Data analysis for concentration measurements in the nonlinear

Various methods for the determination of concentrations In the nonlinear response region of Ion-selective electrodes are de- veloped. Data analysis Is...
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Anal. Chem. 1984, 56, 141-747

cyanide system, a pretreated glassy carbon electrode offers no advantage over a clean electrode, and in fact the increased background current a t pretreated electrodes would be a disadvantage.

ACKNOWLEDGMENT Scanning electron micrographs were obtained with the help of Dan Neufeld of the University of South Dakota School of Medicine. The help of John Evans, Ed Bowdin, and Ganapathy Swami of the University of Minnesota is greatly appreciated. Registry No. Fe, 7439-89-6; 02, 7782-44-7; KNOB,7757-79-1; C, 7440-44-0; NADH, 58-68-4; NAD, 53-84-9; hydroquinone, 571-60-8; catechol, 120-80-9; 123-31-9;1,4-dihydroxynaphthalene, ascorbic acid, 50-81-7;ferrocyanide,13408-63-4;1,4-benzoquinone, 106-51-4; 1,4-naphthoquinone,130-15-4;ferricyanide, 13408-62-3. LITERATURE CITED Engstrom, R. C. Anal. Chem. 1982, 54, 2310. Gunasingham, H.; Fleet, E. Analyst (London) 1982, 107, 896. Rice, M. E.; Galus, Z.; Adams, R. N. J. Electroanal. Chem. 1983, 743, 89. Gonon, F. G.;Fombarlet. C. M.; Euda, M. J.; Pujol, J. F. Anal. Chem. 1981, 53, 1386. Falat, L.; Cheng, H. Y. Anal. Chem. 1982, 5 4 , 2111. Van Rooijen, H. W.; Poppe, H. Anal. Chim. Acta 1981, 130, 23. Ejelica, L.; Parsons, R.; Reeves, R. M. Croat. Chem. Acta 1980, 53. 211. Elaedel, W. J.; Jenkins, R. A. Anal. Chem. 1974, 4 6 , 1952. Elaedel, W. J.; Jenkins, R. A. Anal. Chem. 1975, 4 7 , 1337. Elaedel, W. J.; Mabbott, G. A. Anal. Chem. 1978, 50, 933.

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Wlahtman. R. M.; Paik, E. C.; Eorman, S.; Dayton, M. A. Anal. Chem. 19?8, 5 0 , 1410. Eggll, R. Anal. Chlm. Acta 1978, 9 7 , 195. Evans, J. F.; Kuwana, T.; Henne, M. T.; Royer, C. P. J. Nectroanal. Chem. 1977, 8 0 , 409. Hallum, J. V.; Drushel, H. V. J. Phys. Chem. 1958, 6 2 , 110. Jones, I. F.; Kaye, R. C. J. Nectroanal. Chem. 1989, 2 0 , 213. Drushel, H. V.; Hallum, H. V. J. Phys. Chem. 1958, 62. 1502. Epstein, B. D.; Dalle-Molle, E.; Mattson, J. S. Carbon 1971, 9. 609. Blurton, K. F. Nectrochlm. Acta 1973, 78, 889. Laser, D.; Ariel, M. J. Electroanal. Chem. 1974, 52, 291. Evans, J. F.; Kuwana, T. Anal. Chem. 1977, 4 9 , 1632. Myers, Robert E. Ph.D. Thesis, Duke University, 1980. Strasser, V. A. Master's Thesis, Universlty of South Dakota, 1983. Adamson, A. W. "Physical Chemistry of Surfaces", 4th ed.; Wiiey: New York, 1982; p 341. Taylor, R. J.; Humffray, A. A. J. Electroanal. Chem. 1973, 42, 347. Willman, K. W.; Murray, R. W. Anal. Chem. 1983, 55, 1139. Riggs, W. M.; Parker, M. J. "Methods of Surface Analysis", Czanderna A. W., Ed.; Elsevier: Amsterdam, 1975; p 112. Evans, J. F.; Kuwana, T. Anal. Chem. 1979, 57, 358. Miller, C. W.; Karweik, D. H.; Kuwana, T. Anal. Chem. 1981, 53, 2319. Anderson, J. E.; Tailman, D. E.; Chesney, D. J.; Anderson, J. L. Anal. Chem. 1978, 50, 1051. Hale, J. M. "Reactions of Molecules at Electrodes"; Hush, N. S., Ed.; Wiley-Interscience: New York, 1971; p 247.

RECEIVED for review August 1, 1983. Accepted October 24, 1983. This work was supported in part by a Northwest Area Foundation Grant of the Research Corporation and by the University of Minnesota NSF Regional Instrumentation Facility, Grant No. NSF CHE-7916206. This work was presented in part at the 1983 ACS Annual Meeting in Washington, DC.

Data Analysis for Concentration Measurements in the Nonlinear Response Region of Ion-Selective Electrodes Ravi Jain and Jerome S. Schultz* Department of Chemical Engineering, The University of Michigan, Ann Arbor, Michigan 48109

Various methods for the determination of concentratlons In the nonllnear response region of Ion-selectlve electrodes are developed. Data analysls Is done with a computer program based on Marquardt's method for determlnlng the nonllnear parameters. For the multiple standard addltlon method, It Is shown that a high degree of correlatlon between the model and the experlmental data Is needed for accurate results. Nonlinear callbratlons do give good results, even when the correlatlon between the model and the data Is only falr. The regression procedures that minlmlze the least-squares error may not work well because they may demand a degree of correlatlon not possessed by the orlglnal data. Measurements on the chlorlde and the nitrate electrodes are Included to compare various methods. The methods discussed In this paper can be easlly Implemented and can substantlally Increase the worklng range of Ion-selective electrodes wlthout sacrlflce of accuracy.

During the past 2 decades developments in the area of ion-selective electrodes have made them viable alternatives for measuring various ions as well as the gases that are converted to ions in solution. Although electrodes can respond

to changes in concentrations until a statistical limit of detection is reached ( I ) , measurements are usually limited to the linear region of response. The lower limits of detection depend on different factors for different electrodes. For halide and similar solid-state electrodes, this limit is determined by the solubility of the membrane; for electrodes employing liquid ion exchangers, the limit is determined by the distribution of the exchanger between the membrane and the solution phase. Other factors, such as the impurities in the reagents and the interference from other ions in the solution, also affect the detection limit, as discussed by Midgley (1). Midgley (1) also points out that the detection limit for most electrodes extends to far lower concentrations than the linear or Nernstian response limit. The concentrations in the linear region of electrode response are normally measured with either a calibration curve or the method of known standard addition. Multiple additions of a standard solution can also be performed to improve the accuracy (see Mascini (2) for a review of these techniques). For most cases these techniques work reasonably well. Measurements in the nonlinear region of electrode response can be made to fully utilize the capabilities of the electrodes. Among the few published methods for doing this are the graphical calibration methods and the methods by Midgley (3), Parthasarathy et al. (4), and Frazer et al. (5). Because

0003-2700/84/0356-0141$01.50/00 1984 American Chemical Society

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ANALYTICAL CHEMISTRY, VOL. 56, NO. 2, FEBRUARY 1984

of inherent errors associated with nonlinear graphical interpolation, the graphical methods are not very accurate. The methods by Midgley and Parthasarathy et al. involve assumptions about the electrode slope and standard electrode potental which lead to error in concentration measurements. These methods are mostly concerned with the elucidation of the mechanism of electrode response in the nonlinear region and, in their description, results on the accuracy of the techniques are not included. A method has been suggested by Midgley (6) for measurements in the region of limiting linear response, at concentrations near the detection limit, but it has a very limited range of applicability. The method by Frazer et al. ( 5 ) is the only available method which can estimate concentrations in the nonlinear region to an acceptable degree of accuracy. However, because of the requirements of complex and expensive experimental equipment and extensive computing time, it is not suitable for routine measurements. In our present work various methods for measurements in the nonlinear region, based on the theory for nonideal response, are developed. Two of these involve multiple additions of a standard solution to the unknown solution and have been used previously for measurements in the linear region (7,8). The third method involves preparation of a calibration curve by making multiple additions of a standard solution to a solution of known concentration. The unknown concentration is determined by using the calibration curve. Although conceptually similar to the method of Frazer et al. this method uses potential functions based on theoretical considerations, rather than arbitrary spline functions used by Frazer’s group. Our approach significantly reduces the data required for the calibration curve, and the resultant computations can be completed in a matter of seconds on a minicomputer or personal computer. Another advantage of our approach is that all the proposed procedures can be performed with tools regularly used in ion-selective electrode potentiometry. The implementation of our methods requires a nonlinear regression procedure which can converge to solution, even with poor starting guesses. For this purpose, a robust computer program based on Marquardt’s method (9) is used. Measurements on chloride and nitrate electrodes are included for comparing various methods, and it is shown that the nonlinear calibration method, with parameters determined by Marquardt’s method, works best. The concentrations predicted by this method are compared with those obtained using the shift method of Frazer et al. (5). Implications of the results obtained in the nonlinear region on the analysis of data in the linear region are also discussed.

DEVELOPMENT OF THE METHODS Electrode Response in the Nonlinear Region. Following Midgley (I), the emf of an ion-selective electrode in the nonlinear region of response can be expressed as

E = Eo + S log (C + s + b, + z b i )

(1)

1

where E” is the standard electrode potential, S is the electrode slope in millivolts per decade, C is the determinand concentration, s is the contribution to the determinand from the electrode material, b, is the concentration of determinand due to the impurities in the reagents, and bi is the interference effect of the ith interfering species. Equation 1can alternately be written as

E = E”

+ S g(C, K , Bi, B,)

where g incorporates the functionality expressed by eq 1. This functionality depends on the mechanism of nonideal response, examples of which are given later. K in eq 2 is either the solubility product of the electrode material or the distribution

coefficient of ion exchanger, depending on whether the electrode is a solid-state electrode or a liquid-membrane electrode, Bi is the contribution from the interfering ions, and B , is the contribution from the impurities. Methods Available in the Literature for Nonlinear Region. Available methods for measuring concentrations in the nonlinear region include graphical methods, methods which use potential functions incorporating nonideal behavior, and a method which uses spline functions for the calibration curve. 1 . Graphical Methods. A commonly used graphical method involves preparing a calibration curve by measuring potentials at several points in the nonlinear region of response. The concentrations are then determined by graphical interpolation. Another is the Gran plot titration method. An example of low level chloride measurements using Gran’s method is discussed in ref 10. 2. Methods Based on the Nonideal Behavior of the Electrodes. For this class of methods the potential functions discussed by Midgley (3)and Parthasarathy et al. ( 4 ) can be used to determine the cause of the nonlinear behavior. Once a proper potential function has been found, it is used to determine the concentrations. This normally requires a graphical procedure, though a regression procedure can also be used. Midgley’s method assumes that the electrode slope and the standard electrode potential determined in the linear region can be used for the nonlinear region. Parthasarathy et al. also use values from the linear region, though they do take into account the effect of measurement errors on these values. 3. Method Based on Nonlinear Calibration with LeastSquares Splines. In this method proposed by Frazer et al. (5),a nonlinear calibration curve is prepared by making additions to a blank, and an unknown curve is prepared by making additions to the unknown solution. By use of a nonlinear pattern matching procedure and several replicate measurements, a set of estimates of the unknown concentration is produced. Then with a set of artificial intelligence rules, it is decided which of the estimates of the unknown concentration can be accepted. Both methods 1 and 2 have some shortcomings. The graphical calibration may be acceptable for occasional samples and when very accurate determinations are not required. There is inherent error associated with the graphical interpolations which use nonlinear curves. Graphical methods are also not suitable for a large number of samples because they cannot be automated. Gran’s plot procedures, which are also graphical, may be in substantial error if the electrode slope varies more than 1mV from the slope used in preparing Gran’s plot paper (2). Methods of Midgley and Parthasarathy et al. can lead to erroneous conclusions because of the assumption about the electrode slope and the standard electrode potential. This is shown later in this paper. The major problem with the method of Frazer et al. is difficulty in implementation. Also Frazer uses empirical models for nonideal behavior which requires a considerable amount of data handling and hours of computations, making it unsuitable for most measurements in the nonlinear region with ion-selective electrodes. Proposed Methods for Measurements in the Nonlinear Region. The proposed methods for concentration measurements in the nonlinear region are based on potential functions incorporating nonideal behavior. The procedure for data analysis uses a nonlinear regression method developed by Marquardt (9). I. Multiple Standard Additions Method. Multiple additions of a standard solution are made to the unknown solution. If the unknown concentration is C, the initial solution volume is V,, concentration of the standard solution is CI, and the volume of the ith additon is Vi,then the solution con-

ANALYTICAL CHEMISTRY, VOL. 56, NO. 2, FEBRUARY 1984

centration after the ith addition, Ci, can be calculated by use of eq 3. i

(3)

The potential function in eq 2 in general contains six unknown parameters, E”, S, K , C , Bi, and B,, if an unknown concentration is being measured. E o can be eliminated if potentials a t two different concentrations are known. If the emf is El a t concentration C and E , at concentration Ci,then writing eq 2 for E l and Ei and subtraction gives

This procedure reduces the unknown parameters by one. Equation 4 can be rewritten as AEi

= h(S, C , K , Bi, B,, Vi).

h in eq 5 incorporates the functionality expressed by the right-hand side of eq 4. In eq 5, V, and AE, are known. The unknown concentration, C, along with the parameters S, K , B,, and B, can be determined by using a nonlinear regression procedure. This method is an extension of Brand and Rechnitz’s (7)work t~the nonlinear region of electrode activity. II. Nonlinear Calibration Method. Multiple additions of a standard solution are made to a solution of known concentration. The potential function expressed by eq 2 can be used. If the potential is E , at a concentration Ci, eq 2 can be written as

Ei = E”

+ S g(C;, K , Bj, B,)

Both Ci and E , in eq 6 are known. The Civalues can be calculated with eq 3 because the value of C is known. The parameters S, K, B,, and B, can be found by using a nonlinear regression procedure. The potential function with these parameter values and the value of electrode potential in the unknown solution can be used to determine the unknown concentration. III. Methods Based on Maximizing the Correlation Coefficient or Minimizing the Least-Squares Error i n the Regression Procedure. One such method is that prposed by Horvai et al. (8) for the linear region of electrode response. In this case, it is assumed that the unknown concentration lies between two limits. The value of the concentration that minimizes the least-squares error is found by using an interval halving method. The experimental procedure for this method is the same as in method I. In the linear region of electrode response, the least-squares error can be found by using an analytical solution. However, in order to calculate the least-squares error for the nonlinear region, the parameters have to be determined (using eq 6) at each point in the interval halving method by use of nonlinear regression. A method which maximizes the correlation coefficient can be developed along similar lines. Examination of the proposed methods reveals that in each case we have several unknown parameters imbedded in a highly nonlinear function. Before any of these methods can be used, a robust computational procedure for determining the nonlinear parameters is needed. Marquardt’s method (9) for determining the nonlinear parameters is one such procedure. Marquardt’s Method for the Determination of Nonlinear Parameters and the Computer Program. For the proposed methods either eq 4 or eq 6 is used, depending on the method. If we denote the unknown parameters as cy1, ...,

143

an ( n = number of parameters), functionality expressed by eq 4 or 6 as f , dependent variable as y,, and the independent variable as x,, eq 4 or 6 can then be rewritten as follows:

Y , = f, .*.,an,x , ) (7) So the problem involves finding unknown parameters a l , ...,a, in the nonlinear equation, given a set of values of x, and y,. Nonlinear regression procedure for the determination of nonlinear parameters, as suggested by Marquardt (9),is used. The procedure minimizes a least-squares objective function formed by using eq 7 . The method is a combination of the Gauss-Newton method, the method used by Brand and Rechnitz (7), and the steepest descent method (11) and has very good convergence properties. Our experience suggests that a pure Gauss-Newton method would usually not converge to the final solution for methods I to I11 proposed in this paper. Marquardt’s method, on the other hand, works very well because initially it behaves like the steepest descent method which directs the solution in the direction of minimization of the least-squares objective function. The steepest descent method has a slow rate of convergence and, as the solution approaches the true solution, the method behaves more like the Gauss-Newton method. The Gauss-Newton method, of course, converges rapidly when near the final solution. Details of Marquardt’s algorithm can be found in ref 9. We have implemented Marquardt’s algorithm in a computer program called MARQDT. The computer program consists of five routines for setting up and solving equations for nonlinear regression. The user must supply a routine which defines the potential function and a routine (only for method 11) which defines concentration explicitly in terms of the electrode potential and the parameters in the potential function. The computer program can be obtained from the authors. It includes a listing in FORTRAN and a user’s manual which contains examples on the use of the program and brief discussions of various routines. Application of Marquardt’s Method for Finding the Appropriate Potential Function. This procedure involves the selection of various possible potential functions, based on previously reported work in the literature. For a given method, the best potential function is that which provides both a high degree of correlation and parameter values which are physically realistic. If any parameter values, such as electrode slope or solubility product, are known a priori, the parameter values obtained should be close to the known values. If a potential function gives parameter values which are physically unrealistic or too far away from the known values, it is rejected, even if the correlation coefficient is high. Once an appropriate potential function has been found, it can be used to compare the accuracy of various proposed methods by using solutions of known concentrations. Experimental measurements on chloride and nitrate electrodes were used for this purpose.

EXPERIMENTAL SECTION Apparatus. Electrode potential measurementswere made with an Orion (Orion Research Inc., Cambridge, MA) Model 801A mV/pH meter. An Orion solid-state chloride electrode (Model 94-17) along with a double junction reference electrode (Orion Model 90-02) was used for chloride determinations. For nitrate determinations, an Orion liquid-membrane nitrate electrode (Model 93-07) and Model 90-02 reference electrode were used. Microliter quantities of the standard solutions were dispensed with Eppendorf pipets. All computer calculationswere performed on Amdahl 5860 computer at The University of Michigan. Reagents. All chemicals were of analytical grade. In measurements with chloride electrode, NaCl was used to prepare the standard solutions, NaN03 was used as the ionic strength adjustor (ISA),and 10% potassium nitrate was used as the outer junction filling solution for the reference electrode. In measurements with nitrate electrode, potassium nitrate was used for the standard solutions and ammonium sulfate was used as the ISA and the

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Table I. Comparison of Various Computational Methods for Determination of Chloride by Chloride Electrode Data Initial Concn = 2 x M, V , = 100 mL, CI = 10.' M Vi, mL

i

1

0

2 3 4 5 6 7

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

8

9 method I1 (eq 9 ) S (mV/decade) C (mol/L) x

K ' (molZ/L2) x 10'" E"'

a

2.133 -8.09

outer junction filling solution for the reference electrode. All solutions were prepared with deionized water (resistivity greater than 2.0 MO cm). Experimental Procedure. Low level chloride and nitrate solutions were prepared by serial dilution of 0.1 M solutions. For chloride determinations, 1mL of 1 M NaN03 was added to 100 mL of solution to maintain a constant background ionic strength in the case of both the calibration solution and the unknown solution. Small additions of a concentrated chloride solution (with same ionic strength as the background ionic strength) were made to the solution, and the potentials were recorded. This procedure kept the ionic strength of the solution at a constant level. The same procedure was followed for the nitrate electrode except for the ISA which was 0.04 M (NHJ2S0,. The measurement range M, and for the for the chloride electrode was 5 X lo4 M to M. All meanitrate electrode the range was lo4 M to 5 X surements were made at 25 "C in well-stirred solutions. Potential Functions for Chloride and Nitrate Electrodes. Chloride. In the absence of determinand impurities and interferences, the electrode potential of a solid-statechloride electrode, incorporating the activity coefficient, is given by eq 8

+ S log (yC/2 + (y2C2/4+ K ) l l 2 )

(8)

All symbols in eq 8 were defined earlier, except y, which is the activity coefficient. A slight rearrangement of eq 8 gives

E = E"'

+ S log (C/2 + (C2/4 + K')ll2)

(9)

where E"' = E"

+ S log y

(loa)

and

K' = K / y 2

(lob)

Equation 9 was prposed by Bardin (12) and was also shown to be valid by Midgley ( 3 ) for various cases with negligible impurities and interferences reported in the literature. If the effect of interferences on the electrode response cannot be neglected, the potential function is given by eq 11. E = E"'

+ S log (C/2 + (C2/4 + K')1/2+ B J

(11)

If a small amount of determinand is present in the reagent blank, the response of the electrode is given by eq 12.

E = Eo'

+ S log [(C + BJ/2 + ((C + BJ2/4 + K')1/2]

method I (eq 13)

-55.22

B i(mol/L) x l o 7 % error corr coeff 0.999969 least-squares error 0.0529 The concentration interval used for calculations: 5 X

E = E"

242.3 231.0 222.5 216.4 21 1.4 207.3 203.7 200.5 197.9

Results method I1 (eq 11)

-54.94

lo5

Ei,mV

(12)

-54.71 1.941 2.196

1.975 -9.03 8.15

2.95 0.999941 0.0529

0.999970 0.0527 M to 5 X

method 111' (eq 9 ) -55.07 2.03 2.103 -8.54 1.50 0.999969 0.0531

M.

To analyze low level chloride measurement data, eq 9,11, and 12 were used for methods I1 and 111. The equation corresponding to eq 9 for method I is given as follows:

The equations for other cases and method I can be written similarly. Nitrate. Based on theoretical derivation, Kamo et al. (13) showed that eq 9 can also be used for liquid-membrane electrodes incorporating a sparingly soluble ion exchanger, if K in eq 9 is taken to mean the exchanger distribution coefficient. Both eq 9 and 11were used to analyze the low-level nitrate data.

RESULTS AND DISCUSSION The accuracy of various computational methods discussed earlier was evaluated by use of the computer program and the experimental data on chloride and nitrate electrodes. The graphical methods are not included, for the reasons mentioned earlier. Brand and Rechnitz's (7) data on chloride and lead electrodes in the linear region of response are also analyzed in light of the results obtained with measurements in the nonlinear region. Analysis of Low Level Chloride Data. For measurements with the chloride electrode, the electrode was calibrated in the linear region prior to the measurements in the nonlinear region. A value of -57.4 mV/decade was found for the electrode slope, S, and a value of -7.3 mV was found for the standard electrode potential, E"'. Table I lists the experimental data for multiple additions of a M chloride solution to 100 mL, 2 x M chloride solution. The electrode manufacturer suggests the limit of linear response for this electrode as 2 X M, so these measurements are in the nonlinear region. The data were analyzed first by using Midgley's (3)method and eq 9. Midgley suggests that if the nonlinear behavior is due totally to the solubility effect, a plot of [lO(E-E"')'s - C/2l2 vs. C2 should be a straight line with a slope of 0.25 and an intercept equal to K', the solubility product. Midgley also assumes that the values of E"' and S found in the linear region apply in the nonlinear region. A linear regression between [lO(E-Eo')/s- C/2I2 and C2, using data in Table I, gave a cor-

ANALYTICAL CHEMISTRY, VOL. 56, NO. 2, FEBRUARY 1984

145

Table 11. Comparison of Various Computational Methods for Determination of Chloride by Chloride Electrode Data M, V, = 100 mL, CI = l o - * M Initial Concn = 5 x E,, mV Vi, mL i 252.3 244.3 237.8 232.5 228.1 224.1 220.8 217.8 215.1 212.6

0

2 3 4 5 6 7 8 9

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

10

method I1 (eq 9 ) S (mV/decade) C (mol/L) x l o 6 K ’ (molz/LZ)X l o l o E”’

Bi(mol/L) x l o 5

-51.13 1.849 7.39

Results method I1 (eq 11) -64.47 0.068 -41.03 2.205

% error

corr coeff least-squares error a

0.999874 0.193

The concentration interval used for calculations:

0.999992 0.0122 M to

relation coefficient of 0.999 66. The slope of the straight line was 0.981, and the intercept on the y axis gave a value of K ’ = 11.97 X mo12/L2(Kk,, = 1.93 X mo12/L2). These results indicate that the data in Table I are not described by eq 9. However, this is not the case. The problem with Midgley’s method lies in the assumption that the values of S and Eo’found in the linear region can be applied in the nonlinear region. This assumption fails because (1) the electrodes are inherently less stable in the nonlinear region, leading to larger measurement error compared to the linear region, and (2) the assumption that the electrode membrane is in chemical equilibrium with the solution next to it is not completely true. In well-stirred solutions, a concentration bounary layer exists near the electrode surface which prevents achievement of complete equilibrium. To test this hypothesis, eq 9 and method I1 were used to determine the parameters, S,K, and E O ’ , using the nonlinear regression program. The values are listed in Table I. An electrode slope of -54.94 mV/decade, a value of Eo’= -8.09 mo12/L2were obtained. mV, and a value of K’ = 2.13 X A correlation coefficient >0.9999 was found between the data and the model. The value of K’is within 10% of the literature value a t this ionic strength. These results suggest that the data are fitted well by eq 9. The value of the electrode slope found by the nonlinear regression is different than the value in the linear region. This explains the reason for the failure of Midgley’s method. Equation 11was used to determine the effect of interferences on electrode response. A value of B, = 8.15 X M was found which indicates that the effect of interferences on electrode response is negligible. Other parameters for this case are listed in Table I. Use of eq 12 and the nonlinear regression program gave a value of B, = -4.04 X M which indicates that the effect of determinand blank on electrode response is also negligible. These results suggest that eq 9 can indeed be used for chloride measurements in this experiment. The possibility of determining an unknown concentration by a multiple standard addition method (method I) was explored. Por this purpose eq 13 was used with the data listed in Table I. The values of the parameters S, C, and K were determined by use of the computer program and the results

method I (eq 13)

method IIIa (eq 9)

-54.81

-57.67

11.7

10.0

1.531 134.0 0.999893 0.0965

1.741 -5.77 100.0

0.999951 0.0743

l o - * M. are listed in Table I. The solution concentration was determined to within 3% using this method, and a correlation coefficient >0.9999 was found. Horvai’s method (method 111)was also used to analyze these data. The search interval for minimizing the least-squares error was taken to be 5 x lo4 M to 5 X M. At each point in the interval halving method, method I1 and eq 9 were used to determine the least-squares error. The results in Table I indicate that for this particular set of data, Horvai’s method also works well (% error 2.2 X M), which is unlikely. Equation 9 is applicable in this case also and the subNernstian response is probably caused by nonequilibrium conditions prevailing near the electrode surface. The results in Table I1 for methods I and I11 indicate that both of these methods are in substantial error. The error in all experiments, except for the one in Table I, using methods I and 111, ranged from 20% to 210%. The correlation coefficient (determined using method 11) in all cases was greater than 0.999 but less than 0.9999. The reason for the success of method I for the data in Table I and its failure for other cases lies in the fact that the correlation coefficient for the other cases is not as high as for the data in Table I. This conclusion is supported by the fact that for all cases other than the one in Table I, the percent error in prediction by method I decreased as the correlation coefficient increased. The reason for the failure of method I also applies to method 111. Method I11 also failed because it tries to achieve a degree of correlation not possessed by the original data. For the data in Table 11, the correlation coefficient using method I11 is 0.999 95 while

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Table 111. Comparison of Results Obtained by Single Point Method and Shift Method for a Chloride Electrode starting % error concn for by single % error calibration unknown point by shift shift paracurve, M concn, M method method meter, mV 2 X 10.’ 2 X 10.’ 1 X 10.’ 1x10-5 5 X10-6

4X 6X 2X 3x 7X

10.’ 10.’

3.50

lo-’ 10-5

8.65 7.60

1.00 11.2

2.05 1.47 6.25 8.30 13.7

-0.066 -0.040 -0.086 -0.0018 -0.035

the correlation coefficient found by the nonlinear calibration method is 0.999 87. Similar results were obtained for other data sets using method 111. Though the correlations in most cases are not high enough for methods I and 111, method I1 can be used to measure concentrations accurately in all cases. The maximum and the average deviations from the calibration curve in 10 experiments conducted between 5 X M and M, using method 11, were 3.8% and 0.56%, respectively. Thus a very high degree of correlation between the data and the model is not required for method 11. To determine the unknown concentration using method 11, calibration curves were prepared by use of the method outlined earlier. The electrode potential in the unknown solution along with the calibration curve is used to determine the unknown concentration. We call it “single point method” because it uses only the electrode potential in the unknown solution. To determine unknown concentrations using the method of Frazer et al., additions of a chloride solution of known concentration were made to the unknown solution to prepare an unknown curve. Only the shift method was used because it takes into account the offset in data. For this purpose eq 10 of Frazer et al. was used to find the concentration which minimizes the squared difference between the calibration and the unknown curve. The minimization was done by using a one-dimensional search procedure employing a combination of the goldensection method and successive parabolic interpolation. A computer program for this method is discussed by Forsythe et al. (14). The results of various experiments to determine unknown concentration are listed in Table I11 and include errors in prediction by the single point method (this paper) and shift method (ref 5 ) . No significant difference in the predictions by the shift method and single point method is seen. In most cases the values predicted by the single point method are acceptably accurate. Single point method is preferable in those cases where it works because of its simplicity. The reason that the two methods give similar results is that the shift parameter, which is a measure of offset in data, is small in all cases listed. The shift parameter method may have utility in determining the quality of data. A large value of the shift parameter indicates electrode malfunction, and data in that case should be discarded. The shift method is provided as one of the options in the computer program MARQDT. We believe that the single point method in combination with the shift method to determine electrode stability (and hence the quality of data) would suffice for measurements in most cases. A large number of null (unacceptable) results returned in Frazer’s experiments are probably due to the following reasons. (1)A time of less than 1s was allowed between additions in measurements with the electrodes. Such an interval is too short to obtain stable readings, especially in the nonlinear region where an equilibrium phenomenon is involved (equilibrium between electrode membrane and the solution next to it). The inherent response time for most solid-state electrodes to changes in

Table IV. Nitrate Determination by Nitrate Electrode and Nonlinear Calibration Method Initial concn = 2 x

i

Data M, V , = 101 mL, CI Vi, mL

1

0

2 3 4 5 6 7

0.2 0.2 0.2 0.4 0.4 0.4 0.4 1.0

8

9 10 11

1.0 1.0

Results method I1 ( e s 9) S (mvidecade) -50.08 5.05 K ’ (mol’/L’) X 10” E”’ 38.43 Bi (mol/L) x l o 6 0.99995 corr coeff

=

l o w 3M

E,, m V

293.3 290.2 281.4 284.6 279.8 275.6 271.8 268.7 262.3 257.1 252.9 method I1 ( e s 11) -48.11 10.46 45.48 -4.22 0.99997

concentration is more than 1 s. (2) Frazer discards the beginning portion of the unknown curve. However, experience with the shift method suggests that it is the most important part of the unknown curve in the pattern matching procedure and should be included in calculations. To avoid the drift problems encountered by Frazer’s group in this region, the electrode system should be allowed to come to equilibrium with the unknown solution before any measurements are made; then the additions should be made. Finally, if one decides to use the artificial intelligence method of Frazer et al., it would be easier to apply with the use of a calibration curve based on nonideal electrode behavior. The reason is that the amount of data required to prepare the calibration curve is significantly reduced. Good correlations were found with less than 10 points, using theoretical models discussed in this paper; Frazer’s group used about 300 points with their empirical models. Analysis of Low-Level Nitrate Data. Measurements with a nitrate electrode (Orion Model 93-07) in the nonlinear region of response were made to see if the concepts applicable to the solid-state chloride electrode could be extended to liquid-membrane electrodes, such as the nitrate electrode. The limit of linear response for this electrode, according to the manufacturer, is 2 X M. Measurements were made by making multiple additions of M nitrate solution to a solution containing 2 X lo4 M nitrate. The experimental data for this case are given in Table IV. Based on experience with the chloride electrode, only method I1 was used with the nitrate electrode. Both eq 9 and 11 were evaluated, but eq 11predicted unreasonably high interference and was, therefore, rejected. The results in Table IV indicate that the data can be fitted by eq 9 with a correlation coefficient of 0.999 95. A value of -50.1 mV/decade was found for the electrode slope, and a value of 5.05 x lo-” mo12/L2 was found for the exchanger distribution coefficient. The value of the electrode slope suggested by the manufacturer is -56.0 mV. No estimates are available for the exchanger distribution coefficient. The average and maximum deviations from the calibration curve ranged from 1.08% to 3.45%. The reason for the subNernstian slope is the same in this case as with the chloride electrode; Le., the solution next to the membrane is not in complete equilibrium with the electrode material. The results

ANALYTICAL CHEMISTRY, VOL. 56,

NO.2,

FEBRUARY 1984

147

indicate that method 11,as well as the concepts developed for a solid-state electrode, can be used to analyze nonlinear response data for a liquid-membrane electrode similar to the nitrate electrode. Applications to the Analysis of Multiple Standard Addition Data in the Linear Region of Electrode Response. The conclusions derived from the measurements in the nonlinear region of electrode response are equally applicable to the linear region. The computer program M Q D T can be used for the analysis of data by the multiple standard addition method (method I) and the calibration curve method (method 11). The appropriate potential functions for these two cases are given by eq 14 and 15.

In conclusion, we have developed a simple and general purpose computational technique for the analysis of ion-selective electrode data in both linear and nonlinear regions of response. The analysis of experimental results indicates the nonlinear calibration method to be the most accurate and generally applicable. The computer program for our work was run on a large computer, but the program is fairly small and can be run on a minicomputer or a programmable calculator (see Clare (15) for a calculator program on nonlinear regression using the Gauss-Newton method). Electrode potentiometry in the nonlinear region can be carried out easily, with relatively unsophisticated instrumentation, using the techniques described in this paper.

Ei - El = AEi = S log (Ci/C)

(14)

E = Eo + S log C

(15)

ACKNOWLEDGMENT We wish to acknowledge the very helpful discussions with Mark Meyerhoff of the Chemistry Department at The University of Michigan.

Chloride and lead data of Brand and Rechnitz (7) were analyzed by using methods I and 11. For the chloride electrode, a very high correlation coefficient of 0.999 983 was found by the calibration method, suggesting that the multiple standard addition method (method I) is likely to work well for this case. Indeed, method I and eq 14 gave a correlation coefficient of 0.999 946 and an error in concentration of less than 1%. However, for the lead data of Brand and Rechnitz, the calibration curve method gave a correlation coefficient of only 0.998 66. This value and earlier experience with nonlinear response data for the chloride electrode suggest that a multiple standard addition method will not work in this case. For the lead data the multiple standard addition method gave a correlation coefficient of 0.999 84 (a value higher than-that possessed by the data) and an error in the concentration of about 25%. The correlation coefficients in the linear region of electrode response are usually not as low as for the lead data of Brand and Rechnitz. Experience in the linear region with chloride and nitrate electrodes suggests that a high degree of correlation (>0.9999) usually is found for the data. As expected, the multiple standard addition method gave satisfactory results for most cases involving these electrodes.

LITERATURE CITED (1) (2) (3) (4)

(5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)

Mldgley, D. Ion-Sel. Nectrode Rev. 1981, 3 , 43-104. Masclnl, M. Ion-Sel. Electrode Rev. 1980, 2, 17-71. Mldgley, D. Anal. Chem. 1977, 4 9 , 1211-1218. Parthasarathy, N.;Buffle, J.; Monnler, D. Anal. Chim. Acta 1974, 6 8 , 185-1 96. Frazer, J. W.; Balaban, D. J.; Brand, H. R.;Robinson, G. A.; Lannlng, S. M. Anal. Chem. 1983, 55, 855-881. Midgley, D. Analyst (London) 1880, 105, 417-425. Brand, M. J. D.; Rechnitz. G. A. Anal. Chem. 1970, 42, 1172-1177. HONal, G.; Domokos, L.; Pungor, E. Fresenius’ Z.Anal. Chem. 1978, 292, 132-134. Marquardt, D. W. J . Soc. Ind. Appl. Math. 1983, 1 1 , 431-441. “Analytical Methods Guide”; 9th ed.; Orion Research Inc.: Cambridge, MA, 1978; 33-34. Marquardt, D. W. Chem. Eng. Prog. 1059, 55(6), 65-70. Bardin, V. V. Zavod. Lab. 1962, 2 8 , 910-913 (see Ind. Lab. (Engl. Trans/.) 1982, 2 8 , 967-969, for English translation). Kamo, N.; Hazemoto, N.; Kobatake, Y. Talanta 1977, 2 4 . 111-115. Forsythe, G. E.; Malcolm, M. A.; Moler, C. B. “Computer Methods for Mathematical Computations”; Prentlce-Hall: Englewood Cliffs, NJ, 1977; Chapter 8. Clare, B. W. Chem. Eng. 1982, 89 (17), 83-89.

RECEIVED for review May 11,1983. Accepted October 19,1983. This work was partially supported by National Science Foundation Grant No. NSF/CPE 78-13316.