K. L. ELMORE, J. D. HATFIELD, R. L. DUNN,AND A. D. JONES
3520
Dissociation of Phosphoric Acid Solutions at 25"l
by K. L. Elmore, J. D. Hatfield, R. L. Dunn, and A. D. Jones Division of Chemical Development, Tenwsses V d e y Authority, Wilson Dam, Alabama
(Received May 3,1966)
Degrees of dissociation of phosphoric acid solutions, calculated from the vapor pressure, conductance, and pH of the solutions, pass through a minimum at 1 m acid as the concentration is increased. This phenomenon is attributed to the presence in the acid of a species more acidic than the monomer Hapod; the more acid species is assumed to be the dimer H6PzOswhich ionizes to yield HsPzOs-. On the basis of this model, an expression is derived that expresses the degree of dissociation of phosphoric acids in the concentration range 0.1 to 10 m as a function of the concentration and the pH.
Knowledge of the thermodynamics and kinetics of phosphate syst,emsis essential to an understanding of the reactions involved in the manufacture of phosphatic fertilizers, the equilibration of fertilizers with soil systems, and the utilization of soil phosphates by growing plants. One application of such knowledge would be in studies of the kinetics of the reaction of phosphoric acid with rock phosphate in the preparation of phosphate fertilizers. Complete understanding of this reaction requires knowledge of the degree of ionization of phosphoric acid over the concentration range from dilute to concentrated solutions. Attempts were made to determine the degree of ionization of phosphoric acid at concentrations up to 90% HaP04from vapor pressure12 conductancela and pH (measured with the saturated salt bridge)4 data on 0.1 to 10 m phosphoric acid solutions at 25'. Unpublished TVA data obtained from measurements of pH in cells without liquid junction were used also. For the cells without liquid junction Ag-AgClINaCl(ms), HaPOl(m*) lH2-Pt
erroneous in phosphoric acid solutions of high ionic strengths, but it apparently is satisfactory for solutions not greater than 1 m phosphoric acid, as shown in Figure 1. The degree of ionization CY was calculated by reiteration and smoothed by the method of least squares. The pH was calculated as pH = -log (crmay*). In 0.1 m H3P04containing 0.0002 m NaC1, CY was determined to be 0.2948, which agrees acceptably with 0.2896 as determined in this paper. The cell with the saturated salt bridge was Hg-HgCIIKCl(sat), H8PO41H2-Pt and the pH was calculated directly from the equation -1ogaH = pH = (E - 0.2442)/0.059156
(2) Since the diffusion coeEcient of phosphoric acid solutions over the concentration range 1 to 10 m does not change greatly,' the liquid junction of these cells was assumed to be constant. The cell constant, E o = 0.2442, was determined in a 0.1 m HC1 standard solu~~
the e.m.f. is given as E = 0.2224 - 0.059136 log axmi
(1)
where aEkL = ~ A y J . " y C L , ms is the concentration, m, of NaC1, m,i is the concentration, m, of H3P04, p is the ionic strength = CYVLA ms. It was assumed that the mean ion activity coefficient, Y* = in dilute phosphoric acid solutions was equal to that in hydrochloric acid solution^^^^ of the same ionic strength. This assumption undoubtedly is
+
G,,
The Journal of Physical Chemistry
~~~
(1) Presented in part before the Division of Physical Chemistry a t the 145th National Meeting of the American Chemical Society, Sept. 8-13, 1963 (Abstracts, p. 34T). New York, N. Y., (2) K. L. Elmore, C. M. Mason, and J. H. Christensen, J . Am. Chem. SOC.,68, 2528 (1946). (3) C. M. Mason and J. B. Culvern, ibid, 71, 2387 (1949). (4) A. J. Smith and E. 0. Huffman, Chem. Eng. Data Ser., 1, 99 (1956). (5) H. 5. Harmed and R. W. Ehlers, J. Am. Chem. SOC.,5 5 , 2179 (1933). (6) G. Qerlof and J. W. Teare, ibid., 59, 1855 (1937). (7) 0. W. Edwards and E. 0. Huffman, J . Phys. Chem., 63, 1831 (1959).
DISSOCIATION OF PHOSPHORIC ACIDSOLUTIONS AT 25”
~
o
-W oI T HWITHOUT SATD. L i KC! a u m BRIDGE JUNCTION
,
~
~
3521
~
~
The constants K1 and KOare defined as
l
0.2
I 2 -JCONCN., E, OF H ~ P O +
0
3
Figure 1. Dissociation of phosphoric acid.
tion which has a pH close to that of a 1 m H3P04solution. This value was also checked with other standard buffer solutions. The measured values of the conductance of H3P04 solutions3 do not agree with the theoretical model of Fuoss8 for simply associated 1-1 electrolytes in the dilute range. The anomalies can be explained, however, if it is assumed that the acid contains additional species such as the triple ion HsPzOs- and the dimeric acid HeP208. Bassettg also has concluded that the system P20s--H20contains a dimeric orthophosphate species, and Selvaratnam and SpirolO explain the properties of phosphoric acid on the basis of the existence of the triple ion HsP20s-. The method of Wooster,l’ slightly modified and made more rigorous, was used with the conductance data to evaluate the parameters of the triple ion for concentrations up to 1M by the equation
where X = 1 - A/Ao, n = Xo/Ao, C = moles per liter of total phosphorus, A = equivalent conductance, A. = limiting conductance of the ions H + and H2P04-, Xo = limiting conductance of the ions H+ and HsP2Os-, f+ = mean activity coefficient of all ions, from Debye’s limiting law, b = mobility coefficient of ions, from Onsager’s limiting law, Kl = a H + a H 2 P O , - / a H a P 0 , , and KO = aEsP?Os-/aHaPO,aHaPo,-. The mobility coefficient was tested also in the equation (4) where y is the mole ratio H2P04-/total P and 7 3 is the mole ratio H2PsOs-/total P. According to the electroneutrality equation H + = YC t- y3C or A = b(y&
H+
=
HzP04-
y3AO)
+ HsP20s-
(5)
in which the activity coefficient of the undissociated monomeric acid is assumed to be unity. In eq. 3, the equivalent conductance is expressed in terms of four parameters (Ao, KI, KO,and n or XO/AO) and three variables (b, fit,and X ) . Isovariance contours of Z(CAobsd - CAcslcd)2 are defined by the equation 0.13797110
+ 4518.4K1 - 12.915k +
87.185n = 161.326
=k
{ (r)ii8;1153 Z
(8)
where k = l/Ko and r = CAobed - CAealcd, which was derived from a study of the effects of the four parameters on the variance surface by use of the central composite design12 and by reduction of the empirical surface to canonical form.13 The limiting equivalent conductance, A. = 382.88, was determined independently from the conductance of the ions H+ and H2P04- as calculated from published conductances of HCl,14 NaC1,15 and Na2HP04,a and K1 was taken as 0.007107, as reported by Bates.le Setting = 0.01153, eq. 8 then becomes 87.185n
- 12.915k = 76.3876
(9) for which many pairs of values of n and k will give equally good fits to the conductance data. The limiting conductance of H2P04- is 33.01 mhos, and it is expectcd that that of HSP20e-is between 25 and 33 mhos, so that n is between 0.98 and 1.00. The (8)R. M . Fuoss, J. Am. Chem. SOC.,80, 3163 (1958); 81, 2659 (1959). (9) H. Bassett, J . Chem. SOC.,2949 (1958). (10) M. Selvaratnam and M. Spiro, Trans. Faraday SOC.,61, 360 (1965). (11) C. B. Wooster, J . Am. C h m . SOC.,59, 377 (1937). (12) G. E. P. Box and I(.B. Wilson, J. Rou. Statistical Soc., B13,l (1951). (13) G. E. P. Box in “Design and Analysis of Industrial Experiments,” 0. L. Davis, Ed., Hafner Publishing Co., New York, N. Y., 1954, Chapter 11. (14) R.H.Stokes, J . Phys. Chem., 65, 1242 (1961). (15) T. Shedlovsky, A. S. Brown, and D. A. MaoInnes, Trans. Electrochem. SOC.,6 6 , 165 (1934). (16) R.G.Bates, J . Res. NatE. Bur. Std., 47, 127 (1951).
Volume 69, Number 10 October 1966
K. L. ELMORE, J. D. HATFIELD, R. L. DUNN,AND A. D. JONES
3522
value of KO is then between 1.2 and 1.4; the value of 1.263 was selected because it is near the middle of this range, and, as will be shown in this paper, it is consistent with the properties of phosphoric acid. When KO = 1.263, the limiting conductance of H5P208- is 30.50 mhos. The pH data from cells without liquid junction indicated that the apparent degree of ionization, a, of phosphoric acid passes through a minimum at 1 m acid and then increases with increasing concentration (Figure 1). This observation leads to the hypothesis that an acid stronger than H904 is present in the system; the stronger acid is assumed to be the dimer HePzOs which yields the triple ion HsP208- on ionization. Evidence for the existence of the dimer H,3P20sand the triple ion H5P20s- is presented in Figures 2 and 3. In Figure 2, the units on the ordinate (d In u,,/d In uH) - 1, in which au is the activity of undissociated H3P04,were obtained by differentiation of the logarithmic form of the expression for KI; the units are numericaly the same as d In UH,FO,/d In UH, but those in the figure are more readily obtained from measurable quantities. The units on the abscissa result from the following mathematical operations that were made in attempts to determine reasonable values for a. The equation UH
=
amyH
NUMBERS ON POINTS DENOTE CONCN.. m OF H 3 P 0 4
I
1.0
0.5
0
I
m-a, 2 a,, -n
1.0
~
Figure 2. Relation between d In aanpor/d In a= and activity coefficient of H f in phosphoric acid solutions.
0.I
(10)
in which m is the concentration, molality, of total phosphorus (here considered entirely monomeric), contains two known quantities (for each concentration of acid), UH (as determined by eq. 2) and m, and two unknown but interdependent quantities, CY and YH. Equation 10 may be written (~H/YH)
Adding m terms gives
UH
- am = 0
(11)
to both sides of eq. 11 and collecting
m(1 -. a) - aH(1 -
~/YH)
=
m
- UH
(12)
which may be considered the equation of the straight line AP in Figure 4 in the system with coordinates 1 a and 1 - 1/YH. When 1 - a = 1, 1 - l / r ~ = 1, and the line must pass through P. The area of triangle AOP is (m - aH)/kH; the reciprocal of this area, 2 u ~ / ( m- UH),which is a function of the activity coeficient of the hydrogen ion, is the unit on the abscissa in Figure 2. The marked change in the direction of the curve in Figure 2 is taken to indicate the presence of the stronger acid species. The concentration, 2.1 m, of acid at which this change occurs is that of the acid in which a saturated solution of Ca(H2P04)2is the invariant-point solution at 25' with which The Journal of Physical Chemistry
-I
0
25
50
EOUIVALENT CONDUCTANCE A
75
Figure 3. Relation between equivalent conductance and pH in the system Ca0-Pz06-Hz0.
both Ca(H2P04)2.H20and CaHPO4 are in eq~ilibrium.~' The si,ignificance of 2.1 m acid is shown also in Figure 3 in which is plotted pH vs. equivalent conductance, A, for solutions in the system Ca0-P206-H20.4 The S curve at the right of the figure is for acid solutions containing no calcium; the curve has the same shape as that of the titration of a strong acid with a base. Since (17) K. L.Elmore and T.D. Farr, I d . Eng. Chem., 32, 580 (1940).
DISSOCIATION OF PHOSPHORIC ACIDSOLUTIONSAT 25"
-
3523
the value of 1 - a at C is m(m - a H ) / ( m 2 4which
I- I
YH
aHz)
from
aH
aH
1+-
m
Equation 14 fits the data for acid concentrations below l m so well that it appeared that it might also fit the data for more concentrated acid solutions. Values of a for phosphoric acid solutions then were calculated through use of a modification of the equation of Van Rysselberghe and Eisenberg,'* and the results were checked against those obtained from eq. 14. By definition
K1 =
aHaHIPo4-
(15)
aEsPo,
and Figure 4. Plot of the dehition of am
equivalent conductance is proportional to the concentration of the ionized species, the H + concentration along this curve is decreased (and the pH is increased) by dilution, rather than by neutralization; the inflection point is at 2.1 m acid, and the pH of this acid is 0.52. If the assumption that the acid contains the strong acid species HdPzOs is valid, it may then be assumed that the inflection of the acid curve is at the point a t which the ratio of activities aHsP,O8-/aHe1o8 is Unity and at which pH = pK = 0.52. Then H&Os K4
E H + + H82O8-
= (E+>(HsP2Os-)/(HsP20s) =
aHaPO4-
+aHsPaO8-
[Hf] = fHaPO4-
(17)
fEsPzOr
Since Xo is nearly equal to &, it is assumed that fH~P04-
fHsPaO8-
It then follows that
[H+lf-
=
~HSO~-
= f-
+
aHsP20s-
(174
However, since
antilog -0.52 = 0.3 (13) The diagram in Figure 4 is useful in arriving at reasonable estimates of a for each concentration of acid. Although eq. 12 is taken as the equation of a line, it is apparent that there is a different line AP for each concentration of acid and that only one point on this line will satisfy eq. 8 for this concentration. Values of a for acid concentrations up to 1 m were available from pH data measured in cells without liquid junction and from conductance data3s4;the values of a determined by these methods agreed Within 2%. When values of Q! and the corresponding values of YH were plotted in Figure 4, each point was close to C, which, for each concentration, is the intersection with AP of a line through the origin perpendicular to AP. In Figure 4,
(15) and
then
(18) P. Van Rysselberghe and S. Eisenberg, J. Am. Chem. Sac., 61, 3030 (1939); 62,461 (1940).
Volume 69, Number 10 October 1966
K. L. ELMORE, J. D. HATFIELD, R. L. DUNN,AND A. D. JONES
3524
Hence
[H'l'f~f-
=
+
K ~ ~ H ~ PK oo ~~H [~ P~ O ~ ] (18) 1.2
and =
KlaHsP04[1
+
KOaHaPO41
(19) [H+I2 When concentrations are expressed in moles per liter and fnf-
[H+] = aC
I CA
0.4
(20) 0.0
eq. 19 becomes
- 0.4 or
+ +
log fHf- = log K ~ ~ H ~ P o ~ log [I KOaHaPO41
-0.81 0.4
- 2 log a c
(22)
The value of log f* for strong electrolytes has been expressed by Va:n Rysselberghe and EisenbergL8as logji. =
+ 2.20626 X 10-3aaC +
-''*50914 0.3286afl
2.62692 X 10-6aeC2 (23) in which a is t,he distance of closest approach, 8.,of the ions. By definition fHf- = f&2-On substituting a c for weak electrolytes for of strong electrolytes, eq. 23 Decomes 1.01822/aC 4.41252 X log fEf- = 1 0.3286adaC
c
-
+
+
+
10--3~3(aC) 5.25384 X 10-6a6(aC)2 (24) Equating the right-hand terms of eq. 22 and 24 and inserting the value 4.25 for a yields, for the model of phosphoric acid containing both H2P04- and the triple ion HsPzOa-, the equation
where au = aHaPOa,the activity of the undissociated monomer acid. The degree of ionization a (expressed as a = mH/m, in which mH is the molality of H+ and m is the molality of total P, here assumed to be entirely monomeric) calculated from conductance data for concentrations up to 1m H3P04ais approximately equal to a calculated from eq. 14 when all is computed from the pH (pH = -log aH) measured in cells with the saturated KC1 bridge.4 The J O U Tof~Physical Chemistry
1
I
t
I
0.8
1.2
1.6
2.0
J
m 93 Figure 5. p H of phosphoric acid solutions a t 25".
Table I: pH of Phosphoric Acid Solutions a t 25" Activity
Conon. of Hap04 Molality, Molarity,
of un-
a,
dissood.
oalod. from
HIPOI,
au
-pH-
Calod. from eq. 25 (smoothed) eq. 14
m
C
0.1000
0.20oo
0.0993 0.1977
0.0769 0.1605
0.28963 0.22547
0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0,9000 1.0000 1,5000 2,0000 2.5000 3.0000 3,5000 4.0000 4.5000 5,0000 5.5000 6.0000 6.5000 7.0000 7.5000 8.0000 9.0000 10.0000
0.2951 0.3917 0.4875 0.5823 0.6763 0.7694 0.8617 0,9532 1.3983 1.8239 2,2313 2.6211 2.9950 3.3534 3.6958 4.0260 4.3431 4.6469 4.9416 5.2207 5.4936 5.7605 6.2411 6,7074
0.2493 0.3417 0.4373 0.5360 0.6377 0.7425 0.8504 0,9615 1.5690 2.2750 3.0960 4.0510 5.1650 6.4610 7.9680 9.7210 11.7600 14.1200 16.8600 20.0300 23.7100 27.9600 37.7900 48.9700
0.19990
0.18616 0.17787 0.17263 0.16927 0.16718 0.16598 0.16543 0,16853 0.17671 0.18732 0.19937 0.21240 0.22600 0.24004 0.25419 0.26844 0.28260 0.29650 0.31042 0,32391 0,33683 0.35990 0.37553
Obsd.
1.589 1.391 1.289 1.179 1.105 1.042 0.986 0.936 0.890 0.847 0.670 0.528 0.407 0.301 0.207 0.121 0.043 -0.030 -0.096 -0.159 -0.217 -0.271 -0,322 -0.370 -0.457 -0.516 ~~
1.608 1.407
Diff.
0.019 0.016
1.279 0.010 1.183 0.004 1.104 -0.001 1 . n. ~-n- .nnFi - . ._ 0.977 -0,009 0.924 -0.012 0.876 -0.014 0,832 -0.016 0.648 -0.022 0.504 -0.024 0.384 -0.023 0,280 -0.021 0.188 -0.019 0.105 -0.016 0.030 -0.013 -0.039 -0.009 -0.102 -0.006 -0.160 -0.001 -0.214 0.003 -0.265 0.006 -0.312 0.010 -0.356 0.014 -0,434 0.023 -0.487 0,019
___
Std. dev.
0.015
Probable error
0.010
When values of a,calculated from eq. 25 with values of au obtained from vapor pressure data,2 are substituted in eq. 14 (with conversion of concentration from molarity to molality), the values of UH convert to values
DISSOCIATION OF PHOSPHORIC ACID SOLUTIONS AT 25'
of pH that agree with a probable error of kO.01 unit with the measured values for acid concentrations in the range 0.1 to 10 m HaP04. The density data of Christensen and Reedlg were used in the conversion of the concentration units. Equation 14 has been tested only with phosphoric acid solutions, and with these it gives highly consistent results. The equation is based on the assumption that there are no appreciable liquid-junction effects in the pH measurements, and its applicability to phosphoric Table 11: Calculated Distribution of Species in Phosphoric Acid Solutions
Conon. of
Aotivity of undi8-
Degree of
HaPO4,
dissoon.,
m
au
a
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 9 .o 10.0
0.077 0.161 0.249 0,342 0.437 0.536 0.638 0.743 0.850 0.962 1.569 2.275 3.096 4.051 5.165 6.461 7.968 9.721 11.76 14.12 16.86 20.03 23.71 27.96 37.79 48.97
0.2896 0.2255 0.I999 0.1862 0.1779 0.1726 0.1693 0.1672 0.1660 0.1654 0.1685 0.1767 0.1873 0.1994 0.2124 0.2260 0.2400 0.2542 0.2684 0.2826 0,2965 0.3104 0,3239 0.3368 0.3599 0.3755
socd.
H' 0.0290 0.0451 0.0600 0.0745 0.0889 0.1036 0.1185 0.1337 0.1494 0,1654 0.2528 0.3534 0.4683 0.5981 0.7434 0.9040 1.0802 1.2710 1.4764 1,6956 1.9273 2.1729 2.4293 2.6946 3.2391 3.7553
-
3525
acid solutions indicates that it may be used to calculate the degree of ionization of other weak acids. The agreement between the observed pH data and those calculated from eq. 14 is shown in Figure 5 and Table I. By assuming that the activity coefficients of the two negative ions are equal and that similarly the activity coefficients of both the monomeric and dimeric unionized acids are equal and making suitable substitutions in the total phosphorus and electroneutrality equations containing the ionization constants K1, KO, and K4, we can calculate the concentration of each of the species in each concentration of acid whenever [am]or [H+] is known from eq. 14. The following equations have been derived by the above procedure and the results of the calculations are shown in Table 11.
Concn., -m HiT'o4- HsPzOa- HsPOs HsP2Os 0.0264 0.0375 0.0456 0.0520 0.0573 0.0618 0.0656 0.0690 0.0720 0.0747 0.0848 0.0912 0.0954 0.0978 0.0988 0.0987 0.0976 0.0957 0.0931 0.0900 0.0864 0.0826 0.0785 0.0742 0.0665 0.0598
0.0026 0,0076 0.0144 0.0224 0.0316 0.0418 0.0529 0.0647 0.0774 0.0907 0.1680 0.2622 0.3729 0.5003 0.6446 0.8053 0.9825 1.1752 1,3833 1.6056 1.8408 2.0903 2.3508 2.6204 3.1726 3.6955
0,0682 0.1459 0.2223 0.2969 0,3696 0.4402 0.5089 0.5755 0.6401 0.7026 0,9849 1.2157 1 ,3952 1.5247 I . 6059 1.6430 1.8400 1.6081 1.5373 1.4498 1.3467 1.2316 1.1117 0.9923 0.7830 0.6393
0.0002 0.0007 0.0017 0.0031 0.0049 0.0072 0.0099 0.0130 0.0166 0.0206 0.0471 0.0844 0.1318 0.1884 0.2530 0.3238 0.3986 0,4754 0.5515 0,6245 0.6926 0.7526 0.8041 0.8463 0.9026 0.9549
ru+i
Itm-
1
K4
a,
2Koa, [H+]1 Koa,
+
K4
a,[H3PO4]
(19) J. H. Christensen and R. B. Reed, Ind. Eng. Chem., 47, 1277 (1966).
Volume 69,Number 10 October 1966
