ~The * ~ membranes used in the present work were AMFion C-103 membranes manufactured by the American Machine and Foundry Co. The 233U022+ and 22Na+tracers were absorbed from water onto the membranes mounted in the equilibration cells. After absorption of the tracers, the water was removed and the cells were dried. The two solutions, between which the distribution of tracers was to be measured, were placed in compartments on opposite sides of the membrane. The cells were then agitated in a constant-temperature bath for 5, 3, and 2 hr a t 25.0 f 0.1, 35.0 f 0.1, and 50.0 f O J " , respectively. (Preliminary kinetic studies had indicated that equilibrium with respect to cation interchange was established within these periods.) Less than 0.3% of the sulfate was transferred across the membranes during measurements with the most concentrated solutions studied (0.05 M ammonium sulfate in s and 0.15 M ammonium perchlorate in p). Studies of sulfate transport were not made with more dilute solutions, but past experience indicated that anion transport should be less in these cases. Solutions of ammonium perchlorate, ammonium sulfate, and their mixtures were prepared by accurate dilution of analyzed stock solutions. The final solutions were acidified to pH -5. The stock solutions, which were prepared by the neutralization of reagent grade acids with reagent grade ammonium hydroxide, were analyzed by passing them through cation-exchange columns in the hydrogen form and titrating the effluent acid. The relative concentrations of sodium ion were determined by counting the y activity of 2zNain a welltype y-scintillation counter in conjunction with an RIDL solid-state scaler and timer. The precision was about 0.3%. Samples were mounted for counting 2saU022+a activity by one of three methods, depending on the nature of the solutions. Acid solutions that contained no salts were evaporated to dryness on platinum plates. Dilute solutions of the ammonium salts were evaporated to dryness on stainless steel plates, and the ammonium salts were decomposed by heating the plates carefully on a hot plate. This decomposition however gave very erratic results for concentrated solutions. For more concentrated solutions, the 2aaU022+ was coprecipitated with a small amount of natural uranium (0.7 mg) as ammonium diuranate, The Journal of Physical Chemhtry
which was filtered as a uniform layer on a submicron membrane filter. The filter was mounted on a flat plate and the a activity was determined. Corrections were applied for the activity of the natural uranium. Dstails of this procedure will be described in a subsequent publication. All samples were counted in an a-scintillation counter in conjunction with an RIDL scaler and timer. The over-all precision was about f 1%.
Results and Discussion Table I shows the distribution of U02*+and Na+ between solutions of constant ammonium sulfate concentrations in side s, but varying ammonium perchlorate concentrations in side p a t 25". This test determined the effect of the difference in ionic strengths across the membrane on the distribution of the ions. The value of R was surprisingly constant in both sets of data. A variation in R of about 10% in the first set of measurements and 5% in the second set would be expected if d in the Debye-Hiickel equation is the same for all species in both solutions. Apparently, variations in d were just sufficient to compensate for the other effects of differences in ionic strength and thereby rendered p approximately constant. The results show that small variations in the ionic strength of the two solutions caused by osmosis will have little effect on R. Since osmosis changed the sulfate concentrations slightly, corrections were made as described previously.
Table I: Distribution of U0z4+and Na+ between Solutions of Constant Sulfate and Varying Perchlorate Concentrations at 25" Side p
NH4CIO4, M
0.20 0.15 0.10 0.10 0.075 0.050
Side B
("4)804, M
Nap+/Nss+
UOd UOz,pl+
R
0.0518 0.0505 0.0500 0.0257 0.0250 0.0250
1.866 1.476 0.985 1.895 1.502 0.998
4.37 7.18 16.04 2.78 4.39 9.94
15.22 15.64 15.56 9.98 9.90 9.90
Table I1 contains the results of a series of distribution measurements at 25, 35, and 50" in which the ionic strength was held constant at 0.15 M in both solutions while the sulfate concentration in solution s was varied. Values of Q1 and Q2 and their standard deviations were obtained by a least-squares fit of eq 10 to the data. Data from Table I for 0.05 M ammonium sulfate were also included in the calculation.
DETERMINATION OF STABILITY CONSTANTS BY DONNAN MEMBRANE EQUILIBRIUM
Table 11: Distribution of UOz*+and Na*+between Solutions of Ammonium Perchlorate and Mixtures of Ammonium Sulfate and Ammonium Perchlorate at Constant Ionic Strength ( p = 0.15 M ) Sor*-, M
&I
Nap+/Na.~+
0.0513 0 . a402 0.0301 0.0200 0.00980 0.00787 = 180 f 5.4
Q2
UO%I/UO&p'+
25 " 1.409 1.330 1.225 1.130 1.067 1.048 = 2090 f 18.0
7.59 6.82 5.41 4.35 2.62 2.30 Qz/Qi
=:
(R - I)/ (Sod*-)
274 275 237 228 203 194 11.6 f 1.1
35" 0,0504 1.476 8.62 0.0400 1.363 7.61 1,267 6.36 0.0300 0.0200 1.169 4.74 0.0100 1.095 2.87 Qz = 218.8 f 3.8 Qz = 2767 f 110 &*/&I
i=
356 329 307 274 244 12.6 f 0.48
50"
QI
0.0502 1.486 12.63 0.0400 1.367 11.34 0.0300 1.269 9.35 0.0200 1.176 6.67 0.0100 1.092 3.78 = 314.7 f 13 Qz = 4639 f 410 Q2/Q1
548 500 454 407 361 = 13.9 f 1.4
Table I11 contains the results of measurements made at 25" and lower ionic strengths. Values of &I were calculated with eq 12 using the value of Q2/&1 determined at ionic strength 0.15 M . The constancy of Q1a t the higher ionic strengths indicates the validity of the assumption that Q 2 / Q 1 is independent of ionic strength.
1275
Table IV contains a summary of &I determined at 25" and various ionic strengths together with calculated values of KI from eq 13. The value for d for which K1 was independent of the ionic strength was 6.0 A. The errors shown are standard deviations. The results of less detailed studies at 35 and 50" are also shown. Table IV : Equilibrium Constants in Ammonium Sulfate-Ammonium Perchlorate Mixtures P
91
Ki
0.15 0.10 0.075 0.050 0.025 0.010
25" 180 f 5.4 219 f 5.7 274 f 4.0 324 f 13 451 f 24 630
1390 f 28 1350 f 35 1440 f 21 1380 f 56 1390 f 74 1370
0.15 0.10 0.05
35" 219 f 3.8 297 420
1750 1890 1850
0.15
50" 315 f 13
2700 f 110
Table V contains a summary of average values of K1 and KZat the temperatures shown. The constants at the higher temperatures were also calculated with 11 equal to 6.0 A. K 2 in each case was calculated from KI and values of Q 2 / Q 1were determined a t p = 0.15 M . Values of AH1 and AH2 were calculated from a leastsquares fit of log K vs. l/T°K. Table V: Equilibrium Constanb at Various Temperatures
Table 111: Distribution of UOz2+and Na+ between Solutions of Ammonium Perchlorate and Mixtures of Ammonium Sulfate and Ammonium Perchlorate at Various Ionic Strengths and 25"
T,OC
K1
K2
25.0 35.0 50.0
1390 f 46 1830 f 70 2700 f 110
16,100f1600 23,100f1200 39,700 f 3500
-1
=
5.10 f 0.43 kcal/mole
= 6.98 f 0.91 kcal/mole
0.100 0.100 0.100 0.050 0.050 0.050 0.025 0.025 0.025 0.010
0.0340 0.0203 0.0101 0.0169 0.0121 0.00705 0.00838 0.00653 0.00503 0.00333
1.443 1.214 1.095 1.463 1.289 1.149 1.479 1.338 1.233 1.501
5.51 4.50 2.83 3.47 3.42 2.58 2.23 2.35 2.32 1.41
221 224 213 318 339 315 422 457 474 630
Equilibrium constants in acid media were determined at 25 and 35' by measuring the distribution of uranyl and sodium ions between solutions of sulfuric acid and perchloric acid. The results are shown in Table VI. The perchloric acid concentration was calculated to give approximately the same ionic strength as the sulfuric acid based on previously measured Volume 71, Number 6 April 1967
RICHARD M. WALLACE
1276
degrees of dissociation of the bisulfate ion.2 The concentration of free sulfate ion and the ionic strength of each sulfuric acid solution were calculated from the concentration of sulfuric acid, and the degree of dissociation, a, of the bisulfate ion determined from the distribution of sodium ion is described in ref 2. The equilibrium constants were not determined under conditions for which the ionic strength was held constant while the sulfate concentration was varied; hence no independent determination of Q2/Q1 was obtained. Therefore, in the calculation of Q1, the value of Q2/Q1 was assumed to be the same as that obtained at the corresponding temperature with solutions of the ammonium salts. The values of K1 shown in Table VI were calculated with d = 7.OA. Table VI : Equilibrium Constants Determined in Sulfuric Acid Solutions Side p
Side B
HCIOI, M
HBOI, M
0.1464 0.0767 0.0405 0.0221 0.0118
0.0967 0.0469 0.0207 0.0102 0.00502
1.218 1.253 1.330 1.357 1.381
0.1505 0.0814 0.0399 0.0221 0.0120
0,1012 0.0505 0.0211 0.0102 0.00503
1.253 1.300 1.404 1.420 1.456
uo2,./ a
&I
K1
25 ' 4.66 4.00 2.90 2.28 1.86
0.230 0.306 0.471 0.594 0.698
209 316 435 494 696
1390 1540 1580 1400 1590
35O 4.70 4.10 2.85 2.30 1.86
0.186 0.240 0.348 0.523 0.634
274 426 577 637 867
1850 2100 2030 1800 1970
Nap/NaB UOllPl+
Calculation of K1 from the acid data with d equal to 6.0 A gave 1590 f 90 and 2090 f 160 at 25 and 35", respectively, compared with 1390 46 and 1830 f 70 obtained with ammonium salt solutions. Although the agreement is fair, differences in the average values are larger than the experimental errors. Extrapolation of the acid data with d equal to 7.0 A gave 1500 f 98 and 1950 f 120 for K1 at 25 and 35", respectively. Since d is not necessarily the same in both media, the two sets of results are considered to be in substantial agreement. The absence of large differences in KI indicates no appreciable hydrolysis of uranyl ion at tracer level concentrations in solutions of ammonium salts at pH 5. This conclusion agrees with the observations of Rush, Johnson, and Kraus,* who found that the hydrolytic species of uranyl ion were largely polymeric. The stability constants determined by the mem-
*
The Journal of Physical Chemistry
brane equilibrium method are self-consistent, and the derived value of 6.0 A for d is reasonable. Comparison with measurements by other methods is difficult because of the wide variety of media and the procedures employed for extrapolation and because of the lack of agreement between the various measurements. Three earlier studies were performed at 25" in solutions sufficiently dilute to allow reliable extrapolation to infinite dilution. Davies and Monk* determined K1 = 910 by a spectrophotometric method; Kraus and Nelsonlo determined K1 = 530 and K2 = 16,000 from anion-exchange studies; and Brown," et al., determined K1 = 1700 from conductivity measurements. From other studies carried out at higher ionic strengths, Ahrland12 obtained Q1 = 50 and Q2 = 350 from potentiometric measurements, and Q1 = 56 and Q2 = 450 from spectrophotometric studies at 20" with p = 1, while Matsuola obtained Q1 = 65 and Q2 = 200 from spectrophotometric studies at 25" with p = 1. Allene extrapolated Ahrland's data to infinite dilution and determined K1 = 960 and K 2 = 7700 from the spectrophotometric data, but calculated K1 = 580 and K2 = 3450 from his own solvent extraction data at 25". Allen used d = 7 A in the Debye-Huckel formula for his extrapolation. Ahrland's value for Q1 at 20"' when corrected to 25" using the value of AH1 found in this study, agrees with Matsuo's value at 25". Extrapolation of the corrected Q1 at 25" to infinite dilution with d = 6 A results in K1 = 1460which agrees with the value found in this study. Similar treatment of Ahrland's value for Q2 at p = 1 yields K2 = 12,500, which is in fair agreement with the present results. In this study the values (average between 25 and 50") of AH, and AH2 were calculated to be 5.10 and 6.98 kcal/mole, respectively. These values agree fairly well with the values of Lietzke and St~ughton,'~ who obtained AH1to be 2.35-5.19 kcal/mole a t 25" and 9.209.45 kcal/mole at 50" from measurements of the solubility of silver sulfate in uranyl sulfate solutions. Their values of AH2were 3.8-6.14 kcal/mole at 25" and 10.3110.58 kcal/mole at 50". (8) R. M. Rush, J. S. Johnson, and K. A. Kraus, Inorg. Chem., 1 378 (1962). (9) E. W. Davies and C. B. Monk, Trans. Faraday Soc., 5 3 , 442 (1957). (10) K. A. Kraus and F. Nelson, "The Structure of Electrolytic Solutions," W. J. Hamer, Ed., John Wiley and Sons, Inc., New York, N. Y.,1959,Chapter 23. (11) R. D.Brown, W. B. Bunger, W. L. Marshall, and C. H. Secoy, J. Am. Chem. Soc., 76, 1532 (1954). (12) S. Ahrland, Acta Chem. Scand., 6 , 1151 (1951). (13) 8. Matsuo, J. Chem. SOC.Japan, 81, 833 (1960). (14) M.H.Lietzke and R. W. Stoughton, J . Phys. Chem., 64, 816 (1960).
