LITERATURE CITED ( 1 ) K. V. Sarkanen, 6. F. Hrutflord, L. N. Johanson, and H. S. Gardner. Tappl, 53, 766 (1970). (2) K. Anderson, Sven. Papperstid., 73, l(1970). (3) B. F. Hrutfiord and J. L. McCarthy, Tappi, 50, 82 (1967). (4) M. J. Matteson, L. N. Johanson, and J. L. McCarthy, Tappi, 50, 86
(1967).
(5) W. T. McKean, Jr., 6.F. Hrutfiord, and K. V. Sarkanen, Tappi, 48, 699 (1965). (6) T. T. C. Shih, B. F. Hrutfiord, K. V. Sarkanen, and L. N. Johanson, Tappi, 50, 630 (1967). (7) R. F. Cashen and H. D. Bauman. Tappi, 46, 509 (1963). (8) V. F. Felicetta and J. L. McCarthy, Tappi, 42, 162A (1959). (9) P. Dutoll and 0.V. Welsse, J. Chem. Phys., 9, 608 (1911). (IO) W. D. Treadwell and L. Welss, Helv. Chim. Acta, 2, 680 (1919). ( 1 1 ) P. B. Borlew and T. A. Pascoe. Pap. TradeJ., 122, 31 (1946). (12) M. W. Tamele, V. C. Imine, and L. B. Ryland, Anal. Cbem., 32, 1002 (1960). (13) M. W. Tamele, L. B. Ryland, and R. N. McCoy, Anal. Chem., 32, 1007 (1960). (14) M. W. Tamele, L. 6.Ryland, and V. 0.Imine, hd. Eng. Chem., Anal. Ed., 13, 618 (1941). (15) E. Bllberg and P. Landmark, Nor. Skoglnd., 15, 221 (1961). (16) T. T. Collins, Jr., Pap. TradeJ., 129, 23 (1949). (17) I. M. Kolthoff and N. H. Furman, “Potentiometric Titrations”, John Wlley and Sons Inc.. New York, 1931, p 110. (18) Anon., Tappl Standard T-625 ts-64 (1964), 5 pp.
(19) J. E. Olsson and 0. Samuelson, Sven. Papperstidn., 68, 179 (1965). (20) J. L. Swartz and T. S. Light, Teppl, 53, 90 (1970). (21) E. Bllberg, Nor. Skogind., 13, 307 (1959). (22) M. S. Frant and J. W. Ross, Jr., Tappl, 53, 1753 (1970). (23) P. 0. Bethge, M. Carlson. and R. Radestrom, Sven. Papperstidn., 71, 864 (1968). (24) V. F. Feilcetta. Q. P. Penlston, and J. L. McCarthy, Tappi, 36, 425 (1953). (25) F. E. Murray, L. G. Tench, and C. D. Eamer, 6th Paper Ind. Air and Stream Improvement Conference, April 13-15, 1971, Quebec City, Canada. (26) J. Papp. Cellul. Chem. Techno/., 5, 147 (1971). (27) J. Papp. Sven. Papperstidn., 74, 310 (1971). (28) J. H. Karchmer, Anal. Chem., 29, 425 (1957). (29) F. E. Murray, Tappi, 42, 761 (1959). (30) F. E. Murray, Pulp Pap. Mag. Can., 89, T26 (1968).
RECEIVEDfor review November 25, 1974. Accepted May 13, 1975. This paper is dedicated to the 60th birthday of Dr. Karl Kratzl, Professor, Institute of Organic Chemistry, Vienna, Austria. The financial assistance of the National Research Council of Canada to the senior author during various phases of these and related experiments is gratefully acknowledged.
Simultaneous Potentiometric Determination of Precise Equivalence Points and pK Values of Two- and Three-pK Systems 1.N. Briggs and J. E. Stuehr Department of Chemistry, Case Western Reserve University, Cleveland, Ohio 44 106
Equations are developed for the preclse and slmultaneous determlnatlon of pK values, equivalence polnts, and proton purlty In the potentlometrlc determlnatlon of dl- and trl-bask weak aclds. A computer technlque based upon nonlinear regresslon Is described which automatlcally finds the optlmum parameters. The procedure Is based on exact mole balance relatlonshlps and takes Into account the presence of any strong acld or base, as well as the changes In lonlc strength which occur during the course of a tltratlon. Analysls of several representatlve systems lndlcates that experlmental pH-volume curves can be reproduced to withln a standard devlatlon of 0.003 pH unlt or better.
In a multi-pK system, the individual pK’s will be effectively independent of each other if they are separated by a t least 2.7 units (I).In such cases, each can be evaluated individually by methods developed for single pK systems. However, very often successive pK’s are not sufficiently separated for such procedures to be applicable. Some current methods of obtaining pK’s in such situations are based on the fact that m titration points in a system yield m simultaneous equations. If there are n pK’s, then n titration points will yield a single value for each of the n pK’s. Several combinations can be used from the set of m titration points and the average results can be taken as the most probable values. For example, if there are two pK’s in a system, pairs of simultaneous equations (2) can be solved for pK1 and pK2. Hendrickson (3) has suggested a method by which this principle can be used to obtain any number of pK’s. However, in practice, difficulties may arise when 1916
such methods are used, primarily because random errors can cause the equations to be inconsistent. An alternative approach is to cast the equation into the form of a straight line which yields the constants from its slope and intercept. This approach was used by Speakman ( 4 ) to evaluate the pK’s of dibasic acids. I t was extended to three pK systems by Tate et al. (5). They assumed that the third pK had little effect on the lower regions of the titration curve, and obtained first estimates of the lower pK’s from these points. These estimates were then used with points in the upper region to obtain the third pK. The last value was then substituted into a rearranged form of the functional equation to obtain a better value of the first pK. The process was repeated until the values of the pK’s showed no further changes. A key feature to the above methods is that they all involve the assumption that the concentration of the substance under study has been previously determined. We have already discussed the difficulties involved in such determinations (6). These difficulties become even greater as the separation between pK’s decreases. The method which we present here does not require such prior knowledge. We have shown (6) that if each point in the titration curve is used to calculate the pK of a monoprotic weak acid, the resulting values will be constant only if the correct equivalence volume (V,) is used. If V , is incorrect, the resulting values show a systematic drift with increasing pH. More correctly, the proper criterion (7) for the best choice of the titration parameters is the closeness with which calculated pH values agree with the measured values. In either case, the relevant equations are relatively simple and
ANALYTICAL CHEMISTRY, VOL. 47. NO. 12, OCTOBER 1975
the treatments are such that the effects of small random errors ordinarily encountered in carefully performed titrations are distributed over the entire curve. In this paper, we extend our earlier treatments to systems containing two or three weak protons. As in the earlier work, the treatment will find the best pK values, end point, volume, and proton stoichiometry for a system of any charge. In addition, it will be generalized to accommodate the ionic strength changes which occur during the titration.
THEORETICAL The following set of equations applies to any system of n pK's: CA = [A]
+ [HA] + . . . [HnA]
[HI = ~ C -A[HA] - . . . n[HnA] - [B]
(1)
+ [OH]
(2)
The ratios (fk) may be calculated by the following relations based on an empirical activity coefficient equation of Davies (8) log fk = [2(b - n k) - 11 log fo
+
0.31 - 0.21021
(7)
where n is the number of pK's. The net effect is to convert the value of Kk a t any ionic strength to that at I = 0.1. Since we are calculating ratios of activity coefficient functions at slightly different ionic strengths, the exact functional equation used is not important. This is not true when we need an actual activity coefficient value for the proton. The use of mixed K values allows the use of aH values everywhere in the treatment except in the quantity R , which involves actual concentrations of H+ and OH(see Equation 9). We use the Debye-Huckel equation for YH:
(3)
where the following abbreviations are used: CA = analytical concentration of the acid, [A] = equilibrium concentration of the unprotonated form of the acid, [HI = equilibrium concentration of protons, aH = activity of hydrogen ions, [Hn-k+lA] = equilibrium concentration of the protonated forms of the weak acid, Kk = kth reciprocal mixed ionization constant of the weak acid, CB, = initial concentration of the base, [B] = equilibrium concentration of base in solution, V = volume of base added, Vt = total volume of the solution, V, = theoretical volume of base per proton required to neutralize the weak acid, and [OH] = equilibrium concentration of hydroxyl ions. Equations 1-4 can be combined and rearranged to give different functional equations for different values of n. Before doing so, we must address one further problem, viz., the ionic strength changes which occur during the titration of a multi-pK system. In the titration of a monoprotic weak acid in a medium of 0.1M ionic strength, the latter will remain virtually constant if the titrant is also 0.1M. For an uncharged polyprotic acid, the ionic strength will remain constant only up to the first end point. Beyond this, the ionic strength will gradually increase because of the formation of higher charged ions. If, for example, the charge on the completely deprotonated system is -3, then at the third end point, the ionic strength of the system will have increased over the initial value by 6 C,. Typically C, will be about 5 X 10-3M, yielding a potential increase of 0.03M in the ionic strength. Dilution effects will reduce this increase somewhat. Nevertheless, if the background ionic strength is kept relatively low in order that CH values can be estimated from measured aH values, the change in ionic strength can easily be 10-20%. In the present treatment, corrections to the constants due to small changes in the ionic strength are taken into account by the quantities fk, which are defined as the ratios of the reciprocal mixed ionization constants, at any point in the titration to their values at 0.1M ionic strength, e.g.,
If -b is the charge on the fully deprotonated acid, the ionic strength of the solution can be calculated from the following equation
+
+
+
+
+
2 1 = [HI b 2 C ~ [B] [OH] 2[S] (1- 2b)[HA] + (4- 4b)[HzA] . . . ( n 2- 2nb)[HnA] (6)
+
where [SI is the concentration of the supporting electrolyte.
where A and B are 0.509 and 0.33 for water at 25 "C and I is taken to be 9.48 A. The latter corresponds to YH = 0.83 at Z = 0.1, a number which is supported by theory and experiment (9) insofar as tests can be made. One could of course treat I as an additional variable. It is our experience however that the data analysis is not often sensitive to 8. In addition, other sources of experimental error (such as pH electrode calibration over the wide range under consideration and variations in liquid junction potentials) are much more significant. We are now in a position to develop specific equations for two- and three-pK systems. Two-pK Systems. For a system having two ionizable protons in the pH region under study, Equations 1-4 and 7 can be rearranged to
where R = ([HI - [OH])VJCB,. AV, as introduced earlier (6), is a volume correction which takes into account the presence of any strong acid in the sample at the beginning of the titration. (A negative A V corresponds to the presence of strong base.) This correction, ordinarily quite small, represents a displacement of the volume origin. The procedure for obtaining the best values of K1, K2, V,, and AV from these equations involves the following steps: (1) Calculate, from Equation 6, the ionic strength at each titration point. An estimated value of Ve is used to calculate CA,[HA], [HzA],etc. (2) Estimate AV and K2. A visual estimate of the pK from the titration curve is sufficient. (3) Find the values of K1 and V, which correspond to these estimates by means of Equation 9 with data points from the first pK region and the first few points of the second pK region. (4) Calculate point-by-point K2 and K1 values in turn, using data from the appropriate pH regions and the values of V, and AV. ( 5 ) Calculate U H for each point on the titration curve with the estimate of AV, and the values of K1, K2, and V,. Equation 9 is biquadratic because of the dependence of R upon [HI2. (6) Convert aH to paH and find the standard deviation from the experimental pH values. Vary V , by small
ANALYTICAL CHEMISTRY, VOL. 47, NO. 12, OCTOBER 1975
1917
STOP. a =
5. b
-
AV
Flgurr 1. Schematic diagram of the computational steps in obtaining the best fk parameters. Steps involving iteration of the ionic strength corrections and the varylng of V. (step 6) are not shown
amounts until a minimum in the standard deviation is obtained. (7) Repeat steps 2-6 using new estimates of A V and Kz until the smallest standard deviation in pH is found. The procedure for finding the best parameters is readily carried out by a computer. A program has been written which begins with rough estimates of AV and K2 and varies them systematically until the most probable values of all the parameters are found. These values are then printed out together with a list of the pa^ values calculated at each titration point and a list of pK’s corresponding to each point of each pK region of the titration curve. The procedure is illustrated diagrammatically in Figure 1. The program begins with the estimates of AV and K2 and carries out the first six steps outlined above. A t the end of the first cycle, a small arbitrary increment is added to the initial estimate of AV and the procedure (steps 1-6) is repeated. At the end of subsequent cycles, the increment by which A V is changed is determined by comparison of the standard deviation of that cycle with that of the preceding cycle. If the standard deviation decreases, the previous increment is used. If it increases, the previous increment is multiplied by -0.45. If there is no appreciable difference, AV is returned to its original value and Kz is changed. The process of varying AV is repeated with this new value of K2. New values of K Z are obtained in the same manner as those of AV. When changes of both K2 and AV make no appreciable difference in the standard deviations of succeeding cycles, the iterative process is stopped. The results at this point are based on slightly incorrect ionic strengths. More accurate values of the ionic strength at each point are now computed from the converged constants. Repetition of the minimization procedure yields the final best-fit parameters. Three-pK Systems. In a three-pK system, there are five unknown quantities: K1, K2, K3, V,, and AV. The functional equation now becomes:
1918
The procedure for evaluating the constants is very similar to that described for two-pK systems, with the following differences: the additional species H3A is now taken into account (step 1);an initial estimate of K3 is needed (step 2); and point-by-point pKs values are computed from data in the third pK region (step 4). The results at the end of this procedure are based on an estimated value of K3 as well as on ionic strengths calculated from an estimated V,. Greater accuracy is achieved by replacing these estimates with the latest values and repeating the minimization procedure. The values of V,, AV, K1, K2, and K3 which yield the lowest standard deviation in pH are chosen as the most probable values.
EXPERIMENTAL Citric acid was obtained from Fisher Scientific, adenosine 5’monophosphoric acid (AMP) from Sigma Chemicals, and nicotinamide-adenine dinucleotide phosphate (NADP) from Nutritional Biochemicals, all in the purest forms available. AMP and NADP were stored desiccated below 0 O C . Sample solutions were prepared on the same day that the titrations were performed. The dry crystals were dissolved in doubly distilled water and the ionic strength was adjusted to 0.1M by adding a sufficient quantity of KNO:, to citric acid and AMP and (Me)dNBr to NADP. Aliquots of each system (10-15 ml) were titrated with 0.1N KOH delivered by a 2-ml Gilmont microburet. pH measurements were made by a Corning 101 digital pH meter/electrometer in conjunction with a glass electrode (Fisher 13-639-3) and a reference electrode (Orion 90-01 with 90-00-01 equitransferent filling solution). The titration cells were thermostated a t 26 0.02 OC by means of a Lauda K2R circulator system. Details of the procedure are given in an earlier publication ( 6 ) .
RESULTS AND DISCUSSION To test the method, synthetic titration curves were constructed for both two- and three-pK systems. Selected values of V,, AV, and the pK’s were used to calculate the pH that would correspond to various volumes of 0.1M base when added to a 10-ml solution of hypothetical weak acid. Small random errors were then superimposed on the assumed volumes and calculated pH’s. The “data” so generated were evaluated by the computer program for the bestfit values of the parameters. Ten data points from each pK region were typically used for a given system. The program had no difficulty in correctly interpreting the synthetic
ANALYTICAL CHEMISTRY, VOL. 47, NO. 12, OCTOBER 1975
Table I. Titration Data and pKmValues for AMP, NADP, and Citric Acida V, ml
PHb
PKm
PaHC
Adenssine monophosphoric acid
0.050 3.291 0.100 3.396 0.150 3.502 0.200 3.606 0.250 3.711 0.300 3.816 0.3 50 3.922 0.400 4.033 0.450 4.149 0.500 4,276 4.417 0.550 0.900 5.847 0.950 5.983 1.ooo 6.112 1.050 6.231 6.350 1.loo 1.150 6.474 1.200 6.606 1.250 6.754 1.300 6.933 p K l m = 3.894. pKzm = 6.277. V, ml. Std dev = 0.00221 pH unit.
3.290 3.396 3.503 3.609 3.713 3.818 3.923 4.032 4.147 4.272 4.411 5.846 5.983 6 .lo9 6.230 6.349 6.472 6.604 6.753 6.93 5 = 0.71377 ml.
3.896 3.893 3.893 3.890 3.891 3.891 3.892 3.894 3.895 3.898 3.899 6.278 6.277 6.279 6.278 6.277 6.278 6.278 6.277 6.275 A V = 0.00054
h i c o t i n a m i d e adenine dinucleotide phosphatee
0.060 0.080 0.100 0.120 0.140 0.160 0.180 0.200 0.220 0.240 0.260 0.280 0.300 0.320 0.340 0.360 0.380 0.660 0.680 0.700 0.720 0.740 0.760 0.780 0.800 0.820 0.840 0.860 0.880 0.900
3.373 3.431 3.493 3.549 3.609 3.667 3 -727 3 -783 3.840 3.898 3.956 4.018 4.080 4.146 4.211 4.284 4.355 5.883 5.952 6.018 6.094 6.1 59 6.230 6.290 6.353 6.426 6.500 6.571 6.650 6.730
3.373 3.432 3.492 3.550 3.609 3.666 3.724 3.782 3.839 3.898 3.957 4.018 4.080 4.145 4.213 4.285 4.363 5.883 5.956 6.025 6.093 6.158 6.224 6.289 6.355 6.423 6.493 6.567 6.646 6.732
3.874 3.872 3.876 3.872 3.875 3.875 3.878 3.876 3.875 3.874 3.873 3.874 3.874 3.875 3.872 3.873 3.866 6.253 6.250 6.246 6.255 6.254 6.260 6.255 6.252 6.257 6.261 6.258 6.258 6.252
p K l m = 3.874. pKzm = 6.255. V, = 0.52040 ml. A V = -0.0154 ml. Std dev = 0.00342 pH unit. C i b i c acidd
0.200 0.240 0.280 0.320 0.360
2.901 2.938 2.976 3.015 3.056
2.901 2.938 2.976 3.016 3.056
3.044 3.043 3.042 3.042 3.043
0.400 3.098 3.098 3.044 0.440 3.140 3.140 3.042 0.480 3.184 3.184 3.043 0.520 3.229 3.229 3.042 0.560 3.276 3.276 3.043 0.600 3.325 3.324 3.045 0.640 3.374 3.373 3.045 0.680 3.425 3.423 3.046 1.160 4.111 4.111 4.469 1.200 4.169 4.168 4.470 1.240 4.225 4.225 4.468 1,280 4.281 4.282 4.468 1.320 4.337 4.338 4.468 1.360 4.393 4.394 4.468 1.400 4.448 4.449 4.467 1.440 4.504 4.504 4.468 1.480 4.558 4.560 4.466 1.520 4.614 4.615 4.467 1.560 4.671 4.671 4.468 1.600 4.727 4.728 4.468 1.640 4 -784 4.785 4.467 5.442 5.798 2.080 5.442 2.120 5.502 5.502 5.798 2.160 5.563 5.562 5.799 2.200 5.623 5.623 5.799 2.240 5.683 5.684 5.798 2.280 5 .'I46 5.745 5.800 2.320 5.809 5.808 5.800 2.360 5.874 5.873 5.800 2.400 5.940 5.939 5.799 2.440 6.009 6.010 5.798 2.480 6.083 6.084 5.798 p K l m = 3.043. pKzm = 4.468. pKsm = 5.799. V, = 0.92414 ml. AV = 0.02705 ml. Std dev = 0.00089 pH unit. a at 25" and 1 = 0.1; * measured; calculated from Equation 9 or 10; CB,= 0.1000M, V , = 15 ml; e Cg, = 0.1013M, V , = 10 ml.
data with superimposed random errors, even when successive pK values were within one unit. The procedure was applied to three real systems; AMP5' and NADP (two pK's) and citric acid (three pK's). If successive pK's are sufficiently close together such that one or more of the end points are poorly defined (e.g. citric acid), then data in the vicinity of those end points may be included. Otherwise, one should avoid using data in regions of rapidly changing pH. For AMP, the activity coefficient functions, f l and f 2 , remain almost exactly constant a t unity during the first pK region; during the second, f l decreased from 0.999 to 0.997; f 2 from 0.997 to 0.992. The ionic strength remained constant up to the first end point, after which it gradually increased to 0.104. Similar behavior was found for f l , fz, and f3 for citric acid; the largest variation was in the third pK region as f3 decreased from 0.971 to 0.958. By the last point in Table 11, the ionic strength had increased to 0.113M. We conclude that equations of the Debye-Huckel type can make the appropriate small corrections to the activity coefficient functions with negligible error. The results of the treatment are displayed in Tables I and 11. At the bottom of Table I are given the best fit pa-
ANALYTICAL CHEMISTRY, VOL. 47, NO. 12, OCTOBER 1975
1919
point-by-point-if all other parameters were known. The point-by-point values are calculated after the whole minimization procedure is complete. The best-fit pK’s therefore Temp. ‘C Medium, DK, R K ~ P K ~ Reference are not averages of the individual values. AMP If one subtracts 0.081 (= -log YH) from each pK, the resulting concentration pK’s agree very well with literature 25 0.1 (CH,),NBr ... 6.40 a values under the same conditions of temperature, medium, 25 0.1 KC1 .. 6.30 a and ionic strength (see Table 11). 25 0.1 KNO, 3.81 6.21 b Citric acid is a particularly interesting example in that 25 0 . 1 NaC10, ... 6.14 C the three pK’s are contained within a 2.7-unit span. As a 6.31 d 25 0 . 1 KN03 3.98 consequence, the titration curve is practically linear until 6.20 present 3.81 25 0 . 1 KNO, the third end point is approached. The three pK’s extractCitric acid ed by the computer program are constant to f0.002 unit (or less) over the entire pH range utilized. Part of the high 4.39 5.67 e 0.1 K C 1 2.96 8-20 precision in this system is due to the great buffering capac2.94 4.44 0.1 NaNO, 30 5.82 f ity provided by three overlapping pK’s. Unfortunately, 4.35 0 . 1 NaC10, 2.87 20 5.68 g there is considerable disagreement in the literature con4.38 5.68 h 0 . 1 NaC10, 2.96 20 cerning the pK’s of citric acid. Okac and Kolarik (IO)found 2.79 4.30 5.65 i 0 . 1 KNO, 25 that the values were relatively insensitive to temperature 4.36 2.88 0 . 1 (CH,),NCl 25 5.84 j over the range 8-20 OC. Otherwise, individual determina3.08 4.39 5.49 k 0.1 KC1 20 tions at I = 0.1M differ by as much as 0.35 unit for pK3 5.72 present 2.92 4.39 0 . 1 KNO, 25 (Table 11).From the average of the seven determinations of a E. R. Tucci, E. Doody, and N. C . Li, J. Phys. Chem., 65, 1570 each ionization constant published at I = O.lM, one calcu(1961). * M. M. Taqui Khan and A. E. Martell, J. Am. Chem. SOC., lates the pKc values 2.93,4.37, and 5.70. These are remark84, 3037 (1962). C H. Sigel and H. Brintzinger, Helu. Chim. Acta, ably closeto the values we obtain by the present treatment. 47, 1701 (1964). d M. M. Taqui Khan and A. E. Martell, J. Am. Chem. SOC.,89,5585 (1967). e Ref. 10. f R. C. Warner and I. Weber, These results show that the method described is capable J. Am. Chem. Soc., 75, 5086 (1953). g E. Campi, G. Ostacoli, M. of determining the equivalence point and pK’s with a high Meirone, and G. Saini, J. Inorg. Nuclear Chem., 26, 553 (1964). degree of precision. The procedure is based on exact mole C. F. Timberlake, J. Chem. SOC.,5078 (1964). K. S. Rajan and balance equations and takes into account the possible presA. E. Martell, Inorg. Chem., 4, 462 (1965). S. S. Tate, A. K. ence of any strong acid or base, as well as the small changes Grzybowski, and S. P. Datta, J . Chem. SOC., 3905 (1965). K . K. Tripathy and R. K. Patnaik, J. Indian Chem. Soc., 43,772 (1966). in ionic strength which occur during the course of the titration. A copy of the computer program used in this study may be obtained by writing to the senior author (J.E.S.). rameters: V,, AV, pKm’s as well as the standard deviation in pH. The third column of Table I gives the calculated ACKNOWLEDGMENT ~ Q Hfor, each point, from the best fit parameters. The stanThe authors express their appreciation to J. MuirheadH less than dard deviation of the quantity pH - ~ Q was Gould for helpful discussions. 0.0035 unit for all three systems; for citric acid, it was less than 0.001, The obtainable precision is not necessarily a reLITERATURE CITED liable measure of the accuracy of the results. The computer (1) A. Albert and E. P. Serjeant, “Ionization Constants of Acids and Bases”, analysis is based upon the assumptions that the electrodes Methuen,London, 1962, p 51. are correctly calibrated and respond in the theoretical (2) Ref. 1, p 52. (3) S. Hendrickson, Anal. Blochem., 24, 176 (1968). manner, and that the liquid junction potentials are either (4) J. C. Speakman, J. Chem. Soc., 855 (1940). negligible or remain constant from standardization through (5) S. S. Tate, A. K. Grzybowski, and S. P. Datta. J. Chem. SOC., 3905 (1965). titration. Any errors resulting from non-fulfillment of these (6) T. N. Briggs and J. E. Stuehr, Anal. Chem., 48, 1517 (1974). conditions will be assimilated into best-fit parameters. (7) L. Meites, J. E. Stuehr. and T. N. Briggs, Anal. Chem., 47, 1485 (1975). These, of course, are experimental, as opposed to computa(8) C. W. Davies, “Ion Association”, Butterworths,London 1962, p 41. (9) R. G. Bates, “Determination of pH. Theory and Practice”, John Wiley & tional, problems. Sons, New York, 1964, p 53. The last column in Table I1 merits some comment. Here (IO) A. Okac and 2 . Kolarik, Collect. Czech. Chem. Common.,24, 1 (1959). are displayed “individual” pK values calculated from the RECEIVEDfor review March 31, 1975. Accepted June 19, best-fit values of V, and AV as well as the other pK’s. 1975. This work was supported by the NIH in the form of a Thus, the constancy of these values is an indication of the research grant to JES (GM-13116). precision with which each single pK could be calculated-
Table 11. Concentration pK Values for AMP and Citric Acid at I = 0.1
.
1920
ANALYTICAL CHEMISTRY, VOL. 47, NO. 12, OCTOBER 1975
