Computer Analysis of Data from Potentiometric Titrations Using Ion-Selective Indicator Electrodes Arthur F. Isbell, Jr.,l and Robert L. Pecsok Department of Chemistry, University of Hawaii, Honolulu, Hawaii 96822
R. H. Davies and J. Howard Purnell Department of Chemistry, University College of Swansea, Swansea, Glam. SA2 8PP, United Kingdom
Ion-selective electrode potentiometry is plagued by the necessity of frequent electrode calibration. Standard addition, precipitation, and complexation titrimetry eliminate prior electrode calibration and theoretically enhance precision. However, titration curves can be distorted by the narrow linear response ranges and irrelevant ion interferences typical of many ion-selective electrodes. A computer program, TITRATE, analyzes only the meaningful data and computes the analyte concentration, the fraction of ideal Nernstian response, and the electrode formal potential. In addition, standard deviations, a comprehensive error analysis, and data which facilitate plotting both sigmoid and linear titration curves are computed. The versatility of ion-selective electrode potentiometry is enhanced by permitting the analysis of ions for which electrodes exist plus species which rapidly form precipitates or complexes with such ions. Examples of three argentometric titrations using a silver ion-selective electrode are presented.
Following the discovery of the hydrogen ion-selective glass electrode a t the beginning of the century by Cremer ( I ) and Haber and Klemensiewicz ( 2 ) , progress in the development of ion-selective electrode potentiometry was slow. During the twenties, Hughes ( 3 ) , Horovitz ( 4 , 5 ) , and Schiller (6) developed glass electrodes with some selectivity for ions other than hydrogen ion. In 1934, Lengyel and Blum (7) introduced a sodium ion-selective glass electrode and glass electrodes which showed some selectivity for potassium and lithium ions. Starting in the late fifties when Eisenman and coworkers (8) developed glass electrodes selective for several monovalent cations, progress in the field has proceeded a t an ever-accelerating rate. At present, ion-selective electrodes are available for the analysis of a wide variety of cations and anions (9-11). The ion-selective electrodes available a t present are not as well behaved as the glass p H electrode. Problems of uncertain formal potential, potential response drift, and response to ions other than the ion of interest have yet to A u t h o r t o w h o m a l l correspondence s h o u l d b e d i r e c t e d . Prese n t address, A n a l y t i c a l Research Laboratories, P.O. Box 369, M o n r o v i a , C a l i f . 91016. (1) M . Cremer, Z. Bioi. (Munich). 47, 562 (1906). (2) F. Haber and Z. Klemensiewicz, Z. Phys. Chem. (Leipzig), 67, 385 (1909). (3) W . S. Hughes, J. Amer. Chem. SOC., 44, 2860 (1922). (4) K. Horovitz, Z. Phys., 15, 369 (1923). (5) K . Horovitz, Z. Phys. Chem. (Leipzig), 115, 424 (1925). (6) H. Schiller, Ann. Phys. (Leipzig), 74, 105 (1924). (7) 6.Lengyel and E. Blum, Trans. Faraday Soc., 30, 461 (1934). (8) G . Eisenman, D. 0. Rubin, and J. U. Casby. Science, 126, 831 (1957). (9) E. C. Toren, Anal. Chem., 40, 402R (1968). (10) E. C. Toren and R. P. Buck, AnalChem., 42, 284R (1970) (11) R. P. Buck,Anai. Chem., 44,270R (1972).
be solved. Therefore, the error in ion activity, when measured by direct potentiometry, can be quite large. Accuracy and precision can be increased by using an ion-selective electrode as the indicator electrode for a potentiometric titration. The need to calibrate the electrode is eliminated because the potential change rather than absolute magnitude is relevant. Therefore, the formal potential, the gotantial response, the liquid junction potential, and the activity coefficient are unimportant. The advantages of titrimetric potentiometry over direct potentiometry can be nullified by erroneous determination of the equivalence point. The problems and errors inherent in the equivalence point determination can be eliminated by adapting a nonlinear least squares treatment to fit the proper form of the Nernst equation to the experimental titration data. The necessary calculations are much too tedious to be completed manually, so a modern computer system is required. Brand and Rechnitz (12) have used a similar approach in their study of ion-selective electrode potentiometry by standard addition methods. Meites and Meites (13) have successfully applied a brute-force least squares curve-fitting technique to the data of Brand and Rechnitz. This study describes a computer technique which locates the equivalence point in precipitation and complexation titrations in which either the analyte or titrant is elertroactive. By using data points in a region where the activity ratio of electroactive and interfering ions is favorable, misleading data are excluded. Standard addition titration data can be analyzed also. In addition, this technique computes other valuable information such as the electrode formal potential, the fraction of ideal Nernstian response, standard deviations, and a comprehensive error analysis while simultaneously decreasing experimental and data analysis times. THEORY Electrochemistry. The theoretical relationship between the electrode potential, E, and the electroactive ion concentration, C, is given by the Nernst equation. In actual practice, the form of the Nernst equation is modified somewhat:
E
E‘
=
EO’
+ f ( R T / n F ) In C + E , - E,. + f ( R T / n F ) In y
=
(1’)
E’
(2)
in which E‘ is the electrode formal potential, f, the fraction of ideal Nernstian response which the electrode exhibits, EO’, a complex function of ion-selective electrode parameters including contributions from the inner reference electrode, the inner phase boundary potential of the electrode membrane, and the asymmetry potential, E,, the liquid junction potential, E,, the reference electrode (12) M. J. D. Brand and G. A. Rechnitz, Anal Chem , 42, 1172 (1970) (13) T.Meitesand L. Meites, Talanta, 19, 1131 (1972).
A N A L Y T I C A L C H E M I S T R Y , V O L . 45, NO. 14, D E C E M B E R 1 9 7 3
2363
potential, and y, the activity coefficient of the electroactive species. During the course of a potentiometric titration, E, and y are assumed to remain constant. The expressions defining C differ for the five different titration cases to be considered: (a) standard addition; (b) precipitation of an electroactive analyte; (c) complexation of an electroactive analyte; (d) precipitation by an electroactive titrant; and (e) complexation by an electroactive titrant. During a standard addition titration, electroactive analyte A of unknown concentration CO is titrated by addition of the same ion of known concentration C':
C =
+ C'V V @+ V
COVO
(3)
in which VO is the initial analyte volume and V is the titrant volume added. For a precipitation titration during which electroactive analyte A of charge magnitude a is titrated by an electropassive precipitating agent P of concentration C', the species A$, is precipitated. The mass balances during the titration are
+ V ) + p(mo1es A,P,) = CoVo C,(Vo + V ) + a(mo1es A,V,) = C'V
C(Vn
(4)
(5)
in which C, is the concentration of P. The solubility product constant, K s p ,is given by K , , = (C)'(Cc,)" (6) Combining Equations 4 through 6 gives
Before the equivalence point, the second term on the right side of Equation 7 is usually negligible compared with the first term permitting the approximation,
C =
cove - (p/a)C'V V" + v
(8)
At or after the equivalence point, C is usually so small that other ions in solution affect the electrode potential, thus making the data misleading. The experimental details and the derivation of the concentration expression for the complexation titration of an electroactive analyte A by an electropassive complexing agent P are analogous to those for a precipitation titration. Before the equivalence point, an approximation analogous to that made for the precipitation titration case gives Equation 8. Similarly, data taken a t or after the equivalence point are usually misleading. For a precipitation titration of electropassive analyte P by electroactive titrant A , potential readings taken before or a t the equivalence point are usually erroneous due to electrochemical interference by irrelevant ions. By interchanging the expressions on the right side of Equations 4 and 5 and repeating the derivation making appropriate approximations for the region after the equivalence point,
C =
C'V - (p/a)COVO V@ v
+
V = V ' - V,
(10)
in which V' is the total volume of titrant actually added. Combining Equations 9 and 10,
C =
C7V' - V ) - ip/a)COVO vo+ V' - v
ill)
Therefore, two basic expressions relate E to V for all five titration cases discussed:
E=E'+f
(liF')
-
In (COV" ~
- RC'V)
+v
V@
V" + V' -
v
(12)
(13)
Equation 12 governs precipitation and complexation titrations of an electroactive analyte ( B = p / a ) and standard addition titrations ( B = -1). Equation 13 covers precipitation and complexation titrations by an electroactive titrant ( B = p / a ) . Mathematics. The procedure used for adjusting parameters E', f, and Co to fit a set of experimental variables E and V is based on a rigorous nonlinear least squares treatment described initially by Deming (14) and applied to curve fitting by Wentworth (15, 16). Most least squares routines assume that error is present in the dependent variable but not in the independent variable. Wentworth's generalized treatment makes no such assumption. Both variables are expressed in an error function, F, the average value of which approaches zero as better estimates for E', f, and Co are made: in which C is defined by either Equation 8 or 11. In order to facilitate computations, a linear approximation of Equation 14 is expressed as a first order Taylor's Series about the point ( E ,V,VO, C', T,E',f, Co) using estimates for the values of E ' , f, and CO:
in which the F' terms are the partial derivatives of F with respect to the subscripted variables and parameters, the R terms are the variable residuals (defined as the difference between the experimental and theoretical values), and the 1terms are the corrections to be subtracted from the parameter estimates to give improved estimates. Values for I E ' , A{, and I C 0 are calculated using a least squares adjustment of Equation 15 with the restriction that F equals zero. Cnlike many curve-fitting procedures, Wentworth's least squares treatment permits the weighting of each experimental observation. As a result, the significance of each observation is inversely proportional to the magnitude of the error in that observation. The least squares procedure minimizes S, the sum of the squares of the weighted residuals:
(9)
Equation 9 is also valid for a complexation titration of an electropassive analyte by an electroactive titrant. Because the region of interest follows the equivalence point, the location of which is not known a priori, computational analysis must begin with the data recorded a t the end of the titration and proceed in reverse order toward 2364
the equivalence point. The computation, therefore, involves the hypothetical removal of increments of titrant from the titration vessel. V in Equation 9 must be redefined in terms of the volume of titrant "removed," V,:
= minimum
(16)
(14) W. E. Deming, "Statistical Adjustment of Data," John Wiley and Sons, Inc., New York, N.Y.. 1943. (15) W. E. Wentworth. J . Chem. Educ., 42, 96 (1965). (16) W. E. Wentworth. J . Chem Educ., 42, 162 (1965).
A N A L Y T I C A L C H E M I S T R Y , V O L . 45, N O . 14, D E C E M B E R
1973
Table I. Total Absolute Error Inherent in Volumetric Glassware Volume, mi Type of glassware
Buret
10 25
Volumetric pipet
Volumetric flask
50 5 10 25 50
Absolute error,
0.029 0.039
PRINT
READ AND
@
RESULTS"
>Z
DATA
INVERT
0.056
ELECTKJACTNE?
0.087 0.080
100
0.200
250
0.300 0.500 0.800 1.500
2000
I
i
ml
0.045 0.080 0.150
25 50
500 1000
* DATA EXIST?
CALCULATE INITIAL
0.125
EOUIVALENCE
in which the u terms are the estimated experimental errors in the observed variables. For the present application, the magnitudes of the experimental errors are assumed to remain constant throughout the duration of the experiment. The minimization of Equation 16 uses the method of Lagrange undetermined multipliers (15) resulting in a set
I
I
I
PLOTTING 1
Figure 1. Flow chart of
(17)
18)
19)
V
program T I T R A T E
experimental error being the standard deviation from the mean. One experimental error is recorded for all potential readings recorded during a single titration. Estimates of the random errors inherent in the volume of titrant added and the analyte volume can be taken directly from Table I (18). The absolute errors summarized in Table I include the parallax error in measuring the miniscus plus the maximum permissible deviation in equipment according to official U.S. Bureau of Standards calibration instructions. (Reference 18 includes a more comprehensive table with calibration instructions from several countries.) The experimental error in the titrant concentration can be estimated using the propagation-of-errors equation:
(20)
+ (g/l)'a,?+ ( g / u ) % ) < ?
Equations 17 through 19 can be solved for SCO, If,and LIE' by finding the inverse matrix of the coefficients ( 1 3 ) . Computer Program. Program TITRATE has been written to perform the nonlinear least squares calculation. A general Fortran IV language compatible with most computer systems (17) has been used. A flow chart of TITRATE is shown in Figure 1. All input data are easily obtained with the possible exception of the estimates of experimental error. For the least squares treatment to be valid, all systematic error must be accounted for before attempting to estimate the random experimental error. If knowledge of the parameter standard deviations is unimportant, only the experimental errors in the electrode potential and the buret readings are necessary. The random error in the electrode potential can be approximated by determining the magnitude of the smallest unit which can be read on the potential-measuring device. If repeated readings of the same potential vary randomly, the mean reading should be recorded with its
in which 1 is the volume, g, the mass, u, the molecular weight, and the u terms are estimated experimental errors. The error in the mass is about 0.2 mg for the usual analytical balance. The error in the molecular weight can be estimated as one unit in the last significant digit reported. The volumetric error can be obtained from Table I. The titration solution temperature is the mean temperature recorded during the titration. The standard deviation of the temperature from the mean value is the estimated experimental error. An estimate of the experimental error in the solubility product or formation constant is relatively unimportant because its magnitude affects only the calculated equivalence point data based on the assumptions leading to Equations 12 and 13. Any reasonable estimate will suffice.
(17) E. I . Organick, "A Fortran I V Primer." Addison-Wesley Publishing Co., I nc. Reading, Mass., 1966.
(18) K . Eckschlager, "Errors Measurement and Results in Chemical Analysis," Van Nostrand Reinhold Co., Ltd., London, 1969.
~
rJ(.,
=
Vag?
~~~
Wl
A N A L Y T I C A L C H E M I S T R Y , V O L . 45, N O . 14, D E C E M B E R 1 9 7 3
(22)
2365
A. Experiment 4 Precipitation titration by electroactive titrant Solubility product constant of precipitate = 1.80E-10 T 4.0 E-1 1 (mol/l.)**2.0 Stoichiometric ratio of species forming precipitate, electroactive/electrooassive = 1.0/1 .O Charge on electroactive ion = f 1 Titrant concentration = 1.034 E-01 T 2.1 E-04 mol/l. Analyte volume = 20.00 T 0.050 ml Titration solution temperature = 299.6 T 0;3O0K Electrode linear resoonse range = 1.O E-01 to 1.O E-05 mol/l. Experimental data Electrode potential, mv
Data point
Buret reading. ml
1
31 .O T 0 20
0.06 T 0,022
19
297 2 T 0 20
4.49 T 0.022
Results Analyte concn mol/
Del Cl
5.008D-03 5.508D-03
5.008D-04
Fraction Nernstian response 1.oooo 0.9000
-1.OOOD-01
3.4030-09 6.410D-03 T 4.450E-04
Data points used
Formal potential.
Del FN
Del EF
mv
406.22 386.22
6-19
-6.433D-08
-1.814D-06 405.38 F 6.928E 00
0.9735 T 6.629E-02
7-19
B. Error analysis Relative errors FN: 6.810E 00%
CI: 6.942E 00 %
EF: 1.709E 00%
Covariance matrix elements Sigma(EF*FN) = 4.583D-01 Sigma(EF*CI) = -2.731D-03 Sigma(FN*CI) = -2.688D-05
Sigma (EF*EF) = 4.799D 01 Sigma(FN*FN) = 4.395D-03 = 1.9800-07 Sigma(CI*CI) Variance contributions Data point
19
Total to CI From E From S V From V I From CT From TK Total to FN From E From S V From V I From CT From TK Total to EF From E From S V From VI From CT From TK
6 781 E-08 1 722E-08 4 090E-08 2 477E-09 2 163E-09 5 053E-09 1 275E-05 3 239E-06 7 890E-06 4 658E-07 4 067E-07 9 501 E-07 1575D-01 4 000E-02 9 4970-02 5 753D-03 5 023D-03 1 173D-02
16
13
10
7
026E-07 005E-08 601 E-08 807E-09 484E-09 275E-09 2 970E-05 2 907E-06 2 489E-05 5 230E-07 4 294E-07 9 477E-07 4 086D-01 4 000E-02 3 424D-01 7 195D-03 5 907D-03 130413-02
1 022E-07 4 862E-09 9 337E-08 1 237E-09 9 514E-10 1830E-09 5 313E-05 2 527E-06 4 852E-05 6 428E-07 4 944E-07 9 512E-07 8 412D-01 4 000E-02 7 681D-01 1 0180-02 7 827D-03 1 506D-02
1.020E-07 1.518E-09 9.850E-08 7.290E-10 5.251E-10 7.060E-10 1.371 E-04 2.041 E-06 1.324E-04 9.799E-07 7.059E-07 9.489E-07 2.687D 00 4.000E-02 2.595D 00 1.921 D-02 1.384D-02 1.86OD-02
6.566E-11 1.034E-07
1 1 8 1 1 3
Best line for plot of E v s BR Electrode potent al mv
C. Equivalence point
1.028E-07 3.289E- 10 2.211E-10 5.010E-11 1.952E-03 1.239E-06 1.940E-03 6.207E-06 4.174E-06 9.456E-07 6.301 D 01 4.000E-02 6.261D 01 2.003D-01 1.347D-01 3.052D-02
Buret reading. ml 1.300 T 4.436E-01 1.500 T 0.0
1 2 3 4 6 7 1261E01 230 92 F 6 375E 00
.I 297 20 T 2 909E-01
4.490 7 0.0
Best line for plot of E v s log ( C A ) Slope 57 86 F 3 941 E 00 mv Intercept 405 38 F 6 928 E 00 mv Original data modified for plot of E vs log (CA) Electrode potential : mv
CA
230.5 7 0.20
9.656E-04 T 5.981E-04
297.2 7 0.20
1.350E-02 7 4.040E-04
log iCA) -3.0152 F 1.255E-02
- 1 8696 7 5 319E-04
Figure 2. T I T R A T E o u t p u t ( c o n d e n s e d ) : A. basic routine; B. e r r o r analysis; C. plotting d a t a routine Symbols: C/, initial analyte concentration: FN, fraction of Nernstian response; EF, electrode formal potential: DEL, change between successive iterations; E. electrode potential; S V , volume of titrant added: V / , initial analyte volume; CT, titrant concentration: TK, temperature; BR, buret reading: C A , electroactive ion concentration
2366
A N A L Y T I C A L C H E M I S T R Y , VOL. 45, N O . 14, D E C E M B E R 1973
The input data required by the basic TITRATE routine include a, p (if applicable), T, VO, C’, K s p or K f (if applicable), the upper and lower concentration limits of ideal electrode response, and E us. buret reading data points with their estimated experimental errors. Optional input data include initial guesses for Co, f, and E‘ as well as uv0, UC,, U T , and the error in either K s p or K f (if applicable). Basically, TITRATE makes initial parameter estimates from the first two data points and corrects these estimates through successive iterations. Initial parameter estimates either can be supplied externally as data or can be calculated internally by assuming ideal Nernstian response (f = 1). Co is calculated by eliminating E’ from either Equation 12 or 13 using the initial two experimental data points. E’ is found by substituting f and Co into either Equation 12 or 13 evaluated using the initial data point. L E ‘ , A f , and LCo are determined by substituting the required partial derivatives into Equations 17 through 19 and finding the inverse matrix. Because the first order Taylor’s Series expansion of Equation 14 is not exact, these parameter corrections are only approximate. Therefore, the magnitudes of the corrections must be damped to prevent problems due to extreme over-correction. Several correction iterations are usually required before satisfactory parameter estimates are found. During each iteration, TITRATE uses only those data points which were recorded in the specified concentration region of linear electrode response and in regions where the appropriate mathematical expression, Equation 12 or 13, is valid. For precipitation and complexation titrations, validity is tested by calculating the equivalence point titrant volume, V e p ,and excluding all data points a t or beyond V,, uv. For addition titrations, all data points within the electrode linear response region are valid. If fewer than three data points are valid, computation is terminated with an “Inaccurate Results” warning. This technique ensures that the maximum number of valid data points are utilized. Convergence of the correction process is considered satisfactory when each parameter correction is less than its respective parameter standard deviation. Oscillatory convergence problems associated with not having a fixed number of data points can occur. Therefore, constraints have been included in TITRATE to eliminate these convergence problems. An adjustable limit on the number of parameter correction iterations is included to terminate computation with an “Inaccurate Results” warning, should convergence not occur. A condensed example of the basic TITRATE output is shown in Figure 2A. An echo check of all pertinent input data is printed to provide a permanent record. The results of each least squares iteration are printed including the parameter estimates, corrections, and standard deviations, plus a list of which data points were used during each iteration. Figures 2B and 2C illustrate the output from the comprehensive error analysis and plotting data routines, respectively. TITRATE includes two optional routines in addition to the basic least squares routine. The first option, a comprehensive error analysis, requires, as additional input, UV,,, UC’, UT, and the error in either K S P or K f (if applicable) in addition to the input required by the basic routine. The condensed output of the comprehensive error analysis is shown in Figure 2B. The relative errors in the final estimates of Co, f , and E’ are calculated. The covariance matrix elements are required for propagation-oferror calculations when Co, f, and E’ are used in further
+
calculations. Propagation-of-error equations are solved to determine the contributions from the five sources of experimental error, E, V, VO, C’, and T, to the standard deviations of Co, f, and E‘. These error contributions are calculated a t each point where meaningful data were recorded during the titration. With this valuable information, the precision of any parameter can be increased by minimizing the experimental error in whichever variable contributes the greatest magnitude of parameter imprecision. Likewise, precision can be maximized by recording the majority of data in regions where the best parameter precision results (Le., far from or near the equivalence point). In this manner, no time is wasted minimizing experimental error which contributes little to the parameter imprecision. The second option, the plotting data routine, requires no additional input data. This routine calculates a specified number of E us. buret reading and log C data points using the final estimates of Co, f, and E‘. With the exception of the equivalence point, the region in which data are calculated coincides with the region from which data were used to calculate the final parameter estimates. This permits the graphical comparison of the experimental data with the best fit sigmoid and linear curves. Any “bad” experimental data points can be readily isolated in this manner. The propagated error for each calculated data point is included if the error analysis routine has been specified. Condensed output from the plotting data routine is shown in Figure 2C. A copy of TITRATE is available from Dr. Isbell upon request.
EXPERIMENTAL A series of argentometric titrations was devised to demonstrate t h e versatility of TITRATE. A Beckman 39046 sodium-selective electrode was used as a silver-selective indicator electrode because it responds selectively t o silver ion in t h e presence of sodiu m ion. A Beckman Century SS-1 p H meter was used to measure t h e potential of the indicator electrode with respect to a Beckman 39400 calomel reference electrode. Because t h e reference electrode leaked a small amount of chloride ion, it was isolated from the analyte solution by a potassium nitrate-agar bridge. All solutions were prepared by dissolving dried, reagent-grade salts in triple-distilled water. All titrations were performed with simplicity and speed as prime considerations. No attempt was made t o maintain a cons t a n t ionic strength. Approximately equal increments of titrant were added throughout each titration instead of reducing the increment volumes near the equivalence point. After each addition of titrant, t h e solution was stirred for 30 seconds a n d then allowed to rest for 90 seconds before t h e electrode potential was read. T h e d a t a from each titration were processed by an IBM 360/65 computer system.
RESULTS AND DISCUSSION Three of the five possible types of titrations treated by TITRATE have been demonstrated: a standard addition titration of silver ion, a precipitation titration of silver ion by chloride ion, and a precipitation titration of chloride ion by silver ion. Table I1 shows the results of these titrations. Titration 1 is a standard addition titration of silver ion. This type of titration is advantageous to a single point determination because it requires no prior electrode calibration and the accuracy of the determination is improved. During the titration, the main concern was the addition of approximately equal titrant increments, ignoring any titrant drops on the buret tip. The result is a deliberate increase in the experimental error of the buret reading in order to see what effect such an increase has on the results.
A N A L Y T I C A L C H E M I S T R Y , VOL. 45,
NO. 14,
D E C E M B E R 1973
2367
Table 11. Summary of Experimental Data and Results
Titration 1 . Standard addition of silver ion Data Titrant concentration 1.034 f 0.002 X 1 0 - l M Initial analyte volume 20.00 f 0.05 ml Titration solution temperature 301.5 f 0.2 OK 0.2 m V Electrode potential error Buret reading error 0.04 ml Number of data points taken 16
250--
E
200--
2:
t
$ 3. -
Results
150--
Analyte concentration
C 3
Found: 1.042 f 0.066 X 10-3M Taken: 1.034 f 0,010 X 10-3M 90.69 f 1.77 456.69 i 1.70 mV 16 1.89 sec
Nernstian response, % Electrode formal potential Number of data points used Execution time Titration 2. Precipitation of silver ion by chloride ion Data Solubility product constant 1.8 f 0.4 X 10-lo Titrant concentration 6.420 f 0.030 X 10-3M Initial analyte volume 20.00 f 0.05 ml Titration solution temperature 299.0 f 0.2 "K Electrode potential error 0.2 mV Buret reading error 0.02 ml Number of data points taken 19
loo--
50-
O--
Results
Analyte concentration Nernstian response, % Electrode formal potential Number of data points used Execution time
Found: 1.043f 0.022 X 10-3M Taken: 1.034 f 0.010 X 1 0 - 3 M 109.20 f 4.58 462.64 f 7.67 m V 14
1.37 sec
Titration 3. Precipitation of chloride ion by silver ion
Figure 3. Graphical methods of equivalence point determination A . tangent bisection; 6. first
derivative; C. second derivative: D. Gran
plot
Another aliquot of the same analyte used in Titration 1 is the analyte in Titration 2 , a precipitation titration of silver ion by chloride ion. During this titration, the buret tip was below the titration solution surface, thus eliminating the error due to titrant drops hanging from the buret tip. Titration 3 is the inverse of Titration 2--i.e., a precipitation titration of chloride ion by silver ion. A comparison of the results of Titrations 1 and 2 shows t h a t the deliberate buret reading error introduced in Titration l has a negligible effect on the calculated analyte concentration. The calculated electrode formal potentials are the same within experimental error. A percentage of Nernstian response less than 100% is expected because concentrations rather than activities are used in the calculations and because many ion-selective electrodes respond in a non-ideal fashion. However, the high percentage calculated in Titration 2 might be fortuitous because of the statistically small number of data points used in the calculation. Titration 3 is a n example of how the range of applicability of a n ion-selective electrode can be extended. The analyte is chloride ion for which no ion-selective electrode happened to be immediately available. Because silver ion rapidly forms a relatively insoluble precipitate with chloride ion, a silver ion-selective electrode can be used as an indicator electrode during the precipitation titration of chloride ion by silver ion. The results of Titration 3 show excellent agreement between the calculated and known analyte concentrations. The differences in electrode formal potential between the three titrations can be attrib2368
Data Solubility product constant 1.8 f 0.4 X 10-lo Titrant concentration 1.034 f 0.002 X 10-lM Initial analyte volume 20.00 f 0.05 ml Titration solution temperature 299.6 f 0.3 O K Electrode potential error 0.2 m V Buret reading error 0.02 ml Number of data points taken 19 Results Analyte concentration Found: 6.410 f 0.445 X 10-3M Taken: 6.420 f 0.050 X 10-3M 97.35 f 6.63 Nernstian response, % Electrode formal potential 405.38 f 6.93 m V N u m b e r of data points used 16 Execution time 1.04 sec
Table 111. Comparison of Methods for Equivalence Point Determination Method
Actual values TITRATE
Tangent bisection 1st Derivative 2nd Derivative Gran
Analyte Buret reading concentration, at E.P.. ml M X lo3
Error, %
1.30 1.30
6.42 6.42
1.32
6.51
1.5
1.37 1.37 1.27
6.77 6.77 6.26
5.4 5.4 2.4
0.1
uted to different solution ionic strengths and electrode formal potential drift over the two-week period during which the different titrations were performed. Data from Titration 3 are plotted in four different ways in Figure 3. Figure 3A is a conventional plot of electrode potential us. buret reading showing the 19 data points recorded. Because the data points were recorded a t approxi-
ANALYTICAL C H E M I S T R Y , VOL. 45, NO. 14, DECEMBER 1973
mately equal volume increments rather than concentrating data point density in the vicinity of the equivalence point, finding the equivalence point (assumed to be the inflection point) would be subject to considerable error. Figure 3A was drawn knowing the position of the equivalence point a priori, so the curve shape is probably more accurate than would have been possible had the equivalence point not been known. Figure 3A illustrates an alternative method for locating the equivalence point in titrations involving 1:1 precipitates or complexes only. This consists of drawing tangents to the extremities of the titration curve and the bisecting these tangents. The intersection of the curve with the bisecting line defines the equivalence point. The success of this technique is diminished by the asymmetry of titration curve 3A. Figures 3B and 3C are the manually calculated first and second derivatives of titration curve 3A. Again, the lack of data points near the equivalence point results in large uncertainties concerning the exact shapes of curves 3B and 3C. Figure 3 0 illustrates the curves obtained when the titration data are manually modified and plotted by the method of Gran (19, 20). Ideally, two straight lines should intercept the buret reading axis a t the equivalence point. The distortion of the titration curve before the equivalence point (data points 1 through 6) is shown dramatically in Figure 3 0 . The extremely low silver ion concentration in this region causes almost meaningless electrode response. In contrast, the straight line resulting from the data points recorded aft,er the equivalence point indicates that the system is well behaved in this silver ion concentration region. From Figure 2C, the silver ion concentration is seen to be between 9.7 x 10-4 and 1.4 x 10-2M, well within the region of linear electrode response. However, the method of Gran assumes ideal Nernstian response whereas, in reality, the results of computation by TITRATE indicate that the response is only 97% ideal. Therefore, the intercept, value and the equivalence point do not coincide. Table I11 lists the results of the various methods for determining the equivalence point. The method using TITRATE is clearly superior. Several conclusions can be drawn from the list of variance contributions in Figure 2B. Taking the estimate for the analyte concentration, Co, as an example, the variance contribution totals for data points 19 through 7 are relatively constant. Therefore, no accuracy could be gained by recording data in any specific region. If minimization of the standard deviation of the electrode formal potential were to be attempted, data would not be recorded near the equivalence point because the variance contribution from data point 7 is 400 times greater than that for data point 19. As deduced from a closer examination of the variance contributions to Co, the major source of error is the buret (19) G. Gran, Acta Chem. Scand., 4, 559 (1950). (20) G. Gran, Analyst (London), 77, 661 (1952).
reading. Therefore, in order to minimize the standard deviation of Co, the experimental error in the buret reading should be minimized before any other source of experimental error is attacked. Also, the contribution of error from the buret reading increases in the vicinity of the equivalence point in direct contrast to all other sources of error. Therefore, by minimizing the buret reading experimental error, the total variance contribution will decrease in the vicinity of the equivalence point, thus making the recording of data nearer the equivalence point advantageous.
CONCLUSIONS The use of ion-selective indicator electrodes in potentiometric titrations simplifies end-point determinations for many titrations and permits some titrations which have not been really practical in the past. Extending the range of titration types to include complexation and precipitation titrations adds to the versatility of the method, especially for the analysis of a species for which no ion-selective electrode is available. The analysis of such titration data by TITRATE permits an accurate end-point determination for all titrations in which a few of the data points lie in a concentration region where meaningful potentials can be recorded. Thus, even for the many precipitation and complexation titrations in which the concentration of the electroactive ion a t the equivalence point is so low that the electrode response is meaningless, TITRATE can compute an accurate equivalence point. The use of a computer for a least squares data analysis cannot produce accurate results from bad data. Although ion-selective electrodes are being improved rapidly, interferences from irrelevant ions still occur and must be avoided if reliable results are to be obtained. Schultz and Carr have studied the effects of interferences on the equivalence-point determination for both precipitation (21, 22) and complexation (23, 24) titrations. These studies give quantitative titration error estimates which are valuable to know before performing a titration. For analyses in which speed is a critical factor and electrode calibration is no problem, direct potentiometry is probably the best method of analysis. However, if accuracy rather than speed is critical, electrode calibration is troublesome, or if no ion-selective electrode exists but the unknown rapidly forms a complex or precipitate with an ion for which an electrode exists, then the method described above is ideal. The analysis time is not extreme, and all data analysis is performed by computer. Execution time for the complete analysis of data for one titration is usually between 1 and 2 seconds on an IBM 360/65 computer making the cost for computer time minimal. Received for review February 8, 1973. Accepted July 2, 1973. (21) F. A . Schultz,Anal. Chem., 43, 502 (1971). (22) P. W. Carr, Anal. Chem., 43,425 (1971). (23) F. A . Schultz, Anal. Chem., 43, 1523 (1971). (24) P. W . Carr, Anal. Chem., 44, 453 (1972).
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