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On the evaluation of autoprotolysis constants of amphiprotic solvents from electrochemical measurements
MICROCHEMICAL
JOURNAL
39, 303-307 (1989)
On the Evaluation of Autoprotolysis Constants of Amphiprotic Solvents from Electrochemical Measurements GUSTAVOG.GONZALEZ,AGUSTIN G. ASUERO,'ANDANGELES HERRADOR Department
of Analytical
Chemistry,
Received November
University
of Sevilie, 41012 Seville, Spain
17, 1988; accepted December
15, 1988
A modified procedure for the evaluation of autoprotolysis constants from electrochemical techniques which takes into account deviations from the Nemstian slope is given. The method has been applied to the literature data given for several practical systems in order to illustrate it. The error propagation is obtained along with the normal calculation of the parameters. Q 1989 Acadetttic Press, Inc.
Knowledge of the autoprotolysis constants of pure amphiprotic or mixed aqueous organic solvents is of vital interest in analytical chemistry because it gives direct information concerning the extension of pH scale (I) which is closely related to the feasibility of nonaqueous acid-base titrations, being in addition the frost step in the potentiometric evaluation of acidity constants (2). A variety of procedures has been applied in order to evaluate autoprotolysis constants, electrochemical methods (3-6) being undoubtedly the most widely applied. In this paper a method which takes into account deviations from the Nemstian slope is devised. In order to check its usefulness the method suggested has been applied to several practical systems reported in the analytical literature. A detailed error analysis is also included. THEORY
The titration of a strong acid in a given medium (either added as a solution of appropriate concentration in the solvent of interest or generated coulometrically at the anode) with a strong base, at a fixed ionic strength, may be followed through potential measurements with an indicator electrode sensitive to solvated proton, and an appropriate potentiometric device. The electrode potential (in mV) is given by E, = e + 59.16 logJSHz+) and I$ = Z$ - 59.16 logIS-1, where the subscripts a and b refer to the acidic or alkaline range of pH at 25”C, and the superscript ’ refers to the formal standard potential of the indicator electrode. r To whom correspondence
should be addressed. 303 0026-265X/89 $1.50 Copyright 0 1989 by Academic Press, Inc. All righis of reproduction in any fom reserved.
304
GONZALEZ,
ASUERO,
AND
HERRADOR
Some authors compute an average value of e (or @ from the various values of electrode responseE, (or EJ and the lyonium (or lyate) concentrations (known by, e.g., potentiometric or coulometric titrations). Then, the cologarithm of the autoprotolysis constant K, = (H$+)(S-) is calculated as (69) pK, = (,!$ - eM9.16 - 2 log&, f* being the mean ionic activity coefficient, which approximates individual ionic activity coefficients and which is a function (7) of the permittivity of the medium (D), the temperature (7”)in kelvin, and the ionic strength (a, 1.825 X lo6 @W)-3nI’” -kf*
=
1 + 251.45
(~yp2
11”
*
(4)
However, a question arises. The applications of Eqs. (1) and (2) assumean ideal Nernstian response of the electrode, which may not be a realistic approach. For this reason, a modification which takes into account deviations from Nemstian response has been devised. Several computer programs for use in potentiometry include a non-Nemstian correction factor, e.g., in the procedure used to calibrate the electrode (IO). The changes in potential readings during the titration are in fact given by E, = I$ + k, loglSHz+l
(5)
and z$ = J!# - kb lOgIs-
(6)
in the acidic and basic ranges of the pH scale, respectively. Then, the plots of potential versus logi SH2+1 and log1S-1 allow one to estimate both e and k,, and e and kb, respectively, as the intercept slope of the corresponding straight lines obtained by single linear regression. Here we have assumed that ISH2+I and IS-1 are well known and can be considered free from error. In any case, it is quite reasonable to assumethat the calculation of (SH2+l and I S-1 can be made with extreme accuracy compared to the error of potential measurements(working with an automatic device). In consequence, the following expression, which takes into account only the experimental data obtained in the course of titration, can be easily derived: p& = (e - e)&
- W, + kb)&l log fz.
(7)
In the limit case, k, = kb = 59.16 and expression (7) reduces to (3). By applying the random error propagation law (11) to this expression (and assumingf, is free from error) we obtain
AUTOPROTOLYSIS
CONSTANTS
OF AMPHIPROTIC
SOLVENTS
305
Thus the contribution of the slope variance and the covariance between the slope and the intercept to the variance of the pK, is considered. The covariance of measurementscan be as important as the variances (12) and both contribute significantly to the total analytical error. The random error propagation law applies to Eq. (3) gives instead s2(pKs)= (59.16)-2 (x2(@ + s’(Z$)
(9)
leading to an underestimation of the error involved in the evaluation of pK,. In any case, and before application of Eq. (3) to calculate the pK, value of a given amphiprotic solvent, the slopes of the corresponding straight lines used in the evaluation of e and E(3b must be statistically tested for Nemstian electrode response by meansof the Student t test. On applying this test we assume(13) that the residuals are normally distributed. APPLICATIONS
The theory developed has been applied to the experimental data reported in the analytical literature by a number of authors. The unmodified method applied to the data of Zikolov et al. (6) and Glab ef al. (8) for the evaluation of the autoprotolysis constants of ethylene glycol gives the results compiled in Table 1. The standard deviations of I$ and tib were calculated by means of (i = a,b)
where e refers to the mean value, and n is the number of observations. The standard deviation of pK, values was obtained from Eq. (9). TABLE 1 pK, Values for Ethylene Glycol Obtained by the Unmoditied Method
e e
-bft PK,
GIab“
Glabb
589.80 f 0.35 -308.70 f 0.35 0.23 15.65 2 0.008
410.40 + 0.22 -468.10 f 0.10 0.36 15.57 k 0.004
a Coulometric method (one compartment). b Coulometric method (two compartments). c Potentiometric method.
Zikolov’ 486.50 + 0.21 -410.80 f 0.16 0.28 15.72 + 0.004
306
GONZALEZ,
ASUERO, AND HERRADOR
The procedure devised in this paper when applied to the experimental data of Zikolov et al. (6) and Glab et al. (8) led instead to the results included in Table 2. The experimental values of the Student t test (14), t exp= Iki - 59.16//&)
(i = a,b),
(11)
are lower than the t-tabulated values at a 95% confidence level for the twocompartment coulometric method of Glab et al. (8) and potentiometric data of Zikolov et al. (6). Nevertheless tab C texp for the one-compartment coulometric method of Glab et al. (8) at a 95% confidence level, but tb,., > texp at a 98% confidence level (ka) and 99.4% level (kb). This high-confidence level can generate o-type errors (15), being a signitlcant effect possibly missed, and a true hypothesis rejected. A slight difference between the formal standard potentials was obtained by applying both methods. The difference between the pK, values evaluated by both methods is less than 0.1 pK, unit in two cases (8), but greater than 0.4 pK, unit in the work of Zikolov et al. (6). However, the application of expression (8) leads, as expected, to standard deviations of pK, values much higher than those obtained by means of the unmodified procedure. A strong correlation, as pointed out by Mandel and Lining (16), exists between the estimated slope and the intercept of a straight line obtained by least-squarescalculation. In some cases, the slopes in the acid and alkaline ranges are far from the theoretical value of 59.16 and the application of the Student t test led to uncommonly large values for the t statistic value; e.g., Velinov et al. (9) titrated 50 ml H,O + 20 ml of 0.0921M NaOH with 0.0906 M HCl at Z = 1, with the aim of checking the electrode response. The slope values in the acid and alkaline ranges as evaluated by us from the experimental data reported by those authors were 50.30 rt 1.32 and 67.91 f 1.49 in the acid and TABLE 2 pK, Values for Ethylene Glycol as Calculated by the Modified Procedure
e F a :,a Mb ..p” tCXP b COP CO@ DK.
t
Glab”
Glabb
593.08 + 1.07 -316.71 60.06 -c 2 0.33 1.74
408.93 + 1.47 - ,468.65 ? 1.04 58.59 2 0.59 59.36 + 0.36 0.9995 (12) 0.9999 (7) 0.97 0.64 -0.86 -0.37 15.49 + 0.070
61.66 0.9998 f (12) 0.51 0.9998 (7) 2.72 4.90 -0.36 -0.90 15.21 f 0.096
Zikolov” 487.52 + 0.70 - 411.65 ” 1.33 59.55 ” 0.29 59.61 + 0.54 0.9999 (6) 0.9998 (6) 1.34 0.83 -0.20 -0.72 15.64 + 0.130
Note. The superscripts a and b refer to the acid and alkaline ranges of the pH scale. r(n), Correlation coefficient (number of observations). o Coulometric method (one compartment). b Coulometric method (two compartments). c Potentiometric method.
AUTOPROTOLYSIS
CONSTANTS
OF AMPHIPROTIC
SOLVENTS
307
alkaline ranges, respectively. The following t values were obtained by applying Eq. (ll), t = 6.71 (for kJ and t = 5.87 (for k,,). This means that converse to the result of Velinov et al., electrode responses were bad indeed. ACKNOWLEDGMENT The authors (A.G.A. and M.A.H.) acknowledge the support provided by the “Direccibn General de Investigacidn Cienthica y TCcnica de Espatia (DGICYT)” through Project PB-86-0611.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. II. 12. 13. 14. 15. 16.
Serjeant, E. P. Potentiometry and Potentiometric Titrations. Wiley, New York, 1984. King, E. J. “Acid-Base Equilibria.” Pergamon, Oxford, 1%5. Beans, H. L.; Oakes, E. T. J. Amer. Chem. Sot., 1920,42, 2116-2131. Clever, H. L. .I. Chem. Ed., 1%8,45, 231-235. Muney, W.; Coetzee, J. F. J. Phys. Chem., 1%2, 66, 89-%. Zikolov, P.; Astrug, A.; Budevsky, 0. Talanta, 1975, 22, 511-515. Galus, M.; Glab, S.; Grekulak, G.; Hulanicki, A. Talanta, 1979, 26, 169-170. Glab, S.; Hulanicki, A. Talanta, 1981, 28, 183-186. Velinov, G.; Zikolov, P.; Tcharakova, P.; Budevsky, 0. Tafanta, 1974, 21, 163-168. Meloun, M.; Havel, J.; Hogfeldt, E. Computation of Solution Equilibria, a Guide to Methods in Potentiometry, Extraction and Spectrophotometry. Ellis Horwod, Chichester, 1988. Asuero, A. G.; Gonzalez, A. G.; de Pablos, F.; G6mez-Ariza, J. L. Talanta, 1988, 35, 531-537. Kowalski, B. R. Anal. Chem., 1980, 52, 122R-131R. Draper, N. R.; Smith, H. Applied Regression Analysis, 2nd ed. Wiley, New York, 1981. Kennedy, J. B.; Neville, A. M. Basic Statistical Methods for Engineers and Scientists, 2nd ed. Harper & Row, New York, 1976. Laitinen, H. A.; Harris, W. E. Chemical Analysis, an Advanced Text and Reference, 2nd ed. McGraw-Hill, New York, 1975. Mandel, J.; Lining, F. J. Anal. Chem., 1975, 29, 743-749.
JOURNAL
39, 303-307 (1989)
On the Evaluation of Autoprotolysis Constants of Amphiprotic Solvents from Electrochemical Measurements GUSTAVOG.GONZALEZ,AGUSTIN G. ASUERO,'ANDANGELES HERRADOR Department
of Analytical
Chemistry,
Received November
University
of Sevilie, 41012 Seville, Spain
17, 1988; accepted December
15, 1988
A modified procedure for the evaluation of autoprotolysis constants from electrochemical techniques which takes into account deviations from the Nemstian slope is given. The method has been applied to the literature data given for several practical systems in order to illustrate it. The error propagation is obtained along with the normal calculation of the parameters. Q 1989 Acadetttic Press, Inc.
Knowledge of the autoprotolysis constants of pure amphiprotic or mixed aqueous organic solvents is of vital interest in analytical chemistry because it gives direct information concerning the extension of pH scale (I) which is closely related to the feasibility of nonaqueous acid-base titrations, being in addition the frost step in the potentiometric evaluation of acidity constants (2). A variety of procedures has been applied in order to evaluate autoprotolysis constants, electrochemical methods (3-6) being undoubtedly the most widely applied. In this paper a method which takes into account deviations from the Nemstian slope is devised. In order to check its usefulness the method suggested has been applied to several practical systems reported in the analytical literature. A detailed error analysis is also included. THEORY
The titration of a strong acid in a given medium (either added as a solution of appropriate concentration in the solvent of interest or generated coulometrically at the anode) with a strong base, at a fixed ionic strength, may be followed through potential measurements with an indicator electrode sensitive to solvated proton, and an appropriate potentiometric device. The electrode potential (in mV) is given by E, = e + 59.16 logJSHz+) and I$ = Z$ - 59.16 logIS-1, where the subscripts a and b refer to the acidic or alkaline range of pH at 25”C, and the superscript ’ refers to the formal standard potential of the indicator electrode. r To whom correspondence
should be addressed. 303 0026-265X/89 $1.50 Copyright 0 1989 by Academic Press, Inc. All righis of reproduction in any fom reserved.
304
GONZALEZ,
ASUERO,
AND
HERRADOR
Some authors compute an average value of e (or @ from the various values of electrode responseE, (or EJ and the lyonium (or lyate) concentrations (known by, e.g., potentiometric or coulometric titrations). Then, the cologarithm of the autoprotolysis constant K, = (H$+)(S-) is calculated as (69) pK, = (,!$ - eM9.16 - 2 log&, f* being the mean ionic activity coefficient, which approximates individual ionic activity coefficients and which is a function (7) of the permittivity of the medium (D), the temperature (7”)in kelvin, and the ionic strength (a, 1.825 X lo6 @W)-3nI’” -kf*
=
1 + 251.45
(~yp2
11”
*
(4)
However, a question arises. The applications of Eqs. (1) and (2) assumean ideal Nernstian response of the electrode, which may not be a realistic approach. For this reason, a modification which takes into account deviations from Nemstian response has been devised. Several computer programs for use in potentiometry include a non-Nemstian correction factor, e.g., in the procedure used to calibrate the electrode (IO). The changes in potential readings during the titration are in fact given by E, = I$ + k, loglSHz+l
(5)
and z$ = J!# - kb lOgIs-
(6)
in the acidic and basic ranges of the pH scale, respectively. Then, the plots of potential versus logi SH2+1 and log1S-1 allow one to estimate both e and k,, and e and kb, respectively, as the intercept slope of the corresponding straight lines obtained by single linear regression. Here we have assumed that ISH2+I and IS-1 are well known and can be considered free from error. In any case, it is quite reasonable to assumethat the calculation of (SH2+l and I S-1 can be made with extreme accuracy compared to the error of potential measurements(working with an automatic device). In consequence, the following expression, which takes into account only the experimental data obtained in the course of titration, can be easily derived: p& = (e - e)&
- W, + kb)&l log fz.
(7)
In the limit case, k, = kb = 59.16 and expression (7) reduces to (3). By applying the random error propagation law (11) to this expression (and assumingf, is free from error) we obtain
AUTOPROTOLYSIS
CONSTANTS
OF AMPHIPROTIC
SOLVENTS
305
Thus the contribution of the slope variance and the covariance between the slope and the intercept to the variance of the pK, is considered. The covariance of measurementscan be as important as the variances (12) and both contribute significantly to the total analytical error. The random error propagation law applies to Eq. (3) gives instead s2(pKs)= (59.16)-2 (x2(@ + s’(Z$)
(9)
leading to an underestimation of the error involved in the evaluation of pK,. In any case, and before application of Eq. (3) to calculate the pK, value of a given amphiprotic solvent, the slopes of the corresponding straight lines used in the evaluation of e and E(3b must be statistically tested for Nemstian electrode response by meansof the Student t test. On applying this test we assume(13) that the residuals are normally distributed. APPLICATIONS
The theory developed has been applied to the experimental data reported in the analytical literature by a number of authors. The unmodified method applied to the data of Zikolov et al. (6) and Glab ef al. (8) for the evaluation of the autoprotolysis constants of ethylene glycol gives the results compiled in Table 1. The standard deviations of I$ and tib were calculated by means of (i = a,b)
where e refers to the mean value, and n is the number of observations. The standard deviation of pK, values was obtained from Eq. (9). TABLE 1 pK, Values for Ethylene Glycol Obtained by the Unmoditied Method
e e
-bft PK,
GIab“
Glabb
589.80 f 0.35 -308.70 f 0.35 0.23 15.65 2 0.008
410.40 + 0.22 -468.10 f 0.10 0.36 15.57 k 0.004
a Coulometric method (one compartment). b Coulometric method (two compartments). c Potentiometric method.
Zikolov’ 486.50 + 0.21 -410.80 f 0.16 0.28 15.72 + 0.004
306
GONZALEZ,
ASUERO, AND HERRADOR
The procedure devised in this paper when applied to the experimental data of Zikolov et al. (6) and Glab et al. (8) led instead to the results included in Table 2. The experimental values of the Student t test (14), t exp= Iki - 59.16//&)
(i = a,b),
(11)
are lower than the t-tabulated values at a 95% confidence level for the twocompartment coulometric method of Glab et al. (8) and potentiometric data of Zikolov et al. (6). Nevertheless tab C texp for the one-compartment coulometric method of Glab et al. (8) at a 95% confidence level, but tb,., > texp at a 98% confidence level (ka) and 99.4% level (kb). This high-confidence level can generate o-type errors (15), being a signitlcant effect possibly missed, and a true hypothesis rejected. A slight difference between the formal standard potentials was obtained by applying both methods. The difference between the pK, values evaluated by both methods is less than 0.1 pK, unit in two cases (8), but greater than 0.4 pK, unit in the work of Zikolov et al. (6). However, the application of expression (8) leads, as expected, to standard deviations of pK, values much higher than those obtained by means of the unmodified procedure. A strong correlation, as pointed out by Mandel and Lining (16), exists between the estimated slope and the intercept of a straight line obtained by least-squarescalculation. In some cases, the slopes in the acid and alkaline ranges are far from the theoretical value of 59.16 and the application of the Student t test led to uncommonly large values for the t statistic value; e.g., Velinov et al. (9) titrated 50 ml H,O + 20 ml of 0.0921M NaOH with 0.0906 M HCl at Z = 1, with the aim of checking the electrode response. The slope values in the acid and alkaline ranges as evaluated by us from the experimental data reported by those authors were 50.30 rt 1.32 and 67.91 f 1.49 in the acid and TABLE 2 pK, Values for Ethylene Glycol as Calculated by the Modified Procedure
e F a :,a Mb ..p” tCXP b COP CO@ DK.
t
Glab”
Glabb
593.08 + 1.07 -316.71 60.06 -c 2 0.33 1.74
408.93 + 1.47 - ,468.65 ? 1.04 58.59 2 0.59 59.36 + 0.36 0.9995 (12) 0.9999 (7) 0.97 0.64 -0.86 -0.37 15.49 + 0.070
61.66 0.9998 f (12) 0.51 0.9998 (7) 2.72 4.90 -0.36 -0.90 15.21 f 0.096
Zikolov” 487.52 + 0.70 - 411.65 ” 1.33 59.55 ” 0.29 59.61 + 0.54 0.9999 (6) 0.9998 (6) 1.34 0.83 -0.20 -0.72 15.64 + 0.130
Note. The superscripts a and b refer to the acid and alkaline ranges of the pH scale. r(n), Correlation coefficient (number of observations). o Coulometric method (one compartment). b Coulometric method (two compartments). c Potentiometric method.
AUTOPROTOLYSIS
CONSTANTS
OF AMPHIPROTIC
SOLVENTS
307
alkaline ranges, respectively. The following t values were obtained by applying Eq. (ll), t = 6.71 (for kJ and t = 5.87 (for k,,). This means that converse to the result of Velinov et al., electrode responses were bad indeed. ACKNOWLEDGMENT The authors (A.G.A. and M.A.H.) acknowledge the support provided by the “Direccibn General de Investigacidn Cienthica y TCcnica de Espatia (DGICYT)” through Project PB-86-0611.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. II. 12. 13. 14. 15. 16.
Serjeant, E. P. Potentiometry and Potentiometric Titrations. Wiley, New York, 1984. King, E. J. “Acid-Base Equilibria.” Pergamon, Oxford, 1%5. Beans, H. L.; Oakes, E. T. J. Amer. Chem. Sot., 1920,42, 2116-2131. Clever, H. L. .I. Chem. Ed., 1%8,45, 231-235. Muney, W.; Coetzee, J. F. J. Phys. Chem., 1%2, 66, 89-%. Zikolov, P.; Astrug, A.; Budevsky, 0. Talanta, 1975, 22, 511-515. Galus, M.; Glab, S.; Grekulak, G.; Hulanicki, A. Talanta, 1979, 26, 169-170. Glab, S.; Hulanicki, A. Talanta, 1981, 28, 183-186. Velinov, G.; Zikolov, P.; Tcharakova, P.; Budevsky, 0. Tafanta, 1974, 21, 163-168. Meloun, M.; Havel, J.; Hogfeldt, E. Computation of Solution Equilibria, a Guide to Methods in Potentiometry, Extraction and Spectrophotometry. Ellis Horwod, Chichester, 1988. Asuero, A. G.; Gonzalez, A. G.; de Pablos, F.; G6mez-Ariza, J. L. Talanta, 1988, 35, 531-537. Kowalski, B. R. Anal. Chem., 1980, 52, 122R-131R. Draper, N. R.; Smith, H. Applied Regression Analysis, 2nd ed. Wiley, New York, 1981. Kennedy, J. B.; Neville, A. M. Basic Statistical Methods for Engineers and Scientists, 2nd ed. Harper & Row, New York, 1976. Laitinen, H. A.; Harris, W. E. Chemical Analysis, an Advanced Text and Reference, 2nd ed. McGraw-Hill, New York, 1975. Mandel, J.; Lining, F. J. Anal. Chem., 1975, 29, 743-749.