Experimental Evaluation of Indeterminate Error in Height-Widthand Height Only Measurements of Chromatographic Peaks D. L. Ball1 and W. E. Harris Department of Chemistry, University of Alberta, Edmonton, Alberta, Canada
H. W. Habgood Research Council of Alberta, Edmonton, Alberta, Canada The four basic errors in the measurement of area of a chromatographic peak by height-width integration were isolated experimentally. The standard deviations (taken as measurements of the indeterminate errors) were: error in placement of t h e base line, about 0.010 cm; error in measurement of height f r o m the base line, about 0.012 cm; error i n positioning of a measuring instrument at a n intermediate height, about 0.021 cm; and error in measurement of peak width, 0.0080 c m for sharp peaks and increasing as peaks become flatter. From these four basic errors in measurement the resultant errors in area were calculated according to t h e theory previously developed and found to agree well with t h e observed standard deviations i n calculated areas for peaks of various shapes and sizes. The optimum peak shape for peaks of a given area has a ratio of height to widthat-half-height of about 3 to 5. For a peak of 15-cm2 area the standard deviation a t this optimum i s about 0.5%. For peaks of a given shape the relative e r r o r i n area decreases according to t h e square root of t h e peak area. For peaks of a given height the e r r o r deAA A
instrument at an intermediate height y with an error of standard deviation A y ; and reading the peak width, w, at the intermediate height with an error of standard deviation Aw. Peak area A is calculated from the measured values of h and w using the expression A = C,hw, (1) where C, is a constant whose value varies according to the fractional height r at which the peak width is measured. The total effect of the four errors on the calculated peak area can be expressed ( I ) by a general error equation
44 A =
J(A+)2
+ (53)'
+
($)'
+
( 9 ) 2
(2)
Equation 2 involves four terms associated with the four basic errors and substitution of the appropriate expressions (I) for each of the terms gives
( 1 - r)
creases with increasing width u p t o a limiting value of about 3- to 4-cm width. In general, peaks should be recorded so that peak height i s near the maximum allowable by t h e chart paper and so that the width i s about 3 to 4 cm. The use of peak height alone was also examined, theoretically and experimentally. Comparison of t h e peak-area and peak-height methods shows that i n t e r m s of indeterminate errors peak height i s a much more precise measurement technique. Because of this inherently better precision, control of determinate errors i s worthy of intensive effort.
The primary intention of this study is to examine experimentally the validity and application of this error expression. This has been done by evaluating large numbers of independent measurements made under controlled conditions on peaks of nearly ideal Gaussian shape. An additional objective is the development and experimental examination of the total error incurred in measuring only peak height. EXPERIMENTAL
Present address, Selkirk College, Castlegar, British Columbia, Canada.
The experimental approach to this problem involved a large group of observers making measurements on a set of idealized symmetrical chromatographic peaks of various sizes and shapes. The width measurements were all made at half height ( r equal to 0.5). The measurements were statistically evaluated to give standard deviations which were taken as values of the indeterminate errors. The standard deviation in area as well as the standard deviations in the four basic operations were determined and the observed standard deviations in area were then compared with the values theoretically predicted by Equation 3. Symmetrical gaussian or near-gaussian peaks of seven different shapes and two different sizes (15 and 1.5 cm2) were used. The peaks ranged in shape from a ratio of height to width-at-half-height of 35 (sharp) to 0.089 (flat). One set of peaks was printed without base lines and a second set had base lines drawn in. The details concerning preparation of the peaks, line thickness, ink color, paper quality, and so forth are given elsewhere (2).
(1) D. L. Ball, W. E. Harris, and H. W. Habgood, Separation Sci., 2, 81 (1967).
(2) D. L. Ball, Ph.D. Thesis, University of Alberta, Edmonton, Alberta, Canada, 1967.
IT HAS BEEN SHOWN ( I ) that there are four independent sources of indeterminate error in the determination of areas of chromatographic peaks by the common method of measuring peak height h and width w. (In this previous paper a typographical error occurred in Equation 21, in the last bracket of which l / w , should be replaced by hlw,.) These four errors are associated with the four operations involved in the determination (Figure 1): placing the base line, which involves an error whose standard deviation we designated AB; measuring the peak height h from the established (presumed) base line with an error or standard deviation of Ah; measuring and marking the distance, and positioning the measuring
VOL 40, NO. 1, JANUARY 1968
129
The participants in the study were groups of students enrolled in a course in analytical chemistry. Many of the critical measurements were made by one group of 150, comprising the top 85 of a class. The participants were of a generally homogeneous background and virtually none had had prior experience in gas chromatography; as a group they took part in the project with alert intelligence. Experienced technicians with the same kind of initial instruction, might be expected to produce more precise measurements than such a student group but the general relations and conclusions would still hold. Measurements were made with the centimeter edge (smallest division 0.1 cm) of a high quality triangular scale. Each student was given written instructions regarding the measurement procedure: Work consistently from the line edges, realize that lines printed on a ruler or scale occupy a finite fraction of the smallest scale interval, and use this finite fraction as an aid to the eye in making the interpolation to one tenth of the smallest scale interval. Briefly, the height of a peak was measured, a mark was drawn parallel to the base line at half height, and the width of the peak was then measured at this half height. Further details about the measurement techniques and conditions and the procedure for rejection of invalid data are described elsewhere (2).
P
RESULTS AND DISCUSSION Evaluation of the Four Basic Errors.
The four individual measurement errors were estimated partly from a study of the measurements reported by the observers in the course of measuring peak heights and widths and partly from the results of separate experiments as described below. The base-line error, AB, was obtained from an examination of the base lines drawn in by the observers. The base-line positions were directly measured relative to reference crosses located near the base line on each peak by means of a Hensoldt measuring magnifier (smallest division 0.01 cm). The standard deviation of the distance from the reference cross for each of the various peaks is reported in Table I as the base line error AB. The average value for AB is about 0.010 cm. The error in peak height, Ah, was evaluated directly from the peak height measurements of the peaks with printed base lines, Because the base lines were already established, the reported height measurement was not affected by AB, Values for the measurement error Ah for each peak were obtained simply by computing the standard deviation of the reported peak-height values. The results are given in Table I as Ah, with an average of about 0.012 cm. The error A y is difficult to isolate experimentally. It is defined as including the effects of the errors in reading and
Figure 1. Schematic diagram of a gaussian peak to illustrate the four errors associated with the four basic operations 1, The error AB associated with placing the base line; 2, the error Ah associated with measuring the height from the presumed base line; 3, the error Ay associated with placing the measuring scale parallel to the base line for the width measurement; 4, the error Aw associated with measuring the distance between the sloping sides of the peak at the position of the measuring scale. The inserts attempt to show the details of the various steps. For example, inserts 3 refer to the establishment of an intermediate height at which the width is to be measured. From the presumed base line a distance rh is measured and the attempted intermediate height corresponds to this distance. The intermediate-height line that is actually drawn will differ from this attempted line and, furthermore, the imaginary line along which the measuring scale is positioned for the width measurement will differ from the drawn line. The contributions to Ay are shown as two bands of uncertainty of different widths around the presumed base line and the attempted intermediate-heightline
marking the intermediate-height line and the error in positioning the ruler at that line (Figure 1 ) . Again, examination of peaks with base lines supplied eliminates any confusion from the AB factor. The half-height line drawn in on these peaks, however, is affected by errors in the height measurement, that is an error of Ah/2 is carried over into the at-
Table I. Standard Deviationsa (cm) Representing the Basic Measurement Errors AB9 Ah9 A y , and Aw hlwo.6
35
15
4.8
AB Ah AY Aw
0.0095 0.013 0.023 0,0080
0.0067 0.010 0.021 0.0080
0.0079 0.011 0.020 0.0081
0.0089 0.0084 0.020 0.0092
AB Ah AY Aw
0.011 0.019 0.021 0.0080
0.011 0.015 0.021 0.0080
0.0089 0.012 0.021 0.0081
15 C M ~ 0.012 0.013 0.021 0.0092
1.2
0.52
0.27
0.089
0.0083 0.0084 0.018 0.013
0.0080 0.011 0.019 0.021
0.0097 0.0086 0.019 0.066
0.013 0.010 0.022 0.013
0.011 0.012 0.020 0.021
0.014 0.016 0.020 0.066
1.5 C M ~
Each value of AB, Ah, and Ay for 1.5-cm2 peaks results from the evaluation of about 60 measurements; each value for 15-cm2 peak from about 80 measurements. The value for Am (0.0080 cm) required for calculating Aw was based on about 150 measurements. 5
130
ANALYTICAL CHEMISTRY
0 100
10
n v
1
I
I
I
I
I
I
0.1
1
PEAK SHAPE,
0.01
hW /,
I
0
Figure 3. Relative error in area as a function of peak shape
Figure 2. Relation between reading error and angle of slope, CY
Closed points are directly observed experimental relative standard deviations in area for peaks of 1.5 cm2,and open points for peaks of 15 cm.2 Solid lines represent the values of the relative error calculated from the data of Table I by use of Equation 3. Height-width ratios greater than 40 for peaks of 15 cm2(arrow on lower curve) have no practical meaning because the peak height would exceed the 25-cm width of ordinary chart paper
90
60
30
ANGLE, u in degrees
Points are experimentally observed mean standard deviations. Line is value of Equation 4 with Am equal to 0.0080 cm
tempted line (Figure 1). Consequently, it is included in the standard deviation of the drawn line. The error in positioning the scale is obviously not included in the drawn line. Thus to obtain Ay, the observed standard deviation in the half-height line for peaks with base lines supplied should be reduced (statistically) by one half the standard deviation in the height and increased by the standard deviation in ruler positioning. The positioning error was estimated, as described below, to be 0.015 cm. Using this value along with the above Ah values (Table I) and the observed standard deviations in placement of the line at half height yields the values of Ay listed in Table I. The average for Ay is about 0.021 cm. A separate set of experiments was designed to isolate the error associated with the measurement of peak widths. In these experiments the gaussian peaks were replaced by pairs of straight lines with varying angles of inclination to the horizontal, cy. The distance between each pair of lines was measured at a position fixed by two firm positioning pins (2). The variation in the measured distance corresponds to the error A w as already defined. The standard deviations for several pairs of parallel lines ( cy equal to 90") were 8.3, 8.1. 5.2, 7.8, 10.9, 8.6, 9.2, and 6.1 X 10-3 cm. and for pairs of lines with cy equal to 10" the standard deviations were 46, 30,42, 39, and 59 X 10-3 cm. The mean values for the above sets of standard deviations as well as for those at other angles ( 2 ) are shown in Figure 2. The error Aw can be considered to be composed of two bands of uncertainty illustrated in Figure 1 as A w / f i i n magnitude. The effective width of this band in the direction of measurement will increase according to l/sin cy. Thus if Am is taken as the minimum error as found for parallel lines, Aw as a function of the angle cy should be given by Aw = Am/sin a = A m d l
- cot%
(4)
The line in Figure 2 is calculated by Equation 4 using a value for Am of 0.008 cm. This relationship is distinctly better than that proposed on the basis of preliminary data ( I ) . The value of A w therefore depends on how sharply the sides of a chromatographic peak slope at the position of measurement. Equation 5 can be re-expressed (see Equations 8, 9, and 18 of ref. l and note that u equals 0.4248 w0.5) in terms of a gaussian peak as
is the peak width at half height. Substitution of a where value of 0.0080 cm for Am leads to the values for A w listed in Table I. As mentioned above, the values for Ay include the positioning error. The positioning error was evaluated by additional measurements of the type described in the preceding paragraph but with the ruler position established by two short lines rather than by the fixed pins. The standard deviations of these measurements are shown in Table I1 as total error, that is the statistical sum of the measuring and positioning errors, and may be compared with the mean standard deviations of the measurements at the same angles with positioning pins shown as A=. Table I1 demonstrates, as might be expected, that for parallel lines the erorr is not significantly different from the Am value of 0.0080 established in Figure 2. Only when the angle decreases can an error caused by vertical scale movement be expected and observed. Quantitatively, a small vertical displacement of the scale (that is, in a direction perpendicular to its length) by an amount Ap will lead to a difference in the distance measured along the scale between the inclined lines, equal to Ap cot cy. Since the error in ruler placement is independent of the reading error the total effect of the A w and positioning errors can be expressed by statistically adding A p cot a to the quantity on the right side of Equation 4 to give VOL 40, NO. 1, JANUARY 1968
131
Table 11. Measuring Error and Positioning Error in the Measured Distance Between a Pair of Lines of Slope CY Angle, a 90 70 50 30 20 15 10 5 Aw, cm= 0.0080 0.0105 0.0087 0.015 0,022 0.027 0.043 0.072 0.007 0.013 0.017 0.032 0.062 0.038 0.072 0.15 Total error, cmb Ap, cmc ... 0.028 0.016 0.016 0.021 0.006 0.010 0,010 a Each value is the mean of the standard deviations of at least five groups of independent measurements by 18 to 20 different observers of distances between different pairs of lines that were randomized in terms of the terminal digit. * Each value is the result of a single set of independent measurements made by 18 to 20 different observers using the same scale on one particular pair of lines. Calculated from Equation 6.
Total error
=
dAm2(1
+ cot2a) + Ap*(cot2a)
The magnitude of the standard deviation associated with vertical placement of a ruler in the absence of positioning pins, that is, Ap, can be estimated by rearranging and solving Equation 6 using Am equal to 0.0080 cm and the observed values of total error in Table 11. The resultant calculated values of A p are also given in Table 11; the mean is 0.015 cm which may be compared to a value of 0.008 cm for Am. Relative Error in Area by Height-Width Measurements. The indeterminate errors in area by the height-width method were obtained from the areas of each base line-free peak as calculated from the height and width values reported. The relative standard deviations AAIA are shown in Figure 3 as the experimental points. The error in area measurements calculated from the reported height and width measurements in peaks without base lines may be compared with the error that should be predicted from Equation 3 by use of the four independently determined basic errors as given in Table I. The predicted errors are shown as the lines in Figure 3. The general nature of the agreement between observed and predicted errors is gratifying and demonstrates that one should be able to use the general error expression of Equation 3 with confidence. One conclusion from the results illustrated in Figure 3 is that for peaks of a given area there is an optimum peak shape. This confirms the prediction of the theoretical analysis of errors in height-width measurements (1). For this optimum shape (Figure 3) the value of h/wa.6 lies at about 3 to 5. The original analysis assumed as a limiting case that AB, Ah, A y , and Am were all equal and this led to an expected optimum of 2 to 3. The experimental values for the basic shape of errors AB, Ah, and A y listed in Table I differ somewhat from each other and from Am, but by no more than a factor of 2 or 3. Because the difference is small, the position of the experimentally observed optimum is not significantly different from that expected from the simplified model (1). A second conclusion of this study, at least from the practical viewpoint, is the confirmation that for peaks of a given shape the relative error in area is greater for peaks of small area than for those of large area. The dependence of error on peak area as given in Equation 3 is according to the square root of the inverse of the area for peaks of the same shape. In practice, therefore, one should strive for peaks of maximum area. Although not obvious from Figure 3, another conclusion is that for peaks of a given height the error decreases to a limiting value as the width of the peak increases. Equation 3 has been combined with the data of Table I to produce the curves shown in Figure 4 for several peak heights. The error obviously decreases sharply with increasing peak width for sharp peaks, and the decrease in error for peaks that are already wide is negligible. Figure 4 leads to the somewhat surprising practi132
ANALYTICAL CHEMISTRY
4
(6)
LLI
5> 1 0
0.1
1
10
PEAK WIDTH, wo,5
100 (cm)
Figure 4. The relation between relative error and peak width for peaks of several heights as indicated Calculated from the data of Table I by use of Equation 3
cal conclusion that the peak width for the limiting error is about 3 to 5 cm for peaks of all reasonable heights. Beyond a peak width of about 5 cm, the advantage of increasing the area is just balanced by the disadvantage of measuring the widths of wide peaks. In practice (2), the usual peak height is about 5 to 10 cm. If peaks of this height have adequate width, then Figure 4 indicates that one can expect to measure the peak area with an error of less than 1 %. From the preceding paragraphs it is clear that improved precision of peak integration can be expected when height and width (up to a point) are increased-that is, when peaks of larger area are produced. I n practical chromatography height may often be increased by decreasing the signal attenuation or improving the sensitivity of the entire detection system. It is worthwhile therefore to set the recorder attenuator to give maximum peak height within the limitations of the signal and of the chart width. Increasing the sensitivity of the recorder changes the peak shape towards a higher height-to-width ratio. The change in h/wO.s will modify the effects of the increased peak area to an extent depending on the exact values of h / ~ ~ This . ~ . increase in peak height is especially favorable for peaks of low hlw0.5 since the error is decreased both because one is dealing with a larger peak and because the peak shape moves toward the optimum of curves such as in Figure 3. Obviously, one cannot indefinitely increase the recorder sensitivity; the peak height is limited by the width of the chart paper (normally 25 cm). Another limiting factor in increasing recorder
h
Figure 5. Relative error in height as a function of peak shape or peak height Closed points are directly observed experimental relative standard deviations in height, AH/h, for peaks of 1.5 cmz, open points for peaks of 15 cm2. Solid lines represent values of relative error obtained from AB and Ah (Table 1) by use of Equation 7
$ Y
-c
4
' I a
-
e
0
E
E2
n W
100
10
PEAK SHAPE, sensitivity is the increase in noise level, which will increase all the basic errors except Ay. Increasing the gain for a narrow peak may also introduce a determinate error in the recorder because of the recorder time constant being too large for the pen to adequately track a high narrow peak. Another practical way to increase precision is to increase peak widths by increasing the recorder chart speed. In this case one can expect an advantage only if the peak is initially narrower than about 5 cm, that is if the peak already has a moderate or high value for h / ~ , , . ~ .As a general rule nothing is to be gained by increases in speed beyond this point and there may well be a disadvantage from larger determinate errors within the recorder system. The choice of proper chart speed is complicated in isothermal chromatography by the continually varying peak width with retention volume. In programmed temperature gas chromatography the choice of the best chart speed is straightforward in that peak widths are relatively constant throughout an analysis. In programmed temperature gas chromatography the best chart speed is one that gives peak widths at half height of about 3 to 4 cm. In the first paper (I) the possibility was raised that the baseline uncertainty AB may increase with increasing peak width, hence tending to favor sharper peaks. There is no evidence for this in the present measurements on idealized noise-free peaks (Table I). This does not rule out the possibility of such an effect arising in real peaks particularly with long term, baseline noise. As pointed out, it is expected that base-line noise will shift the optimum toward sharper peaks and at the same time increase errors for peaks of all shapes. The actual values reported in this paper are, of course, based on the experience of a particular group of observers working with idealized noise-free peaks. Each person will have his own level of precision as indicated by a set of values for the various measurement errors which will probably differ somewhat from the values for this group. The relative importance of each of the measurement errors in terms of its contribution to the error in the calculated area
0.1
1
0.01
h/w,
was considered in the previous paper ( I ) . For example, Figure 5 of that paper showed that if the measurement errors were all equal (that is, AB = Ah = Ay = Am) then with narrow peaks the error in width is the most serious error, with wide peaks the errors in base-line placement and in establishment of the intermediate height are of major and roughly equal importance, and in no case is the error in measuring height of appreciable significance in the final area. These qualitative conclusions are modified for the case of the present measurements only to the extent that the effect of the error in establishing the intermediate height is of even greater importance for wide peaks because the basic measurement error Ay is roughly twice the other measurement errors. An increase in the noise level of a chromatographic trace can be expected to increase the relative standard deviation of an area measurement because AB, Ah, and Aw would all become larger. The value of Ay would not be expected to be dependent on the noise level because as it is defined it is solely an internal operation. The combination of steps (measuring, marking, and positioning) included in Ay is independent of the exact shape of the peak. This is also part of the justification for treating these steps as a unit rather than, say, including the positioning error with Aw. It is implicit in Equation 3 that the parameter r , which is the height chosen at which to measure peak width, has an optimum value. The complex manner in which this parameter affects the error in calculated area is the subject of a study to be reported later. Relative Error in Peak Evaluation by Height Measurements Alone. Quantitative measurements of chromatographic peaks based on peak height involve only two experimental steps: drawing the base line and measuring the height from the base line. The errors resulting from these two operations are illustrated in steps 1 and 2 of Figure 1. The total error in the height measurement AHis therefore a direct result of the combined effect of these two independent errors. Hence the total relative error in peak height is expressed simply as the sum of the variances VOL. 40, NO. 1, JANUARY 1968
133
AH
h =
+(
y ) z
(7)
where A H represents the total error incurred in the measurement of peak height and Ah represents the error in measuring the height from the presumed base line. From the measurements of the peak heights of the peaks without base lines a direct value for the error in height measurement AH was available simply by cc,nputing the standard deviations. The relative standard deviations AH/h are shown in Figure 5 as the experimental points. The experimentally observed error in peak height may be compared with the error calculated from Equation 7 using the independently determined basic errors AB and Ah of Table I. The lines in Figure 5 are these calculated values, It is appropriate to compare the precision of height-width integration with peak-height measurement (Figures 3 and 5). Clearly, for peaks of a given height-to-width ratio, the relative indeterminate error is always smaller if only peak height is used rather than peak area. Particularly for peaks of large height-to-width ratio, the advantage in measuring the height only is enormous. The inset of Figure 5 indicates that the error to be expected in measurement of peak height is to a good approximation inversely proportional to the height and almost independent of peak shape for the idealized noise-free peaks that were measured. In practice, separate measurements that were made on less ideal peaks suggest that for two peaks of the same height where one is a sharp spike (height-to-width ratio greater than about 20), the height of the sharp spike is subject to a slightly increased error. A precision of 2 ppt should be readily attain-
able provided peaks of moderate height are measured. For example, the data of Figure 5 indicate that for a peak 10 cm high the error in height measurement was 0.13 %. Measurements of peak area have generally been preferred for quantitative work. This preference may be due to a lack of appreciation of the inherently lower precision in these measurements and also to a recognition that measurements of peak height are sensitive to instrumental and operational variations such as sample injection, column temperature, and column efficiency. For example, if two samples of the same size are injected rapidly in one case and slowly in another, the peak heights may be grossly different although the areas will be the same. Peak height is affected also by fluctuations in column temperature, which impose severe requirements on instrument design. In addition, although peak area is more sensitive to variations in gas flow rate than is peak height, the stabilization of flow rate is easier than stabilization of temperature. The detailed measurements of this present study, however, support the general conclusion often reached in practical work that with good experimental control and stability of a chromatographic system maximum quantitative precision can be obtained from measurements of peak height. The necessary degree of experimental control is often approached in repetitive and control analyses. In general, the gains in precision potentially possible through the use of height measurements should justify intensive efforts to achieve the necessary high quality of experimental performance and control. RECEIVED for review August 16,1967. Accepted November 1, 1967. Financial support to D.L.B. by the National Research Council of Canada is gratefully acknowledged.
Determination of Olefinic Unsaturation by Bromination James S. Fritz and Garth E. Wood’ Institute for Atomic Research and Department of Chemistry, Iowa State University, Ames, Iowa 50010 The direct titration of various unsaturates in 85% acetic acid-lO% water-5% carbon tetrachloride with bromine in glacial acetic acid is described. Titrations are performed both with and without utilization of a mercury(l1) chloride catalyst. I n the absence of catalyst, compounds containing an isolated olefinic linkage can be determined in the presence of compounds which possess either strongly electron-withdrawing substituents allylic to the double bond or a double bond conjugated with a carbonyl group. An indirect spectrophotometric method for the determination of small amounts of unsaturation is also described.
THE MOST WIDELY applicable methods for determination of organic unsaturation involve the reaction of bromine with the unsaturated compound. A mercury(I1) salt is usually added to catalyze the reaction, Both direct and indirect procedures have been utilized. A broad picture of these methods is presented by Polghr and Jungnickel(1). A major difficulty is that in the presence of excess bromine many com1
Present address, Celanese Chemical Co., P. 0. Box 2768,
Corpus Christi, Texas 78403.
(1) A. PoIgCr and J. L. Jungnickel, “Organic Analysis,” Vol. 111, Interscience, New York, 1956, pp. 229-55.
134
e
ANALYTICAL CHEMISTRY
pounds undergo substitution side reactions which may cause high results. Attempts have been made to correct for the substitution side reactions by determining the hydrobromic acid that is produced (2). An extrapolation procedure has been proposed to correct for side reactions such as substitution which may occur to a minor extent and cause high results (3). Direct titration methods offer several advantages over indirect procedures. No excess bromine is present until after the end point is reached; therefore, side reactions are less likely to occur. Sweetser and Bricker (4)developed a spectrophotometric method for titration of olefins and other substances with tribromide ion at wavelengths ranging from 270 to 360 mp. Miller and DeFord (5) adapted this method for electrically generated bromine. Their results were almost all low and often in poor agreement with results obtained by analyzing the same unsaturated compounds by standard methods. Leisey and Grutsch (6) obtained satisfactory results for trace unsaturation using coulometrically generated (2) Ibid.,p. 231. (3) Ibid.,p. 237.
(4) P. B. Sweetser and C . E. Bricker, ANAL.CHEM., 24,1107 (1952). (5) J. W. Miller and D. D. DeFord, Ibid.,29,475(1957). (6) F. A. Leisey and J. F. Grutsch, Zbid.,28, 1553 (1956).
