SimuIt a neous Determination of Precise EquivaIence Points and pK Values from Potentiometric Data: Single p K Systems T. N. Briggs and J. E. Stuehr Department of Chemistry, Case Western Reserve University, Cleveland, Ohio 44 106
Linear equations based upon exact mole balance relationships are developed for the precise and simultaneous determination of pK values and equivalence points in the potentiometric determination of monobasic weak acids. The treatment also takes into account the presence of any excess strong acid or base in the sample being titrated. A computer technique is described which automatically finds the optimum parameters based upon the entire titration curve. The advantages of the new treatment are: first, it involves no mathematical approximations; second, the equations utilized are linear over the entire titration range, and third, p K values and equivalence points are evaluated simultaneously. Results are presented for several monobasic systems.
Although there are many practical applications involving the use of pK’s which do not require very accurate values, there are other cases in which a high degree of accuracy is essential. For example, in a recent study of divalent metal ion interactions with several phosphates and nucleotides, Frey and Stuehr ( I ) found t h a t variations as small as 0.005 unit in the pK of the phosphate or nucleotide yielded significant changes in the computed metal-ion binding constants. A major source of‘ the uncertainties in pK values results from difficulties in precisely establishing equivalence points in potentiometric titrations. The experimentally determined end point is often not well-defined, depending on the pK of the system. Butler (2) has discussed the subject of “titration errors” and has supplied expressions for estimating them from the “sharpness index,” which is defined as the magnitude of‘ the slope of the titration curve a t the equivalence point. Such titration errors can be avoided by obtaining the equivalence point from the major portion of a titration curve rather than from the data in the vicinity of the end point. The result will then be independent of the sharpness index. This could be done if the pK were known; but an accurate value of the equivalence point is needed to obtain an accurate value of the pK. T h e situation can be resolved by determining both parameters simultaneously. We present a method by which this can be accomplished and give examples to show the high degree of precision t h a t can be obtained. The method has also proved to be very accurate when applied to synthetic titration curves. Several methods have been used to determine equivalence points in potentiometric titrations (3-15). Anfalt and (1) (2) (3) (4) (5) (6) (7) (8) (91 (10) (11)
C. M. Frey and J. E. Stuehr, J. Amer. Chem. Soc., 94, 8898 (1972). J. N. Butler, J. Chem. Educ., 40, 65 (1963). C. F. Tubbs, Anal. Chem.. 26, 1670(1954). R. Kohn and V . Zitko. Chem. Zvesti, 12, 262 (1958). I. M. Kolthoff and E. 6. Sandell, “Textbook of Quantitative Inorganic Analyses.” Macmillan. New York. N.Y., 1949. F. L. Hahn, Fresenius’Z. Anal. Chem., 163, 169 (1958). J. M. ti. Fortuin, Anal. Chim. Acta, 24, 175 (1961). S. R. Cohen. Anal. Chem., 38, 158 (1966). G. Gran. Acta Chem. Scand., 4, 559 (1950). C. Liteanu and D. Cormos, Talanta, 7, 18 (1960). J. F. Herringshaw, Analyst(London).87, 463 (1962).
Jagner have made a survey of the methods most frequently used (16). They divided these into four classes and concluded that systematic errors decrease, while precision increases, in the following order: 1) methods based on sigmoid curves ( 3 , 4 ) , 2) differential methods based on AE/AV plots (5-81,3) differential methods based on A V / A E plots (9-111, and 4) methods based strictly on mole balance and equilibrium equations (12, 13).The fourth class includes a) titration to a preselected emf value, b) multiparameter refinement methods, and c) Gran’s second method (12) with the Ingman-Still extension (13).The authors point out t h a t electrode potential reproducibility puts a limit on the possible precision of (a) while (b) involves complicated equations. Gran’s second method neglects contributions due to the self ionization of water. This is taken into account by Ingman and Still, but their method requires accurate prior knowledge of the pK. Johansson and Perhrsson ( 1 4 , 15) by-pass this last requirement by using points only after the equivalence point. Their functional equation is given as:
v
=
v, + K ,
’
(v, +
V)/(C,,
-
C,)
where CH = equilibrium concentration of hydrogen ions, V, = theoretical volume of base required to neutralize the weak acid, V = volume of base added, VO= initial volume of the solution, and CBP = initial concentration of base. A plot of V us. K,( VO V ) / ( C ~ O - C His) made, yielding V , as the intercept. Inspection of this equation shows that if V were altered by a constant amount, a straight line could still result. Although V , would now be in error, there would be no indication of this. In practice, such errors do arise as a result of small amounts of strong acid or base being present in the sample a t the start of the titration. This is especially likely when the compounds are salts or acid salts since the crystals do not always precipitate stoichiometrically. If there are two or more pK’s and the end points are sharply defined, the volume of base required to neutralize any such excess strong acid will be the difference between the volume of base taken to titrate the sample to the first end point and the volume used between the first and second end points. This difference, which will be called AV, will be negative if some excess strong base is present a t the start of the titration. Unfortunately, in the majority of cases in which a compound has more than a single pK, one or more of the end points may be poorly defined so that it is difficult to obtain an accurate value of AV. In single pK systems, AV cannot be determined directly. In order to further appreciate the importance of this AV, especially when dealing with substances of relatively high equivalent weight, let us consider the case of‘ nicotinamide adenine dinucleotide (NAD) which has an equivalent weight of 663. It can be seen that 1 gram-equivalent weight of a sample which is 99% pure can contain 6.63 grams of an
+
(12) (13) (14) (15) (16)
G. Gran, Analyst(London),77,661 (1952). F. lngman and E. Still, Talanta, 13, 1431 (1966). A. Johansson. Analyst(London),95, 535 (1970). A. Johansson and L. Pehrsson, Analyst(London),95, 652 (1970) T.Anfalt and D. Jagner, Anal. Chim. Acta, 57, 165 (1971).
A N A L Y T I C A L C H E M I S T R Y , VOL. 46, NO. 11, SEPTEMBER 1974
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sium hydrogen phthalate. This particular base concentration ensures t h a t the ionic strength stays constant (or very nearly so) during the entire titration. I n s t r u m e n t a t i o n . Samples were placed in 20-ml jacketed cells thermostated a t 25 O C . T h e titrant was delivered in small increments from a 2-ml Gilmont microburet which was withdrawn between additions in order to minimize diffusion. High purity nitrogen was bubbled through the solution to exclude carbon dioxide. Light-absorption was minimized by adding a red colored material to the water circulating through t h e cell jackets. (Some of t h e compounds are sensitive to light in t h e alkaline p H region.) A glass electrode (Fisher 13-639-3) and an Orion reference electrode (Model 90-01) were used with a Corning 101 digital meter t o measure t h e p H after each addition of titrant. This combination of electrodes and p H meter proved t o be very stable and sensitive to changes in pH. Nevertheless, the electrodes were always restandardized a t the start of each titration against Sargent Welch buffers.
0
1
2
THEORETICAL For single p K systems, the following equations apply:
3
aH(V+ R - A V ) x l 0 4
c,
Figure 1. Graphical determination of Ve. 1V, and pK,
=
c, + c,,
for a synthetic
system The values used in constructing the synthetic curve were: V, = 1.1255, 1 V 0.07522, plc“ = 3.712, and Go= 0.09. Curves a-eresult from determinations based on Equation 5e and assumed AV values of 0.125, 0.095, 0.07522, 0.040, and 0.00, respectively. Curve cis the only straight line and is based upon t h e correct value of A V
impurity. If this impurity were HCl, 6.63 grams would represent approximately 0.18 mole, resulting in an 18% error in the equivalence point. Many compounds of biological importance have relatively high molecular weights and are either supplied as salts or prepared by methods which make it quite possible for the compounds to be contaminated by strong acids. Passage through an ion exchange column often does not ensure the stoichiometry of the eluted compound. It seems, therefore, that in order t o obtain accurate values of the p K and equivalence point, A V must be taken into account. This is not done in any of the methods currently in use. T h e method of determining pK’s and equivalence points which we propose is based strictly on mole balance equations and therefore falls into the fourth class of Anfalt and Jagner. No approximations are made in deriving the equations. The dissociation of water as well as dilution effects are precisely accounted for and the presence of any strong acid or base prior to the titration is determined. T h e pK and the equivalence point are determined simultaneously, and no knowledge of either is required beforehand in order to determine the other. T h e only restrictions on the method are the accuracy with which the p H meter and electrodes can be calibrated and the constancy of the liquid junction potentials. Since the titrations are carried out in a medium of constant ionic strength, the results are valid only under these conditions (e.g., I = 0.1M). One can, of course, carry out measurements a t several ionic strengths and obtain the thermodynamic p K by the appropriate extrapolation.
EXPERIMENTAL Materials. Nicotinamide adenine dinucleotide (NAD), acetyl pyridine NAD, and thiamine HC1 were obtained from Sigma as t h e free acids in t h e highest purity available (quoted purity >99% for NAD). They were stored in a desiccator below 0 “C and used without further purification. Sample solutions were prepared on t h e same day t h a t the titrations were performed. T h e dry crystals were dissolved in doubly distilled water and the ionic strength was a d justed to 0.1M by means of (alk)dNBr. Ten-milliliter aliquots were titrated with 0.1M (Fisher certified) K O H standardized us. potas1518
A N A L Y T I C A L CHEMISTRY, V O L . 46,
NO. 11,
where Co is the analytical concentration of the weak acid, CBOis the initial concentration of the base, CB, CH, COH, CHA,and CA are the equilibrium concentrations of the cation of the base, proton, hydroxide ion, acid, and the anion of the acid and U H is the activity of hydrogen ions. K1 is the reciprocal mixed ionization constant of the acid, i.e. l/Kam, where K,” is the mixed ionization constant defined as a H C A / C H A . V , and V, are, respectively, the total volume of the solution and the theoretical volume of base required to neutralize the weak acid. With suitable manipulations, the above equations yield, without approximations:
+
+
V, (5a) V + R - A V = -a,(V R - AV)KI or its equivalent form: K , = (Ve - V - R A\’)/(V + R - AV)% (5b)
+
where V is the volume of base added, AV is the volume of this neutralized by any strong acid present, and R = (CH C O H ) vt/cBW
+
Equation 5a is of the form y = mx 6. Therefore if AV were known, a graph of the left hand side of Equation 5a us. -aH( V R - AV) would yield values of K1 and V, directly as the slope and intercept of a straight line. Since AV is not known, several plots can be made using different estimates of AV. That which gives the best straight line is taken as correct, and K1 and V , are determined from this line. This method was tested on a synthetic system having pKm = 3.712, V, = 1.1255, and AV = 0.0752. From these values, a synthetic titration curve (pH us. volume) was generated. T h e “data” so generated were treated uia Equation 5a with a variety of AV values. Five of the resulting sets of points are shown in Figure 1. As expected, only the correct AV value yields a straight line for all the points in the titration. Curves a and b show the curvature obtained when 6V is too large; d and e, when AV is too small. The curvature is most apparent far from the equivalence point, Le., nearer the beginning of the titration. Near the equivalence point, each line appears to be almost linear. Actually, the slopes of all lines except c are not constant. Each yields a series of pK values which display a systematic drift. The computed equivalence points (intercepts) for curves a, b, d, and e are characterized by quite large errors. Curve c, on the other hand, yielded the precise p K and V , values which were assumed in generating the data.
SEPTEMBER 1974
+
.
fl
F i t data t o eq(5a) by l e a s t squares
c I
,
Compute Ki
I r IIf SD,
I
\I
R
.r”i
I
2 SD, , , Stop
Figure 2. Schematic diagram of the computational steps in obtaining V,, A V, and K,
While this method gives good results, it is very time consuming. A computer program was therefore constructed to determine the best values of K1, V,, and A V using Equations s a and 5b. T h e program generates many pairs of V , and K1 values by substituting different estimates of AV into Equation Sa, and determines the best straight line by the method of least squares. T h e assumed value of AV and the resulting value of V , are then used with Equation 5b to generate a set of K1 values corresponding to the set of titration points. The mean value of each set and the standard deviation are also determined. The most probable values of K 1 , V,, and A V are those corresponding to the set of K i values which gives the lowest standard deviation. In principle, the most probable values of V,, AV, and K1 can be found by using a very large number of estimates of AV. However, the results can be achieved more rapidly if these are chosen in a systematic manner. Figure 2 is a diagrammatic representation of the computational procedure (A copy of the complete program may be obtained by writing to the senior author, J.E.S.). Unless there is reason to expect otherwise, the first estimate of A V can be set a t zero. T h e value of the initial estimate of AV has no bearing on the final values, only on the speed with which they are obtained. T h e second estimate is made by adding an arbitrary increment to the first. Subsequent estimates are obtained by adding a calculated increment to the preceding estimate. Its value depends on a comparison of the standard deviations of successive sets of Kl’s. The standard deviation of any set of Kl’s can be (a) smaller than, (b) greater than, or (c) virtually equal to the standard deviation of the preceding set. If (a) is true, the new increment is made equal to the preceding increment. If (b) is true, the new increment is obtained by multiplying the preceding increment by the factor -0.45. T h e effect of this procedure is to change the values of A V in the direction which gives smaller standard deviations. If (c) is true, the procedure is terminated and the current values of V,, AV, and K1 are taken to be the most probable values. T h e procedure described has a t least five advantages over previously published methods. First, it optimizes the values of quantities (V,, AV,K1) involved in an exact therm o d y n a m i c t r e a t m e n t . (It is necessary to distinguish between mathematical treatment, which involves exact relationships with no approximations, as opposed to the actual carrying out of an experiment. The latter involves several assumptions or restrictions, such as proper electrode standardization, constant liquid junction potentials during the course of the titration, and the use of a n approximate value
for the activity coefficient of the hydrogen ion.) Second, the method differs from other multivariable treatments in that the parameters are treated not as independent, but rather as inter-related variables. T h a t is, the optimum values of AV and V , are giuen by the best values of K1, which is the only parameter t h a t is optimized. Third, the experimenter has a table of point-by-point pK values printed out for the course of the whole titration. He can determine by simple inspection whether the values exhibit a drift or vary randomly, and how much each point differs from the average value. I t is important to be able to examine the data over the course of the titration, because even a relatively small drift in p K values may be symptomatic of serious errors in the determination. Fourth, since the technique is not dependent upon data near the end point, the attainable precision is independent of the sharpness index. Finally, the value of AV is a useful measure of the “proton purity” of the compound being titrated, in the sense that it determines how much strong acid or base was present a t the start of the titration. For a pure acid or acid salt, of course, A V should be statistically indistinguishable from zero. A particularly simple situation results if in fact A V = 0 (no strong acid or base present in the untitrated sample). In t h a t instance, Equation 5a becomes
V
+R
=-
%(V
+
+
(6) R)Ki + V, R ) now yields K1 and V ,
A single graph of V i-R us. aH( V directly, utilizing points in the entire titration curve.
RESULTS AND DISCUSSION The computer program described above was used to determine the pK’s and equivalence points of NAD, benzoic acid, acetyl pyridine NAD, and thiamine HCl. Typically, 20 data points, corresponding to volumes between 5% and 90% of the equivalent volume, were used. Data in the end-point region of a titration are often the least accurate; they were not used in the analysis. The results, listed in Table I, show the high degree of precision that is obtainable. T o ascertain t h a t results obtained by this method were indeed independent of the sharpness index, a series of synthetic curves ( p H us. volume) were generated for pKm values between 1.7 and 12.7 with the same A V and V, values. Random errors were then placed on the pH values (0 to f 0.002 unit) and on the volumes (0 f 0.0002 ml). The resulting numbers were then analyzed by the program outlined above for pKm, AV,and V,. For each system, we used all points in the titration curve except those in regions of
A N A L Y T I C A L C H E M I S T R Y , VOL. 46, NO. 11, SEPTEMBER 1974
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Table 11. Results f o r S y n t h e t i c Systems
Table I. Potentiometric Titration D a t a f o r Monobasic Acidsn pHC
Vh
PK
Vb
Benzoic acidd 3.430 3.555 3.675 3.791 3.904 4.011 4.122 4.230 4.349 4.474 4,620 4.788
0.05 0.10 0.15
0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60
4.117 4.116 4.115 4.115 4.116 4.114 4.117 4.114 4.117 4.116 4.119 4.113
CBo = 0.1. A v = -0.009, ve
0.7414. pKm = 4.116 zt 0.003. (15-ml sample)
Assumed valuesa
PH‘
PK
NADd 0.05
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
2.850 2.927 3.002 3.080 3.154 3,230 3.303 3.375 3.445 3.516 3.586 3.656 3.727 3.799 3.874 3.950 4.036 4.124 4.225 4.337
3.675 3.679 3.677 3.678 3.675 3.676 3.675 3.675 3.674 3.675 3.676 3.676 3.677 3.677 3.678 3.676 3.679 3.677 3.677 3.673
C B ~= 0.0991. AV = 0.05501. = 3.676 zt 0.003.
V, = 1.1332. pK”
Acetyl pyridine NADe 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36
3.187 3.241 3.298 3,355 3.410 3.466 3.526 3.589 3.648 3,709 3.772 3.839 3.911 3.986 4.063 4.148 4.244 4.362
3,673 3.673 3.675 3.676 3.673 3.670 3.672 3.676 3,674 3.673 3.672 3.673 3.675 3.677 3.674 3.672 3.670 3.675
C B ~= 0.101. AV = -0.0152. Ve = 0.4589. pKm = 3.674 zt 0.0035.
Thiamine HCIe 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36
3.809 3.953 4.081 4.188 4.276 4.358 4.438 4.513 4,583 4.644 4.705 4.767 4.829 4.890 4.947 5.005 5.064 5.133
4.896 4.902 4.909 4.908 4.900 4.897 4.899 4.902 4.904 4.900 4.89e 4.899 4.902 4.904 4.902 4.900 4 .e99 4,906
C B ~= 0,101. AV = -0.0056. V, = 0.5837. pKm = 4.901 0.0065.
+
a At 25 O C and I = 0.1 M; volume of added base; a H converted to CH by Y H = 0.83; supporting electrolyte = (Et)rNBr; e supporting electrolyte =
pKm
1.7 1.7 2.7 5.7 8.7 11.7 12.7
V,,ml
0.9* 0.9 0.9 0.9 0.9 0.9 0.9
Best fit values
AV, ml
pKm
V e ,ml
AV, ml
0.0 0.0 0.0 0.0 0.0 0.0 0.0
1.698 1.691 2.698 5.700 8.700 11.700 12.706
0.902 0.926 0.902 0.900 0.900 0.900 0.906
-0.002 -0.024 -0.002 0.000 0.000 0.000
0.001
Unless otherwise stated, the titrant was 0.1M base, V u = 10 ml (correaponding to CJ = 0.009N) at 25’ and an ionic strength of 0.1. Random errors, as described in the text, were put on all pH and V values. Titrant = 1.OM base, VU= 10 ml.
The pattern of the curves is similar to that of the synthetic system described above. Only the center line yields p K values which do not drift either upward or downward during the course of the titration. As in the synthetic system, if one were to ignore points far from the equivalence point for curves, a, b, d , and e, one would obtain results that showed only small drifts in the pK. However, the errors in V , would be relatively large. This shows the possible error associated with the methods that use only points in the vicinity of the equivalence point. As pointed out above, the determination of metal binding constants is very sensitive t o errors in the p K and concentration of the acid. This sensitivity can in fact be used to test the accuracy of our results. T h e coenzyme NAD was titrated in the presence of Ni2+ a t two different ratios of metal to NAD and the data were analyzed for the complexes NiNAD and Ni(NAD)*. T h e metal-ion binding constants of these two complexes are defined as follows
=
K,L
[NiNAD] [Nil[ NAD]
and KML2
=
[NiNAD2]
[NiNAD][ NAD]
where the charges have been omitted. If M o and Lo are used to represent the analytical concentrations of Ni2+ and NAD, and L represents the equilibrium concentration of the deprotonated form of NAD, then these constants may be obtained from the following equation ( I )
(MehNBr.
rapidly changing pH. T h e latter includes the end-point region for low p K systems. For very weak acid (pH > g), data through the end-point region can (and for greatest accuracy should) be used. The results are shown in Table 11. We notice first that all systems returned, to a high degree of accuracy, the values of the parameters used to generate the data. More importantly, about the same accuracy was obtained for the hypothetical very weak acid (pKm = 12.7) as for the strong acid (pK = 1.7). T h a t is, the accuracy of the method is independent of the “sharpness index” of the system being titrated. T h e reason for this is obvious: the method does not depend a t all on the inflection a t the equivalence point, but rather uses the constancy of the computed p K to fix the equivalence point. T h e sensitivity of the data to a “best fit” is the same for a high p K as for a low pK. In addition, the pK and equivalence point of NAD were determined graphically by means of Equation 5a. Figure 3 shows the computed curves based on five estimates of AV. 1520
9
where A = L , / L - (1
+
a&)
.
Ideally, a graph of Y us. X should be a straight line, with data from different MolLo ratios superimposing. The results are plotted in Figure 4 for the value of A V as deduced from Figure 3. Points having = 1:1 are represented by filled circles and those with M0:Lo = 2 : l by open circles. T h e plots for both metal to ligand ratios do superimpose and are straight lines as well. This indicates that the values of K1 and V , used are correct. Additional sets of points based on different values of A V (and consequently K1 and V,) have also been plotted. In the upper curves, AV is taken as zero. This results in a value of V , that is too high. The plots give lines which curve upward and d o not superimpose for different M,&o ratios. The lower sets are based on a value of SV that is too large. Similar results are obtained except that the plots curve downward.
ANALYTICAL CHEMISTRY, VOL. 46, NO. 11, SEPTEMBER 1974
lo/
0
1
2
3
aH(V+ R-AV)x104
x x10-2
Figure 3. Graphical determination of the p K and equivalence point in a titration of NAD with 0.09991MKOH according to Equation 5a The curves correspond to the following values of ml: (c)0.075 ml; (d) 0.035 ml; and (e)0.00 ml
A k ( 8 ) 0.1 15 ml: (6)0.085
Figure 4. Graphical determination of the binding constants for the interaction of Ni2+ with NAD Data are graphed via Equation 7 for pKm and V. values computed (Equation 5)for the following A Vvalues: 0.00 ml (upper curves): 0.075 ml (center line): 0,115 ml (lower curves). Open symbols correspond to I%& = 2 , closed symbols to &/Lo = 1
The discrepancies noted in determining metal-ion binding constants are not so evident in the determination of pK's. Figure 3 shows that variations in AV result in only small changes in the slope for a large part of the titration curve. One would observe then only a small drift in the pK values calculated for each point on the titration curve. However, the effect of AV on the intercept (Le., V,) is quite significant. T h i s means t h a t a small drift in p K values could correspond t o a relatively larger error in t h e concentration of t h e sample. I t shows the danger in accepting as correct an average of pK values that show even a small drift over the titration range. For example, the manual prepared by Albert and Serjeant (27) contains a number of workedout potentiometric pK determinations. A table of pK values for one of these, boric acid, displays a progressive drift of 0.06 pK unit. Application of our method to their results clearly indicates an error in the concentration of boric acid, i.e., in the equivalence point. When this is corrected, the data yield pK values with no trend and constant t o 5 0.01 2 unit over the entire range. In conclusion, we have demonstrated the importance of considering the possibility that some strong base or acid
may be present when determining the concentration of a weak acid. The method we developed to take this into account inherits the degree of accuracy already credited to other methods based on mole balance equations. In addition, it takes into account all the common factors, one or more of which has been overlooked in other methods; yet it remains simple. The use of a large number of points over virtually the entire titration curve tends to minimize the effect of random errors. In contrast to conventional methods which use the inflection a t the end point, this method uses the constancy of the p K values to locate the equivalence point. At the same time, the treatment reduces the only sources of systematic errors to the concentration of base and to the accuracy with which the pH system can be calibrated. The former, however, can be determined to a high degree of accuracy. This treatment is being extended to multiple pK systems.
(17) A. Albert and E. P. Serjeant, "Ionization Constants of Acids and Bases,'' Methuen, London, 1962, p 33.
1974. This work was supported by the NIH in the form of a research grant to JES (GM-13116).
RECEIVEDfor review January 30, 1974. Accepted May 20,
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