ANALYTICAL CHEMISTRY, VOL. 51, NO. 8, JULY 1979
matrix effects could also be minimized by "limited viewing". Le., observing the zone immediately above the surface of a filament-type atomizer. Here again, for reaction 1 and K l , thermodynamics would predict minimal change in the relative amount of free metal with respect to the system which does not employ limited viewing. since the ratio of the partial pressures should be nearly constant a t all heights, assuming that the diffusion rates of metal and interferent are not significantly different. Using reaction 2 and the respective expression for K 2 ,the expected increase in the partial pressure of M and C immediately off the rod would suggest a decreuse in the relative concentration of M. Conversely kinetic control of these processes would qualitatively account for the observed reduction in the interference effects in ail cases cited above. For facile reactions, atomizer designs which permit long residence times or samples with relatively high concentrations of interferents, thermodynamic data and the assumption of equilibrium control should be useful in predicting the extent of gas phase interactions. For example, this has been used successfully in explaining several cases where a signal entzuncement w a observed as a result of adding an "interferent" to the sample ( 1 3 ) . On the other hand, it may be necessary to employ gas phase kinetics in order to completely understand reactions which are slow andior are being observed a t low concentrations.
LITERATURE CITED ( 1 ) R. E. Sturgeon, C. L. Chakrabarti. and C. H. Langford, Anal. Chem., 48, 1792 (1976).
1209
S. L. Paveri-Fontana. G. Torsi, and G. Tessari, Anal. Chem., 46, 1032 ( 19 74). G. Torsi and G. Tessari, Anal. Chem., 47, 839 (1975). G. Torsi and G. Tessari, Anal. Chem., 47, 842 (1975). D. J. Johnson, B. L. Sharp, and T. S.West, Anal. C > k m .47, , 1235 (1975). G. Torsi and G. Tessari, Anal. Chem., 48, 1318 (1976). W. C. Gardiner, Jr.. "Rates and Mechanisms of Chemical Reactions", W. A. Benjamin, New t'ork, 1969. A. Fontign, W. Felder, and J. Houghton, "XIV Symposium (International) on Combustion", Pittsburgh, Pa., 1975, p 775. W. Felder and A. Fontijn, J . Chem. Phys. W. Felder, R. Gould, and A. Fontijn, J . Chem. Phys., 685 (1977). A. Fontijn, S. Kurzius, and S.Houghton, "XV Symposium (International) on Combustion", Pittsburgh, Pa., 1973, p 167. J. F. Alder and T. S. West, Anal. Chim. Acta. 51, 365 (1970). R. H. Eklund and J. A. Holcombe, Anal. Chim. Acta, in press. S. G. Salmon and J. A. Holcombe, Anal. Chem.. 51, 648 (1970). K. H. Stern, J . Phys. Chem. Ref. Data, 1, 747 (1972). G. S. Vizard and A. Wynne, Chem. lnd. 196 (Feb. 7, 1959). M. P. Bratzel, Jr., and C. L. Chakrabarti. Anal. Chim. Acta, 63,1 (1973). R. E. Honig and D. A. Kramer, "Vapor Pressure Curves of the Elements", Radio Corp. of America, Princeton, N.J.. 1969. R . E. Sturgeon and C. L. Chakrabarti, Specfrochim. Acta, Part B , 32, 231 (1977). D. Alger, R. Anderson, I . Maines, and T. West, Anal. Chim. A c t a 57, 271 (1971). J. Aggett and T. West, Anal. Chim. Acta. 55, 349 (1971). R. Anderson. H. Johnson, andT. West, Anal. Chim. Acta, 57, 281 (1971).
RECEILXII for review August 14, 1978. Accepted April 2 , 1979. This material is based upon the work supported by the National Science Foundation under Grant No. CHE78-15438. One of us (J.E.S.)wcluld like to express our appreciation to the National Science Foundation for financial assistance through the NSF-URP program
Theoretical Assessment of Accuracy in Dual Wavelength Spectrophotometric Measurement K. L. Ratzlaff" Department of Chemistry, Northern Illinois University, DeKalb, Illinois 60 1 15
D. F. S. Natusch" Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523
Equations are presented describing the contributions of absorbing and scattering interferents, cell errors, stray light, optical bandpass, multiple scattering, and multiple internal reflections to the accuracy of dual wavelength spectrophotometric (DWS) measurement. It is established that cell errors are substantially reduced in DWS measurement with respect to single wavelength spectrophotometric measurement (SWS)? but the other sources of error may be significant. A scattering interferent produces error that cannot be reduced instrumentally. Otherwise, error may be diminished by employing secondary wavelength isolation, narrow slit widths, and an intense source.
Modern ratiometric spectrophotometry, as performed using instruments commonly referred to as "double beam spectrometers" may be carried out utilizing two distinct modes of operation. T h e mcire common of these is emplo:,-ed in conventional Single Wa\ elength Spectrophotometry (SWS) in which the log ratio of light intensities passing through 0003-2700/79/035 1-1209$01 .OO/O
separate sample and referenw cells is determined and presented in terms of the analyte absorbance a t a single wavelength. A fundamentally different principle is. however, employed in Dual b'avelength Spectrometry (DLVS) in which two beams of different wavelengths pass through a siiigle sample cell. In this case. the difference between the sample absorbances a t the two wavelengths is recorded. The main operational advantage of DM S is its ability to distinguish the absorbance of an analyte in the presence of severe spectral interferences such as are produced by additional absorbing or scattering species in the sample. In such circumstances, which are frequently encountered in samples of biological origin, the use of SLVS is prohibited so that quantitative analyses can be performed only in the DWS mode. It is appropriate. therefore, to establish the precision and accuracy associated with D\VS measurement. Previous studies ( 1 ) have shown that under certain conditions the precision of ULVS meascirernent may be superior to that of SLVS measurement m e n in the absence of interferents. Acceptable precision can also be achieved over a limited absorbance range (0.1 t o 2.5) when interterents 1979 American Chemical Society
1210
ANALYTICAL CHEMISTRY, VOL. 51, NO. 8, JULY 1979
contributing as much as 2.0 absorbance units are present. However, no similar assessment of the accuracy of true ratiometric DWS measurement has been made. It is the object of this paper, therefore, to present equations which describe the extent to which all recognized sources of error contribute to the overall accuracy of DWS measurement. T h e results are then used to establish instrumental design parameters and operating conditions which should be employed to achieve maximum accuracy.
BASIS O F DWS MEASUREMENT T h e primary application of DWS is to chemical systems in which an absorbing or scattering interferent is present. The ohserced absorbances Ai’, A2‘ at the two wavelengths XI,h2 are then related to the true absorbances A I , A , and the scattering, D,, D2, by = Ai AS’ = A z AI’
+ Di + D2
(1)
Normally. the wavelengths XI, h2 are chosen such that D,= in which case the net absorbance difference 1A (= A i A,) is independent of scattering effects. Furthermore, from Beer’s law AA = ‘4,- A2 =
(6, -
eJbC, = (1 - $)t,bC,
(2)
where el, e?, b, and C, are molar absorptivities, path length, and analyte concentration, respectively. 3 is the absorptivity ratio, t 2 / t l ( I ) . T h e absorbance difference, 1 A , between the two waveIo2)and lengths A, and h2, is also related to the incident (Ioi, transmitted (Il,Z2) light intensities by the expression LA = -log ( I l / I J
+ log K
(3)
where K = Iol/Ioz. K is t,hus a constant for a particular pair of wavelengths. Its magnitude is determined by the difference between the incident light intensities at each wavelength which are due to differences in incident, light source output. instrumental light transmission, or detector response a t the wavelengths. In most conventional DU’S spectrotneters, K would be expected to be close to unity ( I ) . Since several of the errors considered in the following sections are contributed by optical characteristics of a DWS spectrometer, it is convenient to express 1.4 by a single function which explicitly contains values of light intensity, analyte absorbance, and interferent optical density at the two wavelengths. Such a n expression can be obtained by combining Equations 1-3 arid is given ( I ) by
CALCULATION OF ERRORS Equations describing the percentage error cmtributed to DWS measurement by each source of inaccuracy are presented in the following sections. Some of these are Ptraightforward extensions of equations already developed to describe the accuracy of SWS measurement. In such cases, the deriuations differ only in the fact that in DWS the analyte attenuates the light intensity a t two wavelengths (A1 and A), whereas in SiYS only the sample beam is attenuated by the analyte. It should be stressed, however. that although only simple changes in the equations may result, their influence on the accuracy of measurement may be very significant. Previous papers (&4) dealing with the accuracq of DWS measurement have considered only the error produced when the quantity I , - 1% was computed as an approximation of the log ratio. -log ( I , / 1 7 ) . Since this error resulted from the limitations of early analog signal processing circuitry. it is nul considered herein.
The general instrumental system considered has been defined in detail in a previous publication ( I ) . It is important to reiterate, however, t h a t the following discussion encompasses errors associated with differential UV-visible-IR spectrophotometers as well as with instruments designated as dual wavelength, double wave!ength, or two wavelength spectrophotometers. T h e sources of error considered can be conveniently divided into two categories. viz., those due to chemical nr physical interferents in the sample and those due to instrumental nonidealities. These latter include errors related to sample cell quality and positioning, stray light and signal offset, excessive bandpass, and nununiformity of path length due to multiple scattering or multiple internal reflection. In each case the percent error, %A, is defined by
1 A ’ - AA Rd = ---x 100% AA
13)
where LA and 1 A ’are, respectively, the true absorbance difference (as defined by Equation 2) and the observed absorbance difference (as measured according to Equation 1). Sample Interferences. Chemical I n t r r f e r e n t s . As pointed out previously, the primary advantage of DWS arises from its nature as a difference measurement. Thus. in the presence of a chemical interferent whose absorption spectrum overlaps that of the analyte, judicious choice of the wavelengths XI and X 2 can yield exact cancellation of the optical densities (absorbances) D 1 , D , (Equation 1). Where such cancellation is achieved, the interferent will contribute no error to 1A. A number of quantitative analyses based on this premise are reported in the literature (6512). Where cancellation is not exact, however. the percent error %/Cc, is given by
where -, = D,/D,, which is analogous t o the definition of B. It i s apparent from Equation 6 that, even in the case of relatively poor cancellation, the DWS error will be very much less than that associated with SWS (= D/,4 X 100%). It should, however: be noted that the constraints on the choice of ,\! and h2 required to achieve maximum discriiiiination against a chemical interferent may result in poor DWS precision in cases where A h is large (I). Physical Znterferents. Dual wavelength spectrophotometry is frequently employed to reject error contributed by an interl’erent which scatters light. Turbid solutions and T I L plates represent extreme examples. In such cases. it is usually assumed that the turbidity is the same a t both wavelengths. T h e resulting optical density of wavelength hl is given by D1
= -log
( ~ ~ / = ~ g 7 1 ~6 c)~
(7)
where ct is the molar Concentration of the light scattering material and T~ is its associated turbidity constant (13). The constant, T , is, however: a function of wavelength ( 1 4 ) according to the relation _ I - kX‘ (8) where I? is a function of the size distribution and refractive index of the scattering particles arid t is a function whose value decreases with increasing particle size and depends also on particle shape (15). For particles of diameter less than h/’20, Rayleigh jcattering prevails and t = 4 ; for larger particles t is normally less than 1 although cases where t > 4 have been reported ( 1 6 ) . Irrespective of the actual values of k and 1 , however, the turbidity constant. T , monotonically decreases as h increases so that the optical densities UI and DI, can never be identical
ANALYTICAL CHEMISTRY, VOL 51, NO. 8, JULY 1979
-_______
-~
by employing small wavelength differences and operating a t long wavelengths where all else is equal. Consideration of the errors presented in Table I1 suggests that accurate DWS measurements are limited to situations where values of LA are large, and where D , is small. Indeed, measurement of 1 A values less than 0.1 absorbance unit is essentially prohibited in all but ideal circumstances of small wavelength differences, long wavelength absorption, and very low scattering. Even measurements in the range 1 A = 0.1 t o 1.0 absorbance unit require the use of small wavelength differences for achievement of acceptable accuracy. Perhaps most noteworthy is the fact that for highly scattering systems such as TLC plates where D1 > 1 the inaccuracy of DWS measurement is so great as to invalidate absolute results. Thus, while DWS can usefully be employed to cancel out the absorbance of chemical interferents present in a sample, its utility for discriminating against material which scatters light is much more limited. Instrumental Nonidealities. Cell Errors. Both the positioning and quality of sample cells can give rise to systematic errors in spectrophotometric measurement. Three sources of error can be distinguished. (i) Cell Transmittance. For accurate SWS measurement the optical transmission characteristics of the sample and reference cells must be closely matched; otherwise a net absorbance results (18). However, since both light beams in DWS follow the same light path through a single sample cell, transmittance imperfections are unimportant to the extent that they are independent of wavelength. (ii) Refraction. In SWS, refraction of light may result in the transmitted beams from sample and reference cells striking different areas of the photocathode surface. If these areas have different quantum efficiencies, a systematic error will occur (18,19). In DWS, however, both beams follow the same light path so that error will occur only if the photocathode quantum efficiency is significantly different for the two wavelengths employed; this difference is unlikely to be significant under normal operational conditions. (iii) Cell Rotation. If the sample cell is rotated about its axis so that the incident light beam strikes its front face at an angle a / 2 f 0 then the effective sample path length, b', is increased such that
Table I. Fraction of Scattering Not Cancelled in DWS Measurement (1 - 7 ) Ah A,
t
300 400 500 600 700 300 400 500 600 700 300 400 500 600 700 300 400 500 600 700
1.0 1.0 1.0
1.0 1.0 2.0 2.0 2.0 2.0 2.0 3.0 3.0 3.0 3.0 3.0 4.0 4.0 4.0 4.0 4.0
2 0.007 0.005 0.004 0.003 0.003
5 0.016 0.012 0.010 0.008 0.007 0.033 0.025 0.020
0.013 0.010 0.008
0.007 0.006
0.016
0.014
0.020 0.015 0.012
0.048 0.037 0.029 0.010 0.025 0.009 0.021 0.026 0.064 0.020 0.048 0.016 0.039 0.013 0.032 0.011 0.028
10 0.03 2 0.024 0.020 0.016 0.014 0.063 0.048 0.039 0.033 0.028 0.094 0.071 0.058 0.048 0.042 0.123 0.094 0.076 0.064 0.055
20 0.06 2 0.048 0.03 8 0.032 0.028 0.121 0.093 0.075 0.063 0.055 0.176 0.136
50 0.143
0.111 0.091 0.077 0.067 0.265 0.210 0.174 0.148 0.129 0.370 0.298 0.249 0.213 0.187 0.460 0.376 0.317 0.273 0.241
0.111
0.094 0.081 0.228 0.177 0.145 0.123 0.107
1211
a t two different wavelengths. Consequently, there is always an absolute error, 6,, equal to the difference (Dl - D J whenever light scattering material is present. T h e magnitude of this error can be evaluated by combining Equations 7 and 8 to yield
D , = kXl-tbc, and
D., = kX,-lbc, and taking their difference such that
It can also be seen that > reduces to (X1/X2)' and 6,/D, = (1 7). It is apparent from Equation 9 that 6 , may have a variety of values depending on the values of XI, A?, t , and D1.In this regard it is useful t o recognize that the quantity (1 - -0 represents the fraction of the optical density D , , which is uncompensated between the two wavelengths. Values of this quantity are presented in Table I for several values of wavelength, wavelength difference, and t likely to be encountered in practice. The choice o f t values is based on the results of measurements, performed in our laboratory, which showed that t varied from 1.6 to 3.5 for scattering interferents such as colloidal silica, sonicated a-L-lecithin, milk, soap, and TLC plates (17). T h e data in Table I show that cancellation of error due to turbidity is most effectively achieved in DWS ~
b ' = b/cos 0 The observed DWS absorbance from Equation 2 thus becomes
L A ' = 1 A sec 0 and gives rise to an error, %&, given by
706,= (sec 0 - 1) X 100% For 0 = 1,3, and 5 degrees, respectively, %6* is 0.015%, 0.14%, and 0.38%. It is apparent, therefore, that path length error is likely to be of little consequence. StraS Light a n d Signal Offset Error. The sources and error contributions of stray light have been considered by several
Table 11. Percent Relative Error for Selected Values of A A , D , , and (1 - y)
(1 -
- ~ _ _ _ _ _ _ _
0.01
0.005 5.0
0.05 0.10
AA
0.20 0.50 1.00 1.50 2.00
D , = 0.1 -____0.010 0.050 0.100
10.0
50.0
100.0
1.0
2.0
10.0
20.0
0.5 0.25 0.10 0.05 0.03 0.025
1.0
5.0 2.5
10.0
0.5 0.2 0.1 0.07
0.05
1.0 0.5
0.33 0.25
5.0 2.0 1.0
0.67 0.5
7 )
______
D , = 0.5
-~
~~
0.005 25.0 5.0 2.5 1.25
0.010 50.0 10.0
5.0 2.5
0.5
1.0
0.25 0.17 0.125
0.5 0.33 0.25
0.050 250.0 50.0 25.0 12.5 5.0 2.5 1.7
1.25
0.100
500.0 100.0 50.0 25.0 10.0 5.0 3.3 2.5
D , = 1.0
____
0.005 50.0
100.0
0.050 500.0
10.0
20.0
100.0
1000.0 200.0
10.0
50.0 25.0
100.0 50.0
10.0
20.0
5.0 3.3 2.5
10.0
5.0 2.5 1.0 0.5 0.33 0.25
0.010
5.0 2.0 1.0
0.67 0.50
0.100
6.7 5.0
1212
ANALYTICAL CHEMISTRY, VOL. 51,
NO. 8,
JULY 1979
authors for SWS measurement (18,20-26). The effect of any signal offset due to amplifier offset or to any dark current which is uncompensated is effectively the same. In all cases a quantity of radiation may be considered to be added to each intensity measurement. (In the case of an electronic signal offset, however, this quantity may in fact be negative.) In order to consider the effect of stray light, the observed DWS absorbance difference, hi’is,most usefully expressed, by combining Equations 1-4, in the form
Stray light can be included simply by adding fractional intensities al,a2 to the incident beams whose nominal wavelengths are X,I X2. The stray light will then be attenuated by the sample in the same way as other incident radiation even though the stray light wavelengths may differ considerably from X1, As. I t is appropriate, therefore, to generalize the wavelength dependences of the observed analyte absorbance and the scattering or absorbance due to the interferent by writing A j a ( X ) and DJd(A). T h e factors f a ( A ) and f d ( X ) thus take account of the effective absorbance or scattering of light a t each wavelength. Equation 10 then becomes
Absorbance at Wavelengtr 1
I
,
r‘/
AAs’ =
-lor[
(
10-(Alfa(Xd + DlfdOi) +
a1
10-(Adala(Xs) + Dild(XJ)
+ Dlfd(Az) +
N2
10-(Ade.(XJ + DLfd(Xs))
10-(Alfa(Xz)
,
I
ccc
j.,;
where 24; is the DWS absorbance difference in the presence of stray light and the subscripts 1, 2, s refer to the nominal incident beam wavelengths XI, As, and the wavelengths of the stray light, respectively. Several simplifying assumptions can now be made. First, it is reasonable to assume that al 0 and y = 1,
A,
c2s -253
t
occ
I 8353
,
1
I
,
I
100
153
233
i60
303
Absorbance at Wavelength 1
Figure 2. Error due to optical bandpass. D , = 0.0; shown for various values of p (Equation 21). (A) Theoretical calibration curve for the bandpass t o peak-width ratio, s / w of 0.4. Curve R represents zero error. (8)Percent relative error where s/w = 0.4. (C) Percent relative error where s I w = 0.04
and t h a t the dependence of these errors on absorbance is strongly influenced by the value of 6. In order to illustrate the effect of the value of s / w , the error is also shown for S / U = 0.04 DWS measurement. Path Length Errors. A measurement error results if the radiation transmitted by the sample is the sum of light rays which travel different lengths through the sample. At least three examples appear in the literature: viz. (1) a parallel beam passing through a cylindrical cell ( I S 2 3 ) ;( 2 ) light being repeatedly reflected off cell faces and other optics so t h a t a fraction passes through the cell more than once (18,23,33);
-
m;
p > 0;y = 1;SA,’
-+
(1 -@)A,
(25)
and the error tends to zero. Of the cases listed above, two are pertinent to common instrumentation and are considered in detail as follows. (i) Multiple Scattering by Turbid Solutions. Light entering a turbid solution will be scattered by the suspended particles. Multiple scattering of the light increases the path length travelled by the beam, and consequently the probability for absorption of a photon is increased. The probability that a photon will travel path length b is given by f ( b ) . Where the solution is sufficiently turbid to be diffusely illuminated, the classical Kubelka-Munk equations (35, 36) may be used to compute the transmittance; because these equations are quite unwieldy, several attempts at writing simpler approximations have been made (37-39). However, these equations are not applicable to solutions which are not sufficiently turbid to be diffusely illuminated. For the case at hand, an expression for f ( b ) will not be derived; rather, a functional form of f ( b ) is chosen subject to the two following constraints: (a) the function must be continuous between the cell length, bo. and infinity; (b) the function must vanish a t infinity. The expression f ( b ) = b-4 was chosen where y is greater than zero and could itself be a function of D,. As long as the two constraints are observed, the actual choice of f ( b ) is not critical to this discussion; the
ANALYTICAL CHEMISTRY, VOL. 51, NO. 8, JULY 1979
1215
. . . . . . . . - - - .
:SC' :^c
2
,
'e\
06
~, I \
:5c
133
23-8
153
I
I
25,:
203
Absorbarce at $gavelength 1
Figure 4. Error due to multiple internal reflection. Percent relative error where the fraction reflected, f , is 0.04 and D , = 0.0; shown for various values of p
of attention for SWS measurement (18, 23, 33). For DWS, Equation 22, rewritten to give the apparent absorbance in the presence of these multiple internal reflections, AAr', becomes ( 1 - fl)lO-(A1+D1) + f l f 2 10-3(A1+D1)
AAr' = -log ( 1 - fl)1o-i&l+?D11
+ f1f2
(27)
10-3(PA1+.7Dd
The percent error, %6,, reduces to
706,= -((1 - /3)A
\." , .- -_
, "
=.
,
,dc
5:
,
I
2ci
25,:
3'1
Absorbance at Waveleigth 1
Figure 3. Error due to multiple scattering. D , = 0.0, y = 1.0; shown for various values of 13 (Equation 26). (A) Theoretical calibration curve where curve R represents zero error. (B) Percent relative error
same general predictions are obtained irrespective of the definition. Substituting b 9 for f ( b ) in Equation 22,
rmb-iiO~ib,bO)iA1+Dl)db
Equation 26 is plotted in Figure 3A for 4 = 1.0; curve R represents measurement with no error. For each of the curves drawn from Equation 26, the slope is greater than that of R as A , approaches zero; this is predicted by Equation 23. For A I > 2 and = 0, the curve shows the constant offset predicted by Equation 24 while for Al > 2 and p > 0, the curves converge with R as predicted by Equation 25. I t is interesting to observe that in spite of the arbitrariness of the choice of the form of f ( b ) ,the curves generated from Equation 26 bear a striking resemblance to those obtained from the Kubelka-Munk equations when the degree of turbidity approaches the lower limit to the applicability of those equations (30);the agreement confirms the validity of this independent and less complex approach. (ii) Multiple Internal Reflections. At any interface in an optical path such as t h a t created by a cell wall, a fraction of the light will be reflected. If after the light passes through the cell a fraction, f l , is reflected back through the cell, and of that quantity a fraction, f2, is reflected again in a forward direction, the total amount of incident light reaching the detector will be the sum of that passing through the cell once and that passing through the cell thrice; further passes are insignificant. This problem has received a significant amount
.I.
1-f
In a cell containing a liquid, the most significant interface is that of glass and air for which f l = f 2 = 0.04; in Figure 4, %6, is plotted for these values. The error is very small and probably is insignificant. However, Goldring et al. (18)state that the fraction f l may reach 0.2 for some reflections off slit jaws, other components, or the photomultiplier envelope; the quality and cleanliness of components will also affect this figure. In the case where f l is 0.2 but f 2remains 0.04, the error at A, = 0.1 and p = 0.0 is 1.676, nearly an order of magnitude larger for f l = 0.04; consequently it may be significant. Summation of Errors. T h e errors in the various sections may be summed linearily to give the total error in the measurement. However, it must be stressed that the magnitude of the error depends markedly on instrumental characteristics and measurement parameters; for one instrument or set of measurement conditions, an error may predominate which is insignificant for another set of conditions. In order to summarize the data, the range of error expected under various conditions for a given type of error is presented in Table 111. Values for scattering interferents are taken from Table 11. Cell error is estimated from the "cell transmission flicker factor" values previously reported (41,42); lower and upper limits result from computation with complete correlation and no correlation, respectively, of the flicker error a t the two wavelengths. Stray light is expected to vary from the often reported for holographic minimum of cy1 = cr2 = grating monochromators, to a maximum of = a2 = 10 The range of slit width to peak width ratios was estimated at 10 to 0.4. Finally for path length errors for the cases with no interferent or a case I interferent, multiple internal reflections are the source of error; f 2 is assumed to be 0.04, and f l is taken to have a range of 0.04 to 0.2.
'.
DISCUSSION The conditions for DWS measurement resolve into three sets of conditions as seen in Table 111. Where there are no
1216
0
ANALYTICAL CHEMISTRY, VOL. 51, NO.
8, JULY 1979 -
Table 111. Expected Range of Errors %S where 0 = 0.3‘; values of minimum, maximum, typical
AI
0.1 0.5 1.0 1.5 2.0
Dl
uncompensated scatteringb
0.0
___
0.0 0.0 0.0 0.0
___ _____
---
cell errorsC
stray lightd
optical band passe
path length
n, 4., 0.8 n, 0.9, 0.2 n, 0.4, 0.1 n, 0.3, 0.1 n, 0.2, n
n, 0.8, - 0.1 n, -15., -0.15 n, --3., -0.4
n, -13., n n, -12., n n, -lo., n n, -8., n n, - 7 . , n
0.2, 1.5, 0.2c 0.1, 0.5, 0.1
-
11, -7.,
n,
-0.8
- 14., - 2.
n, 0.15, n n, n, n n, n, n
Case I 0.1
0.5 1.0 1.5
2.0 0.1 0.5 1.0 1.5 2.0
0.5 0.5 0.5 0.5 0.5 1.0 1.0
1.0 1.0 1.0 Case I1
n, n, n, n, n, n, n, n, n, n,
4.,0.8 0.9, 0.2 0.4, 0.1 0.3, 0.1 0.2, n 4., 0.8 0.9, 0.2 0.4, 0.1 0.3, 0.1 0.2, n
n, -- 2., - 3. n, -4., -5. n, -9., -1. n, - 18.,- 2.5 -1.,-28., -5.6 n, -7., -0.8 n, -12., - 1 . 5 n, - 22., - 3. -0.1, -34., - 7 . - 0.2, - 44., - 24.
n, n, n, n, n, n,
n, n, n, n,
-13., n -12., n -lo., n -8., n -7., n - 13., n -12., n -lo., n 8., n - 7., n -
2., 150., 35. n, -0.8 -0.1 n, - 13.,n 7., l o o . , - - p n, 4., 0.8 n, -12., n 2., 30., n, 0.9, 0.2 n, - 1 . 5 , p -0.2 0.4, 20., 7. n, 0.4, 0.1 n, - 3 . , -0.4 0.2, 15., 4. 0.8, 13., . n, -lo.,n n, - 7.2, - 0.8 n, - 8 , n 1.5 0.2, lo.,3. 0.3, 5., - n, 0.3, 0.1 n, - 7., n 2.0 n, - 14., - 2. 0.2, 4., 0.1, 7., 2. n, 0.2, n 0.1 13., 200., - 4., 300., 70. n, -0.8, -0.1 n, --13., n n, 4.,0.8 0.5 0.8, 40., 14. n, - 1.5, - 0.2 4.,60., - n, - 1 2 . , n n, 0.9, 0.2 1.0 0.4, 30., 7 . n, 0.4, 0.1 1.6, 25., - n, - 3 . , --0.4 n, -lo.,n n, - 8., n 1.5 0.3, 20., 5. n, 0.3, 0.1 n, -7.2, -0.8 0.6, lo.,. n, - 14., - 2. 0.5, 8., - n, - 7 . , n 0.2, 14., 4. 2.0 1.0 n, 0.2, n a Entry n indicates negligible error ( 0.1 and cell errors are expected to be significantly less for DWS measurement than for SWS measurement, one may expect DWS measurement to be more accurate than SWS measurement a t low concentrations. It must be stressed, however, that this advantage is obtained only when the cell errors are near equal at both wavelengths, and that condition is best achieved when AX is small. When Al is greater than about 0.5, stray light will dominate unless a double monochromator or a monochromator with a holographic grating is used; since when /3 > 0 stray light is greater in DWS measurement than in SWS measurement, the latter will be more accurate when simple monochromators are employed. The main advantage of DWS measurement is, however, its ability to discriminate against interferents, and when interferents are present it serves no useful purpose to compare with SWS measurement. I t is important, however, to note any significant source of error. For measurement in the presence of a case I, molecular, absorbing interferent, stray light error dominates. I t is apparent that high-accuracy measurements require special attention to stray light error; this necessitates a dispersive monochromator and stray light filter combination or a double dispersive monochromator and/or a holographically-produced grating. For a case I1 interferent which attenuates the beam by scattering light, the serious measurement errors are those caused by the interferent itself; Le., they are caused by uncompensated scattering and by multiple scattering. In Table 111, the typical values listed for the former result from a AX which is only 5 nm when AI = 400 nm at t = 4 (in Equation
*
~
9) or is 20 nm when A1 = 700 nm a t t = 1. Clearly, an assumption that the scattering a t both wavelengths is approximately equal (y = 1.0) cannot be made if the scattering ( D J is greater than 0.2 and AI is less than 1. For a case I1 interferent, observable nonlinearity in the calibration curve must also be expected. Here we are able only to predict the general shape of the curve and suggest that the error when A, < 0.5 may be in the range of 1% to 10% or greater. The information in Table I11 suggests the optimization of the operational parameters wavelength difference (AX) and concentration. The value of AX determines the absorptivity ratio, p, upon which depends stray light error, and this error increases with p ; consequently, there is an advantage in maintaining 3 < 0.5. For a peak with a width a t half height of 30 nm, the value of AX would have to be a t least 15 nm. However, two other factors affect the choice of AX in the opposite direction since both uncompensated scattering and cell errors increase with AA. Where no interferent is present and A I < 0.5, cell error may be the maximum error if AX is not small enough so that the error is correlated at the two wavelengths and thereby cancelled. For highest accuracy, AX should therefore be set less than about 20 nm where all else is equal. Where scattering interferents produce a value of D1 greater than 0.5, the error due to uncompensated scattering is severe and AX must again be kept low so that the error, computed from Table I and knowledge of the expected range of scattering, will be kept small. The concentration of analyte may also be optimized; this is particularily true for measurements in the presence of absorbing (case I) interferents. If both Al and D1 are greater than about 0.5, the total error will generally be dominated by stray light error. Dilution of the sample will then lead to lower
ANALYTICAL CHEMISTRY, VOL. 51,
error until Al is reduced to about 0.2 at which point cell errors may become significant. When reducing the concentration prior to measurement. it is also necessary to be aware of the effect on the precision of the measurement; considering an instrument whose precision is shot-noise (square root uncertainty) limited, the relative standard deviation in the measurement of 1 A passes through a minimum in the region of A , = 0.9 when all other factors are equal ( I ) . However, when A , and D1 are simultaneously reduced, this minimum occurs a t a lower value of A , ; calculating from Equation 11 in Ref. 1, the minimum will occur a t about A , = 0.5 when A , = D , and at about AI = 0.2 when AI = 0.25 D,. The dilution technique may also have value when a scattering interferent is present although the stray light error is less severe in this case than it is when an absorbing interferent is present. Throughout this section, the error due to large optical bandpass has been assumed to be typically small; consequently, attention should be drawn to the possibility of exceptions. DWS instruments, operated as derivatike spectrometers for gas analysis in the ultraviolet region, may be subject to the maximum error shown in that column of Table I11 since bands in gas phase spectra are typically narrower than those in solution spectra. Narrowing the slit for a bandpass less than 0.1 nm as will often be required reduces the photosignal; therefore a high-intensity source is mandatory.
CONCLUSIONS As a result of the preceding treatment and discussion, the following general conclusions may be drawn regarding the relative accuracy of dual wavelength spectrophotometric measurement. (1) Errors due to optical artifacts in the cell (cell errors) may be substantially reduced by utilization of DWS rather than SWS measurement. Consequently cells of lower optical quality may be tolerated for DWS than for SWS measurement. ( 2 ) Errors due to the scattering of an interferent, although substantially reduced by DWS, cannot be completely eliminated in a single DWS measurement. Either some knowledge of the scatterer is necessary in order to make an accurate correction or AX must be made very small in order to reduce the error. (3) An intensely scattering system may cause calibration curve nonlinearity due to multiple scattering. (4) Stray light must be carefully controlled, especially in the presence of a molecular interferent in which case the error may typically be on the order of several percent: strav light
NO. 8,
JULY 1979
1217
error may cause calibration curves with a maximum. ( 5 ) When the total error is dominated by stray light and both the analyte absorbance, A,, and interferent absorbance, D,, are greater than about 0.5, improved accuracy and precision may be achieved by dilution of the sample.
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RECEIVED for review March 8, 1976. Resubmitted January 3, 1978. Accepted March 7, 1979.
