J. S. ELLIOT, R. F, SHARP AND L. LEWIS
686
P is the algebraic sum of gas desorbed and leaked out. In the first-order case P
=
1
- e-KI
- bC
so T
P dT
(14)
When C is large, P is small, so we write
In the second-order case
P =
Vol. 62 Ze-EIT CE(1 KZ)2
+
In Fig. 2, PC/Z is plotted versus T for the firstand second-order cases. As with the case without a leak, the curves are the same at low T. In the first two cases, the values of K and E may be obtained from the temperature and slope at the inflection point. In the last two cases, the temperature at the maximum and the half width may be used. These may be calculated simply. However, one should use the full curve to determine the order of the reaction and we suggest that, to obtain accurate values of K and E, the whole curve should be fitted using trial values of K and E.
THE APPARENT DISSOCIATION CONSTANTS OF PHOSPHORIC ACID AT 38’ AT IONIC STRENGTHS FROM 0.1 TO 0.5‘ BYJAMESS. ELLIOT,ROBERT F. SHARPAND LEONLEWIS From the Poliomyelitis Respiratory and Rehabilitation Center, Fairmont Hospital, San Leandro, California; The Division of Urology, Department of Surgel.y, University of California School of Medicine, San Francisco, California; The Department of Medicine, Stanford University School of Medicine, S a n Francisco, California Received J a n u a r y 22, 1868
The present study was undertaken as part of a comprehensive investigation of the urinary chemistry involved in the formation of renal phosphatic calculi, a common and serious problem in severely paralyzed persons. Factors affecting the solubility of calcium phosphate are of the utmost importance, not only in the etiolog of renal calculus disease, but also in the formation of bone, teeth and other biological phosphatic structures. Since the sofubility of calciuni phosphate depends in part on the concentration of HP04‘ and k’04*, and since the concentration of these ions cannot be directly determined, but must be derived from calculations employing the dissociation constants of phosphoric acid, accurate knowledge concerning these factors is of considerable importance. Although the dissociation constants of phosphoric acid are reasonably well known in dilute solution, little is known regarding the apparent dissociation constants at higher ionic strengths. Since the ionic strength of urine ranges from 0.1 to 0.5 with an average of 0.32, the present study was undertaken in order to determine the effects of higher ionic strengths on the dissociation constants of phosphoric acid. Sodium phosphate solutions were prepared according to the method of Bjerrum and Unmack, sodium chloride was added to increase the ionic strength, and the pH of the solution mixtures was determined a t 38”. From the data so obtained, the apparent dissociation constants were calculated, and equations were derived relating the apparent dissociation constant to the ionic strength of the solution.
Introduction The present study was undertaken as part of a comprehensive investigation of the urinary chemistry involved in the formation of renal phosphatic calculi, a common and serious problem in severely paralyzed persons. Factors affecting the solubility of calcium phosphate are of the utmost importance, not only in the etiology of renal calculus disease, but also in the formation of bone, teeth and other biological phosphatic structures. Since the solubility of calcium phosphate depends in part on the concentration of HP04- and PO4’, and since the concentration of these ions cannot be directly determined, but must be derived from calculations employing the dissociation constants of phosphoric acid, accurate knowledge concerning these factors is of considerable importance. The first and second thermodynamic constants have been fairly well established and their variations with changing ionic strengths from P of 0 to 0.1 have been adequately studied. Bates2 has published recently the results of a comprehensive study of the first dissociation constant and has (1) Aided by a grant from the National Foundation for Infantile Paralysis. (2) R. G. Bates, J . Research Natl. BUT.Standards, 47, N o . 3, 127 (1950, Research Paper 2238.
discussed the difficulties involved in t,his determination. Bates and Acree2 have made a similar study of the second dissociation constant. The literature has been reviewed in both publications. On the other hand, data regarding the third dissociation constant of phosphoric acid is rather meager. Latimer* estimated the thermodynamic dissociation constant to be 1 X 10-l2. Kugelmass6 studied the third dissociation constant at 38’ with a range of ionic strength from 0.013 to 0.115, and derived a value of 1.48 X 10-l2. Bjerrum and Unmacke studied all three dissociation constants a t 37’. The first and second constants were calculated in the range u = 0 to 0.10, and the third in the range p = 0 to 0.17. They calculated pK2 to be 12.180 at 37’, and related the dissociation constant to varying ionic strengths with the equation pK’ = 12.180 - 2.575 flu 2.63~. Since the ionic strength of urine ranges from 0.1 to 0.5 with an average of 0.32,7the present study
+
(3) R. G . Bates and S. F. Acree, ibid., 30, 129 (19431, Research Paper # R P 1524. (4) W. M. Latimer, “The Oxidation States of the Elements and Their Potentials in Aqueous Solution,” Prentioe-Hall, Ino., New York, N. Y., 1952. (5) I. N. Kugelnmss, J . Biochem., 93, 587 (1929). ( 6 ) N. Bjerruin and A. Unrnack, Det Kg1. Danske Videnskabernee Selskab. Mathematisk-fysiske Meddelelser, 1X 1 (1929).
APPARENT DISSOCIATION CONSTANTS OF PHOSPHORIC ACID
June, 1958
687
TABLE I Composition of solution: 1 part solution A and 4 parts solution B giving a molar ratio of NaH2P04/HCl = 2.0. log
(a-)
(Na +) molesfl.
molesh.
PH
0.1000
0.0800
.1940 .2940 .3940 .4940
.I740 .2740 .3740 .4740
2.26 2.23 2.20 2.18 2.16
PfH
(H+)
d,ii
P
(HePOa-)
(HrPOa)
(HrPOn-) (HsPOa)
0.092 0.00679 0.1068 0.3268 0.02679 0.01321 .02741 .00741 .2014 .4488 ,01259 .lo0 .02794 ,01206 ,100 .00794 ,3020 .5495 ,02817 .01183 .092 .00817 .4022 .6342 .7088 .02843 ,01157 .OS6 .00843 .5024
0.307 .338 .364 .377 .390
PKI’
1.953 1.892 1.836 1.803 1.770
Mi’
+
Ad,ii
2.124 2.126 2.123 2.134 2.140
TABLE I1 Composition of solution: 1 part solution A and 6 parts solution B giving a molar ratio of NaHzP04/HC1 = 3.0.
0.1000 .1965 ,2965 .3965 .4965
0,07143 ,1679 .2679 .3679 ,4679
2.49 2.45 2.42 2.40 2.38
0.092 0.00400 0.1040 0.3225 0.03257 0.01029 ,00447 .2010 .4483 .03304 ,00982 .lo0 ,03336 ,00950 . l o 0 ,00479 .3013 .5489 .4014 .6336 .03349 .00937 .092 .00492 .00508 .5016 .7082 .03365 .00921 .086
was undertaken in order to determine the effects of higher ionic strengths on the dissociation constants of phosphoric acid. Experimental Sodium phosphate solutions were prepared according to the method of Bjerrum and Unmack, sodium chloride was added to increase the ionic strength, and the pH of the solution mixtures was determined. Anhydrous dibasic sodium phosphate prepared as a special buffer salt by Fisher Scientific Company was the source of all phosphate used in the solutions. One-tenth molar hydrochloric acid was prepared from Baker reagent grade concentrated hydrochloric acid by the constant boiling method. The sodium chloride employed was Baker reagent grade. Both the sodium phosphate and the sodium chloride were dried for two hours a t 115’. All water used was triply distilled and freshly boiled. The compositions of the solutions were: Solution A was prepared by adding 5.845 g. of NaCl to 100 ml. of 1 M HC1 and diluting to 1 liter with distilled water. This solution was 0.10 M with respect to sodium. Solution B was prepared by adding 7.098 g. of dibasic sodium phosphate to 50 ml. of 1 M HC1 and diluting to 1 liter with distilled water. This solution was 0.10 M with respect to. NaHzP04, and 0.05 M with respect to NaC1. Solution C consisted of 7.098 g. of dibasic sodium phosphate dissolved in water, and diluted to a volume of 1 liter. This solution was 0.05 M with respect to Na2HP04. Solution D consisted of 0.1 M sodium hydroxide prepared free of carbonate and standardized against potassium acid phthalate. For convenience, all solutions were prepared a t room temperature. The error introduced by the assumption that molar concentrations at 38’ are equal to those a t 25” is much less than the uncertainty of the pH measurement. The pH of the solution mixtures was determined by means of the Beckman model GS pH meter using the Beckman general purpose glass electrode. All measurements were made anaerobically in the Beckman Blood Electrode assembly which was immersed in a constant temperature bath at 38”. These buffer solutions were employed as pH standards. 1. For the pH range 2-4, 0.01 M potassium tetroxalate with a pH of 2.16 at 38’ and 0.05 M potassium acid phthalate having a pH of 4.03 at 38’ were employed. 2 . For pH determinations in the vicinity of 7.0, the Beckman buffer solution having a pH of 6.97 a t 38” was employed. 3. For pH determinations in the range 10-11, Beckman buffer solution #3505 with a pH of 9.89 at 38” and 0.01 M trisodium phosphate having a pH of 11.38 a t 38” were employed . I n the higher pH ranges correction was made for sodium ion error by employing the Beckman nomograph #225-N. pH values for the buffers are in accord with National Bureau of Standards data.8 (7) J. 8 . Elliot. R. F. Sharp and L. Lewis, unpublished data.
0.500 ,527 .545 .554 .563
1.990 1.923 1.875 1.846 1.817
2.158 2.157 2.161 2.177 2.187
The First Dissociation Constant of Phosphoric Acid.-&’ was calculated from the HendersonHasselbalch equation, pK1’ = pH - log (HzP04-)/ (H3P04). The pH of the solution mixtures and other data needed to calculate pK1’ are given in Tables I and 11. In order to obtain the molarities of phosphoric acid and primary phosphate ion from which the buffer ratio is computed, the activity coefficient of HC1 and the hydrogen ion concentration must be known accurately. Feldmang has calculated the variation of the activity coefficient of hydrogen ion with changing ionic strengths up to u = 0.5 employing the equation
The mean values determined by Feldman were employed for the pfH values in Table I and Table 11. pH has been assumed to approximate paH in accordance with the procedure followed a t the National Bureau of Standards.lo The concentration of hydrogen ion was derived from the equation pcH = pH - pjH. The ionic strength is defined as one-half the sum of the molar concentration of each ion multiplied by the square of its charge. I n order to calculate the buffer ratio (H2P04-)/(H3P04),the concentrations of HzP04and H3P04were determined by ( H2POa-) = [HzP04-]
- (HaPO4)
and (HaP04) = [HCl] - (H+)
where [ ] = initial concn. ( ) concn. after equilibrium is established
+
The values for pK1’ A p< were calculated according to the procedure followed by Bjerrum and Unmack. The value for A of 0.5220 was taken from Bates.io The for pK1’ A d@in Table I when plotted against p may be expressed by the equation,
+
(8) Letter Circular LC 993, Standardization of PH Measurements Made with the Gloss Electrode, Natl. Bureau of Standards, 1950. (9) I. Feldman, Anal. Chem., 38, 1859 (195G). (10) R. G. Bates, “Electrometric pH Determinations.” John Wiley and Sans, New Y o r k , N. Y . , 1954.
J. S. ELLIOT, R. F. SHARP AND L. LEWIS
688
+
pK1' = 2.128 - 0.522cii 0 . 0 0 4 ~ When the same values in Table I1 are plotted, the expression pK1' = 2.145 - 0.522dji 0.0778~is derived. The average of these values is pKl' = 2.137 - 0.522Cji 0.041p. If pK1' is related only to p< as done by Sendroy and Hastings, l1 variation in the apparent dissociation constants with changing ionic strength may be expressed as pKr' = 2.110 - 0.485 fi (1A + 4B) pK1' = 2.129 - 0.448 4j.i (1A + 6B) giving an average of
+
+
pK1' = 2.120
- 0.467 p
