Anal. Chem. 1996, 68, 3973-3978
Derivative Analysis of Potentiometric Titration Data To Obtain Protonation Constants Jian-Feng Chen, Yuan-Xian Xia, and Gregory R. Choppin*
Department of Chemistry, The Florida State University, Tallahassee, Florida 32306-3006
A methodology is described to calibrate glass electrodes and to analyze potentiometric titration data to calculate protonation constants. The analysis uses the variation of dV/dpH with titrant addition in terms of two physical parameters which involve the concentrations of H+, OH-, and HmA. The data for titration of acetic acid and 8-hydroxyquinoline in 0.10-5.0 m NaCl media are analyzed by this method to obtain the stoichiometric protonation constants of the acids, the ionization constants of water, and the parameters s and b in the pH electrode calibration equation, pcH ) spHm + b, where pcH ) -log [H+], and pHm is the pH meter reading. The measurement of pH by conventional glass electrodes is a continuing problem in the study of the chemistry of concentrated electrolyte solutions.1-3 Although pH reference solutions can be used as primary or secondary standards for pH electrode calibration for dilute solutions, their value for concentrated solutions is limited by the significant differences in the activity coefficients of the reference solutions and the test solutions. In addition, the liquid junction potential for the common glass combination electrodes causes further problems.4-8 The latter problem is increased if solutions of salts other than potassium chloride are used to avoid precipitation in the interface. Various methods9-14 have been proposed to calibrate glass electrodes to correct for the liquid junction potential. To avoid concerns related to the activity coefficients in concentrated solutions, often attention is devoted to the measurement of pcH ()-log [H+]). A simple and convenient approach in this case is to measure several solutions of known concentrations of strong acid or base, assuming complete deprotonation in the solutions. However, there remains the difficulty of obtaining reproducible meter readings (pHm) in the neutral range. In low (pH < 4-5) and high (pH > 9-10) pH regions, linear correlations are obtained, but often these linear calibration curves for the two regions do not coincide. To extend the calibration over the entire (1) Byrne, R. H.; Breland, J. A. Deap Sea Res. 1989, 36 (5), 803. (2) Byrne, R. H.; Baldo, G. R.; Thompson, G.; Chen, C. T. A. Deap Sea Res. 1988, 35, 1405. (3) Baldo, G. R.; Morris, M. J.; Byrne, R. H. Anal. Chem. 1985, 57, 2564. (4) Kristensen, H. B.; Salomon, A.; Kokholm, G. Anal. Chem. 1991, 63, 885A. (5) Knauss, K. G.; Wolery, T. J.; Jackson, K. J. Geochem. Cosmochim. Acta 1990, 55, 1519. (6) Mesmer, R. E.; Holmes, H. F. J. Solution Chem. 1992, 21, 725. (7) Bates, R. G. Determination of pH; Wiley: New York, 1973. (8) May, P. M.; Williams, D. R. Talanta 1982, 29, 249. (9) Hedwig, G. R.; Powell, H. K. J. Anal. Chem. 1971, 43, 1206. (10) Avdeef, A.,; Bucher, J. Anal. Chem. 1978, 50, 2137. (11) Irving, H. M.; Miles, M. G.; Petti, L. D. Anal. Chim. Acta 1967, 38, 475. (12) Powell, H. K. J.; Curtis, N. F. J. Chem. Soc. B 1966, 1205. (13) Yamazaki, H.; Sperline, R. P.; Freiser, H. Anal. Chem. 1992, 64, 2720. (14) Slyke, D. D. J. Biol. Chem. 1922, 52, 525. S0003-2700(96)00138-2 CCC: $12.00
© 1996 American Chemical Society
pH range, a common practice is to use buffer systems which consist of a weak acid or mixture of several weak acids.9-12 An advantage of the use of a buffer system is that the hydrogen ion concentration is not affected by acid impurities, dissolved carbon dioxide, etc. An alternative procedure involves the use of spectroscopic indicator dyes13 for calibrating glass electrodes, in which the ratio of the absorbence of the acidic form to the basic form of an indicator is used to calculate pcH from known acid deprotonation concentration constants. Both of the latter methods are successful over the entire pH range for the purpose of the calibration. However, their use in potentiometric titration has limitations. First, the value of pcH depends on knowledge of the acid deprotonation concentration constants; second, it is difficult to ensure reproducibility of the liquid junction potential in different experiments. The purpose of this paper is to propose a new methodology to analyze potentiometric titration data to obtain simultaneously information on the pHm-pcH calibration as well as on the protonation constants. Two physical quantities, Γ and Q, are defined to characterize the pH (or pcH) change of a test solution upon addition of a small amount of titrant solution. A plot of Q vs pHm or pcH can be used to reflect the protonation (or deprotonation) process. The results of analysis by this method are reported for two acids in NaCl media over a range of ionic strength. THEORY The proton mass balance during titration of a weak acid (HmA) can be written as
n j ) m - [(V - V0)(COH° - CH°) + V0([OH-]0 - [H+]0) + V([H+] - [OH-])]/[V0CHmA°] (1) and
n j)
∑i[H A] i
CHmA
(2)
V0 and V are respectively the initial volume and the total volume, [H+]0 and [OH-]0 the initial concentrations of the proton and the hydroxide ions in the test solution, CH° and COH° the concentrations of the proton and the hydroxide ions in the titrant solution, and CHmA° the initial concentration of the acid. The calibration equation is defined as
pcH ) spHm + b
(3)
Following the work of Slyke14 on buffer capacity of acid or base Analytical Chemistry, Vol. 68, No. 22, November 15, 1996 3973
solutions, a general equation can be derived by combining eqs 1-3 and taking the first derivative with respect to pHm:
Γ ) (COH° - CH° - [OH-]0 + [H+]0) + (n j0 - n j )CHmA
0.4343 dV (4) sV dpHm
Γ ) Q + [H+] + [OH-]
(5)
Q)H ˆn j CHmA
(6)
Γ is defined as
and Q as
H ˆ is an operator equal to d/d ln [H+]. As a “normalized” quantity, Γ is a measure of the test solution and is independent of the titration conditions. It can be calculated directly from experimental data if the concentration of the titrant is much higher than the sum of [H+]0 and [OH-]0 in the test solution. If this condition is not met, Γ can be obtained by an iterative calculation. The usefulness of the quantity Q derives from the fact that Q is a function of the concentration and the protonation constants of the ligand being titrated as well as of the proton concentration. The value H ˆ nj can be shown to have the relation
H ˆn j)
A(ii) - A(ij) A(11)
(7)
and m m
A(kl) )
∑∑klβ β [H
+ i+j
i j
]
(8)
i)0 j)0
where βi are the overall protonation constants (β0 ) 1 and βi ) i ∏j)1 Ki). If the interval between the two adjacent protonation constants (∆ log K ) log βi - log βi-1) is more than 1.5 (the case for the acids investigated herein), this equation can be simplified to m
H ˆn j)
∑
i)1 (K
Ki[H+] + [H ]) +
i
2
(9)
Equation 9 indicates that Q is a quasi-parabolic function of pcH, with each peak corresponding to a protonation and the peak maximum value, Qmax ) 0.25CHmA for pcH ) log Ki.14 This indicates the nature of the graphical pattern observed in analyzing the potentiometric titration data by this approach. EXPERIMENTAL SECTION Reagents. 8-Hydroxyquinoline (HOxn, Aldrich, ACS grade), sodium acetate (Aldrich, ACS grade), and 2-amino-2-(hydroxymethyl)-1,3-propanediol (TRIS, Fisher, reagent grade) were used without purification. The stock solutions were prepared by dissolving required amounts of the reagents in corresponding NaCl solutions. To completely dissolve the HOxn, the solutions were stirred for several days. The stock solutions of sodium 3974
Analytical Chemistry, Vol. 68, No. 22, November 15, 1996
hydroxide and hydrochloric acid were 1.00 M standard solutions (Fisher). All other reagents used were analytical grade. Measurement of pcH. The meter readings, pHm, were obtained with an Accumet 950 pH/ion meter equipped with a Corning Semimicro combination glass electrode. The outer sleeve of the reference cell of the electrode was filled with saturated NaCl solution. The electrode was immersed in a saturated NaCl solution and pretreated with 0.1 M HCl solution prior to use so that a stable and fast response could be assured. The glass electrode was calibrated to the NBS scale using two standard pH buffer solutions: 0.05 M potassium biphthalate buffer at pH 4.00 at 25 °C (Fisher) and 0.05 M potassium phosphate monobasicsodium hydroxide buffer at pH 7.00 at 25 °C (Fisher). Potentiometric Titration. The potentiometric titration were conducted in a 50 mL jacketed cell controlled to 25.0 ( 0.1 °C with an Isotemp constant temperature circulator system. The cell was capped with a rubber stopper with a single hole (diameter of 1 mm) open to air. Nitrogen gas was bubbled through the cell solution to remove dissolved CO2. Titrant solutions (1-30 mM NaOH in NaCl) were delivered to the cell via Teflon tubing by a Schott Gerate automatic buret. A variable volume of titrant was added to obtain an approximately constant increment of pHm in the cell. The relative error of the buret was less than (0.2% for titrant additions of larger than 1.0 mL and was estimated to be less than (10% for the addition at microliter level. The pHm readings were recorded at a fixed time interval (1 min) after addition of the titrant volume. In the buffer regions, a steady pH meter reading was reached within 30-40 s for the small-volume addition used. In less buffered regions (e.g., pcH 6-8), times longer than 1 min are required in order to obtain a constant reading. However, the data in these regions have no significance for the calculation of the quantities to be determined. The uncertainties in both volume and pH meter reading in the less buffered regions produce some scatter in the plots of Q or Γ vs pHm but have little effect on the peak position and, hence, on the log K values. Calculation. A spreadsheet program coded in the Quattro Pro (Borland version 5.0) software was developed to process potentiometric titration data. The data fittings were performed using the Optimizer function of the software. Values of dV/dpHm were calculated with eq 10, which was derived from the Lagrangian polynomial:15
( ) ( (
)
1 dV 1 = V + dpHm i pHm-1 - pHmi pHm-1 - pHm+1 i-1
)
1 1 + V + pHmi - pHm+1 pHmi - pHm-1 i
(
)
1 1 V (10) pHm+1 - pHmi pHm+1 - pHm-1 i+1
RESULTS AND DISCUSSION The potentiometric titration data were processed to calculate the electrode calibration parameter, b, the protonation constants, log Ki, and the ionization products of water, pKw. This was achieved by adjusting the three parameters to minimize the deviation, σ ) (∑(Qi(exp) - Qi(cal))2/N)1/2 (N is the number of data points) between the Q values derived with eqs 4-6 using (15) Berden, R. L.; Fatres, J. D. Numerical Analysis, 5th ed.; PWS Publishing Corp.: Boston, MA, 1993; p 159.
Table 1. Potentiometric Titration Data of 1.0 mM Acetic Acida vol
pHm
vol
pHm
vol
pHm
vol
pHm
0.000 0.260 0.540 0.772 0.996 1.248 1.486 1.726 1.936 2.162 2.344 2.542 2.748 2.954 3.184 3.404 3.506 3.624 3.722 3.820 3.920 4.020 4.090 4.162 4.230 4.268 4.308 4.350 4.390 4.424 4.448 4.470 4.494 4.514 4.538
1.814 1.848 1.885 1.918 1.953 1.993 2.034 2.079 2.120 2.168 2.209 2.259 2.315 2.379 2.460 2.552 2.602 2.666 2.725 2.794 2.875 2.972 3.052 3.148 3.257 3.326 3.405 3.496 3.589 3.669 3.727 3.781 3.839 3.890 3.948
4.564 4.588 4.612 4.636 4.656 4.676 4.690 4.706 4.724 4.742 4.768 4.786 4.806 4.826 4.846 4.872 4.890 4.908 4.930 4.938 4.946 4.952 4.964 4.968 4.974 4.978 4.982 4.984 4.988 4.990 4.992 4.994 4.998 5.000 5.002
4.011 4.069 4.128 4.186 4.234 4.283 4.318 4.357 4.404 4.451 4.520 4.570 4.629 4.693 4.760 4.858 4.933 5.019 5.139 5.192 5.249 5.298 5.404 5.447 5.518 5.573 5.634 5.670 5.746 5.785 5.838 5.887 6.009 6.086 6.174
5.004 5.006 5.010 5.012 5.014 5.016 5.018 5.020 5.024 5.026 5.028 5.030 5.032 5.040 5.050 5.060 5.068 5.084 5.094 5.104 5.112 5.120 5.126 5.134 5.140 5.148 5.158 5.166 5.174 5.182 5.190 5.200 5.208 5.216 5.222
6.285 6.427 6.870 7.337 7.817 8.106 8.296 8.423 8.607 8.674 8.748 8.804 8.909 9.014 9.157 9.271 9.345 9.461 9.524 9.579 9.619 9.654 9.679 9.712 9.732 9.762 9.794 9.817 9.842 9.864 9.885 9.904 9.925 9.947 9.958
5.230 5.238 5.248 5.260 5.268 5.276 5.286 5.296 5.308 5.318 5.340 5.370 5.406 5.430 5.490 5.544 5.590 5.636 5.732 5.810 5.890 5.952 6.074 6.196 6.312 6.416 6.524 6.628 6.734 6.850 6.978 7.102 7.214 7.344 7.610
9.977 9.993 10.013 10.035 10.050 10.062 10.080 10.094 10.113 10.129 10.159 10.196 10.238 10.264 10.320 10.365 10.402 10.433 10.492 10.535 10.575 10.604 10.653 10.696 10.734 10.765 10.795 10.822 10.846 10.872 10.898 10.920 10.943 10.966 11.005
a 15.0 mL of 1.0 mM acetic acid (10.0 mM HCl as excess acid) was titrated with 30.0 mM NaOH. T ) 25.0 °C, I ) 1.0 m (NaCl). Volume is given in milliliters.
experimental data and those calculated by eqs 7 and 8. Since eqs 4-5 contain the quantities [H+] and [OH-], [H+]0 and [OH-]0, which require calculation from the parameters to be determined, an iterative approach was used to reach a “self-consistent” result. In systems where the ligand being titrated is sufficiently concentrated or its protonation occurs in the neutral region, the value of Q is less dependent on the three parameters. The initial guess for the parameters was made with the help of graphs of Γ as a function of pHm. An assumption of s ) 1 was made in the calculation for convenience, as it was found that, when s was a variable, values very close to 1 resulted (see Table 2b). Two systems consisting of 1.00 mM acetic acid in 0.10-5.0 m NaCl and 1.00 mM 8-hydroxyquinoline in the same background electrolyte were investigated. For each titration, at least 150 points were taken, and the pHm interval for successive titrant additions was 0.03-0.1 pH unit in the buffer regions of the test solutions. This allows the value of dV/dpHm to be calculated with high accuracy by eq 10. Table 1 presents a representative set of titration data (V0 ) 15.0 mL, 1.0 mM acetic acid in 1.0 m NaCl, 30 mM NaOH as titrant). Figure 1a,b shows respectively the corresponding plots of Γ and Q against pHm. No systematic error was observed between the experimental data and the values calculated from eqs 5-8. Calibration of Glass Electrode for Hydrogen Concentration. The calibration parameters calculated from the acetic acid
Figure 1. (a) Plot of Γ vs pHm (15 mL of 1.0 mM acetic acid in 1.0 m NaCl titrated with 30.0 mM NaOH). (b) Plot of Q vs pHm corresponding to (a).
system are listed in Table 2, together with the protonation constants of acetate and the ionization products of water. Figure 2 presents two sets of the electrode calibration parameter b (see eq 3), respectively from acetic acid and 8-hydroxyquinoline titrations. Despite a deviation of (0.04 pH unit, both sets of data exhibit a consistent variation with background electrolyte concentration. Negative values of b were obtained for concentrations below ∼0.05 m, while b f -∞ as the ionic strength approached zero. This contrasts with the values of 0.088 reported by Hedwing and Powell9 and 0.2269 by Avdeef and Bucher10 at zero ionic strength for the combination glass electrodes filled with KCl solution. A semiempirical equation,
0.509xI C b ) (0.305 ( 0.021) - log + 0.208 log m + m 1 + 1.5xI 0.156m + 0.0026m2 (11) was obtained to correlate the calibration parameter b with the concentrations, m (in molality) and C (in molarity), of NaCl. After correction for liquid junction potential, the proton activity coefficients calculated with the values of b agreed well with those calculated by the Pitzer formalism.16 Two tests were made to evaluate the precision of the present method. First, the calibration parameters were calculated using the titration data over either the entire pHm region or the HAc/ Ac- buffer region (3 < pHm < 6). As listed in Table 2b, the maximium deviations in the calibration parameters from the two sets of data are (0.019, which is within the uncertainty of (0.02 of the electrode for pHm measurements. The agreement for log (16) Pitzer, K. S. Activity Coefficients in Electrolyte Solutions, 2nd ed.; CRC Press: Boca Raton, FL, 1991; p 86.
Analytical Chemistry, Vol. 68, No. 22, November 15, 1996
3975
Table 2. Values Calculated for Acetate-Sodium Chloride System and Comparison of the Calibration Parameters, the Protonation Constants of Acetate, and the Ionization Constants of Water at 25.0 °C
Table 3. Protonation Constants of 8-Hydroxyquinoline in NaCl Solutions at 25.0 °C log K1 NaCl, m
(a) Values Calculated for Acetate-NaCl Systema log K pKW NaCl, m
b
pw
lit.b
pw
lit.(27)
0.10 0.30 0.50 1.00 2.00 3.00 5.00
0.002 ( 0.001 0.062 ( 0.022 0.137 ( 0.004 0.236 ( 0.019 0.438 ( 0.007 0.643 ( 0.010 1.063 ( 0.004
4.562 ( 0.001 4.511 ( 0.003 4.499 ( 0.001 4.515 ( 0.002 4.623 ( 0.001 4.770 ( 0.003 5.130 ( 0.001
4.56c 4.51c 4.50c 4.50d 4.61d 4.76d
13.83 ( 0.02 13.71 ( 0.05 13.73 ( 0.01 13.70 ( 0.02 13.78 ( 0.02 13.93 ( 0.02 14.30 ( 0.01
13.780 13.701 13.718 13.966 14.291
0.02 0.10 0.30 0.50 1.00 2.00 3.00 5.00
log K2
pwa
lit.
pwa
lit.
9.811 ( 0.003 9.643 ( 0.034 9.564 ( 0.012 9.563 ( 0.015 9.612 ( 0.040 9.651 ( 0.014 9.722 ( 0.016 10.004 ( 0.004
9.82b,c
4.984 ( 0.008 4.920 ( 0.033 4.923 ( 0.014 4.941 ( 0.010 5.021 ( 0.008 5.284 ( 0.002 5.438 ( 0.019 5.823 ( 0.019
4.94b,c 5.10d 5.08d 5.08d 5.22e
9.60e 10.05f
5.85f
a 15.0 mL of 1.00 mM 8-hydroxyquinoline (10.0 mM HCl as excess acid) titrated with 30.0 mM NaOH. b I ) 0.01 M. c Reference 28. d Reference 29. e Reference 30. f Reference 19.
(b) Comparison of Calibration Parameters, Protonation Constants of Acetate, and Ionization Constants of Water at 25.0 °C test Ie test IIf NaCl, m
∆b
∆ log K
s
∆b
0.10 0.30 0.50 1.00 2.00 3.00 5.00
-0.019 -0.006 -0.011 -0.007 0.001 -0.003 -0.003
-0.001 0.008 0.002 0.001 -0.001 -0.002 -0.001
0.981 ( 0.002 1.001 ( 0.002 0.991 ( 0.003 1.006 ( 0.008 1.018 ( 0.004 1.012 ( 0.004 1.005 ( 0.006
0.046 -0.004 0.018 -0.015 -0.038 -0.020 -0.008
∆ log K
∆pKw
-0.019 -0.016 -0.013 -0.02 -0.006 -0.09 -0.001 0.03 0.008 0.16 0.007 0.11 0.014 0.06
a 15.0 mL of 1.00 mM acetate (10.0 mM HCl as excess acid) titrated with 30.0 mM NaOH. Data in the entire pHm region were fitted. The errors given in all the tables are the random errors estimated as the standard deviation (1σ). b Interpolated from literature data for NaCl media. c Reference 17. d Reference 18. e Test I: data in pHm region of 3-6 were used. ∆b, ∆ log K, and ∆pKw are the differences between the values obtained in the test approaches and those listed in Table 2a. f Test II: the slope s was used as an adjustable parameter in the calculations, and data in the entire pHm region were used.
Figure 2. Ionic strength dependence of the calibration parameters, b.
K values is also quite satisfactory (within (0.008). In the second test, calculations were made to include the slope, s, in the calibration equation as an adjustable parameter. As summarized in Table 2b, an average value of s ) 1.002 ( 0.018 was obtained from the 21 sets of experimental data. The deviations between the log K values thus obtained and those assuming s ) 1 are within (0.02. However, larger discrepancies in pKw values were observed. The values obtained for s ) 1 agree better with the literature data for pKw, supporting the approximation s ) 1. A scatter of (0.03 pH unit was observed for the calibration parameters obtained from triplicate experiments, which may reflect electrode drift, variation in liquid junction potential, or an effect of the glass surface. The drift in electrode response seems 3976 Analytical Chemistry, Vol. 68, No. 22, November 15, 1996
critical from our observations; i.e., the potential of the electrode immersed in a standard solution shifted from one experiment to another in a range of about 0.1-5 mV. Greater drifting was observed after the electrode was immersed in highly alkaline solution, probably due to the deterioration of the glass surface. Therefore, an error can be introduced if the calibration is conducted prior to the titration for measuring an acid constant. However, a simultaneous determination of the calibration parameter and the acid constant from the same set of titration data presumably eliminates the effect of electrode drift. The commonly used approaches, based on a direct pHm-pcH correlation, to calculate the calibration parameter b attach equal importance to each experimental point, which may not be valid. A solution in the neutral pH region has less buffer capacity, and the pH can be altered relative easily by many factors, such as acidic impurities, dissolved CO2, etc. Moreover, the ionization product, pKw, of water must be known for the calculation of pcH in the alkaline region. However, it may not be always available for concentrated electrolyte solutions. In the proposed approach, through use of the quantity Γ, a weight proportional to the buffer capacity is given to each data point, resulting in a significant reduction in the importance of the data in less buffered regions. Protonation Constants. The protonation constants of acetate calculated from the 21 sets of titration data are listed in Table 2a. Robertis et al.17 reported data for 0.04-1.0 M NaCl solutions and Belevantsev et al.18 for 1.0-4.0 M solutions. The deviations between our data and those interpolated from the literature values are within (0.013. The protonation constants of 1,8-hydroxyquinoline are listed in Table 3. The values of log K2 ) 5.823 ( 0.019 and log K1 ) 10.004 ( 0.004 for 5.00 m NaCl solution show satisfactory agreement with the data from spectrophotometry.19 Our data at low ionic strength are also in reasonable agreement with those in the literature. A general equation used to describe the ionic strength dependence of stoichiometric protonation constant has the form20 (17) Robertis, A. D.; Stefano, C. D.; Rigano, C.; Sammartano, S.; Scarcella, R. J. Chem. Res. (S) 1985, 42. (18) Belevantsev, V. I.; Mironov, I. V.; Peshchevitskii, B. I. Russ. J. Inorg. Chem. 1982, 27, 29. (19) Xia, Y. X.; Chen, J. F.; Choppin, G. R. Talanta, in press. (20) Grethe, I.; Fuger, J.; Lemire, R.; Muller, A.; T-Gegu, C. N.; Wanner, H. Chemical Thermodynamics of Uranium; NEA-TDB; OECD: Paris, France, 1991.
xI log K ) log KT + 0.509∆Z2 + ∆I 1 + 1.5xI
(12)
Table 4. Results of the SIT Analysis of the Protonation Constants log KT
∆Z2 )
∑ i
∆ )
Im
-∆a
pw
lit.b
σ
0.1-3.0 0.1-5.0 0.1-3.0 0.1-5.0 0.1-3.0 0.1-5.0
0.155 ( 0.005 0.167 ( 0.004 0.185 ( 0.013 0.183 ( 0.007 0.117 ( 0.005 0.126 ( 0.004
4.763 ( 0.005 4.753 ( 0.017 4.865 ( 0.032 4.867 ( 0.029 9.845 ( 0.012 9.837 ( 0.016
4.757
0.005 0.016 0.026 0.029 0.015 0.026
acid
and
νiZi2(product) -
∑
HAc
νiZi2(reactant)
(13)
i
∑ν (i,NaCl)(product) - ∑ν (i,NaCl)(reactant) i
i
H2Oxn+ HOxn
i
4.92 9.82
i
(14) where Zi is the electrical charge of species i, νi the corresponding stoichiometric number in the protonation reaction, and (i,NaCl) the interaction coefficient between species i and background electrolyte NaCl. Plots of log K - 0.509∆Z2[I1/2/(1 + 1.5I1/2)] versus I are linear for both of the test systems up to 5.0 m. The values of log KT and ∆ were obtained from the linear regression and are listed in Table 4. The deviations between the calculated and experimental log K values are within (0.03. Agreement of the protonation constants, log K1T, of acetate and 8-hydroxyquinoline with the corresponding literature values is satisfactory. Our constant, log K2T, for 8-hydroxyquinoline seems slightly lower than the literature value; however, a curvature was observed in the plot of the second protonation constant as a function of ionic strength at I < 0.5 m. Applying a curve-fitting to the corresponding data, the thermodynamic constant of oxine can be calculated to be 4.92 ( 0.01, which is in excellent agreement with the literature value. These results demonstrate the precision of the present methodology in processing titration data for different ionic strengths. An algorithm used in numerous computer programs (e.g. BEST,21 MICMAC,22 PROTAF,23 MAGEC,8 and MINIQAUD24) is the minimization of ∑wi(pHi(exp) - pHi(cal))2 or ∑wi(Ei(exp) - Ei(cal))2. The method to evaluate the weighing factor wi is, thus, one of the major differences among these programs. For example, the BEST program defines wi ) 1/(pHi+1 - pHi)2 to lessen the influence of less accurate pHm values in poorly buffered regions.21 Such definitions make the weighing factor highly dependent on the data structure, and an increase in the interval between two adjacent points can modify significantly the weighing factor. Therefore, equal additions of titrant solution seem necessary to obtain a valid result. In the proposed approach, the weight of each data point is proportional to the buffer capacity of the test solution, so the data in poorly buffered regions have less contribution. As pointed out by Martell and Motekaities,21 inaccurate standardization of reagents could be a source of considerable error in log K calculation when using a program such as BEST. The BEST calculation becomes increasingly sensitive to the concentrations of excess acid and titrant when the ligand concentration decreases, especially if deprotonation occurs in the neutral pcH range. This can be illustrated in Table 5, in which three sets of triplicate titration data serve as examples and the concentrations used in calculation were not standardized. As can be seen from (21) Martell, A. E.; Motekaities, R. J. Determination and Use of Stability Constants; VCH Publishers, Inc.: New York, 1988. (22) Laouenan, A.; Suet, E. Talanta 1985, 32, 245. (23) Fournaise, R.; Petitfaux, C. Talanta 1987, 34, 385. (24) Sabatini, A.; Vacca, A.; Gans P. Talanta 1974, 21, 53.
a ∆(HAc) ) (HAc,NaCl) - (H+,NaCl) - (Ac-,NaCl); ∆(H Oxn+) 2 ) (H2Oxn+,NaCl) - (H+,NaCl) - (HOxn,NaCl); ∆(HOxn) ) + b (HOxn,NaCl) - (H ,NaCl) - (Oxn ,NaCl). Reference 31.
Table 5. Protonation Constants of Acetate Calculated from Different Methods BEST NaCl, m
pw
aa
1.0
4.510 ( 0.004 4.518 ( 0.002 4.520 ( 0.003 4.772 ( 0.001 4.762 ( 0.007 4.769 ( 0.004 5.129 ( 0.024 5.129 ( 0.011 5.129 ( 0.008
4.72 4.54 4.72 4.86 4.86 4.88 5.32 5.26 5.37
3.0 5.0
bb
lit.c
4.64 4.64 4.64 4.86 4.79 4.87 5.17 5.16 5.18
4.50 4.76
a Estimated errors are (0.03. b Estimated errors are (0.01. c Reference 18.
Table 6. Calibration Parameter and Equilibrium Constant Measured Using Different Concentrations of Acetate (0.10 m NaCl at 25.0 °C) CHAc, µM
b
log K
pKw
1000a
0.001 0.002 0.003 -0.018 -0.029 -0.029 -0.010
4.572 4.563 4.562 4.581 4.578 4.579 4.627 4.57d 4.633 4.56d 4.642 4.57d
13.806 13.831 13.844 13.786 13.760 13.755 13.792
130b 60c
-0.013 -0.015
13.786 13.779
a 15.0 mL of acetate (10.0 mM HCl as excess acid) titrated with 30 mM NaOH. b 15.0 mL of acetate (1.0 mM HCl as excess acid) titrated with 3.0 mM NaOH. c 15.0 mL of acetate (1.0 mM HCl as excess acid) titrated with 1.5 mM NaOH. d Value after correction for the protonation of dissolved CO2. Total concentration of carbonate, 6 µM, log K ) 6.12.25
the table, a random error of roughly (0.006 was obtained from our approach, while the BEST program calculation gave a value of (0.1 when the original concentrations of the excess acid and acetic acid were used. An error of (0.03 was reached with BEST after the concentrations and the pcH correction parameters were refined to further minimize the standard deviation. However, the log K values are higher than those from our approach by an average of 0.12 logarithm unit. (25) Thurmond, V.; Millero, F. J. J. Solution Chem. 1982, 11, 447. (26) Izaguirre, M.; Millero, F. J. J. Solution Chem. 1987, 16, 827.
Analytical Chemistry, Vol. 68, No. 22, November 15, 1996
3977
Figure 3. Plot of Q vs pHm (15.0 mL of (a) 130 and (b) 60 µM acetic acid in 0.01 m NaCl medium titrated with 30-1.5 mM NaOH). Dashed line, without correction for carbonic acid; solid line, with correction for carbonic acid.
The influence of the inaccurate standardization of reagents on log K calculation is greatly reduced in the proposed approach. In the definition equations of Γ and Q, the reagent concentrations are linear factors of the two quantities and have less influence on (27) Busey, R. H.; Mesmer, R. E. J. Chem. Eng. Data 1978, 23, 175. (28) Paljk, S.; Klofutar, C.; Krasovec, F.; Suhac, M. Microchim. Acta 1975, 2, 485. (29) Mason, J. G.; Lipschitz, I. Talanta 1966, 13, 1462. (30) Janjic, T. J.; Pfendt, L. B. Talanta 1992, 39 (1), 55. (31) Martell, A. E.; Smith R. M. Critical Stability Constants; Plenum Press: New York and London, 1989; Vol. 6, p 273.
3978 Analytical Chemistry, Vol. 68, No. 22, November 15, 1996
the peak position (or log K) in the plot of Q vs pHm (pcH). Subsequently, an accurate log K value can be obtained using lower ligand concentrations. Table 6 reports the results of titrations using 1000, 130, and 60 µM acetate. The values of b and pKw from the first two sets of experiments are in good agreement within experimental error. However, the log K values obtained using 60 µM acetate are higher than those that remained by an average of 0.06 logarithm unit. This has been ascribed to the influence of dissolved CO2. In Figure 3, a shoulder is apparent at pHm ≈ 6 and a weak peak at pHm ≈ 9, which can be related to the deprotonation of carbonic acid. Significant improvement of the fit has been obtained by assuming 6 µM carbonate present and using the log K values at µ ) 0.1 from Thurmond and Millero.25 The log K values of acetate, after correction for the deprotonation of carbonic acid, agree well with the first two sets of data (see Table 6). To assess the lowest ligand concentration accessible, titration data using 20 µM TRIS in 1.0 m NaCl as test solution and 1.5 mM NaOH as titrant were processed. A log K value of 8.278 ( 0.010 was obtained, which compares well with the value of 8.281 reported by Izaguirre and Millero.26 In conclusion, the proposed method of analyzing potentiometric titration data has the advantages of graphic representation, internal pH electrode calibration, and high accuracy. It is particularly useful in protonation studies of the acids or bases which have low solubilities or are present in concentrated electrolyte media. Investigations of the application of this method to polyprotic acids, such as oxalic acid, citric acid, and EDTA, are underway. ACKNOWLEDGMENT This work was performed as part of the Waste Isolation Pilot Plant (WIPP) Actinide Source Term Program, supported at Sandia National Laboratories by the United State Department of Energy under Contract DE-AC04-94AL85000, and at Florida State University under Contract AH-5590. Received for review February 13, 1996. Accepted August 3, 1996.X AC960138T X
Abstract published in Advance ACS Abstracts, October 1, 1996.
