Selection of Optimum Ranges for Photometric Analysis Hubert L. Youmans* and Van H. Brown' Department of Chemistry, Western Carolina University, Cullowhee, N.C. 28723
Because a is essentially constant over a range of less than 1% T , AC can be defined as
Statistical photometric error was defined as r
-1
Ac = (log T, - log T,,) ab
When plotted against YO T, it proved an adequate basis for evaluating a spectrophotometer, comparing instruments, and determiningthe concentration range of an analytical method for minimizing random error effects. Photometric error Is a function of the number of replicates, decreasingas n increases; the shape of the error curve also changes with n; hence, the error curve for an analysis should be based on the degree of replication of the analyses. Other variables affect the error curve, making it specific for an instrument and the operating conditions.
Although Ringbom-Ayres ( 1 )plots have fallen into general disuse for determining optimum concentration ranges for photometric analysis, photometric or relative concentration error plots have not ( 2 , 3 ) .In the usual method of calculating these, assumption that the error in transmittance, AT, is constant employs the fallacy that led to abandonment of Ringbom-Ayres plots. Alternate analytical solutions are equally dubious or of limited use because of the variety and source of transmittance errors ( 4 ) .Photometric error measurements and curves should take into account the randomness of transmittance fluctuations. Photometric error is defined in theoretical discussions (4-6) as dC/C, equivalent to d In C. Sometimes d In C and d log C appear to be confused or used interchangeably. This is relatively unimportant, so long as the same definition is used in comparing instruments or data. Photometric error curves or equations do not represent constant errors that can be corrected for, but only likely maxima or averages that have some statistical probability. This leads to statistical methods for determining optimum operating ranges. Whitehead ( 7 ) , for example, derived three equations for fitting absorbance data to linear, quadratic, and cubic curves to be used as the basis of calculating photometric error. A general method of determining photometric error as easily as the dC/C vs. T plot seems desirable. McBryde (8)derived the expression dC _ dT
(0.434)2
T log T ( $ $ / a b )
but because of the approximations involved, the error does not differ sufficiently from functions obtainable when Beer's law holds exactly and A T is constant (9). For experimental evaluation, photometric error is defined (9) as AC/C, an incremental definition analogous to the differential d In C. Since A = -log T = abC -1
C = A/ab = -log T = log T(-llab) ab
(2)
Present address, Deering-MillikenCompany, La Grange, Ga. 1152
ANALYTICAL CHEMISTRY, VOL. 48, NO. 8 , JULY 1976
=1 log Tav
(3) ab T, where av and m stand for average and minimum readings of transmittance a t a fixed solution concentration. With a assumed constant across the range of fluctuations of T for a given concentration, even though a linear calibration curve may not be obtained for a photometric system, 1 log Tav C log Tav AC = =A/C T, A T, Dividing this by C and using A = -log T,, gives
AC 1 Tav Statistical photometric error = - = logC -logTaV T,
(4)
(5)
An estimated population standard deviation of T can be calculated from multiple measurements. This can be corrected for bias with Student's t ( 1 0 , l l ) .Then
By using the confidence limit T, instead of the upper transmittance limit, T h , a = 0.05 is retained for AC/C. At higher 0 and Its/ T,,, concentrations, however, when T a v T,, iT, becomes a negative quantity with an indeterminate logarithm. This is not important, though, because it occurs at transmittances beyond the range of interest. Crawford ( 4 ) states that ". . . the precision range of absorbancy may be obtained in infinitely many ways." The method of this paper seems a simple way when optimum operating range under experimental conditions is determined or when instruments are compared. In using it, however, the operator should remember that, as Crawford implies, he determines to some degree what photometric error includes. A simple example of this is that the operator may choose to include or exclude wavelength repetitive setting. +
EXPERIMENTAL Apparatus. One Beckman DU, one Beckman DB, and six Bausch & Lomb Spectronic 20 spectrophotometerswere used in this study. All data given in this report, however, were obtained with the same Spectronic 20. This instrument had no constant power supply, which increased point scatter on the photometric error curves. Reagents. Stock solutions of chromium nitrate, cobalt nitrate, and
potassium chromate were diluted to give solutions with percent transmittances in the range of about 0.5-98. Tris(o-phenanthroline)iron(II)solutions were prepared by the procedure commonly used to prepare calibration curves for colorimetric analysis of iron ( 1 2 ) . Solutions of copper-EDTA complex werede by diluting, with 0.050 M EDTA, a stock solution 0.020 M in copper sulfate and 0.050 M in EDTA. Stock solution and diluent had been adjusted to pH 7.0' with 10%NaOH. Distilled water was the solvent throughout. These solutions were chosen for study because they were known to be stable over long periods of time. Some gave linear A vs. C plots, others gave decreasing values of absorptivity with increasing concentration. Procedure. Except for KZCr-04 solutions at 426 nm, all transmittances were measured at maxima of the absorbance curves. Solutions minus the metal of the absorbing species were used to set 100% T .
Transmittances of solutions were measured 3 to 50 times, according to the purpose of the experiment.Although Hiskey (9) recommended
deviations of T over the range 0-100% T indicated that both Ringbom-Ayres plots and curves based on'Equation 9 can be grossly misleading. The derivative
Table I. Variations of 70 Tav and s with n for K,CrO, Solutions at 426 nm 250 ppm
9 8 PPm
n
% Tav
5 10 15 20 25 30 35 40 45 50
16.9 16.8 16.8 16.8 16.8 16.8 16.9 16.9 16.9 16.9
S
0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
% Tav
S
4.1 3.8 3.8 3.8 3.8 3.7 3.7 3.7 3.7 3.7
0.4 0.5 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4
T log T
AlnC=-
error.
%T,, and estimated population standard deviation,s , W F -e calculated for each solution by the forn ulas
s was divided by 6and multiplied by t , obtained from t tables (11). The degrees of freedom, f , equaled n - 1. A value of 0.05 was selected for a;it is the probability level most commonly used in analytical
chemistry. From T,, and td&, statistical photometric error was calculated using Statistical photometric error = _ _ _1l o g %Tav (8) -logTav %T, Values of photometric error were plotted vs. %T,, to obtain photometric error curves. Calculations were done with an IBM System 360/30 computer and a Monroe 1655 programmable calculator.
RESULTS AND DISCUSSION This work was initiated to compare statistically derived photometric error curves with Ringbom-Ayres plots and photometric error curves based on 0.434 A T T log T
0.434
dT
(10)
could be extended by curve fitting to systems that do not follow Beer's law, as we have done using Meites brute force (14) and matrix methods, and the general equation
that when determining AT each measurement include the operations of emptying, refilling,and repositioningthe cuvette, error associated with emptying and refilling appears to be operator error. Therefore, once the blank and sample cuvettes had been adequately rinsed and filled, sample transmittances were measured by setting 6% T with the beam block and 100%T with the blank, then inserting the sample to obtain % T. The cycle was repeated with the same sample and blank solutions for the desired number of replications. Once the wavelength was set, it was not changed while transmittances of a complete set of solutions were obtained. This eliminated a possible source of operator
AC/C =
dln C -=-
dT
AT
could be used for determining photometric error. A In C could be determined as a measure of the variability of T and the slope of the In C vs. T curve, but fitting nonlinear curves to mathematical equations took much computer time and the residual error often was quite large. Besides, the problem of measuring the independent variable remained. There thus does not appear to be a simple analytical relationship between T and AT that applies to practical spectrophotometers. Equation 8, however, does not depend upon a relationship between T and AT. Because it also combines simplicity with mathematical correctness for the chosen instrumental conditions, it is a flexible and useful device for studying random variables of spectrophotometry as functions of analyte concentration. Estimates of Population Parameters. Photometric error is a measure of how far experimental T can deviate from its true value. If the error is normally or uniformly distributed, the population mean and standard deviation are sufficient to describe the variability of T . Mean and standard deviation are never known, since n is potentially infinite. But very good estimate's of both are available a t the value of n when %Ta, and s become essentially invariant. Table I shows this occurs when n E 35. The data agree with Freund's (15)statement that Student's t and normal distributions are practically equivalent when the degrees of freedom exceed 30. A necessary condition of this is that T,, and s be constant, then =
ts zs T,, f -= T,, f -
6
6
for confidence limits based on the two distributions. When values of %Tavand s are good approximations of population parameters over a range of concentrations, values of percent average deviation
(9)
with A T constant. After curve smoothing, there was some agreement of predicted working ranges. The spectrophotometers used were of the type that supposedly ( 2 , 1 3 )gives a constant AT. But the considerable inconstancy of standard
d In C
a=
5[%Ti- %Tad
i= 1
(13)
n
also should be constant. This is because the variability of
!%Ti- %Tadhas a grosser effect on s than on a. Table I1 shows that neither s, I tslv'nl, nor d is constant over a 0-100% range
Table 11. Variation of s, I t s / f i I and awith % T,, K,CrO, at 370 nm % Tav s
3 -Its/rnl
0.9 0.3 0.2 0.1
1.9 0.4 0.3 0.1
3.5 0.4 0.3
0.1
13.9 0.4 0.3 0.1
22.3 0.4 0.4 0.2
41.4 0.7 0.6 0.2
48.5 0.9 0.7 0.3
61.6 0.6 0.4 0.2
75.4 1.0 0.8 0.3
88.4 0.8 0.6 0.2
92.6 0.8 0.6 0.2
96.6 0.8 0.6 0.3
Cu-EDTA complex at 710 n:n % T,, s
a
Its/Ji?l
4.2 0.1 0.1 0.03
7.1 0.4 0.4 0.1
12.2 0.2 0.1
0.1
14.7 0.2 0.2 0.1
25.8 0.4 0.2 0.1
34.8 0.4 0.3 0.1
54.0 0.4 0.3 0.1
59.4 0.6 0.5 0.2
73.9 0.8 0.6 0.3
86.1 0.7 0.6 0.2
89.1 0.7 0.5 0.2
94.8 0.9 0.8 0.3
97.3 1.0 0.6 0.3
ANALYTICAL CHEMISTRY, VOL. 48, NO. 8, JULY 1976
1153
Table 111. Variation of l t / f i I with n at a = 0.05
%T
Figure 1. Photometric error curves for K2Cr04 in distilled water mea-
sured at 370 nm ( 0 ) n = 35, ( x ) n = 20
X
I
d
”
/
I %T
Figure 2. Photometric error curve for CU-EDTA complex in 0.050 M
EDTA solution at pH 7.0, measured at 710 nm with n = 3
I /
i
6L
I 4 P
1
0
0
4
IO
”
,
20
“
I
30
0
I
40
50
60
70
80
90
100
%T
Figure 3. Photometric error curve for CU-EDTA complex in 0.050 M
EDTA solution at pH 7.0,measured at 710 nm with n = 35
of T . Each point was replicated thirty-five times. Only one significant figure is justified for s and I tsldnl. I ts/dnl and d also do not give values over the range of Table I1 proportional to T or 6 T . These variations were considered in the theoretical paper of Crawford ( 4 ) ,along with AT as a constant. Effect of Replication. Maximum possible error and optimum working range depend upon the number of replicates. Dependence of error size is apparent in the definition AT = ts/&. Dependence of error and operating range on n is seen in Figure 1and Figures 2 and 3. 1154
ANALYTICAL CHEMISTRY, VOL. 48, NO. 8, JULY 1976
n
ItlJirI
20 30 35 50 60
0.468 0.373 0.343 0.284 0.258
n
70 80 90 100 120
It/&l
0.238 0.222 0.209 0.198 0.180
Photometric error is a function of several variables, including n. Instruments can be compared a t constant n , but no instrument can be said to have a single photometric error curve because each curve varies with n and the wavelength a t which transmittances are measured (Figures 1,2, and 3). Figure 1 shows that as n increases, photometric error decreases and the apparent operating range increases. With large numbers of replicates I t / 6 1 decreases more slowly as n increases (Table 111),but eventually it would approach zero. At high values of n, however, photometric error would change very slowly with increasing n. n = 100 might be arbitrarily chosen for comparing instruments at a stated wavelength. With AutoAnalyzers, this amount of data could be readily collected; manually collecting each value of T , however, required about one minute in this study. n = 35 seems more reasonable than n = 100, although n = 20 was satisfactory for much of our work. T,, and s are invariant a t n = 35 and the required number of transmittance measurements for a curve can be manually collected in one day. For determining operating ranges for routine analyses, however, n should be of the order of replication used in the analysis. Confidence limits of photometric error curve points and the analyses would then be of the same or similar populations. Such curves as Figures 2 and 3 show that the predicted optimum operating range is much narrower with n = 3 than with n = 35. Scatter is also greater, requiring more points for a useful curve. As expected, photometric error is greater with n = 3. And although the curve indicates that optimum range extends below 10% T , transmittance in this region can be read to only two significant figures, unless scale expansion or a high resolution readout device is used (16).Many instruments thus are unsuitable for precise analyses a t high concentrations. Sources of Photometric E r r o r . As previously indicated, randomness of AT derives from many sources. Ingle and Crouch ( 1 6 ) and Rothman, Crouch, and Ingle ( 1 7 )have extensively studied signal-to-noise.(S/N) ratio in spectrophotometry as a means of evaluating precision of measurements. Their work is especially useful in showing why a photometric error curve changes as instrumental parameters are changed. Among the more important instrumental parameters that determine photometric error are photocathodic current, noise bandwidth, readout variance, flicker factor, photomultiplier gain, secondary emission factor, equivalent cathodic dark current, and average gain of photomultiplier tube (16). Adjustability of these parameters varies with different instruments and determines optimal operating conditions for a particular instrument. Since S/N ratio determines precision, and thus photometric error, a large number of photometric error curves apply to any spectrophotometer. Photocathodic current varies over several orders of magnitude as wavelength and slit width are changed. As a result, as many variables as instrumentally possible should be equivalent when comparing spectrophotometers. Another conclusion from S/N theory is that when optimizing a n operating range for a n analytical method, the photocathodic current should be maximized within resolution, linear absorptivity and photodecomposition limits. Photometric error applies to a single instrument under definite operating conditions. Reduction of photometric error
inherent to the instrument should utilize S/N theory. Many factors exterior to the instrument, however, can influence a photometric error curve. For instance, most of the dispersion of the curves of Figures 1-3 is attributable to fluctuations of temperature and line voltages. The statistical photometric error function thus evaluates a system. The function may be limited to combined use with S/N theory to evaluate an instrument, compare several, or establish a n optimum concentration range for an analytical method. Or the system could be extended to include noninstrumental sources of random errors in spectrophotometric analysis. Also, decline in the performance of a n instrument might show up better on a photometric error curve as new points were added than on the usual quality control chart.
LITERATURE CITED (1) G. H. Ayres, Anal. Chem., 21, 652 (1949). (2) H. H. Willard, L. L. Merritt, Jr., and J. A. Dean, “Instrumental Methods of Analysis”, 5th ed., Van Nostrand, New York, 1974,pp 92-97.
(3)R. A. Day, Jr., and A. L. Underwood, “Quantitative Analysis”, 3rd ed.. Prentice-Hall, Englewood Cliffs, N.J., 1974,pp 325-328. (4)C. M. Crawford, Anal. Chem., 31, 343 (1959). (5) H. Von Halban and J. Eisenbrand, J. Proc. Roy. Soc. (London), 116, 643 (1927). (6)F. Twyman and G. F . Lothian, Roc. Phys. Soc. (London),116, 643 (1933). (7)D. Whitehead, Talanta, 20, 193 (1973). (8)W. A. E. McBryde, Anal. Chem:, 24, 1639 (1952). (9)C. F. Hiskey. Anal. Chem., 21, 1440 (1949). (IO)H. L. Youmans, “Statistics for Chemistry”, Charles E. Merrill Publishing Company, Columbus, Ohio, 1973,pp 24-27. (11)G. W. Snedecor and W. G. Cochran, “Statistical Methods”, 6th ed.. Iowa State University Press, Ames, Iowa, 1967,pp 59-62, 549. (12)R. A. Day, Jr., and A. L. Underwood, “Quantitative Analysis Laboratory Manual”, 3rd ed., Prentice-Hall, Englewood Cliffs, N.J.. 1974,p 128. (13)L. Cahn, J. Opt. Soc. Am., 45, 953 (1955). (14)T. Meitesand L. Meites. Talanta, 19, 1131 (1972). (15)J. E. Freund. “Mathematical Statistics”, Prentice-Hall. Englewood Cliffs, N.J., 1962,p 202. (16)J. D. Ingle, Jr., and S. R. Crouch, Anal. Chem., 44, 1375 (1972). (17)L. D. Rothrnan, S . R. Crouch, and J. D. Ingle, Jr., Anal. Chem., 47, 1226 (1975).
RECEIVEDfor review December 12, 1975. Accepted March 24, 1976.
Bulk Analyzer for Grade Determination of Iron Ore Fines by Backscatter of Cobalt-60 Gamma Radiation Ralph J. Holmes CSIRO Division of Mineral Physics, P.O. Box 124, Pori Melbourne, Victoria 3207, Australia
An accuracy of f0.7% Fe (95 % confidence intervals) can be achieved for bulk samples of normal production ore dried to less than 1% free moisture. Apart from a preliminary crushing of lumpy ores to -6-mm particle size, no elaborate sample preparation is required. Analysis time is 10-20 min or less, depending on the size of the scintillation detector crystal used. Provided Fe ore grade is all that is sought, the technique is competitive with routine wet chemical analysis, which is subject to large sample preparation errors.
Backscattered y radiation reaching a scintillation detector from a medium is a function both of the composition of the scattering medium and its bulk density. Numerous attempts to measure ore grade and ash content of coals by means of the so-called selective gamma-gamma method have been reported in which only the low energy scattered radiation carrying information on chemical composition is monitored (1-6). T o improve on this technique and overcome the inevitable effect of changing density, Blymentsev and Migunov (7) proposed the measurement of a parameter P (hereafter referred to as P,) which gave a linear relationship when plotted against the grade of an iron ore. P, is defined by the ratio
P, =
Intensity of scattered y rays in the high energy region of the spectrum Intensity of scattered y rays in the low energy region of the spectrum
(1)
The spectral energy regions defined in the above ratio are separated by a n energy threshold E,. Czubek (8, 9) studied the P , function theoretically and demonstrated its dependence on Zeq,the “equivalent atomic number” of the medium, its dependence on E , and its essential independence of changes in bulk density. The “equivalent or effective atomic number” of a medium characterizes the “average” chemical
composition of the medium, and, in the special case of high grade iron ore where a heavy element (Fe) is dominant, Z, and consequently P , are directly correlated with the Fe concentration. I t should be noted, however, that in this case the method can only be employed to measure Fe. Impurities in iron ore such as silica and alumina cannot be determined by this technique. Some knowledge of the other ore constituents is also required before the technique can be applied to a particular iron ore. For example, an ore which is high in manganese (or other elements with high atomic number) is not suitable since Mn will report approximately as Fe. However, this is not the case with the Western Australian Pilbara iron ores considered here. Charbucinski, Eider, Mathew, and Wylie (10)studied the P , factor in detail in relation to iron ores, and applied it to the grade determination of Pilbara iron ores in blast holes and in exploration holes. Using this technique, the m e a n ore grade in blast holes could be determined with 95% confidence intervals of f0.8% Fe, Le., considerably better than by most current mining techniques based on sampling and chemical assay. The chemical error, it should be noted, is considerably diminished when assays are averaged over a large number of drill holes. The present work involves the application of the y backscatter technique to the dry basis grade determination of bulk samples (25-30 kg) of iron ore. Samples from various locations in a crushing plant are brought to a central laboratory, given a preliminary crushing if necessary to reduce the particle size to -6 mm, and then analyzed using the P , technique. The grade of individual samples can be measured in this way to an accuracy of f0.7% Fe (95% confidence intervals) with savings in sample preparation (since it is largely avoided) and analysis time. EXPERIMENTAL Sensor Assembly. The source, detector, and sapple configuration for the P, backscatter measurement was as shown in Figure 1.In the
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