ionization currents, the photometric mode seems capable of greater sensitivity. This might be the result of geometrical considerations, because electrode geometry can be a rather critical factor in determining the sensitivity limits of a detector system. The results of this investigation have shown that the measurement of sodium emission provides a detector system which is highly sensitive to halogen-containing compounds. Since photometric detection does not affect the measurement
of ionization currents, it is easy to record the enhanced ionization currents and photocurrents simultaneously. The system can also be easily converted to the ordinary flame ionization or flame photometric modes by removal of the salt-coated probe. RECEIVED
for review December 11, 1967. Accepted March
19,1968.
Selection of Optimum Position for Measuring Widths of Chromatographic Peaks with Minimum Error in Area D. L. Ball' and W. E. Harris Department of Chemistry, Uniuersity of Alberta, Edmonton, Alberta, Canada
H. W. Habgood Research Council of Alberta, Edmonton, Alberta, Canada Relations previously developed relating the four individual measurement uncertainties in height-width measurements to the resultant error in area were used to examine the effect of variations in the value of r , the fractional height at which the width is measured, on the precision of the determination of area. The optimum fractional height is shown to depend on peak shape. I n general, sharp Gaussian peaks should be measured for width close to the base line, and flat, broad peaks should be measured close to the commonly used half height. I n practice, a single fractional height is desirable. If a full range of smooth Gaussian peaks is normally encountered, then the best single value of r to choose is 0.25.
choice of fractional height on the errors resulting from each of the four measurement steps involved in the height-width method. The area A of a Gaussian peak is calculated from the measured values of height h and width wr by the formula
where C, is a constant for a given fractional height r and is given by
c, = ! 2
MEASURING THE AREA under a Gaussian or near Gaussian peak is a common operation in gas chromatographic analysis. For a given method of integration, many factors affect the precision of the area determination. In this paper the most commonly used method of integration, the height-width technique, is considered. In particular, the problem of choice of optimum position at which to measure the width is examined in detail. As shown in previous studies (I, 2 ) there are four independent sources of indeterminate error arising out of the four operations in the height-width method. The choice of fractional height at which the peak width is measured enters into and contributes to the final error in a complex manner. Our object is to consider the optimum position for measuring peak width and how this optimum position is affected by peak shape and peak area. This question has been partly treated by Said and Robinson (3). Their analysis was concerned primarily with the effects of uncertainties in measuring peak height. The present study evaluates the effect of Present address, Selkirk College, Castlegar, British Columbia, Canada
(1) D. L. Ball, W. E. Harris, and H. W. Habgood, Separation Sci., 2, 8 1 (1967). (2) D. L. Ball, W. E. Harris, and H. W. Habgood, ANAL.CHEM., 40, 129 (1968). (3) A. S. Said and M. S . Robinson, J. Gas Chromatog., 1(9), 7 (1963).
(1)
A = C,hwr
In ( l l r )
(See Reference 3 for a development of this expression.) The intermediate height y at which the peak width is to be measured is chosen to be some fraction r of the total peak height so that r is equal to y/h. As previously explained ( 2 ) , the calculation of peak area according to Equation 1 depends on h and w,, but to obtain values for these two parameters requires four separate operations. Each of these operations introduces an error directly or indirectly into the computed result. First a base line must be located and drawn under the peak. The standard deviation of this measurement we have designated AB. The peak height h is then measured from this established base line and the associated standard deviation is Ah. Next, the position of the intermediate height y (equal to rh), is located. The standard deviation in locating y is A y . Finally, the peak width wr is measured at position y with a standard deviation in this measurement of Aw. In a previous study (2) numerical values obtained by a particular group of observers for AB, Ah, and A y were about 0.010, 0.012, and 0.021 cm and the values for A w were found to depend on the angle a between the peak sides and the scale used to measure width according to the relation Aw
=
Am(1
+C O ~ ~ C ~ ) ~ * ~
(3)
with Am equal to 0.008 cm. These four errors occur independently, hence their total effect is obtained by adding them in a statistical sense according to the following generalized error expression (2)) VOL. 40, NO. 7, JUNE 1968
1 113
AA A
AB 2 In ( l l r )
2r In ( l l r )
Equation 4 may be modified by substituting for Aw from Equation 3 , expressing cot In l / r ) , and removing the quantity l / h to the outside of the square root to give
CY
for a Gaussian peak ( I ) as w 4 4 r h -
! ! ! = h! d A B 2 { 1 A
+ ('
Term associated with
AB (base line)
- r,
}2
2r In ( l l r )
+ Ah2 ( 1
-
{
L}2 2r In-'(1/r) + Am2 +
2 In ( l / r )
Ah (height)
}2
[(i) +I'>-'(
100
10
1
PEAK SHAPE,
1
10
Factor
Base line
Figure 1. Relative standard deviation in area as a function of peak shape for varying fractional heights r at which peak width is measured Curves were calculated from Equation 5 with values for AB, Ah, Ay, and Am of 0.010,0.012,0.021, and 0.008 cm. Left side, for peaks of area 1.5 cm*,right side, for peaks of area 15 cm2
The contributions of the width factors from Tables I and I1 are combined in Table I11 as the square root of the sum of the squares for peaks of given shapes. The over-all value of the width factor for sharp peaks depends primarily on the shape factqr and for flat peaks depends primarily on the factor involving r. For peaks of intermediate shape, both factors are important. As far as the peak width term is concerned, for sharp peaks the width obviously should be measured near the base line.
Height
I - r (1
+
m
r
(1 - m
11 14
Width
r
1 2r In Ilr
4r In llr
3.34 2.17 1.44 1.36 1.44 1.65 5.27
1.67 1.09 0.72 0.68 0.72 0.82 2.64
)
1
Value of factor
Value of r 0.05 0.10 0.25 0 . 3 7 (e-l) 0.50 0.61 0.9
Intermediate height
1
)
3.21 2.96 2.08 1.86 1.72 1.65 1.53
ANALYTICAL CHEMISTRY
0.01
h/w,,
Table I. Values of Each Factor Involving Fractional Height r in Equation 5 , Showing the Effect of Changing r Term associated with
(5)
Aw (width)
AY (intermediate height)
Variation of Error with Fractional Height and Peak Shape. According to Equation 5, the relative error in area depends on r , which occurs in each of the first three terms and in the last part of the Aw term. That an optimum r exists and varies with peak shape can be best appreciated by examining each term in detail. Table I illustrates the effect of varying r on each of the factors in which it occurs. Examination of the values in Table I shows that the first (base line) term must continually decrease as r increases. Consequently, from this point of view, measuring the peak width near the top of a peak is preferable. The second (peak height) term goes to zero at the inflection point of the peak; hence from this point of view, the peak width should be measured at 0.61 of peak height. The third (intermediate height) term which has the same form as that developed by Said and Robinson (3) goes to a minimum when r is lie of the peak height, thus favoring measurement of width at this lower point. Finally, the r portion of the width term also passes through a minimum at l i e of the peak height, which once again favors a lower point for the peak width measurements. The llh and h/w, portions of Equation 5, constitute the peak-shape factors in the over-all error expression. If we consider only the l / h factor ahead of the square-root sign, to minimize error the peak height should be at a maximum. In contrast, the hjw, factor of Equation 5 reveals that not only does this factor favor flat peaks, but the error dependent on it also decreases when peak width is measured near the base line ( r small, hence w, large). Table I1 shows values of this shape factor for three peaks of different shapes: sharp (h/wo. = l o ) , intermediate (h/wo.5 = l ) , and flat (hiw0.5 = 0.1). The values in this table are large for sharp peaks and, as expected, for a given peak the value increases as r increases.
4r In l / r
0.83 0.78 0.64 0.50 0.28 0 3.74
The base line, height, and intermediate height factors of Table I and the width factor of Table I11 can be combined with AB, Ah, A y , Am, and also h in Equation 5 to given the over-all relative error in area. The dependence of this relative error on the fractional height is obviously complex. The base line term minimizes error when the width is measured near the peak maximum. The height and intermediate height terms minimize error when the width is measured at intermediate points. Finally, the width term minimizes error in the case of sharp peaks when the width is measured near the base, and with increasingly flatter peaks, the minimum error is at a position approaching lje. The height term is probably the least important of all four terms because the value of the height factor and the value of Ah are both relatively small. The other terms may all be large and significant under particular conditions. Thus, the total net error that occurs in the four measurement operations of the height-width technique results from a complex and competitive interplay of the various factors. One way to assess the effect of changing the fractional height at which width is measured is to calculate the relative error in area according to Equation 5 using the values mentioned earlier of AB, Ah, Ay, and Am along with the numerical values of Tables I and 111 for the various factors. Figure 1 shows the relative error AA/A as a function of the index of peak shape hiw0.6 for peaks of 1.5 and 15 cm2 area. The unusual direction of the abscissa scale was chosen so that sharp peaks (high hiw0.5) would be to the left, analogous to the early peaks in an isothermal chromatogram. Each curve corresponds to a particular value of fractional height as labelled in the figure. It becomes apparent that to minimize the relative error for sharp peaks of any given area, the operator should measure the width close to the base line. This requirement is reasonable because the larger width value obtained near the peak base reduces the relative error occurring in this particular measurement. As the peaks flatten, the optimum r increases to a limiting value of around 0.5. For any given peak area, a curve of minimum relative error as a function of peak shape using the appropriate optimum fractional heights would form an envelope to the family of curves as illustrated in Figure 1. Thus, the minimum relative error for any given peak area, requires both an optimum peak shape and an optimum fractional height at which to measure width. Figure 1 , for example, illustrates that for peaks of a given area this optimum corresponds to a peak with height approximately five or six times the width at half height and with area determined by a width measured at quarter height (r equal to 0.25). For a given peak shape, the best value of the fractional height is the r value for that particular curve which touches the abovementioned envelope. The choice of r is least critical for peaks of an intermediate shape; here any value between about 0.2 and 0.6 would be equally satisfactory. In the first paper of this series, a model was proposed in which AB, Ah, and A y were all assumed equal to Am. This model yields a set of curves similar to those in Figure 1, indicating that the conclusions reached here are not highly sensitive to the particular values chosen for the basic errors. The experimentally determined Ay value is approximately twice the values for the other basic errors, and the effect has been to shift the family of curves slightly in the direction of sharper peaks as compared with the curves calculated for equal errors. Equation 5 has been verified not only qualitatively but quantitatively for a fractional height of 0.5. Corresponding experimental curves have not been determined for other values of r but a somewhat less exhaustive study has confirmed the relative positions of the different r curves. These experiments
Table 11. Values for Shape Factor h/w, for Sharp, Intermediate, and Flat Peaks hlWo.bn 10 1 0.1 Value of r Value of h/w, 0.05 0.10 0.25 0.50 0.90
3.6 5.5 7.1 10.0 25.8
0.36 0.55 0.71 1.00 2.58
0.036 0.055 0.071 0.100 0.258
The ratio of height to width-at-half-height is a convenient index to peak shape. Table 111. Over-All Width Factor from Equation 5 for Sharp, Intermediate, and Flat Peaks ’ h/wo s 10 1 0.1 , Value of factor + (&r)2]o. Value of r
[(k)’
0.05 0.10 0.25 0.37 (e-1) 0.50 0.61 (e--1/2) 0.90
4.0
5.6 7.1 8.3 10.0 12.5 25.8
1.71 1.22 1.01 1.07 1.23 1.50 3.69
1.67 1.09
0.72 0.68 0.73 0.84 2.65
consisted of independent measurements by a separate group of 180 observers. The idealized chromatographic peaks described earlier ( 2 ) were used, but cheaper boxwood metric rulers were used instead of triangular scales (hence, the basic errors would be larger). Widths at several positions on the peaks were measured with the aid of a chart of the type described by Harris and Habgood (4). The chart had several lines for rapid selection of several fractional heights. The number of replications of any single measurement was smaller than in the previously reported series of measurements (2) but the results agreed at least semiquantitatively with Figure 1. The detailed values are given elsewhere (5). In the earlier study(2) to generally evaluate the indeterminate errors in height-width integration and to validate the generalerror expressions developed, the width readings were taken at half height. The present work indicates that the standard deviations would have been somewhat lower if an r value of 0.25 had been used. The general conclusions, however, would have been the same although there would be slight modifications in detail. The principal modification would be that when peak widths are measured at 0.25 of the height, minimum error is attained with slightly sharper peaks than when widths are measured at 0.5 of the height. Increasing Base Line Uncertainty with Peak Width. Because of base line drift and noise in a recorder trace, allowance may have to be made for an increase in base line uncertainty AB as peaks broaden (I). If base line uncertainty increases with increasing peak width, the first or base line term in Equation 5 will assume greater importance, and the values of the second column in Table I will have even greater significance. For peaks of a given shape, base line noise will favor a larger value for r , but for peaks of given area the optimum
(4) W. E. Harris and H. W. Habgood, “Programmed Temperature Gas Chromatography,” John Wiley and Sons, New York, 1966, p 219. (5) D. L. Ball, Ph.D. thesis, 1967,University of Alberta, Edmonton. VOL. 40, NO. 7 , JUNE 1968
1 1 15
shape for minimum error will shift toward sharper peaks (1). This shift in turn suggests some decrease in optimum r. The over-all effect of increasing base line uncertainty will be to increase the error with which the peak area can be measured, but not to significantly change the fractional height at which the peak width is measured. The above analysis was derived for Gaussian peaks. As real peaks frequently show tailing, which increases the width near the base, a slightly larger value of r may be preferable to that predicted for this generalized case. Practical Choice of Fractional Height. It is clear that no single value of fractional height r will produce minimum relative error in area for all peak shapes and conditions. The choice of r for peaks of small area is more critical than for those of large area. Similarly, the choice of r for sharp peaks is more critical than for flat peaks. Nevertheless, from a practical point of view, a single fractional height (or a restricted number of fractional heights) at which to measure the width of all peaks in a chromatogram is desirable. The
choice of a single r necessarily involves a compromise to minimize the relative error in area for varying peak shapes. This compromise will depend in part upon the measurement conditions and analytical requirements of the individual laboratory, Considering Figure 1, however, we can see that if a full range of smooth Gaussian peaks is normally encountered, then a single value of r should be chosen near 0.25. Such a choice means a slight loss in precision for flat peaks, but a considerable gain for sharp peaks. For those types of analyses yielding smooth, sharp, well resolved peaks, r would be better chosen in the neighborhood of 0.1. Under conditions of serious peak tailing, r would be better chosen at perhaps a value as high as the popular fractional height of 0.5. RECEIVED for review January 15, 1968. Accepted February 29, 1968. Paper presented in part at the 153rd National Meeting of the American Chemical Society at Miami Beach, Fla., 1967. Financial support to D. L. B. by the National Research Council of Canada is gratefully acknowledged.
Application of Gas Chromatography to Qualitative and Quantitative Copolyamide Analysis Anthony Anton Carothers Research Laboratory, Textile Fibers Department, Experimental Station, E. I . du Pont de Nemours and Co., Inc., Wilmington, Del. A method is described for the gas chromatographic separation of the diacids (as the dimethyl esters) recovered from hydrolyzed copolyamides prepared from a single diamine-i.e., hexamethylenediaminewhich will give both qualitative and quantitative results. The method requires only 50.2-gram samples, and has a relative error in accuracy of 15%. The per cent 6 nylon in a copolyamide must be determined by difference, and with copolyamides made from more than one diamine, a calibration curve for each diamine must be prepared as well as for the diacids.
TECHNIQUES PREVIOUSLY DESCRIBED for the analysis of copolyamides are either time-consuming or give only qualitative or quantitative information, but not both. For example, paper chromatography, infrared spectrometry, and pyrolytic analysis cannot be used to quantitatively establish the concentration ratio in such copolymers as 66/610 or 66/610/612 nylon. Even if the components are known, a melting point-composition relationship is not reliable. In the 66/610 case for example, a melting point of 210 “C characterizes both a 60/40 and a l0/90 copolyamide composition (1). DTA, however, can be used to determine whether the sample is a block or random copolymer. The titration of diacids or diamines recovered by extraction or ion exchange from an acid hydrolyzate may produce quantitative data but a tedious separation and characterization is required to establish the nature of the copolyamide (2). The method described in this paper, which involves the gas chromatographic resolution of the polymer hydrolyzate, (1) W. E. Catlin, E. S. Czerwin, and R. H. Wiley, J . Polymer Sci., 2, 412-19 (1947). (2) Haslam and Willis, “Identification and Analysis of Plastics,” Chap. 4, Van Nostrand, New York, 1965.
1 1 16
ANALYTICAL CHEMISTRY
will give both qualitative and quantitative results. The liberated diacids in the hydrolyzate are esterified with BF3methanol and the diesters are recovered in diethyl ether, dried, gas chromatographed, and the retention time is measured to identify the corresponding diacid. A second hydrolyzate is made caustic, extracted with n-butanol which is then removed by atmospheric distillation, and the residue is gas chromatographed to identify the diamine. Assuming one diamine is present, the copolyamide cornposition can be determined by converting the peak height ratio of each diacid to the corresponding polymer weight. The method is applicable to 0.05- to 0.2-gram samples and has a relative error in accuracy of 10 %. Similar techniques for identifying carboxylic acids in plasticizers, in polyesters fibers, and in alkyd coating resins have been described (3-5). In these techniques, isophthalic, terephthalic, adipic, and sebacic acids have been recovered and converted to methyl esters from their products via transesterification with sodium methoxide and separated by gas chromatography. However, these techniques are not applicable to polyamides which are relatively resistant to aqueous caustic hydrolysis. EXPERIMENTAL
Apparatus. An F & M Model 1609 with a flame ionization detector was used for all gas chromatographic separations. Diacids were separated on a 6-ft, ‘i4-inch 0.d. X 3/16-in~h i.d. stainless steel column packed with 60/80-mesh acid-washed Chromosorb W (a product of Hewlett-Packard) (3) G. G. Esposito and M. H. Swann, ANAL.CHEM., 34,1048 (1962). (4) S . J. Jankowski and P. Garner, ibid., 37, 1709-11 (1965). (5) D. F.Percival, ibid., 35, 236 (1963).
