- Email: [email protected]
PKPOT, a program for the potentiometric study of ionic equilibria in aqueous and non-aqueous media
AIVAHTICA CHIME4
ACTA
ELSEVIER
Analytica Chimica Acta 317 (1995) 75-81
PKPOT, a program for the potentiometric study of ionic equilibria in aqueous and non-aqueous media Jo& Barbosa, Dolores Barr&, Jo& Luis Belt& Departament
*, Victoria Sanz-Nebot
de Quimica Analitica, Universitat de Barcelona, Avgda. Diagonal 647, 08028 Barcelona, Spain Received 23 March 1995; revised 31 July 1995; accepted 9 August 1995
Abstract The least-squares computer program PKPOT has been developed to run on a PC-compatible computer, for lthe study of ionic equilibria from potentiometric data. It allows for the refinement of equilibrium constants in systems described by up 5 components and 20 complex species A,B,C,D,H,. Other parameters (standard potentials of electrode, reactartt concentrations, etc.) can also be refined. The program can deal with up to ten titration curves. It provides several statistical tests, as also graphics presentations of data and residuals distribution. The program allows for the determination of s@ichiometric formation constants (at fixed ionic strength), or thermodynamic constants; in this case, corrections for changes on the activity coefficients are taken into account. The data analysed can be given as volume/e.m.f. or volume/lpH. Several application examples are given, including titrations in aqueous and non-aqueous media. Keywords:
Potentiometry;
Ionic equilibria;
Computer
program
1. Introduction At present, there are many programs developed for the determination or refinement of dissociation and complex formation constants from potentiometric titration data. Most of them [l-8] have been designed for equilibria in aqueous medium, or it is assumed that the activity coefficients are constant (this is, the constants determined are defined in terms of concentration of reactants); some programs allow for correction of activity coefficients in aqueous medium [9]. There are also programs for the acidbase dissociation in non-aqueous medium [lo], in-
* Corresponding
author.
0003-2670/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0003-2670(95)00400-9
eluding the development of a series of equations and procedures which allow for a complete description of dissociation, ion-pair formation and homoconjugation equilibria in these media [ll-131. The main difference between the different procedures used for the study of ionic equilibria in aqueous and non-aqueous media is owed to the activity coefficients, as in most equilibria in aqueaus medium, a background electrolyte is added to maintain constant the ionic strength (its concentration ranging from about 0.1 to 3.0 M); this is also allowed in some water-ethanol or water-dioxane mixtures, but not in solvents of low dielectric constant (as tert.butyl alcohol, anhydrous acetic acid or tetrahydrofuran) where the solubility of electrolytes is very low, or they can modify the equilibria because of the formation on ion-pair associates with the reactants;
76
J. Barbosa et al. /Analytica
in these cases, as also in absence of background electrolyte, the ionic strength of the system changes when running the potentiometric titration, and a correction of activity coefficients can be necessary. The program described in this paper, PKPOT, is a non-linear least squares program designed to take into account the correction of activity coefficients, allowing the refinement of thermodynamic equilibrium constants in non-aqueous solvents. Of course, it can be also used for the determination of stoichiometric constants (in concentration terms) by making equal to the unity the activity coefficients. PKPOT has been applied to several systems to show its possibilities. The application examples are: protonation and complex formation equilibria of glycine with Ni2+ ions in aqueous medium, acidbase and ion-pair formation equilibria of diazepam in acetic acid medium, and dissociation constants of phthalic acid in 30% (w/w) acetonitrile-water medium.
2. PKPOT
program
This program is written in Turbo-PASCAL for use on personal computers (386 with math coprocessor or higher). PKPOT is based on menu-driven selections for interactive operation, in a similar way to STAR program [14]. As in this, the user has direct access to several graphics routines for the presentation of data and results; the plotting facilities include the drawing of experimental and calculated titration curves and the residuals distribution. These graphics can be obtained directly on the computer screen, as also in HP-GL code (output to a file in this case). The present version of PKPOT can handle up to ten potentiometric titrations simultaneously, each containing a maximum of 100 data pairs volume/e.m.f. or volume/pH. The systems studied can be defined by 5 components (A, B, C, D, H), forming up to 20 complex species A,B,C,D,H,, apart from the free components. PKPOT allows for the refinement of global parameters of the system, as the formation constants of the species or the liquidjunction potentials, but also local parameters of each titration, as the standard potential of the electrode, its slope or the concentration of reactants. Some characteristics of PKPOT are described below.
Chimica Acta 317 (1995) 75-81
2.1. Mass balance calculations
The procedure complex used in PKPOT is a modified version of a iterative method described previously [15], in order to account for the changes in ionic strength (I) of the solution along the titration. This procedure uses two sets of constants, thermodynamic formation constants (pL> and stoichiometric formation constants ( pi). The former are considered as the ‘working’ constants, and the pi values are recalculated iteratively in each titration point according to: logP,“=log&-S&logf* In this equation, S& is a term related with the charges of the k species and those of the free components, and log f + is calculated after the limiting Debye-Hiickel relationship: logf,=
-(AJZ)/(l+a,ZVZ)
Where the A and a,B parameters are supplied as input to the program. It is assumed that a,B is the same for all the ionic species in solution. For the determination of stoichiometric constants, the A value is set equal to zero. The program starts solving the mass balances for log p: = log pi, the ionic strength of the solution is calculated and a new set of log pi values is obtained. The procedure is repeated iteratively until the system is solved. 2.2. Refinement procedure The refinement method is based on the iterative Gauss-Newton non-linear least-squares algorithm [16], by numerical differentiation, allowing for the refinement up to ten parameters simultaneously. The minimized function (U) can be defined as the unweighted sum of squares residuals in e.m.f. or in pH. In the first case it takes the form:
’
=
ntit
np
C
C
i=l
j=l
(Eijexp
-Eijcak)2
Where ntit is the number of titrations, and np the corresponding number of experimental points in each titration. Eij exp indicates the measured e.m.f., and Eij
J. Barbosa et al. /Analytica
ca~c the calculated obtained from:
e.m.f.;
for pH electrodes
it is
Where Eo and gi are the standard potential of the electrode and its slope, respectively, for the ith titration, u,;~~,~ the calculated activity for hydrogen ion in the jth point in the ith titration, ju+ and j,,the liquid junction potentials in acid and alkaline medium, and K,, the autoprotolysis constant of solvent. If pH measurements are made instead of e.m.f., the values of EP and g, are set to 0 and - 1, respectively, for all titrations. The iterative procedure is repeated until the relative change in U is less than 0.01% or the number of iterations exceeds a predefined value (15 in this case). Alternatively, in the case of divergence in the refinement process, the ‘shifts’ of the parameters are optimized according to Wentworth [17]. 2.3. Statistical
tests
PKPOT provides several statistical parameters test the reliability of the regression process, and the evaluation of the fit of the proposed model experimental data [18]. The most important is the standard deviation residuals (s), given in mV, calculated as: s = { U/(tp
to for to of
71
Chimica Acta 317 (1995) 75-81
results to an output file or to the computer screen. This output includes the description of the system, statistical tests, the last set of refined parameters and the experimental and calculated data, together with the errors and the computed ionic strength. It contains also the calculated concentrations of aP1 species for all data points.
3. Application examples The program has been applied to several pH-metric data sets described in the literature [7,19,20]. The systems studied, described below, include thermodynamic and stoichiometric equilibrium constants, in aqueous and non-aqueous media. The experimental conditions are given in Table 1. 3.1. (I) Ni2 ‘-glycinate-H
’ system
This system was studied in aqueous medium with 1.0 M NaCl as background electrolyte at 25°C 171, so stoichiometric constants are calculated. The data set includes three potentiometric titrations for the determination of the two protonation constants of glycinate ion (Ia), and five titrations for the Nil’glycinate complex formation constants (Ib; the species formed are NiGly+, NiGly, and NiGly, ). In this example, acidic solutions containing glycine and
- 1 - nr)}“2
Where tp indicates the total number of experimental points, and nr the number of parameters refined simultaneously. The program gives also the residual mean (that should be equal to zero), the mean residual (the mean of the absolute values of residuals) and the Hamilton R-factor (in %). Other valuable parameters for the examination of the distribution of residuals are the Pearson’s x2 test (it should be less than 14.07 for seven degrees of freedom at 95% confidence level), and the skewness and kurtosis tests (which should be equal to 0 and 3, respectively). 2.4. Output of results Apart from the graphics presentation of data and errors, PKPOT provides a complete listing of the
Table 1 Initial concentration solutions (ml)
of reactants
(M) and initial volume
of the
Titration number: 1 System la : 0.0122 C glY 40 v, System Ib: 0.0120 c @Y 0.0025 C 42 YY’ System II: C H*Pht 0.00567 20 V” System III: 0.00387 Ck? 25 VU
2
3
0.0265 40
0.0318 40
0.0108 0.0053 49
0.0216 0.0053 49
0.00572 20
0.00572 20
0.00376 2.5
0.00428 25
4
5
0.0333 0.0087 30
0.0241 0.0119 44
J. Barbosa et al. /Analytica Chimica Acta 317 (1995) 75-81
78
4
3
Fig. 1. Experimental and Ni*+ -glycinate system.
calculated
titration
curves
for
the
Ni2+ ion (when necessary) were titrated with sodium hydroxide solution. The formation constants refined by PKPOT are in good accordance with those obtained previously, as indicated in Table 2, together with the statistical tests given by this program. The experimental and calculated titration curves for the determination of complex formation constants are plotted in Fig. 1, show-
Table 2 Equilibrium
constants
and statistical
Equilibrium
Fig. 2. Species concentration PKPOT.
5
6
7
-kfi
9
for titration 2 of Fig. 1, computed by
ing a good fit of the model to experimental data. The computed concentrations for a single titration in this system is given in Fig. 2. 3.2. (II) Acid-base equilibria of phthalic acid In this example, the dissociation constants of phthalic acid were determined in 30% (w/w) acetonitrile-water medium [19]. Solutions containing ph-
parameters log K a
log K b
s(mV)’
x2
R (o/o)
Skewness
Kurtosis
9.657 + 0.003 12.070 + 0.005
9.667 + 0.003 12.087 + 0.004
0.334
9.19
0.541
0.007
4.582
5.620 f 0.005 10.346 + 0.014 13.708 f 0.039
5.628 f 0.006 10.364 f 0.007 13.752 + 0.010
0.620
9.25
0.54
- 0.460
2.953
- 3.60 f 0.03 - 6.86 + 0.03
- 3.67 + 0.04 - 6.90 + 0.03
0.279
9.20
0.27
0.491
3.111
- 3.64 * 0.03 - 6.74 + 0.04
0.685
16.31
0.58
0.735
3.652
-5.12 f 0.09 - 6.87 + 0.11 -5.58 + 0.15
2.851
20.05
0.53
0.249
3.578
System la : Gly-+ Gly-+
H+ + HGly 2H+ =: HaGly+
System Ib : Gly-+Ni2+ + NiGly+ 2Gly- + Ni2+ + NiGly, 3Gly- + Ni2+ + NiGly,
System Ha: H,Pht Y= H++HPhtHPht - + H++Pht*-
System IIb: H, Pht + HPht- + System III: HClO, + B+HAcO BBClO, *
H++HPhtH++Pht’H+ + ClO,+ HB++AcOHB + + ClO,-
- 4.75 f 0.30 - 6.92 + 0.03 - 5.85 * 0.06
a Equilibrium constants, as given in Refs. [7,19,20]. b Equilibrium constants obtained by PKPOT. Confidence ’ Standard deviation of residuals.
intervals are calculated
as three times the standard deviation given by the program.
J. Barbosa et al. /Analytica
Chimica Acta 317 flW5)
19
75-81
the program: all the parameters are worse when no correction is made, mainly in the standard deviation of residuals and in the Hamilton R-factor. The differences are also apparent in Fig. 5A and B (corresponding to system IIa and IIb, respectively), where the distribution of residuals are plotted: Fig. 5B shows a trend on residuals, not found in Fig. 5A, which indicates a systematic error that can be attributed to changes in the stoichiometric constants when running the titration. Otherwise, the results obtained for system IIa are in good accordance with those previously reported [19].
Fig. 3. Titration curves of phthalic acid with 0.1 M potassium hydroxide, in 30% (w/w) acetonitrile-water medium. The calculated curves are indicated as continuous lines.
thalic acid and 0.0004 M KC1 were titrated with potassium hydroxide at 25’C. Three titrations were treated simultaneously with PKPOT, in order to refine the equilibrium constants. We have used this system to compare the results obtained when the activity coefficient correction is applied in the determination of equilibrium constants. In system IIa, this correction was made, leading to the determination of thermodynamic constants. In system IIb, stoichiometric constants were determined (no correction was made). In both cases, the equilibrium constants were refined simultaneously together with the electrode standard potentials. As PKPOT gives the overall formation constants of the species (log p1 = 6.895 f 0.005 and log pZ = 10.568 f 0.011 in system IIa), we have obtained the corresponding dissociation constants (as log K,) to compare the results with those given in Ref. [19] (see Table 2). The titration curves corresponding to this system are plotted in Fig. 3. The comparison of results obtained in systems 1Ia and IIb indicates that the correction of the activity coefficient is necessary in this case, as shown in the Fig. 4, where the calculated logarithm of the mean activity coefficient is plotted versus the experimental point for each titration. The different results can be observed also in Table 2, not only in the values of the constants, but also in the statistical tests given by
3.3. (III) Equilibrium constants of diazepam in acetic acid This system is described by the protonation constant of diazepam (B) and the formation constant of the ion-pair HB+ . ClO,. The equilibrium constants were determined by titrating solutions of diazepam with perchloric acid in pure acetic acid [20]. In this case, stoichiometric constants are calculated, as the changes on the mean activity coefficient are negligible (less than 0.02 logarithmic units). Calculations with PKPOT have been carried out over three potentiometric titrations, allowing for the refinement of these equilibrium constants, as also the dissociation constant of perchloric acid in acetic acid medium. The good fit of the model to experimental data is shown in Fig. 6. The overall formation constants
Fig. 4. Changes acid titrations.
on the activity
coefficient
(log f i: ) in phthalic
80
J. Barbosa et al./AnaIytica
Chimica Acta 317 (1995) 75-81
obtained by the program are given below: B+H++HB+ ClO,+
H+=
for the different
equilibria
log p = 7.000 f 0.036 HCIO,
log /3 = 5.12 & 0.030
B+H+ClO,*HB+CIO; log p = 12.583 + 0.039
-0.6
1
-250.0
n -1500
-50.0
500 e rn.f
As in the previous example, we have derived these results to be comparable with the basicity and ion-pair dissociation constants given in Ref. [20]. Table 1 shows that both results are in accordance.
(nl”)
4. Discussion
1
-1.5
-250 0
I -150.0
50.0
-50.0 0.m.f.
(mv)
Fig. 5. Residuals distribution in e.m.f., obtained after the determination of protonation constants of phthalate ion. (A) With activity coefficient correction (thermodynamic constants calculated), and (B) without correction (stoichiometric constants calculated).
7000
emI. (-
Fig. 6. Titration curves of diazepam acid medium.
with perchloric
acid in acetic
As noted before, PKPOT can be used for the refinement not only of formation constants, but also the other parameters which define the potentiometric titrations (standard potentials of electrodes, concentration of reactants, etc.). However, the user should be aware of the refinement of some of these ‘local parameters’ together with the formation constants because of their strong interdependence, mainly when the formation constants are calculated from a single titration. In these cases, the concentration of reactants should be exactly known before the treatment of data, as well the electrode parameters or the autoprotolysis constant of solvent. In practice, the refinement of these local parameters is intended for the minimization of systematic errors. In this sense, the plotting facilities of PKPOT are a valuable tool in the detection and interpretation of error concentrations or in standard potentials [18]. Other possibilities of PKPOT include runtime procedures for the selection of the data points included in the calculations, and for adding or deleting species from the model. This latter feature is valuable for testing different models for a given system, as can be used without interrupting the program execution; the statistical parameters are also useful in testing different models. These features, together with the different possibilities for data treatment (in aqueous or non-aqueous medium, pH or e.m.f. data, and refinement of thermodynamic or stoichiometric formation constants) make PKPOT a versatile program for the evaluation of stability constants from potentiometric
J. Barbosa et al. /Analytica
titration data. The program, together with sample data files, is available upon request from the authors.
References [II
L.G. Sill&, Acta Chem. &and.,
18 (1964) 1085.
Dl A. Sabatini, A. Vacca and P. Gans, Talanta, 21 (1974) 53. [31 P. Gans, A. Sabatini and A. Vacca, J. Chem. Sot., Dalton Trans., (1985) 1195. 141 A. Zuberbiiler and T.A. Kaden, Talanta, 29 (1982) 201. [51 R.J. Motekaitis and A.E. Martell, Can. J. Chem., 60 (1982) 725. [61 R.M. Alcock, F.R. Hartley and D.E. Rogers, .I. Chem. Sot., Dalton Trans., (1978) 115. Anal. Chim. Acta, 181 (1986) [71 A. Izquierdo and J.L. Belt& 87. (81 F. Gaizer and 1.1. Kiss, Talanta, 41 (1994) 419. 191 I. Ting-Po and G.H. Nancollas, Anal. Chem., 44 (1972) 1940.
Chimica Acta 317 (1995) 75-81
DO10. Budevsky,
81
T. Zikolova and J. Tencheva, Talanta, 35 (1988) 899. (111 M. Ro&, Anal. Chim. Acta, 276 (1993) 211. 1121M. Rests, Anal. Chim. Acta, 276 (1993) 223. [131M. Rests, Anal. Chim. Acta, 285 (1994) 391. [141J.L. Belt&. R. Codony and M.D. Prat, Anal. Chim. Acta, 276 (1993) 441. [151 A. Izquierdo and J.L. BeltrBn, J. Chemom., 3 (1988) 209. iI61 E. Durand, Solutions Numeriques des Equations Algbbriques, Tome II: Systtmes de Plusieurs Equations, Masson, Paris, 1972. (171 W.E. Wenhvorth, J. Chem. Educ., 42 (1965) 96. of [181 M. Meloun, J. Have1 and E. Hiigfeldt, Computation Solution Equilibria: a Guide to Methods in Potentiometry, Extraction and Spectrophotometry, Ellis Horwood, Chichester, 1988. [191 J. Barbosa and D. Barr6n, Analyst, 114 (1989) 471. and V. Sanz-Nebot, Anal. Chim. I201 J. Barbosa. J.L. Belt& Acta, 288 (1994) 271.
ACTA
ELSEVIER
Analytica Chimica Acta 317 (1995) 75-81
PKPOT, a program for the potentiometric study of ionic equilibria in aqueous and non-aqueous media Jo& Barbosa, Dolores Barr&, Jo& Luis Belt& Departament
*, Victoria Sanz-Nebot
de Quimica Analitica, Universitat de Barcelona, Avgda. Diagonal 647, 08028 Barcelona, Spain Received 23 March 1995; revised 31 July 1995; accepted 9 August 1995
Abstract The least-squares computer program PKPOT has been developed to run on a PC-compatible computer, for lthe study of ionic equilibria from potentiometric data. It allows for the refinement of equilibrium constants in systems described by up 5 components and 20 complex species A,B,C,D,H,. Other parameters (standard potentials of electrode, reactartt concentrations, etc.) can also be refined. The program can deal with up to ten titration curves. It provides several statistical tests, as also graphics presentations of data and residuals distribution. The program allows for the determination of s@ichiometric formation constants (at fixed ionic strength), or thermodynamic constants; in this case, corrections for changes on the activity coefficients are taken into account. The data analysed can be given as volume/e.m.f. or volume/lpH. Several application examples are given, including titrations in aqueous and non-aqueous media. Keywords:
Potentiometry;
Ionic equilibria;
Computer
program
1. Introduction At present, there are many programs developed for the determination or refinement of dissociation and complex formation constants from potentiometric titration data. Most of them [l-8] have been designed for equilibria in aqueous medium, or it is assumed that the activity coefficients are constant (this is, the constants determined are defined in terms of concentration of reactants); some programs allow for correction of activity coefficients in aqueous medium [9]. There are also programs for the acidbase dissociation in non-aqueous medium [lo], in-
* Corresponding
author.
0003-2670/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0003-2670(95)00400-9
eluding the development of a series of equations and procedures which allow for a complete description of dissociation, ion-pair formation and homoconjugation equilibria in these media [ll-131. The main difference between the different procedures used for the study of ionic equilibria in aqueous and non-aqueous media is owed to the activity coefficients, as in most equilibria in aqueaus medium, a background electrolyte is added to maintain constant the ionic strength (its concentration ranging from about 0.1 to 3.0 M); this is also allowed in some water-ethanol or water-dioxane mixtures, but not in solvents of low dielectric constant (as tert.butyl alcohol, anhydrous acetic acid or tetrahydrofuran) where the solubility of electrolytes is very low, or they can modify the equilibria because of the formation on ion-pair associates with the reactants;
76
J. Barbosa et al. /Analytica
in these cases, as also in absence of background electrolyte, the ionic strength of the system changes when running the potentiometric titration, and a correction of activity coefficients can be necessary. The program described in this paper, PKPOT, is a non-linear least squares program designed to take into account the correction of activity coefficients, allowing the refinement of thermodynamic equilibrium constants in non-aqueous solvents. Of course, it can be also used for the determination of stoichiometric constants (in concentration terms) by making equal to the unity the activity coefficients. PKPOT has been applied to several systems to show its possibilities. The application examples are: protonation and complex formation equilibria of glycine with Ni2+ ions in aqueous medium, acidbase and ion-pair formation equilibria of diazepam in acetic acid medium, and dissociation constants of phthalic acid in 30% (w/w) acetonitrile-water medium.
2. PKPOT
program
This program is written in Turbo-PASCAL for use on personal computers (386 with math coprocessor or higher). PKPOT is based on menu-driven selections for interactive operation, in a similar way to STAR program [14]. As in this, the user has direct access to several graphics routines for the presentation of data and results; the plotting facilities include the drawing of experimental and calculated titration curves and the residuals distribution. These graphics can be obtained directly on the computer screen, as also in HP-GL code (output to a file in this case). The present version of PKPOT can handle up to ten potentiometric titrations simultaneously, each containing a maximum of 100 data pairs volume/e.m.f. or volume/pH. The systems studied can be defined by 5 components (A, B, C, D, H), forming up to 20 complex species A,B,C,D,H,, apart from the free components. PKPOT allows for the refinement of global parameters of the system, as the formation constants of the species or the liquidjunction potentials, but also local parameters of each titration, as the standard potential of the electrode, its slope or the concentration of reactants. Some characteristics of PKPOT are described below.
Chimica Acta 317 (1995) 75-81
2.1. Mass balance calculations
The procedure complex used in PKPOT is a modified version of a iterative method described previously [15], in order to account for the changes in ionic strength (I) of the solution along the titration. This procedure uses two sets of constants, thermodynamic formation constants (pL> and stoichiometric formation constants ( pi). The former are considered as the ‘working’ constants, and the pi values are recalculated iteratively in each titration point according to: logP,“=log&-S&logf* In this equation, S& is a term related with the charges of the k species and those of the free components, and log f + is calculated after the limiting Debye-Hiickel relationship: logf,=
-(AJZ)/(l+a,ZVZ)
Where the A and a,B parameters are supplied as input to the program. It is assumed that a,B is the same for all the ionic species in solution. For the determination of stoichiometric constants, the A value is set equal to zero. The program starts solving the mass balances for log p: = log pi, the ionic strength of the solution is calculated and a new set of log pi values is obtained. The procedure is repeated iteratively until the system is solved. 2.2. Refinement procedure The refinement method is based on the iterative Gauss-Newton non-linear least-squares algorithm [16], by numerical differentiation, allowing for the refinement up to ten parameters simultaneously. The minimized function (U) can be defined as the unweighted sum of squares residuals in e.m.f. or in pH. In the first case it takes the form:
’
=
ntit
np
C
C
i=l
j=l
(Eijexp
-Eijcak)2
Where ntit is the number of titrations, and np the corresponding number of experimental points in each titration. Eij exp indicates the measured e.m.f., and Eij
J. Barbosa et al. /Analytica
ca~c the calculated obtained from:
e.m.f.;
for pH electrodes
it is
Where Eo and gi are the standard potential of the electrode and its slope, respectively, for the ith titration, u,;~~,~ the calculated activity for hydrogen ion in the jth point in the ith titration, ju+ and j,,the liquid junction potentials in acid and alkaline medium, and K,, the autoprotolysis constant of solvent. If pH measurements are made instead of e.m.f., the values of EP and g, are set to 0 and - 1, respectively, for all titrations. The iterative procedure is repeated until the relative change in U is less than 0.01% or the number of iterations exceeds a predefined value (15 in this case). Alternatively, in the case of divergence in the refinement process, the ‘shifts’ of the parameters are optimized according to Wentworth [17]. 2.3. Statistical
tests
PKPOT provides several statistical parameters test the reliability of the regression process, and the evaluation of the fit of the proposed model experimental data [18]. The most important is the standard deviation residuals (s), given in mV, calculated as: s = { U/(tp
to for to of
71
Chimica Acta 317 (1995) 75-81
results to an output file or to the computer screen. This output includes the description of the system, statistical tests, the last set of refined parameters and the experimental and calculated data, together with the errors and the computed ionic strength. It contains also the calculated concentrations of aP1 species for all data points.
3. Application examples The program has been applied to several pH-metric data sets described in the literature [7,19,20]. The systems studied, described below, include thermodynamic and stoichiometric equilibrium constants, in aqueous and non-aqueous media. The experimental conditions are given in Table 1. 3.1. (I) Ni2 ‘-glycinate-H
’ system
This system was studied in aqueous medium with 1.0 M NaCl as background electrolyte at 25°C 171, so stoichiometric constants are calculated. The data set includes three potentiometric titrations for the determination of the two protonation constants of glycinate ion (Ia), and five titrations for the Nil’glycinate complex formation constants (Ib; the species formed are NiGly+, NiGly, and NiGly, ). In this example, acidic solutions containing glycine and
- 1 - nr)}“2
Where tp indicates the total number of experimental points, and nr the number of parameters refined simultaneously. The program gives also the residual mean (that should be equal to zero), the mean residual (the mean of the absolute values of residuals) and the Hamilton R-factor (in %). Other valuable parameters for the examination of the distribution of residuals are the Pearson’s x2 test (it should be less than 14.07 for seven degrees of freedom at 95% confidence level), and the skewness and kurtosis tests (which should be equal to 0 and 3, respectively). 2.4. Output of results Apart from the graphics presentation of data and errors, PKPOT provides a complete listing of the
Table 1 Initial concentration solutions (ml)
of reactants
(M) and initial volume
of the
Titration number: 1 System la : 0.0122 C glY 40 v, System Ib: 0.0120 c @Y 0.0025 C 42 YY’ System II: C H*Pht 0.00567 20 V” System III: 0.00387 Ck? 25 VU
2
3
0.0265 40
0.0318 40
0.0108 0.0053 49
0.0216 0.0053 49
0.00572 20
0.00572 20
0.00376 2.5
0.00428 25
4
5
0.0333 0.0087 30
0.0241 0.0119 44
J. Barbosa et al. /Analytica Chimica Acta 317 (1995) 75-81
78
4
3
Fig. 1. Experimental and Ni*+ -glycinate system.
calculated
titration
curves
for
the
Ni2+ ion (when necessary) were titrated with sodium hydroxide solution. The formation constants refined by PKPOT are in good accordance with those obtained previously, as indicated in Table 2, together with the statistical tests given by this program. The experimental and calculated titration curves for the determination of complex formation constants are plotted in Fig. 1, show-
Table 2 Equilibrium
constants
and statistical
Equilibrium
Fig. 2. Species concentration PKPOT.
5
6
7
-kfi
9
for titration 2 of Fig. 1, computed by
ing a good fit of the model to experimental data. The computed concentrations for a single titration in this system is given in Fig. 2. 3.2. (II) Acid-base equilibria of phthalic acid In this example, the dissociation constants of phthalic acid were determined in 30% (w/w) acetonitrile-water medium [19]. Solutions containing ph-
parameters log K a
log K b
s(mV)’
x2
R (o/o)
Skewness
Kurtosis
9.657 + 0.003 12.070 + 0.005
9.667 + 0.003 12.087 + 0.004
0.334
9.19
0.541
0.007
4.582
5.620 f 0.005 10.346 + 0.014 13.708 f 0.039
5.628 f 0.006 10.364 f 0.007 13.752 + 0.010
0.620
9.25
0.54
- 0.460
2.953
- 3.60 f 0.03 - 6.86 + 0.03
- 3.67 + 0.04 - 6.90 + 0.03
0.279
9.20
0.27
0.491
3.111
- 3.64 * 0.03 - 6.74 + 0.04
0.685
16.31
0.58
0.735
3.652
-5.12 f 0.09 - 6.87 + 0.11 -5.58 + 0.15
2.851
20.05
0.53
0.249
3.578
System la : Gly-+ Gly-+
H+ + HGly 2H+ =: HaGly+
System Ib : Gly-+Ni2+ + NiGly+ 2Gly- + Ni2+ + NiGly, 3Gly- + Ni2+ + NiGly,
System Ha: H,Pht Y= H++HPhtHPht - + H++Pht*-
System IIb: H, Pht + HPht- + System III: HClO, + B+HAcO BBClO, *
H++HPhtH++Pht’H+ + ClO,+ HB++AcOHB + + ClO,-
- 4.75 f 0.30 - 6.92 + 0.03 - 5.85 * 0.06
a Equilibrium constants, as given in Refs. [7,19,20]. b Equilibrium constants obtained by PKPOT. Confidence ’ Standard deviation of residuals.
intervals are calculated
as three times the standard deviation given by the program.
J. Barbosa et al. /Analytica
Chimica Acta 317 flW5)
19
75-81
the program: all the parameters are worse when no correction is made, mainly in the standard deviation of residuals and in the Hamilton R-factor. The differences are also apparent in Fig. 5A and B (corresponding to system IIa and IIb, respectively), where the distribution of residuals are plotted: Fig. 5B shows a trend on residuals, not found in Fig. 5A, which indicates a systematic error that can be attributed to changes in the stoichiometric constants when running the titration. Otherwise, the results obtained for system IIa are in good accordance with those previously reported [19].
Fig. 3. Titration curves of phthalic acid with 0.1 M potassium hydroxide, in 30% (w/w) acetonitrile-water medium. The calculated curves are indicated as continuous lines.
thalic acid and 0.0004 M KC1 were titrated with potassium hydroxide at 25’C. Three titrations were treated simultaneously with PKPOT, in order to refine the equilibrium constants. We have used this system to compare the results obtained when the activity coefficient correction is applied in the determination of equilibrium constants. In system IIa, this correction was made, leading to the determination of thermodynamic constants. In system IIb, stoichiometric constants were determined (no correction was made). In both cases, the equilibrium constants were refined simultaneously together with the electrode standard potentials. As PKPOT gives the overall formation constants of the species (log p1 = 6.895 f 0.005 and log pZ = 10.568 f 0.011 in system IIa), we have obtained the corresponding dissociation constants (as log K,) to compare the results with those given in Ref. [19] (see Table 2). The titration curves corresponding to this system are plotted in Fig. 3. The comparison of results obtained in systems 1Ia and IIb indicates that the correction of the activity coefficient is necessary in this case, as shown in the Fig. 4, where the calculated logarithm of the mean activity coefficient is plotted versus the experimental point for each titration. The different results can be observed also in Table 2, not only in the values of the constants, but also in the statistical tests given by
3.3. (III) Equilibrium constants of diazepam in acetic acid This system is described by the protonation constant of diazepam (B) and the formation constant of the ion-pair HB+ . ClO,. The equilibrium constants were determined by titrating solutions of diazepam with perchloric acid in pure acetic acid [20]. In this case, stoichiometric constants are calculated, as the changes on the mean activity coefficient are negligible (less than 0.02 logarithmic units). Calculations with PKPOT have been carried out over three potentiometric titrations, allowing for the refinement of these equilibrium constants, as also the dissociation constant of perchloric acid in acetic acid medium. The good fit of the model to experimental data is shown in Fig. 6. The overall formation constants
Fig. 4. Changes acid titrations.
on the activity
coefficient
(log f i: ) in phthalic
80
J. Barbosa et al./AnaIytica
Chimica Acta 317 (1995) 75-81
obtained by the program are given below: B+H++HB+ ClO,+
H+=
for the different
equilibria
log p = 7.000 f 0.036 HCIO,
log /3 = 5.12 & 0.030
B+H+ClO,*HB+CIO; log p = 12.583 + 0.039
-0.6
1
-250.0
n -1500
-50.0
500 e rn.f
As in the previous example, we have derived these results to be comparable with the basicity and ion-pair dissociation constants given in Ref. [20]. Table 1 shows that both results are in accordance.
(nl”)
4. Discussion
1
-1.5
-250 0
I -150.0
50.0
-50.0 0.m.f.
(mv)
Fig. 5. Residuals distribution in e.m.f., obtained after the determination of protonation constants of phthalate ion. (A) With activity coefficient correction (thermodynamic constants calculated), and (B) without correction (stoichiometric constants calculated).
7000
emI. (-
Fig. 6. Titration curves of diazepam acid medium.
with perchloric
acid in acetic
As noted before, PKPOT can be used for the refinement not only of formation constants, but also the other parameters which define the potentiometric titrations (standard potentials of electrodes, concentration of reactants, etc.). However, the user should be aware of the refinement of some of these ‘local parameters’ together with the formation constants because of their strong interdependence, mainly when the formation constants are calculated from a single titration. In these cases, the concentration of reactants should be exactly known before the treatment of data, as well the electrode parameters or the autoprotolysis constant of solvent. In practice, the refinement of these local parameters is intended for the minimization of systematic errors. In this sense, the plotting facilities of PKPOT are a valuable tool in the detection and interpretation of error concentrations or in standard potentials [18]. Other possibilities of PKPOT include runtime procedures for the selection of the data points included in the calculations, and for adding or deleting species from the model. This latter feature is valuable for testing different models for a given system, as can be used without interrupting the program execution; the statistical parameters are also useful in testing different models. These features, together with the different possibilities for data treatment (in aqueous or non-aqueous medium, pH or e.m.f. data, and refinement of thermodynamic or stoichiometric formation constants) make PKPOT a versatile program for the evaluation of stability constants from potentiometric
J. Barbosa et al. /Analytica
titration data. The program, together with sample data files, is available upon request from the authors.
References [II
L.G. Sill&, Acta Chem. &and.,
18 (1964) 1085.
Dl A. Sabatini, A. Vacca and P. Gans, Talanta, 21 (1974) 53. [31 P. Gans, A. Sabatini and A. Vacca, J. Chem. Sot., Dalton Trans., (1985) 1195. 141 A. Zuberbiiler and T.A. Kaden, Talanta, 29 (1982) 201. [51 R.J. Motekaitis and A.E. Martell, Can. J. Chem., 60 (1982) 725. [61 R.M. Alcock, F.R. Hartley and D.E. Rogers, .I. Chem. Sot., Dalton Trans., (1978) 115. Anal. Chim. Acta, 181 (1986) [71 A. Izquierdo and J.L. Belt& 87. (81 F. Gaizer and 1.1. Kiss, Talanta, 41 (1994) 419. 191 I. Ting-Po and G.H. Nancollas, Anal. Chem., 44 (1972) 1940.
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T. Zikolova and J. Tencheva, Talanta, 35 (1988) 899. (111 M. Ro&, Anal. Chim. Acta, 276 (1993) 211. 1121M. Rests, Anal. Chim. Acta, 276 (1993) 223. [131M. Rests, Anal. Chim. Acta, 285 (1994) 391. [141J.L. Belt&. R. Codony and M.D. Prat, Anal. Chim. Acta, 276 (1993) 441. [151 A. Izquierdo and J.L. BeltrBn, J. Chemom., 3 (1988) 209. iI61 E. Durand, Solutions Numeriques des Equations Algbbriques, Tome II: Systtmes de Plusieurs Equations, Masson, Paris, 1972. (171 W.E. Wenhvorth, J. Chem. Educ., 42 (1965) 96. of [181 M. Meloun, J. Have1 and E. Hiigfeldt, Computation Solution Equilibria: a Guide to Methods in Potentiometry, Extraction and Spectrophotometry, Ellis Horwood, Chichester, 1988. [191 J. Barbosa and D. Barr6n, Analyst, 114 (1989) 471. and V. Sanz-Nebot, Anal. Chim. I201 J. Barbosa. J.L. Belt& Acta, 288 (1994) 271.