pyrimidine compounds found in this study to be amenable to analysis by coulometric bromination. The only real assumptions involved would be that of 100% current efficiency and that a single, stoichiometric overall bromination reaction were occurring. By combined coulometric-spectrophotometric analysis of solutions of highly purified nucleotides, absorptivity values obtained would be independent of calculated or corrected nucleotide molecular weight values. The present, direct spectrophotometric method requires assay for, or assumptions concerning, metal ion content of charged nucleotides and hydrated or adventitious water content, processes which in themselves usually result in uncertainties on the order of f2-4% ( I , 17). The limiting factor in determining absorptivity values, for example, would probably be the calibration of the spectrophotometric system itself. Efforts are now under way in this laboratory along these lines.
(2) E. Volkin and W. E. Cohn, in "Methods of Biochemical Analysis", D. Glick. Ed., Vol. 1, Interscience, New York, 1954, pp 287-305. (3) G. H. Beaven, E. R. Holiday, and E. A. Johnson, in "The Nucleic Acids", E. Chargaff and J. N. Davidson, Ed., Vol. 1, Academic Press, New York, 1955, pp 493-553. (4) J. T. Stock, "Amperometric Titrations", Interscience. New York, 1965. (5) J. J. Lingane, "Electroanalytical Chemistry", 2nd ed., Interscience. New York, 1958, pp 267-295, 484-616. (6) J. T. Stock, Anal. Chem., 46, 1R (1974). (7) D. G. Davis, Anal. Chem., 46, 21R (1974). (8) G. Marinenko, in Nat. Bur. Stand. Tech. Note, 583, 4-38 (1973). (9) D. G. Marsh, D. L. Jacobs, and H. Veening, J. Chern. Educ., 50, 626 (1973). (10) D. H. Evans, J. Chern. Educ., 46, 613 (1969). (11) J. Biol. Chem., 241, 527 (1966). (12) "Handbook of Biochemistry", H. A. Sober, Ed., 2nd ed., The Chemical Rubber Co., Cleveland, OH, 1970, pp A-12-15. (13) A. M. Moore and S. M. Anderson, Can. J. Cbem., 37, 590 (1959). (14) D. J. Brown, "The Pyrimidines", Interscience, New York. 1962, pp 172-174, 346. (15) H. L. Wheeler and 1.B. Johnson, J. Biol. Chem., 3, 183 (1907). (16) G. E. Hilbert and E. F. Jansen, J. Am. Chern. Scc., 56, 134 (1934). (17) P.L Biochemicals, Inc., Milwaukee, WI, Circular OR-10 (1969).
LITERATURE CITED
RECEIVEDfor review October 28, 1974. Accepted January 27, 1975. Acknowledgment is made to the Donors of The Petroleum Research Fund, administered by the American Chemical Society, for the support of this research.
(1) National Academy of Sciences-National Research Council, "Specifications and Criteria for Biochemical Compounds", Publication 719, Washington, DC. 1960, pp P-1 to P-43: Third edition, 1972, pp 149-183.
Titrimetric Applications of Multiparametric Curve-Fitting: Locations of End Points in Amperometric, Conductometric, Spectrophotometric, and Similar Titrations John G. McCullough' Department of Chemistry, Grand Valley State College, Allendale, MI 4940 1
Louis Meites Department of Chemistry, Clarkson College of Technology,Potsdam, NY 13676
A multiparametric curve-fitting procedure is described for locating the end point of a titration from data obtained by amperometry, conductometry, spectrophotometry, or any other technique that gives rise to a segmented titration curve. The procedure consists of fitting the data to a discontinuous function involving four parameters and consisting of two straight lines that intersect at the end point. The computer program by which this fit Is accomplished includes two point-rejecting routines. One discards points that deviate from the line segments because they lie in a region of curvature around the equivalence point; the other dlscards outliers anywhere on the curve. The result is Identical in both principle and effect with that of a very careful graphical treatment.
Much interest has been evinced during the past few years in the use of nonlinear regression techniques for interpreting titration-curve data. This can be traced back to the extensive and successful use, by Sill6n and his coworkers, of such techniques to elucidate the compositions and stabilities of metal ion-hydroxide complexes, but follows that precedent at a distance of so many years that it can be regarded as a new development. Anfalt and Jagner ( I ) were the first to recognize the potentialities of these techniques, although the complexity of the system with which they were concerned was so great
that they did not implement their suggestion. Ingman, Johansson, Johansson, and Karlson ( 2 ) , employing data obtained in prior titrations of the separate acids, were able in this way to deduce the compositions of binary mixtures of formic, acetic, and propionic acids from data obtained in potentiometric titrations; this was a striking achievement because the thermodynamic pK values for acetic and propionic acids differ by only 0.12 unit, so that there is no point of maximum slope at or near the equivalence point of the stronger acid. Brand and Rechnitz ( 3 ) had found that copper(I1) could be determined with good accuracy and reliability by applying a nonlinear regression technique to data secured with a copper-selective potentiometric indicator electrode by a multiple-standard-addition procedure; and Isbell, Pecsok, Davies, and Purnell ( 4 ) showed that a similar determination of silver(1) gave as good results as could be obtained by titration and that the end point of a potentiometric silver-chloride titration with a silver-selective indicator electrode could be located more accurately by nonlinear regression than by any other technique. Barry and Meites ( 5 ) showed that locating the equivalence point of a potentiometric titration of acetate ion with hydrochloric acid by nonlinear regression was about ten times as precise as inflection-point location in the ordinary way at high concentrations, and remained feasible even at dilutions so extreme that the inflection point had disappeared. Barry, Meites, and Campbell ( 6 ) further showed that the necessity for prior standardization of the reagent ANALYTICAL CHEMISTRY, VOL. 47, NO. 7 , JUNE 1975
1081
I
”/ o / O
/
/
I
L/o/ume o f m i d . cm3 Figure 1. Conductometric titration curve obtained in the titration of 150 cm3 of 3.3 X 10-4M acetate with 0.002F hydrochloric acid.
The significance of the solid circles is described in the text
in such a titration can be eliminated by evaluating the concentration of the reagent along with that of the solution being titrated. In related work, Meites and Barry (7) combined nonlinear regression with deviation-pattern recognition (8) to permit the classification of a base as mono-, di-, tri-, . . .-functional on the basis of such data, and Campbell and Meites (9) automated the scheme so as to obtain a fully computerized elucidation of the curve. All these developments have ignored the segmented titration curves obtained in amperometric, conductometric, spectrophotometric, and similar titrations. This paper describes a simple and reliable least-squares procedure for locating the end points of these curves. It is convenient to regard such a curve as belonging to one of two classes. In one, either because the equilibrium constant Kt for the reaction that proceeds from left to right during the titration is large, because the solution being titrated is fairly concentrated, or because the data do not include any points very close to the equivalence point, the curve consists of two sharply defined line segments and shows no evidence of a region of curvature between them. In the other, not all of these conditions are satisfied and, as a result, a transitional curved region does appear. Using an -shaped amperometric titration curve of the first class as an example, one of us (IO)argued some years ago against the use of least-squares techniques for locating the end points of amperometric titrations, and described a modified graphical technique that gave results very much more precise than the conventional one by placing the greatest weight on the points closest to the equivalence point. In a more recent paper, we (11)have shown how this technique can be applied to curves having other shapes. It is based on the idea that the straightforward procedure of fitting the data on each branch of the curve to a linear equation, followed by calculating the volume a t the point of intersection of the two corresponding lines, leaves the analyst defenseless against adventitious errors that are prominent on graphical plots and that are, therefore, easily rejected if such plots are constructed. The argument is even more forceful when applied to a curve of the second class, such as the one shown in Figure 1. In such a case, there is nothing to be said in favor of an a priori decision that some points belong to one line segment and others to another, and any such decision misrepresents the chemistry of the system and limits the accuracy and precision that can be attained. The modified procedure (11) is at a disadvantage here because it emphasizes the points that are least useful because they lie in the region of 1082
ANALYTICAL CHEMISTRY, VOL. 47, NO. 7, JUNE 1975
curvature. What is wanted is a procedure that, like a careful analyst dealing with a curve of this sort, will reject points within this region and ignore outliers elsewhere. Such a procedure is described below. It is applicable to segmented titration curves of either class and of any form. It is easier and faster than the classical graphical approach illustrated by Figure l ; it eliminates the purely graphical errors inherent in that approach; and its accuracy and precision are always at least comparable to, and sometimes very much better than, those attainable by a very skilled and careful chemist employing that approach. I t is unquestionably superior to the modified graphical-analytical approach (11) for curves of the second class, but there is little to choose between them for curves of the first class, where the present procedure consumes more computer time but eliminates the final graphical step and the reliance on the analyst’s judgment that it entails. Liteanu et al. (12-16) discussed end-point location by fitting the two branches of the curve to separate linear equations and taking the end point as the point a t which the lines thus defined intersect, and considered the rejection of points with suspiciously large deviations ( 1 5 ) but on a basis rather different from that described here. The errors involved in such procedures were analyzed by Rosenthal, Jones, and Megargle (17), and the establishment of confidence intervals for them was discussed by Kotrly et al. (18). The various graphical procedures that have been proposed for dealing with cases where curvature around the equivalence point is extensive were reviewed by Vrestal and Kotrly (19).
EXPERIMENTAL Most of the experimental work was done a t Grand Valley State College and has been described elsewhere (11).
COMPUTATIONS The computations were done at Clarkson College of Technology. Some were performed on a Digital Equipment Corp. PDP/8I minicomputer operated in EduSystem 25 BASIC; others were performed on a PDP/8E operated in EduSystem 50 BASIC. In either case, the user area employed was always 4096 words, which would have sufficed to permit execution with approximately 130 data points if so large a number had been available: the actual number of points on a single titration curve was usually about 16. The computer program that was employed was based on a general multiparametric curve-fitting program that has been described elsewhere (20), and to which several important additions were made. One of these additions was a subroutine that described the discontinuous function to which the f i t was performed. This may be written in the form S’ = S’* S’ S’*
- Sl(U* - u )
+ s2(u - u * )
u
u Iu*
(la)
> u*
Ob)
In these equations, u, the independent variable, represents the volume of reagent added, while S’ denotes the dependent variable defined by
S‘= S(v0 + U ) / V O
(2)
where S is the measured signal (limiting current, conductance, absorbance, etc.) and v0 is the volume of solution titrated. The superscript asterisks in Equations l denote values at the equivalence point, and the symbols s1 and sz denote the slopes of the two line segments represented by the straight lines in Figure 1. The correction for dilution, which is described by Equation 2, is applied to each value of S as it is entered, and ini-
tial rough estimates of the four parameters that have to be evaluated are computed from the equations s1
(34
(3b) - S’N.,)/(U.V - u N . z ) = [(s’,- 51U2) - (S’N-2 - SzUN.z)]/(SZ - S I ) (3‘2) S’* = S’, + S l ( U * - u,) (3d) s2
U*
= (S’, - S’1)/(u2 - u1)
PlJN
= (S’N
where the subscript numbers on the right-hand sides denote the ordinal number of the experimental points, N in number, so arranged that each u,, > u,,.~, and where t is the integer that is next larger than N/4. These initial values are then used to compute a value of S’ a t each experimental point in turn; the squares of the deviations of the computed values from the experimental ones are summed, and the esu*, S I , and s2 are timated values of the four parameters S’*, adjusted in successive cycles so as to minimize this error sum. Another addition to the basic program, and the one that is the heart of the present procedure, is a subroutine that comes into play after computation has proceeded for long enough to obtain a reasonable f i t from the above initial estimates, which may be very crude, and then again a t regular intervals until execution is terminated. This subroutine mimics the judgment that is exercised by the chemist in discarding points that lie in a region of curvature close to the equivalence point and outliers elsewhere. Before the structure of this subroutine is described, it is appropriate to attempt t o reconstruct the thought processes of the chemist confronted with data like those in Figure 1. The curvature around the end point is prominent, 27 cm3 are discarded a t and data in the range 2 1 Iu I once. On laying a straightedge along the points in the range 3 5 u 5 15 cm3, it becomes apparent that S’ is too high a t u = 18 cm3, and this point is consequently disregarded as well. On this basis, the straightedge is moved to conform to u 5 1 2 cm3, and now the point the points in the range 3 I a t u = 15 cm3 is observed to lie above the line and is disregarded. A similar process at the other end of the curve leads to the rejection of the points a t u = 30 and 33 cm3, and possibly also to that of the one a t u = 36 cm3, though the last of these lies so little above a line through the following points that many analysts might finally decide against rejecting it. When the line segments are finally drawn through the surviving points, which are denoted by the solid circles, it may be observed that two points (at u = 6 and 45 cm3) have been retained although their deviations from the line segments are not appreciably smaller than the deviations that led to the rejection of the points a t u = 15 and 36 cm3. A double standard is clearly being applied. Observing the curvature around the end point predisposes the chemist to reject the points a t u = 18 and 30 cm3; having rejected these, he is further predisposed to believe that the region of curvature extends as far as u = 15 cm3 on the one side and as far as u = 33 cm3 on the other, and so on until no evidence of continued curvature can be discerned on very close inspection. However, points like the one a t u = 45 cm3 are likely to be retained, unless their devintions are quite large, because the retention of a point closer to the equivalence point predisposes the analyst to judge those deviations in the light of the presumed standard error of measurement rather than in that of the systematic error associated with the region of curvature. The subroutine for rejecting points operates in the following way. It begins by locating the two points that bracket the best value of u* so far obtained. For each of those points, it employs the current estimates of u*, SI*, SI, and
IIIFF. ‘TTU. .
DE’,].
..
FEHIIr’
Figure 2. Initial input and final printout for data obtained in a titration of 150 c m 3 of 0 . 0 3 3 M a c e t a t e with 0.2Fhydrochloric acid
provided by the main portion of the program and the current estimate u’ of the standard deviation from regression to evaluate the quantity [SGN(s2 - Sl)](S’calcd S’measd)/U’, where SGN(s2 - S I ) = +1 if s2 > s1 or -1 if sa < S I , and rejects the point if this quantity exceeds 1. (The choice of the value 1 in this criterion is discussed below.) There are two ways in which this can happen: either, if s2 > S I , S’calcd must exceed S’measd by at least u’ or, if s2 < S I , S’caIcd must be smaller than S’measd by a t least u’. In either event, the difference between S’calcd and S’measd must have the sign that would reflect curvature around the equivalence point to lead to rejection on this basis. When this has been done, the values of the ratio I S’calcd - S’measdl /u’ are computed for all of the points, including those that bracket the equivalence point, and each point for which this ratio exceeds 2.5 is rejected, regardless of whether S’calcd exceeds, or is smaller than, S l m e a s d . Computation in the ordinary way (20) is then resumed. On a later cycle, the subroutine will be recalled unless a satisfactory fit is found and execution is terminated before then. The second of these criteria guards against outliers some distance away from the equivalence point and (although detailed analysis is impossible because no fundamental significance can be ascribed to u ’ ) is much more stringent than it may appear to be. This is partly because the value of u’ decreases continuously as the f i t proceeds, even if no point is ever rejected, and may substantially exceed the actual standard error from regression, u, until the computation is nearly done; and partly because the value of (I’ is further increased above that of u as long as any point in the region of curvature remains to be rejected. The first criterion is deliberately made much more lenient for deviations in the direction corresponding to the anticipated systematic error. The numerical values employed in these criteria were selected on the basis of fits to the data obtained in conductometric titrations of acetate with hydrochloric acid a t various levels of concentration. For the first criterion, which is applied to the points closest to the equivalence point, a plot of the final standard deviation from regression against the value of [SGN(s2 - SI)](S/ca]cd - S’measd)/u’ above which a point was rejected always had a flat minimum extending from 0.75 to 1.25, and the effects of variations within this range amounted to the rejection or retention of points, like s2
ANALYTICAL CHEMISTRY, VOL. 47, NO. 7 , JUNE 1975
1083
Table I . Results Obtained E:id-point volume
Solution t i t r a t e d
OAc', 3 3 m F 3.8mF 0.33mF
As(II1)"
Reagent
HC1, 0 . 2 F 0.2F 2mF KBrO,
h k t h o d a n d conditlons
Present x e t h o d
Conductometric Conductometric Conductometric Amperometric. in 1F HC1-
25.29 ( 2 / 1 6 ) " 2.561 (61'17)' 24.98 (8/16)' 37.804 ( 0 / 1 8 ) * 43.371 (1/18)* 36.604 ( 0 / 1 6 ) * 41.098 (0/22)* 4.124 ( 2 / 2 0 ) * 3.949 (0/26)*
4 F KBr
Fe(I1)
KBrO,
Amperometric, in 1F HC1-
Cu(I1)
V(I1)
Mo(V1)'
V(II)
Amperometric. in 1F KC1 Amperometric, in 1F Na,C,HjOT, pH 7.1
Graphical method
Modified graphical m e t h o-analvtical d (11)
25.22 ( 2 / 1 6 ) * 2.566 (6-7/17)* 24.8 (7-8/16)* 37.80 c 0.005* 43.365 = 0.Olb 36.60 * O . O l b 41.10 i 0 . 0 2 b 4.120 I 0.005 3.96 i 0.02
25.4 + 0.2 2.53 -t 0.02 24.85 * 0.25 37.802 0.002 43.38 * 0.01 36.622 = 0.007 41.107 I 0.005 4.124 + 0.004 3.946 + 0.003
*
4 F KBr
+ 0.05 (2/16)"
5.101 (2/16)*
5.15
5.77 ( 0 / 1 1 ) * 4.21 (0/8)* 5.72 ( 0 / 9 ) *
5.75 = 0.1 4.29 = 0.08 5.70 + 0.08
* The numerator gives the number of points rejected (or. for the graphical method, ignored in drawing the line segments); the denominator gives the total number of points originally obtained. For these four titrations. the mean deviation of the ratio of end-point volume to weight of As203 is *0.1670. From expanded plots as described in Ref. ( 1 2 ) . For these three titrations, the mean deviation of the ratio of end-point volume to volume of Mo(V1) solution titrated is +0.817~. the one a t u = 36 cm3 in Figure 1, about which different but equally skilled chemists might make different decisions. For the second criterion, which rejects points afflicted by large random errors, changing the value of /Scaled S'measd/o' from 2 to 3 had no effect in any of the cases we tried, but decreasing it to 1.5 led to the occasional unwarranted rejection of one of the points farthest away from the equivalence point. Typical input to, and output from, the program in interactive operation are shown in Figure 2. No other information is required of the operator except for the coordinates of the data points, which are combined with the program before execution is begun. The following conditions and restrictions should be observed: 1) The data points should be equally, or almost equally, spaced along the volume axis to avoid undue weighting of any particular region. 2) There should not be more than one end point within the range of volumes considered. 3 ) The values of S should be measured in such a way (e.g., a t constant sensitivity) that their absolute errors can be assumed to be randomly distributed; changes of scale or sensitivity during any one titration should be avoided. (A briefly annotated hard-copy listing, in BASIC, of the program ENDPOI may be obtained by remitting $20.00, to defray the costs of handling, duplication, and postage, to the Computing Laboratory of the Department of Chemistry, Clarkson College of Technology, Potsdam, NY 13676. A copy of the parent curve-fitting program CFT3, accompanied by a 167-page manual of explanation and instruction and hard-copy listings in BASIC, POLYBASIC, and FORTRAN-I\', may be obtained a t the same time for a total remittance of $42.50.)
RESULTS AND DISCUSSION Table I compares the results with those obtained by careful graphical analysis and, for some of the titrations, with those obtained by the modified graphical-analytical method ( 1 1 ) . The values resulting from the graphical procedures are accompanied by crude estimates of the purely graphical uncertainties. The present procedure yields endpoint volumes in close agreement with those obtained by 1084
ANALYTICAL CHEMISTRY, VOL. 47, NO. 7, JUNE 1975
the other procedures, and comparison of the numbers of points rejected shows that it employs criteria very similar to those tacitly applied in careful graphical treatments. Obviously, this procedure is not a panacea for the ills that can beset segmented-curve titrations. I t cannot produce an accurate or precise value of the end-point volume in the face of systematic chemical or instrumental errors, if the angle between the two line segments is very obtuse, if the data are poor, or if the restrictions stated above are ignored. However, it can relieve the chemist of the time and tedium involved in graphical treatments of the data while exercising much the same judgment that those require and, if coupled with automatic addition of reagent and digital data acquisition, should make completely automatic segmented-curve titration possible without entailing the sacrifice of accuracy that this has involved in the past.
LITERATURE CITED (1) T. Anfalt and D. Jagner, Anal. Chim. Acta, 57, 1965 (1971). (2) F. Ingrnan, A. Johansson, S. Johansson, and R. Karlson, Anal. Chim. Acta, 64, 113 (1973). (3) M. J. D. Brand and G. A. Rechnitz, Anal. Chem., 42, 1172 (1970). (4) A . F. Isbell, Jr., R. L. Pecsok, R. H. Davies, and J. H. Purnell, Anal. Chem., 45, 2363 (1973). ( 5 ) D. M. Barry and L. Meites. Anal. Chim. Acta, 68, 435 (1974). (6) D. M. Barry, L. Meites, and B. H. Campbell, Anal. Chim. Acta, 6 9 , 143 (1974). (7) L. Meites and D. M. Barry, Talanta, 20, 1173 (1973). (8) L. Meites and L. Lampugnani. Anal. Chem., 45, 1317 (1973). (9) E. H. Campbell and L. Meites. Talanta, 21, 393 (1974). (10) L. Meites, "Polarographic Techniques", lnterscience Publishers, New York, 2nd ed., 1965, pp 481-484. (11) L.Meites and J. G. McCullough, submitted for publication. (12) C. Liteanu and D. Cormos, Talanta, 7, 18 (1960). (13) C. Liteanu and D. Corrnos, Acta Chem. Acad. Sci. Hung., 27, 9 (1961). (14) C. Liteanu and D. Corrnos, Rev. Roumaine Chem., 10, 381 (1965). (15) C. Liteanu, Rev. Chim. Acad. Rep. Pop. Roumaine, 7, 291 (1962). (16) C. Liteanu and E. Hopirtean, Studia Univ. Babes-Bolyai, Ser. Chem., 11, 135 (1966). (17) D. Rosenthal. G. L. Jones, Jr.. and R. Megargle, Anal. Chim. Acta, 5 3 , 141 (1971). (18) P. Jandera. S. Kolda, and S. Kotrly, Talanta, 17, 443 (1970). (19) J. Vrestal and S. Kotrly, Talanta, 17, 151 (1970). (20) L. Meites, "The General Multiparametric Curve-Fitting Program CFTB", Computing Laboratory, Department of Chemistry, Clarkson College of Technology, Potsdam, NY, 2nd ed., 1974.
RECEIVEDfor review November 18, 1974. Accepted February 5, 197.5.
