2956
C. W. CHILDS
Equilibria in Dilute Aqueous Solutions of Orthophosphates by C. W. Childsl Department of Medical Chemistry, John Curtin School of Medical Research, Australian National University, Canberra, Australia (Received February 17, 1969)
Aqueous solutions containing orthophosphoricacid (0.002 to 0.012 mol l.-') and potassium nitrate (0.15 mol l.-l) have been titrated potentiometrically with potassium hydroxide (0.9 mol l.-l) at 37". The data are consistent H ~ ( P O ~ ) Zand ~ - ,H3(P04)2-, with the presence in solution of the hydrogen-bonded dimeric species H~(PO~)Z', as well as the usual monomeric species. The effects of possible errors arising in the calculations of hydrogen ion concentrations from pH values have been considered. The values obtained for the practical formation constants of two of the dimers were K(H2P04- HzP04= 5.6 2.0 1. mol-' and K(HP042- HzP04-
*
+
Ha(P04)2*-) = 2.7 i 2.5 1. mol-l.
Introduction The properties of dilute solutions of orthophosphoric acid and the alkali metal mono- and dihydrogen orthophosphates have usually been interpreted only in terms of the successive dissociations Hap04 HP04'-
+ HzP04H + + HP04'H + + P04'-
H+
(1)
(2)
(3)
However, Selvaratnam and Spiro have shown2 that the concentration variation of the transference numbers of aqueous solutions of orthophosphoric acid at 25", for concentrations less than 0.1 mol l.-l, can be explained by assuming equilibrium 1 together with the presence of the hydrogen-bonded species Hh(P04)2-. They obtained a value of 3 1. mol-' for the thermodynamic formation constant for HzP04-
+ H3P04
Hh(P04)z-
(4)
and showed that such complex formation is consistent with literature values of the conductances of dilute and concentrated orthophosphoric acid solutions. Other authors3 have also suggested that H6(P04)2exists in aqueous solutions. Further, Hb(P04)2- could act as a base by adding a proton to form He(P04)2, a species which is consistent with the conductances of concentrated solutions of orthophosphoric acida213 The present paper reports measurements at 37" over the pH range 2.2-10.1, with the cell (cz), KNOa glass electrode 1 H3PO4(c~),KOH (0.15 mol L-l) 1 KCl (satd) calomel electrode
I
where c1 (0.002-0.012 mol 1.-1) and cz are small compared with 0.15 mol 1.-'. The data are consistent with the presence of the species Hs(P04)2-, H4(P04)z2-, and H3(P04)23- in the solutions and have enabled approximate values of the formation constants for HzPO4-
+ &Pod-
The Journal of Physical Chemistry
H4(Po4)22-
(5)
+
and HzP04-
+ HP0d2-
H3(P04)z3-
(6)
to be obtained. The value of the formation constant for equilibrium 4 is particularly sensitive to the assumptions involved in the analysis. Values for the three practical dissociation constants of orthophosphoric acid have been obtained.
Experimental Section A . Materials and Solutions. Boiled-out glass-distilled water was used in the preparation of all solutions. B.D.H. AnalaR orthophosphoric acid was used without further purification to prepare a stock solution which was standardized (with a precision of 0.15%) by the method described by B r ~ n i s h o l z . ~Stock potassium nitrate and potassium hydroxide solutions were prepared as previously de~cribed.~AnalaR potassium hydrogen phthalate and sodium tetraborate were used without further purification for the preparation of 0.05 mol 1.-' standard buffer solutions (pH values 4.029 and 9.082, respectively,6 at 37"). All concentrations are expressed in moles per liter at 37". B. Potentiometric Measurements. All measurements were made at 37 0.05". The experimental apparatus and procedure have been de~cribed.~The Agla micrometer syringes and all volumetric glassware were calibrated with water. The estimated precision of all of the pH measurements is *0.005 pH unit. I n all cases the standard buffer readings after a titration agreed within 0.005 pH unit with the initial settings.
*
(1) Address from Nov 1969: Canada Centre for Inland Waters, Department of Energy, Mines and Resources, P.O. Box 5050,Burlington, Ontario, Canada. (2) M. Selvaratnam and M. Spiro, Trans. Faraday SOC.,61, 360 (1965). (3) K. L. Elmore, J. D. Hatfield, R. L. Dunn, and A. D. Jones, J . Phys. Chem., 69,3520 (1965). (4) G. Brunisholz, Helv. Chim. Acta, 30,2028 (1947). (5) C. W.Childs and D. D. Perrin, J . Chem. Soc., A , 1039 (1969). (6) D. J. Alner, J. J. Greczek, and A. G. Smeeth, ibid., 1206 (1967).
EQUILIBRIA IN ORTHOPHOSPHATE SOLUTIONS
2957 C . Data. The pH titration data are shown in Table
Table I : Data from p H Titrations a t 37"
I.
(Initial Concentrations in mol 1.-1; Titers in ml.) Titer
PH
Titer
PH
Titer
PH
potassium nitrate, 0.15; (1) Orthophosphoric acid, 2.389 X initial volume, 50.21 ml; titrant potassium hydroxide, 0.9195 7.040 0.060 3.035 0,139 5.606 0.219 5.954 0.229 7.200 0.149 0.070 3.103 7.397 0.159 6.174 0.239 0.079 3.183 0,169 6.348 0.249 7.674 0.089 3.280 0,259 8.235 0,099 3,402 0,179 6.497 9.445 0.189 6.634 0.269 0,109 3.574 0.279 9.864 0.199 6.764 0.119 3.858 0.289 10.079 0,209 6.898 0.129 4.645 (2) Orthophosphoric acid, 3.588 X potassium nitrate, 0.15; initial volume, 50.21 ml; titrant potassium hydroxide, 0.9195 0.339 7.139 0.119 3.034 0,229 6.029 7.256 0,239 6.170 0.349 0.129 3.096 7.391 6.289 0,358 3,168 0.249 0.139 7.562 0.259 6.395 0.368 0,149 3.264 0.378 7.802 0.159 3.361 0.269 6.493 8.242 0.279 6.584 0.388 0,169 3.501 9.279 6.674 0.398 3.707 0.289 0,179 9.772 0,299 6.761 0.408 0.189 4.093 9.906 0.309 6.849 0.413 0.199 5.061 0.418 10.010 0,319 6.940 0.209 5.591 0.423 10,092 0.329 7.036 0.219 5.852 (3) Orthophosphoric acid, 5.980 X potassium nitrate, 0.15 initial volume, 50.21 ml; titrant potassium hydroxide, 0.9195 0.239 3.030 0.398 6.162 0.558 7.092 0.249 3.084 0,408 6.236 0.568 7.157 0.259 3.145 0.418 6.304 0,578 7.227 3 216 0.428 6.367 0.588 7.303 0.269 0.279 3.297 0.438 6.428 0.598 7.387 0.608 7.484 0.289 3.397 0.448 6.485 0.299 3.527 0.458 6.542 0.618 7.598 0.309 3.710 0.468 6 596 0.627 7.742 0.319 4.020 0.478 6.650 0.637 7.938 0.329 4.704 0.488 6.703 0.647 8.253 0.339 5.282 0.498 6.755 0.657 8.951 0.349 5 561 0.508 6.809 0.667 9.587 0.358 5.742 0.518 6.862 0.677 9.874 0.368 5.876 0.528 6.918 0.687 10.049 0.378 5.986 0.538 6.973 0.690 10.083 0.388 6.079 0.548 7.031 (4) Orthophosphoric acid, 11.96 X 10-3; potassium nitrate, 0.15; initial volume, 50.21 ml; titrant potassium hydroxide, 0.9225 0.000 2.207 0.360 2.610 0.720 5.815 0.020 2.223 0.380 2.644 0.740 5.937 0.040 2.239 0.400 2.680 0.760 6.037 0.060 2.256 0.420 2.721 0.780 6.125 0.080 2.273 0.440 2.764 0.800 6.203 0.100 2.291 0.460 2.812 0.820 6.274 0.120 2.312 0.480 2.864 0.840 6.340 0.140 2.330 0.500 2.924 0.860 6.402 0.880 6.462 0.160 2.350 0.520 2.990 0.180 2.371 0.540 3.068 0,900 6.520 0.200 2.393 0.560 3.160 0.920 6.575 0.220 2.416 0.580 3.276 0.940 6,630 0.240 2.440 0.600 3.434 0.960 6.683 0.260 2.464 0.620 3.672 0.980 6.737 0.280 2.490 0.640 4.170 1.000 6.790 0.300 2.618 0.660 5.056 1.020 6.844 0.320 2.546 0.680 5.443 1.040 6.899 0.340 2.577 0.700 5.661
Results Estimated values of the equilibrium constants were refined using the computer program SCOGS' as previously described.6 All of the values for the equilibrium constants were calculated as practical quotients and the concentrations (deinvolving (Hf) = noted by brackets) of all other species. As examples
(7) for equilibrium 2 and
K (Hs(pod23
-> = [Ha(P04)z3-]/[HzP04-] [HP04'-]
(8)
for equilibrium 6. Possible errors in the results, due to variations in ionic strength during titrations, are considered below. Potassium ions and nitrate ions were assumed not to form complexes with any other solute species. Data from all of the titrations were treated simultaneously. For the purpose of mass balance equations, [H+] was estimated from the p H measurements by6
- log F
(9)
- A E j + ApH
(10)
p[H] = pH where log F = log f
f is the activity coefficient of hydrogen ions, A E j is the residual liquid junction potential, and ApH is the sum of any systematic errors in the pH measurements. The value of F is assumed to be constant for all of the measurements. As log f is expected to be the major term on the right-hand side of eq 10, for a first estimate F was taken to be equal to f = 0.75 as calculated from for systems the equation given by Bates.8 Previou~ly,~ similar to that reported here, the value of F was adjusted to 0.77 f 0.05 on the basis of minima in a residual least-squares parameter as F was varied. However, the minimum for the present system (see Table 11) lies about F = 0.82 when equilibria 1 to 6 are all included in the analysis. If the data are analyzed in terms of equilibria 1, 2, and 3 only, the minimum is slightly displaced to about F = 0.85. Here calculations have been carried out for F in the reasonably wide range 0.75 to 0.90, at intervals of 0.05, to assess the effects on the results of the uncertainty in F. An error of 10.075 in F = 0.825 corresponds to about *0.04 in p [HI. Table I1 lists the refined values of the equilibrium constants for F = 0.75, 0.80, 0.85, and 0.90. Limits shown are the sums of the standard deviations of the (7) I. G. Sayoe, Talanta, 15, 1397 (1968). R.G.Bates, J. Res. Nat. Bur. Stand., A , 66, 179 (1962).
(8)
Volume 75, Number 9
September 1969
C. W. CHILDS
2958 over-all stability constants (as refined by the computer program) used in obtaining -the values. The standard deviation in titer' (SDT) is a measure of the abilitv of the refined values to reproduce the experimental data. For F = 0.90 the inclusion of equilibrium 4 did not improve the fit to the data, the estimated value of K(Hs(P04)z-) was decreased with successive cycles of refinement by the computer, and the calculated concentrations of H5(P04)2-became negligibly small. For all other values of F the inclusion of equilibria 4,5, and 6 in the analysis improved the least-squares fit to the data. For example, for F = 0.80, the value of 103SDT was 2.42 ml when equilibria 1, 2, and 3 only were considered, but was 1.14 ml when equilibria 4,5, and 6 were considered as well. The inclusion of equilibria involving H6(PO& and Hz(P04)24-did not improve the fit to the data, and the values for their equilibrium constants failed to converge for any of the values of F tried. For Ha(P04)2this was consistent with Selvaratnam and Spiro's estimate2 of 0.02 1. mol-' for the equilibrium constant of
K(H~(~O~)Z
