Environ. Sci. Technol. 1992, 26, 1844- 1846
Automatic Electrode Correction in Binding Site Analysis James R. Kramer," Plerre Brassard, and Pamela V. Collins
Department of Geology, McMaster University, Hamilton, Ontario L8S 4M1, Canada In a previous paper ( I ) , we showed the validity of using a linear programming (LP) optimization procedure to obtain binding site densities and dissociation constants. This procedure assumes that multiple sites in a heterogeneous matrix can be represented by additive monoprotic or monometal sites. Since then, this technique has been applied successfully to determine H+ binding sites on humic matter (2,3),on kaolinite (4), and for titration data from the US. National Acid Precipitation Program (NAPAP) (5). For aquatic acid (dis)association studies, the electroneutrality expression is used to obtain binding sites: [baseli - [acid]; + [H+Ii- [OH-Ii = C[ligand-Ii - [ANC] (1) where [baseIi and [acidIi are the base and acid titrant added for the ith data set. For the j t h dissociation constant, K,, [ligand-Ii = Cjaij, aq = Kj/(Kj + [H+Ii),and [ANC] is the initial difference in cations and anions that do not react with [H+] and [OH-]. All terms are adjusted for dilution as required. Equation 1 is made linear with respect to [H+]by choosing a discrete interval and limits for Kj so that Y = -can,+ C1al Czaz+ ... Cna, (2) where Y = [base] - [acid] + [H+] - K,/[H+] and C, is the derived ANC. [H+] is determined using a glass electrode which has been calibrated either by titration of a blank solution or by using appropriate calibration buffers. Both calibrations follow an equation of the sort emf = a, a, log [H+],where emf is the meter reading and a, and al are fitting parameters for a constant temperature and ionic strength. This analysis considers the bias involved in electrode calibration at high and low pH's which would add error to the binding site equation. Specific ion electrodes and more specifically pH electrode systems show a bias deviation from a linear log [H+] response at both low and high pH's. Figure 1shows this effect as a deviation from a linear log [H+]fit for a precise titration of distilled water at constant ionic strength and temperature. This deviation has been attributed to a junction potential effect dependent upon H ion concentration (6). Large OH- concentrations also induce the same effect. [The electrode selectivity coefficient for H ion is still high enough at high pH (low [HI) and the highest ionic strength to suppress possible deviation from Nernstian linearity.] Nonlinear iterative curve-fitting techniques have been proposed to adjust deviations from linear log [H+] response (7). The deviation from log [H+] linearity at low and high pH's is a major problem in this binding site analysis and many other calculations because the expressions manipulate [H+]or [OH-] terms, which become significant at these extremes. For example, stability constants obtained at low or high pH by potentiometric methods could be affected by this deviation from linearity if [ligandIi,eq 1,is not large compared to the electrode error. Examination of eq 1 reveals this effect. Thus a small logarithmic bias can result in a large concentration bias, which will result in artifacts in the analysis of binding sites. Using the procedure de-
+
+
* To whom correspondence should be addressed. 1844
Environ. Sci. Technol., Voi. 26, No. 9, 1992
veloped previously ( I ) , we find that a major artificial binding site can be induced as a minimum or maximum boundary pK as shown for sodium carbonate (Figure 2a) and citric acid (Figure 3a). The approach used to compensate for this electrode bias is to consider deviation from log [H+] linearity a linear function of [H+] and [OH-] with coefficients & and &, respectively. The reformulation of eq 1 and 2 becomes Y = [base] - [acid] = &[OH-] - P1[H+]- [ANC] + C[ligand-1; (3) which in linear form becomes
y = BzKw/[H+l - P,[H+I -
emc+ c,a, + czaz + ... Cna,
(4)
Thus the LP optimization routine gives values for P,, &, C,, C1, C2,... C, from input of titrant-pH data and a value of K, at a specific temperature and ionic strength. The procedure assumes that the electrode response is fitted precisely to data in the mid-pH range (e.g., buffers at p H s 4,7, and 10) so that deviations are random and small. The effect of correction of the [H+]and [OH-] terms would be influenced by the low and high pH values, that is, where they are large values. At the same time, corrections of [H+] and [OH-] will have little influence on eq 4 for the midrange pH values due to the relatively small values of [H+] and [OH-]. Equation 4 is solved by the dual simplex linear programming optimization method used previously by us (1). We used the computer program of Best and Ritter (8) for the primal and dual solutions. Examining eq 4, the LP primal problem is defined as follows: Minimize lCeil such that ti + CCjaij - C,, - &[H+]i + &K,/[H+]i I Yi i=l,m ci - CCjaij
+ ,C + pl[H+]i - &K,/[H+]i
1 -Yi
i=l,m CjIO j=l,n where ti is the error for the ith term. The dual problem is as follows: Maximize ICYiI such that
CUiaij+ YzWj = '/zCaij CUi[H+Ii- f/zWj = 1/2C[H+]i
j =1, n
j =n
+1
CUiKw/[H+li+ yzWj = f/zCKW/[H+li j = n
XUi = m / 2
j =n
+2
+3
where (Vi ... Wi)is the solution vector in the dual transformation. A similar series of LP equations can be developed for a metal titration where a mass balance expression (see ref 1, eq 5) is employed in place of eq 4. Testing with Simulated and Real Data The Tanager autotitrator system was used for simulation tests and titrations. First titration data were simulated without and with an electrode calibration bias. In addition, these data were assessed with/without a random unbiased
0013-936X/92/0926-1844$03.00/0
0 1992 American Chemical Society
i
10
0
1
0051
I -*
;
-011
3
-0 15'
2
4
3
5
6
8
7
9
10
12
11
PH
PK
Figure 1. pH electrode calibration deviation from linearity of emf = a log [H'] for distilled water at 20 "C and 0.05 N KN03 ionic strength. Plot has been normalized to average zero deviation between pH 5 and 9 to obviously show the bias of electrode response. Some of the deviation at high pH may be due to COPcontamination in base tltrant.
a.
+
1
I
2
3
ML".I:
1-
co
4
PK
Figure 3. LP reduction of data for (a, top) uncorrected and (b, bottom) corrected electrode data for citric acid. See Figure 2 for explanation. Table I. Summary of Recovered and Expected Values with LP Reduction of Titration Data
PK
+2
Na2C03 found expected
060.6 ;
'
30
C,
U
E
06-
2
C
55 0
s 02
t
i....;-...;;1\' 1
0 04 4-
5
\
0
3
10
+
-
4
5
8
7
B
D
10
11
0
PK
Figure 2. LP reduction of data for (a, top) uncorrected and (b, bottom) corrected electrode data for NaPC03. The curve with (+) is for A Y , whlch is the difference between YaCt,, and Ye, from eq 4.
error added to the pH's of the simulated data. For a perfectly linear log [H+]electrode calibration simulation, PI and Pz should be one with recovery of the pK-concentration spectra in the modified LP determination. Analysis of the simulation set with electrode bias at the low and high pH's should recover values of PI, P2 # 1. In addition, the pK-concentration spectrum should be recovered without modification or introduction of artifacts. All of these tests were achieved by analyzing simulated data with/without an electrode bias. Experimental data were assessed using Na2C03 and citric acid. Both studies were carried out a t an ionic strength of 0.05 N KNOBat 20 "C. Concentrations of 1 mmol/L Na2CO3 and citric acid were titrated with a Tanager autotitration system. The citric acid solution was adjusted to a pH of lo+. Both solutions were titrated at 0.1 pH increments to a pH of less than 3, with additions made after stability was reached. The LP analysis was
(mequiv/L) uncorrected corrected 617
Pz
PKI uncorrected concn (mequiv/L) PKI corrected concn (mequiv/L) PK, uncorrected concn (mequiv/L) PKZ corrected concn (mequiv/L) PK3 uncorrected concn (mequiv/L) PK3 corrected concn (mequiv/L)
citric acid found expected
2.10 1.92 1.2349, 0.8659
2.00 2.00
3.56 -0.062 1.1049, 0.9228
0.0 0.0
6.25 0.966
6.29 1.00
3.14 0.505
3.05 1.00
6.23 0.941
6.29 1.00
3.04 0.844
3.05 1.00
10.08 1.01
10.01 1.00
4.44 0.992
4.49 1.00
10.10 1.05
10.01 1.00
4.43 0.991
4.49 1.00
5.81 0.917
5.93 1.00
5.79 1.09
5.93 1.00
carried out between pK 3 and 11and for a pK interval of 0.2. The analysis was done both with and without electrode correction. Citric acid was chosen as it has a number of low pKs, requiring titrations to acidic conditions. Similarly, sodium carbonate has a high pK site. Figure 2 shows the results for sodium carbonate, and Figure 3 shows the result for citric acid. Table I summarizes various fundamental parameters. Concentrations are obtained by summation of adjacent pK peaks. The pK is obtained from the concentration-weighted average of adjacent ICs. The "expected" pK's of Table I are obtained for Na2C03 at 20 "C from Harned and Davies (9). The p K s at 20 "C Environ. Sci. Technol., Vol. 26, No. 9, 1992
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Environ. Sci. Technol. 1992, 26, 1846-1847
for citric acid are adjusted from values at 25 “C (IO),assuming AH and AS are constant over the small temperature difference. All “expected” pK’s were adjusted for ionic strength using the Guntelberg expression (11). Figures 2a and 3a show that an anomalous boundary pK was obtained for both solutions when the conventional LP technique was used. The citric acid boundary artifact is quite significant. In both cases, however, when the electrode calibration was invoked, the boundary artifacts disappeared, and concentrations and pK values similar to those expected were obtained (Figures 2b and 3b). Table I shows that the electrode effect on the recovery of the carbonate pK‘s is small both for concentration and pK displacement. The electrode effect for the first pK of citric acid, however, is large particularly for the modification of the concentration. This is undoubtedly due to the first pK being in the area where one would expect an electrode bias.
Conclusion We have shown that a LP expression, modified to consider a linear bias of electrode response at low and high pH values can correct for the bias and produce acceptable acid dissociation constants for sodium carbonate and for citric acid. Furthermore, simulation runs show that data without electrode bias are also recovered. We therefore recommend the use of this modified algorithm for the general case.
Registry No. Na2C03,497-19-8; H’, 12408-02-5; citric acid, 77-92-9.
Literature C i t e d (1) Brassard, P.; Kramer, J. R.; Collins, P. V. Enuiron. Sci. Technol. 1990,24, 195. (2) Kramer, J. R.; Brassard, P.; Collins, P. V. Water, Air, Soil Pollut. 1990, 46, 199. (3) Clair, T. A,; Barlocher, F.; Brassard, P.; Kramer, J. R. Water, Air, Soil Pollut. 1990, 46, 205. (4) Kramer, J. R.; Collins, P. V.; Brassard, P. Mar. Chem. 1991, 36, 1. (5) Kramer, J. R.; Brassard, P.; Collins, P. V.; Clair, T. A.; Takats, P. In Organic Acids in Aquatic Ecosystems: Perdue, E. M., Gjessing, E. T., Eds.; John Wiley and Sons, Ltd.: London, 1990; pp 127-139. (6) Ingman, G.; Johannsson, A.; Johannson, S.; Karlson, R. Anal. Chin. Acta 1973, 64, 113. (7) May, P. M.; Williams, D. R. In Computational Methods for the Determination of Formation Constants; Leggett, D. J., Ed.; Plenum Press: New York, 1985; pp 37-70. (8) Best, M. J.; Ritter, K. Linear Programming; Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1985; Chapter 6. (9) Harned, H. S.; Davies, R. J. Am. Chem. SOC.1943,65,2030. (10) Martell, A. E.; Smith, R. M. Critical Stability Constants; Vol. 3, Plenum Press: New York, 1977; p 161. (11) Stumm, W.; Morgan, J. J. Aquatic Chemistry, 2nd ed.; Wiley-Interscience: New York, 1981; p 135. Received for review March 30,1992. Revised manuscript received May 6, 1992. Accepted May 11, 1992.
CORRESPONDENCE Comments on “Stoichiometry and Kinetics of the Reaction of Nitrite with Free Chlorine in Aqueous Solutions” SIR: The recent paper by Diyamandoglu, Marifias, and Selleck ( I ) contains an error in its conclusion that the reaction between nitrite and free chlorine proceeds by reaction between molecular HN02 and HOC1. This assumption was based on their observation that the rate of nitrite oxidation is proportional to [H+], whereas the equation proposed by Lister and Rosenblum (2) was perceived to contain no pH dependence: dA/dt = -kA[HOCl] (4) Here A is the total nitrite concentration. In fact, this equation does contain a pH dependence in the range studied (pH 9.7-11.4), where the HOCl ionization constant KA is much larger than [H+],as seen by their eq 6: [HOCl] = [H+]C/([H’] + KA) [H+]C/KA (6) where C is the total free chlorine concentration. Substituting eq 6 into eq 4 yields dA/dt = -kAC[H+]/K, (4’) This equation is completely consistent with Lister and Rosenblum’s suggestion that the reaction occurs only between nitrite ion and undissociated HOC1. Indeed, if the reaction were to occur between the undissociated forms of both acids as proposed, one would expect a second-order 1846
Environ. Sci. Technol., Vol. 26, No. 9, 1992
dependence on [H+]in the pH range studied (well above both ionization constants), contrary to the result observed. Furthermore, the proposal of Diyamandoglu et al. is contrary to general experience in aqueous oxidation chemistry that protonation of oxidants tends to make them more reactive while protonation of reductants reduces their reactivity. For example, reaction between ammonia and chlorine occurs only between unprotonated NH, and undissociated HOCl (3), and oxidation of nitrite by ozone occurs only with nitrite ion and not with HN02 (4). Kinetically it is impossible to distinguish between a mechanism involving HOCl + NO, and one involving OCl+ HN02 using the current data at high pH. However, chemical intuition and Occam’s razor tell us that the former mechanism is much more likely. The authors in fact use the latter mechanism as a basis for the calculation of Itl, rather than a reaction between two neutral molecules. Assuming the first mechanism is correct and a pKA of 7.58 for HOCl at 20 “C, I calculate a rate constant of about 1.0 X lo5 M-’ s-l from the authors’ data for reaction between molecular HOCl and NO2- ion. This is a much more reasonable value than a kl of 1.9 X lo9 M-’ s-l calculated for OC1- + HN02 because the rate constant is expected to be well below the diffusion-limited value. The two pathways can be distinguished by kinetic measurements in the pH range 3.5-7, where the first mechanism predicts an observed rate essentially independent of pH but the second predicts a maximum in rate at about pH 5.3. Registry No. Chlorine, 7782-50-5; nitrous acid, 7782-77-6; hypochlorous acid, 7790-92-3.
0013-936X/92/0926-1846$03.00/0
0 1992 American Chemical Society
