Theoretical Assessment of Precision in Dual Wavelength Spectrophotometric Measurement K. L. Ratzlaff’ and D. F. S. Natusch*2 School of Chemical Sciences, University of Illinois, Urbana, Illinois 6 180 1
Equations are presented which describe the contributions of several sources of independent, square root, proportional, and constant error to dual Wavelength spectrophotometric (DWS) measurement. I n the absence of interfering specks in solution, the principal sources of imprecision are usually photon shot noise, source flicker, and readout noise and DWS measurement may be more precise than conventional single wavelength spectrophotometric measurement because of reduction of sample presentation errors. I n the presence of interferents, independent and square root errors increase markedly but proportional error is unaffected. For high precision measurement, it is generally desirable to employ a high intensity light source in conjunction with suitable modulation.
The technique of dual wavelength spectrometry (1, 2) represents a fundamental departure from conventional ratiometric single wavelength spectrophotometry. In single wavelength spectrophotometry (SWS), a light beam consisting of a single narrow wavelength band passes through separate sample and reference cells and the transmitted intensities are processed as separate sample and reference signal channels. In dual wavelength spectrometry (DWS), however, two light beams having distinct narrow wavelength bands pass through a single sample cell and the transmitted intensities at each wavelength are processed as separate signals ( 3 ) . The main advantages claimed for DWS over SWS are the improved selectivity, sensitivity, and accuracy of analyte absorption measurement in the presence of severe spectral overlap from interferents which absorb or scatter light at the analytical wavelengths (3-8). In addition, claims of improved precision have been made ( 4 ) although these claims have not been substantiated quantitatively. It is the object of this paper, therefore, to present equations describing the contributions of each source of error in DWS to the overall precision of measurement. Assessment of the magnitude of each contributed error is then used to establish measurement precision under different operating conditions, and criteria are presented for the design and operation of high precision DWS instrumentation. Similar assessments have already been published for SWS (9-13) and these are used as a basis for comparing the two types of measurement. B A S I S OF T H E DWS MEASUREMENT DWS measurement involves determination of the absorbance difference, LA, between absorbances A I and A? measured at two different wavelengths, X1 and X2. Since Beer’s law holds,
A A = A i -A2 =
(€1
-~ 2 ) b ~
(1)
where el, e2, b, and c are molar absorptivities, path length, and analyte concentration. The absorbances A I and A2 can be ‘Present address, Department of Chemistry, Northern Illinois University, DeKalb, Ill. 60115. *Present address, Department of Chemistry, Colorado State University, Fort Collins, Colo. 80523. 2170
ANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
expressed in terms of incident and transmitted light intensities such that
I1 12
A A = -log--
+ log-Io 1
(2)
Io2
where 11, 12, are transmitted intensities and lo’,Io2are incident intensities a t wavelengths X1 and X2, respectively. Equations 1 and 2 are thus the DWS equivalents of the Beer’s law expression and the definition of absorbance in conventional SWS. The second term in Equation 2 takes account of any differences in measured incident light intensity due to differences in source output, wavelength discrimination, detector response, or instrumental gain between wavelengths X1 and As. For most conventional spectrometers employing continuous, nonpulsed, light sources and stable detectors this second term is essentially constant for given X1 and X2 and is close to zero. Since Equation 2 has the log ratio form of SWS absorbance, it is apparent that the advantage of instrumental drift rejection is retained in DWS. For applications of DWS to equilibrium methods of analysis, two wavelengths are normally chosen at which the absorbance or scattering of the interferent is equal but the absorbances of the analyte are different. In this situation the measured absorbances, Al’, A i , at wavelengths X1 and X2 are related to the analyte absorbances, A’, A 2 , by the relations
Ai’=A,+D AZ’=Az+D
(3)
(4)
where D is the optical density (absorbance or scattering) due to the interferent. Since the measured value of AA is independent of D, Equation 1 can be applied directly to calculate the analyte concentration. CALCULATION OF P R E C I S I O N Equations describing the percent relative standard deviation (RSD) of the DWS measurement contributed by each source of noise or uncertainty are presented below. Many of these are straightforward extensions of equations already developed for SWS measurement (12) and differ only in the fact that consideration of error contributions to signals at two distinct wavelengths is necessary. Since most significant errors are associated with the optical components of a spectrometer we have, in conformity with precision studies for SWS (12),considered error contributions to the photoanodic signal. For this purpose the DWS absorbance, AA, is expressed as
where Eol is the photoanodic signal voltage produced by the incident radiation intensity, Io’, a t wavelength X1, and
Eol is related to the photoanodic current, iol, produced by Io1 and the associated resistance, Rf,of the OA current-to-voltage
converter by Eol = -iol Rp Equation 5 can be obtained by combination of Equations 2-4, 6, 7. Application of propagation of error mathematics to Equation 5 leads to a generalized expression for the percent relative standard deviation, % u u , in AA due to any source of random noise. Thus
"9 t
t
where u1 and uz are the standard deviations associated with the signals a t each wavelength. The total percent RSD from all sources of noise can be obtained by taking the square root of the sum of the squares of '3% uL4 contributed by each source. For the purpose of the following discussion we assume that the determination of K , which actually "zeroes" the instrument, is performed with significantly higher precision than any individual measurement of LA. Even if K were to be determined each time IIand I2 are measured, it can be shown that the overall precision would never be changed by more than a factor of 2 . Therefore, we have chosen to consider the uncertainty in K to be negligible. In practice, different types of noise give rise to only four functional forms of Equation 8 so that the overall precision is effectively determined only by those sources which predominate for each functional category. These categories are determined by the dependence of the standard deviations u1, uz (Equation 8) on photomultiplier signal intensity. In considering the various sources of noise, it is useful to distinguish between those whose energy spectra are frequency dependent (black noise) and those which are frequency independent (white noise). This is because frequency dependent noise can be more effectively reduced by modulation techniques than is the case for white noise. Similarly, pattern noise can often be effectively reduced insofar as it is often highly correlated within the signal time frame. Independent Error. When the standard deviation of the signal intensity is independent of the magnitude of the signal, the associated error is called Type I (9-12) or Independent (13)error. Sources of independent error include dark current shot and excess noise, digitization noise, thermal or Johnson noise, and noise associated with readout devices reading in units of AA. This last source of noise, although signal independent, gives rise to a different functional form of Equation 8, due to the fact that it is not introduced until after logarithmic conversion; it will be considered separately in a later section. T h e standard deviation, (TI, associated with signal independent noise is the same a t both wavelengths so Equation 8 can be written
K-1102q
lj2
x 100%
(9)
Figure 1 shows a plot of (9% uu)1 vs. AI from Equation 9 for several representative values of the molar absorptivity ratio, ,f3, in the absence of absorbing or scattering interferents (i.e., D = 0). A representative value of the ratio U I / Eof~1~x has been chosen for illustration of independent error; however, choice of other values merely moves the precision curves along the Y axis without changing their shape. Curve A in Figure 1, for which p = 0 and K = 1, also represents the SWS situation under the same conditions and can be obtained by replotting figures presented in previous work (9, 10, 12) on logarithmic axes. I t is apparent that
where e is the electron charge and id, m, and CY are, respectively, the photomultiplier dark current, average gain, and relative variance of the gain. Af is the noise equivalent bandwidth of the signal amplifier-readout system and R f the resistance of the OA current-to-voltage converter. uexis the standard deviation due to excess dark current noise. Dark current noise, which is primarily frequency independent excess noise, is usually the predominant source of independent error in spectrophotometric measurement. Rothman e t al. (12) report a standard deviation of 10-l' A, which corresponds to u I / E o l in the range 10-5-10-6,for dark current noise. Generally, the higher value is realized only when narrow slit widths, weak sources, or poor detection response ANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
2171
a t the wavelength of interest are encountered. Digitization Noise. Where photomultiplier signals are processed in the digital domain, conversion can introduce an uncertainty (or pattern noise) of plus or minus the quantization level whose magnitude depends upon the resolution of the analog-to-digital converter. In this situation, the uncorrelated standard deviation due to digitization noise is u~ = q / & (14-16). Unless low resolution analog-to-digital conversion is employed, the resulting independent error is generally small compared to that due to dark current. Johnson Noise. This consists of so-called “white” thermal noise in amplifier resistances. The standard deviation, 01, due to Johnson noise is (4kTR,Af11/’where Rf and Af are as defined for Equation 10 and k and T a r e Boltzmann’s constant and the absolute temperature, respectively (10,17). Under most normal operating conditions, Johnson noise makes a negligible contribution to independent error; however, it may become comparable with dark current noise where very broad bandwidths (Af) are employed. This condition is rarely encountered. S q u a r e Root Error. When the standard deviation of the signal intensity is proportional to the square root of the magnitude of the signal the associated error has been termed Type I1 (9-12) or Square Root (13) error. This error is due only to the shot effect which is the result of frequency independent statistical fluctuations in the photoelectron pulse rate (9-13,18-20). The standard deviation in signal intensity, us,due to shot noise is equal to [2meRf2Afiol (1 + where the variables are as defined for Equation 10. In terms of the photomultiplier reference voltage, Eol,the standard deviations due to shot noise in signals derived from wavelengths X1 and X2 are thus
u l = ( 2 r n e R f A f E o l 1 0 - ( D +) A ‘I2 ~) u = ( 2 m e R A f K E , 1O - ( D + pA
1
)
)
’
Inserting these standard deviations into Equation 8 gives the percent RSD due to shot noise square root error as
1ODl2 = 2.303(1 - @ ) A l
(”/.
1 1 1 2
K-1lo@”qj
x 100%
(11)
Figure 2 shows a plot of ( % u& vs. Al from Equation 11 for several representative values of /3 and values of D = 0, m = 2 X lo4,Rf = 10 MQ, Af = 10 Hz, and Eol = 1 V. This last value corresponds to the production of 1.5 X lo6 photoelectrons during the period of measurement of XI. Such a value is considered to be representative of modern DWS and SWS spectrometers since photoelectron pulse rates generally lie in the range 1 X lo6 to 1 x lo9 s-’ and measurement into 5 x IO-’ s. tervals range from 1 x Curve A in Figure 2, for which /3 = 0 and K = 1, represents the SWS situation (9-13). As in the case of independent error, it can be seen that (70ulA)s is essentially proportional to (1 - p)-’; however the influence of an interfering species (Le. D > 0) is significantly smaller than for independent error. The absorbance corresponding to minimum square root error is obtained by numerical evaluation of the transcendental equation
1.152 A1(1OAl + p K ’ l O P A l ) = 10Al + K - 11()PA 1 which is the derivative with respect to Al of Equation 12. For K = 1 and 0 5 p 5 0.9, (70uU)s is a minimum between values of A l of 0.86 and 0.99 which is approximately twice the corresponding absorbance for independent error as expected 2172
ANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
loo[
t
3 9
B
lo[\
t 0
1
1
3
Absorbance
Figure 2. Square root error. Curves A, B, C, and D are for values of fi of 0.0, 0.25, 0.50, and 0.75, respectively, where K = 1. These curves are computed for E,,,-’ (2 meR,Af) = 5 X lo5 (Equation 11) from consideration of Equations 9 and 11. Proportional Error. When the standard deviation of the signal intensity is directly proportional to the magnitude of the signal, the associated error has been termed Type I11 (9-12) or Proportional (13)error. Such noise, which includes light source flicker, beam modulation noise, and noise resulting from variations in sample presentation, can have substantidy different contributions to DWS and SWS. Uncertainty introduced by the finite resolution of absorbance readout devices has the same functional form as proportional error and is included in this section even though the actual noise is independent of signal intensity. The standard deviation of signal intensity due to proportional noise is equal to .$& (10) where ip is the photoancdic current, R, the OA circuit resistance, and is the so-called “flicker” factor. f is thus the standard deviation per volt and depends on bandwidth for most sources. Writing the product i,R, as a voltage gives the standard deviations in signals a t wavelengths X1 and h2 as 01
= ~lEO1lO-(D+A’)
u* = ~ZKEO1lO-(D+@Al) Inserting these standard deviations into Equation 8 then gives the percent RSD for proportional error as
It can be seen from Equation 12 that ( % u u ) p is exactly proportional to (1- @)-’and inversely proportional to Al so that no minima occur. Also, by contrast with independent and square root errors, proportional error is independent of the presence of an interfering species. Figure 3 shows a plot of ( % u l ~ ) pvs. A l from Equation 12 for several values of p and a value of El = E2 = 4 x The flicker factor value chosen for illustration is representative of that due to light source flicker (10, 12). Individual sources of proportional error are considered below. Light Source Flicker. Proportional errors in DWS are usually dominated by that due to light source flicker which is a type of ‘ ‘ l / f ’ noise, so-called because of the inverse
Absorbance
Figure 3. Proportional error. Curves A, B, C, and D are for values of p of 0.0, 0.25, 0.50, and 0.75, respectively, where K = 1. These (Equation 12) curves are computed for E , = t 2 = 4 X
frequency dependence of its power spectrum (21). The contribution of source flicker to the percent RSD is obtained simply by writing E, = = E2 in Equation 12 where E, is the is flicker factor for the light source. Since t, = 4 X representative of flicker factors reported (10, 12) for a conventional tungsten lamp, the curves in Figure 3 reasonably describe the theoretical light source flicker in DWS when a tungsten lamp is employed. In practice, however, it is advantageous to use higher intensity light sources such as a tungsten iodide lamp, a xenon arc, or a CW laser. Such high intensity sources are generally more difficult to regulate than weak sources so that larger light source flicker factors may be encountered. Nevertheless, the effective reduction of source flicker noise by modulation and of shot noise by increased Eol can result in improved precision. B e a m Modulation Noise. The technique of mod.ulation is often used specifically to combat l / f noise in SWS. In the case of DWS, however, modulation is intrinsic to instrumental operation. Where a single DWS detector is employed, the transmitted beams are alternately monitored and any l / fnoise or drift which is correlated within the time frame of the modulation cycle is rejected. Where separate detlectors are used to monitor each wavelength continuously l/f noise is completely correlated and rejected although differential drift between detectors may become significant. While modulation may reduce the influence of correlated l / f noise (e.g., light source flicker) it can also contribute pattern noise a t the modulation frequency. In terms of the classification used herein, the contribution of this noise depends upon the way in which modulation is achieved. Two alternative methods can be distinguished. The first, or beam modulation, employs two distinct wavelength isolation devices whose fixed wavelength outputs are alternately sampled. The second, or wavelength modulation, employs a single wavelength isolation device whose output is wavelength modulated. Noise generated by beam modulation contributes to proportional error but that generated by wavelength modulation contributes to constant error which is discussed in a later section. Beam modulation error is given by Equation 1 2 in which the flicker factors f l and t2 now refer to the uncertainties
associated with beam modulation at each wavelength. These are determined by the way in which modulation is achieved, the quality of the multiplexing optics, and the mode of signal processing employed. Experimental measurements of flicker factors due to beam modulation are not available; however, they would need to be comparable to or greater than about 4 x for beam modulation noise to exceed that due to conventional light source flicker. Beam modulation in DWS is generally accomplished mechanically by means of a rotating sector or a rotating or vibrating mirror ( 3 ) . If one assumes that modulation flicker factors are determined primarily by the uncertainty associated with mechanical movement, it seems unlikely that their for low frequency (6100 Hz) magnitude will exceed 4 X operation. Higher frequency ( - 1 kHz) operation may, however, give rise to larger flicker factors as a result of vibration. Sample Presentation. Variations in sample presentation also give rise to proportional error. Such variations originate from vibration and from irreproducibility of the sample cell position in the cell holder and of the cell holder itself (12). Signal imprecision then results from (i) variations in sample and cell transmission characteristics, (ii) variations in path length, and (iii) variations in photocurrent which occur when the refracted light beam strikes regions of the photocathode surface having different quantum efficiences. Mechanisms (i) and (iii) both produce proportional errors but mechanism (ii) gives rise to a constant error discussed in a later section. Since, in DWS measurement, both beams pass through the same sample cell and are detected b y the same photomultiplier, sample presentation errors will be correlated, and thus rejected, to the extent that they are independent of wavelength. Variations in cell transmission are due to reflection, which is essentially independent of wavelength, and to refraction and scattering which, although wavelength dependent (22, 23), are relatively constant over the short wavelength interval normally employed in DWS. Furthermore, under most operational circumstances, the photocathode response is relatively constant a t the two wavelengths. Consequently, the uncorrelated fractions of the flicker factors El, t2 in Equation 1 2 will be small so that proportional error due to sample presentation is negligible in DWS. By contrast, Rothman et al. (12) have shown that flicker factors due to sample presentation in SWS are commonly as large as (and often larger than) El = 6 X (Ez = 0) which results in percent RSD values approximately an order of magnitude greater than those depicted by curve A (0= 0 ,K = 1) in Figure 3. It is apparent, therefore, that sample presentation errors, which are frequently the limiting proportional error in SWS, are unlikely to be limiting in DWS. Readout Resolution. The finite resolution of absorbance readout devices such as analog meters, digital displays, and recorders can introduce considerable uncertainty into spectrophotometric measurement (12). Although the associated standard deviation, urn,is independent of signal intensity the percent RSD, given by
has the same functional form as proportional error. This is because uncertainty is contributed in the final electronic stage of the spectrophotometer after logarithmic conversion. Readout resolution error has been discussed for transmittance measurements by Rothman et al. (12) who have shown that it can severely limit the precision of SWS measurement. Thus, an analog meter can contribute errors which are 10 to 70 times greater than those depicted in Figure 3. Errors contributed by 31/2 and 4lI2 digital panel meters, ANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
2173
respectively, were shown (12)to be 3 to 14 and 0.3 to 1.4 times as great as those in Figure 3. Although not considered by Rothman e t al. (12), most analog recorders would also be expected to contribute substantial readout error. This error can, of course, be minimized by employing scale expansion and by reading the recorder trace only after several time constant periods have elapsed. I t can be concluded, therefore, that readout errors will constitute the limiting error having proportional form in both SWS and DWS unless a high resolution readout device or scale expansion is employed. I t will be noted that, for readout errors, exactly the same considerations apply to DWS as to SWS except that the percent RSD is increased by a factor of (1 - @)-I in DWS for a given value of A l . Constant Error. Where the noise or uncertainty associated with a signal is proportional to molar absorptivity or path length, the resulting error will be termed Constant error. Such errors are due to noise associated with wavelength modulation and with variations in sample path length and are described by separate error functions. Wavelength Modulation Noise. Where the output of a single wavelength isolation device is modulated to produce alternate beams of wavelength X1 and X2, the standard deviation of the absorbance, A , a t each wavelength is given by yMA where y is the uncertainty in wavelength determination and MA and MD are the slopes of the absorbance and interferent curves a t the wavelength in question. Assuming that y is the same at both wavelengths the total standard deviation of AA is
where the subscripts refer to wavelengths X1 and h2. The percent RSD in AA due to wavelength modulation noise is then
Equation 14 shows that when interfering species are absent ( D = 0), or when the values of MDl and M D are ~ small, the percent RSD due to wavelength modulation noise is independent of A I but may be strongly dependent on the absorptivity ratio @. When interferents are present and the values of MA1 and MA2 are substantially less than those of M D , and M D 2 however, , the form of Equation 14 approaches that for proportional error at small A I and tends towards a constant value a t large AI. Wavelength modulation can be achieved by means of an oscillating grating, mirror, or slit in the wavelength isolation device. The last of these methods is commonly employed in differential UV-visible spectrometers which constitute a s u b s e t of DWS instruments. The maximum value of y in Equation 14 will then be the measured standard deviation of either wavelength. These have been determined (16, 24) for the cases of an oscillating slit (y = 0.01 nm at 546.1 nm), an oscillating refractor plate (y = 0.001 nm a t 253.7 nm), and an oscillating grating (y = 0.0004 nm at 283.3 nm). Values of the slopes MAl, M A 2in Equation 14 range from 0.15 to 0.004 nm-l and are most commonly around 0.05 nm-l. Using this value for both MA1and M A 2and choosing y = 0.01 nm, @ = 0.5, and D = 0, Equation 14 gives a maximum constant error of 0.12%. By comparison with proportional and square root errors in DWS, a constant error of this magnitude is only marginally significant. It appears reasonable, therefore, to assume that 2174
ANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
wavelength modulation error is usually insignificant except, perhaps, where a noisy modulation technique is employed in conjunction with narrow absorption peaks and small AX. Such conditions are most likely to occur in differential spectrometers. Path Length Variation. Variations in the length of the light path passing through different samples can occur when a cell is tilted so that the angle of incidence is 7r/2 f 0 rather than 7r/2 (12). The measured DWS absorbance from Equation 1 is thus
AA =
(€1
-
€2)
bc
SeC
6
Since all values of 0 lead to a positive error, random path length variations give rise to a systematic error which contributes to the accuracy rather than the precision of measurement. Summation of Errors. The precision of the overall DWS measurement can be closely approximated by considering only those errors which predominate in each functional category. For most conventional DWS spectrometers, these will be dark current noise, photon shot noise, source flicker, and wavelength modulation noise. For SWS spectrometers, however, it has been shown that sample presentation uncertainty normally replaces source flicker as the limiting proportional error(12). It must be stressed, however, that individual spectrometer components and operating conditions differ so that the relative magnitudes of different noise contributions may vary considerably. For example, readout uncertainty or beam modulation noise may predominate in the proportional category where a readout device of limited resolution or a noisy modulation technique is employed. In order to illustrate how the predominant sources of error in each functional category contribute to the overall DWS measurement precision, each is evaluated as a function of absorbance in Table I, both in the presence and absence of an interferent. The limiting contributions to SWS measurement are presented for comparison. The data in Table I are calculated for @ = 0 and K = 1. For dark current noise. gI/EO1 = and lo6 photoelectrons are assumed to arrive during the measurement interval for the purpose of calculating shot noise error. A sample presentation flicker factor of (1 =6X is employed for SWS and a source flicker factor is used for DWS. Consideration of of F1 = E2 = 4 X constant errors and proportional error due to limited readout resolution is omitted since the former are small and independent of absorbance and the latter can be made negligible by scale expansion or choice of a high resolution readout device. It should be noted, however, that constant wavelength modulation error may become significant near the absorbance minimum (Table I) when no interfering species are present. DISCUSSION Examination of the data presented in Table I and in Figures 1-3 shows that the relative contributions of the different types of noise vary considerably as a function of absorbance in both DWS and SWS. This variation is particularly apparent for DWS measurement in the presence of the interferents due to the fact that independent and square root errors are increased by factors of loD and 10D’2respectively, while proportional error is unaltered. Thus, although independent dark current noise normally makes only a minor contribution to the overall DWS precision it can become a significant error when an interferent is present. As pointed out previously, the intrinsic precision of SWS is usually superior to that of DWS under comparable conditions due to the fact that the effective absorptivity (el - €2) is usually smaller in DWS. Nevertheless, the advantage gained by using a single sample cell, and thereby reducing sample presentation errors, can result in DWS having greater overall
Table I.
Relative Percentage Error, % u A , for SWS A, = 0.01
Source Dark current Photon shot Sample presentation
D 0
-
0
0.06 6.2 26.1
( x(%UA)2)1’2
0
26.8
0.05
0.1
0.2
0.5
1.0
1.5
2.0
2.5
3.0
0.01 1.3 5.2
0.007 0.65 2.6
0.004 0.35 1.3
0.003 1.8 0.52
0.004 0.14 0.26
0,009 0.17 0.17
0.02 0.22 0.13
0.06 0.31 0.10
0.15 0.46 0.09
5.4
2.7
1.4
0.55
0.30
0.24
0.26
0.33
0.50
Relative Percentage Error, Sb U A , for DWS A, =
Source Dark current Photon shot Source flicker
D 0 0
-
0 0.5 1.0 1.5 2.0 2.5 3.0
(Z(%UA)2)*’2
0.01 0.06 6.2 2.5 6.7 11.3 19.7 35.0 62 111 204
0.05
0.01 1.3 0.49 1.4 2.3 4.0 7.1 12.7 22.8 42.1
0.1
0.007 0.65 0.25 0.70 1.2 2.1 3.7 6.6 11.8 21.8
0.37 0.63 1.1 2.0 3.5 6.3 11.8
precision that SWS. It should be stressed, however, that this precision advantage is manifest only in situations where the absorptivity ratio, p, can be made close to zero without resorting to large AA. In such situations DWS may be the method of choice for performing high precision measurements a t low absorbances. The main advantage of DWS is, however, its ability to discriminate against interferents. In this situation the precision attainable is highly dependent on the relative magnitudes of AI and D as illustrated in Table I. While a rather high relative dark current noise level has been chosen for the purpose of illustration in Table I, it is apparent that high precision operation over a wide absorbance range is limited to samples for which D is quite small (0.5-1.0). Furthermore, operation at even moderate precision is likely to be impractical for D > 2.0 except in cases where a limited intermediate absorbance range is acceptable. The precision data presented in Table I point out those sources of error which must be minimized in order to improve DWS measurement precision under different conditions. They also suggest several operational and instrumental parameters which can be optimized. Operationally, the only parameters which are readily adjustable are the absorbances of the analyte and interferent and the wavelengths employed. Thus, dilution or concentration of the sample (which maintain the ratio A 1 / Dconstant) can result in a small but significant improvement in precision. A much greater effect may be achieved by variation of the wavelength difference, AA, so as to minimize the absorptivity ratio, p. In this regard there is considerable flexibility in the choice of wavelengths when the interferent is a scattering material; however, for an absorbing species, this choice is largely determined by spectral constraints (23). While i t is desirable to employ a wavelength difference which achieves @ 0, it should be stressed that large LA may result in there being a significant uncorrelated fraction of the wavelength uncertainty in each beam. In such circumstances, the proportional sample presentation error increases substantially and is enhanced where thermal gradients, mechanical instabilities, or flow cell cavitation effects occur. Consequently, it is not always justifiable to assume that sample presentation errors are negligible in DWS. It is noteworthy that differential spectrometers, which operate with a very small wavelength difference, maintain p very close to unity. Consideration of Equation 8 shows that the precision of such instruments must generally be poor, being greatest a t the point(s) of maximum slope of the absorbance
-
0.5
1.0
0.003 1.8 0.05
0.004 0.14 0.02
0.18 0.32 0.56 1.0 1.8 3.3 6.3
0.14 0.25 0.46 0.82 1.5 2.9 6.3
0.2 0.004 0.35 0.12
1.5 0.009 0.17 0.02 0.17 0.30 0.53 0.97 1.9 4.1 10.6
2.0
2.5
3.0
0.02 0.22 0.01
0.06 0.31 0.01
0.15 0.46 0.008
0.22 0.39 0.72 1.4 3.08 7.9 22.8
0.31 0.60 1.12 2.5 6.3 18.2 55.8
0.48 0.93 2.04 5.3 15.2 46.5 146
curve and lowest at the absorbance peak. Furthermore, some of the wavelength modulation methods employed may contribute substantial wavelength modulation noise which is also maximized at the absorbance peak. Such considerations raise serious questions about the advisability of using differential UV-visible spectrometers for quantitative determinations. This is particularly pertinent to their current use for measuring gaseous air pollutants. A second approach to improvement of DWS precision involves modification of those spectrometer components which contribute the limiting sources of noise in each category. In this regard it is important to emphasize that reduction of independent and square root errors (which depend on D ) is generally preferable to reduction of proportional error in situations where interferent levels are substantial. It is equally important to consider the frequency spectra of the limiting noise contributions since frequency dependent noise can often be significantly reduced by suitable modulation whereas frequency independent white noise cannot. It is generally desirable, therefore, to increase frequency dependent noise relative to white noise. The foregoing considerations dictate that high light source intensity should be actively sought in improving DWS measurement precision. Thus, the greatly enhanced signal voltage, Eel, reduces the contribution of dark current noise (and to some extent shot noise) to the signal while additional source flicker noise (which is l/f noise) can be effectively reduced by modulation. Also, since source flicker contributes a proportional error, its contribution does not increase with
D. Probably the most obvious, though most frequently neglected, source of error in both DWS and SWS is the resolution of the readout device. Since this error may often limit the overall measurement precision, it is necessary to employ either a 4l/2 digital panel meter or scale expansion where high precision measurements are required.
CONCLUSIONS The following general conclusions can be drawn regarding the precision of dual wavelength spectrophotometric measurement: 1. The principal sources of imprecision in DWS are usually photon shot noise, source flicker, and readout noise when sample interferents are absent. 2. The use of intense light sources in conjunction with suitable high frequency modulation (1CC-1ooO Hz) is generally desirable for high precision DWS measurement. ANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
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3. In the absence of interferents, the precision and detection limits of SWS and DWS are normally comparable. DWS may, however, be more precise than SWS where the latter measurement has substantial associated sample presentation error. 4. In the presence of interferents, DWS precision is markedly reduced because of increase of independent and square root errors. In particular, dark current noise may become significant under these conditions. 5 . The use of very small wavelength differences, as in differential spectroscopy, results in poor measurement precision. While the above conclusions are generally valid, it must be stressed that the different functional dependences of errors of different types on A I and D lead to a complex overall dependence which will vary with individual instruments and measurements. T o some extent, therefore, it is appropriate to consider each instrument and each type of measurement as a special case. LITERATURE CITED (1) (2) (3) (4) (5) (6)
B. Chance, Rev. Sci. Instrum., 22, 634 (1951). B. Chance, Methods Enrymol., 24, 322 (1972). T. J. Porro, Anal. Chem., 44 (4), 93A (1974). S . Shibata, M. Furukawa, and K. Goto, Anal. Chim. Acta, 46, 271 1969). S Shibata, M. Furukawa, and K. Goto, AnalChim. Acta, 53, 369 1971). S.Shibata, K. Goto, and Y. Ishiguro, Anal. Chim. Acta, 62. 305 1972).
(7) (, 8,) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21)
(22) (23) (24)
R . L. Sellers, G. W. Lowry, and R. W. Kane, Am. Leb, March (1973). S. M. Gerchakov. Saecfrosc. Lett.. 4. 403 (19711. J. D.Ingle, Jr., and's. R . Crouch, Anal Chem., 44, 1375 (1972). J. D. Ingle, Jr., and S. R . Crouch, Anal Chem., 44, 785 (1972). J. D. Ingle, Jr., Anal. Chem., 45, 861 (1973). C. D. Rothman, S . 3. Crouch, and J. D. Ingle, Jr.. Anal. Chem., 47, 1226 (1975). H. L. Pardue, T. E . Hewitt, and M. J. Milano, Clin. Chem. ( Winston-Salem, N . C . ) . 24, 1023 (1974). P. C. Kelly and G. Horlick, Anal Chem., 45, 518 (1973). K . Steiglitz, "An Introduction to Discrete Systems", Wiley, New York, N.Y., 1974. K . L. Ratzlaff. K . R. O'Keefe, and E. F . S . Natusch, manuscript in preparation. V i . J . McCarthy, in "Spectrochemical Methods of Analysis", J. D. Winefordner, Ed., Wiley, New York, N.Y., 1971, pp 493-518. RCA Photomuttiplier Manual, RCA E!ectronic Components, Harrison, N.J., 1970. H. V. Malmstadt, M. L. Franklin, and G. Horlick, Anal. Chem.. 44 (8), 63A (1972). M. L. Franklin, G. Horlick, and i-l.V. Malmstadt, Anal. Chem., 41, 2 (1969). H. V. Malmstadt, C. G. Enke, S. R. Crouch, and G. Horlick, "Optimization of Electronic Measurements", W. A. Benjamin, Menlo Park, Calif., 1973. W . Heller, H. L. Bhatnager, and M. Nakagaki, J Chem. Phys., 36, 1163 (1962). K. L. Ratzlaff 2nd D.F.S. Natusch, submitted to Anal. Chem. J. D.Defreese. Ph.D. Thesis, University of Illinois, Urbana, IN, 1975.
RECEIVED for review January 19, 1976. Resubmitted December 20,1976. Accepted August 22,1977. This work was supported by the National Institute of Environmental Health Sciences, NIH, under Grant No. 7-R01 ES 01472- 01.
Supply and Removal of Sample Vapor in Graphite Thermal Atomizers Wim M.
G. T.
van den Broek and Leo de Galan"
Laboratorium voor Analytische Scheikunde, Technische Hogeschool, P. 0. BOX 5029, Delft, The Netherlands
The timedependent supply and removal of sample atoms have been measured separately for two commercial graphite thermal atomizers. I n either type the release of the atoms from the graphlte wall Is determlned by the wall temperature and descrlbed by an Arrhenlus-type rate constant. With common heatlng rates, the equivalent tlme constant Is about 1 8. The removal of the atoms from the cell depends on the type of atomizer used. I n the Varian Techtron mlnlfurnace, dlffuslon Is the domlnatlng process and the equlvalent time constant is less than 70 ms. I n the larger Perkln-EIPner furnace operated under flow conditions, convection leads to equally small time constants. I f , however, the argon flow is slopped completely, dlffudon and, to some extent, expansion raise the time constant to about 1 s. Theoretical analysis and experimental measurement show that only In the latter case some 25% of the sample can be contained in the cell. Under common operating conditions, this efficlency is less than 10%.
The great sensitivity and the correspondingly low limits of detection of the thermal atomizers in atomic absorption spectrometry stem from the ability to contain a substantial amount of the analyte in the observation zone for a finite period of time. The basic condition towards the realization of this goal has first been formulated by L'vov ( 1 ) : the rate of supply of the analyte into the observation region must a t least be equal to its rate of removal from this zone. There is no doubt that constricted, furnace-type atomizers fulfil this 2176
ANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
condition better than open filament-type atomizers. This advantage partly explains the waning interest in the latter type atomizers in favor of the enclosed cylindrical graphite cuvettes. Of the different designs described in the literature or commercially available, two extremes have beccme popular and typical: a Varian Techtron minifurnace with a lengthldiameter ratio of 3 and a Perkin-Elmer furnace with a length/diameter ratio of 6. However, a t present very little is known about the timedependence of the analyte supply into and its removal from either type of furnace. Henceforth, their efficiency in containing the analyte inside the graphite enclosure i s equally unknown. I t is true that the mechanism of sample release from the graphite wall has received increased attention in the past few years (2-4), but the time-dependence and the shape of the transient signal has been considered in only a few papers. Fuller ( 5 ) assumes a very simple model consisting of two exponentials to describe the release and removal of copper in a Perkin-Elmer HGA-70 graphite furnace. He concludes from curve fitting his transient absorbance signal that the rate of removal exceeds the supply rate 3-20 times, depending on the atomization temperature. A much more thorough study has been described by Tessari, Torsi, and Paveri-Fontana (6-8), who describe the supply of the analyte by an Arrhenius-type rate constant and relate the removal of the analyte to diffusion and convection. Unfortunately, these investigations refer t o a home-made open graphite rod atomizer and their data cannot be used for the enclosed commercial furnaces. Recently, Sturgeon and Chakrabarti
