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The use of glass electrodes for the determination of formation constants—I A definitive method for calibration
0039-9140/82/040249-08$03.00/O Copyright 0 1982 Pergamon Press Ltd
Jcrlunru.Vol. 29. PP. 249 to 256, 1982 Printed in Great Britain.All rights reserved
THE USE OF GLASS ELECTRODES FOR THE DETERMINATION OF FORMATION CONSTANTS-I A DEFINITIVE
METHOD
PETER M. MAY*
and
FOR
DAVID
CALIBRATION
R. WILLIAMS,
Department of Chemistry, UWIST, Cardit?: Wales and PETER
W.
LINDER
and
RALPH G. TORRINGTON
Department of Physical Chemistry, University of Cape Town, Rondebosch, South Africa 7700 (Received 2 July 1981. Accepted 9 October 1981)
Summary-The
computer program, MAGEC (Multiple Analysis of titration data for Glass Electrode Calibration), described can optimize simultaneously any or all of the titration parameters pertinent to the calibration of glass electrodes. In particular, the protonation constants of a ligand and the glasselectrode parameters may be determined simultaneously from a given single set of titration data. A method is described for obtaining definitive values of these parameters, involving cyclical treatment of titration data by means of MAGEC and another program such as MINIQUAD. The use of MAGEC for evaluating the performance of a glass electrode or as an analytical tool for determining, with high precision, the concentrations of acid and base solutions, is also described.
three decades of extensive effort directed towards obtaining unequivocal formation constants for ligand-metal systems by glass-electrode potentiometry, there remain worrying uncertainties concerning the accurate calibration of the electrodes. Originally, glass electrodes were calibrated against buffers of specific pH but, although this approach remains valid in work where only a relative scale of acidity is required, it is unsatisfactory nowadays in metalligand equilibrium studies. Indeed, owing to the difficulties involved in relating pH values to hydrogen-ion activities, IUPAC has introduced an operational definition of PH.’ Even if a specific mathematical relationship between the dejned pH and the hydrogen-ion activity of a buffer is assumed, difficulties remain in inferring the hydrogen-ion concentration values of a test solution, such as would be required in the determination of the protonation constants of a ligand and the formation constants of ligand-metal complexes. First, the hydrogen-ion activity coefficients are not the same in the buffer and in the test solutions, because of their different ionic compositions. Attempts to overcome this difficulty by applying various types of theoretical or empirical correction2-g introduce some inherent inaccuracy. *O*l1 Moreover, differences in liquidjunction potential obtained with the buffer and with the test solution impose further uncertainty.10v1 l These problems with the use of buffers in the calibration of glass electrodes are exacerbated when high concentrations of background electrolyte are used for Despite
* To whom correspondence
should be addressed. 249
controlling the ionic strength of the test solutions. potentiometric Thus, researchers who employ methods to measure formation constants have long used solutions of known hydrogen-ion concentration instead.‘* This is often done by titrating strong acid solutions with strong alkali and plotting the emf at each point against the corresponding values of log[H+], where [H’] denotes the free hydrogen-ion concentration. The procedure aims to obtain a linear calibration curve for the electrode system. There are several reasons why an ideal response is not observed in practice, however. Chiefly, it is found that unless there is a sufficient excess of acid or alkali to ensure that the solution is concentration-buffered, small errors become significant. It has, for example, been shown that the presence of glass itself causes a large deviation from linearity,13 a fact that can probably be attributed to adsorption of hydrogen ions onto the glass surface or to dissolution of minute amounts of the glass. Another important factor is the imperfect behaviour of glass electrodes in alkaline solution: many of the types of glass used for electrode manufacture become increasingly sensitive to metal ions, especially sodium ions, above -log[H+] = 11.0.14 The effect of hydrogen-ion concentrations on liquid-junction potentials also tends to cause some deviation at high concentrations of hydrogen ion. Accordingly, there is a restricted range of free hydrogen-ion concentration over which strong-acid us. strong-base titration data are suitable for calibration purposes. In the authors’ experience, the most nearly linear response occurs in the -log[H+] ranges 2.3-2.9 and 10.8-11.3. Data collected inside these
250
PETER M. MAY et
limits with most research equipment can yield a level of precision that is adequate for the determination of formation constants for biological systems. What is less satisfactory, however, is that the critical measurements with most systems lie outside the calibration range. Thus the hydrogen-ion concentrations must be procured by extrapolation. This makes them sensitive to small errors in the analytical concentrations of the acid. Also, it is well known that the standard potential of the glass membrane varies from day to day (due to asymmetry effects) and that it is difficult to reproduce liquid-junction potentials with adequate precision. These factors can be significant even from one experiment to another.“‘“” Thus internal calibrations of the electrode, performed in the test solution itself; are highly desirable.
In principle, there are two main ways of achieving this goal. The first applies to solutions in which any strong acid added is completely dissociated and gives a correspondingly increased free hydrogen-ion concentration. It is then possible to calibrate by a series of constant additions. Alternatively, in titrations of weak acids or bases, free hydrogen-ion concentrations can be calculated at various points in the titration from the protonation constants, provided these are accurately known. Very precise calibrations can be made in this way but they are of limited use. Indeed, the object of many titrations is to measure the protonation constants. In these cases a number of parameters, including glass-electrode properties and one or more equilibrium constants, must be determined simultaneously. Straightforward solutions are very rarely available so general optimization techniques must be employed. This paper describes an approach for the optimization of some or all of the parameters required for calibrating glass-electrode systems and shows how protonation constants of a ligand can, if desired, be determined simultaneously. The relevant algorithms are incorporated in a Fortran program named MAGEC (Multiple Analysis of titration data for Glass Electrode Calibration), the coding for which is available to interested readers on request.* THEORY
Consider an electrochemical cell in which a test solution surrounding a glass electrode is in electrical contact with a reference electrode through a salt bridge. It can be represented as follows:
* Supplementary Publication PMM No. 10 (26 pages), on application to Department Administrator, Department of Chemistry, UWIST, Cardiff, Wales, enclosing E10.00 (EEC countries) or f 15.00 (elsewhere), to cover postage and packing.
al.
The boundaries, g and I, respectively, indicate the glass membrane and the liquid junction at the interface between the salt bridge and the test solution. There are four contributions to the measurable potential difference between the two reference electrodes.“” Two arise from the reference electrodes themselves. They will have opposite signs and will usually be of comparable magnitudes. Most importantly, their contribution will be independent of the composition of the test solution and so may be represented as a fixed combined potential, E,. On the other hand, the potential differences generated across the boundaries of g and 1 will depend on the activities of all the chemical species on either side of them. If these boundary potentials are represented by E, and E,, the measured emf of the cell is given by the equation. E cell = E, + E, + E,
(1)
In the case of the liquid junction, considerable changes in the composition of the test solution are required to alter El significantly, so, for the time being, this will be considered as a constant. Glass electrodes, in general, are found experimentally to exhibit a Nernstian response over a wide range of concentration.‘6 Equation (1) can be rewritten as E ..,,=E,+E,+$+~ln(H+)
(2)
where Ei is the standard glass-electrode potential at unit activity of hydrogen ions, R is the universal gas constant, T is the absolute temperature, F is the Faraday constant and {H+ } represents the hydrogenion activity. As long as the ionic strength of the test solution remains constant, the free hydrogen-ion activity, {H+ }, can be expressed in terms of concentration. Hence, equation (3) is obtained by putting s = 2.303RT/F and collecting together all the constants (including the hydrogen-ion activity coefficient and factors arising from it) as EEons,: E ecu - Em,
+ slog[H+]
(3)
In circumstances where strong-acid us. strong-base titrations are to be used in the calibration of an electrode system (i.e., in the approximate pH ranges 2.3-2.9 and 10.8-l 1.3), MAGEC uses a subroutine entitled CALIBT. CALIBT first analyses the data by the method of Gran.“*” Since the potentiometric data are transformed into a linear form, this analysis gives a good indication of glass-electrode performance and also yields an end-point that is independent of the slope and intercept used in equation (3). Furthermore, if extrapolation of the data from before the end-point produces a value significantly lower than that obtained from data after the end-point, it is possible that an alkaline titrant may have become contaminated with carbon dioxide from the atmosphere.” The end-point obtained from the Gran extrapolations
251
Formation constants-I
RUN PROGRAM MAGEC )
Fix: APKW, ACIOB. SLOPE ACIOV. EZERO
Refine:
I
A
Is the
x value?
V
STOP
parameter is very highly correlated with the concentration of alkali in the burette, i.e., a small error in its value will cause a significant deviation in the apparent slope of the electrode response, because it is used to convert the hydroxide-ion excess into free hydrogenion concentrations, and so it manifests itself in the optimized alkali concentration. Another way of looking at this is that either the titrant concentration or K, (but not both) can be determined by finding the value which yields the most ideal least-squares slope. To accommodate those situations in which K, is uncertain, CALIBT permits the user to vary the estimate systematically. The recommended procedure for using CALIBT is illustrated in Fig. 1. As pointed out earlier, it is often necessary to determine E,,,,, and the ligand protonation constants simultaneously. In such circumstances, two further equations in addition to equation (3) are applicable. These are the mass-balance equations, (4) and (5) for the ligand and for protons.
~~= [L] + C C PS,CLI~CH+I’ G = CH’I + ~~r&CLlpCH’I’. P
symbols =
apparent dissociation constant of water (KU)
ACIOV
=
titrand acid concentration (negative for alkali)
AC109
=
titrant acid concentration (negative for
EZERO
=
electrode intercept (Econst)
q
electrode slope
APKW
SLOPE
(5)
r
Here, TL and Tu are the total concentrations of ligand and protons, respectively; [L] and [H’] are the free alkali) Concentrations of linand and protons, respectively; 8, is the equilibrium constant for the general reaction, =
(4)
pL + rH + = L,H,
Fig. 1. provides an independent check of the CALIBT optimization of the strong acid concentration referred to below. Further processing of strong-acid us. strong-base titrations is divided into three stages. To begin with, the input concentration values are used to calculate free hydrogen-ion concentrations at each point, and a linear least-squares fit is performed on the data before and after the end-point and over the entire range. This first analysis is used mainly for comparison with subsequent output. Because of the effect of relatively small errors in the concentrations of titrant and titrand, the least-squares straight line does not normally possess a Nernstian slope, so the concentration of the titrand is varied slightly until the slope from the data before the end-point becomes Nernstian. It is then possible to adjust the titrant concentration in a similar manner, but on the basis of the whole range of data. Of course, to maintain the same end-point, a corresponding change in the titrand concentration also needs to be made. In this way, very close agreements between the calculated and observed values for the emf at each point can be achieved. A factor that critically affects the refinement of the titrant concentrations in the final stage is the value used for the dissociation constant of water, K,. This
It is clear that (3), (4) and (5) constitute only two independent equations for each titration point. TL and TH may be expressed in terms of the initial volume of the titrant, the volume of each added increment of titrant and K,. E is measured at each titration point. It follows that for n titration points, there are 2n independent equations containing (pr + n + 2) unknowns, namely, the two electrode parameters E,,,, and s, pr /l-values, and n free-ligand concentrations. Thus, in principle, the 2n equations may be solved simultaneously to yield E,,,, and the b-values from a single set of titration data, provided that the number of titration points is sufficiently large. The values of TL and TH are subject to analytical error; the value of K, is sometimes uncertain, which imposes additional uncertainty on TH. Thus, in a sense, TL, TH and K, may be regarded as additional unknowns. In the analysis of data obtained for the titration of monoprotic ligands, the main routine in MAGEC utilizes a number of approximation formulae, for example, the Henderson-Hasselbalch equation, -log[H+]
= -logK,
+ log-
[salt] [acid]’
to solve for the free hydrogen-ion concentration at each point in the titration. Analysis of the data then
252
PETERM. MAY et al.
0
START
k
,
t
7
RUN PROGRAM MINIQUAD (all titrations)
1
Fix: APKW. ACIOV, LIGIV, ACIDB, EZERO, SLOPE Refine:
LBETA
1 RUN PROGRAM MAGEC (individual titrations) Fix:
LBETA, APKW.
Refine:
ACIDV, LIGIV, ACIDB, SLOPE
EZERO
I
Symbols LBETA =
ligand protonation constants
APKW
=
apparent dissociation constant of water
ACIDV
q
titrand acid concentration (negative for alkali)
ACIDB
=
titrant acid concentration (negative for alkali)
LIGIV
=
ligand concentration in vessel
EZERO =
electrode intercept
SLOPE
electrode slope
=
Fig. 2. follows a similar approach to that described above for strong acids and strong bases. On the whole, however, this is not very satisfactory because it is much more difficult to know when the solution can be considered to be reasonably well buffered. Accordingly, the main analysis applied by MAGEC to all titrations involving ligands is one of general optimization of parameter values by using an objective function based on titration volumes. The sum of squared residuals is minimized by the simplex method introduced by Nelder and Mead.20 Any of the parameters EEOO.,,s, the /I-values, TL, TH and K, can be flagged for refinement, so the specific procedure is left largely in the hands of the user. Usually, the requirement is to find the value for E,,,,, in each of a series of titrations with the ultimate objective of determining the protonation constants of a ligand.
ing a CALIBT analysis to the titration data for a strong acid vs. a strong base. An additional advantage of this step stems from the possibility of using the CALIBT analysis for checking the reliability of an individual glass electrode. Lifetimes of glass electrodes differ considerably and it is often difficult to detect the first signs of deteriorating performance. In the MAGEC analysis of strong-acid vs. strong-base titration data, however, the appearance of sudden and marked increases in the titration-volume residuals signals the imminent demise of the electrode. A second useful criterion of incipient electrode unreliability is a tendency for discrepancies to develop between the end-points determined by CALIBT minimization of the Nernstian response and by Gran plot. 2. The E,,,, value obtained in step 1 should be used in conjunction with data obtained for titration of a ligand (perhaps acidified with mineral acid) with a strong base, in a suitable program for refining the values of the protonation constants for the ligand. The authors favour MINIQUAD2’ for this purpose. Individual titrations (i.e., either replicate titrations or titrations differing in the total concentration of ligand) should be processed together to yield global values for each protonation constant. 3. The ligand-titration data and the protonation constants from step 2 are processed by the main routine of MAGEC to yield refined E,,,,, values. 4. Steps 2 and 3 are repeated cyclically until convergence to a satisfactory degree is obtained. 5. If desired the values of E,,,,, and the protonation constants may be improved further by refining the values of TH, TL and K, in a series of subsequent cycles. Care must be taken, however, to ensure that the calculation is not performed with more degrees of freedom than are warranted by the accuracy of the data. If this precaution is neglected, all optimization procedures can give rise to spurious results. The number of parameters which can be refined with safety from data with a given experimental precision depends on the particular system being investigated. Thus, it is important to ensure that the refinement includes only those parameters for which optimization leads to a sufficient improvement in the sum-of-squares function that is to be minimized. The flow diagram of Fig. 2 illustrates steps (2)-(4).
RESULTS
An example of the calibration of a glass electrode by strong-acid vs. strong-base titration THE USE OF MACEC
IN PRACTICE
In our experience, the calibration of an electrode system is best carried out in the following steps. 1. Even when the calibration is needed outside the pH ranges 2.3-2.9 or 10.8-11.3, it is recommended that an initial estimate of E,,,,, be obtained by apply-
A mixture of hydrochloric acid (10.00 ml, co. 5OmM and sodium chloride (20.00 ml, 212.0mM) was titrated with sodium hydroxide (lOO.OmM) and sodium chloride solution. The sodium chloride was used to maintain the ionic strength (ca. 0.15M) as constant as possible. The solution was maintained at 37.0 * 0.1”.
253
Formation constants-l Table 1. Strong-acid us. strong-base titration
Before CALIBT optimization Using only the data before the end-point (47 points) (i) electrode intercept, ml/ (ii) electrode slope, mV (iii) overall standard deviation, mV (io) number of residuals greater than 0.1 mV Usina all the buffered data (6fpoints) (ii electrode interced.. mV (ii) electrode slope, mV (iii) overall standard deviation, mV (iv) number of residuals greater than 0.1 mV
After optimization of the acid concentration
362.4 f 0.15 62.48 & 0.07 1.4 x 10-l 26
360.7 &-0.04 61.53 _?r0.02 4.1 x 1o-2 0
360.4 + 0.10 61.53 : 0.02 5.0 x 10-r 41
360.6 + 0.05 61.50 i 0.01 2.6 x 10-r 15
Notes (i) The optimized initial acid concentration of 16.59mM compares with an expected value of 16.67mM and yields an end-point of 4.978 ml. (ii) A slope of 61.50 mV corresponds to pK, = 13.310. In practice, the calculation could be repeated using pK, = 13.305 to obtain a better agreement with the Nernstian slope (61.54 mV).
A Gran-plot analysis for MAGEC gave end-points of 4.977 f 0.002 and 4.97 f 0.03 ml for the acid and alkaline pH data respectively. The CALIBT optimization analysis is summarized in Table 1. An example of the internal calibration of a glass electrode in titration of a ligand solution Duplicate titrations, using different electrode systems, were performed on a solution of acetic acid (20.00 ml, ca. 20mM) with 0.150M sodium hydroxide. A background concentration of perchlorate ions (0.150M) was maintained and the solution was kept at 25.0 + 0.1’. The protonation constant and the initial
Table 2. Acetic acid vs. strong base titrations
PK Electrode Intercept mV [acetate], mM
Titration 1
Titration 2
4.522
4.524
- 367.5 20.83
- 379.9 20.81
concentration of the acetic acid as well as E,,,,, for each electrode system were refined simultaneously by MAGEC. A Nernstian slope of 59.16 mV per log[H+] was assumed throughout. The results are shown in Table 2. MINIQUAD analysis of these titrations, using the electrode intercepts obtained by MAGEC and an initial value of [acetate] = 20.82 mM, yielded a refined pK, value 4.523 + 0.0004 with a sum of squared residuals of 2.3 x lo-* and a MINIQUAD R factor of 0.0008. An example of the internal calibration of a glass electrode in a student titration of ligand solution In a fourth-year exercise, a pair of students titrated a solution of glycine (25.00 ml, 9.17mM) in hydrochloric acid (219OmM). A background concentration of chloride ions (l.OOM) was maintained and the solution was kept at 25.0” + 0.1. Table 3 indicates the progress of the MINIQUADMAGEC cycling refinement.
Table 3. Honours students’ titration: glycine OS.strong base initial values PKW P&I P&Z Electrode Intercept, mV [glycine], mM [NaOH], mM Sum of squared residuals
13.69 2.4 9.6 400.0 9.171 49.65 151.3
Refined values after first MAGEC run (29 iterations)
382.1
0.4289
Refined values after 5 MINIQUADMAGEC cycles 13.63 2.318 9.583 381.7 9.176 49.63 0.04859
254
PETERM. MAY et al. Table 4. Honours students’ titration: glycine US.strong base
Titre volume, ml
Observed emf, mV
0.000 0.020 0.040 0.060 0.381 0.875 1.358 1.838 2.318 2.792 3.255 3.721 4.177 4.632 5.076 5.513 5.943 6.368 6.781 7.181 7.575 7.957 8.322 8.778 9.013 9.328 9.618 9.881 10.122 10.311 10.479 10.602 10.703 10.779 10.838 10.881 10.920 10.952 10.975 10.998 11.019 11.039 11.059 11.079 11.099 11.133 11.173 11.224 11.286 11.365 11.458 11.574 11.708 11.866 12.039 12.241 12.437 12.664 12.908 13.166 13.428 13.693 13.964 14.237 14.510 14.767 15.022
273.30 273.10 273.05 273.00 271.80 269.70 267.60 265.50 263.30 260.90 258.60 256.10 253.60 250.90 248.10 245.20 242.20 239.00 235.60 232.10 228.40 224.40 220.30 215.80 211.00 205.80 200.20 194.30 187.50 181.00 173.60 166.70 159.40 152.40 145.20 138.40 130.70 121.50 112.30 99.40 77.40 36.50 - 26.00 -53.40 - 66.50 - 80.70 -91.30 - 101.30 - 109.70 -117.90 - 125.10 - 132.40 - 138.90 - 146.00 - 151.40 - 158.70 - 164.10 - 169.70 - 175.10 - 180.70 - 186.20 - 191.60 - 197.00 - 202.50 - 208.40 -214.20 - 220.00
Initial residuals, ml 3.048 3.061 3.053 3.045 2.966 2.870 2.770 2.661 2.552 2.459 2.349 2.244 2.133 2.027 1.923 1.817 1.706 1.596 1.490 1.380 1.265 1.153 1.035 0.832 0.804 0.692 0.586 0.483 0.389 0.311 0.242 0.189 0.146 0.112 0.086 0.067 0.048 0.034 0.023 0.012 0.002 -0.010 -0.012 0.002 0.015 0.041 0.071 0.115 0.162 0.222 0.284 0.358 0.426 0.521 0.565 0.678 0.732 0.773 0.790 0.800 0.795 0.773 0.735 0.688 0.648 0.613 0.582
Residuals after 1st MAGEC run, ml 0.076 0.098 0.092 0.086 0.060 0.059 0.053 0.036 0.022 0.030 0.016 0.013 0.001 0.000 0.001 0.001 - 0.003 -0.002 0.005 0.008 0.007 0.013 0.010 -0.075 0.011 0.013 0.015 0.011 0.013 0.011 0.011 0.010 0.008
- 0.022 - 0.024 - 0.026 - 0.027 - 0.025 - 0.026 - 0.024 - 0.025 -0.023 -0.027 -0.015 - 0.034 0.000 0.004 0.004 - 0.005 -0.007 -0.010 -0.017 - 0.032 -0.051 -0.061 - 0.072 -0.092
Final residuals, ml 0.012 0.035 0.029 0.023 -0.001 0.002 0.000 -0.013 - 0.022 -0.009 -0.019 -0.017 -0.024 -0.021 -0.015 -0.012 -0.011 - 0.007 0.004 0.010 0.012 0.020 0.019 - 0.065 0.022 0.024 0.026 0.021 0.023 0.020 0.019 0.017 0.015 0.012 0.011 0.010 0.007 0.006 0.005 0.002 -0.000 -0.008 -0.016 -0.018 -0.019 - 0.020 -0.021 -0.018 -0.017 -0.013 -0.013 - 0.009 -0.010 0.004 -0.011 0.026 0.033 0.035 0.029 0.029 0.029 0.023 0.010 -0.006 -0.014 -0.021 -0.037
Formation constants-I
255
Table 4.-continued
Titre volume, ml
Observed emf, mV
Initial residuals, ml
15.276 15.521 15.761 16.003 16.248 16.508 16.778 17.068 17.378 17.713 la.00
-226.10 - 232.20 - 238.20 - 244.00 - 249.40 - 254.70 - 259.50 - 264.00 -268.10 - 272.00 - 275.00
0.571 0.590 0.642 0.730 0.849 1.019 1.221 1.463 1.734 2.057 2.363
A Nernstian slope of 59.16 mV per log [H’] was assumed throughout. The refinement, within the first run, by MAGEC of the electrode intercept to within 0.4 mV of the final value, is impressive. This is in spite of the use of approximate acid dissociation-constant values. Table 4 shows how the magnitudes and distributions of the residuals have been improved.
DlSCUSSlON
The accurate experimental determination of hydrogen-ion concentrations is important in many areas of chemistry, often because the measurements provide information about other chemical species which occur in solution. Despite this, considerable differences exist in the way in which the electrode systems are calibrated. For the determination of formation constants, many investigators still employ the pH-scale, as evidenced by the plethora of publications describing how pH measurements can be adjusted to yield values for hydrogen-ion concentration.2-9 A standardized approach, based directly on hydrogen-ion concentrations, is clearly needed. This would facilitate both the comparison of formation-constant values and elimination of errors inherent in the use of the pHscale. In this respect, numerical analysis of titration data, to determine parameters such as E,,,,, would appear to have many advantages. The results presented here demonstrate the efficacy of MAGEC in the calibration of a glass-electrode system both in the strong-acid us. strong-base titration range and in the hydrogen-ion concentration range where the interaction of a given ligand with protons or a metal ion is significant. A strong advantage lies in the ability of MAGEC together with MAGECMINIQUAD cycling to solve this problem even when the protonation constants of the ligand are not known a priori. Thus, E,,,, is optimized for a given titration, and any change in the liquid-junction potential is incorporated into the particular E,,,,, value
Residuals after 1st MAGEC run, ml -0.109 -0.123 -0.135 -0.146 -0.160 -0.169 -0.181 -0.194 -0.214 -0.231 -0.240
Final residuals, ml - 0.047 -0.051 - 0.050 -0.045 -0.040 - 0.024 -0.007 0.012 0.029 0.053 0.081
obtained. Small variations in the liquid-junction potential arising, for example, from poor reproducibility or concentration effects, cease to be a problem. This is a significant improvement over less general approaches such as that of Gaizer and Puskh.” A further marked advantage is derived from the MAGEC (CALIBT) analysis of strong-acid us. strongbase data in assessing the reliability of a given glass electrode. Yet another feature of the MAGEC (CALIBT) routine lies in the analytical facility whereby the concentration of a given strong acid or strong base solution can be determined more precisely than by the Gran method. As far as the main routine in MAGEC, as applied to ligand-proton equilibrium, is concerned, the text describes how, in principle, the parameters EEO”S,,s, /?-values, total concentrations of ligand and protons and K,, can all be determined provided there are sufficient titration points. In practice, however, limitations arise, mainly from the correlation of errors between two or more of the parameters. This results from an effect in which small differences in the values of these particular parameters have similar (increasing) effects on the function to be minimized. Thus, the errors in the parameters concerned tend to compensate each other, leading to false mathematical solutions. The difficulties arising from this source may be restricted by the MAGEC-MINIQUAD cycling procedure, which provides for only one or a selected few of the parameters to be estimated by MAGEC. While work on MAGEC has been in progress, another program with similar objectives has appeared in the literature.23 Like MAGEC, this permits the optimization of any or all of the titration parameters. However, it does not offer the additional facilities simultaneously. Hence, the two programs partially overlap and partially complement one another.
Acknowledgements-P.W.L. and R.G.T. gratefully acknowledge generous grants from the South African C.S.I.R. and the Council of the University of Cape Town. Honours students Valerie Mizrahi, Sybella Meltzer and Pedro Monteiro, are thanked for the data used in Tables 3 and 4.
PETER
256
M.
REFERENCES
1. Manual of Symbols and Terminology for Physicochemical Quantities and Units, 1973, International Union of Pure and Applied Chemistry, Oxford, 1975. 2. W. A. E. McBryde, Talanta, 1969, 94, 337; Analyst, 1971,%,739.
3. C. W. Childs and D. D. Perrin, J. Chem. Sot. (A), 1969, 1039. 4. G. R. Hedwig and H. K. J. Powell, Anal. Chem., 1971, 43, 1206. 5. H. S. Dunsmore and D. Midgley, Anal. Chim. Acta, 1972, 61, 115. 6. E. W. Baumann, ibid., 1973, 64, 284. 7. M. Bos and W. Lengton, ibid., 1975, 76, 149. 8. R. G. Bates, in Analytical Chemistry. Essays in Memory of Anders Ringbom, E. Wiinninen, (ed.), p. 23. Pergamon Press, Oxford, 1977. 9. A. K. Covington, Anal. Chim. Acta, 1981, 127, 1. 10. H. M. Irving, M. G. Miles and L. D. Pettit, ibid., 1967, 38,415.
11. G. Mattock and D. M. Band, in Glass Etectrodesfir Hydrogen Ion and Other Cations, G. Eisenman (ed.), Chapters 2, 9. Dekker, New York, 1967. 12. G. Anderegg, in Proc. Summer School on Stability Con-
MAY et al.
stants, (Bivigliano, Florence, Italy, June 1974). P. Pao-
letti, R. Barbucci and L. Fabbrizzi (eds.), p. 11. 13. D. R. Williams, in Proc. Summer School on Stability Consrants. (Bivieliano. Florence, Italy. June 1974), P. Paoletti, R.‘Barbucci and L. Fabbrizzi (eds.), p. 125. 14. K. Schwabe, Advan. Anal. Chem. Instrum., 1974, 10, 495.
15. M. F. G. Camoes and A. K. Covington, Anal. Chem., 1914,46, 1547. 16. H. Rossotti, The Study of Ionic Equilibria. Longmans, London, 1978. 17. A. Albert and E. P. Serjeant, The Determination 01 lonisation Constants, Chapman & Hall, London, 1971. 18. G. Gran, Acta Chem. Stand., 1950, 4, 559; Analyst, 1952, 77, 661.
19. F. J. C. Rossotti and H. Rossotti, J. Chem. Educ., 1965, 42, 375. 20. J. A. Nelder and R. Mead, Comput. J., 1965, 7, 308. 21. A. Sabatini. A. Vacca and P. Gans. Talanta, 1974, 21, 53; P. Gans, A. Sabatini and A. Vacca, Inorg. Chim. Acta. 1976, 18, 237. 22. F. Gaizer and A. Puskls, Talanta, 1981,28, 565. 23. G. Arena, E. Rizzarelli, S. Sammartano and C. Rigano, ibid., 1979, 26, 1.
Jcrlunru.Vol. 29. PP. 249 to 256, 1982 Printed in Great Britain.All rights reserved
THE USE OF GLASS ELECTRODES FOR THE DETERMINATION OF FORMATION CONSTANTS-I A DEFINITIVE
METHOD
PETER M. MAY*
and
FOR
DAVID
CALIBRATION
R. WILLIAMS,
Department of Chemistry, UWIST, Cardit?: Wales and PETER
W.
LINDER
and
RALPH G. TORRINGTON
Department of Physical Chemistry, University of Cape Town, Rondebosch, South Africa 7700 (Received 2 July 1981. Accepted 9 October 1981)
Summary-The
computer program, MAGEC (Multiple Analysis of titration data for Glass Electrode Calibration), described can optimize simultaneously any or all of the titration parameters pertinent to the calibration of glass electrodes. In particular, the protonation constants of a ligand and the glasselectrode parameters may be determined simultaneously from a given single set of titration data. A method is described for obtaining definitive values of these parameters, involving cyclical treatment of titration data by means of MAGEC and another program such as MINIQUAD. The use of MAGEC for evaluating the performance of a glass electrode or as an analytical tool for determining, with high precision, the concentrations of acid and base solutions, is also described.
three decades of extensive effort directed towards obtaining unequivocal formation constants for ligand-metal systems by glass-electrode potentiometry, there remain worrying uncertainties concerning the accurate calibration of the electrodes. Originally, glass electrodes were calibrated against buffers of specific pH but, although this approach remains valid in work where only a relative scale of acidity is required, it is unsatisfactory nowadays in metalligand equilibrium studies. Indeed, owing to the difficulties involved in relating pH values to hydrogen-ion activities, IUPAC has introduced an operational definition of PH.’ Even if a specific mathematical relationship between the dejned pH and the hydrogen-ion activity of a buffer is assumed, difficulties remain in inferring the hydrogen-ion concentration values of a test solution, such as would be required in the determination of the protonation constants of a ligand and the formation constants of ligand-metal complexes. First, the hydrogen-ion activity coefficients are not the same in the buffer and in the test solutions, because of their different ionic compositions. Attempts to overcome this difficulty by applying various types of theoretical or empirical correction2-g introduce some inherent inaccuracy. *O*l1 Moreover, differences in liquidjunction potential obtained with the buffer and with the test solution impose further uncertainty.10v1 l These problems with the use of buffers in the calibration of glass electrodes are exacerbated when high concentrations of background electrolyte are used for Despite
* To whom correspondence
should be addressed. 249
controlling the ionic strength of the test solutions. potentiometric Thus, researchers who employ methods to measure formation constants have long used solutions of known hydrogen-ion concentration instead.‘* This is often done by titrating strong acid solutions with strong alkali and plotting the emf at each point against the corresponding values of log[H+], where [H’] denotes the free hydrogen-ion concentration. The procedure aims to obtain a linear calibration curve for the electrode system. There are several reasons why an ideal response is not observed in practice, however. Chiefly, it is found that unless there is a sufficient excess of acid or alkali to ensure that the solution is concentration-buffered, small errors become significant. It has, for example, been shown that the presence of glass itself causes a large deviation from linearity,13 a fact that can probably be attributed to adsorption of hydrogen ions onto the glass surface or to dissolution of minute amounts of the glass. Another important factor is the imperfect behaviour of glass electrodes in alkaline solution: many of the types of glass used for electrode manufacture become increasingly sensitive to metal ions, especially sodium ions, above -log[H+] = 11.0.14 The effect of hydrogen-ion concentrations on liquid-junction potentials also tends to cause some deviation at high concentrations of hydrogen ion. Accordingly, there is a restricted range of free hydrogen-ion concentration over which strong-acid us. strong-base titration data are suitable for calibration purposes. In the authors’ experience, the most nearly linear response occurs in the -log[H+] ranges 2.3-2.9 and 10.8-11.3. Data collected inside these
250
PETER M. MAY et
limits with most research equipment can yield a level of precision that is adequate for the determination of formation constants for biological systems. What is less satisfactory, however, is that the critical measurements with most systems lie outside the calibration range. Thus the hydrogen-ion concentrations must be procured by extrapolation. This makes them sensitive to small errors in the analytical concentrations of the acid. Also, it is well known that the standard potential of the glass membrane varies from day to day (due to asymmetry effects) and that it is difficult to reproduce liquid-junction potentials with adequate precision. These factors can be significant even from one experiment to another.“‘“” Thus internal calibrations of the electrode, performed in the test solution itself; are highly desirable.
In principle, there are two main ways of achieving this goal. The first applies to solutions in which any strong acid added is completely dissociated and gives a correspondingly increased free hydrogen-ion concentration. It is then possible to calibrate by a series of constant additions. Alternatively, in titrations of weak acids or bases, free hydrogen-ion concentrations can be calculated at various points in the titration from the protonation constants, provided these are accurately known. Very precise calibrations can be made in this way but they are of limited use. Indeed, the object of many titrations is to measure the protonation constants. In these cases a number of parameters, including glass-electrode properties and one or more equilibrium constants, must be determined simultaneously. Straightforward solutions are very rarely available so general optimization techniques must be employed. This paper describes an approach for the optimization of some or all of the parameters required for calibrating glass-electrode systems and shows how protonation constants of a ligand can, if desired, be determined simultaneously. The relevant algorithms are incorporated in a Fortran program named MAGEC (Multiple Analysis of titration data for Glass Electrode Calibration), the coding for which is available to interested readers on request.* THEORY
Consider an electrochemical cell in which a test solution surrounding a glass electrode is in electrical contact with a reference electrode through a salt bridge. It can be represented as follows:
* Supplementary Publication PMM No. 10 (26 pages), on application to Department Administrator, Department of Chemistry, UWIST, Cardiff, Wales, enclosing E10.00 (EEC countries) or f 15.00 (elsewhere), to cover postage and packing.
al.
The boundaries, g and I, respectively, indicate the glass membrane and the liquid junction at the interface between the salt bridge and the test solution. There are four contributions to the measurable potential difference between the two reference electrodes.“” Two arise from the reference electrodes themselves. They will have opposite signs and will usually be of comparable magnitudes. Most importantly, their contribution will be independent of the composition of the test solution and so may be represented as a fixed combined potential, E,. On the other hand, the potential differences generated across the boundaries of g and 1 will depend on the activities of all the chemical species on either side of them. If these boundary potentials are represented by E, and E,, the measured emf of the cell is given by the equation. E cell = E, + E, + E,
(1)
In the case of the liquid junction, considerable changes in the composition of the test solution are required to alter El significantly, so, for the time being, this will be considered as a constant. Glass electrodes, in general, are found experimentally to exhibit a Nernstian response over a wide range of concentration.‘6 Equation (1) can be rewritten as E ..,,=E,+E,+$+~ln(H+)
(2)
where Ei is the standard glass-electrode potential at unit activity of hydrogen ions, R is the universal gas constant, T is the absolute temperature, F is the Faraday constant and {H+ } represents the hydrogenion activity. As long as the ionic strength of the test solution remains constant, the free hydrogen-ion activity, {H+ }, can be expressed in terms of concentration. Hence, equation (3) is obtained by putting s = 2.303RT/F and collecting together all the constants (including the hydrogen-ion activity coefficient and factors arising from it) as EEons,: E ecu - Em,
+ slog[H+]
(3)
In circumstances where strong-acid us. strong-base titrations are to be used in the calibration of an electrode system (i.e., in the approximate pH ranges 2.3-2.9 and 10.8-l 1.3), MAGEC uses a subroutine entitled CALIBT. CALIBT first analyses the data by the method of Gran.“*” Since the potentiometric data are transformed into a linear form, this analysis gives a good indication of glass-electrode performance and also yields an end-point that is independent of the slope and intercept used in equation (3). Furthermore, if extrapolation of the data from before the end-point produces a value significantly lower than that obtained from data after the end-point, it is possible that an alkaline titrant may have become contaminated with carbon dioxide from the atmosphere.” The end-point obtained from the Gran extrapolations
251
Formation constants-I
RUN PROGRAM MAGEC )
Fix: APKW, ACIOB. SLOPE ACIOV. EZERO
Refine:
I
A
Is the
x value?
V
STOP
parameter is very highly correlated with the concentration of alkali in the burette, i.e., a small error in its value will cause a significant deviation in the apparent slope of the electrode response, because it is used to convert the hydroxide-ion excess into free hydrogenion concentrations, and so it manifests itself in the optimized alkali concentration. Another way of looking at this is that either the titrant concentration or K, (but not both) can be determined by finding the value which yields the most ideal least-squares slope. To accommodate those situations in which K, is uncertain, CALIBT permits the user to vary the estimate systematically. The recommended procedure for using CALIBT is illustrated in Fig. 1. As pointed out earlier, it is often necessary to determine E,,,,, and the ligand protonation constants simultaneously. In such circumstances, two further equations in addition to equation (3) are applicable. These are the mass-balance equations, (4) and (5) for the ligand and for protons.
~~= [L] + C C PS,CLI~CH+I’ G = CH’I + ~~r&CLlpCH’I’. P
symbols =
apparent dissociation constant of water (KU)
ACIOV
=
titrand acid concentration (negative for alkali)
AC109
=
titrant acid concentration (negative for
EZERO
=
electrode intercept (Econst)
q
electrode slope
APKW
SLOPE
(5)
r
Here, TL and Tu are the total concentrations of ligand and protons, respectively; [L] and [H’] are the free alkali) Concentrations of linand and protons, respectively; 8, is the equilibrium constant for the general reaction, =
(4)
pL + rH + = L,H,
Fig. 1. provides an independent check of the CALIBT optimization of the strong acid concentration referred to below. Further processing of strong-acid us. strong-base titrations is divided into three stages. To begin with, the input concentration values are used to calculate free hydrogen-ion concentrations at each point, and a linear least-squares fit is performed on the data before and after the end-point and over the entire range. This first analysis is used mainly for comparison with subsequent output. Because of the effect of relatively small errors in the concentrations of titrant and titrand, the least-squares straight line does not normally possess a Nernstian slope, so the concentration of the titrand is varied slightly until the slope from the data before the end-point becomes Nernstian. It is then possible to adjust the titrant concentration in a similar manner, but on the basis of the whole range of data. Of course, to maintain the same end-point, a corresponding change in the titrand concentration also needs to be made. In this way, very close agreements between the calculated and observed values for the emf at each point can be achieved. A factor that critically affects the refinement of the titrant concentrations in the final stage is the value used for the dissociation constant of water, K,. This
It is clear that (3), (4) and (5) constitute only two independent equations for each titration point. TL and TH may be expressed in terms of the initial volume of the titrant, the volume of each added increment of titrant and K,. E is measured at each titration point. It follows that for n titration points, there are 2n independent equations containing (pr + n + 2) unknowns, namely, the two electrode parameters E,,,, and s, pr /l-values, and n free-ligand concentrations. Thus, in principle, the 2n equations may be solved simultaneously to yield E,,,, and the b-values from a single set of titration data, provided that the number of titration points is sufficiently large. The values of TL and TH are subject to analytical error; the value of K, is sometimes uncertain, which imposes additional uncertainty on TH. Thus, in a sense, TL, TH and K, may be regarded as additional unknowns. In the analysis of data obtained for the titration of monoprotic ligands, the main routine in MAGEC utilizes a number of approximation formulae, for example, the Henderson-Hasselbalch equation, -log[H+]
= -logK,
+ log-
[salt] [acid]’
to solve for the free hydrogen-ion concentration at each point in the titration. Analysis of the data then
252
PETERM. MAY et al.
0
START
k
,
t
7
RUN PROGRAM MINIQUAD (all titrations)
1
Fix: APKW. ACIOV, LIGIV, ACIDB, EZERO, SLOPE Refine:
LBETA
1 RUN PROGRAM MAGEC (individual titrations) Fix:
LBETA, APKW.
Refine:
ACIDV, LIGIV, ACIDB, SLOPE
EZERO
I
Symbols LBETA =
ligand protonation constants
APKW
=
apparent dissociation constant of water
ACIDV
q
titrand acid concentration (negative for alkali)
ACIDB
=
titrant acid concentration (negative for alkali)
LIGIV
=
ligand concentration in vessel
EZERO =
electrode intercept
SLOPE
electrode slope
=
Fig. 2. follows a similar approach to that described above for strong acids and strong bases. On the whole, however, this is not very satisfactory because it is much more difficult to know when the solution can be considered to be reasonably well buffered. Accordingly, the main analysis applied by MAGEC to all titrations involving ligands is one of general optimization of parameter values by using an objective function based on titration volumes. The sum of squared residuals is minimized by the simplex method introduced by Nelder and Mead.20 Any of the parameters EEOO.,,s, the /I-values, TL, TH and K, can be flagged for refinement, so the specific procedure is left largely in the hands of the user. Usually, the requirement is to find the value for E,,,,, in each of a series of titrations with the ultimate objective of determining the protonation constants of a ligand.
ing a CALIBT analysis to the titration data for a strong acid vs. a strong base. An additional advantage of this step stems from the possibility of using the CALIBT analysis for checking the reliability of an individual glass electrode. Lifetimes of glass electrodes differ considerably and it is often difficult to detect the first signs of deteriorating performance. In the MAGEC analysis of strong-acid vs. strong-base titration data, however, the appearance of sudden and marked increases in the titration-volume residuals signals the imminent demise of the electrode. A second useful criterion of incipient electrode unreliability is a tendency for discrepancies to develop between the end-points determined by CALIBT minimization of the Nernstian response and by Gran plot. 2. The E,,,, value obtained in step 1 should be used in conjunction with data obtained for titration of a ligand (perhaps acidified with mineral acid) with a strong base, in a suitable program for refining the values of the protonation constants for the ligand. The authors favour MINIQUAD2’ for this purpose. Individual titrations (i.e., either replicate titrations or titrations differing in the total concentration of ligand) should be processed together to yield global values for each protonation constant. 3. The ligand-titration data and the protonation constants from step 2 are processed by the main routine of MAGEC to yield refined E,,,,, values. 4. Steps 2 and 3 are repeated cyclically until convergence to a satisfactory degree is obtained. 5. If desired the values of E,,,,, and the protonation constants may be improved further by refining the values of TH, TL and K, in a series of subsequent cycles. Care must be taken, however, to ensure that the calculation is not performed with more degrees of freedom than are warranted by the accuracy of the data. If this precaution is neglected, all optimization procedures can give rise to spurious results. The number of parameters which can be refined with safety from data with a given experimental precision depends on the particular system being investigated. Thus, it is important to ensure that the refinement includes only those parameters for which optimization leads to a sufficient improvement in the sum-of-squares function that is to be minimized. The flow diagram of Fig. 2 illustrates steps (2)-(4).
RESULTS
An example of the calibration of a glass electrode by strong-acid vs. strong-base titration THE USE OF MACEC
IN PRACTICE
In our experience, the calibration of an electrode system is best carried out in the following steps. 1. Even when the calibration is needed outside the pH ranges 2.3-2.9 or 10.8-11.3, it is recommended that an initial estimate of E,,,,, be obtained by apply-
A mixture of hydrochloric acid (10.00 ml, co. 5OmM and sodium chloride (20.00 ml, 212.0mM) was titrated with sodium hydroxide (lOO.OmM) and sodium chloride solution. The sodium chloride was used to maintain the ionic strength (ca. 0.15M) as constant as possible. The solution was maintained at 37.0 * 0.1”.
253
Formation constants-l Table 1. Strong-acid us. strong-base titration
Before CALIBT optimization Using only the data before the end-point (47 points) (i) electrode intercept, ml/ (ii) electrode slope, mV (iii) overall standard deviation, mV (io) number of residuals greater than 0.1 mV Usina all the buffered data (6fpoints) (ii electrode interced.. mV (ii) electrode slope, mV (iii) overall standard deviation, mV (iv) number of residuals greater than 0.1 mV
After optimization of the acid concentration
362.4 f 0.15 62.48 & 0.07 1.4 x 10-l 26
360.7 &-0.04 61.53 _?r0.02 4.1 x 1o-2 0
360.4 + 0.10 61.53 : 0.02 5.0 x 10-r 41
360.6 + 0.05 61.50 i 0.01 2.6 x 10-r 15
Notes (i) The optimized initial acid concentration of 16.59mM compares with an expected value of 16.67mM and yields an end-point of 4.978 ml. (ii) A slope of 61.50 mV corresponds to pK, = 13.310. In practice, the calculation could be repeated using pK, = 13.305 to obtain a better agreement with the Nernstian slope (61.54 mV).
A Gran-plot analysis for MAGEC gave end-points of 4.977 f 0.002 and 4.97 f 0.03 ml for the acid and alkaline pH data respectively. The CALIBT optimization analysis is summarized in Table 1. An example of the internal calibration of a glass electrode in titration of a ligand solution Duplicate titrations, using different electrode systems, were performed on a solution of acetic acid (20.00 ml, ca. 20mM) with 0.150M sodium hydroxide. A background concentration of perchlorate ions (0.150M) was maintained and the solution was kept at 25.0 + 0.1’. The protonation constant and the initial
Table 2. Acetic acid vs. strong base titrations
PK Electrode Intercept mV [acetate], mM
Titration 1
Titration 2
4.522
4.524
- 367.5 20.83
- 379.9 20.81
concentration of the acetic acid as well as E,,,,, for each electrode system were refined simultaneously by MAGEC. A Nernstian slope of 59.16 mV per log[H+] was assumed throughout. The results are shown in Table 2. MINIQUAD analysis of these titrations, using the electrode intercepts obtained by MAGEC and an initial value of [acetate] = 20.82 mM, yielded a refined pK, value 4.523 + 0.0004 with a sum of squared residuals of 2.3 x lo-* and a MINIQUAD R factor of 0.0008. An example of the internal calibration of a glass electrode in a student titration of ligand solution In a fourth-year exercise, a pair of students titrated a solution of glycine (25.00 ml, 9.17mM) in hydrochloric acid (219OmM). A background concentration of chloride ions (l.OOM) was maintained and the solution was kept at 25.0” + 0.1. Table 3 indicates the progress of the MINIQUADMAGEC cycling refinement.
Table 3. Honours students’ titration: glycine OS.strong base initial values PKW P&I P&Z Electrode Intercept, mV [glycine], mM [NaOH], mM Sum of squared residuals
13.69 2.4 9.6 400.0 9.171 49.65 151.3
Refined values after first MAGEC run (29 iterations)
382.1
0.4289
Refined values after 5 MINIQUADMAGEC cycles 13.63 2.318 9.583 381.7 9.176 49.63 0.04859
254
PETERM. MAY et al. Table 4. Honours students’ titration: glycine US.strong base
Titre volume, ml
Observed emf, mV
0.000 0.020 0.040 0.060 0.381 0.875 1.358 1.838 2.318 2.792 3.255 3.721 4.177 4.632 5.076 5.513 5.943 6.368 6.781 7.181 7.575 7.957 8.322 8.778 9.013 9.328 9.618 9.881 10.122 10.311 10.479 10.602 10.703 10.779 10.838 10.881 10.920 10.952 10.975 10.998 11.019 11.039 11.059 11.079 11.099 11.133 11.173 11.224 11.286 11.365 11.458 11.574 11.708 11.866 12.039 12.241 12.437 12.664 12.908 13.166 13.428 13.693 13.964 14.237 14.510 14.767 15.022
273.30 273.10 273.05 273.00 271.80 269.70 267.60 265.50 263.30 260.90 258.60 256.10 253.60 250.90 248.10 245.20 242.20 239.00 235.60 232.10 228.40 224.40 220.30 215.80 211.00 205.80 200.20 194.30 187.50 181.00 173.60 166.70 159.40 152.40 145.20 138.40 130.70 121.50 112.30 99.40 77.40 36.50 - 26.00 -53.40 - 66.50 - 80.70 -91.30 - 101.30 - 109.70 -117.90 - 125.10 - 132.40 - 138.90 - 146.00 - 151.40 - 158.70 - 164.10 - 169.70 - 175.10 - 180.70 - 186.20 - 191.60 - 197.00 - 202.50 - 208.40 -214.20 - 220.00
Initial residuals, ml 3.048 3.061 3.053 3.045 2.966 2.870 2.770 2.661 2.552 2.459 2.349 2.244 2.133 2.027 1.923 1.817 1.706 1.596 1.490 1.380 1.265 1.153 1.035 0.832 0.804 0.692 0.586 0.483 0.389 0.311 0.242 0.189 0.146 0.112 0.086 0.067 0.048 0.034 0.023 0.012 0.002 -0.010 -0.012 0.002 0.015 0.041 0.071 0.115 0.162 0.222 0.284 0.358 0.426 0.521 0.565 0.678 0.732 0.773 0.790 0.800 0.795 0.773 0.735 0.688 0.648 0.613 0.582
Residuals after 1st MAGEC run, ml 0.076 0.098 0.092 0.086 0.060 0.059 0.053 0.036 0.022 0.030 0.016 0.013 0.001 0.000 0.001 0.001 - 0.003 -0.002 0.005 0.008 0.007 0.013 0.010 -0.075 0.011 0.013 0.015 0.011 0.013 0.011 0.011 0.010 0.008
- 0.022 - 0.024 - 0.026 - 0.027 - 0.025 - 0.026 - 0.024 - 0.025 -0.023 -0.027 -0.015 - 0.034 0.000 0.004 0.004 - 0.005 -0.007 -0.010 -0.017 - 0.032 -0.051 -0.061 - 0.072 -0.092
Final residuals, ml 0.012 0.035 0.029 0.023 -0.001 0.002 0.000 -0.013 - 0.022 -0.009 -0.019 -0.017 -0.024 -0.021 -0.015 -0.012 -0.011 - 0.007 0.004 0.010 0.012 0.020 0.019 - 0.065 0.022 0.024 0.026 0.021 0.023 0.020 0.019 0.017 0.015 0.012 0.011 0.010 0.007 0.006 0.005 0.002 -0.000 -0.008 -0.016 -0.018 -0.019 - 0.020 -0.021 -0.018 -0.017 -0.013 -0.013 - 0.009 -0.010 0.004 -0.011 0.026 0.033 0.035 0.029 0.029 0.029 0.023 0.010 -0.006 -0.014 -0.021 -0.037
Formation constants-I
255
Table 4.-continued
Titre volume, ml
Observed emf, mV
Initial residuals, ml
15.276 15.521 15.761 16.003 16.248 16.508 16.778 17.068 17.378 17.713 la.00
-226.10 - 232.20 - 238.20 - 244.00 - 249.40 - 254.70 - 259.50 - 264.00 -268.10 - 272.00 - 275.00
0.571 0.590 0.642 0.730 0.849 1.019 1.221 1.463 1.734 2.057 2.363
A Nernstian slope of 59.16 mV per log [H’] was assumed throughout. The refinement, within the first run, by MAGEC of the electrode intercept to within 0.4 mV of the final value, is impressive. This is in spite of the use of approximate acid dissociation-constant values. Table 4 shows how the magnitudes and distributions of the residuals have been improved.
DlSCUSSlON
The accurate experimental determination of hydrogen-ion concentrations is important in many areas of chemistry, often because the measurements provide information about other chemical species which occur in solution. Despite this, considerable differences exist in the way in which the electrode systems are calibrated. For the determination of formation constants, many investigators still employ the pH-scale, as evidenced by the plethora of publications describing how pH measurements can be adjusted to yield values for hydrogen-ion concentration.2-9 A standardized approach, based directly on hydrogen-ion concentrations, is clearly needed. This would facilitate both the comparison of formation-constant values and elimination of errors inherent in the use of the pHscale. In this respect, numerical analysis of titration data, to determine parameters such as E,,,,, would appear to have many advantages. The results presented here demonstrate the efficacy of MAGEC in the calibration of a glass-electrode system both in the strong-acid us. strong-base titration range and in the hydrogen-ion concentration range where the interaction of a given ligand with protons or a metal ion is significant. A strong advantage lies in the ability of MAGEC together with MAGECMINIQUAD cycling to solve this problem even when the protonation constants of the ligand are not known a priori. Thus, E,,,, is optimized for a given titration, and any change in the liquid-junction potential is incorporated into the particular E,,,,, value
Residuals after 1st MAGEC run, ml -0.109 -0.123 -0.135 -0.146 -0.160 -0.169 -0.181 -0.194 -0.214 -0.231 -0.240
Final residuals, ml - 0.047 -0.051 - 0.050 -0.045 -0.040 - 0.024 -0.007 0.012 0.029 0.053 0.081
obtained. Small variations in the liquid-junction potential arising, for example, from poor reproducibility or concentration effects, cease to be a problem. This is a significant improvement over less general approaches such as that of Gaizer and Puskh.” A further marked advantage is derived from the MAGEC (CALIBT) analysis of strong-acid us. strongbase data in assessing the reliability of a given glass electrode. Yet another feature of the MAGEC (CALIBT) routine lies in the analytical facility whereby the concentration of a given strong acid or strong base solution can be determined more precisely than by the Gran method. As far as the main routine in MAGEC, as applied to ligand-proton equilibrium, is concerned, the text describes how, in principle, the parameters EEO”S,,s, /?-values, total concentrations of ligand and protons and K,, can all be determined provided there are sufficient titration points. In practice, however, limitations arise, mainly from the correlation of errors between two or more of the parameters. This results from an effect in which small differences in the values of these particular parameters have similar (increasing) effects on the function to be minimized. Thus, the errors in the parameters concerned tend to compensate each other, leading to false mathematical solutions. The difficulties arising from this source may be restricted by the MAGEC-MINIQUAD cycling procedure, which provides for only one or a selected few of the parameters to be estimated by MAGEC. While work on MAGEC has been in progress, another program with similar objectives has appeared in the literature.23 Like MAGEC, this permits the optimization of any or all of the titration parameters. However, it does not offer the additional facilities simultaneously. Hence, the two programs partially overlap and partially complement one another.
Acknowledgements-P.W.L. and R.G.T. gratefully acknowledge generous grants from the South African C.S.I.R. and the Council of the University of Cape Town. Honours students Valerie Mizrahi, Sybella Meltzer and Pedro Monteiro, are thanked for the data used in Tables 3 and 4.
PETER
256
M.
REFERENCES
1. Manual of Symbols and Terminology for Physicochemical Quantities and Units, 1973, International Union of Pure and Applied Chemistry, Oxford, 1975. 2. W. A. E. McBryde, Talanta, 1969, 94, 337; Analyst, 1971,%,739.
3. C. W. Childs and D. D. Perrin, J. Chem. Sot. (A), 1969, 1039. 4. G. R. Hedwig and H. K. J. Powell, Anal. Chem., 1971, 43, 1206. 5. H. S. Dunsmore and D. Midgley, Anal. Chim. Acta, 1972, 61, 115. 6. E. W. Baumann, ibid., 1973, 64, 284. 7. M. Bos and W. Lengton, ibid., 1975, 76, 149. 8. R. G. Bates, in Analytical Chemistry. Essays in Memory of Anders Ringbom, E. Wiinninen, (ed.), p. 23. Pergamon Press, Oxford, 1977. 9. A. K. Covington, Anal. Chim. Acta, 1981, 127, 1. 10. H. M. Irving, M. G. Miles and L. D. Pettit, ibid., 1967, 38,415.
11. G. Mattock and D. M. Band, in Glass Etectrodesfir Hydrogen Ion and Other Cations, G. Eisenman (ed.), Chapters 2, 9. Dekker, New York, 1967. 12. G. Anderegg, in Proc. Summer School on Stability Con-
MAY et al.
stants, (Bivigliano, Florence, Italy, June 1974). P. Pao-
letti, R. Barbucci and L. Fabbrizzi (eds.), p. 11. 13. D. R. Williams, in Proc. Summer School on Stability Consrants. (Bivieliano. Florence, Italy. June 1974), P. Paoletti, R.‘Barbucci and L. Fabbrizzi (eds.), p. 125. 14. K. Schwabe, Advan. Anal. Chem. Instrum., 1974, 10, 495.
15. M. F. G. Camoes and A. K. Covington, Anal. Chem., 1914,46, 1547. 16. H. Rossotti, The Study of Ionic Equilibria. Longmans, London, 1978. 17. A. Albert and E. P. Serjeant, The Determination 01 lonisation Constants, Chapman & Hall, London, 1971. 18. G. Gran, Acta Chem. Stand., 1950, 4, 559; Analyst, 1952, 77, 661.
19. F. J. C. Rossotti and H. Rossotti, J. Chem. Educ., 1965, 42, 375. 20. J. A. Nelder and R. Mead, Comput. J., 1965, 7, 308. 21. A. Sabatini. A. Vacca and P. Gans. Talanta, 1974, 21, 53; P. Gans, A. Sabatini and A. Vacca, Inorg. Chim. Acta. 1976, 18, 237. 22. F. Gaizer and A. Puskls, Talanta, 1981,28, 565. 23. G. Arena, E. Rizzarelli, S. Sammartano and C. Rigano, ibid., 1979, 26, 1.