Analyst April 1983 Vol. 108 pp. 457-463 457 Evaluation of Equivalence Points in the Potentiometric Titration of Mixtures of Halides Duncan Thorburn Burns Binod K. Maitin and Gyula Svehla Department of Analytical Chemistry Queen’s University of Belfast Belfast BT9 5AG The accurate and precise location of the equivalence points in the potentio-metric titration of halide ions can be achieved numerically by using Gran’s method. The results have been shown to be better by this method than those obtained on the same titration data by the differential methods of Kolthoff of Fortuin and of Hahn. Keywords ; Equivalence points ; potentiowaetric titration ; halide ions ; Gran’s method Simple graphical methods for the location of equivalence points are tedious and prone to sub-jective errors.Methods involving mathematical calculations of the equivalence points with-out plotting the titration curves are relatively rapid and convenient to make using a pro-grammable calculator. In addition they yield results with improved accuracy and precision. One of the most useful numerical methods for end-point location is that due to Gran,l which uses functions that linearise sigmoidal potentiometric titration curves. For each type of titration two functions were derived one for the region before and the other for the region after the equivalence point. For example for argentimetric titrations using a silver indicator electrode the appropriate form of the Gran functions F and FA for the regions before and after the equivalence points respectively are and where Vo is the initial volume of the solution being titrated VT is the volume of titrant added and E is the measured potential (volts).Each function is linearly dependent on the volume of the titrant and both become zero at the equivalence point. Various modifications have been Anfalt and Jagnerg have given estimates of the precision and accuracy expected from a variety of numerical and graphical procedures to evaluate equivalence points from potentio-metric titration data. Gran’s method and the multi-parameter method are expected to give the least systematic errors and highest precisions. Because the expected precisions attainable were similar the comparatively simpler Gran’s method was chosen and has been compared with alternative methods involving simple calculations such as those of Kolthoff and co-workers,1°-12 Fortuin13 and Hahn.14 These latter three methods make use of the three or four largest potential steps obtained in the equivalence-point region resulting from the addition of equal volumes of titrant.Each method is based on the assumption that the stoicheiometric equivalence point and the inflection point of the titration curve are coincident. which all involve computer-based calculations. Experimental Mixtures containing 0.04 mmol of each halide ion (for binary) and 0.03 mmol of each halide ion (for ternary mixtures) were titrated with 0.2 M silver nitrate solution using a silver billet indicator and a mercury - mercury(1) sulphate reference electrode as described e1~ewhere.l~ An AGLA micrometer syringe burette and an Orion 901 Microprocessor Ionalyser were used for measurement of potentials.A Metrohm Titroprocessor 636 Autotitrator 635 was also used in the fixed volume addition and fixed waiting time modes; the volume - potential data for the entire titration were printed out and used for equivalence-point calculations.16 Results and Discussion The first series of results concerned titrations of single halides. The silver nitrate titrant was Standardisations and standardised using pure inorganic halides (re-assayed gravimetrically) 458 THORBURN BURNS et al. EVALUATION OF EQUJVALENCE Analyst VoZ. 108 blank determinations from the paper carrier and reagents were carried out separately for each halide as described elsewhere.16 Each method of calculation of the equivalence point was applied systematically to each set of titrations for the standardisation the blank evaluation and for the assay.Several organic compounds containing chlorine bromine or iodine were analysed by poten-tiometric titration with silver nitrate solution after oxygen flask combustion. The results calculated by each method are summarised in Tables 1-111. Of the three differential methods (Kolthoff’s Fortuin’s and Hahn’s methods) only Fortuin’s method gave good results for all three halides. Kolthoff’s method was satisfactory for chloride and bromide but not for iodide. Hahn’s method gave similar results to that of Kolthoff except for the assay of hexachlorobenzene where precision was affected by a single low result. The increase in apparent errors must arise from the method of data treatment because the compounds assayed were pure or reference materials.This view is in accord with that of Ebel TABLE I COMPARISON OF RESULTS OBTAINED BY DIFFERENT METHODS OF EQUIVALENCE-POINT EVALUATION FOR CHLORIDE Chloride found yo Chloride ReDlica- I L \ Compound l-Chloro-2,4-dinitro-benzene-Mean . . Standard deviation . . chloride*-Apparent error . . S-Benzylthiuronium Mean . . Standard deviation . Apparent error . . p-Chlorobenzoic acid*-Mean . . Apparent error . . Standard deviation . . (pufity expected 4 99.5%, purity by GLC 99.44%) Hexachlorobenzene*-Mean . . * . Apparent error . Standard deviation . . calculated tibn % No. Kolthoff 17.50 1 17.’53 2 17.57 3 17.65 4 17.53 5 17.57 .. 17.57 . . +0.07 . . f0.05 17.49 1 17.63 2 17.48 3 17.42 4 17.48 5 17.39 6 17.38 . . 17.46 ,. . . -0.03 . . f0.09 22.65 1 22.59 2 22.63 3 22.59 4 22.76 5 22.70 6 22.64 . . 22.64 . . -0.01 . . f0.08 74.70 1 74.47 (74.28)t 2 74.37 3 74.57 4 74.45 . . 74.47 * . . . -0.23 * . . . f0.08 (+0.19) t Fortuin 17.55 17.61 17.66 17.56 17.51 17.58 + 0.08 f 0.06 17.64 17.52 17.48 17.50 17.44 17.43 17.50 +0.01 fO.08 22.65 22.64 22.60 22.82 22.74 22.55 22.67 + 0.02 fO.10 74.52 74.41 74.72 74.40 74.51 - 0.19 (+0.23)t f0.15 Hahn 17.52 17.59 17.54 17.48 17.53 17.53 + 0.03 f 0.04 17.44 17.42 17.46 17.51 17.46 17.41 17.45 f 0.04 22.62 22.69 22.69 22.77 22.69 22.52 22.66 +0.01 f0.08 - 0.04 74.82 74.71 74.67 72.99 74.30 - 0.40 (+0.02)t f0.87 Gran before 17.62 17.70 17.68 17.62 17.61 17.65 + 0.15 f0.04 17.47 17.48 17.61 17.55 17.59 17.49 17.53 + 0.04 f 0.06 22.60 22.70 22.70 22.72 22.69 22.63 22.67 + 0.02 f0.05 74.68 74.77 74.86 74.76 74.77 +0.69 (+ 0.69) t f 0.07 Gran after 17.54 17.69 17.62 17.66 17.54 17.55 + 0.06 f 0.03 17.54 17.53 17.50 17.54 17.42 17.45 17.50 + 0.01 k0.05 22.61 22.56 22.55 22.64 22.63 22.59 22.60 - 0.05 f 0.04 74.39 74.25 74.15 74.50 74.32 -0.38 (+ 0.09) t f 0.15 * Microanalytical-reagent grade BDH Chemicals Ltd.Poole Dorset. t The values in parentheses are calculated on the basis of results obtained by gas - liquid chromatographic analysis of the sample; the errors calculated on the basis of these results are shown in parentheses April 1983 POINTS I N POTENTIOMETRIC TITRATION OF HALIDES TABLE I1 COMPARISON OF RESULTS OBTAINED BY DIFFERENT METHODS OF EQUIVALENCE-POINT EVALUATION FOR BROMIDE 459 Compound Bromobenzoic acid*-Mean . . Apparent error . . Standard deviation . . N-Bromosuccinimide t-Mean . . ,. Apparent error . . Standard deviation . . Bromoacetanilide$-Mean . . Apparent error . . Standard deviation . . . . . . Bromide found % Bromide P A 1\ calculated Replication % No. 39.75 1 2 3 4 5 6 7 .. . . 44.89 1 2 3 4 5 6 . . 9 . 37.33 1 2 3 4 5 Kolthoff 39.79 39.80 39.84 39.80 39.72 39.66 39.65 . . 39.74 . . -0.01 . . f0.08 44.88 44.71 44.57 44.77 44.83 44.76 . . 44.75 . . -0.14 . . fO.11 37.31 37.30 37.26 37.28 37.28 . . 37.29 . . -0.04 . . f0.02 Fortuin 39.91 39.97 40.13 39.84 39.88 39.70 39.64 39.87 +0.12 f0.16 44.97 44.52 44.35 44.64 44.70 44.53 44.62 - 0.27 f0.21 37.36 37.27 37.33 37.24 37.18 37.28 f0.07 -0.05 Hahn 39.87 39.96 40.07 39.80 39.83 39.66 39.60 39.83 + 0.08 f0.16 44.93 44.75 44.70 45.00 44.76 44.81 44.83 - 0.06 f0.12 37.32 37.24 37.27 37.31 37.34 37.30 f0.04 - 0.03 Gran before 39.93 40.20 39.98 39.80 39.94 39.47 39.65 39.84 + 0.09 f0.25 44.89 44.42 44.41 44.70 44.74 44.60 44.63 -0.26 kO.19 37.37 37.31 37.33 37.23 37.13 37.27 - 0.06 hO.10 Gran after 40.03 40.09 40.14 39.86 39.80 39.67 39.67 39.89 +0.14 f0.19 44.90 44.67 44.38 44.6 1 44.80 44.73 44.68 -0.21 k0.18 37.35 37.31 37.32 37.26 37.14 37.27 - 0.06 f 0.08 * Microanalytical-reagent grade BDH Chemicals Ltd.t Relative molecular mass 177.99; melting-point 180 “C. Aldrich (99%) material recrystallised five $ Relative molecular mass 214.07 ; melting-point 169 “C. BDH Chemicals Ltd. (99-101%) material times from benzene. recrystallised five times from ethanol.and Seuringl’ who refer to these differential methods as approximate and suggest that the errors arise because only a few results near the equivalence points are used. Potential readings near the equivalence points are subject to “potential transfer errors,” which arise from slow attainment of solution equilibria and of the electrode potential and from time constants of the measuring circuits. Here it was noted that “potential transfer errors” were more frequent in the titration of iodide probably owing to the very abrupt potential change near the equivalence point. In an attempt to improve the precision and accuracy of the results Gran’s method was then applied to data before and data after the equivalence point using a programmable calculator (Texas TI59) with a printout.A program was devised to calculate the Gran’s functions and the line of linear regression with volume for a variable number of data points using a least-squares fit. After entering the initial volume of titrand (V,,) for a calculation based on data prior to the equivalence point the next stage was to enter data potential (E in volts and volume of titrant VT in millilitres) for 8-10 points considered to be prior to the equivalence point. The data were treated system-atically in order to calculate the equivalence point from the points diagnosed as being before the equivalence point. The equivalence point (ie. the volume where Gran’s function = 0) and the correlation coefficient were calculated from the first three Such errors occurred only occasionally with chloride and bromide.The procedure was as follows 460 THORBURN BURNS et U l . EVALUATION O F EQUIVALENCE AfiUhySt VOJ. 108 data points and printed out. The next data point was then included and the equivalence point and Correlation coefficient re-calculated from the new regression line using all of the four points and so on until all the data points were incorporated into the series of least-square fits. The equivalence point was normally determined using the condition that the equivalence volume must be greater than the volume co-ordinate of the last acceptable data point. In addition to this criterion the values of the correlation coefficients for each calculation printout were inspected. Normally the correlation remains close to unity for the points prior to the equivalence point and decreases when points beyond the equivalence points are included.Occasionally the correlation coefficient decreased for the last data point accepted as prior to the equivalence point owing to “potential transfer errors” as described earlier in such instances this point was rejected and the equivalence point calculated from the previous point. TABLE I11 COMPARISON OF RESULTS OBTAINED BY DIFFERENT METHODS OF EQUIVALENCE-POINT EVALUATION FOR IODIDE Compound P-Iodonitrobenzene*-Mean . . Apparent error . . Standard deviation . . N-Iodosuccinimidet -Mean . . Apparent error . . Standard deviation . . o-Iodobenzoic acid:-Mean . . Apparent error . . Standard deviation . . Iodide calculated, YO 50.96 56.40 .. 51.16 f . Replication No. 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 . . Iodide found yo f h \ Kolthoff 50.54 50.88 50.74 50.66 50.62 . . 50.69 . . -0.27 . . f0.13 56.42 56.88 56.70 56.93 57.03 56.79 . . 56.79 . . +0.39 . . f0.21 51.80 51.87 51.85 51.84 51.67 . . 51.81 . . +0.65 . . fO.08 Gran Gran Fortuin Hahn before after 50.92 50.39 50.77 50.92 50.95 51.14 51.39 60.88 51.07 50.49 50.80 61.11 50.95 50.32 51.23 50.83 50.93 50.46 51.34 50.66 50.96 50.56 51.11 50.86 f0.06 f0.33 f0.30 k0.20 56.37 56.51 56.81 56.42 56.45 56.69 56.81 56.19 56.42 66.58 57.12 56.19 56.58 56.79 56.94 56.52 56.62 56.93 57.06 56.64 56.46 56.57 57.03 56.10 0.00 -0.4 +0.15 -0.10 56.48 56.68 56.96 56.34 hO.10 f0.16 f0.13 f0.22 51.41 51.84 52.02 51.48 51.45 51.88 51.81 51.40 51.38 51.55 51.95 51.15 51.42 51.82 51.87 51.37 51.25 51.63 51.91 51.18 +0.08 +0.28 +0.56 -0.06 51.38 51.74 51.91 51.32 +0.22 +0.58 +0.75 tO.16 kO.08 h0.15 fO.08 f0.14 * Relative molecular mass 249.01 ; melting-point 172-173 “C ; Aldrich material recrystallised five times t Relative molecular mass 224.99; melting-point 203-204 “C.Hopkin and Williams laboratory-from ethanol. reagent grade material recrystallised five times from dioxan - tetrachloromethane. Microanalytical-reagent grade BDH Chemicals Ltd. For calculations based upon data after the equivalence point any pre-equivalence-point data are rejected on the basis that the volume co-ordinate of the point entered must not be less than the equivalence point calculated; before it was occasionally necessary to reject a point on the grounds of a reduced correlation coefficient.Fig. 1 shows the computer printout of such an evaluation process. The left-hand column contains the data as they were fed in; Vo is the initial volume of the sample in the cell 75.00 cm3; V the volume of the titrant added in the first instance 0.15 cm3; E is the corresponding electrode potential -0.32 V and S is th April 1983 POINTS I N POTENTIOMETRIC TITRATION OF HALIDES 461 counter for the data pairs. In the right-hand column the first figure is the intercept the second the slope of the Gran function versm titrant volume plot calculated by linear-regression analysis followed by the value of the equivalence point (V,) and the coefficient of correlation [(R) in this instance always negative as we have a negative slope] and finally the counter N is again printed out.From the values it is easy to select V = 0.2064 cm3 as the final result. FLP-?& 75. 00 a. 15 -0.321 I . 0. ! 6 -0. 116 2 0. !7 -0. ?!U 3. U. 13 -0. 301 4. 0. 19 -0. 269 5. 0. 20 -0. 266 6 . 0.21 -0. a37 7. 0. 22 -0.230 a. REF 1. Printout for Fig. point evaluation. VE = 0.206 4 ml. 'dE R N 'd E R N VE R N 'VE R ti YE R N VE R N end-The results for these calculations are shown in Tables 1-111. For chloride and bromide both data before and after the equivalence point yielded satisfactory results.Although differences in precision and accuracy obtained by using both sets of data were small the results using data after the equivalence point appeared to be better. For hexachlorobenzene both precision and accuracy were relatively poor. Examination of a sample of the hexachlorobenzene by gas chromatography using an integrator revealed that the compound was slightly impure (99.44%) and the result obtained by data after the equivalence point confirmed this view. The results for iodide using data after the equivalence points were distinctly better than those obtained using data prior to the equivalence point. Further results by Gran's method were better than those obtained by the differential methods except for that of Fortuin which gave comparable accuracy and precision.The second series of experiments concerned the analysis of organic compounds containing bromide and chloride. Standardisations were carried out using mixtures of individually weighed pure organic compounds containing single halides. The blank determinations were carried out using the procedure as described before.15 Each method of calculation was again applied systematically to each set of titrations for standardisation blank determination and for the assays. The data obtained using Gran's method for data prior to the equivalence point were clearly superior to the differential methods including that of Fortuin which was previously found to be satisfactory for the single halide titrations. The use of the before equivalence-point data is also supported by the correlation-coefficient data.It was found that for bromide the correlation coefficients are close to unity only for data points prior to equivalence points while for chloride (second equivalence point) there was no significant difference between correlation results using before and after the equivalence-point data. As the chloride equivalence volume is determined by the difference of the second and first equivalence points the same calculation criterion was used for both the equivalence points and values closer to the true values were obtained. These results show the importance of the method of data examination and that for single halide determinations satisfactory results can be obtained using either Fortuin or Gran's method but for mixtures of chloride and bromide Gran's method is superior (e.g.see Table IV). The results are summarised in Table IV 462 THORBURN BURNS et al. EVALUATION OF EQUIVALENCE Artalyst VoZ. 108 TABLE IV COMPARISON OF RESULTS OBTAINED BY DIFFERENT METHODS OF EQUIVALENCE-POINT EVALUATION Bromide found yo Chloride found % L I A \ I i Replica- Kolt- Gran Gran Kolt- Gran Gran Compound tion No. hoff Fortuin Hahn before after hoff Fortuin Hahn before after 1- [l-( 4-Bromophenylmethy1)-4-~i~eridinvll-~-chloro-2-~ tri-fl;okmeth$lj-l If-benzimi‘dazole (CaoHmBslFaNtJ*- 1 17.14 17.06 17.34 17.01 17.13 7.42 7.47 7.37 7.48 7.47 2 17.12 17.02 17.38 17.07 17.09 7.41 7.45 7.36 7.45 7.36 - . . - - - . - - 4 16.90 16.90 K s i i637 16.84 7.49 7.49 7.58 7.48 7.59 5 17.11 17.02 17.29 17.07 17.18 7.45 7.50 7.41 7.47 7.47 6 16.74 16.79 16.86 16.92 16.81 7.54 7.50 7.38 7.47 7.47 7 16.90 16.90 16.83 16.94 17.10 7.48 7.48 7.41 7.48 7.35 8 17.12 17.03 17.19 17.05 17.28 7.55 7.58 7.46 7.59 7.47 9 16.96 16.94 16.74 16.90 16.86 7.47 7.49 7.62 7.49 7.68 10 17.02 16.99 17.38 16.94 17.10 7.48 7.49 7.34 7.48 7.48 Mean .. . . . . . . . . 17.01 16.97 17.12 16.99 17.05 7.47 7.49 7.43 7.49 7.47 Standarddeviatiod‘ . . . . . f0.13 h0.08 f0.27 &-0.06 50.16 f0.05 f0.03 f0.10 f0.04 *0.08 Apparent error +0.11 +0.07 +0.22 $0.09 +0.15 -0.03 -0.01 -0.07 -0.01 -0.03 N-( 4-Bromophenyl)-N‘-( 2-chloro-1 20.64 20.68 20.50 20.69 20.66 9.24 2 20.37 20.40 20.59 20.57 20.48 9.32 3 20.40 20.46 20.59 20.61 20.50 9.26 4 20.49 20.56 20.58 20.64 20.92 9.20 5 20.35 20.40 20.54 20.58 20.47 9.29 6 20.36 20.39 20.65 20.55 20.44 9.27 9 20.38 20.43 20.57 20.65 20.47 9.38 8 20.39 20.44 20.69 20.58 20.43 9.26 9 20.40 20.46 20.53 20.59 20.38 9.23 Mean .. . . 20.42 20.47 20.57 20.61 20.53 9.27 Standard deviatioi’ . . . . . . *0.09 fO.09 f0.04 f0.04 f0.17 k0.05 Apparent error -0.24 -0.19 -0.09 -0.05 -0.13 +0.10 9.24 9.30 9.21 9.16 9.26 9.26 9.37 9.22 9.21 9.25 k0.06 - 0.08 9.34 9.24 9.27 9.10 9.22 9.18 9.08 9.20 9.15 9.04 9.14 8.99 9.29 9.24 9.15 9.05 9.21 9.16 9.26 9.17 9.16 9.08 9.24 9.19 9.22 9.17 9.29 9.16 9.20 9.17 +0.01 +0.03 0.00 fO.11 f0.04 kO.09 a-Bromo-fi-chloroacetophenone (ClC,H,COCH,Br) 3- 1 34.65 34.43 34.72 34.50 34.64 15.00 15.13 15.07 15.03 15.16 2 34.03 34.18 34.30 34.21 34.20 15.37 15.28 15.03 15.38 15.13 3 34.44 34.34 35.23 34.31 34.61 15.14 15.21 14.87 15.22 15.13 4 34.42 34.30 35.35 34.33 33.77 15.04 15.11 14.78 15.13 15.42 5 33.83 33.94 34.08 34.25 34.08 15.34 15.27 15.03 15.19 15.15 6 34.74 33.50 34.82 34.38 34.80 15.11 15.22 15.08 15.24 15.15 7 33.92 34.02 34.21 34.17 34.03 15.45 15.40 15.46 15.39 15.24 8 33.66 33.75 33.96 33.91 33.86 15.30 15.25 15.02 15.31 15.10 9 33.82 33.93 34.05 34.07 34.12 15.28 15.23 15.32 15.27 15.05 Mean .. . 34.17 34.04 34.52 34.24 34.23 15.23 15.23 15.07 15.24 15.17 Standard deviation’ . . f0.40 f0.30 50.52 h0.17 f0.36 f0.16 f0.09 k0.21 k0.12 kO.11 Apparent error -0.05 -0.18 +0.30 +0.02 +0.01 +0.05 $0.05 -0.11 +0.06 -0.01 * BCR CRM 073. Calculated bromide l6.90% chloride 7.50%.t BCR CRM 071. Calculated bromide 20.66% chloride 9.17%. 3 Calculated bromide 34.22y0 chloride 15.18%. Aldrich 98% material recrystallised five times from ethanol. The reasons for the over-all superiority of the Gran-functions procedure to evaluate the equivalence point over that of Fortuin are two-fold and arise from a consideration of the propagation of errors and the reliability of the data used in each example. Gran’s method is based directly on potential measurements at discrete values of volumes added ; Fortuin’s method relies on the ratio of difference between three successive potentials measured in the vicinity of the equivalence point after addition of equal volume increments of the titrant which compounds any random potential errors four-fold. A further advantage of Gran’s method is that each point is independent of each other.Gran’s method is applied to data away from the equivalence-point region where “potential transfer errors” can arise whilst Fortuin’s method requires data points that are necessarily from the equivalence-point region. Both methods rely on achieving a Nernstian electrode response which with mixed halides only applies to data well before the equivalence point due to co-precipitation problems. Hence Gran’s procedure remains valid for mixtures as well as for single halides. The over-all procedure described here was also applied to evaluate equivalence points in the titrations of iodide - chloride mixtures18 and of binary and ternary mixtures of halides using a microprocessor controlled aut otit rat or.1 April 1983 POINTS IN POTENTIOMETRIC TITRATION OF HALIDES References 463 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. Gran G. Analyst 1952 77 661. Liteanu C. and Cormos D. Talanta 1960 7 25. Liteanu C. and Cormos D. Talanta 1960 7 32. Ingman F. and Still E. Talanta 1966 13 1431. McCallum C. and Midgley D. Anal. Chim. Actct 1973 65 155. Midgley D. and McCallum C. Talanta 1974 21 723. Magallanes J . F. and Caridi A. F. Anal. Chim. Acta 1981 133 203. Mascini M. “Ion Selective Electrode Reviews,” Volume 2 Pergamon Press Oxford 1980 p. 17. Anfalt T. and Jagner D. Anal. Chim. Acta 1971 57 165. Kolthoff I. M. and Laitinen H. A. “pH and Electrotitrations,” John Wiley New York 1944, Kolthoff I. M. and Sandell E. B. “Textbook of Quantitative Inorganic Analysis,” Macmillan, Kolthoff I. M. and Furman N. H. “Potentiometric Titrations,” Second Edition John Wiley New Fortuin J . M. H. Anal. Chim. Acta 1961 24 175. Hahn F. L. FreseniusZ. Anal. Chem. 1958 163 169. Thorburn Burns D. and Maitin B. I<. Analyst 1983 108 452. Thorburn Burns D. and Maitin B. K. J . Indian Chem. Soc. in the press. Ebel S. and Seuring A. Angew. Chem. Int. Ed. Engl. 1977 16 157. Thorburn Burns D. and Maitin B. K. unpublished data. p. 110. New York 1952 p. 488. York 1949 pp. 95-96. Received August 6th 1982 Accepted November 17th 198