Comparison of the Precision of Normal and Precision Spectrophotometric Techniques J. D. Ingle, Jr. Department of Chemistry, Oregon State University, Corvallis, Ore. 97331
A general theory for the relative instrumental precision of normal and precision (transmittance ratio, trace analysis, and ultimate precision) spectrophotometric absorbance measurements is presented. The theory presented takes into account both reading error and electrical noise (photocurrent shot noise, dark current shot noise, and source flicker noise). Equations are developed which indicate the gain in relative precision expected when a precision spectrophotometric technique is used to expand the transmittance scale in order to reduce the reading error. In some cases, the relative precision is reduced by scale expansion. The precision techniques are shown to be most advantageous for measurement of high absorbance solutions.
Previous theories dealing with the relative concentration error in precision spectrophotometric techniques are not always valid since only reading error is considered. Many instruments are readout limited such that scale expansion may improve the relative precision. However, as expansion is increased, the precision of absorbance measurements will eventually be limited by noise. The theory and equations presented in this paper consider both the effect of reading error and system noise on the normal and precision spectrophotometric techniques. The equations developed can be used to evaluate the relative concentration error and to evaluate the gain in precision, compared to the normal spectrophotometric method, expected for a given scale expansion which results from the use of precision spectrophotometric technique. If absorbance measurements are limited by the resolution of the readout device, a number of alternatives are available for increasing the measurement precision. Basically, these alternatives involve modification of the instrument to reduce the effect of reading error. A low resolution readout device such as an analog meter can be replaced by a high resolution readout device such as a 3 to 5 digit digital meter. Some instruments provide the capability of calibrated electronic scale expansion to allow display of part of the transmittance or absorbance scale on the readout device. Many modern instruments use logarithmic ratio circuits to provide direct absorbance readout with readout resolution of O.OOO1 absorbance unit in some cases. Direct absorbance readout is particularly advantageous a t high absorbances since the relative effect of readout error continually decreases with absorbance unlike for transmittance readout instruments (1). Finally, at low light levels, photon counting can be used for absorbance measurements (1-4) to eliminate reading error in essence. All of the above techniques for increasing the readout resolution could be classified as precision techniques be(1) J. D. Ingle, Jr., andS. R . Crouch, Anal. Chem.. 44, 1375 (1972). (2) H. H. Ross,Anal. Chem., 38,414 (1966). (3) M. L. Franklin, G. Horlick and H. V. Malmstadt. Anal. Chem., 41, 2 (1969). ( 4 ) K. C. Ash and E. H. Piepmeier, Anal. Chem., 43, 26 (1971).
cause they can result in an increase in measurement precision if readout error is limiting. Ingle and Crouch (1) have recently discussed the enhancement of measurement precision in absorbance measurements for the above techniques. In this paper, precision molecular absorption spectroscopy will refer only to those techniques in which one or both ends of a linear readout scale are defined by means of appropriate reference solutions that contain finite concentrations of the analyte. The three precision techniques [often called differential techniques (5)] normally identified are the transmittance ratio (5-13), trace analysis (14), and ultimate precision (5, 14-17) methods. In all three cases, a particular part of the transmittance range is expanded in order to reduce the relative error due to reading the scale and, hence, to reduce the relative concentration error (5). It should be noted that the terminology “ultimate precision” can be misleading since it does not necessarily provide the highest precision (5). An increase in the precision or decrease in the relative concentration error in precision spectrophotometric techniques relative to the normal spectrophotometric technique depends on the validity of the assumption that all measurements are readout limited. However, previous authors (15, 18, 19) have indicated that other sources of errors such as noise may affect the precision in addition to reading error. Recent signal-to-noise ratio (S/N) treatments of normal molecular absorption measurements (1, 20, 21) have quantitatively taken into account the effect of noise on the relative precision of measurements. Particularly with modern stable instruments, high resolution readout devices, and/or electronic scale expansion, absorbance measurements may not always be readout limited. The “chemical” scale expansion resulting from use of the precision spectrophotometric techniques enhances the probability that the measurements will be noise limited. The purpose of this paper is to develop a general expression which indicates the effect of reading error and noise on the relative concentration error for both normal and precision spectrophotometric techniques. Under cer(5) J. W. O’Laughlin and C. V . Banks, “Encyclopedia of Spectroscopy,” G. L. Clark, Ed., Reinhold, New York, N.Y., 1960, pp 1gS33. (6) C. F. Hiskey, Anal. Chem., 21, 1440 (1949). (7) C. F Hiskey and II. Firestone, Anal. Chem., 24, 342 (1952). (8) C. F. Hiskey, J. Rabinowicz, and I . Young, Anal. Chem., 22, 1464 (1950). (9) C. F. Hiskey and I. Young, Anal. Chem., 23, 1196 (1951). I O ) I. G.YoungandC. F. Hiskey,Anal. Chem., 23,506 (1951). 11) G. H. Ayres, Anal. Chem., 21, 652 (1949). 12) R. Bastian, Anal. Chem., 21, 972 (1949); 23, 580 (1951): 25, 259 (1953). 13) R . Bastian, R . Weberling, and F. Balilla, Anal. Chem., 22, 160 (1950). 14) C. N. Reilley and C. M. Crawford, Anal. Chem., 27, 716 (1955). 15) C. M. Crawford, Anal. Chem., 31, 343 (1959). 16) C. V. Banks, J. L. Spooner, and J. W. O’Laughlin, Anal. Chem., 28, 1894 (1956). (17) C. V. Banks, J. L. Spooner, and J. W. O’Laughlin, Anal. Chem., 30, 458 (1958). (18) L. Cahn. J. Opt. SOC.Amer., 45, 953 (1955). (19) H. K. Hughes, Appl. Opt., 2,937 (1963). (20) J. J. Cetorelli. W. J. McCarthy, and J. D. Winefordner, Anal. Chem., 45,98 (1968). (21) J. J. Cetorelli and J. D. Winefordner, Talanta, 14, 705 (1967).
ANALYTICAL CHEMISTRY, VOL. 45, NO. 6 , MAY 1973
861
Table I. Signal Voltages before and after “Chemical” Scale Expansion
Before expansion: f b = ibmR = signal voltage before expansion with distilled water reference or reagent blank in the sample cell, V i b = photocathodic current for blank solution, A €1 = E b T l = ilmR = signal voltage before expansion with “high” reference solution in sample cell, V il = photocathodic current for “high” reference s o b tion, A TI = 10-LbC!= transmittance of “high” reference t = molar absorptivity of analyte species, I mole-’ cmb = cell path length, cm c1 = concentration of analyte in ”high” reference solution, moles I . - l E , = i,mR = E b T , = signal voltage before expansion with standard or unknown sample in t h e sample cell, V is = photocathodic current for sample solution, A T s = 1 0 - f b C s = transmittance of sample solution cs = concentration of analyte in sample solution, moles I. Eo = iomR = &To = signal voltage before expansion with the “low”reference solution in the sample cell, V io = photocathodic current for low reference solution, A
= transmittance of “low” reference solution, A co = concentration of analyte in “low” reference soiution, moles I.- l E d = idmR = dark current signal voltage before expansion, V id = effective cathodic dark current, A After expansion: f b ‘ = g E b = blank signal voltage after expansion, v g = (m’R’)/(mR) = expansion factor, dimensionless m‘ = current gain of photomultiplier necessary for expansion, dimensionless R’ = O A feedback resistance after expansion, ohms €1’ = gE1 = “high” reference signal voltage after expansion, To =
10-‘bco
V E,’ = gEs = sample signal voltage after expansion, v €0’ = gEo = “low” reference signal voltage after expansion, V Ed‘ = g E d = dark current voltage after expansion, V
tain conditions, this expression can be reduced to three limiting cases in which measurements are either photocurrent shot noise, source flicker noise, or readout limited. This expression is used to compare the precision techniques to the normal spectrophotometric method. This comparison indicates that for a given instrument and set of instrumental parameters used for a particular analysis, chemical scale expansion may result in anywhere from a significant increase in relative precision to a slight decrease in relative precision. The gain in instrumental precision derived from chemical scale expansion depends on the relative magnitudes of the source flicker noise, the photocurrent shot noise, and the readout resolution. The precision techniques are shown to be more advantageous for low transmittance measurements because the relative effect of noise in relation to readout error is reduced. It should be stressed that the equations developed in this paper only take into account instrumental precision and, hence, represent an estimate of the best precision that can be obtained for a solution of given analyte concentration. In many cases, t h e experimentally determined precision may be less than predicted because of sampling or cell positioning imprecision. 862
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The accuracy of determination of the concentration of an unknown depends not only on the precision with which the absorbance of the unknown is measured, but also the precision with which the absorbances of standards used to prepare the calibration curve are measured as well as the numerous chemical and instrumental factors (22, 23) that can cause error. Chemical scale expansion may also introduce other sources of error or more error than in the normal technique. The accuracy of the reference solutions used to define the readout scale and cell path length differences may increase the error. Scale expansion (except the transmittance ratio technique) generally requires a greater amount of time to set the end points of the scale and make a sample voltage measurement because of the reiterative procedure necessary to adjust the gain and suppression. Hence, chemical scale expansion is more likely to be susceptible to drifts (e.g., electrical or temperature). Use of precision spectrophotometric method may provide certain advantages compared to the normal technique because a relative measurement is involved. O’Laughlin and Banks (5) have discussed how the relative error may be reduced due to compensation of systematic error. Factors such as variation in the molar absorptivity with time or wavelength can be compensated for, particularly if the reference and sample solutions are approximately equal in concentration. Because the factors that affect the relative precision and accuracy of absorbance measurements are numerous and complex, it is difficult to evaluate critically the merits and disadvantages of the normal and precision techniques. The study presented in this paper makes the comparison with the criterion of instrumental precision for transmittance readout instruments.
RELATIONSHIP OF READOUT TO SAMPLE CONCENTRATION The system to be described for dc measurements is a modern single-beam spectrophotometer with a stabilized tungsten or deuterium source, a monochromator, a sample cell compartment, a photomultiplier transducer with stabilized power supply, a high quality operational amplifier (OA) current-to-voltage converter, and a linear readout device. In order to perform “chemical” scale expansion in the precision spectrophotometric techniques, it must be assumed that the instrument has provision for varying the gain of the photomultiplier and amplifier and provision for full scale current or voltage suppression. In general, the signal voltage E in volts from the output of the OA current-to-voltage converter due to transmission of radiant power by a given solution in the sample cell is given by
E
=
imR
where m = current gain of the photomultiplier, dimensionless; i = effective photocathodic current, A; and R = feedback resistance of OA, ohms. The magnitude of i can be related to parameters of the instrument utilized and to properties of the chemical system studied (I). Transmission of radiant power by a given solution in the sample cell produces a readout R, on the readout device. The magnitude of R, depends on the boundary conditions used to set the end points of the scale. It will be assumed that full scale on the readout device corresponds to R, = 1 and (22) L. S. Goldring, R . C. Hawes, G. H. Hare, A. 0. Beckman and M. E. Stickney, Anal. Chem., 25, 869 (1953). (23) E. J. Meehan, in “Treatise on Analytical Chemistry,” I . M. Kolthoff and P. J. Elving, Ed., Part I , Vol. 5, John Wiley and Sons, New York, N.Y., 1964, pp 2753-2803.
Table II. Equations and Conditions Expansion factor, g
Sample concentration, c s
Method
Normal
cs = - ( i / q o g T ,
Transmittance ratio or
cS =
(8)
-(l/tb)log(RsTi)
(9)
Suppression voltage
1
Ed
(TI)-’
Ed
’
high
absorbance Trace analysis or low
absorbance Ultimate precision or double reference
0 corresponds to R, = 0. In Table I, four signal voltages and the dark current voltage are defined prior to and after (indicated by a prime) scale expansion. It is assumed here that the chemical system, spectral bandpass, and stray light are such that Beer’s law is valid over the absorbance interval studied. The effective cathodic dark current and effective photocathodic current are assumed to be independent of the photomultiplier gain. Before scale expansion ( i . e . , for normal spectrophotometric analysis), the gain of the photomultiplier and OA and the suppression control are adjusted so that Rx corresponds to 1 and 0 for the blank signal voltage, E b , and the dark current signal voltage, E d , respectively. In precision spectrophotometric techniques, scale expansion is accomplished by increasing the photomultiplier and amplifier gain and by adjusting the suppression control such that one or both of the reference solutions are used to set R, = 1 and R , = 0. Expansion can also be achieved by increasing the photocathodic current by widening the slits. In this paper, it is assumed that before scale expansion, the maximum allowable spectral bandpass has been chosen to meet the requirements of resolution and to prevent errors due to polychromatic radiation. Thus the photocathodic current for any sample or reference solution is the same before and after expansion. For the most general case, ultimate precision, the “high” reference solution is used to set R, = 1, and the c * l ~ reference ~7’ to set R x = 0. Thus after scale expansion, the sample solution gives a readout Rs defined by Equation 2:
where E s + d = E‘s + E r d ; E f i + d = E’I + E ‘ d ; and EO+^ = E‘o E ‘ d . Substitution of the relationships in Table I into Equation 2 and rearrangement yield Equation 3 for cs:
+
RELATIVE STANDARD DEVIATION FOR DETERMINATION OF cs Application of propagation of error mathematics to Equation 3 yields an expression for the relative standard deviation of cs
where ucs = standard deviation in cs, moles/l.; and U R ~= standard deviation in R,, dimensionless. The quantity ucs/cswill denote the relative concentration standard deviation or the relative concentration error. The latter term is less descriptive but is often found in previous treatments. Thus the relative standard deviation for determining the sample concentration depends on the precision of the measurement of R,, the sample transmittance T,, and the magnitude of the expansion factor g. Since we are concerned only with instrumental precision, imprecision in the concentration of the reference solutions or in the cell path length are not considered. These factors would affect the accuracy of a determination but not the instrumental precision. Equation 12 is an approximation since differentials are replaced by finite errors (standard deviations). However, if T s > U R ~ Equation , 1 P i s a good approximation that only breaks down at extremely low transmittances.
STANDARD DEVIATION OF R, To use Equation 12, U R must ~ be determined. In most treatments U R ~is assumed to be constant and independent of Rs. However, recent treatments of normal molecular absorption spectrometry (1, 20, 21) with S / N theory have shown that this assumption is not always valid. Equation 2 indicates that R, is a function of three independent variables: E I S + d , E ’ l + d , and E ’ O + d . Thus application of propagation of error mathematics requires that Equation 2 be differentiated with respect to each of these variables with the result
(3) where all terms are defined in Table I. The equations defining Rs and c, for analysis, transmittance ratio, and normal spectrometric methods can be derived from Equations 2 and 3. These expressions are shown in Table I1 along with the defining conditions, the expansion factor, and the suppression voltage required for each spectrophotometric method.
where ( U ’ s + . d ) ’ , ( r ’ l + d ) ’ , and ( U ’ O + ~ are ) ~ the variances in volts squared for the measurement of E ‘ s + d , E ‘ l + d , and E ’ O + d , respectively. Equation 13 is an approximation since ANALYTICAL C H E M I S T R Y , VOL. 45, NO. 6, M A Y 1973
863
Table 111. Definitions of Variance Terms
+
a 2 = ( u p 2 ) s ( u p 2 ) s e= c 2rn2eiR2Af(l b2 = ( u p 2 ) ,= C'
=
(0d')q
+
+
(r~d2)sec
+ a)
(A.5)
=
+
(A.6)
= ( ~ d 2 ) t or2 = 2rn2eidR2Af(l a )
+ + ur2
(A.7)
= variance in photocurrent signal voltage due to quant u m noise, V2 = 2rn2eiR'At
e = charge of an electron, coulomb Af = noise equivalent bandwidth of amplifier readout sys-
tem, sec-' = variance in photocurred signal voltage d u e to secon-
dary emission noise, V2 a = relative variance in photomultiplier gain due to
secondary emission, dimensionless (usually 0.1
< 0.5)
> ab2A + bb2B, then Equation 15 reduces to
-[(ar/g)?+ (ad2)t]%[l + R,2 EbT,lnT,
+ (R, - 1)”h
(20)
Usually the dark current shot noise variance will be smaller than the reduced readout variance ( u r / g ) 2 . Thus it will be assumed unless otherwise stated that ( a r / g ) 2>> ( f f d 2 ) twhich allows reduction of Equation 20 to
Thus Equation 21 and Case I correspond to the usually assumed conditions and justification for scale expansion: readout limited conditions. Equation 21 indicates that the relative concentration error depends on the relative reading error br/Eb (some fraction of the full scale deflection of the readout device), the factor T,lnT,, P I 2 , and g . For a normal meter or recorder, the relative reading error will typically be 0.0050.001, For higher resolution readout devices or with electronic scale expansion, the relative reading error will be smaller. Under these conditions, it is less likely that Case I will hold since other variance sources become relatively more important as readout variance is reduced. Case I is more likely at lower sample transmittances because A and B decrease more rapidly with T, than C. The factor T,lnT, is found in most treatments of relative concentration error and varies from a minimum of zero a t T, = 0 and Ts = 1 to a maximum of 0.158 a t T, = 0.368. The factor C1/2 varies from a minimum of 1.22 at R, = 0.5 to a maximum of 1.41 a t R, = 0 and R , = 1. In normal spectrophotometry g = 1, while in the precision spectrophotometric techniques, g is greater than 1 as determined by the values of TI and TOchosen. For normal spectrophotometry ( R s = Ts), the relative readout error and T, determine the relative concentration error which has a minimum at Ts = 0.388. The precision techniques are seen, as a first approximation, to reduce the relative concentration error by the expansion factor g or (TI - To)-l. However, for the precision spectrophotometric techniques, Rs # T, and hence [l + RS2 + (R, 1)2]1’2 # [l + Ts2 + ( T , - 1)2]1/2since R, depends on the values of TO and TI. Since the above factor varies from 1.22 to 1.41, it can be shown that the relative precision is increased by 0.86-1.16 g by use of precision spectrophotometric techniques. In general, the gain in precision will be greater than g if R, is closer to 0.5 than T, and less than g if Ts is closer to 0.5 than R,. This depends solely on the choice of reference solutions used to define the readout scale. Note that for large g or small a?, it is possible that ( ( r d 2 ) t > ( f f r / g ) 2 .For this condition a t low transmittances where A and B are small, Case I may exist under dark current shot noise limiting conditions. For this situation, a c s / c c from Equation 20 is independent of the expansion factor g . Also a t low transmittances, negative deviations from Beer’s law (caused for example by stray light or polychromatic radiation) are expected to decrease the relative precision (6). Case 11. If ab2A >> (c’b)2c/gz + bb2B, Equation 15 reduces to
-(%)
[T,
+ Rs2T, + ( R , - l)2T# T , lnT,
(22)
This case will result under a few sets of conditions. First, at relatively small blank photocathodic currents, the relative standard deviation for measuring the blank signal voltage, ( a b / E ) , is large. This is likely under conditions where it is necessary to use small slit widths (Le., small spectral bandpasses) and/or to use wavelengths where response characteristics are unfavorable. Second, if a higher resolution readout device is used, readout variance may be negligible before expansion. Third, although readout variance, a r 2 , may be relatively important before expansion, it may be insignificant after expansion since expansion reduces the contribution of ( r r 2 by a factor g2. Also Case I1 is more likely at high transmittances where A is large. Case I1 applies to conditions where photon counting can be utilized as previously discussed ( I ) . Equation 22 indicates that the relative concentration error depends on the magnitudes of (ab/&), (T,lnT,), and A . The factor TslnT, varies from 0 to 0.158 as in Case I. The factor a b / & ,equals [2m2eibR2Af(1+ a ) ] 1 / 2 / i b m Ror 6.39 X 10-10(ib)1/2 for If = 1 Hz and a = 0.275. Thus to 6.4 x 10-6 over the ab/Eb varies from about 6.4 x normally expected reference photocathodic current range of 10-16 to 10-8 A. Note that Equation 22 is independent of the expansion factor g since both the signal and the standard deviation of the signal increase the same relative amount with expansion. Differences in the relative concentration error between normal and precision techniques are reflected in the value of A1/2. In the normal technique, A112 = (T, + T,z)l/2 and a c s / c s has a minimum at Ts = 0.109. In the transmittance ratio technique for which To = 0 and R, = T,/T1, A1/2 = (T, + T,2/T1,1/2which is always greater T,)I12. Thus the transmittance ratio method than (T, always gives worse precision than the normal technique under photocurrent shot noise limiting conditions. This conclusion has also been reached by Hughes (19). In a similar manner, the magnitude of A112 can be used to compare the normal spectrometry to the trace analysis and ultimate precision methods. It is found that the relative concentration error of these two precision techniques is from 29% less to 15% more than that of the normal technique. The exact value depends on the magnitudes of T,, TO,and TI. Case 111. If bb2B > ab2A ( C ’ b ) ’ c / g 2 , Equation 15 reduces to
+
+
This case generally will occur at high photocathodic currents where the readout variance is small or the relative readout variance has been reduced by scale expansion. Also Case I11 is more likely at higher transmittances where B is large. The value of ucs/c,-depends on (bb/&), T,lnT,, and B . The factor T,lnT, varies in the same manner as for Case 1 and 11. The factor (bo/&) is equal to the source flicker factor ,$.For light sources normally used in molecular absorption work, ,$,and hence (bb/Eb), varies from to 10-5. The magnitude of B depends on T,, TI, and TO. As for Case 11, (rcs/cs is independent of g . Thus in the comparison of normal and precision techniques, any differences in the relative concentration error are due to differences in the magnitude of B112. In the normal technique B1/2 = fl T, and u c s / c s increases as T, approaches zero as long as Case I11 is valid and stray light is negligible. The transmittance ratio technique yields the same precision as the normal technique since B112 is idenA N A L Y T I C A L C H E M I S T R Y , V O L . 45, NO. 6, M A Y 1973
865
R
5
'O
I
0
0
-0"0
-
0.5
'VI
b"
b
0.1
0.1
O'O'
1I 1
i.0
M Z b .
as
8
0.8
I
*
0.7
as
*
0.5
,
.
0.4 0.3
I
0.1
I
ni
,
0.0
TS
l
I
I
1.0
0.0
0.8
'
0.7
Figure 1. Relative concentration error vs. transmittance for the
limiting cases with the normal method ( a ) Case I, ( C b / E b ) = 0.001, g = 1 (6) Case 1 1 , ( a b / E b ) = 0.001, g = 1 ( c )Case I l l , ( b b / E b ) = 0.001,g= 1
tical. The trace analysis and ultimate precision methods can be shown by comparison of B1/2 to yield a relative precision equal to or up to 15% better than the normal method. DISCUSSION Dependence of ucs/cs and F on Transmittance and Variance Terms. Figures 1 and 2 show plots of ocs/cs in per cent us. T s for the three limiting cases, where it is assumed that ( o r / g ) 2 >> ( u d 2 ) t . The curves in Figure 1 correspond to the normal spectrophotometric technique while the curves in Figure 2 correspond to 10-fold chemical scale expansion. Thus in Figure 2, the two intervals 100-90% T and 10-0% T represent the trace analysis and transmittance ratio techniques, respectively, and the remaining 10% T intervals represent various cases of the ultimate precision method. The oscillatory nature of the curves d, e in Figure 2 is due to variation of the factor C, A, or B in Equations 21, 22, or 23, respectively. The curves in Figure 1 and 2 demonstrate that the dependence of ucs/cs on the sample transmittance is quite different for each case. Comparisons of curves b and e and c and f indicate that chemical scale expansion affects the relative concentration standard deviation only slightly for Cases II and 111, respectively. Since ab and b b are the same in Figures 1 and 2, any differences are due to the differences between A and B before and after expansion as discussed in the previous section. Curves a and d are seen to be approximately identical. The factor (Cb/&) is ten times smaller in Figure 1 (curve a ) than in Figure 2 (curve d ) . Therefore, 10-fold scale expansion increases the precision by about a factor of 10 for Case I without readout limited conditions. In other words, 10-fold expansion is approximately equivalent to replacement of the original readout device with one that has a factor of 10 better resolution. Definition of the limiting cases is somewhat arbitrary since no case can be completely valid in real life. However, the classification is useful in that it provides insight into the different factors that affeot the precision. Also 886
*
A N A L Y T I C A L CHEMISTRY, VOL. 45, NO. 6, M A Y 1973
/
0.6
1
0.5
I
0.4
I
j
0.3 0 1
'
I
01
00
TS
Relative concentration error vs. transmittance for the limiting cases with chemical scale expansion Figure 2.
( d ) Case 1, (e) Case 1 1 , (t) Case 1 1 1 ,
(Cb/Eb)
= 0.01, g = 10
(ab/&)
= 0.001, g = 10 = 0.001,g = 10
(bb/Eb)
the limiting cases provide reasonable estimates of the precision under conditions where one type of variance is considerably larger than other variances for a particular transmittance range of interest. For intermediate cases in which two or more variance terms must be considered, Equation 15 can be used to construct plots of the relative concentration error us. transmittance. The relative concentration error will always lie in the region bounded by Cases I and III. Figure 3 shows plots of the acs/cs in per cent us. T s for some intermediate cases. Curves a and b are for Case I before and after 10-fold scale expansion, respectively, where it assumed that ( a r / g ) 2 >> Curves c-f are for cases in which various amounts of photocurrent shot or source flicker noise must also be considered. For all the plots, the often assumed readout resolution of 0.2% is used. Clearly, the gain in precision and the optimum transmittance range predicted by the usually assumed Case I (curve b ) is not realized if electrical noise is also considered. Figure 3 illustrates that Case I is always approached at lower transmittances so that scale expansion is more advantageous under these conditions unless dark current shot noise becomes limiting. Also note that the effect of source flicker noise decreases more rapidly than the effect of photocurrent shot noise as the transmittance decreases (curves d and e). Figure 4 indicates the increase in precision realized with scale expansion by plotting F (Equation 19) us. transmittance. Curves a, i, and h correspond to Cases 1-111, respectively, while curves b-g correspond to intermediate cases. Note that these plots are independent of the absolute values of the group variance terms and only depend on the relative magnitudes. It can be seen that as the magnitudes of the photocurrent shot noise and/or the source flicker noise approach the magnitude of the readout resolution before expansion, the improvement in precision is considerably decreased. However, at higher absorbances, significant improvement is still realized. Note also that under Case I11 limiting conditions, no or slight improve-
0 . 0 1 , 1.0
0.02 O’05!
/ I I I I ( I I I I I
1.0
0.8
0.0
0.7
0.8
0.5
0.4
0.3
0.2
0.1
0.0
0.9
I
0.8
e
I
0.7
0.6
I
0.5
0.4
0.3 0.2
I
,
0.1
0.0
Figure 4. Factor by which the relative concentration error is reduced by 10-fold chemical scale expansion ( a ) C b >> a b + b b ( b )C b = 1 0 ( a b z + bb’)”’, a b = b b (C) c h = 2bh > >ab ( d ) Cb = 2ab >> bb ( e )c b = (ab’ b b 2 ) ” ’ , a b = b b (1) C b = ’/zbb >> a b (9)Ch = %ah >> b b
+
( h ) a h >> b h ( i ) b b >> a b
ment in precision occurs over the whole transmittance range. For Case 11, the precision is generally decreased by scale expansion as the transmittance decreases. This occurs because, under shot noise limiting conditions, the relative variance in E‘1 and E’o increases as T I and TO decrease. Calculation of acs/csand F. The relative concentration standard deviation and the improvement in precision from scale expansion depend on the particular analysis performed and instrument and instrumental parameters used. More specifically, F depends on the relative magnitudes of c b , ab, and bb or the relative magnitudes of the readout resolution, the photocurrent shot noise, and source flicker noise. Equations 15 and 19 and Figures 3 and 4 illustrate that the scale expansion provided by the precision techniques results in a significant improvement in precision for Case I or readout limited conditions. Under conditions of Case I1 or I11 or a mixture of these cases, scale expansion provides differences in the relative precision of only &30%. For situations in which both readout error and noise must be considered, the net improvement in relative precision derived by chemical scale expansion will be dependent on the relative magnitudes of a b , b b , and c b , and the transmittance range of interest. The advisability and benefits of scale expansion can be empirically evaluated without calculations as follows. The instrument is set up for an ordinary transmittance measurement with the sample to be determined in the sample cell so that the readout indicates E,. If noise is not evident on the readout device, then scale expansion will provide a significant reduction in u c s / c s .However, if noise or fluctuations in E, are noted on the readout device, then scale expansion may provide some improvement although only moderate expansion will be advantageous. The actual gain in precision can be calculated from Equation 19.
+ Cb +cb
Equations 15 and 19 are put into a form such that the feasibility of scale expansion can be readily ascertained before scale expansion is attempted. Once the wavelength and slit width for an analysis is chosen, the blank soluticn is placed in the sample cell and the photomultiplier gain and/or the amplifier gain ( i e . , R ) are adjusted to yield full scale deflection on the readout device. The group variances, a b 2 , b b 2 , and C b 2 , are then evaluated by use of Equations A.5-A.7. Knowledge of the transmittance range of interest permits evaluation of A, B, and C from Equations 16-18. Equation 15 can then be used to evaluate u c S / c sbefore or after expansion, or Equation 19 can be used to determine the gain in precision expected for a given expansion. Equation 19 can also be used to calculate F for different values of g (or TI and TO)to determine a t what point further expansion results in little gain in precib b 2 ) , then scale expansion is expected sion. If c b 2 2 ( a b 2 to provide a significant gain in instrumental precision. However, if c b 2 is considerably less than ( a b 2 + bo2), then scale expansion will not provide significant gain in precision at high transmittances. At lower transmittances, however, significant gain in precision may be realized since the relative effect of photocurrent and source flicker noise is reduced.
+
APPENDIX Equation 13 can be simplified to Equation A . l by use of the following relationships: E’I = T l E ‘ b , E‘, = T s E ’ b , E’o = T O f l b , and (E”, - E’I)~/(E’I - E ’ O )=~( R , UR. = # - [ ( d q + d , 2
+
+ (R, - 1 ) ’ ( ~ ’ o + d ) ’ I ~
R,2(dl+d)*
(AS) The three variance terms in Equation A.l can be written as follows. (~’,+d)’
= LZ~’
+
b,’
+ c,’
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(A.2) 867
where 0 2 , b2, and c2 arrange the total variance ipto three groups which are proportional to i, proportional to i2, and independent of i, respectively, and the subscripts s, 1, and 0 denote sample, high reference, and low reference group variance terms, respectively. The group variance terms are defined in Table 111, Equations A.5-A.7. The group variance terms for molecular absorption spectrometry have been discussed previously ( I ) , and only the significant terms are included here. Variance terms due to other sources (e.g., photomultiplier flicker noise, Johnson noise, amplifier noise) can be added to the appropriate group variance terms according to their dependence on the photocathodic current. The group variance terms defined in Equations A.2-A.4 are found by substituting the appropriate photocathodic current, is, il, or io, respectively, for i in Equations A.5-A.7. If a photodiode is used in place of a photomultiplier, then m = 1 and a = 0 in the equations defining the variance terms in Table I11 and the signal voltages in Table I. Under these conditions amplifier noise and Johnson noise may be relatively important (24) and their contributions should be added to the c2 variance terms. Unlike other noise variance terms, Johnson and amplifier noise variance are not proportional to g2. Substitution of the group variance terms defined by Equations A.2-A.4 into Equation A.l yields
The magnitude of the group variance terms depends on concentration of the sample and reference solutions involved in a particular analysis. Equation A.8 can be placed in a more useful form by relating all group variance terms in Equation A.8 to the blank group variance terms after expansion (denoted by subscript b and a prime) with the following relationships. (24) J. D. Ingle. Jr., and
868
S.R. Crouch, Anal. Chem., 43,1331 (1971)
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However, even the magnitude of (a'b)2, (b'b)2, and ( c ' * ) ~ varies depending on the magnitude of scale expansion in a given analysis. Thus, it is convenient to relate all variances to the variance terms of the blank before expansion (denoted by subscript b ) . These can be measured before expansion and are constant for analysis of a given species. From the definitions of the group variance terms (Equations A.5-A.7), it can be shown that
where it is assumed that the effective cathodic dark current is independent of the photomultiplier gain m. Also for photocurrent and dark current shot noise terms, it must be assumed that N is independent of g. Actually a decreases slightly and id increases slightly as m increases. Note that cb2 = ( c ' b ) 2 for the normal spectrophotometric technique. Use of all the above relationships in Equation A.8 and the relationship E'b = gEb yields
ACKNOWLEDGMENT The author wishes to thank S. E. Ingle for many helpful discussions and suggestions during the preparation of this paper. Received for review July 26, 1972. Accepted December 8, 1972. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, and the Oregon State University General Research Fund for partial support of this research.
