THE PRECISION OF POTENTIOMETRIC MEASUREMENTS. G. Horvai, K. Toth and E. Pungor. Institute for General and Analytical Chemistry,

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1 THE PRECISION OF POTENTIOMETRIC MEASUREMENTS G. Horvai, K. Toth and E. Pungor Institute for General and Analytical Chemistry, Technical University of Budapest, Hungary 1111 Widespread use of ion-selective electrodes and the growing possibility of using sophisticated computer technique in conjunction with the electrodes has made necessary to investigate the degree of precision attainable by potentiometric techniques. Results obtained by applying error propagation statistics to various potentiometric methodologies, like calibration, single and multiple standard addition, titration, etc. have been critically reviewed. Before the 1960-s' analytical potentiometry was mainly limited to ph and redox potential measurement. The development of various ion-selective sensors, sensitive to a wide range of ionic and nonionic species, brought about a renaissance of potentiometry. Electrodes and modern ph-mv meters are now used in most analytical laboratories for routine measurements and they have found wide acceptance in industrial monitoring, too. Much experience has been gathered in the use of these electrodes as manifested in the great number of publications dealing with them. The possibilities for directly sensing newer and newer ionic species by electrodes seem to have almost reached their limits by now. Recently developed sensors are either improved versions of earlier types or indirect sensors, i.e. they are based on one of the known sensors, but they can measure a second species through an interposed chemical or biochemical reaction. Many applications of ISEs have been described for solving practical analytical problems. When compared with alternative methods, ISE methodologies are often found to be fast, simple and avoidable of troublesome interferences. On the other hand, the precisi-on of the ISE methods is not always satisfactory. Time has arrived when the precision of ISE measurements has to be studied more closely. This has been recognized by a number of authors who have very recently and independently published their studies on this subject1)-12). Earlier publications on this matter are rather scarce13)-18) The sources of experimental error in potentiometric measurements may arise in different parts of the measuring system, vis. in the ph-mv meter, the electrodes, solutions and standards. Such potential sources of error have exhaustively been reviewed recently by Durst19). Some of these errors can be avoided by careful experimental work while the others should be decreased as much as possible by suitable data processing.

2 BUNSEKI KAGAKU Vol. 30 (1981) The potentials of error reduction by statistical processing of the data have enormously increased in recent years. Many laboratories and industrial control systems are now equipped with on-line working computers which can easily handle the data e.g. by averaging, curve smoothing, curve fitting, etc. The computer may also control the recalibration cycle of the measuring system. Dedicated new ph-mv meters are incorporated with their own computing facilities based on microprocessors. The possibilities of sophisticated data processing being given, we have to see to what extent they can be used for improving the quality of our experimental results. Another task is to set limits for the allowable error, so that the computer can accept or reject the results. The potentiometric error It is difficult to make any general statement about the nature of the potentiometric errors. The reason seems to be quite obvious: the errors caused by the instrument, by the electrodes and by the standardization process are roughly in the same order of magnitude so that the main source of error may be any of these, depending on the actual experimental set-up. This may be the reason why almost no author has tried to assign the error to a single source. On the other hand, most authors agree that if a given experimental procedure is repeated at a number of times, the measured e.m.f. (E) will be randomly distributed with standard deviation se. It should be stressed here that se may largely depend on the experimental circumstances. For example, if the electrodes are removed from solution and dried after each reading, the error of the e.m.f. may be considerably larger than in the case when subsequent readings are taken in the same solution without removing the electrodes. According to this remark, se may be much less in a standard addition measurement than with the two-point calibration method (see below), even if the same devices and solutions are used. se could in principle depend on the measured concentration and consequently on the e.m.f. (E) itself. Most authors agree, however, that apart from extremely low or high concentrations, se will be independent of E. In other words, the absolute error of the e.m.f. is regarded constant in this range. E.m.f. drift has been experienced quite often. It is, however, usually omitted from the discussion of potentiometric error, because in most cases it can be corrected for. The potentiometric error is primarily manifested in the erroneous e.m.f. reading. The above remarks were concerned with this error. The analyst is, however, more interested in the error of the sample concentration which has been calculated from the e.m.f. readings. The error in the calculated sample concentration will depend not only on se, but also on the specific mathematical relation between the measured e.m.f. -s and the sample concentration. This relation depends on the adapted measurement technique (e.g. standard addition or potentiometric titration) and on the concentrations and volumes of the solutions used in the measurement. General expressions for the standard deviation of the sample concentration will be reviewed below together with the conclusions that can be made on this basis.

3 The scheme of the calculations In the following, the sample concentration will be denoted by cx and its standard deviation by Scx E In accordance with the earlier remarks it will he assumed that the error of e.a.f. measurements can be expressed by their standard deviation SE, and se is independent of E. The aim of the calculations is, as already stated, to express s c as a function of se and the experimental parameters. This can be done by using propagation of error statistics: (1) where the E.-s are measured e.m.f.-s. To obtain the derivatives in this equation, one tries to express first cx as a function of the Ei-s. This expression will be different for the different measuring techniques, like two-point calibration, standard addition, etc. In some cases, e.g. multiple standard addition, cx is calculated from the Ei-s by nonlinear regression methods, hence there is no explicit expression for cx. For these cases no general solution of the problem has been found but some typical experiments have been discussed. Discussion of various measuring techniques A number of different potentiometric measuring techniques have been worked out and claimed to have specific advantages under certain circumstances20)-27). These techniques spread from the most conventional calibration method to potentiometric titrations. As it has been stated before, the relationship between sc and se depends among others on the measuring technique chosen, thus the need isxfelt to give here short definitions of the more important techniques before the respective mathematical results will be shown. 1) Direct measurement after multipoint calibration. Several (n) standard solutions of concentration ci(i=1,2,...n), dispersed over the expected sample concentration range, are used for calibration of the electrode. The sample concentration cx is determined from the potential of the electrode measured in the sample, by using the least-squares calibration line. For the sake of simplicity we shall discuss here only a limiting case of this method, when an infinite number of calibration points is taken, distributed over a large enough interval to include all sample concentrations. In this case there will be no error in the determination of the calibration line and thus, the only source of error is the error in the potential measurement in the sample. It will be assumed here, as in all subsequent cases, that the calibration line can be described by the following equation: where E' and S' are constants. If this equation is written for the sample concentration cx and solved for cx,one obtains: (2) (3)

4 BUNSEKI KAGAKU Vol. 30(1981) By derivation of this expression: (4) Introducing this result into Eq. (1): (5) or for the relative standard deviation of c x : (6) This result indicates that the relative error of the concentration measurement is independent of c x. 1 mv standard deviation in the e.m.f. reading causes about 4 percent error in the calculated concentration of a monovalent ion and about 8 percent in the concentration of a divalent ion. This well-known relation holds, however, only if the constants of the calibration line, E' and S', are very reliable, i.e. are measured practically without any error. Ebel et al.10) discuss the same method with n calibration points, i.e. when E' and S' are measured with a certain error. They have treated the statistical errors in E' and S' as uncorrelated. Since this assumption does not hold, their conclusions cannot be readily accepted. 2) Direct measurement after two-point calibration. This is a subcase of method 1) with n = 2, i.e. with the use of only two calibrating solutions of concentrations c1 and c2' This subcase is treated here separately, because of its practical importance: many automated potentiometric analyzers are working on this principle. A derivation similar to the previous one yields the following result: (7) Comparison of Eq. (7) with Eq. (6) shows that the relative standard deviation has increased here by the square-rooted factor. This factor is obviously greater than unity; by closer examination it is found to vary between sqr 1.5 and sqr 2, assuming that the sample concentration is bracketed by the standards. This means that compared with a completely error-free calibration the precision will decrease at most by a factor of 1.4 (in a statistical sense, of course). This explains why the two-point calibration method is usually found to be quite satisfactory. Eq. (7) has been independently deduced by Svehla2) and Horvai et al.3)6) Heidecke et al.11) have also set out with the problem. Although their statistical treatment is not rigorous, they obtained a good approximate result which agreed quite well with their experimental results. 3) Standard addition or known addition. With this method the e.m.f. is measured in the sample (c x'vx) at first; then a certain amount of standard solution (cst' Vst)

5 is added to the sample and the new e.m.f. is recorded. The relative standard error of cxhas been found3)6) to be: (8) Almost the same result has been found by others1)7); the factor 2 which these authors have found instead of sqr 2 seems to be erroneous. Eq. (8) can be used to find the optimum amount of standard solution. By increasing the amount of standard, i.e. cst Vst' the relative error of cx decreases; this decrease is very abrupt until c st Vst < cx Vx but is only very slight if cst Vst > 3 c V.Hence it is advisable that c st Vst be at least 2 cx Vx but it need not exceed 3 cx Vx. The standard solution is usually much more concentrated than the sample, i.e. the change of volume during the addition is negligible. This means that the sample concentration should be increased by a factor of 2 to 3. The standard addition method requires the knowledge of S' from a separate experiment. Thus, the value of S' can also be in error. If the standard deviation of S' is ss, then the total error in cx will be described by: (9) The effect of the error in S' has been correctly deduced by Ratzlaff7) and by Horvai et al.3)6), whereas the result of Mascini1) seems to contain a slight error. 4) Double known addition. This method starts as a standard addition, but the first addition of standard is followed by a second one. With this method one does not have to know E' or S' from separate measurement; from the three e.m.f. readings (in the sample and after both additions) one can calculate the three unknowns: E', S' and c x. This method does not appear to have gained any practical importance. Its precision has been studied3)6), however, because this allows to make conclusions on the precision of the more interesting multiple standard addition method4). For the results and their discussion the quoted references may be consulted. 5) Multiple standard addition. This method is again a standard addition method, this time with more additions than necessary for calculating c x. There are two important versions of this method. In the first one S' is known from a separate experiment and the evaluation is usually made by the so-called Gran's method1)28) In the second one neither E' nor S' are known in advance and the evaluation is made by nonlinear regression. Since in both cases the number of e.m.f. readings is more than the necessary minimum for calculating c x, regression methods (graphical or numerical) have to be used. Analysis of the statistical behaviour of both versions 4)6)13)-15) gave some s urprising results. Buffle et al. have have proved that in the Gran type linear regression, suitable weights should be used to avoid distortion of the results. Thus, the usual graphical evaluation is also incorrect from the statistical point of view. They have also shown that if the separately measured value of S' is itself also in error, this may greatly increase the error in c x. If S' is not measured separately and therefore nonlinear regression is used for the calculation of c x 4)29)30), the error in cx may be surprisingly high. This has

6 BUNSEKI KAGAKU Vol. 30(1981) been attributed to the nonlinear character of this regression problem. A recent study31) on nonlinear regression in analytical chemistry seems to support this idea, 6) Analate addition. This is the reverse of standard addition, i.e. the potential is measured first in the standard (c st'vst) and then the sample (cx, Vx) is added and the potential is read again. Here3)6): (10) and by the same reasoning as with the standard addition method cxvx should be 2 to 3 times as much as cstvst. Whether this is achieved by keeping the ratio of Vx to Vst high, or by using a low concentration standard solution, can be decided by the experimenter. 7) Known subtraction. The e.m.f. is measured first in the sample (cx, Vx), then an analytical reagent which reacts quantitatively with the analyte (cr, Vr) is added and the e.m.f. is read again. The concentrations are expressed in equivalents per volume (cx Vx > cr Vr ). The relative error of the sample concentration is (11) This result3)6)8)is very remarkable, since the bracketed expression can approach zero by increasing the amount of reagent until it becomes nearly equivalent to the amount of sample. This means that the effect of the error in the c.m.f. reading may become negligible. Known subtraction can, in fact, be regarded as a singlepoint titration (see below). 8) Potentiometric titration. A reagent of concentration cr is added in increments A Vi to the sample (cx, Vx) and the equilibrium potential after each addition is measured. The concentrations are expressed in equivalents per volume. The total Potentiometric titrations can be evaluated by graphical or by numerical, i.e. digital, methods. Digital methods have been reviewed by Ebel et al.24)32), who have distinguished approximation and mathematical methods. Approximation methods are used to locate the inflection point on the titration curve, which is not necessarily the equivalence point. In contrast, the so-called mathematical methods are used to locate the equivalence point on the basis of a mathematical description of the titration curve. The distinction between approximation and mathematical methods is not sharp. Approximation methods are actually based on the mathematical description of the titration curve, too, but usually with many neglections which are not always explicitly stated. These neglections cause that approximative methods, although they have very general character, cannot be used in a number of situations, e.g. when there is no inflection point on the titration curve33) or when the titration curve is unsymmetrical 34) By reviewing the respective literature one will have the impression that approximative methods apply only a few titration points in the neighbourhood of the

7 equivalence point, whereas mathematical methods use a much larger portion of the titration curve for the evaluation. It is believed that using a great number of titration points, more or less far away from the equivalence point, will increase both the accuracy and the precision of the titration. This seemingly obvious concept deserves further consideration5)6). It has been shown that the relative location of the titration points on the titration curve is a very important factor in deciding their usefulness for the calculation of the equivalence point. As a consequence, it has been proved that by application of the mathematical methods to only one, two or three titration points near the equivalence point, very precise results can be obtained, and using more points of the titration curve may be quite superfluous. Conclugion The above discussed statistical results may serve as a guideline in choosing the optimal potentiometric technique for a given purpose. It should be clearly seen, however, that precision is only one of the many factors that have to be considered before such a choice. Selectivity, accuracy and other factors have to be cleared first. When this had been done it is desirable to study the reproducibility of the e.m.f. at different concentration levels. This will enable the experimenter to use the above discussed results for finding the most suitable method and estimating the precision attainable with it. References 1) M. Mascini: Ion-sel. Electr. Rev., 2, 17 (1980). 2) G. Svehla : "Electroanalysis in Hygiene, Environmental, Clinical and Pharmaceutical Chemistry", Edited by W. F. Smyth, p. 21 (1980), (Elsevier, Amsterdam). 3) G. Horvai, E. Pungor : Anal. Chim. Acta, 113, 287 (1980). 4) G. Horvai, E. Pungor : Anal. Chim. Acta, 113, 295 (1980). 5) G. Horvai, E. Pungor : Anal. Chim. Acta, 116, 87 (1980). 6) G. Horvai : "Critical Study of Potentiometric Measuring Techniques", (1979), (Candidate's Thesis, Budapest). 7) K. L. Ratzlaff: Anal. Chem., 51, 232 (1979). 8) H. Matsushita, N. Ishikawa : Nippon Kagaku Kaishi, 1975, ) S. Ebel, E. Glaser, H. Mohr : Fresenius' Z. Anal. Chem., 293, 33 (1978). 10) S. Ebel, E. Glaser, A. Seuring : Fresenius' Z. Anal. Chem., 291, 108 (1978). 11) G. Heidecke, G. Stork, J. G. Schindler, M. V. Gulich, W. Schmid, H. Maiser, H. -O. Lindt, D. Sailer : Fresenius' Z. Anal. Chem., 301, 406 (1980). 12) E. Still, Talanta, 27, 573 (1980). 13) J. Buffle, N. Parthasarathy, D. Monnier : Anal. Chim. Acta, 59, 427 (1972). 14) J. Buffle : Anal. Chim. Acta, 59, 439 (1972). 15) N. Parthasarathy, J. Buffle, D. Monnier : Anal. Chim. Acta, 59, 447 (1972). 16) S. L. Burden, D. E. Euler : Proc. Indiana Acad. Sci., 82, 167 (1973). 17) S. L. Burden, D. E. Euler : Anal. Chem., 47, 793 (1975). 18) T. Eriksson : Anal. Chim. Acta, 58, 437 (1972).

8 BUNSEKI KAGAKU Vol. 30 (1981 ) 19) R. A. Burst : "Ion-selective Electrodes in Analytical Chemistry" Edited by H. Freiser, p. 311 (1978), (Plenum, New York)., Vol. 1, Chap.5, 20) R. A. Burst (Ed.) : "Ion-selective Electrodes", National Bureau of Standards Spec. Publ. No. 314, (1969). 21) G. J. Moody, J. D. R. Thomas : "Selective Ion-sensitive Electrodes", (1971), (Merrow, Watford). 22) P. L. Bailey : "Analysis with Ion-selective Electrodes", (1976), (Heyden, London). 23) K. Camman : "Das Arbeiten mit ionenselektiven Elektroden", 2nd Ed., (1977), (Springer, Berlin). 24) S. Ebel, W. Parzefall: "Experimentelle Einfuhrung in die Potentiometrie", (1975), (Verlag Chemie, Weinheim). 25) N. Lakshminarayanaiah: "Membrane Electrodes", (1976), (Academic Press, New York). 26) R. P. Buck, Anal. Chem., 46, 28R (1974) ; 48, 23R (1976) : 50, 17R (1978). 27) J. Koryta : Anal. Chim. Acta, 61, 329 (1972) ; 91, 1 (1977). 28) A. Liberti, M. Mascini : Anal. Chem., 41, 676 (1969). 29) M. J. D. Brand, G. A. Rechnitz : Anal. Chem., 42, 1172 (1970). 30) G. Horvai, L. Domokos, E. Pungor : Fresenius Z. Anal. Chem., 292, 132 (1978). 31) L. M. Schwartz : Anal. Chim. Acta, 122, 291 (1980). 32) S. Ebel, A. Seuring : Angew. Chem., 89, 129 (1977). 33) D. M. Barry, L. Meites : Anal. Chim. Acta, 68, 435 (1974). 34) W. Bartscher : Fresenius' Z. Anal. Chem., 297, 132 (1979). Keyword phrases Ion-selective sensor, statistical processing of data, potentiometric error, multipoint calibration, standard addition, potentiometric titration. (Received Nov. 9, 1981)