Signal-to-noise ratio theory of fixed-time spectrophotometric reaction

Effect of noise and rate computational approach on the precision of rate measurements. J. D. Ingle , M. J. White , and Eric D. Salin. Analytical Chemi...
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Signal-to-Noise Ratio Theory of Fixed-Time Spectrophotometric Reaction Rate Measurements J. D. Ingle, Jr.,1 and S . R. Crouch Department of Chemistry, Michigan State University, East Lansing, Mich. 48823 The effect of noise on the precision of automated spectrophotometric rate measurements is discussed. Exact and simplified signal-to-noise ratio (S/N) expressions are developed for the fixed-time rate measurement approach. The precision of a rate measurement i s shown to be highly dependent on the magnitude of the photocathodic current and source flicker as well as the measurement time. For fast reaction rate measurements with a stopped-flow apparatus, the relative precision is primarily limited to greater than 1% by photocurrent shot noise, Slow reaction rate measurements are shown to be limited by source flicker noise in many cases.

DURING THE PAST FEW YEARS numerous applications of reaction rate measurements to quantitative chemical analysis have been made (2-3). A major factor which has contributed to the increased popularity of reaction rate methods has been the development of automated instrumentation for the computation of reaction rates. Analog ( 4 , 5 ) , digital (6, 7), hybrid (a), and computer-based systems (9-11) have all recently been described for the automation of reaction rate measurements. Automated rate computation systems have become increasingly sophisticated, and the precision of rate measurements on simulated transducer outputs has increased to the 0.01-0.1 level. However, the precision of individual reaction rate measurements on real chemical systems has not been increased significantly, and relative standard deviations of 1-3z are typical (6, 7), particularly when rates are measured using spectrophotometric reaction monitoring techniques. In many of the methods, ensemble-averaging increases the precision, which, coupled with the observation that the rate computation systems are a t least a n order of magnitude more precise than overall spectrophotometric rate measuring systems, suggests that random errors in either the reaction cell (errors in sample and reagent introduction, mixing, and temperature control) or in the spectrophotometric monitoring system, or in both are responsible for the imprecision,

z

Present address, Department of Chemistry, Oregon State University, Corvallis, Ore. 97331. (1) H. B. Mark, Jr., and G. A. Rechnitz, “Kinetics in Analytical

Chemistry,” Wiley-Interscience, New York, N.Y., 1968. (2) G. G. Guilbault, ANAL.CHEM.,42, (5), 334R (1970). (3) H. L. Pardue, in “Advances in Analytical Chemistry and Instrumentation,” Vol. 7, C. N. Reilley and F. W. McLafferty,

Ed., Wiley-Interscience, New York, N.Y., 1968, pp 141-207. (4) G. E. James and H. L. Pardue, ANAL.CHEM., 40, 796 (1968). (5) E. M. Cordos, S. R. Crouch.andH. V. Malmstadt, ibid.,p 1812. ( 6 ) R. A. Parker, H. L. Pardue, and B. G. Willis, ibid., 42, 56 (1970). (7) J. D. Ingle, Jr., and S . R. Crouch, ibid., p 1055. (8) S. R. Crouch, ibid., 41, 880 (1969). (9) G. E. James and H. L. Pardue, ibid.,p 1618. (10) G. P. Hicks, A. A. Eggert, and E. C. Toren, Jr., ibid., 42,729 (1970).

In this report, signal-to-noise ratio (S/N) theory is utilized to evaluate the precision to be expected from automated reaction rate measurements when noise in the spectrophotometric monitoring system is the limiting factor. The two most popular methods of computing reaction rates, the fixedtime method and the variable-time method (12), are considered. Specific expressions are developed for spectrophotometric fixed-time measurements. Measurements on fast reactions (measurement times 1 sec) are both discussed. The optimization of measurement system parameters for improving the precision of fixed-time rate measurements is discussed based upon the theoretical equations developed. GENERAL CONSIDERATIONS

One of the major limitations of spectrophotometric reaction rate measurements compared to equilibrium-based measurements is that the S/N of the measurement is inherently smaller since only a portion of the total available signal is utilized. Because measurements are made on a dynamic signal, the noise equivalent bandwidth of the measurement system cannot be reduced indefinitely to improve the S/N without distortion. Since the S/N indicates the instrumental precision with which a measurement can be made, a knowledge of the factors which affect the S/N can be extremely useful in evaluating the expected precision and in optimizing experimental conditions. The two integral methods of obtaining rate data, the fixedtime and the variable-time method, are most often used when system noise is the limiting factor. The basic components of the automated spectrophotometric rate measurement system considered here are a single beam spectrophotometer with a thermostated reaction cell, a signal modifier (current-tovoltage converter and/or logarithmic amplifier), and the rate computation system (fixed-time or variable-time). The output of the signal modifier, M , is considered to be the electrical quantity from which rate information 1s to be extracted. For the integral rate measurement approaches, the desired information is encoded in the value of A M / A t , which is measured by holding either the numerator (variabletime method) or denominator (fixed-time method) constant. The S/N of an integral rate measurement can be found by consideration of the response and noise characteristics of the spectrophotometric reaction monitoring system, the signal modifier system, and the rate computation system. Although several different signal-to-noise ratios can be defined, it is most useful to define a S/N in which the measured quantity AM/At is the signal S, and the relative standard deviation in the measured quantity, us, is therms noise. The relative standard deviation in the signal, u s , can be obtained by applying propagation of error mathematics [As] AM' u ~ t ’ ~ ] ’ to AM/At, which yields us = (AM)* (At)

[

(

(11) B. G. Willis, J. A. Bittikofer, H. L. Pardue, and D. W. Margerum, ibid., p 1340.

~

12) W. J. Blaedel and G. P. Hicks in “Advances in Analytical Chemistry and Instrumentation,” Vol. 3, C. N Reilley, Ed.,

Wiley-Interscience, New York, N.Y., 1964, pp 126-40.

ANALYTICAL CHEMISTRY, VOL. 45, NO. 2, FEBRUARY 1973

333

‘ ~

If the ratio of S to us is taken, the signal, S = AM/Ar, cancels, and the signal-to-noise ratio for an integral rate measurement, is given by

In Equation 1, AM = M Z - Ml and At = tz - tl, where M2 and Ml are the signal modifier outputs at times tz and t l , respectively. Also, U ~ and M u ~ A t 2are the variances in AMand At, respectively. It is assumed in Equation 1 that At and AM are independent and that U A M < AMand UAt < At. Equation 1 is a general equation which applies to either of the integral rate measurement approaches. Specific considerations for spectrophotometric fixed-time and variabletime reaction rate measurements lead to simplification of the general equation and are discussed below. Fixed-Time Measurements. In the fixed-time approach to reaction rate measurements, At in Equation 1 is held constant. Modern fixed-time computational systems use highly reproducible time bases derived from crystal oscillators or other highly stable oscillators (7). Thus, the reproducibility of the time base is usually much higher than the reproducibility of measuring the signal modifier output. Therefore, the variance in At, u A t 2can usually be assumed negligible in Equation 1. In this case, S/N for a fixed-time rate measurement, can be reduced to

of the great complexity in developing the probability density functions. Hence, the specific S/N treatment given below is applicable only to spectrophotometric fixed-time measurements. S/N EXPRESSIONS FOR SPECTROPHOTOMETRIC FIXED-TIME RATE MEASUREMENTS Transmittance Monitoring. In many of the spectrophotometric reaction rate methods which have been described, a signal proportional to the transmittance T of the solution is followed. In this case, M in Equation 2 can be replaced by the transmittance, which yields

(4) where AT = TZ- Tl, TZand T I are the transmittances at times tz and t l , respectively, and u T I 2and uTg2are the variances in Tl and T2,respectively. Usually the transmittance is obtained as a voltage ratio E p / E Twhere , Ep is the photocurrent output voltage (voltage from the current-to-voltage converter which follows the photomultiplier or phototube transducer after dark current has been subtracted) with the sample solution in the sample cell and E, is the corresponding photocurrent output voltage with the reference solution in the sample cell. In this case, Equation 4 can be written ErAT (i)Rf

where uaw12 and uMZ2 are the variances in Ml and M2, respectively. The times tl and rz at which the signal modifier output is measured are referred to some starting time to, which is assumed to be precisely equal to the time of initiation of the reaction. Variable-Time Measurements. In the variable-time approach to reaction rate analysis, A M in Equation 1 is held constant. Imprecision in measurements is caused by noise on the signal modifier output, which can turn the signal level sensors on and off at times other than tl and t z , fluctuations in the signal sensor reference levels, and fluctuations in the characteristics of the signal level sensors. For modern, stable, high resolution signal level sensors, such as voltage comparators, this latter source of imprecision, and hence uA1w2,is usually negligible. Under these conditions, the signal-tonoise ratio for a variable-time rate measurement, (S/N)Rp, can be expressed as

(3) where u i I 2and u t g 2are the variances in determining the times tl and tz at which the signal modifier reaches levels M I and M z , respectively. Determination of requires knowledge of the variance in the times tl and t i . Evaluation of these variances involves the development of probability density functions around times tl and t 2 . Low frequency lifnoise components are expected to influence the triggering of the signal level sensors differently from white noise components. Thus, not only the total magnitude of the noise present must be considered, but so must the shape of the noise power spectrum. In addition, the voltage transfer characteristics, the response speed, and hysteresis of the voltage level sensor must be considered. Attempts to develop S/N expressions for spectrophotometric variable-time measurements were unsuccessful because 334

= (UEi2

+

‘JEgz)l‘z

(5)

where El and Ez are the photocurrent output voltage with the reaction proceeding at times tl and t 2 , respectively, and u E , 2 and u~~~are the corresponding variances in El and E?. Equa~ tion 5 follows from Equation 4 because uT12= u E , ’ / E ~and UT^^ = UE22/Erz. Usually in a reaction rate measurement, only one initial setting of’ the 100% T and 0% Tis made. Hence, the variance involved in these operations does not enter Equation 5. However, the variance in the dark current output voltage must be considered since it influences the measurement of E, and E2. The variance in determining the photocurrent output 2 uE22)is given by the sum of the voltage at a given time ( u E 1or group variance terms, u2, b z , c2, which have been previously discussed (13, 14) and are defined here in Appendix I. Equation 5 can, therefore, be rewritten as ETAT (6) (:)Rf = [al’ bl2 c12/2 asz bZ2 C Z ~ / ~ ] ” ~

+ +

+ + +

where the subscripts 1 and 2 denote the value of the group variance terms at times tl and t?, respectively. The c z variances are divided by two because sources of variance independent of the photocurrent affect the measurement of E1 or EZ only once. Because the group variance terms for the sample solution at times tl and f 2 are simply related to the group variance terms for the reference solution, Equation 6 is better expressed in terms of the group variances for the reference solution, ur2, b r 2 ,cT2(14) by use of the relationships u12 = ar2T1,uz2 = aT2T2,b12= bT2Tl2, bZ2= br2T22,and c12 = cZ2= cr2as shown in Equation 7

(13) J. D. Ingle, Jr., and S. R. Crouch, ANAL.CHEM., 44, 785 (1972). (14) Ibid., p 1375.

ANALYTICAL CHEMISTRY, VOL. 45, NO. 2, FEBRUARY 1973

where the subscript r refers to the group variance terms for the reference solution in the sample cell. This is a more convenient form because the group variances for the reference solution can be measured before the reaction is initiated and are independent of the concentration of the analyte. Equation 7 can be used directly to calculate the precision of a reaction rate measurement using transmittance monitoring. To evaluate the signal-to-noise ratio of the measurement, E, must be known, the group variance terms must be calculated or measured, and Tl and T2 must be evaluated for a specific measurement. Equation 7 , however, is not a convenient equation to use for fixed-time rate measurements because T I and T, vary with the concentration of sought-for species. Equation 7 can be greatly simplified by a few approximations, which are discussed below, Approximations to Exact Equation. In the majority of reaction rate analyses where transmittance monitoring is used, an absorbing product is formed from a solution whose initial absorbance is very nearly zero. To use the fixed-time method with linearity with transmittance monitoring, the change in transmittance AT during the measurement time must be less than for the fastest reaction studied (15). Thus, if this latter restriction is combined with the assumption that a product is being followed whose initial absorbance is zero, it is reasonable to approximate TI = T2 = 1, which allows the simplification of Equation 7 to

Zz

)(;

=Rf

(20,'

+

E,AT 2br2

+ cT2)l'*

(8)

The approximations used to obtain Equation 8 imply that the magnitude of the noise remains constant over the small transmittance integral near 100 T where measurements are made. Equation 8 will thus provide a slightly low, but reasonably accurate, estimate to the true signal-to-noise ratio. Equation 8, however, is much easier to work with since only AT varies with the concentration of sought-for species in a particular analysis. If the initial absorbance of the reaction varies with the analyte concentration, as would be the case if the decrease in absorbance of an absorbing reactant were followed, the more exact Equation 7 must be used. In most spectrophotometric measurements, particularly in the case discussed here where measurements are made near 100% T , the Johnson noise, amplifier noise, and phototube flicker noise terms are small compared to other noise sources. Furthermore, the particular rate computation system used in an analysis normally has a finite full-scale voltage setting, Eo, to which the reference voltage E, (10Oz T)is usually adjusted. This limitation may be imposed by the full-scale range of a recorder, voltage-to-frequency converter, analog-digital converter, etc. Thus, if the noise sources discussed above are assumed negligible and the restriction that EO = E, = (iCp),mR/is imposed [where (iC& is the effective photocathodic current with the reference solution in the sample cell], Equation 8 can be written in expanded form as

(a/ =

0.707 AT ( i c p ) , [2eAf(l f a)((iCp), i c d ) f ( i c p ) r 2 ( E 2

+

+

gR2/E02]1'2

(9)

Here, if a photodiode is used in place of a photomultiplier as the radiation transducer, the secondary emission factor a in Equation 9 is zero. ~~

(15) J. D. Ingle, Jr., and S . R. Crouch, ANAL.CHEM.,43, 697 (1971).

The most recent fixed-time instruments (5, 7) utilize integration over the total measurement time by computing the difference in the average voltages of two equal and adjacent time intervals. If rl and t2 are considered the center times of these two intervals, the total measurement time is ZAr, where Ar is the integration time for the measurement of either El or E2. Thus E, and E2 or TI and TZare average values of E or T during the two adjacent time intervals. Since the noise bandwidth of the reaction monitor and signal modifier systems must be much greater than 1/(2Ar) to prevent distortion of the signal modifier output, the noise bandwidth of the total rate measurement system is essentially determined by the integration time of the rate computer [Af = l/(ZAr)]. If this result is substituted into Equation 9, Equation 10 results.

Two limiting forms of Equation 10 are most important in a reaction rate analysis. First, the photocurrent shot noise limit occurs when the dark current shot noise, source flicker noise, and readout noise terms are negligible in Equation 10 compared to photocurrent shot noise. This is likely to, occur when the monochromator spectral bandpass must be small, when integration times are short (At < 1 sec), or when the response and efficiency characteristics of the lamp, monochromator, and photomultiplier are low and yield small photocathodic currents. In this case, Equation 10 simplifies to

If the photocathodic current (incident light level) or integration time is increased, source flicker noise eventually predominates and Equation 10 reduces to

Equations 1&12 all provide important insight in optimizing the signal-to-noise ratio and hence the measurement precision in a fixed-time reaction rate analysis as is discussed in a later section. Absorbance Monitoring. In many fixed-time reaction rate methods, the restrictions reqbired for linearity with transmittance monitoring cannot be tolerated, and direct absorbance measurements must be made. If the signal modifier includes a log-ratio amplifier with a 1 V per decade sensitivity, M in Equation 2 is replaced by the absorbance A , which yields

where A1 and A2 are the absorbances at times rl and r2, and ( T ~ and , ~g A Z are 2 the corresponding variances in the absorbances. Equation 1 3 can be transformed into a useful form in a manner very similar to the development for transmittance monitoring. The group variance terms for direct absorbance readout, u z 2 ,b12,and c z *are similar to those for transmittance readout, except that the readout variance becomes a b 2 type variance rather than a c2 type variance as has been previously discussed (14). The group variance terms for direct absorbance monitoring are defined in Appendix 11. Manipulation of all the group variance terms into group variances for the reference solution yields Equation 14, which

ANALYTICAL CHEMISTRY, VOL. 45, NO. 2, FEBRUARY 1973

335

The majority of reaction rate analyses monitor a n absorbing reaction product whose initial absorbance is near zero. If it is assumed as before that TI = T2 = 1, and that a n integrating fixed-time computation system is used with Af = 1/(2At), Equation 14 becomes

where it is assumed that many of variances terms [(vEp2)pm, u J 2 ,u A m p 2(vEd2)pm, , and v r 2 ]are negligible a t the high photocathodic currents near A = 0. Cases othzr than that considered in the assumptions to Equation 15 can be treated, but the more general expression, Equation 14, must be used. In the signal shot-noise limit, Equation 15 reduces t o

At the source-flicker limit, Equation 15 becomes 12

13

II 10 - L o g (i,p)l

9

8

($)R,

Figure 1. Dependence of S,’K on photocathodic current, integration time, and source flicker noise CL.

Ar = 10 msec, E = 0

h.

Ar =

r. Ar d. Ar e. A f

f. g.

= = =

Ar = At =

/I. Ar i. A t

= =

Ar

=

k. Ar

=

j.

=

2.303 E, AA __ [LZTI’(TI-I Ti-’) f 2bTl2 c,j’(TI-’

+

+

+ T‘-’)]’”

(14)

where the suhscript r again refers to the group variance terms for the reference solution. The relationships between the variances in the absorbance, uA,?and u9%?,and the group variance terms are presented in Appendix 111, along with the relationships which express the group variance terms at times tl and t2 to the group variance terms for the reference solution. Again Equation 14 can be used to calculate relatide precision of a fixed-time reaction rate measurement if the group variances are known, if the reference photocurrent output voltage E, is known, and if the values of T I , T2,and AA are known. Like Equation 7, Equation 14 is inconvenient because these latter values vary with the concentration of the sought-for species. Similar approximations to those used for the transmittance monitoring case lead to a more useful form than the exact expression. Simplified S ’N Expression. Although the restrictions on direct absorbance monitoring for reaction rate measurements are not so severe as those on transmittance monitoring, measurements must still be made over a relatively small absorbance interval if initial rates are being measured (15). 336

(1.64)AAt-I

Equations 16 and 11, and Equations 17 and 12 are equivalent sinceAA = 2.303ATnear T = 1. DISCUSSION

100 msec, [ = 0 1 sec, E = 0 10 sec, E = 0 100 sec, := 0 1 sec, E = lo-+ 10 sec, E = 100 sec, E = 1 sec, E = 10 sec, E = 10-5 100 sec, E = 10-5

is analogous to Equation 7 for transmittance monitoring.

(%,

=

Comparison of Rate Measurements to Normal Molecular Absorption Measurements. In comparing normal molecular absorption measurements to reaction rate measurements using spectrophotometry to monitor the reactions, some basic differences can be noted. First, photocathodic currents utilized in reaction rate methods can be much larger than those encountered in normal molecular absorption work since measurements are usually made near 100% T and larger spectral bandpasses are used. Because of the large photocathodic currents, the dark current will usually be negligible compared to the photocurrent, and icd in Equations 10 and 15 can be dropped. This is equivalent to saying that dark current shot noise is negligible. Large spectral bandpasses can be used because high resolution is not often required and Beer’s law errors due to polychromatic radiation are small over small transmittance intervals near 100% T. Of course, there is a practical limit to how wide the spectral bandpass can be made since nonlinearities can result even for small transmittance changes if slit widths are too large. Also, the average molar absorptivity over the bandpass may decrease as the spectral bandpass is increased. This will cause the absorbance change for a given concentration change to be smaller, and hence reduce the S/N of the rate measurement. Higher readout resolution (or lower readout variance) is needed for reaction rate analysis when compared to normal absorption methods. As discussed previously ( I d ) , the readout variance should always be made negligible for the highest precision. A readability of 0.5% is sufficient for 1-2z precision in the optimum transmittance range of normal absorption measurements, but totally unacceptable for kinetic measurements. Because the readout variance must be very small, it is expected that shot and flicker noise will both contribute significantly to the total measurement variance. In normal molecular absorption spectrometry, noise bandwidths of 0.1-1 Hz or integration times of 1-10 sec are usually

ANALYTICAL CHEMISTRY, VOL. 45, NO. 2, FEBRUARY 1973

sufficient to yield high precision. However, in reaction rate methods, the measurement time is often dictated by the rate of the reaction utilized under optimum conditions and may vary from a few msec to several minutes. Moreover, reaction conditions cannot always be varied to adjust the rate so that measurements are made in the initial stages of a reaction with a particular measurement time. For instance, if it is desired to slow down a pseudo-first order reaction, possibly some reagent concentration can be changed. However, such a change may make the reaction unusable by altering the primary mechanism, causing a deviation from pseudo-first order kinetics, or shifting the equilibrium. Optimization and Plots of Signal-to-Noise Ratios. Equations 10 and 15 reveal that ( S / N ) R fis primarily dependent on the photocathodic current, the measured transmittance change, the noise bandwidth or measurement time, the readout variance, and the flicker factor. Figure 1 shows plots of (S/N)Rf us. photocathodic current for different values of integration time and flicker factor. Typical values of a = 0.275 and AT = 0.01 were assumed in constructing these plots, and the dark current and readout variance were assumed negligible. Curves a-e (dashes) correspond to the shot noise limit for measurement times which vary from 10 msec to 100 sec. This range of integration times covers the normal range that would be encountered in a reaction rate analysis. These curves represent the highest S/N obtainable for a 1 % transmittance change with a given photocathodic current. Curvesf-11 in Figure 1 illustrate the influence of a source flicker factor of on the signal-to-noise ratio. As the photocathodic current (light level) is increased, source flicker becomes important and begins to limit the signal-to-noise ratio. For those typical values of a and AT chosen, a 0.01 % source flicker limits the maximum achievable signal-to-noise ratio to lefs than 100. Curves i-k illustrate the same effect for a source with 0.001 flicker. Figure 1 graphically illustrates the inherently low signal-tonoise ratios of reaction rate measurements. For a large range of the experimental conditions represented in Figure 1, S/N’s are less than 100. For these cases, relative standard deviations of individual measurements are expected to be greater than 1 %. Each of the curves reaches a plateau, which can be calculated from Equations 12 and 17, when source flicker becomes limiting. The source flicker factor is dependent on the measurement time. Because of the l/f character of flicker noise, the relative effect of source flicker noise would not be expected to decrease in direct proportion to the square root of the integration time as does the relative effect of shot noise. For a typical modern spectrophotometric system (Heath EU-701 A Spectrophotometer, Benton Harbor, Mich.), experimental measurements were made of source flicker. For a one-second integration time (noise equivalent bandwidth of 0.5 Hz), the flicker factor was determined to be 0.5-2 X lov4. For a ten-second integration period (Af = 0.05 Hz), the measured flicker factor was approximately the same, which clearly indicates the llfcharacter of source flicker. For integration times below 100 msec (Af 2 5 Hz), flicker noise is expected to be negligible compared to shot noise. It is important to note that once the flicker noise limit is reached for a given transmittance change, increasing either the photocathodic current or reducing the noise bandwidth will not improve the signal-to-noise ratio. Figure 1 indicates the large dependence of the S/N on the measurement time, which as previously mentioned, is often dictated by the rate of reaction utilized under optimum con-

ditions. A reaction rate measurement remains shot noise limited to higher photocurrents as the measuremeni time is decreased since the absolute magnitude of the shot noise is proportional to Two limiting cases for measurement can be identified as discussed below. FASTREACTION KINETICS.For measurements of the rates of rapid reactions (usually in a stopped-flow apparatxs), integration times are expected to be 100 msec or less. Also, due to the small diameter of the reaction cell in a fast mixing apparatus compared to a normal 1-cm cell, the radiant power throughput and, hence, the photocathodic current is often lower than in conventional measurements. Thus, from Figure 1, fast kinetic measurements are very likely to be shot noise limited. This indicates that optimization of (S,”),, can be accomplished by increasing the photocathodic current through the use of the high intensity light sources, large spectral bandpasses and high efficiency optical transfer s) stems to increase the radiant power impingent on the reaction cell. SLOWREACTION KINETICS.For relatively slow reactions for which the rate can be measured with a conventional mixing apparatus, integration times are usually 1 sec or greater. From Figure 1, it is predicted that the relative contribution of shot noise compared to flicker noise will be reduced, and for high photocathodic currents, measurements will become source flicker limited. Effect of Transmittance Interval and Integration Time. Since AT was assumed to be 0.01 in construction of Figure 1, from Equation 10, the S/N for a different transmittance change can be found by multiplying a point on the appropriate curve by ATi(O.01). For a normal tenfold range of analyte concentrations, AT will vary over about an order of magnitude, and AT = 0.01 was chosen as an average value. For a particular reaction and set of conditions, the transmittance or absorbance interval over which the measurement is made is approximately proportional to the integration time, At, because reaction rate curves are approximately linear in the initial stages of a reaction. Thus, in the shot noise limit, for a given reaction is approximately proportional to A t 3 / *and, in the source flicker limit, is proportional to Ar, Therefore, in order to increase measurement precision, AT should be increased by increasing the integration time within the limits imposed by the reaction of interest. It should be noted that Equations 10-12 and 15-17 and Figure 1 apply to fixed-time instruments which use integration over the total measurement time. It is possible to perform the fixed-time measurement such that the measurement time for the transmittance or absorbance at times r , and t 2 is less than At = tz - tl. This case could arise for instance if a small laboratory computer with an A-D converter with an aperture time pml

= variance for photocurrent output voltage, E,, due to quantum noise,

ail2

V2

bit2

= =

c112

= c212 =

( u ~ ~ =~ 2m2ei,,R,2Af’ ) ~

(16) P. M. Beckwith and S. R. Crouch,ANAL.CHEM., 44,221 (1972). (17) P. M. Beckwith and S. R. Crouch, unpublished observations, Michigan State University, East Lansing, Mich., 1971. (18) H. L. Pardue and P. A. Rodriguez, ANAL.CHEM.,39, 901 (1967). (19) H. L. Pardue and S . N. Derning, ibid.,41,986(1969). 338

ari2Ti,a2i2 = aTz2Tz b7l2Ti2,b2z2 = hTz2T22 c7 1

RECEIVED for review July 3, 1972. Accepted October 30, 1972. This work was partially supported by NSF Grant No. GP-18123 and an American Chemical Society, Analytical Division Fellowship sponsored by Perkin-Elmer Corporation.

ANALYTICAL CHEMISTRY, VOL. 45, NO. 2, FEBRUARY 1973