An Error Analysis of Indicator Measurements A. J. Kresge and H. J. Chen Department of Chemistry, Illinois Institute of Technology, Chicago, Ill. 60616
An error function is derived which expresses the dependence of errors in indicator ratios on the value of these ratios. This function, which is supported by experimental data, shows that the uncertainty in I (indicator ratio) is least when the value of I is near unity but that the region of minimum error is broad and errors do not become prohibitively large until log I approaches ~k2.0. An extension of this analysis to consideration of errors in indicator plots shows that points in the vicinity of log I = 1 0 . 7 are the most valuable in fixing the slope of a linear logarithmic plot. Extending the measurements to log I = ca. k1.5, however, involves little sacrifice of accuracy in slope and gives the added benefit of providing more information about the shape of the line.
THEindicator method of measuring acid-base properties has been especially useful in its application to concentrated solutions of strong acids and bases. An early product of this work was the HOacidity function extension of the pH scale ( I ) . More recently, the wide availability of photoelectric spectrophotometers has permitted more careful indicator measurements which have revealed that not one but many acidity functions exist [for a brief summary and guide to the original literature, see reference (2)], and examination of differences between these scales is now providing valuable insight into many properties of concentrated acid and base solutions. These detailed indicator studies have quite naturally raised the question of the accuracy of indicator measurements (3-9). It is generally believed that an indicator ratio, the principal quantity determined in indicator studies, because it is a quotient of two concentrations, can be fixed most accurately when its value is near unity. This has led some investigators to reject from consideration indicator ratios with values beyond some arbitrarily fixed limits, usually 0.1 to 10 (6-8). However, no systematic analysis of the problem seems to have been published. We therefore present some of the conclusions we have reached from a general consideration of the errors involved in indicator measurements. [For a preliminary account of a somewhat different approach to this problem, see reference (9).] Our results can be divided into two parts: errors in the individual values of indicator ratios, and errors in the slopes of indicator plots.
of the acidic and basic forms of the indicator. The first of these two operations usually involves volumetric techniques, whereas the second requires optical absorbance measurements in some region of the ultraviolet or visible spectrum. Because volumetric manipulations can readily be performed with an accuracy considerably greater than absorbance measurements, it would seem justifiable to assume that errors in absorbance measurement constitute the major source of overall errors in indicator ratios. It is likely, moreover, that errors in solution preparation make a constant contribution to overall errors throughout the entire range of values of indicator ratio in any given set of experiments, and, because we shall be primarily interested in changes in the overall error with changes in the indicator ratio, neglect of this minor and possibly insignificant source of error should not affect our conclusions. We shall therefore proceed on the assumption that errors in indicator ratios arise solely from errors in absorbance measurements. Let AB and ABH be the optical absorbances at some fixed wavelength of the indicator in its basic and acidic forms, respectively, and let A be the absorbance at the same wavelength of a solution containing the indicator in both forms. Providing that the Beer-Lambert law is obeyed, the indicator ratio, I , in the latter solution is then given by Equation 1: I = C B H / C B= (AB
(1) L. P. Hammett, “Physical Organic Chemistry,” McGraw-Hill, New York, 1940, p 267. (2) K. Yates and K. A. McClelland, J. Amer. Chem. SOC.,89,
2686 (1967). (3) W. M. Schubert and R. H. Quacchia, ibid., 85, 1278 (1963). (4) A. J. Kresge, G. W. Barry, K. R. Charles, and Y.Chiang, ibid., 84, 4343 (1962). (5) M. J. Jorgenson and D. R. Hartter, ibid., 85, 878 (1963). (6) R. L. Himnan and J. Lang, ibid., 86, 3796 (1964). (7) K. Yates and H. Wai, ibid., 86, 5408 (1964). (8) K. Yates and J. B. Stevens, Can. J. Chem., 43, 529 (1965). (9) R. W. Alder, G. R. Chalkey, and M. C. Whiting, Chem. Comm., 1966, 405. 74
ANALYTICAL CHEMISTRY
(1)
Application of standard methods for the propagation of errors (loa) shows that the error in I, uI,is the following function of the errors in the three measured absorbances, ( u ~ ) BU, A ,and (uA)BH : I UI = [(uA)B2 f (1 I)2(.A)2 f 12(6‘A)2BH1112 (2) AB - ABH
+
+
There is reason (11) to believe that errors in absorbance measurements made with commercial spectrophotometers are constant throughout the useful absorbance range of approximately 0.1 to 1.0 (however, see below). Provided then, that AB, A , and ABH are each measured the same number of times, we can set the three absorbance errors equal to one another: (UA)B = Q A
ERRORS IN INDICATOR RATIOS
Measurement of an indicator ratio requires two rather different operations : preparation of solutions containing the indicator and appropriate acids or bases at known total concentrations, and determination of the relative concentrations
- A ) / ( A - ABH)
=
(UA)BH
(3)
In a series of measurements made with one indicator, however, the quantities AB and ABHenter into the calculation of each indicator ratio; it is therefore sometimes the practice to measure A B and ABHmore times than A . In this situation, the reliabilities of A B and ABH increase in proportion to the square roots of the numbers of times they are determined, and Equation 3 must be modified to Equation 4: (uA)Bp1’2
= UA =
(UA)BHq’”
(4)
Here, p and q are the numbers of independent times AB and respectively, are determined. Combination of Equations 2 and 4 gives:
ABH,
(10) C. A. Bennett and N. L. Franklin, “Statistical Analysis in Chemistry and the Chemical Industry,” Wiley, New York, 1954, (a) p 49; (b) p 222. (11) L. Cahn, J. Opt. SOC.Am., 45,953 (1955).
120
-
I
I I 100
-
I I
I BO
-
I
ba
\
H
0
60-
b’
Figure 1. Indicator ratio error functions
40
-
20
-
Solid line: indicator with both acidic and basic forms absorbing light (Equation 6, AB - A B H = 1.00 and p = q = 1) Broken line: indicator with only basic form absorbing light (Equation 7, A B = 1.00 andp = 1)
_--2
In acidity function studies, the quantity log I is very often of greater interest than 1; Equation 6 gives the error function for this variable: 0.4343(1
+ I ) [ + (1 + z ) +~
uiogI = ___--
(AB - A B H Y
-
CA
(6)
9
This analysis shows that the error in I (or log Z) is inversely proportional to the useful range over which absorbance measurements can be made with any one indicator (AB - ABH). For a given indicator, the error in I is directly proportional to the constant error in absorbance, U A , and is also a complicated function of I , p, andq. The solid line in Figure 1, which U S . log I with (AB is a plot of Equation 6 in the form ulogA[./ - ABH) = 1.00 and p = q = 1, shows that the error in log Z is always greater than the error in absorbance by at least a factor of two. It shows also that the error in log Z is symmetrical about a minimum value occurring at log I = 0.00; this supports the common belief that errors in log I are smallest when the concentrations of the two forms of the indicator are equal. It should be noticed, however, that the minimum in this error function is quite broad: at log I = + l . O , the error is only 3 times the minimum value and at log I = f1S, it is only 8 times the minimum. This indicates that it may be too stringent a requirement to exclude from consideration values of log Z outside the limits i1.0. Calculations based on Equation 6 also show that only modest benefit is gained from measuring A B and ABH more than once. Increasing both p and q from 1 to 5 decreases the minimum error by only 17%, and a further rise in each from 5 to 20 gives only an additional 4% improvement; any further increase results in very little change. The effects are slightly greater at low and at high values of log Z. For example, when log I = e 2 . 0 , the situation p = q = 5 is 29% better than p = q = 1, and going t o p = q = 20 gives an additional 7% improvement. In this region, however, errors in log I have reached a level 25 to 30 times their minimum values, and indicator measurements have lost much of their utility. This analysis has so far considered only indicators for which both acidic and basic forms absorb light at the wavelength
-I
0
, +I
+2
of the measurement-i.e., cases for which neither A B nor AB= is zero. This,is not always the case in practice. It is commonly regarded that indicators for which only one form absorbs light at a given wavelength are superior to those for which such a situation cannot be achieved, and experiments are therefore often carried out under conditions of either A B = 0.00 or ABH = 0.00. The error functions covering these situations are simply special cases of Equations 5 and 6. For example, inserting the conditions ABH = 0.00 and (UA)BH= 0.00 into Equations 2 and 4 gives Equation 7 :
This function, with A B = 1.00 and p = 1, is plotted as the broken line in Figure 1. It may be seen that for negative values of log I, there is essentially no difference between the special case (ABH = 0.00) and the more general situation (ABH > 0.00). Differences do occur at positive values of log I , but these differences are not large. The minimum in the error curve for the special case occurs near log I = f0.1 where it is 1 2 z lower than the curve for the general case; at log 1 = f1.0, the difference between the two curves is 3.5% and at log I = f2.0, 40%. If A B is measured more than once, the positive branch of the curve for the special case does not change very much, but the negative side drops until, with p = ca. 20, it becomes a reflection of the positive branch. The situation can be summarized by saying that the special case in any circumstance is never better than the general case under conditions of both p and q = m ; for practical purposes this is achieved when p = q = 20, and it was shown above that this condition affords only a modest improvement over the general case with p = q = 1. There is one additional feature of an indicator with either A B or A B H = 0.00, however, which makes it intrinsically superior to one for which both AB and ABH are finite. In both cases, the error in log I is inversely proportional to the useful absorbance range over which measurements can be made : A B - A B z (Equation 6) or A B (Equation 7). With an indicator having only one absorbing form (ABH = O.OO), this can be made t o be the entire absorbance range of the spectrophotometer, whereas, with an indicator having two absorbing VOL. 41, NO. 1, JANUARY 1969
75
absorbance errors which are not wholly independent of absorbance values but which tend to decrease slightly as absorbance rises. The agreement on the whole is good, and this, we feel, provides considerable support for the validity of the assumptions incorporated in the error analysis. ERRORS IN SLOPES OF INDICATOR PLOTS
log I
Figure 2. Comparison of error functions with experiment Upper plot: errors in log Z for measurements with o-nitroaniline Lower plot: errors in log Z for measurements with pnitroaniline
forms (AB=and A B both >O.OO), the useful range is necessarily limited by the absorbance of the less-strongly-absorbing form. The benefit to be derived as a consequence of this depends, of course, on the individual absorption characteristics of the indicators used. It is interesting to compare the results of this analysis with experimental data. In the course of our acidity function studies (12), we have had occasion to perform a large number of indicator measurements with two Hammett-type indicators, 0- and p-nitroaniline. For each base, indicator ratios were measured in three different aqueous acids, HC104, HCl, and H2S04; in each acid, some 25 different values of I were determined with each base, each determination was done in triplicate, and absorbances were recorded at three different wavelengths for each independent determination. This provided some 150 different values of log I, each one based on nine separate absorbance readings. The standard deviations of these values, ulogI , are shown plotted against corresponding values of log I in Figure 2; the solid lines were calculated using Equation 6 with a value of uA(ca. 0.0015) which gave the best fit (judged by eye) to the data in the vicinity of log I = 0.00. For both indicators, errors do pass through minimum values near log I = 0.0 and increase beyond this region in both directions as expected. It can be argued that the points in this figure tend to concentrate slightly above the calculated lines for negative values of log I and slightly below the lines for positive values of log I. The discrepancy, however, if significant, is not great and can be accommodated by (12) A. J. Kresge and H. J. Chen, Illinois Institute of Technology, unpublished work, 1968.
76
ANALYTICAL CHEMISTRY
Increasing attention is currently being paid to the way in which indicator ratios change with the acidity of concentrated solutions--i.e., to the slopes of indicator plots (2). It is of some interest, therefore, to extend the error analysis of the previous section to a consideration of errors in the slopes of plots of indicator ratios against some other measure of acidity. The error in the slope of an indicator plot depends on the reliability of the individual indicator ratios used to construct the plot. This dictates use of points in the vicinity of log I = 0.0 where ulogI is least (see Figure 1). The error in the slope depends also on the length of the linear portion of the line over which the slope is measured, and this implies that points covering as wide a range of indicator ratio as possible should be included. These two requirements are, of course, in conflict, for extending measurements much beyond log I = 0.0 in the interest of obtaining a longer line necessarily introduces points with greater error. The best situation will be one involving some compromise between these two requirements ; it is of interest to see what this may be in terms of the optimum range over which indicator measurements should be made. Consider an indicator for which the indicator ratio is a linear logarithmic function of acidity as expressed by the generalized acidity function H: logI
=
a
+ bH
(8)
Assume that H i s known exactly so that the error in the slope, b, is governed solely by errors in a set of values of indicator ratios, It. The error, a b , in the best value of lthe slope obtained by least squares fit of the data, log It and Ht, to Equation 8 is then given by Equation 9 (IOb):
Here, W( are weights which must be assigned to each of the points according to Equation 10 because the errors and therefore the reliabilities of all values of log I a r e not the same. Wt =
1/(UIOG 1 ) t 2
(10)
Combination of Equations 9 and 10 gives Equation 11 :
1 [ z t ( H t - fl>2/(ui,, r)i211’2
(11)
It is useful to recast Equation 11 solely in terms of the variable log I . Application of Equation 8 gives Equation 12: Ht
-B
=
(log It
- log)/b
(12)
and Equation 1 1 is therefore equivalent to Equation 1 3 : Ualb
=
1
[Zt(log It - log
I)”(Ulo,
(13) r)t71’2
Equation 13 expresses in quantitative terms the statement made previously to the effect that the error in the slope of an indicator plot is governed by two factors, (1) the errors in the
Figure 3. Relationship between the error in the slope of an indicator plot and the length of the line (2 log I,,,) for three different distributions of points (I) Five points placed at -log I,, points placed at +log I,, (11)
and five
Eleven points distributed evenly between
-log I,,
and f l o g Imax
(111) Points spaced between -log I,,, flog I,,, at equal intervals of 0.2 log 1
and
individual -points (mag 1)1, and (2) the length of the line, (log I t - log I ) . Either decreasing the errors (qoa -1)i or increasing the dispersion of the points (log Ir - log I) (increasing the length of the h e ) will decrease the quantity u b / b . An immediately apparent consequence of Equation 13 is that the error in the slope will be smallest when all of the points are grouped in two sets of equal size at the ends of the line; this will give maximum values to those parts of the log I ) . summation of Equation 13 which involve (log It At the same time, if the errors in log I are governed by some function such as Equation 6 whose minimum value occurs at log I = 0.00, then the error in the slope will be least when the midpoint of the line is at log I = 0.00. In order to examine this kind of distribution of points in detail, we have evaluated Equation 13 for a series of cases using 10 values of log I , five at each end of a line extending from -log I,,, to +log I,,,. The length of the line was varied by increasing Zmax in regular increments, and values of clog1 were calculated using Equation 6 with p = q = 1 , The results are presented in Figure 3, line I, which is a plot of the relative error in the slope divided by the error in absorbance, U b l b U A , L.S. log I,,,. This shows produces a sharp that lengthening the line (increasing log Z,), decrease in the error at low values of log I,,, and a somewhat more gradual increase in error at high values of log I,,,. The error is a minimum at log Zma, = 10.7. Thus, the factors which govern the error in the slope are balanced when the line extends from log I = -0.7 to log I = f0.7, and the slope may be fixed most accurately by concentrating the points at these two values. The minimum value of ub/bua for this distribution of points is 5.7/(n)1’2, where n is the number of points. For n = 10, uolb = 1&A, and, with an error in absorbance of 0.005, the slope can be ascertained with an accuracy of f1%. This kind of distribution of points, however, can only be used when there is independent evidence that the plot of log I us. H i s indeed a straight line. Since that is rarely the case, this is a highly artificial distribution. Most indicator experiments are done for the dual purpose of fixing an indicator slope and determining whether or not the line is straight, and in this case it is best to distribute the points evenly throughout the range covered. To investigate this situation, we calcu-
--
lated the relative error in the slope of sets of 11 points distributed at equal intervals along a line whose center is log I = 0.00. As before, the length of the line was varied by extending its limits ( i l o g I,,,) in a regular way. Because the point at log I = 0.00, which is common to all of these sets, contributes - nothing to the summation of Equation 13 (log I - log I = O.OO), this is equivalent to using only 10 points, and the results are strictly comparable with those of the previous calculation. Line I1 of Figure 3 shows that spacing the points at equal intervals along the line results in little sacrifice of accuracy in the slope; the minimum value of u b / b u A is now only 20% greater than it is when the points are placed at the ends of the line. This is a small price to pay for added information concerning the shape of the line. Line I1 of Figure 3 shows also that the error in the slope for a regularly-spaced distribution of points rises only very gradually once it has passed the minimum value. This minimum occurs at log I,,, = 1 0 . 9 ; at log I,,, = + 1.5, the error is only 12% greater and at log I,,, = 12.0, 2 6 z greater. It is tempting to conclude from this that the length of the line can be extended almost indefinitely for a gain in information about its shape at little sacrifice of accuracy in the slope. This, however, is misleading, for the reason that Line I1 of Figure 3 rises so slowly at large log Zmex is that points in this region are much less accurate and consequently are weighted very little. The contribution they make to the summation of Equation 13 adds very little to the total provided by points at small +log I . This can be seen more clearly by evaluating Equation 13 for a case in which points at increasingly larger *log I are added regularly to the ends of an indicator plot. Line 111, Figure 3, was calculated by starting with one point at log I = 0.00 and adding points successively in pairs at increments of +0.20 in log I . This gave 11 points (10 effective points, for log I = 0.00 does not contribute to the error) at log Zmax = =t1.O, and Line I11 necessarily has the same value here as Line 11. Beyond this point, Line I11 decreases only gradually and approaches a limiting value between log Zmsx = 1 1 . 5 and h2.0. This indicates that, for the purpose of obtaining the best value of the slope of an indicator plot, making measurements at values of log I less than -1.5 or greater than f1.5 is of little benefit provided that a reasonVOL. 41, NO. 1, JANUARY 1969
* 77
able number of points are already available within this range. This of course does not mean that points outside these limits are valueless for other purposes; they can be very useful, for example, in characterizing a situation with strong deviations from linearity. EXPERIMENTAL
Materials. Commercially available samples of o- and p nitroaniline were recrystallized to constant melting point: o-nitroaniline, mp 71-71.5 “C (from ethanol-water) [lit. (13a) 71.5 “C]; p-nitroaniline, mp 146.5-147 “C (from 9 5 z ethanol [lit. (136) 148 “C]. Acid solutions of appropriate concentrations were prepared by diluting reagent grade HC104, HC1, and H2SO4. Indicator Measurements. All measurements were made using aqueous stock solutions of indicators a t constant concentration, 4.32 x lO-4M in the case of o-nitroaniline and 1.33 x 10-4M in the case of p-nitroaniline. Carefully pipetted 10-ml samples of these indicator solutions and 10-ml samples of aqueous acids were mixed and the resulting solutions were equilibrated for at least 20 minutes with a constant temperature bath operating at 25.0 0.02 “C.
In the range of acid concentrations employed, volume changes upon mixing indicator and acid solutions were less than 0.2 %, and this procedure therefore provided a series of sample solutions of constant total indicator concentration and varying acid concentrations. Absorption spectra of these solutions from 5000 to 3500 A were recorded using 1-cm matched quartz cells and a Beckman Model DK-2 spectrophotometer whose cell compartment was thermostatted by circulating water from the 25 “C bath. Reference solutions were prepared in the same manner as sample solutions except that distilled water was substituted for indicator stock solutions. The acid concentrations of the sample solutions were determined by titration with standard base. Indicator ratios were calculated using Equation 1. Absorbances, A , were estimated to 0.001 unit from the machine traces of absorbance L‘S. wavelengtb; values at 4050, 4100 (absorption maximum), and 4150 A were used for o-nitrotniline and at 3780, 3800 (absorption maximum), and 3820 A for p-nitroaniline. Values of A B were measured in pure water and values of AB=, in acids whose HOvalues were a t least three units more negative than the indicator pK,’s.
*
(13) I. M. Heilbron, H. M. Bunbury, and W. E. Jones, “Dictionary of Organic Compounds,” Vol 111, Oxford University Press, London, 1943,(a) p 86; (b) p 87.
RECEIVED for review July 23, 1968. Accepted November 7, 1968. Research supported by the Petroleum Research Fund of the American Chemical Society under Grant No. 1180-A1, 4 to the Illinois Institute of Technology.
~
An Absorption Spectrometric Study of Molybdogermanic Acid. Methods for the Determination of Germanium Robert Jakubiecl and D. F. Boltz Department of Chemistry, Wayne State University, Detroit, Mich. As the result of a spectrophotometric study of molybdogermanic acid, the conditions required for the formation of a stable form of this heteropoly acid were determined. On the basis of the formation of a stable molybdogermanic acid, four absorption spectrometric methods were developed for the determination A direct ultraviolet spectrophotoof germanium. metric method most suitable for determining 1.2-4 ppm germanium in aqueous solution was developed. By extracting molybdogermanic acid with a (1:4) 1pentano1:diethyl ether mixture, an analogous direct ultraviolet spectrophotometric method having an optimum concentration range of 1 to 3.1 ppm of germanium resulted. An indirect ultraviolet spectrophotometric method based on measurement of the absorptivity of the molybdate resulting from the decomposition of molybdogermanic acid is suitable for the determination of 0.3 to 1.15 ppm of germanium. The molybdate resulting from the decomposition of the heteropoly acid can also be measured by atomic absorption spectrometry at the 313.3 mw resonance line of molybdenum. This indirect atomic absorption spectrometric method has a percentual concentration limit (PCL) of 0.05 ppm of germanium and shows conformity to Beer’s law for the 0 to 1.6 ppm range.
INACIDIC SOLUTION germanium(1V) reacts with molybdenum(VI) to form 12-molybdogermanic acid which has a yellow hue, is quite unstable, and has been used in the colorimetric determination of germanium (1-5). It is also possible to reduce the 12-molybdogermanic acid to the corresponding heteropoly blue (6). Although the heteropoly blue method 1
Present address, Corn Products Co., Moffett Technical Center, Ill. 60502.
Argo,
78
ANALYTtCAL CHEMISTRY
is sensitive, it lacks precision because of the instability of the parent heteropoly acid and the critical nature of the time interval between formation and reduction of the heteropoly complex (7-11). Recently, the instability constant for molybdogermanic acid has been determined to be 1.38 x which indicates higher stability than indicated by previous work (12). Very little information has been published on the atomic absorption spectrometric determination of germanium except that a nitrous oxide-acetylene flame was used and the sensitivity was 1.5 ppm of germanium (13, 14). (1) C. G. Grosscup, J . Amer. Chem. Soc., 52, 5154 (1930). (2) I. P. Alimarin and B. N. Ivanov-Emin, Mikrochemie, 21, 1 I1 936). - -,\ - -
(3) R. E. Kitson and M. G. Mellon, IND. ENG. ~ H E MANAL. ., ED. 16, 128 (1944). (4) F. Chauveau, P. Souchay, and R. Schaal, Bull SOC. Chim. Fr., 1959, 1190. (5) R. A. Chalmers and A. G. Sinclair, Anal. Chim. Acta, 33, 384 (1965). (6) Poluektov, Z. Anal. Chem., 105, 23 (1936). (7) D. F. Boltz and M. G. Mellon, ANAL.CHEM.,19, 873 (1947). (8) E. R. Shaw and J. F. Corwin, ibid., 30, 1314 (1958). (9) L. Erdey and E. Bodor, 2.Anal. Chem., 134,81(1951). (10) F . Lucena-Conde and L. Sant’Agostino, Anal. Chim. Acta, 16, 473 (1957). (11) W. Kemula and S. Rosolowski, Roczniki Ckem., 34, 835 (1960). (12) I. P. Senise and L. Sant’Agostino, Mikrochim. Acta, 1956, 1445. (13) J . W. Robinson, “Atomic Absorption Spectroscopy,” Marcel Dekker, Inc., New York, N. Y., 1966, p 127. (14) M. D. Amos and J . B. Willis, Spectrochim. Acta, 22, 1325 (1966).
