5. 1. Cooke, Jr.
University of Louisville Louisville, Kentucky
I
I
The Ionization of Polyhydric Acids .
Numerical results for phosphoric acid
The ionization of a polyhydric acid frequently appears to many freshmen and sophomores to be an extremely complex phenomenon which is not easily understood. This unfortunate condition can often be traced to the very thorough and logical derivations presented in textbooks. The resulting equations have little meaning at this academic level, however, without some numerical values. The calculations for polyhydric acids are more involved than those for simple acids and require considerable time. For thorough wnil~rehensio~;, ir would br de:irat,le ro h a w thr results of tl~rsccalculations for ,111 sprcics involved :in11over >I wide pH range. Previous authors have considered the mathematical equations for a variety of acids (I-@, mixtures of acids (6) and complex ions (7-9). Phosphoric acid was chosen as an example of a polyhydric acid since it has three equilibria with convenient pK values throughout the normal aqueous pH range. For these same reasons, phosphates are nsed in many buffer solutions and are familiar to most students. The numerical results reported here were obtained with an IBM 1710 (modified 1620) computer using a Fortran I1 program PHACT (PolyHydric Acid Concentration Tabulator) written for the purpose. The author will be happy to make PHACT available to interested parties. Individual values as well as the complete table over the range from 0-14 pH can be obtained by supplying either the pK or K. values and the interested pH values. Complete results for phosphoric and other common polyprotic acids are also available from the author. The author appreciates the use of the computing hboratory facilities of the University of Louisville, and the assistance of Professor Alfred T. Chen of the Engineering Mathematics Dept.
Table 1. Custornaly svmbol
Computer svmbol
ao
H3P04
a,
H2P04-
(12
HP04--
a8
P04---
Z
Z
vz
N
r/C
U/C
620
/
PHACT can also be expanded for more involved processes. PHACT executes the equations found in a number of sources (1-5) and presented in Table 1. Students not familiar with these equations should be encouraged to derive them, a surprisingly easy task in spite of their apparent complexity. Table 1 also defines the symbols customarily, nsed as well as those available from the computer and used in Table 2. More elaborate equations can and should be applied for concentrated solutions. To facilitate such calculs, tions, s d u r s for thr ionic arrengrh vonwnrrxrion ~xtios used ?rc adequnt? are also tahulnted. Thr rqu~~lions a t concentrations normally encountered and illustrate clearly the more important equilibrium principles. Table 2 presents some of the values obtained over the pH range from 0-14 with increments of 0.1 pH unit. The pK values of Buelcenkamp (10) were used. These appear to be the most consistent set containing all three pK values among those cited by Bjerrum (11). In addition, values for the number of hydrogen ions bound per phosphate (Z), the number of hydrogen ions added per phosphate (n),and the ionic strength-concentration ratio (r/C) are also tabulated. Conventional computer notation is used. The relative [Hap011 at pH 7.0, listed as 8.225M6, is read as 8.225 X lo-=. The two figures are graphical presentations of the numerical results of Table 2. Figure 1 is a linear plot of the relative concentrations and emphasizes the behavior of the major species a t each pH. Figure 2 presents the quantities r/C, Z , and n as a function of pH. It should be noted that all of these quantities are dimensionless. It is apparent that Table 2 supplies all of the information contained in all of the figures with even greater precision and is more useful for detailed analysis, but trends and qualitative conclusions are more readily obtained from the figures.
Symbols and Equations Used (1, 2)
Word definition Fraction of total hosphate present as unionized HZ& Fraction of total phosphate present as HBPOIFraction of total phosphate present as HPOaP Fraction of total phosphate present as POlP Average number of hydrogen ions bound per ~hosnhate . Average number of hydrogen ions added per phosphate
.
Ionic strength/concentration
Journd of Chemical Education
.
Equation
[&o+]" + K, tHaO+l*+ K,KdHzO+l + K&Kz K,[HaO+I2 [H,O+I3+ Kx[I40+l2 + K,Kz[HaO+] + KIKsK, K&[HaO +I [HaO+la+ K,[H,O+Ia + R,K*[H,O+l + K A K s K,KA [H,O+I8 + K,[HaO+Ja+ &IGIHaO+] + K,K,Ka [HBO+I'
31H~O+I3f 2K~[Ha0+1'f KIKS[HZO+I IH,O+lS K L [ H ~ O + I ~K,K,tH,O+I K&Ka + [H80+1 - [OH-] (C = total phosphate concentration) C u, 4rrz 9az
+
+ + 9
+
+
Advantages of the Table I t permits detailed study of the change in any variable with a change in any other variable. It indicates all concentrations to the same precision, including the extremely low values frequently approximated aa zero when compared with other species. It, identifies the customarv "hudmarks" with
1
l l 8 L E 2. NUMERICAL R E S U L T S FOR PHOSPHORIC A C I D KA1= 7.585E-03 l A 2 i 6.025E-08 K A 3 i 4.365E-13 2.120 PK2= 7.220 PK3= 12.360 PH H3POb H2PObHPO4-PO+-UIC
Table 3. Reference number
Name
"Landmark" Characteristics
Approximate PH
(1) (#)
F i s t half-neutralization point F i t equivalence point
PKL ' l d p K ~ pKd
(3)
(4)
Second half-neutralization point PK* Second equivalence point 'ldpKz fpKd
16)
Third half-neutralization ~ o i n t
+
PKI
Application of the Table to General Equilibrium Problems Specific values are obtttinable to verify equilibrium trends: Infinitesimally low concentrations at extreme pH values. For example, only 12 PO4-' ions are present in a liter of 0.1 M solution at a pH of 0.0 (a relative concentration of 1.980 X 1o-Pz). Relative changes in concentration. For example, a t very low pH values, an increase in [H+] by one power of ten produces corresponding increases of one power of ten for [HaPOd-I, two powers of ten for [HP04-I, and three powers of ten for
wn.-al >--.. ,.
Changes around pK values. For example, the change from 0.909 to 0.0909 in concentrations between pK - 1 and pK 1, useful in explaining indicator behavior. Changes in concentrations of ions involved in other equilibria. For example, precipitation of insoluble phosphates. Characteristics obtainable for particular solutions are: Concentrations a t a. particular pH. pH of a specific salt solution or mixture. Change in pH caused by addition of salt, strong acid, or strong base. For example, the Van Slyke (1s) buffer index 6 can be closely approximated by the quantity An/ApH. Values for this quantity a t the "landmarb" are also included in Table 3. 6 is a measure of the sharpness of the endpoint as well as the effectiveness of a buffer system (13).
+
Figure 1. p H versus rclotive concentrations of phosphoric acid rpecier Circled numbers refer to Toblo 3.
pH value 2.12
Numerioal behavior nu = a, = 0.5
An
A(dc)
TH-
n 2.5
-0.74
0.57
APE
4.67
a, = max a" = a2
2.0
-0.02
0.02
7.22
a, = a* = 0.5
1.5
-0.57
1.14
a2 = max = ar
1 .O
-0.02
0.04
0.5
-1.09
1.72
9.79 12.36
a x
al = as =
0.5
The fate of added reagent as reflected by n and Z changes, shown graphically in Figure 2. Ionic strength and ionic strength changes, also shown in Figure 2.
Literature Cited LAI~NEN H., A., "Chemical Analysis," McGraw-Hill, Inc., New York, 1960, pp. 35-36. SILLEN,L. G., in "Tre&tise on Analyticd Chemistry," I. M., AND ELVING, P. J., editors, Interscience KOLTHOFP, Publishers, Inc., New York, 1959, Part I, Val. 1, Chap. 8. BUTLER.J. N.. "Ionic Ec~uilibrium."Addison-Wesley - Pub, 1964. lishing Co., &., ~ e a d h g~assachusetts, Rmcr, J. E., "Hydrogen Ian Concentrations," Princeton University, Princeton, New Jersey, 1952,,p. 35. E. P., "Ionizat~onConstants," ALBERT,A,, AND SERIEANT, Methuen, London, 1962, p. 9. KING, E. L., THISJOURNAL, 31, 183 (1954). J. W., AND MILLER,W. W., THIS JOURNAL, SWINNERMN, 36,485 (1959). HUGUS.Z. Z.. in "Advances in the Chemistry of Coordinat.ion ~ O ~ D ~ I I ~ ~ ~ . " K I RS.. S Ceditor. H N E61acmillan R. Co..
Figure 2. p H variation of the averogo number of hydrogen ions added per phosphate, N; the overage number of hydrogen ions bound per phorphate, Z; and the ionic strength contribution per phosphate, PIC. Circled numbers refer to Table 3.
Volume 42, Number 1 1 , November 1965
/
621
(10) BEUKENKMIP, J., RIEMAN, W., AND LINDENBAUSI, S., Anal. Chem., 26,510 (1954). (11) BJERRUM, J. R., SWARZENBACH, G., AND SILLEN.G., "St&bility Constants,'' Vol. 2, Special Publication No. 7, The Chemical Society, London, 1958, p. 58.
622
/
Journal o f Chemical Education
(12) VANSLYKE, D. A,, J. Bid. Chm., 52,525 (1922). (13) BRUCKENSTEIN, S., AND KOLTROFF, I. M., in "Treatise on Analytical Chemistry," KOLTHOFF, I. M., AND ELVING, P. J., editors, Interscience Publishers, Inc., New York, 1959, Part I, Vol. 1, Chap. 11, p. 458.
