”cw-nu.“- -.‘.._.. THEORETICAL AND EXPERIMENTAL CONSIDERATIONS FOR ANALYTICAL RATE MEASUREMENTS Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY JAMES DAVIS IMGLE. IR. 1971 LIBRARY Michigar State Uni". :. sity . This is to certify that the thesis entitled THEORETICAL AND EXPERIMENTAL CONSIDERATIONS FOR ANALYTICAL RATE MEASUREMENTS presented by James Davis IngIe, Jr. has been accepted towards fulfillment of the requirements for Ph.D. degree in_§_b.emj_s_trx_ flég/j K 4.2.1; Stanley R. Crouch Major professor I} .3: 7.1-... Da‘e December 31 1971 I: - r 1' I1)-7 0-7639 .Ib VH'DRPHN ~v of “new... und-~ n. e a \(u . E r‘LIv (9 :HJ ‘, ~ ‘1: «amen 1:i’7' 3‘ 0' r4 :»4 . b". ‘.‘ 5100. II» 50'~vr '.A ’. ‘.‘V tho treu;o. _ _ . . , L, I ” ”95$ .1.“ (’u t_v v. m 3' JUNIIIIY I: “ I - ,. — CI!“ 3“ ' neared Ie mm- , - . . ‘ .thvttlez. . ‘ ‘,”l tO'IICT ‘L "TIA "I; 'i I‘ ' ~ ."‘0 U Pug} ' Lo. fixa‘h'b ‘N ' ‘ lufphor MC?!“ 1- '. . ‘. 3“ ' . ,. -"-‘.' Indy m uu} "Ere ~. th‘s “:9r979‘9' -;: ‘W '4'. )e.-L Y :\¢06I‘IQO ABSTRACT THEORETICAL AND EXPERIMENTAL CONSIDERATIONS FOR ANALYTICAL RATE MEASUREMENTS BY James D. Ingle, Jr. Ari investigation of the factors influencing the accuracy and pre- cision (If automated analytical reaction rate measurements has led to an increased understanding of the fundamental basis of kinetic methods of analysis, a new automated instrument for the computation of reaction rates, and a rapid and precise procedure for the simultaneous analysis of phosphate and silicate in mixtures. A mathematical treatment of fundamental equations has revealed that the choice of rate measurement approach for analysis depends on the type of reaction, the sought-for species, and the signal concentration rela— tionship of the transducer used to follow the reaction. The fixed-time approach is best suited for first or pseudo—first order reactions and determinations of substrate concentrations in enzyme reactions, while the variable-time method is best suited for the analysis of catalysts, in- cluding enzyme activities. A unique signal-to—noise ratio expression is derived for quantita- tive molecular absorption spectrometric measurements. Under certain experimental conditions, this expression can be considerably simplified and three limiting cases identified in which measurements are either readout limited, photocurrent shot noise limited, or source flicker Timited. This treatment reveals that the usual assumption that optimum James D. Ingle, Jr. leasurement precision occurs near 37% T can be grossly in error under certain conditions. Use of signal-to—noise ratio expressions for the evaluation of measurement precision and for optimization of experimental conditions is presented. Direct current and photon counting measurement systems are critically compared in terms of signal-to—noise ratio and other criteria for application to molecular absorption measurements. Signal-to-noise ratio theory is used to evaluate the precision of fixed-time spectrophotometric rate measurements. Under most conditions, fast reaction rate measurements are photocurrent shot noise limited, while experimental data are presented which indicate that slow reaction rate measurements are often source flicker limited. A novel fixed-time instrument is described for reaction rate measure- ments which uses digital computational circuitry. Performance of the instrument on synthetic signals and phosphate samples indicates excellent accuracy and precision. A new kinetic-based procedure for the determination of phosphate and silicate in mixtures is discussed. Analysis is based on the kinetic differences in the formation reactions of the heteropolymolybdates and the reduced heteropoly blues of phosphate and silicate. Data are pre- sented to illustrate the range of applicability of the procedure. A mathematical treatment has revealed that the linearity and signal- to-noise ratio obtained in photon counting systems are dependent on the discriminator level and the magnitude and source of dead time. A com- parison of signal-to-noise ratio expressions for spectrometric measure- ments using photon counting and direct current techniques is presented to provide criteria for choosing between these two light measurement techniques. The inherent signal-to-noise ratio characteristics of James D. Ingle, Jr. " ‘f‘ and vacuum photodiodes are compared to establish criteria \ inch transducer is better in a particular spectrometric .‘v WEONE.’ it ‘ “.4 I - . 1p mrtia? ‘, :‘l! .., I ,. g... .,.- V :1 i" ‘. (WC-w .' 'HUISDW' Department at Tmétr, l97l THEORETICAL AND EXPERIMENTAL CONSIDERATIONS FOR ANALYTICAL RATE MEASUREMENTS By James Davis Ingle, Jr. A THESIS > Submitted to ‘" ‘ Michigan State University ‘ in partial fulfillment of the requirements ’ fur the degree of DOCTOR OF PHILOSOPHY Department of Chanflstry l9?! ' ii W t Thinks on gum ~~ Wv‘dfivg te‘ r: I ‘ f’ 1" “New 9.4m»; '. t W‘d‘”) 331)th : :nl. ’.W1Mn9 I Yefim‘u’f N, _ L Mt’hilea (00".‘1 . “‘5 .vthtir Ql‘J‘hs ‘ ' - . T0 056"). Mom. and Dad m M. ““211an av , 3““ [madam tor crwin‘n “ v 1 Society end 1'1»an Etna: . Mica! So: 1ny Am: “‘3“. Of course. I‘M). u '_- ‘-‘ _.. \y. If» .v. _ “Uh” Ml} smile, for teepim .-,:.-. —... x “I wk~}‘_‘ .'m‘ to! an mount“! an“. "1ka \' ,1: ‘ \ “3.4-, '1 : ~ ' 19 ' “A? ~ ‘ ;‘\$Ju"‘v ‘ — r‘ ‘ .-.' Q 1 ‘ ‘ ‘ - l7} ~ ACKNOWLEDGMENTS This author gratefully acknowledges the guidance, encouragement, and friendship of Dr. S. R. Crouch during the course of this investi- gation. Thanks are given to Dr. C. G. Enke for serving as his second reader, and for providing helpful comments. The fellow members of his research group also deserve acknowledgment for providing suggestions, criticisms, and humorous moments and for maintaining a respectable signal-to-noise ratio except under alcohol flicker limited conditions (Case IV or 4 cases). This author expresses his thanks to Michigan State University for granting him teaching assistantships and fellowships, to the National Science Foundation for providing him a traineeship, and to the American Chemical Society and Perkin Elmer Corporation for awarding him an American Chemical Society Analytical Division Fellowship. Finally, this author wishes to thank his parents for their unfalter- ing support and, of course, his wife Sally, for making everything worth- while, for making him smile, for keeping him out of trouble, and for giving solicited and unsolicited advice. TABLE OF CONTENTS LIST OF TABLES ....................... LIST OF FIGURES ....................... I. INTRODUCTION ..................... II. HISTORICAL ...................... Automated Rate Measurement Systems ........ Rate Computational Approaches .......... Theory of Rate Measurements ........... Kinetic Analysis of Phosphate .......... Kinetic Analysis of Silicate ........... Analysis of Phosphate and Silicate in Mixtures ..................... Precision of Molecular Absorption Measurements . . EXPERIMENTAL ..................... A. Reaction Monitor and Cell ............ B. Signal Modifier and Recorder Output ....... C. Chemicals, Solutions, and Mixing Procedures . . Phosphate Analysis .............. 2. Silicate Analysis .............. 3. Analysis of Silicate and Phosphate in Mixtures ................... 4. Practical Hints ............... FACTORS INFLUENCING THE ACCURACY OF ANALYTICAL RATE MEASUREMENTS .................. A. Introduction ................... 8. Type of Reaction ................. l. Pseudo- First Order Reactions ......... a. General Treatment ............ b. Variable—Time Measurement ........ c. Fixed-Time Measurement .......... d. Experimental Comparison ......... 2. Enzyme Catalyzed Reactions .......... a. General Treatment ............ b. Variable-Time Measurement ........ c. Fixed- Time Measurement .......... Other Catalyzed Reactions .......... C. Characteristics of the Reaction Monitor ..... l. Amperometric and Fluorometric Detection 2. Spectrophotometric Detection ......... 3. Potentiometric Detection ........... Conclusions ................... Page viii —‘O \O‘DNUTU'I 0'1 —l—l Page V. PRECISION OF QUANTITATIVE MOLECULAR ABSORPTION SPECTROMETRIC MEASUREMENTS .............. 38 A. Introduction ................... 38 8. Relationship of Readout to System Parameters .................... 39 l. Current and Pulse Output Expressions for the Photomultiplier ........... 39 2. DC Readout System .............. 45 3. Photon Counting System ............ 46 C. Relationship of Variance to System Parameters .................... 47 D. Relationship of Signal-to-Noise Ratio to System Parameters ............... 52 l. DC Measurement System ............ 52 a. S/N for Photocurrent Measurement ..... 52 b. S/N for Transmittance Measurement . . . . 54 c. S/N for Absorbance Measurement ...... 55 d. Limiting Cases .............. 57 e. S/N Plots ................ 6O 2. Photon Counting System ............ 60 a. S/N Expressions ............. 60 b. Limiting Cases .............. 64 E. Practical Considerations ............. 65 l. Simplification of Group Variance Terms . . . . 65 2. Effect of Reading Error ........... 66 F. Comparison of DC Measurement and Photon Counting ..................... 71 G. Evaluation and Optimization of S/N's ....... 75 l. DC Measurement ................ 75 a. Evaluation of (5/11):c .......... 75 b. Optimization of System Parameters . . . . 76 2. Photon Counting ............... 77 H. Discussion .................... 77 VI. PRECISION OF RATE MEASUREMENTS ............ 80 A. Introduction ................... 80 B. General S/N Considerations ............ 80 l. Fixed-Time Measurements ........... 8l 2. Variable- Time Measurements .......... 82 C. Spectrophotometric Fixed- Time Measurements . . . . 82 l. Introduction ................. 82 2. Transmittance Monitoring ........... 83 3. Absorbance Monitoring ............ 87 4. Comparison to Normal Molecular Absorption Measurements ............ 88 5. Optimization and Plots of (S/N)kf ...... 90 a. Fast Reaction Kinetics .......... 93 b. Slow Reaction Kinetics .......... 93 6. Effect of Transmittance Interval and Integration Time ............... 94 V 7. Effect of Readout Variance .......... D. Variable-Time Measurements ............ FIXED-TIME DIGITAL COUNTING SYSTEM FOR REACTION RATE MEASUREMENTS .............. A. Introduction ................... B. Instrumentation ................... l. Relationship Between Readout and Rate 2. Range and Limitations ............ 3. Timing, Gating, and Counting Circuits 4. Decoder and Readout ............. C. Procedures .................... 1. Circuit Layout and Construction ....... 2. Phosphate Rate Measurements ......... D. Results and Discussion .............. SIMULTANEOUS DETERMINATION OF SILICATE AND PHOSPHATE BY AN AUTOMATED DIFFERENTIAL KINETIC PROCEDURE .................. A. General Considerations and Observations ..... 8. Experimental ................... C. Results and Discussion .............. l. Silicate Determination ............ 2. Analysis of Phosphate-Silicate Mixtures LIST OF REFERENCES ..................... APPENDICES A. PULSE OVERLAP EFFECTS ON LINEARITY AND SIGNAL-TO-NOISE RATIO IN PHOTON COUNTING SYSTEMS . A. Introduction ................. B. Relationship of Readout to Counting System Parameters .............. C. Variance in the Readout ........... D. Signal-to-Noise Ratio Expressions ...... E. Conclusions ................. B. CRITICAL COMPARISON OF PHOTON COUNTING AND DIRECT CURRENT MEASUREMENT TECHNIQUES FOR QUANTITATIVE SPECTROMETRIC METHODS .............. A. Introduction ................. B. Readout Signal Expressions .......... l. Current and Pulse Output Expressions for Photomultiplier ........... 2. DC Readout ................ 3. Photon Counting Readout ......... C. Noise Expressions .............. l. Total Variance .............. 2. Quantum Noise .............. 3. Secondary Emission Noise ......... vi Page 95 95 ll8 ll8 l23 l23 l23 l24 128 Page 4. Signal and Background Radiation Flicker Noise .............. 159 5. Photomultiplier Flicker Noise ...... 160 6. Johnson Noise .............. 161 7. Amplifier Noise ............. 161 8. Readout Variance ............. 161 9. Other Noise Sources in DC Measurement ............... 162 10. Excess Noise ............... 162 D. Signal-to-Noise Ratio Expressions ...... 163 1. DC Measurement System .......... 163 2. Photon Counting System .......... 170 E. Comparison of Photon Counting and DC Measurement Techniques ............ 172 l. S/N Comparison .............. 173 2. Discrimination Against Dark Current . . . 174 3. Reading Error and Direct Digital Processing ................ 176 4. Measurement System Stability ....... 176 5. Use of Modulation Techniques ....... 177 F. Conclusions ................. 178 SIGNAL-TO-NOISE RATIO COMPARISON OF PHOTOMULTIPLIERS AND PHOTOTUBES ......... 181 A. Introduction ................. 181 B. Signal-to-Noise Ratio Expressions ...... 182 C. Discussion .................. 186 "2 ,; Tfi vii 'f 311: l-‘I 'u‘ ‘. -. ' ' :I - LIST OF TABLES Table Page I. Validity of Initial Rate Approximation for First or Pseudo-first Order Reactions ........... 23 II. Error of Variable-Time Approach ........... 25 III. Comparison of Phosphate Determinations by Fixed- and Variable-Time Methods .............. 28 IV. Validity of Pseudo—zero Order Kinetics Approximation for Enzyme Catalyzed Reactions ............ 32 V. Linearity of Transmittance ............... 36 VI. Definitions of Symbols in Equations 5.1 and 5.2 . . . . 40 VII. Definitidns of Dark Current Symbols ......... 44 VIII. Expressions for Variance in Erp’ Erd’ Nop’ and N0d . . 49 IX. Equations and Optimum Conditions Nhen One Groups Variance Term Predominates .............. 58 X. Fixed-time Rate Measurements of Synthetic Slopes ........................ 112 XI. Fixed-Time Rate Measurements of Synthetic Slopes with Superimposed Sine Wave ............. 114 XII. Fixed-Time Reaction Rate Determinations of Phosphate ..................... 115 XIII. Determinations of Silicate .............. 124 XIV. Simultaneous Determinations of Phosphate and Silicate ..................... 126 XV. Observed Photoelectron Pulse Readout with Dead Time Loss .................... 137 XVI. Comparison of Signal, Noise, and Signal-to-Noise Ratio for Photon Counting Circuits .......... 144 XVII. Definitions of Variance Terms ............ 153 .' viii LIST OF FIGURES Figure Page 1. Diagram of Thermostatted Cell Holder and Stirrer Assembly ................... 13 Construction of Thermostatted Cell Block ........ 14 Normalized Signal-to—Noise Ratio Versus Transmittance ..................... 61 Effect of Readout Variance and Photocathodic Current on Signal— —to- Noise Ratio ............ 68 Dependence of S/N on Photocathodic Current, Integration Time, and Source Flicker Noise ....... 91 Block Diagram of Reaction—Rate Measurement ....... 98 Expanded Section of the Initial Part of a Typical Reaction Rate Curve .................. 100 Circuit Diagram of Fixed-Time Rate Computer ...... 104 Waveforms of Sequencing Signals for Timing Circuits ........................ 107 Recorded Reaction Rate Curves for Formation of the Heteropoly Blues of Phosphate and Silicate Measured at 650 nm ....................... 120 Recorded Reaction Rate Curves for Formation of 12-MPA and B-lZ-MSA Measured at 400 nm ......... 122 Effect of Signal Shot and Flicker Noise on the Signal-to—Noise Ratio ................. 166 Effect of Dark and Background Current Shot Noise and Background Flicker Noise on the Signal-to-Noise Ratio ......................... 168 Signal-to-Noise Ratio as a Function of Photocathodic Current ........................ 185 I. INTRODUCTION Kinetic analysis has established itself in the last decade as a powerful technique for analytical determinations. Attesting to this are several recent review articles (1-6), books (7,8), and chapters (9-11) dealing with analytical applications of kinetics. Basically the tech- nique involves monitoring a reaction in which a sought-for species is a reactant. Under suitable conditions and in the initial stages of the re- action, measurement of the rate or an approximation to the rate yields data which can be directly related to the concentration of the sought-for species. A number of reasons can be given for the increasing popularity of kinetic analysis. First, kinetic—based measurements have several unique advantages compared with equilibrium-based measurements. For example, non-stoichiometry or unfavorable equilibrium constants make many reactions unsuitable for equilibrium-based analysis. However, the initial rates of such reactions are often quantitatively related to the concentration of the sought-for species. Kinetic-based methods are faster than equi- librium-based methods because only the initial portion of the reaction is used. A kinetic-based measurement is a relative measurement because only a change in the signal from the reaction monitor is measured. Thus, non- reacting chemical species or instrumental factors that contribute to the total absolute magnitude of the reaction monitor signal do not interfere as they would in an equilibrium-based method. Finally, kinetic-based methods can be much more selective than equilibrium-based methods under l 1." 2 conditions where species react at different rates. Either conditions can be arranged so that the kinetic contributions of two or more species can be completely separated or in situ analysis of two or more species can be performed by the method of proportional equations. A second important reason for the growth of kinetic methods of analy- sis is the development of stable and sensitive reaction monitoring sys- tems. Since the signal-to-noise ratio (S/N) in kinetic methods is nec- essarily lower than in equilibrium methods because a smaller signal change is measured, high sensitivity, low noise characteristics, and stability against drift are all mandatory prerequisites for reaction monitoring systems. Some of these prerequisites have been realized with recent developments such as optical feedback stabilized light sources and low- noise operational amplifiers. Finally, the popularity of kinetic methods has been enhanced by the development of automated rate computational systems which convert the raw data from the reaction monitor into a form directly proportional to the rate or concentration of the sought-for species. In many instances, this has replaced point by point data acquisition and manual reduction of data to a useful form. Automated data acquisition and reduction, particularily with digital circuitry, has considerably increased the speed, accuracy, and precision of the total kinetic analysis. To utilize effectively the advantages of kinetic-based methods, the limitations involved in using the kinetic technique must be well under- stood. To carry out rate measurements during the initial stages of a re- action, the half—life of the reaction studied must lie between about five milliseconds and a few hours. The lower limit is determined by the fast- est uflxing time achievable in a stopped-flow apparatus. For conventional, 3 manual-mixing kinetic instruments, the lower limit for the half-life is about ten seconds. The upper limit is restricted by slow drifts in the reaction monitoring and rate computational systems, which become compara- ble to the measured rate for very slow reactions. Usually rate measure- ments are made within five minutes to utilize the advantage of short an- alysis time and to minimize the effect of instrumental drifts. A second limitation of kinetic methods when compared to equilibrium methods is that the S/N of the measurement is inherently smaller since only a portion of the reaction is utilized for the measurement. Because measurements are made on a dynamic system, the noise bandwidth cannot be reduced indefinitely to improve the S/N without distorting the reaction monitor output. Since the S/N indicates the instrumental precision with which a measurement can be made, a knowledge of the S/N and the para- meters that affect it can be extremely useful in evaluating the total precision of a kinetic measurement and in optimizing experimental con- ditions. A third limitation of kinetic methods is the need to control very carefully the reaction conditions. Temperature, pH, reagent concentra- tion, ionic strength, and any other factors which influence the reaction rate must be carefully regulated or monitored during the reaction. In addition to the above limitations, the effect of the reaction monitor and the rate computational system on the final readout must be well understood. In some cases, the reaction monitor signal is not di- rectly proportional to the concentration of the species being followed. Also, since the true rate of a reaction is an instantaneous quantity which cannot be measured exactly, any means of extracting the rate from the changing reaction monitor signal will be an approximation. Thus, 4 the relationship of the measured rate to the true rate, or the concen— trationtrfthe sought-for species, must be enumerated to evaluate the accuracy of rate measurements. The primary emphasis of the first part of this thesis is directed toward elucidating the effects of instrumental factors on the accuracy and precision of analytical rate measurements. This should lead to a better understanding of the restrictions imposed by the above mentioned limitations of kinetic-based methods and provide information that can be used to optimize instrumental parameters. First, a mathematical treat- ment of different types of reactions reveals the relationship between the measured rate data and the concentration of the sought—for species. This study indicates the effect of the reaction monitor and of the rate compu— tation system on the accuracy. Next, a unique S/N treatment is presented for evaluation and optimization of the precision of normal quantitative molecular absorption measurements. This chapter establishes the ground work for the following chapter in which the precision of rate measure- ments is evaluated using S/N theory. It is hoped that a sound theoretical understanding of the principles involved in kinetic measurements will improve the reliability of routine kinetic analysis and also fundamental kinetic investigations. The latter part of the thesis is devoted to studies more experimental in nature. A novel fixed-time digital counting system for reaction rate analysis is presented. The construction of the instrument and the per- formance of the instrument on synthetic curves and in kinetic measure- ments are discussed. Finally, a new automated differential rate proce- dure is proposed for the determination of silicate and phosphate in mixtures. II. HISTORICAL A. Automated Rate Measurement Systems The basic components of an automated rate measurement system include a reaction cell, a reaction monitor, a signal modifier, and a rate com- putational system. The reaction cell should be well thermostatted and may have provision for automated sample and reagent introduction and mix- ing. The reaction monitor provides an electrical signal related to the concentration of some species involved in the reaction. The most conmon detection techniques employed are spectrophotometric, potentiometric, fluorometric, and amperometric. The signal modifier converts the reaction monitor signal into a form (data domain) and amplitude acceptable by the rate computational system. Signal modifiers include current to voltage converters, linear and log amplifiers, domain converters, and voltage dividers. The rate computational system extracts and computes from the changing signal modifier output a quantity proportional to the rate or to the concentration of the sought-for species. Usually the signal modifier output is also connected to a recorder or oscilloscope so that the reaction can be followed continuously. Manu- al rate data can be obtained from the oscilloscope photographs or recorder chart paper traces. B. Rate Computational Approaches Three basic approaches are utilized in rate computational systems. The most straightforward approach is the derivative or slope method in 5 6 which the derivative of the signal modifier output with respect to time is electronically computed. The result of the measurement is an elec- trical signal proportional to the reaction rate within the time response characteristics of the system. This is usually accomplished with a con- ventional operational amplifier differentiator circuit (12) or by a slope comparison method (13-16). Relatively simple implementation and rapid data acquisition are primary advantages of the derivative technique. Also, for linear reaction monitors, any changes in the rate with respect to time are readily apparent. The primary disadvantage is that derivative circuits are quite susceptible to noise, although time averaging can be used at the output of the differentiator to increase the precision. The two remaining computational approaches, the fixed-time and the variable-time, are integral methods because they measure finite time or concentration changes. Thus, these measurement approaches yield an ap~ proximation to the instantaneous rate since they calculate the average rate over the measurement time. This can be an advantage since the rate computation can be implemented with all—digital circuitry, unlike the derivative approach which necessarily involves some analog computational circuitry. The fixed-time or constant-time approach to reaction rate analysis involves the measurement of the change in the signal modifier output during a preselected time interval. Early work using this approach was directed to continuous flow techniques (17-19), while more recent appli- cations have been with discrete sampling systems (20,21). The most ele- gant of previous fixed-time systems utilized operational amplifiers to integrate the signal modifier output over two equal time intervals (20). The two integrals were subtracted with analog integrators and the 7 resulting voltage difference was shown to be directly proportional to the reaction rate. The fixed-time approach has a number of advantages. Under suitable conditions, the measured signal change is directly proportional to the concentration of the sought-for species. Since the measurement time re— mains constant from sample to sample in a given procedure, this approach can be easily incorporated into a completely automated sample handling and measurement system. Also, because integration techniques can be ap— plied over the total measurement time, high noise immunity results. The primary disadvantage of the fixed-time approach is that it is not well suited for non—linear reaction monitors unless the signal modifier pro- vides linearization. In the variable-time technique, the time required for a fixed sig- nal modifier output change to occur is measured. Usually, this entails two signal level sensors, such as comparators, which turn a timer on and off when the signal modifieroutput passes through two preselected signal levels (22-28). The time interval measured is inversely proportional to the rate or the concentration of the sought-for species under certain conditions. Thus, completely automated variable-time computational sys— tems calculate and display on the readout device the reciprocal of the measured time interval to provide direct rate or concentration readout. The most recent variable-time instrument (28) employs all-digital circuit elements in the computational system and has led to a significant improve- ment in the reliability of automated reaction rate data. The main advantage of the variable-time instruments is that linear reaction monitors are not required. The linearity of the reaction moni- tor is unimportant for a given analysis because the signal change, and hence the concentration change, is held constant. The major limitation ‘ 8 is the increased complexity of the computational system compared to other approaches, since the reciprocal of the time interval must be calculated for the readout to be directly proportional to the concentration of the sought-for species. C. Theory of Rate Measurements As previously mentioned, it is important to have a mathematical treatment which relates the measured quantity to the sought-for quantity. This is imperative for establishing the accuracy of rate measurements, for illustrating potential simplifications, for choosing optimal reaction conditions, for predicting the relative merits of the different rate com- putational approaches, and for developing new reliable means of implemen— tation. This is especially true for the integral rate approaches because the measured quantity is only an approximation to the true rate. A few mathematical treatments have appeared in the literature (7,10,11). The most comprehensive and useful of these is by Pardue am. By considering a completely generalized reaction, Pardue failed to point out the unique situations in which one of the rate computational approaches is superior to the others. Little work has appeared which theoretically treats the precision of automated rate measurements with such techniques as S/N theory. Morrow (29,30) briefly discusses S/N in a stopped-flow system in which measurements are shot noise limited, although the treatment is rather in- complete and ambiguous. As will be shown in a later chapter, rather than defining a traditional S/N for the output of the signal modifier, it is more useful to define a S/N in which the measured quantity (1:2, the rate) becomes the signal. 9 0. Kinetic Analysis of Phosphate Colorimetric procedures for the determination of phosphate are usually based on the formation of 12-molybdophosphoric acid (12—MPA) from phosphate and molybdate in strong acid or the subsequent reduction of lZ-MPA to phosphomolybdenum blue (31-33). These colorimetric proce— dures are plagued by interferences and instability of the yellow or blue color, respectively. Recently, Crouch, Javier, and Malmstadt (34,35) have carried out detailed mechanistic studies of the reactions involved which have cleared up many of the misconceptions about the colorimetric procedures and which have resulted in two new kinetic-based procedures for the determination of phosphate. Both the heteropoly acid and heteropoly blue reactions were found to be first order with respect to phosphate in sulfuric and nitric acid. The initial rates of both reactions were measured and related directly to the phosphate concentration in aqueous solutions and blood serum samples (36-38). Relative errors of 1-2% were obtained. The formation of lZ-MPA is fast and the initial rate must be measured with a stopped- flow apparatus. The reduction of 12-MPA to its heteropoly blue is slower, and the initial rate can be measured with a conventional mixing apparatus. E. Kinetic Analysis of Silicate Quantitative analysis procedures for the colorimetric determination of silicate are often based on the formation of lZ-molybdosilicic acid (IZ-MSA) or the subsequent reduction to its heteropoly blue (31,32,39). These silicate procedures are subject to interferences and instability and are further complicated by the existence of two isomers of 12-MSA "PIC“ have different absorption properties. The B isomer of 12-MSA 10 (B-lZ-MSA) forms when the acid-to-molybdate ratio is greater than 1.5 (40). The 8 form converts to the o isomer (a-lZ-MSA) upon standing. Hargis (41,42) has conducted mechanistic studies of the formation of a- and 8— lZ-MSA and the reduction of u-lZ-MSA to its heteropoly blue. The forma- tion of B-lZ-MSA was shown to be first order with respect to silicate in solutions of perchloric acid, and an analysis for silicate was based on this result (43). F. Analysis of Phosphate and Silicate in Mixtures Colorimetric determination of phosphate and silicate in mixtures is difficult because the heteropolymolybdates of these anions have very similar absorption Spectra (44) as do the heteropoly blues (45). Mixtures of the two anions have generally been treated by separating the two species, by proper control of reaction conditions, or by adding reagents which destroy or prevent the formation of one of the heterOpoly acids. For example, determinations of silicate in the presence of phosphate can involve separation of phosphate prior to colorimetric determination of silicate (39), control of solution pH so that only the heteropoly blue of silicate is formed (46), or destruction of lZ—MPA with a complexing agent such as oxalic acid (47). Phosphate can be determined in the presence of silicate by precipitating the silicate prior to colorimetric determination of phosphate (48), addition of tartaric acid to prevent formation of 12- MSA (47), or selective extraction of 12-MPA into a suitable organic sol- vent (49,50). Simultaneous analysis of both species in mixtures usually involves c°Mbinations of these techniques. Such analyses are time consuming and questionable due to the empirical nature of some of the above mentioned sePal”at‘.ion techniques. In this text, a rapid and accurate simultaneous 11 differential kinetic method which requires no separations is presented for the analysis of both anions. G. Precision of Molecular Absorption Measurements Most treatments of spectrophotometric precision assume that the limiting factor is the uncertainity in reading a linear scale (51). This gives the familiar result that conditions should be arranged so that mole- cular absorption measurements are made near 37% T. More recently (52,53), S/N theory has been used to describe the precision of molecular absorp- tion measurements, although these treatments were only applied at the limit of detection and neglected the effect of reading error. The treat- ment presented later in this text defines a unique S/N for absorbance measurements, which incorporates the variance due to reading error and can be used over the total absorbance range to estimate the measurement precision. This treatment is later extended to spectrophotometric kinetic methods of analysis. III. EXPERIMENTAL A. Reaction Monitor and Cell For all rate studies, a Heath single beam molecular absorption spectrophotometer (EU-701A) was utilized as the reaction monitor. The light source, monochromator, and photomultiplier modules were unmodified. The tungsten light source was used in the optical feedback mode which provided excellent stability over long periods of time. A slit width of 2000 um (4 nm spectral bandpass) was used in all cases to realize the highest S/N as will be discussed later. The sample cell was modified by replacing the existing cell holder and platform with the thermostatted cell holder and magnetic stirrer assembly illustrated in Figure l. The thermostatted block holds a stan— dard 1.00—cm spectrophotometric cell in which is placed a 1/2 by 1/4 inch telflon coated magnetic spinfin (Bel-Art). Construction of the thermo- statted block from 1/16 inch brass is shown in Figure 2. Part b is welded to the top of part a to form the center insert. This insert is sand— wiched between two identical outside shell pieces (part c). Tubing for circulating water through the thermostatted cell block and for driving the magnetic stirrer are brought in through holes drilled in the back of the sample cell module. Although either water or air can be used to drive the magnetic stirrer, water was preferred over air be- cause of less difficulty with vibration. The water flow through the re- action block was controlled by a Tempunit Constant Temperature Circulator (Tecam, model TU-9) which was inserted into a 12 x 12 inch Pyrex jar 12 52232 .8:qu use 33o: :3 8333535 .8 5233 .F 952.“. A60 .EEm mg .H ...\\ Lotzm ozocmoi . __.__,_§..e... ... Joe—n =oU ago: to: 329.8... _. .. . \uozeaéufi Figure 2. Construction of Thermostatted Cell Block 15 filled with distilled water. Experiments were conducted at 23 i 0.1°C. B. Signal Modifier and Recorder Output A Heath Photometric Readout Module (EU-703-3l) which provides either transmittance or direct absorbance readout was used as the signal modi- fier. In the transmittance mode, it basically operates as an operational amplifier current-to-voltage converter with offset capabilities. The 10'6 A input current range was used always,and the photomultiplier gain was adjusted to give 100% T on the readout device for the blank solution. In the absorbance mode, a logarithmic amplifier follows the current-to- voltage converter. The 10'6 A input current range and 0.0-1.0 A unit span were always used, and the photomultiplier gain was used to set 0.0 A units for the blank solution. Although the readout module provides a meter readout, the Heath Universal Digital Instrument was USEd in the digital voltmeter (DVM) mode for more accurate and precise voltage read- out by connecting the DVM probe to the l V output of the readout module. 7 A cur- To follow the progress of reactions on a recorder, the 10- rent output of the readout module was connected to the Heath Log—Linear Current Module (EU-20-28). The recorder was used in the % transmittance mode with the current span and suppression voltage appropriate for the desired scale expansion. This allowed visual monitoring of the reaction rate curves to insure that no abnormalities occurred during a kinetic run and to confirm that measurements were made by the rate computer during the desired portion of the voltage versus time curve. C. Chemicals, Solutions, and Mixing Procedures All chemicals used were analytical reagent grade and required no further purification. Reagent and sample solutions were prepared from 16 distilled water and stored in polyethylene bottles to prevent possible silicate leaching. l. Phosphate Analysis Optimum reagent concentrations for analysis of phosphate by measure- nent of the initial rate of formation of its heteropoly blue have been established by Crouch (38), and very similar solution preparation pro- cedures were used here. A 0.018 M_Mo(VI) solution in 1.48 N H $04 was 2 2MoO4-2H20 in water in a 250—ml volumetric flask. After adding 9.4 ml of concentrated H prepared by dissolving 1.089 g of Na 2504, the solu— tion was diluted to volume, allowed to sit one day, and then filtered through a millipore filter. This solution was replaced after about two weeks. Ascorbic acid, 0.2265 M, was prepared by dissolving 3.990 g of L- ascorbic acid in water and diluting to 100 ml. A stock phosphate solu- tion, 100 ppm in phosphorous, was prepared by dissolving 0.1798 g of KZHPO4 in water and diluting to 1 liter. This stock solution was diluted to provide sample solutions in the 2.5 to 50 ppm P range. The reaction was followed at 650 nm. Two milliliters of acid- molybdate solution and one milliliter of the phosphate sample were added to the reaction cell with pipets and allowed to mix and equilibrate for one or two minutes. The reaction was initiated by injecting 100 pl of the ascorbic acid solution into the cell with a syringe. 2. Silicate Analysis A stock solution of approximately 100 ppm silicon was prepared by dissolving 1.015 g of NaZSiO3-9H20 and diluting to 1 liter. The stock solution was diluted to provide concentrations of 2.5-50 ppm Si. These 17 working solutions were adjusted to pH 3 with H2304. Monomeric silicon units have been shown to be most stable in the 1-3 pH range and above pH 13 (39). This solution should be standardized by conventional gravi- metric techniques for accurate silicate determinations. A 0.155 M_Mo(VI) solution in 1.14 N H2S04 was prepared by dissolving 9.35 g of Na2M004-2H20 in water in a 250-ml volumetric flask. Eight milli- liters of concentrated H2504 were added and the solution was diluted to volume. This solution was allowed to stand one day and was filtered be— fore use. The formation of 8—12-MSA was followed at 400 nm. Two milliliters of acid-molybdate solution were added to the reaction cell with a pipet. The reaction was initiated by injecting 1 ml of the silicate sample with a syringe. 3. Analysis of Silicate ang_Phosphate in_Mixtures Determination of phosphate and silicate in mixtures was accomplished with the procedures just described. The only difference was that a phos- phate and a silicate mixture, made from the 100 ppm stock solutions, was substituted for the pure silicate or phosphate solutions in the appro- priate procedure. 4. Practical Hints Fluctuations caused by air bubbles and particles in the light path of the reaction cell can cause considerable error in measuring reaction rates. To eliminate particles, extremely clean glassware should be used and all reagent and sample solutions should be filtered before use. It was found that air bubbles were often introduced by the syringe used to initiate reactions. To minimize the injection of air bubbles, the syringe l8 plunger should be pushed with a smooth steady motion rather than with a fast abrupt motion. Also the syringe tip should be inserted well into the reaction cell before injection. To prevent condensation, it is important to set the thermostatted bath temperature such that the reaction cell is above room temperature. Especially on hot and humid summer days, condensation can cause erroneous results. IV. FACTORS INFLUENCING THE ACCURACY OF ANALYTICAL RATE MEASUREMENTS A. Introduction The measured quantity in automated kinetic analysis is the time derivative of the signal modifier output (derivative method), the change in the signal modifier output occurring over a constant time interval (fixed-time method), or the time interval necessary for the signal modi- fier output to change by a fixed amount (variable-time method). Usually kinetic analysis is based on the assumption that the measured quantity or readout is linearly related to the concentration of the sought-for species. Quantitative analysis requires calibration of the kinetic sys— tem with standards. Either one standard is utilized to establish a pro- portionality constant between the readout and the concentration of the sought-for species, or two or more standard solutions are employed to con— struct a linear calibration curve. The latter approach is often necessary for cases in which a blank rate exists, such as in catalytic reactions. The accuracy of the analysis of an unknown sample depends on the accuracy of the standard, on the degree of similarity of reaction condi- tions (1, g, temperature, ionic strength) between a sample and the stan- dard, on factors influencing the linearity of the calibration curve, and on the precision of an individual measurement. The absolute accuracy of the standard, sample, or reagent volumes delivered to the reaction cell is unimportant, although sample and standard solution volumes must be identical to minimize error. Because of calibration with a standard, the IQ 1‘ 20 absolute accuracy of the reaction monitor or signal modifier is generally unimportant. However, the absolute characteristics of the reaction moni- tor and signal modifier must remain constant over the time necessary to perform standard and unknown sample measurements. Usually differences in reaction conditions can be minimized by treating standards and samples identically. However, the possibility exists that chemical species present in real samples, but absent in relatively pure standards, may in- fluence the rate and hence affect the accuracy. Although not commonly used, the method of standard additions could compensate for such errors in certain situations. In this chapter, the effect of the rate computational system and the reaction monitor on the relationship between the readout and the concen- tration of the sought-for species is discussed. Since for completely automated kinetic analysis linearity is desired, it is important to know the factors that can cause deviations from linearity and the rate compu— tational system which gives the best linearity. Precision of kinetic measurements is discussed in Chapter VI. B. Type of Reaction In this section, the application of the fixed-time and variable-time methods to commonly used types of reactions for kinetic analysis is ex- amined. Equations are developed to point out the factors which influence the choice of rate computation approach and to describe the errors that can be encountered under certain measurement conditions. The general procedure is to express the concentration of the sought-for species in terms of the measured parameters of the integral computational approaches. l. Pseudo-First Order Reactions a. General Treatment. In many analytical rate methods, conditions are controlled so that pseudo-first order kinetics exist. For irrever— sible cases, the reaction can be written as k R-——aP (4.1) where k is the pseudo-first order rate constant. The concentration of R at any time is given by [R] = [Rloe'kt (4.2) where [R]o is the initial concentration of R. The theoretical rate at any time is found by differentiating Equation 4.2: Ratet = J15? = k[R] = k[R]oe‘kt (4.3) With the integral rate methods, either A[R] is measured for a fixed time interval, At, or At is measured for a constant A[R]. The relation- ship of the measured quantity to initial concentration can be obtained by integrating Equation 4.3 -A[R] = [R]o(e'kt1 - e’ktz) (4.4) where t1 and t2 are the starting and finishing times of the measurement. Here it is assumed that the reaction can be followed by some reaction monitor whose response is linear with concentration or can be electron- ically linearized so that the change in concentration of a reactant or product is related to the measured signal change by a constant. Non- linear response characteristics are discussed in a later section. Equation 4.4 can be solved for the initial concentration of R re- sulting in Equation 4.5. 22 [R10 = -(A[R])/(e"‘t1 - e'ktz) (4.5) If the product is being followed instead of the reactant, a similar equation can be obtained in terms of A[P] by substituting the stoichio- metric relationship between A[P] and A[R] into Equation 4.5. A more con- venient form is obtained if the relationship t2 = At + t1 is substituted into Equation 4.5. [R]o = -A[R]ektl/(l - e'kAt) (4.6) Equation 4.6 expresses [R]0 in terms of the experimentally adjustable and measurable parameters of both of the integral rate approaches. For initial rate methods, pseudo-zero order kinetics are assumed to prevail during the measurement. This assumption requires that the meas- urement be completed while the exponential concentration versus time re- lationship is linear. To determine the range over which this approxima- tion is valid, the exponential can be expanded in a Maclaurin series: e”kAt = 1 -kAt + (kAt)2/2! + ... (4.7) and substituted in Equation 4.6 which yields _ kt 2 ... [R]o - -A[R]e l/(kAt - (kAt) /2! + ) (4.8) For small kAt, Equation 4.8 reduces to [R]o =-A[R]ektl/kAt (4 9) Equation 4.8 predicts a direct linear relationship between [R]o and either -A[R] or the reciprocal of the measurement time (l/At) under the conditions of small kAt. Previous treatments have indicated that the linear relationxhip between [R]o and reciprocal time is accurate to 1% if the measurement is completed before the concentration of R changes by 23 5% (10). If [R]2 is the concentration at the end of the measurement time, t2, and [R]1 is that at time t], the concentration change which can be tolerated with high accuracy can be easily obtained. The value of kAt necessary for a given accuracy is calculated from Equation 4.8, and this result is used to calculate the relative concentration change from Equation 4.10. £n([R]2/[R]]) = -kAt (4.10) Table I presents the error which results from ignoring the higher order kAt terms as a function of the relative concentration change. Table I indicates that for high accuracy, the initial rate approximation is valid for only a very small fraction of the reaction. For 1% accuracy in the initial rate approximation, the relative concentration change must be less than 2%. As will be shown below, this restriction is important in variable-time methods, but unnecessary in fixed-time procedures for first or pseudo-first order reactions. TABLE I. Validity of Initial Rate Approximation For First or Pseudo-First Order Reactions kAt [R]2/[R]1 Relative Change Error in [R]1 % % 0.002 0.9980 0.20 0.10 0.005 0.9950 0.50 0.25 0.010 0.9900 1.00 0.50 0.020 0.9802 1.98 1.00 0.050 0.9512 4.88 2.52 24 b. Variable-Time Measurement. In the variable-time method, the time required for the reaction to proceed by a fixed amount is measured, and the initial concentration of the rate-limiting species is related to the reciprocal of the measured time interval. Thus in Equation 4.6, A[R] is kept constant and at is measured. Since both At and t1 vary depend- ing on the value of [R]o, a non-linear relationship always exists between [R]° and l/At. If A[R] is chosen to represent a small fraction of the reaction, then Equation 4.9 is valid as an approximation and within a certain accuracy, [R]o is directly proportional to l/At. However.in practice, A[R] must be large enough to obtain a measurable signal change. Also, for the smallest error, the concentration of R at t] should be as close to [R]o as mixing times and induction periods allow so that the exponential term in Equation 4.9 approaches one. To illustrate the errors that can result in the variable-time method even when small concentration changes are chosen, consider a pseudo-first '1 and test concentrations in the range order reaction with k = 10"3 sec [R]o = l - 10 mM. If the product, P, is monitored, a reasonable choice of two concentration levels would be [P]1 = 0.01 mM and [P]2 = 0.02 mM. For the lowest concentration of R, 2% of R will have reacted at the end of the measurement interval. The time required for the reaction to pro— ceed from [P]1 to [P]2 can be calculated from Equations 4.11 and 4.12 for each initial concentration of R. [p]1 = [R]o(l - e'ktl) (4.11) [P]2 = [R]o(l - e'ktz) (4.12) The results of these calculations are shown in Table II for l, 5, and 10 mM R. The 5 mM solution is chosen as the standard and used to 25 TABLE II. Error of Variable-Time Approach [R]o taken At l/At [R]0 calculated Error (mM) sec sec-1 (mM) % 1 10.1524 0.09850 0.9879 -1.21 5 2.0060 0.49850 - - 10 1.0015 0.99850 10.0150 +0.15 calculate the other concentrations. It can be seen that an error of 1.21% results for a lO-fold concentration range by using the variable-time approach without considering errors of sampling or measurement. It is also noted that the error is higher than the 0.5% indicated in Table I for a 1% relative change in the concentration of R. This arises because Table I only considers errors due to the magnitude of At, while in an actual analysis both t1 and At change for different initial concentrations. c. Fixed-Time Measurement. In the fixed-time approach, the change in concentration that occurs during a fixed-time interval is measured. Thus in Equation 4.6, t] and At are held constant during the measurement so that absolute linearity holds between the measured quantity, A[R], and [R]o, even if the rate curve is not linear over the measurement interval. This is a basic property of first order reactions. The relative % change in concentration over a fixed-time interval is constant, and the absolute % change is directly proportional to the initial concentration. Calcu- lations of the initial concentration of R with the fixed-time approach, on the hypothetical case presented in the previous section, show abso- lutely no error. Since the measurement is not restricted to the initial 26 part of the reaction, the fixed-time approach has a larger dynamic range than the variable-time method for first or pseudo-first order reactions. Although the above discussion considers pseudo-first order reactions that are irreversible, the same arguments can be applied to a first or pseudo-first order equilibrium of the form: 1 R # P (4.13) “-1 where k_] is the reverse pseudo-first order rate constant. It can be shown (54) that the rate expression can take the form tn WED—3%,, = t(k] + k_]) (4.14) e where [P]e is the equilibrium concentration of the product. This form is analogous to a first order rate equation with [P]e replacing [R]o and (k1 + k_]) replacing k. By using the same procedure that was used to obtain Equation 4.6 from Equation 4.2, it can be shown that the relation- ship between [R]o and A[P] is K k [R10 = (TLE’TLJ'HALPDeHl I k-1)t1/(1 - e‘("1 " k-l’“) (4.15) 1 As before, if At and t] are held constant by using the fixed-time approach, a direct linearity exists between [R]0 and A[P]. d. Experimental Comparison. The analysis of phosphate by spectro- photometric monitoring of the rate of formation of the heteropoly blue was chosen to compare the fixed- and variable-time methods with a pseudo— first order reaction. The spectrophotometric system, reagents, and re— action conditions are described in Chapter III. Absorbance-time curves were recorded on a strip chart recorder, and both variable-time and fixed- time data were manually taken from the same curves. Time intervals of 27 one and two minutes were chosen for the fixed-time method, while absor- bance intervals of 0.005 and 0.01 were chosen for the variable-time method. These intervals were chosen so that similar amounts of phosphate were consumed during the measurement interval by either approach. For the longest time interval, about 5% of the initial concentration of phos- phate was consumed during the fixed-time measurements, while approximately 7% reaction occurred during the 0.01 absorbance unit change for variable- time measurements on the lowest phosphate concentration (2 ppm P). Table III presents the data obtained for the analysis of 5 and 10 ppm P based upon calibration with a 2 ppm standard. Data treated by the fixed- time method for both time intervals show no significant differences from the expected values calculated from the 2 ppm standard (Student "t" test, 95% confidence level). Likewise, the variable-time data for the smaller absorbance interval (0.005A) show no significant errors. However, highly significant errors ("t" test at 99% confidence level) are indicated when the variable-time method is used with an absorbance interval of 0.01. This occurs because Equation 4.9 is a poor approximation to Equation 4.6 for the 2 ppm standard, which leads to positive errors when unknowns of higher phosphate content are measured. The data also indicate that the accuracy and precision of fixed—time data on the pseudo-first order reaction improve as the time interval is increased. In practice, extreme- ly long time intervals would be avoided to decrease the effects of tem- perature and measurement system drifts and possible side reactions. 2. Enzyme Catalyzed Reactions a. General Treatment. Enzyme catalyzed reactions are used analyt- ically to determine both enzyme activities and substrate concentrations. As with the pseudo-first order case, a rigorous treatment of the rate 28 m.~+ wn.o~ o._- om.m ¢.~+ um.m o.~+ o~.m N.Loccm ncaou &.Loccm ouczom ._mm u ._mm mewauanmwcm> mzococqmoca sag muogpmz mew»-w_amwcm> can .c_s _ .ncmncmpm Ema N a co women use mpFDmmc m mo mmmcm>m co m_ cowumcucmucou ucsoe N.o+ mo.o_ o.m+ om.o— o._- mm.v v.m+ NF.m m.coccm uczom &.coccm chom .Pmm A .Pmm a mswuuumxwe maogosamoca sag -vmxwm ma meowumcwememo mpmcamoza mo comegmqsou . << .moo.o u q a u << .moo.o " _< u - . - F 1 pa cwe _ u y a u pa .cwE _ u —p m mg» mmmmu P_m :H cmxmu maococamogn Ema .HHH m4mm3 coumeoccuocos u 0A mLmLmeAm o m + 0A.w A.w m1oA Lo» +111m11 1 AV u :o_pu::m “AAm u : o _ A1A_ Eu sump: “AAm Eu .AmuAAm “Axm use mucmcpco Amzcm GCAEammmV nope: pAAm o.A AV cm .LOmeoczuocoE to mAmcm wocmpqmuum vaom Coll! : 3 I cc mmmAcoAmcmEAu .muwuqo coumsoczuozoe mo couumw :oAmmAEmcmcp u AA V 0e a AoA.AVp2:aAAV 0A Eu cm .copumw coAmmAEmcmcu LoumEocsuocoE 11 A r< V ...- m m m u c m o o a x . u mu A1; A1um N15 m5 c w M AmoA wmmm m AAVzeAA.AV$o11Lrumw111Ac .QEmA :mummczu com A 8 VI A15: N1Eu A1cm A1umm mcogoza .mucmAuML Amcuomam muc30m u A: go .ucmEmAwe :mummcsu mgp co weapmcmasmu LOAOU u A.“ AAVze.w o .mmmAcoAmcmEAu .3ou:A3 820A we souume :oAmmAEmcmLu1 A V AUV A.w AF.Ach.w o .mmmACOAmcmeAn .pcmEmAAe cmpmmcsp VAVAHT>AmmAEm 1 Ah xo E: o— x mwm¢._ o_ x om~._ A Eczop ll U m u A-.. N1... 4 A1 m w 11LEDDLW1 u A2 .quA :mummcap to» m1AAU A15: N158 A1cm 3 .mucmwumc Amcuumqm mucsom u Az N.m vcm A.m mcowumscm cw onnEXm mo cowuwcwwmo .A> m4mApAmcmm LmAAaAAADEouogq II A r-C v U3 E mmmAcoAmcmEAn .mnau co :Amm mmmcm>m mmmAcoAmcmEAu .muocxu uche mo AucmAuAeem :oAuumAAou c A-3 A .AAA>AAAmcmm UAeocAmu AcmAemL 1 AAVo ECAAVo 3 < .Aouumw AAA>AAAmcmm LmAAunAzeouoga II A r< v m Fl A u x mA ucm>A0m mews; mmAumam chnLOmam mo amass: u A A1A monE .mmAumqm mcwncomnm cuA mo :oApmcucmucou u Au A158 A1onE A .mmwumam chnLOmac cuA mo AAA>AAaLomam cone u AAV o A Eu .gumcmA sung AAmu u n on mmmAcoAmcmEAu .AAmu mAaEmm An noncomnm cmzoa pcmAnmc mo :oAuumcm u mmmAcoAmcmeAu .mmumeczm AAmu mAqum um :oAAumAAmL ou wsv AmoA cmzoq ucmAnmc mo :oApumcm u A A A Au. 8 A.UAAV.. A AA.N-VQXAA A - AVA. - AV 1 x mmmAcoAmcwEAn .couume :oAmmAEmcmcp Acosucmasou mAaEmm u AAVmA 2: 18mm mcouona .cmzoa acmAGmc Accuumam ugmAA Accum 1 AApmqV A- A . AA... 15: 3 .cmzoa ucmAnmc Amcuumam acmAA Accum 1 A AV A A150 E: .Loumeoccuocoe mo :oAmcmamAu AmchA Amuocqwumc u cm 2: .mmmqucma Amcuumam u 3cm 1 m A.U.u:ouV .A> m4m 10 ) are discussed in detail in Appendix A. Unless otherwise stated, it will be assumed that photon counting is used 47 only under conditions of negligible pulse overlap so that Equations 5.10 and 5.11 are valid. A useful form for Rap taken from equations developed in the first section is k = 2 Rap nAoTop(Ao)QHW Rd(l-f)(l-aC)K(xO)n[exp(-2.3035;?1(I )C )] (5.12) O l The final expressions for the readout in both the DC measurement and photon counting systems are consistent with those of the general spec- trometric system discussed in section B of Appendix B. The primary dif- ference is that background radiation is absent in a molecular absorption spectrometer. Thus, all the background terms discussed in Appendix B are zero and 1ap = 1as’ Eop = 505’ Rap = Ras’ R0p Ros’ and N0p = Nos“ Note also that the incident signal photon arrival rate, rs, defined in Appendix13 15 just Rap d1v1ded by s(x0). C. Relationship of Variance To System Parameters If the complete spectrOphotometric system is considered, many sources of error can be identified. Excellent discussions Of the chemical and instrumental deviations that affect the accuracy have been presented (51, 60) and will not be considered here. Only random instrumental errors that limit the precision Of making the absorbance measurement will be discussed so that imprecision due to such factors as sampling or cell positioning is not evaluated. A modern,stable, single-beam spectrophotometer is considered such that with suitable warmup time, the stability of the various components is excellent, and undirectional drifts in the light source, the photo- multiplier, the amplifiers, and the readout device are negligible over the time needed to perform an ordinary analysis. The photomultiplier and 48 current-to-voltage converter 0A are assumed to be Operated under condi- tions where errors due to non-linearity are negligible. The errors con- sidered are therefore random and independent, and the total instrumental variance is thus the sum of the different individual variances in the system. A treatment of variance sources in a generalized spectrometric system utilizing DC measurement or photon counting is found in Appendix B. Ex— pressions for the total variance in Er E N , and N0d for a single- p’ rd’ Op beam molecular absorption spectrophotometer are presented in Table VIII. A more detailed discussion and explanation of the variance terms and para- meters utilized in Table VIII is found in section C of Appendix B. Two primary differences exist between the variance expressions for the gen- eral spectrometric system and the variance expressions for molecular absorption spectrometry. First, since background radiation is negligible in molecular absorption, the variance in the background is negligible. Secondly, the signal radiation flicker factor, 5, is denoted the source flicker factor in molecular absorption because signal radiation flicker is solely due to non-ideal fluctuations in the tungsten or deuterium lamp spectral radiance. If a photodiode is used instead of a photomultiplier, a and thus 2 (o) Eop sec unless otherw1se stated, 1t 15 assumed that Erp = E0p and Erd = Eod' If Eop and E0d are modified before reaching the readout device, additional variance terms must be added to Equations 5.13 and 5.19 and the individual equal zero. For Table VIII and for the remainder of the paper, variance terms multiplied by suitable factors to make their contribution apprOpriate for the final readout voltage. For instance, if logarithmic amplification is used after the current—tO-voltage converter so that E = log E then rp op’ 49 AAA.mV AAAmAxe N> .LOpmwmmL xumnvwmw cw mmwoc :OmczomA OH 016 QLm CA. wucwwLm> NU mmmAcoAmcmEAu .guquncmn acmemcammme Lm>o mucmAumc Amcuumam mucsom AcmAA Ao mucmAAm> m>AumAmc Ao poo; mcmzcm Lo LegumA cmeAAA muczom AsmAA u w A no no A new AmA.mV N>.NA m AMV u cmquAA OOADOm AgmAA ou msu m :A mucmAcm> u A NoV no EA new N> .mon: cmeAAA LOAAQAAA350p02a on one u cA mucmALm> n A NOV mmmAcoAmcmEAu .coAmmAEm Acmccoumm ou man :Amm cmAAaAAA350Aoca cA mochLm> m>AumAmA u a AmA.mV AammsaaAAEN 1 no umm mom N> .monc :oAmmAEO Acmucoumm ou man u cA mucmAcm> u A NOV NI .Empmzm psonmmc1LmAAAAqu Ao gunAzucmn Acon>Azcm mon: 1 Ac < .cauA u pcmcczu OAuocumOOAOLQ O>AAOOAAO u QOA < .Ec\amA u pcmccsu OAuozpmuoaosa u QOA AAA.mV AAAAQUAA EN 1 AAAmauAca EN 1 AAAmaaAAEN 1 N> .mon: Eaucmac ow man now :A mucmAcm> u 5A on no no no . nco name a A we Ea mo umm mo 2A3 N. N .x. ANV. A NV. 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UL NLLJ N a c.ua N m.cc N D > .OOA>mc “sonmmc Ao AAAAAnmummc Lo :o_u:A0mmc on man cm :A mucmALm> N m cm :A mucmAcm> o u mon: uaonmmc ou man new :A muceAcm> 1 B0 A > .muA>mn Azovmmc CA umuwcmcmm mon: AmOAcAOmAm 0“ man N m.uc N . .L.nc ANA mV N> N0 + O N> .mon: AOAAAAan on man cm :A mucmAcm> u 0 A0 .A1 A0 mczumcmmsmu muaAomnm ll 1.. acmpmcou :cmENAAom II x A.U.u:ouV .HHH> m4m n A NOV no n opo A no no o noz AA m < + a u A NoV .1 no no no xm zo zo o zo 1 ANV+N+ANvl no noz oA oocoAao> Aoooo 1 mo oo oonc mmooxo on non ooz :A oocoAco> u on moV oo ncz oonc noonooc op won 2 cA mocoAco> u No on A oo oo A ooz NAoA m u A NoV on A oo oo o ooz A m < u z 1 oonc sopcooo op won 2 :A mocoAco> u A NoV no no oo oo oo xo zo zo A zo o z 1 z A N v + N + A N v + A NOV 1 ND A.n.ucouV .AAA> m4ms..+. <42» rp EOp op op op op 2 2 2 2 + OJ + Camp] + Ord + 0log where 0709 is the variance in the readout voltage due to noise in the log amplifier, and the log amplifier is assumed to have an output voltage of l V per decade. The variance due to the readout deserves special mention. For pho- ton counting, ofi d is unity because of :1 count uncertainty inherent in a non-synchronou; digital frequency measurement. This will rarely be Of importance in molecular absorption measurements unless extremely high . 2 absorbances are be1ng measured. For DC measurement systems, 0rd can 2 Often be a significant contributor to the total variance. Usually °rd r will determine the magnitude Of aid. In the case Of an analog readout device (oscillosc0pe, meter, recorder), 0rd,r is a fraction of the total readout scale, which represents the standard deviation in reading a value from the scale. For instance, a normal meter can be read to 10.5% of full scale, which yields a readout variance of 2.5 x 10'5 V2 if full scale is one volt. In a similar manner, a digital readout device has an error due to uncertainty in the least significant digit. Thus for a 3 digit deci- mal readout reading 1 V full scale, the resolution is l mV,which yields ‘5 v2. a readout variance of 10 Analog readout devices with multiple scales have a variable “Ed' 0. Relationship Of Signal-tO-Noise Ratio to System Parameters l. DC_Measurement System a. S/N For Photocurrent Measurement. In a DC system, the S/N for measuring Erp’ (S/N)gc, is given by 53 é)dc = E” 2 EM 1 2 = E”, (5.28) N (32 + b + C2) / (a2 + b2 + c2)1/2 as discussed in section 0 of Appendix B where Ert = total readout voltage, V = Erp + Erd = 1atRf a2, b2, c2 = group variance terms which arrange the total variance into three groups which are proportional to icp’ pro- portional to i2 , and independent of i , respectively cp cp The group variance terms for molecular absorption measurements are defined in the following equations: 2 2 2 2 a = (o ) + (o ) + (o ) (5.29) Eop q Eop sec E0p pm 52 = (a: )f (5.30) 0P 2 2 2 2 2 2 2 c =2[o +11 +11 +(o )+(o ) +10 )1 (5.31) J amp rd E0d q E0d sec E0d pm 2 The individual variance terms in the c group are multiplied by 2 because two measurements (Ert and Erd) are needed to obtain Erp‘ (SN);C gives an estimate of the precision with which a particular photocurrent readout voltage can be measured. For absorbance measure- ments, the information desired is encoded in the measured value of trans- mittance (T), of one over transmittance (l/T), or of absorbance (A) so that it is useful to define S/N's in which T, l/T, or A become the "signalt Since T, 1/T, and A are functions of the ratio of the photocurrent read- out voltage for the reference and sample solutions, the precision of mak- ing an absorbance measurement depends on the precision involved in making the sample and reference solution measurements. For this discussion, it will be assumed that the concentration of the jth species is being determined by measuring the solution absorbance. 54 The reference photocurrent readout voltage, (Erp)r’ 15 equal to Rf(1ap)r’ where (iap)r is the photoanodic current with the reference solution in the sample cell, as given by Equation 5.9 with Cj = 0. The sample photo- current readout voltage, (Erp)s’ 1s likewise equal to Rf(iap)s, where ( ) iap s is the photoanodic current with the sample solution in the sample cell, as given by Equation 5.9 with Cj some finite value. The refractive indices of the sample and reference solutions are assumed to be nearly equal so that the same value of f in Equation 5.9 can be used for both solutions. The only difference in calculating the group variance terms for each voltage measurement is that for the photocurrent dependent individual variance terms, iap is replaced by (iap)r and (ia reference and sample group variances, respectively. p)S for the b. S[N For Transmittance Measurement. Equation 5.32 gives the S/N for a transmittance measurement. dc a2+b2+c2 a2+b2+c2 “”2 S _ T _ s s s r r r (11) - —- 2 + 2 (5.32) m 0T (E ) (E ) rp s rp r where S dc (N) = S/N or the reciprocal of the relative standard deviation m of a transmittance measurement, dimensionless T = transmittance, dimensionless = (Erp)s/(Erp)r = [(Ert)s - ErdJ/[(Ert)r — ErdJ CT = standard deviation Of a transmittance measurement, dimensionless 2 2 2 2 2 2 = TEaS + bS + C5 + ar + br + Cr]1/2 (1:?) (1:7) rp s rp r (Ert)r = total readout voltage with the reference solution in sample cell, V (Erp)r + Erd 55 (Ert)s = total readout voltage with the sample solution in sample cell, V = (Erp)s + Erd as, bi, c: = values of group variance terms with sample solution in sample cell, v2 a3, bi, CE = values of group variance terms with reference solution in sample cell, V2 Note that measurement of e1ther (Erp)s or (Erp)r requ1res two meas- urements because of the presence of dark current. The transmittance may be found by making separate measurements of (E E or by elec- rt)s’ ( rt)r tronically setting Erd to 0% T and (E r to 100% T on the readout device rt) and measuring (Ert)s as the transmittance. In either case, Equation 5.32 is valid since the Operation of setting 0% or 100% T is a measurement. The S/N for a 1/T measurement, (S/N)?;T, equals (S/N)$C since the stan- dard deviation of a l/T measurement, Ol/T’ equals oT/Tz. c. S/N For Absorbance Measurement. The S/N for an absorbance measurement, (S/N)gc, is given by (Ride = 2": 2 2 -2" T 2 2 2 A A aS + bs + cS + ar + 5r + cr]l/2 (5.33) (Efpls (Efp1r where °A = standard deviation of an absorbance measurement, dimensionless = (0'4343)T°1/T = (0'4343)°T/T A = absorbance = -1og T Note that Equation 5.34 has been used to calculate CA (61,62) 0A 2 log(l/T + al/T) - log(l/T) (5.34) To see where Equation 5.34 is valid, oA/(O.4343)T can be substituted for 56 01/1 in Equation 5.34 and the log term expanded, which yields 2 CA CA °A ' 0°4343 (0.4343 “ 2(o.4343)2 l "'> (5'35) Equation 5.34 is valid only for small values of 0A where the second term in Equation 5.35 is negligible (<1% error for 0A <0.0086 or oT/T < 0.02). Equation 5.33 is the most useful S/N form, because the S/N is directly proportional to the absorbance and hence the concentration of the sought-for species. Thus (S/N)gC gives a direct estimate of the relative precision of a concentration determination. Because of the dif- ferent photocurrent dependences of the group variance terms, it is easily 2 2 2 2 2 2 2 shown that aS = Tar, bS = brT , and cS = Cr‘ Us1ng these relat10nsh1ps in Equation 5.33 with some rearrangement yields dc -(E ) 1n T (s) = rp r (5.35) N.A [aE(l + 1‘1) + 25E + ci(l + 1'2)]V2 The value of (S/N)gC is thus dependent on the magnitude of the group variance terms for the reference solution and on the value of transmit- tance. If the log is taken electronically with a log amplifier, and log (Erp)r is set equal to zero, the readout variance term enters differently into the total variance as was indicated in Equation 5.27. Usually for direct absorbance readout, the dark current is considered negligible com- pared to the photocurrent. If the dark current is significant, it must be suppressed before entering the log circuitry to yield an accurate value of A. If it is assumed that the dark current is nulled by using a very sensitive meter to make the readout variance negligible, Equation 5.36 can still be used with the group variance terms for the reference . . . 2 2 2 solut1on be1ng redef1ned as ar,1. br,] r,1 2 _ 2 ar,] - ar (5.37) and c as follows: 57 +(2.303)2(E ) [oEd+ 2 1 (5.38) b Eop r 0log '50“) "SN - 20% (5.39) To find the optimum value of (S/N)gC with respect to transmittance, Equation 5.36 is differentiated with respect to T and set equal to zero, resulting in Equation 5.40. aE(T2 + 1) + 25312 + cE(12 + 1) 1n T = 2 V?_ (5.40) -(arT/2 + Cr) This is a transcendental equation which can be solved numerically for the optimum value of transmittance, Top’ if values of ai, bf, and c: can be calculated or measured. d. Limiting Cases. Under certain conditions one of the three group variance terms may predominate over the absorbance range utilized, which allows considerable simplification of Equations 5.36 and 5.40. Table IX presents these simplified expressions, values for the Optimum transmit- tance (Top), the Optimum absorbance (Aop), and the S/N at T0 p’ (S/N)AC op for the three limiting cases. Note that these limiting cases hold for a specified absorbance range within a given accuracy which can be cal- culated from Equation 5.36. As the transmittance approaches zero, the cE(l + T'Z) term in Equation 5.36 becomes dominant, and in the limit (SN):C always approaches zero. Case I occurs when c3 >> a: + bi, so that (SM):C is limited by factors independent of the photoelectron current such as amplifier-read- out noise, dark current shot noise, or readout variance. From Equation dc . . . 5.41, (S/N)A is directly proportional to (icp)r’ where (i cp)r1s the effective photocathode current with the reference solution in the sample ) ) /m). If 12 cell ((iCp r = (i 1ap r in Equation 5.42 is considered negligi- ble compared to unity, which would result if there were no variance in 581 ocoz ago. A 7 LAAA..A AeALANLNV- AA..NV oonc LoonAA cocoon L on 6 A NAV L Na AAA omau co on.mV LANLNVAmao.oV Noo.o moA.o Amo.mV AA . AVN- 1 AcA A 1A+AVco A¢¢.mV mnr1r14w1111 AcALA LoV- ”cougauouogo cA oonc uozm L no A AVo c- N AA omou NeoAA AAA.» An aoNAeAA AAAcaA .. 8. An..mV _o1NH11NW11w11 «An. A NVAAAN oV on a . ac . on. o a .o=.a.oNa. assANao . aa . Ann o A .oc...AeA¢.ao aaaAoao AN¢.mV AA . AV- . AeA o..m caAoasoo to sea» ocANAer N x A AAAVLO Aon.m coAuooau Ao nLoA ucAuAnAAA AA¢.mV N1A1Nn1m1111 .3 AaA.A .AV- aaAA. oucoALos usen-c .Oonc LOAAAAoeo nco comczon .ucocgau neon :A oonc yogm oucoAgo> Ao «oucaom “canoooonc. c ouAV co oucoAcos so coconcoooo A L 0 ouooAcos oaocu ueocAeoo A anon mmuocAEonoco Eco» mocoAco> ozone woo cos: mcoApAncou EosAuoo non mcoAuooom .xA mAm CE + bi, so that (S/N)gc is limited by variances pr0portional to (i ) such as photocurrent shot noise or photo- cp r multiplier flicker noise, and (5m):C increases with the square root of (‘cp)r' higher S/N than Operation under Case I, the S/N can still be improved by 2 Although operation under conditions where Case II holds provides increasing (i ) until the b variance terms predominate (Case III). At cp r r high absorbances, c§(T’2+ l) is no longer negligible in Equation 5.36 and a combination of Cases I and II results. 2 2 2 If br >> ar + cr, Case III occurs, and (5m):C is limited by vari- ances proportional to (icp)E. Hence Equation 5.47 indicates that (SN):C becomes independent of (icp)r when Case III holds. Equation 5.40 cannot be solved for the optimum transmittance since T0p approaches zero. Thus (5m):c increases linearly with absorbance until stray light, which has been neglected, becomes limiting or [aE(l + T") + c§(l + T'2)] in Equation 5.36 is no longer negligible. Operation under conditions where Case 111 applies represents the highest achievable S/N for a given Spec- traphotometer. 60 The above discussion is equally valid if the log is taken electroni- 2 cally rather than manually, except that the readout variance is a br rather than a c: type variance. Thus, under Case III a direct absorbance measurement can be limited by either readout variance or source flicker. As fOr Case I, the readout resolution should be increased until aid is negligible compared to (a: )f. 0P e. §1§_Plots. For a particular instrument and analysis, a plot Of (SN):c versus transmittance can be made to determine the absorbance range giving Optimum S/N and thus the best precision. From such a plot the expected measurement precision for any concentration analyzed can be evaluated. Figure 3 shows plots of the normalized S/N for an absorbance dc mr’ dc mr 15 the measurement, (S/N)gc/(S/N) versus transmittance, where (S/N) S/N for measuring the reference intensity. Curves l-3 are for limiting Cases I-III respectively. Curves 4 and 5 are for two intermediate cases in which two of the group variance terms must be considered. The normalized (5m):C for any instrument will lie in the area be- tween curves l and 3. The absolute magnitude of (SlingC at a particular transmittance is found by multiplying the normalized (SN):C at the transmittance value of interest by the (S/N)::. For instance, if Case II is valid, the normalized S/N at T0 = o.l09 is 0.695. If the (S/N)§: P is 1000 (the rms shot noise is l/lOOO of the reference signal), (5m):C = 695, and the relative precision of an absorbance measurement at l0.9% T is 0.149%. 2. Photon Counting_System a. S/N Expressions. Analogous S/N expressions can be develOped P for photon counting. The measured S/N, (a) , is given by Equation 5.48, m 61 ll (0 Nu AA Na AF.OV N O N 0 AV AAA ammo mu 0 x mm AA mm AP.OV HH mmmu A mmmu mucmppwsmcmLp mzmsm> oApmm mmvoz-0H-AmcmAm nmNAAmELoz .m mesmwu 62 m ogsmAA .P AUAU AAu NAG ngo .vAu mAu wAu 5A0 .wAu mzu AUA fi A A 4 q A A A fi WI \9\ Q X/ \D ““V\ X/ \x“@ A xnllllxlllluxtllllxu\\\1MT\\\\ 4 a\ o I], u““““ // .1 ||\\usuuuuumv““\‘nw .l o “VIIIMT\\\\O a ’O\ \ N O 0:0 m.nv A2. 0.. v (NIS)/ ”(N/S) Jun 09 ,1»: p. .‘y ~( I Ill l- ".‘p- . u I. . in Q - ‘ I \ ... C E. 63 (%’p = 2 Nate- N03 1 2 = 2 NQQ’ (5'48) m [x + y + z J / [x + y2 + 221172 where N0t = total observed pulse count = NOp + N0d * [Al(Rap + Ratc) + AdRad]t x2, y2, 22 = group variance terms for photon counting which are proportional to Rop’ proportional to RS , and indepen- P dent Of Rop’ respectively. For molecular absorption spectrometry, the group variance terms are defined as: x2 = (cg )q (5.49) on y2 = (ofiop)f + (Ofiop)ex (5.50) 22 = mafiodlex + (afiod)q + afirdl (5.51) P The S/N expressions for transmittance and absorbance measurements, (%9 T and (fiiA, are given by Equagion525.522and 3.53 2 2 S P _ T _ x5 + ys + 2s xr + yr + 2r -l/2 (Ni - g- - [ 2 + 2 ] (5-52) T T (Nop)s (Nop)r (§9P = fl. = '1" T N o 2 2 2 2 2 2 A A [x5 + ys + 2s + xr + yr + ZrJI/Z (Nip)S (Nfip), -(N ) ln T = '2 -l %p r 2 -2 *l/z (5°53) [xr(l + T ).+ 2yr + zr(l + T )] where (N ) (Not)s ' Nod _ Op 5 = T NOp r (Notjr ' Nod x2 + yz + 22 x2 + y2 + Z2 _ S S S l" 7‘ Y‘ OT ‘ T [ 2 + 2 Op)S (Nop)r (N 64 All other terms are as previously defined except that subscripts r and s indicate the reference solution in the sample cell (Cj = 0 in Equations 5.l0 and 5.12) and the sample solution in the sample cell (Cj finite), respectively, and the reference and the sample group variances are re- 2 _ 2 2 _ 2 2 2 = 2 s - Txr, yS - T yr and 25 Zr' respect to T, an equation identical to Equation 5.40 results except that 2 2 2 2 2 2 . ar, br’ and cr are replaced by xr, yr, and Zr’ respectlvely. If pulse overlap is significant, defining the transmittance as lated by x If (S/N)£ is maximized with (Nop)s/(Nop)r may cause difficulty at high photoelectron pulse rates and large absorbances. If pulse overlap in the reference beam is significant and T is small so that (R ) is sufficiently reduced from (R ) to make ap s aP r pulse overlap in the sample beam negligible, there will be a positive error in the measured transmittance. At T near unity, this error should be negligible. Transmittance could be more accurately defined as (Nap)s/ (Nap)r , since pulse overlap in the photomultiplier has been assumed to be negligible. However, this ratio is not measured. b. Limiting Cases. Similar to DC measurement, Cases 1, II, and III 2 2 2 r’ xr’ or yr can result if one of the group variance terms 2 dominate. Expressions and optimum conditions shown in Table IX are the same if 2 2 2 2 2 2 . (Erp)r’ ar, br’ Cr’ are replaced by (Nop)r’ xr, yr, and zr, respectlvely. Case I is unlikely when using photon counting because the readout variance is only 1 count and usually (Nop)r > N0d > 1. Hence in photon counting xi is usually much larger than 2:. Case III (source flicker noise limit) is also unlikely for photon counting because y? terms are dependent on (Rop)r and will only dominate at high pulse rates which are unachievable because of pulse overlap. Case II is most common in photon counting since usually x3 >> y: + '0 I 'i‘ a". .- t . . . . ’- . ' C .1; n 5- d- n 65 23. Thus (S/N)K is limited by quantum noise and is directly proportional to (Nop)r]/2’ If pulse overlap is significant, the (S/N)X increases faster than predicted by Possion statistics as has been discussed in detail in Appendix A. If N0d is not negligible compared to (Nop)r’ a mixture of Cases I and II is likely. (S/N): can be plotted versus T from Equation 5.53 for known values 2 2 2 of yr. xr, and Zr“ Plots of the normalized S/N for an absorbance meas- urement, (S/N):/(S/N);r, are analogous to those shown in Figure 3. E. Practical Considerations l. Simplification 9f_Group Variance Terms The magnitudes Of the group variance terms determine the magnitude of (S/N)?‘c or (S/N): and indicate whether one of the limiting cases dis- cussed in the last section applies. In molecular absorption spectrosc0py, light levels are usually much higher than encountered in other types of spectroscopy. Because of this, photoelectron shot noise is expected to be a significant contributor and many of the individual variance terms will be negligible compared to shot noise variance. . . . 2 2 2 2 As discussed in Appendix B, OJ, Camp, (oEop)pm, and (050d)pm usually negligible compared to other individual variance terms, particu- are larly in molecular absorption spectrometry. With these assumptions, the group variance terms for DC measurement of the reference solution can be simplified to 2 2 2 . 2 a = (o ) + (o ) = 2me(l ) R Af(l + a) (5.54) r Eop q Eop sec ap r f 2 _ 2 = . 2 br - (05 if (5(‘ap)er) (5.55) 0P 2 _ 2 2 2 _ . 2 2 Cr — 2[(OEOd)q + (OEOd)Sec + 0rd] - 4m81adeAf(l + a) + 20rd (5.56) .c’v (I? ..-, .‘.' :n a. _ .F "O. 66 Here it is assumed that y = m in Equations 5.20 and 5.2l. . 2 2 2 . In photon counting, ONrd, (oNop)ex and (ONOd)ex will usually be negligible (see Appendix B) yielding for the group variance terms 2 _ 2 = xr - (ON )q AlRapt (5.57) 0P 2 2 2 y = (o ) = (AR ta) (5.58) r Nop f 1 ap 2_ 2 = * Zr — 2(0N0d)q 2t[AlRatc + AdRad] (5‘59) 2. Effect gf_Reading Error Most traditional S/N treatments for DC measurement systems can be somewhat misleading because they assume that reading error can always be made negligible by scale expansion techniques or high resolution readout devices. However for most practical situations, a particular instrument has a readout device with limited resolution and input voltage range capabilities. Thus for high photocathodic currents, either Rf or m must be reduced to keep the reference solution signal within the range of the readout device. Also the maximum voltage output Of the current to voltage converter may impose a similar restriction. The measured S/N for the reference solution from Equations 5.28, 5.54, 5.55, and 5.56 is given by Equation 5.60 after division by me: dc (1 ) Q) = CR '" 2 2 (5.60) mr O . . 2 . 2 rd l/2 [2eAf(l + a)((lcp)r + Zlcd) + r, (lcp)r + 372?] f where lcd = effective cathodlc dark current = lad/m, A The limitation of the readout device is easily incorporated into thuation 5.60 by requiring that the reference readout voltage equal the fiull scale reading of the readout device, E0, or E0 = (E ) = (i ) me- 67 Solving for me and substituting into Equation 5.60 yields dc (lcp)r )+21’ . 2 2 2 2 r cd) + (1cp)r(€ + 20rd/Eo)] (i) N = (5.6l) mr [ZeAf(l + a)(( . l/2 lcp Equation 5.6T indicates that as (icp)r is increased, the readout variance becomes relatively more important compared to the shot noise terms. At the limit of large (i ) (S/N)%: approaches (g2 + zoEd/Egl'1/2 cp r’ The fact that readout variance becomes more important with (icp)r seem— 2 ingly contradicts the fact that 0rd is a c? type variance. However, the relative contribution of shot noise variance to the full scale reading cp)r’ while the relative contribution of readout variance remains constant with (lcp)r. decreases with (i Figure 4 presents log-log plots of (S/N)g: versus (icp)r for dif- ferent values of did. It is assumed that 05d is determined only by the resolution of the readout device and that Af = l Hz, a = 0.275. icd = ~16 l0 A, E0 = lV and g = l0-4. Curve l corresponds to an instrument having a standard meter or recorder for a readout device. Curve 2 cor- responds to a meter or recorder with l0-fold scale expansion capabilities. Curves 3 and 4 could correspond to digital readout devices with 4 and 5 digit readouts. l3 For photocathodic currents greater than 10' A, curve I shows that with poor readout resolution (S/N)g: is essentially limited by readout -l3 variance. For (i ) < l0 A the measurement is partially shot noise cp r and readout variance limited. As the readout variance decreases, (S/N)g: is shot noise limited to higher photocurrents. Curve 4 is obtained if aid is 10’10 V2 or zero so that over the photocurrent range represented in Figure 4, the readout variance is negligible if Cid :lO'10 V2. Note that scale expansion techniques or current suppression at the 68 Figure 4. Effect of Readout Variance and Photocathodic Current 0n Signal-To-Noise Ratio. X 0rd = 5-10'3 V -4 ‘Ord=5-'|O V E] 0rd = 1.10'4 v =l-l0’5V 0 0rd 69 /‘ :3 - ““::‘| / l/x ——x—x—x—x—x | 2 Log (sunfifr O l l l l J I l4 I3 l2 ll IO 9 3 -Log (icp’r Figure 4 0 ‘ o . ~~r;\r ‘. HI. 1;: r ‘I 5» 2‘ a ‘K A.. ... fl! -‘ d!‘ ffl 7O summing point of the current-tovvoltage converter do not reduce 03d but increase the maximum voltage accepted by the readout device, which de- Ed‘ Reducing the signal voltage creases the relative contribution of'd with a voltage divider before the readout device increases the relative contribution of aid to the full scale readout variance as does reducing Rf. Reducing Af decreases the magnitude of all variance terms except 03d and hence increases its relative contribution. Since (5m):c is a frac- tion of (S/N)$:, Figure 4 clearly illustrates that high resolution read- out devices can greatly improve the precision of molecular absorbance measurements. The above discussion applies to instruments in which the readout is in transmittance. If log circuitry is utilized (S/N)§: will be exactly the same (if O$Og is negligible) unless the readout variance is signifi- cant. Readout variance affects the direct absorbance case differently because Cid enters as a b? rather than c? group variance. Thus if the readout variance is dominant, (g):C will follow the curve for Case III rather than for Case 1. Under conditions where readout variance dominates, the effect of reading error on the precision of an instrument reading in transmittance can be compared with that of an instrument with direct ab- sorbance readout by assuming that the same readout device with readout variance, GEd, is used for both instruments and the only difference is a log circuit providing l V per decade inserted between the 0A and the readout device for the direct absorbance instrument. For the transmit- 2 tance reading instrument, c2 = 2°rd’ and if readout variance predominates, r 2 2 ar and br 2 _ 2 2 ment, br - (2.303) °rd’ and a can be assumed negligible. For the direct absorbance instru- E and c: can be assumed negligible. It is also assumed for the latter case that there is no readout error in sup- pressing Out the dark current before entering the log circuitry. The ,- L r l'. " .r-f‘ - x at" " a .. . . . ... I . o - J c . .. ~‘. f .... . .. 0". . n' o . .E ' 0'. ‘- ' fi . ‘r .. I u . N.- I- h ' U L" ' O .0 In" I I .' u ”if [all r-' 7l equivalent readout variance thus makes a larger contribution in a direct absorbance readout instrument. Setting Equation 5.4l equal to Equation 5.47 with the above values of b3 and CE, shows that transmittance read- out yields a higher (S/N)gC up to an absorbance of 0.32, while direct absorbance yields the higher (S/N)2C if the absorbance is greater than 0.32 because, for direct absorbance readout, the readout variance is a 2 r b term and its contribution decreases with absorbance. Because there is essentially no reading error in photon counting, the S/N always increases with increasing (Rap)r until limited by pulse overlap or source flicker. The measured signal-tO-noise ratio for the reference solution is thus given by combining Equations 5.46, 5.57, 5.58, and 5.59. (§9P = (Nop)r 2 1 N mr [(Nop)r + 2N0d + E (N0p)E] /2 l/2 (5.62) A](R ) t = ap r * [A]((Rap)r + ZRatc) + 2AdRad + t(A1(Rap)r€)2]]/2 F. Comparison of DC Measurement and Photon Counting A critical comparison of DC measurement and photon counting for a general single beam spectrometric system in Appendix B reveals the ad- vantages and disadvantages of both techniques. Only the conclusions relevant to molecular absorption will be presented here. Using Equations 5.35, 5.54, 5.55, and 5.56, (5m):C is given by Equation 5.63 with the restriction me(icp)r = E0. dc (pr = [-(icp)r ln T][2eAf(l + aim ) (l + i") (5.63) cp r )ZIEZ + (ord/Eo)2(l + l'ziii"/2 . —2 . + Zicd(l + T )} + 2(iCp r ~va‘ r.-- . l a” ‘5 72 Likewise, (S/N): is found from Equations 5.53, 5.57, 5.58, and 5.59 to be 5 P —l —2 (a) [-(N > in T][(Nop)r(l + T > + 2(N0d) (l + i > A opr 2 2 -l/2 + 2g (Nap)r] (5.64) [-A (R ) t”2 ln T][A](Rap)r(l + i") + 2(A l ap r 2 ~l/2 aplr) l * -2 lRatc + AdRad)(‘ + T ) + 2t(A]5(R TO compare the S/N advantage of photon counting to DC measurement under equivalent conditions, as discussed in detail in Appendix B, Equation 5.64 is divided by Equation 5.63, where the relationships Af = l/2t and (icp)r = e(Rap)r are used, and it is assumed that A] = Ad = l, iCd = eRad, and aid is negligible. The ratio indicates that photon counting has an advantage of /TE§ if the shot noise terms are dominant compared to source flicker variance. This amounts to an advantage of 5 to 22% because a usually varies from 0.1 to 0.5. Since pulse overlap will usually occur in photon counting systems at light levels below the source flicker limit, under equivalent noise bandwidth conditions the maximum achievable S/N with linearity may be much lower than can be ob- tained with DC techniques. For example, even with a very fast photon counting system possessing a 10 nsec deadtime, pulse overlap limits the maximum pulse rate to lo5 sec-1 for 0.l% linearity (unless instrumental or mathematical linearization is used). In DC operation photomultipliers can be Operated with excellent linearity up to currents correSponding to pulse rates of at least l010 sec-1 if the photomultiplier gain is reduced for high photocathodic currents to keep the photoanodic current in the linear operating range of the photomultiplier. For even higher light levels the photomultiplier can be replaced by a photodiode to extend the linear range for DC techniques even further. Criteria for choosing ‘1' 73 between the two transducers are discussed in detail in Appendix C. To reduce the light level impingent on the photocathode to bring the pulse rate within the linear range of the photon counting system would have serious disadvantages. Reducing the light level by even a factor of two will reduce the S/N by approximately 30%, and thus negate any inherent S/N advantage in photon counting. Discrimination against dark current pulses originating down the dynode chain is expected to provide little or no advantage for photon counting because the photocurrent will normally be much greater than the dark current except possibly at very high absorbances or extremely small spectral bandpasses. Photon counting is not as subject to reading errors, domain convere sion errors or non-linearities as are DC techniques. However,scale ex- pansion methods and high quality, high resolution A-D converters can Off- set these advantages. Although photon counting is inherently more stable than DC measurements, permitting the use of long counting periods to reduce the noise bandwidth, usually no distinct advantage results because light levels are such that adequate S/N's can be obtained in the DC mode with short measurement times. Another advantage of DC detection is that the analog signal can be easily processed by logarithmic amplifiers to give a direct absorbance readout. With photon counting either a D-A converter must be used before logarithmic amplification or an on-line computer must be employed to cal- culate the absorbance. The former approach makes the photon counting system more susceptible to drift and l/f noise, while the latter adds considerable expense. Photon counting methods appear to be most advantageous in molecular 74 absorption methods where small spectral bandpasses are necessary or where extremely low or high absorbances must be measured. In most applications spectral bandpasses of l - 10 nm are used in analysis so that photocathodic currents are usually much larger than can be measured without serious pulse overlap non-linearity in photon counting. In a few cases, such as in the analysis of rare earths (63), spectral bandpasses must be less than l nm for Beer's law to hold,and the reduced light level might make photon counting the advantageous detection technique. In the normal range of absorbances encountered in analysis (0.1 to 1.5 A), photon counting offers little advantage, and may give less pre- cise results compared to DC measurement, if the light level must be re- duced in order to avoid pulse pileup. For cases in which (5m):C is very small (small or large A), it may be advantageous to reduce the light level (i,g,, slit width) and use photon counting with long integration times. Which detection method gives the higher S/N depends upon the photo- cathodic current or pulse rate, which is a function of the combined response characteristics of the light source, the monochromator, and the photomultiplier at the particular wavelength and slit width used for the analysis as was indicated in Equations 5.9 and 5.l2. If the pulse rate for the reference beam is out of the linear range of the photon counting system used because of pulse pileup, then DC measurement should be em- ployed. 0n the other hand, photon counting will give the higher S/N if pulse pileup is not significant for the reference pulse rate. Near the dividing line between photon counting and DC measurement ((Rap)r1 = 10'3), it may be advantageous to reduce the pulse rate slightly to utilize the S/N and other advantages of photon counting. As discussed in Appendix A, 75 the dividing line between DC measurement and photon counting, or the effective linear range Of a photon counting system, may be moved to higher pulse rates if dead time compensation or mathematical correction is used. G. Evaluation and Optimization of Signal-To-Noise Ratios l. Q§_Measurement a. Evaluation of_ S/N)dC. The instrumental precision of a molecular absorption measurement is inversely proportional to the magnitude of the signal-to-noise ratio. The value of (S/N)gC at a particular absorbance dependson the magnitude of the group variance terms. The magnitude of the group variance terms depends on the specific instrument used and on the particular conditions (i,g,, wavelength, slit width) used for the analysis. Under certain conditions, when one of the group variance terms predominates, the expression for (5m):C is considerably simplified as was shown in Table IX. The procedure for determining (SN):C is rela- tively simple. For a particular analysis, the magnitudes of the group variance terms for the reference signal voltage are calculated from Equations 5.54, 5.55, and 5.56. The calculated group variance terms are substituted into Equation 5.32, as was shown in Equation 5.63. From a plot of (Sm):C versus absorbance or transmittance, the instrumental pre- cision of measuring a given concentration is estimated from the value of (5m):c at the measured absorbance. Calculation of the group variance terms requires that the following parameters be known or measured: (i lcp)r, icd’ m, Rf, a, Af, g, and 02d. The degree to which these parameters can be adjusted depends on the par- ticularinstrument. All instruments provide the capability of adjusting the reference readout voltage to meet the requirement that (E ) = E = rp r O (i ch)rmR The photocathodic current, (icp)r’ varies over several orders 76 of magnitude because of the great change in the response functions of the light source, the monochromator, and the photomultiplier with respect to the wavelength and slit width used for different analyses. The noise bandwidth, Af, and readout variance, Cid, are usually fixed for simple 2 inexpensive instruments. For more sophisicated instruments, of and 0rd may be variable, and the flicker factor, 2, must be measured for each noise bandwidth. The equivalent cathodic dark current, icd’ and secondary emission factor, a, are somewhat dependent on the photomultiplier gain, m. b. Optimization gf_$ystem Parameters. Equation 5.63 can also be used to Optimize the precision by maximizing (S/N)gc. For a particular instrument only the parameters (icp)r’ Af, and did are usually variable. The readout variance should always be made insignificant. The exact relationship between (SN):C and noise bandwidth is dependent on the . nature of the noise power spectrum of the source flicker noise. If the flicker noise is white or negligible, (SN):C is directly proportional to (Af)']/2. Usually bandwidths of 1.0 — 0.1 Hz are used to keep measure— ment times relatively short. Finally, the photocathodic current, (icp)r’ should be maximized within the limits set by resolution, Beer's law con- siderations and photodecomposition. Equations 5.1 and 5.9 reveal that the following parameters can be changed to increase (icp)r and improve the S/N: NA0 can be increased by using a more intense light source; Tm(Ao) can be increased through using a large slit width; and S(AO) can be increased by using a photomultiplier with high cathode sensitivity in the wavelength range of interest and high collection efficiency. Equation 5.63 indicates that for a particular instrument and analysis, (icp)r should be increased, if possible, until the absorbance measurement is flicker noise limited over the absorbance range utilized (Case III). 77 2. Photon Counting. If the light level has been maximized for a particular analysis and the pulse rate is in the linear dynamic range of photon counting as es- tablished by the criterion in the last section, calculation of the signal- to-noise-ratio, (S/N)K, is extremely simple. Calculation of the group variance terms from Equations 5.57, 5.58, and 5.59 requires knowledge of only t, (Rop)r (or (Rap)r and A1), R0d (or Ratc’ If the group variance terms are substituted into Equation 5.52, as was * Rad’ A1 and Ad), and g. shown in Equation 5.64, and (S/N)K is plotted versus T, the instrumental precision expected for a particular absorbance can be found graphically. H. Discussion To evaluate the magnitude of the photocathodic reference current, ( ) distilled water in the sample cell, the wavelength region from 200 to icp r’ the spectrophotometer described in Chapter III was used. With 800 nm was scanned using a l nm bandpass (500 um slit width) with either the deuterium or tungsten lamp in its appropriate wavelength regions. The RCA 935 vacuum photodiode, which has the same Spectral response (S-5) as the corrmonly used RCA lP28 photomultiplier was used to measure (icp)r' It was found that (icp)r varied from a maximum value of approximately 1.5 x lO'9 A to a minimum of about 3.0 x mm12 A. The l nm spectral band- pass is smaller than used in many commercial instruments, and thus the photocurrents measured should represent minimum values for typical spec- trophotometers. Also, a l nm spectral bandpass is sufficiently small to prevent errors due to Beer's law deviations in the great majority of molecular absorption analyses. Note that for cases in which an organic solvent is used, (icp)r may be considerably reduced by solvent absorption. 78 From the range of photocurrents found, a number Of conclusions can be drawn. First, the corresponding photoelectron pulse rate for a photon counting system using a 1 nm spectral bandpass over the 200—800 nm wave- 10 1 length region varies from about l x 10 to 2 x 107 sec” . Thus photo- electron pulse rates are out of the linear range for photon counting over the entire UV-visible range. Second, since for the major part of the wavelength region (icp)r exceeded l0']O absorbance measurements will generally be readout or flicker noise limit- A, Figure 4 illustrates that ed. However, in wavelength regions near the minimum photocurrent measured, measurements will be shot noise or readout limited depending on the mag- nitude of the readout variance. If a normal meter or recorder with 0.5% resolution is utilized, measurements will be readout limited over the entire spectral range. Thus it is possible for any of the 3 limiting signal-to-noise ratio cases to be valid depending on the magnitude of the photocathodic current, the readout variance, the flicker factor and the mode of readout used (transmittance or absorbance). This illustrates that the usual assumption that all absorbance measurements are readout limited (Case I) can cause serious error in estimating the precision and in choosing the optimum transmittance range for absorbance measurements. For instruments or analyses in which spectral bandpasses much larger than 1 nm are utilized, (icp)r may be large enough over the entire spec- tral region that absorbance measurements will be readout or flicker noise limited (see Figure 4) and only Case I or Case III will apply. The mag- nitude of the readout variance, 0f the flicker noise, and the mode of readout determine whether Case I, Case III, or a mixture holds. Although the discussion has been directed to quantitative single-beam spectrometry, the same basic theory and conclusions can be applied to 79 double beam scanning instruments. In double beam systems, as well as for quantitative applications involving quite small absorption bandwidths, spectral bandpasses Of O.l nm or less are often desirable. Since the photocathodic current, (i ) Cp r’ is directly prOportional to the square of the slit width, W, measured signals will be 2 orders of magnitude or more lower than values obtained for a 1 nm bandpass. In these cases,photon counting may be applicable and the more desirable detection system. With either photon counting or DC techniques under low light level conditions, the shot noise limit is more likely, assuming insignificant readout variance, and Case II will apply. VI. PRECISION OF RATE MEASUREMENTS A. Introduction The total precision of a kinetic-based analysis is determined by the reproducibility of reaction conditions and by the stability and noise characteristics of the reaction monitor, the signal modifier, and the rate computational system. Reaction condition reproducibility is in- fluenced by the quality of sample and reagent introduction, of mixing, and of temperature control. In this chapter, the precisioncfi’making a rate measurement, as affected by system noise, will be discussed by applying S/N theory to the integral rate measurement approaches. A com- plete characterization of kinetic analysis precision should include the effect of reaction condition reproducibility, although it is difficult to generalize because of the great variety of reactions used. 8. General S/N Considerations Consider a generalized reaction rate measurement system which prO- vides a signal modifier output, 8, that is related to the concentration of the sought-for species. For the integral rate computational approaches, the informatkwidesired is encoded in the quantity AS/At, which is measured by holding either the numerator or denominator constant in the variable- time and fixed-time procedures, respectively. The S/N of the signal modifier output at any time during the reaction can be found from knowl- edge Of the response and noise characteristics of the reaction monitor and signal modifier system. As for the case Of molecular absorption 80 8l spectrometry, it is most useful to define a S/N in which the measured quantity, in this case AS/At, becomes the "signal". The S/N for an integral rate measurement, (S/N)k, is given by 2 2 s 0 0 _ (a) = 2 /At = _£§L§.+ _é£_2] 1/2 (5.1) k AS/At (AS) (At) where AS=52'S] Atztz’t] S1, 52 = magnitude of signal modifier output at t1 and t2, respectively 0 2 = variance in measuring AS/At AS/At 02 = the variance in At At a2 = the variance in AS AS l. Fixed-time Measurements For the fixed-time approach, At is held constant. In most fixed- time measurements, the reproducibility Of the time base is much higher than the reproducibility of measuring the signal change. Therefore, Git can usually be assumed negligible, and the S/N for a fixed-time measure- ment, (S/N)kf, can be expressed as S AS AS H = = (6.2) S .9 l 2 where 2 2 _ ' 0 ' C 0 GS], 032 - variances of measuring 5] and 62, respectively Times t1 and t2 are referred to some starting time t0 which is assumed to be precisely equivalent to the time of initiation of the monitored reaction. 82 2. Variable-time Measurement In the variable-time approach AS is held constant. Imprecision is caused by noise on the signal modifier output which trips the signal level sensors on and off at times other than t1 and t2, and fluctuations in the reference signal level or in the characteristics of the signal level sensors. For modern, stable, high resolution signal level sensors, the latter source of imprecision, and hence 0:5, is usually negligible. Under these conditions, the S/N for a variable-time measurement, (S/N)kv’ is given by At At (S/N) = ——-= (6.3) kV OAt [O 2 + O 211/2 t t l 2 where at2 and at2 are time end point variances for determining t1 and t2, 1 2 respectively. C. SpectrOphotometric Fixed-time Measurement 1. Introduction Since molecular absorption spectrometry is the most commonly employ- ed technique for monitoring chemical reactions, it will be discussed in detail. The equations formulated in Chapter V for normal spectrOphoto- metric measurements can be applied to the general S/N equations develOped in section B of this chapter. The measured signal in spectrophotometric monitoring is proportional to either transmittance or absorbance. As discussed in Chapter IV, either transmittance or absorbance monitoring can be used equally well with the variable-time approach for certain types of reactions. Trans- mittance monitoring can be used with the fixed-time approach only for small transmittance changes. The S/N treatment given here is independent of the absolute accuracy of a given procedure since it indicates only 83 the precision of making a rate measurement. 2. Transmittance Monitoring The basic symbolism used in Chapter V will be used here. The photo- current output voltage from the current-to-voltage converter, Eop’ will be simplified to E. In transmittance monitoring, S in Equation 6.2 is replaced by the transmittance T, which yields _ AT _ AE _ r (5mm ’ (O 2 + O 2)l/2 ‘ (O 2 + O 2)l/2 ‘( 2 + O 2)l/2 (5'4) T2 El E2 Tl where AT = T2 - T], dimensionless l El/Er T2 = Ez/E E], E2 = photocurrent signal output voltages with the reaction T transmittance at t], dimensionless r = transmittance at t2, dimensionless proceeding at t1 and t2, respectively, V Er = photocurrent signal output voltage with the reference solution in the sample cell, V AEzEz-E19V 2 _ 2 2 _ . . . . . 0T - 0E /E — variance in measuring T], dimenSionless l l r 2 _ 2 2 _ . . . . . o- — n /E - variance in measuring T , dimenSionless I2 E2 r 2 2 2 _ . . . . 2 GE , OE — variance in measuring E1 or E2, respectively, V l Normally before a series of kinetic measurements, the lOO% T and 0% T are set and not readjusted. Hence, the variance involved in these two operations does not enter Equation 6.4. However, the variance in the dark current output voltage must be considered since it influences the measurement of E] or E2. The variance in determining the photocurrent signal output voltage 84 at a given time is given by the sum of the group variance terms defined in Equations 5.29-5.31. The only difference is that c2/2 is used in the sum because the measurement Of E involves only one dark current measure- ment. Thus, Equation 6.4 becomes ErAT (S/N)kf = 1/2 (5.5) 2 2 2 + b2 [afi + b? + c$/2 + a + c372] where the subscripts l and 2 denote the group variance terms at time t1 and t2, respectively. The group variance terms are calculated by sub- stituting the photocathodic currents at times t1 and t2, (lcp)1 and (lcp)2, respectively, into the appropriate formula for each individual variance term. Equation 6.5 can be expressed in terms of the group variance terms for the reference SOIUthh as E AT r (S/N)kf = l/2 (5'6) [33(Tl + T2) + b§(if + T?) + oi] These reference group variance terms, denoted by the subscript r, are identical to those described in Chapter V. To evaluate Equation 6.6, it is only necessary to measure Er’ to calculate the group variance terms for the reference solution prior to kinetic measurement, and to know T1 and T2 for a particular kinetic measurement. Equation 6.6 is an exact expression that can be used to evaluate the precision of making a fixed- time measurement. However, Equation 6.6 is not a convenient equation to use since T1 and T2, and hence the denominator, vary with the concentration of the sought-for species. For the majority of kinetic analyses, an absorbing product is formed from a solution whose initial absorbance is zero. For such cases Equation 6.6 can be readily simplified as discussed below. Simplifications may be possible for other cases, although it is 85 more difficult to generalize. To use the fixed-time method with linearity, AT must be less than 2% T for the fastest reaction studied. Even without this restriction, measurement Of the rate in the initial stages of the reaction usually requires AT to be less than 5% T. Thus, it is reasonable to assume that i1 3 T on the formation of an absorbing product. To make measurements in the 2. The majority of spectrophotometric kinetic analyses are based initial stages of such reactions thus requires that measurements be made close to 100% T. Together with the first assumption, this results in the approximation that T1 3 T2 2 l, which allows simplification of Equation 6.6 to ErAT 2 2)l/2 (5'7) r (S/N)k = f 2 (2ar + 2hr + c The approximations used to Obtain Equation 6.7 mean that the magnitude of the noise on the signal modifier output remains constant in the small transmittance interval near l00% T. This assumption provides a slightly low but reasonably accurate estimate of (S/N)kf. Calculation of (S/N)kf is considerably simplified since only AT in Equation 6.7 varies with the sought-for concentration in a particular analysis. If the simplified group variance terms of Equations 5.54-5.56 are = ( used, and the restriction that E0 = E ) me is imposed, Equation r 1cp r 6.7 can be rewritten (O.707)AT(i ) (S/N) _ cp r (6.8) “f [2eAf(l + aiuicpir + icd) + (icp)3(gz + aid/E§)J‘/2 Equation 6.8 assumes that Johnson noise, amplifier noise, and photo- multiplier flicker noise are negligible as discussed in section E of Chapter V. 86 The most recent fixed-time instruments (20,21) utilize integration over the total measurement time by computing the difference between the average voltages of two equal and adjacent time intervals. Let t1 and t2 represent the center times of these two intervals so that the total measurement time is 2At, and At is the integration time for the measure- ment of either E1 or E2. Since the noise bandwidth of the reaction moni- tor and signal modifier must be much less than the total measurement time to prevent distortion of the signal modifier output, the noise bandwidth of the total measurement system is essentially determined by the inte- gration time (Af = l/2At). Use ofthis result in Equation 6.8 yields (0.707)AT(i ) (6.9) (S/N)kf = -l CD r 2 2 2 2 *l/2 [e(l + a)(At) ((icp)r + icd) + (icp)r(€ + ord/Eo)] Under conditions where dark current shot noise, source flicker noise, and readout variance are negligible compared to photocurrent shot noise, Equation 6.9 reduces to (S/N)kf = 1.77 x lo‘9Ai(i + a)-]/2(At)]/2(icp)l/Z (6.l0) Equation 6.l0 represents the highest S/N obtainable for a fixed-time measurement for a given transmittance change, integration time, and photocathodic current under shot noise limiting conditions. For increasing photocathodic currents or integration times, the source flicker limit is reached and Equation 6.9 reduces to 1 (6 ll) (S/N)kf = (0.707)AT5' which indicates the highest S/N obtainable for a given system. Equation 6.ll is also valid under conditions where it is advantageous to use a photodiode instead of a photomultiplier (see Appendix C) and source flicker limiting conditions exist. 87 3. Absorbance Monitoring If the signal modifier includes a log-ratio amplifier providing l V per decade so that the readout is linear in absorbance,S'in Equation 6.2 is replaced by the absorbance A, which yields (S/N)kf = (0A2 iAOA2)1/2 (6.12) l 2 where A] = -1og T1 = absorbance at t1 A2 = -1og T2 = absorbance at t2 AA = A2 - A1 0 2, o 2 = variance in measuring A or A , respectively A1 A2 l 2 Manipulation of Equation 6.12 into a useful form is very similar to the development for transmittance monitoring. The primary difference, as discussed in Chapter V, is that the readout variance becomes a b2 rather 2 than a c type variance and the group variance terms for direct absor- bance readout, defined in Equations 5.37-5.39, must be used. Use of these reference group variance terms in Equation 6.l2 yields (2.303)ErAA -l_ 2 2 -2 42 ) + 2hr,] + cr,](T] + T2 )/21 (S/N)kf = (6.13) ’2* :i l/2 [ar,i(“l + T2 If the simplified group variance terms are used in the equations defining 2 2 2 ar 1, br 1, and cr 1 and it is assumed that T1 3 T2 3 l and that for inte- grating fixed-time systems, Af = l/2At, Equation 6.13 becomes _- l.64AA(iCp)r (6.l4) kf _ [e(1 + a)(At)"((icp)r + ica) + (icp)§(az + (2.3osizofidii‘/2 (S/N) Under the conditions used to obtain Equation 6.10, the photocurrent shot noise limit results,and Equation 6.14 becomes 88 (5mm, = 4.10 x lO-gAA(l + a)-]/2(At)]/2(icp)l/2 (6.15) At the source flicker limit, Equation 6.14 reduces to 1 (S/N)kf = (1.64)AA§- (6.16) Note tha:Equation 6.10 is equivalent to Equation 6.15 and Equation 6.11 is equivalent to Equation 6.16 since AA = (2.303)AT near 100% T. 4. Comparison tg_Normal Molecular Absorption Measurements In comparing normal molecular absorption measurements to kinetic measurements using spectrophotometric monitoring, some basic differences can be noted. Many of the differences and problems encountered in kinetic meaSurements are similar to those found in extremely low absor- bance measurements. First, photocathodic currents utilized in spectrophotometric moni- toring of reactions 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. Due to the large photocathodic currents, the dark current will usually be negligible compared to the photocurrent, and iCd in Equations 6.9 and 6.14 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 negligible over small transmittance intervals near 100% T. Thus, large slit widths can be used with monochromators, and in some cases filters are suitable. Of course, there is a practical limit to how wide the spectral bandpass can be since non-linearities 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 89 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 kinetic measurement. Higher readout resolution, or lower readout variance, is needed for kinetk:analysis when compared to normal absorbance methods. As emphati— cally pointed out in Chapter V, the readout variance should always be made negligible for the highest precision. A readability of 0.5% is suf- ficient for 1-2% precision in the optimum transmittance range of normal absorbance measurements, but totally unacceptable for kinetic measure- ments. 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 other words, Case I or completely readout limited conditions are unlikely. In normal molecular absorbance work, noise bandwidths of O.l-l Hz or integration times of 1-10 sec are usually sufficient to yield reasona— ble S/N's. However, in kinetic methods, the measurement time is Often dictated by the rate of the reaction utilized under Optimum conditions and may vary from 10 msec to 100 sec. Moreover, it is not always possi— ble to vary reaction conditions to adjust the rate so that measurements can be made in the initial stages of a reaction with a particular measure- ment time. For instance, if it is desired to slow down a pseudo-first order reaction, possibly some reagent concentration can be changed. How— ever, 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. 90 5. Optimization ang_flgt§_gj_(S/N)kf Equations 6.9 or 6.15 reveal that (S/N)kf is primarily dependent on the photocathodic current, the measured transmittance change, the noise bandwidth or measurement time, the readout variance, and the flicker factor. Figure 5 shows plots of (S/N)kf versus the 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 Figure 5, and the dark current and readout variance were assumed negligible. Curves a-e correspond to shot noise limiting conditions in which the measurement time varies from 10 msec to 100 sec. This range of inte- gration times covers the normal range that would be encountered in a reaction rate analysis. These curves represent the highest S/N obtaina- ble for a 1% transmittance change with a given photocathodic current. The remaining curves (f-k) illustrate the effect of source flicker noise on the S/N. Figure 5 graphically illustrates the inherently low S/N of kinetic measurements. For a large range of the experimental conditions repre— sented in Figure 5, 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 Equation 6.10, when source flicker becomes limiting. The source flicker factor is dependent on the measurement time. Because of the l/f character of flicker noise, source flicker noise would not be expected to decrease in direct proportion to the square root of the integration time as does shot noise. For the spectrophotometric system discussed in Chapter III, it was found that the flicker factor, 5, was 0.5-2 x 10-4 for a one second integration time. The flicker factor for a 10 sec 91 Figure 5. Dependence Of S/N 0n Photocathodic Current, Integration Time, and Source Flicker Noise *- L‘o -" :‘O-flfDQOD'QJ At At At At At At At At At At At 10 msec, a = 0 100 msec, g = O 1 sec, 6 = 0 10 sec, 6 = O 100 sec, 2 = O 1 sec, 6 = 10"4 10 sec, 6 = 10'4 l00 sec, 2 = l0‘4 1 sec, 5 = 10"5 10 sec, 6 = 10-5 100 sec, 6 = 10-5 92 - Log (icp)r Figure 5 93 integration time was approximately the same as for a 1 sec integration time, which clearly illustrates the l/f behavior. For integration times below 100 msec (AFELS Hz), flicker noise is expected to be negligible. Thus, if the flicker noise limit is reached for a given photocathodic current and measurement time, increasing either iCp or At is not expected to increase (S/N)kf for a particular transmittance change. Figure 5 indicates the large dependence of the S/N on the measure- ment time, which as previously mentioned, is Often dictated by the rate of the reaction utilized under Optimum conditions. A kinetic measurement remains shot noise limited to higher photocurrents as the measurement time is decreased since the absolute magnitude of the shot noise is prO- —1/2 portional to At Two limiting cases for measurement can be identified as discussed below. a. Fast Reaction Kinetics. For measurements of the rate of rapid reactions (usually in a stopped-flow apparatus), 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 l-cm cell, the radiant power throughout and, hence, the photocathodic current are Often lower than in conventional measurements. Thus, from Figure 5, fast kinetic measurements are very likely to be shot noise limited. This indicates that optimization Of (S/N)kf can be accomplished by increasing the photocathodic current through the use of high intensity light sources, large spectral bandpasses and high efficiency Optical transfer systems to increase the radiant power impingent on the reaction cell. b. Slow Reaction Kinetics. For relatively slow reactions for which the rate can be measured with a conventional mixing apparatus, integra- tion times are usually 10 sec or greater. From Figure 5, it is predicted 94 that the relative contribution of shot noise compared to flicker noise will be reduced, and for high photocathodic currents, measurements will be source flicker limited. The experimental work to be described in the next two chapters inr volves integration times of approximately 10 sec. The maximum spectral bandpass of the Heath spectrophotometer, 4 nm (2000 um slit width), was used to Obtain the highest photocathodic current and hence the highest achievable S/N for the instrument. Photocathodic currents of approxi- 9 _ 10'10 A were realized. Since 5 was approximately 10.4’ mately 10' Figure 5 indicates that measurements were made under source flicker limiting conditions. 6. Ejfggt_gf_Transmittance Interval and Integration Time Since AT was assumed to be 0.01 in constructing Figure 5, from Equation 6.9, the S/N for a different transmittance change is found by multiplying a point on the appropriate curve by AT/(0.0l). For a normal ten-fold range of the analyte concentration, AT will vary over about an order of magnitude, and AT = 0.01 is expected to be an average value. For a particular reaction and set of conditions, the transmittance or absorbance interval over which the measurement is made is approximate- ly 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, (S/N)kf is approximately proportional to At3/2 and, in the source flicker limit, (S/N)kf is proportional to At. 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. 95 7. Effect Qf_Readout Variance Although reading error was neglected in Figure 5, it may not be neg- ligible in certain situations. As can be seen from Equation 6.9 or 6.14, readoutvariance has the same effect on (S/N)kf as does flicker noise and causes (S/N)kf to reach a plateau at high photocathodic currents. As will be pointed out in the next chapter, the absolute value of the readout in integrating fixed-time instruments is proportional to Atz. However, the readout variance is constant and independent of At. Thus, under readout limiting conditions, (S/N)kf increases with Atz. D. Variable-time Measurements Evaluation of (S/N)kv requires that the variance in measuring At, oAE,be known. This variance should be directly related to the magnitude of noise present on the signal modifier output. Attempts to determine this relationship were unsuccessful as the problem was much more complex then first anticipated. Determination of o 2 2 and O 2 be calculated. Evaluation of these variances involves developing probability density requires that 0 functions around the turn on and Off times that indicate the probability of a given error in t1 or t2. The variances calculated from these density functions could then be used in Equation 6.3. It has been stated (10) that the S/N remains constant in a given variable-time procedure since the reference signal levels remain con- stant" This statement is only true if the noise present on the signal modifier output is white. For cases in which l/f noise, such as source flicker noise, is important, the S/N will depend on the concentration of the sought-for species. This occurs because At increases with decreasing 96 concentration, which causes the relative contribution of l/f noise to be greater for lower concentrations. Lower frequency and higher amplitude l/f noise components are expected to affect the triggering of the signal level sensor differently from white noise components, which all have the same average amplitude. Thus, not only the total magnitude of the noise present but also the shape of the noise power spectrum must be considered for evaluation of GAE. In addition, the voltage transfer characteristics, the response speed, and hysteresis of the voltage level sensor must be considered. Definitely more study in this area is needed to evaluate measurement precision for variable—time methods and to compare the variable-time and fixed-time methods in terms of signal-to-noise ratio theory. However, some of the conclusions reached in the discussion of the fixed-time approach can be applied to variable-time measurements for the case of spectrOphotometric monitoring. For fast kinetic measurements, the S/N will usually be limited by shot noise and can be optimized by changing parameters to increase the photocathodic current. Rate measure- ments for slow reactions will be largely influenced by source flicker noise. For either case, increasing the measurement time At by increasing the difference between the voltage level sensors will in general increase the S/N. VII. FIXED-TIME DIGITAL COUNTING SYSTEM FOR REACTION RATE MEASUREMENTS A. Introduction In this chapter, a new method is presented for automated rate meas- urements. The instrument described here computes the voltage change over a fixed-time interval by a digital integration procedure, which eliminates all drifts and non-linearities in the computational system. Integration over the total measurement time provides high noise immunity and excellent signal averaging as has been discussed previously (20). B. Instrumentation A block diagram of a complete rate measuring system using the fixed- time digital counting system is shown in Figure 6. The chemical reaction is monitored in a reaction cell by a suitable transducer, whose signal is modified to Obtain a voltage input of the prOper level for the voltage to frequency converter. A spectrophotometer and an Operational amplifier current-to-voltage converter are used in the specific rate measurement system described later. The voltage to frequency converter acts as an interface between the reaction monitor and the fixed-time computer by changing the analog reaction monitor signal to a train of pulses that can be processed by the digital circuits. The fixed-time computer is shown in dotted lines in Figure 6. The up-down counter, which is controlled by 97 98 .mchA umupoo cA czocm smuzaeou mama meAp-nman .Empm»m “cosmcammmz mumm-coAuuuom we seemer xqum .m mcamAA a) lllllllllllllllllllllllllll J _.l /////////// V///////////////// Abooduc 02¢ (wooouo ////////////////// 9:3ch 02:10 ./////////// / ////////////////// _ (UhZDOu M///////////////// M a .23.. .u ”////////////////// w 235.63 a fl 2.: 22:: a 53;: / m ”////////////////// ”/Il/ ///////////// x / / / 3;»... / M ..Mfiwwumw M 5:62. .229... / / 92 w 2 32.6) M 3:22. 20:32. /////////// 2300. t: _ lllllllllllllL _ _ _ _ L 99 the timing and gating circuits, computes the difference in counts between two equal time intervals on the reaction rate curve. This difference in counts, appearing at the up-down counter output, is decoded from binary coded decimal to decimal and displayed on a Nixie tube readout. The mathematical relationship between the digital readout and the output slope of the reaction monitor-signal modifier system is discussed in the following section. 1. Relationship Between Readout and Rate An expanded section of the initial portion of a typical reaction rate curve is shown in Figure 7. During the time represented in Figure 7, pseudo zero-order kinetics are assumed to prevail for the reaction of interest so that the initial rate approximation is valid. The necessary conditions for the initial rate approximation in pseudo-first order and in enzymatic reactions were discussed in Chapter IV. It is also assumed in Figure 7 that the reaction monitor follows solution concentration in a linear manner. It has been previously shown (20) that the lepe of such a plot can be expressed as S = tan a a/At = AA/(At)2 (7.1) where At is 1/2 the total measurement period, AA is the difference in the areas A2 and A1 in Figure 7, and S is the slope in V/sec. The slope can also be expressed as 2 - V;)/At (7.2) where V; and VT'are the average voltages of intervals 2 and 1, respec- tively. In the described instrument, the output of the reaction monitor- signal modifier system, V, is converted to a frequency, f, by a V to F converter according to Equation 7.3 100 < VOLTAGE FROM REACTION MONITOR“ SIGNAL MODIFIER Fir-n, Figure 7. Expanded Section of the Initial Part Of a Typical Reaction Rate Curve 101 f = kCIVI (7.3) where kC is the conversion rate in Hz/V. If Equation 7.3 is solved for V and the result is substituted into Equation 7.2 for VX'and 7;, Equation 7.4 results S = (f2 - T;)/kCAt (7.4) where T; and T; are the average frequencies over the first and second intervals respectively. The readout N, is the number of counts accumu- lated during the second interval minus the number of counts accumulated in interval one, and is expressed by —— N = (Ti—2‘- f1) - At (7.5) If Equation 7.4 is solved for (f; - f?) and substituted into Equation 7.5 the result is N = S - k ° At (7.6) As indicated in Chapter IV, it is not necessary for pseudo-zero order kinetics to prevail if the fixed-time approach is used to extract concentration information from first order reactions. The linear rela- tionship between N and initial concentration is still valid for first order reactions even if the rate curve is not linear during the measure- rnent time. From Equation 4.2, the relationship between the reaction imonitor-signal modifier output, V, and the concentration of species R is given by Equation 7.7 v = om = u[R]o e‘kt (7.7) where u is the transfer function of the reaction monitor-signal modifier System in volts per mole of R. It is easily shown by integrating Equation 7.7 over the two equal 102 time intervals that V‘ - v— = ——9 'ktm - e'kitfi (7.8) Thus (VE’- —;) and hence N will be directly prOportional to [R]0 if t1, t2, and t3 are held constant for a given analysis. 2. Range and Limitations The range and limitations of the instrument can be predicted from Equation 16. 'The readout is directly proportional to the slope, the con- version factor, and the square of one half the measurement period. The computing circuit has an error of :1 count because the timing and sequenc- ing ciruicts are not synchronized with the input signal pulses from the V to F converter (64). For 0.1% accuracy, the slope, conversion factor, and integration period must be manipulated to insure a readout of at least 1000 counts. The degree of manipulation is dictated by the particular reaction of interest and the noise characteristics of the reaction moni- tor-signal modifier system. For slower reactions, where S is small, a large integration period is used to accumulate at least 1000 counts, to average the noise, and to increase the S/N as discussed in Chapter VI. For faster reactions, where S is large, the integration period must be shortened to insure that the rate is measured in the initial stages of the reaction. Equation 7.6 predicts that decreasing the integration period by a factor of ten will decrease the readout counts by a factor of one hundred. The V to F converter chosen should have the highest possible con- version factor and linearity since both affect the accuracy. The accuracy of the instrument also depends on the accuracy of the integration period. Use of a crystal oscillator produces such stable and accurate timing that the accuracy Of the integration period is not the limiting factor. The 611 103 V to F converter imposes another limitation in the range of input vol- tages it can accept. The measurement must be made when the signal from the reaction monitor-signal modifier system is within the input range ' of V to F converter. The maximum counting rate is determined by the propagation delays in the gating and counting circuits and the tracking rate of the V to F converter. Practically, prOpagation delays are not limiting in compari- son to available V to F conversion rates. The maximum count readout is constrained by the number of decades in the up-down counter. In this instrument, the V to F converter on the Heath Universal Digital Instrument (Model EU-805A) was used. It has a conversion rate of 100 KHz/V and a linearity of 0.05%. Thus Equation 7.6 becomes N = 3 ~ l05 . At2 (7.9) The input range is +1 to -1 volts, so that the measurement must be made in a l to 0 or -1 to 0 voltage interval depending on the polarity of the specific reaction monitor-signal modifier system. The counting frequency of the gating and counting circuits is greater than 10 MHz and, because a seven decade up-down counter is used, the maximum readout is 9,999,999. 3. Timing, Gating, and Counting Circuits A circuit diagram of the constant-time computer is shown in Figure 8. Switch S is used to choose the mode of operation,which can be continuous- 2 measurement Or single-measurement. Triggering can be accomplished manu- ally by means of S] as shown in Figure 8, or by the sample introduction system. The pulse introduced by S] triggers the monostable multivibrator whose delay time can be adjusted to start the measurement cycle. This provision allows the user to delay the measurement until mixing is com- plete or any induction period has passed. When the monostable is triggered, 104 rm. J‘Qd..\.i( ( rum-0 IIIITIIIII|||| WWW: A<¥ oom-:mv eeeu LOAAAAAomo cued: - LOAAAAAomo Aepmxeu mAzuoz paoummm AmsAomo spam: :0 mAneumocoz mEAA AmAaon ucm :0pu3m ummmm . mAnmmeCoz use Am Aoq-oow-:mv ocau AaAom spam: - N m Amuoom-:mv etau ao_a-aAAa ¥-n :paa: - ao_a-aA_a A¢0 ~38 A A muh¢u>zou bzulucandua 040:.» A) 7 A. z .5» c .....3 ... 3o . a AS“. 111%? 5.28 ... A -63... s... o 2.. So So a. n baoo 20¢“. 106 its output changes from logic level l to 0 causing the output Of gate 2 tO be at logic level l. The logical l at the output Of gate 2 clears each decade Of the up-down counter to a logical 0 and simultaneously re- sets the scaler. The 1 output Of gate 2 is inverted by gate 8, and the resulting 0 output Of gate 8 simultaneously resets the outputs of the Decade Counting Unit (DCU) to O and sets the Q output of the J—K flip- flOp to 1. After the monostable output returns to logic level l, the timing sequence is begun, as illustrated in Figure 9. The l MHz signal from the crystal oscillator is divided by the scaler by a factor of l0", where n can be varied from l to 7. The scaler output will be designated as the clock. The clock period can thus be varied between l0'6 and l0 sec in decade steps. The first l to 0 transition Of the clock occurs after one period of the selected clock time base. This l to 0 transition causes the flip- flop to undergo a transition from logic level l to 0. The l to 0 transi— tion of the flip-flop causes the A output of the DCU to change from a logical 0 to a logical l. Gate l controls the transmission of pulses from the V to F converter to the computer. When 52 is in the continuous-measurement mode, input b to gate l is Open and does not affect gate Operation. Input c to gate 1 is connected tO the A output of the DCU. Thus the gating Of pulses from the V to F converter is controlled by the A output logic level. The A logic level also controls the transmission Of pulses from the Q (Mltput Of the flip-flop through gate 3. From the first to second l to 0 ‘Ufiansition of the clock, Q is 0 and A is l. During this period, the out- wit:of gate 3 is l, which Opens gate 6 to the pulses from the V to F 107 uwzuc_u mcwewh Lou mfimcmwm maucmscmm +0 mELowm>m3 .m mczmwm . . . . . . . “ Bo: “ 2.52332“ 3.5“ 22.. “ 2.6.33.1" >25 “ . . . _ . _ lib .q , l u - . fl . < . - — - n u . . o - h PDmPDO qudum x0040 O. 0 o b m 0 C m N _ ~40> kP(y) could not be Obtained. At lower acidities or higher moly- bdate concentrations than those used for the phosphate determination, the rate of formation of 8-12-MSA is measurable after the phosphate has re- acted. Figure 11 illustrates the recorded reaction rate curves for the reaction of molybdate with 10 ppm P (curve b) and a mixture of 10 ppm P and 2.5 ppm Si (curve a). It can be seen that the reaction to form 12- MPA comes to equilibrium quite rapidly compared to the silicate reaction. Thus, the formation of lZ—MPA offers little interference in measuring the rate Of formation Of B-lZ-MSA. If the measurement of the rate is made after 12-MPA has formed, but during the initial stages of the sili- cate reaction, Equation 8.2 can be written Rate( (8.4) y) kSi(y)[5103]o Since the formation of the heteropoly blue of phosphate and the formation of B-lZ-MSA follow pseudo—first order kinetics in the initial stages of their respective reactions, it is not necessary to measure either rate under pseudo-zero conditions if the fixed—time approach is utilized. Thus for phosphate determinations, it is only necessary that the absorbance change due to the formation of the silicate heteropoly blue be negligible compared to the total absorbance change in the meas- urement interval. Likewise for silicate determinations, it is only necessary that the absorbance change due to 12-MPA formation be negligible (Kwnpared to the absorbance change due to B—lZ-MSA formation during the measurement period. Conditions were arranged so that the above assump- tions are reasonably valid. ABSORBANCE,400nm Figure 11. 122 ‘TINAE Recorded Reaction Rate Curves for Formation of 12-MPA and u-lZ-MSA Measured at 400 nm (a) 10 ppm P and 2.5 ppm Si (b) 10 ppm P 123 8. Experimental The basic spectrOphotometric reaction rate measurement system was described in Chapter III. The absorbance mode of the photometric read- out module was used for all rate measurements. This is necessary for silicate determinations because the % transmittance range over which the measurements are made varies with the phosphate concentration. Thus non- linearities would result in the % transmittance mode. The fixed-time digital counting system described in Chapter VII was used to yield a direct digital readout Of the reaction rates. Solution preparation, reaction conditions, and mixing procedures were described in Chapter III for both silicate and phosphate analyses. The delay time is chosen so that the measurement starts after mixing is complete and any induction period has passed. For both reactions, the rate computer clock period is chosen so that measurements are made under pseudo-first order conditions (before molybdate, sulfuric acid, or ascorbic acid concentrations change significantly). C. Results and Discussion l. Silicate Determination Since the determination of silicate by measurement of the rate of formation of B-l2-MSA had not previously been reported at the time of this research, initial experiments were done to determine the accuracy which ‘ could be expected for pure Si standards. Under the conditions chosen, the digital rate readout varies linearly with the silicate concentration over the 0-10 ppm Si range. Results of silicate analyses based on one Silicate standard are presented in Table XIII, and the data indicate 2% Or better accuracy. 124 TABLE XIII. Determination of Silicate Silicon Concentration in ppm Digital Taken Foundb Rel. error Readouta % 7980 3 3.05 +1.6 13338 5 5.10 +2.0 20919 8 -— —- 26037 10 9.96 —O.4 aAverages of 5 results bBased on 8 ppm standard: integration time of 40 sec; premeasurement time 45 sec. 2. Analysis of Phosphate - Silicate Mixtures Analysis of phosphate by measurement of the initial rate of 12-MPA formation (37) or the initial rate of heteropoly blue formation has been reported (39). Mixtures of phosphate and silicate were prepared to de- termine if synergistic or other effects existed and to determine the con- centration ranges over which Equations 8.3 and 8.4 are applicable. Results of the simultaneous determinations are shown in Table XIV. These data indicate that phosphate in the O-lO ppm P range can be determined with accuracies of about 1% in the presence Of as much as 50 ppm Si. The data also demonstrate that silicate can be determined with better than 3% accuracy in the presence of up to 10 ppm P. Relative standard deviations were typically 2-4% for both determinations. For both silicate and phosphate analyses, 40 second measurement times and 45 second pre-meas- urement times are used. The 45 second pre-measurement time is sufficient for the reaction to form 12-MPA to come to equilibrium. Once standards Tuave been run, 5 determinations of silicate and phosphate can be made in 125 about 15 minutes. For the highest phosphate concentration shown in Table XIV, a nega- tive error is evident in the determination of silicate. This error is not not attributed to a synergistic effect, but to a "concentration depletion effect," which results because the formation of 12-MPA requires 12 moles of molybdenum for every mole Of phosphate. Thus, at high phosphate con- centrations, the molybdate concentration after reaction with phosphate is substantially lower than the initial concentration. Since the rate of formation of B-lZ-MSA is dependent on the molybdate concentration, the depletion of MO(VI) results in a lower rate and errors in silicate deter- minations in the presence of large amounts of phosphate. Preliminary results Obtained at lower acid and molybdate concentrations were satis- factory for pure silicate solutions. However, the concentration deple— tion effect became important at lower phosphate concentrations. Higher molybdate concentrations were tried to reduce the concentration depletion effect, but the high densities of these solutions prevented efficient mixing and caused noisy reaction rate curves. Silicate determinations are not as accurate as phosphate determina- tions as shown by the data in Table XIV. There are several possible reasons for the lower accuracy in silicate determinations. The concen- tration depletion effect mentioned above can give low results if large amounts of phosphate (>25 ppm) are present. Positive errors can result if the phosphate reaction is not complete before the rate of formation of B—lZ—MSA is measured. The errors due to this latter effect also increase With increasing phosphate concentration. Thus the silicate procedure is recommended only for phosphate concentrations in the range of 0-25 pme. 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> I and AlRap > 1, the variance is given by 2 —A]R ON = A1Ra te —A]R op p apT[l - 2A1RapTe apT] (A.9) If a Poisson distribution is assumed, the variance would be incorrectly given by Equation A.3. Under conditions where AlRapT < 0.044 or 0.014 Equation A.9 further reduces within 1% or 0.1% to 2 _ ON - A R T A.10 Up a) 1) l p Rapt(1 - 3A] For type II circuits, assuming the discriminator passes an average pulse rate of AlRap’ the variance has been shown (81) to be approximately equal to 140 2 _ 0N - A1Rapt/(1 + A 0P 3 1R6 T) (A.11) P which reduces to Equation A.ll within 1% if AlRapT < 0.038 and within 0.1% for AlRapT < 0.012. The Poisson distribution would predict a variance given by Equation A.4. The variance for type Ia circuits is more difficult to calculate unless A1 = l, which is similar to type Ib. However, if AlRapT < 0.044, ofiop is approximated by 2 0N0 = Rapt[A](l - 3Ra T) + (A p 2 - A])R T] (A.12) P Thermionic dark current pulses follow Poisson statistics as do photo- electrons. However, other sources of dark current have been shown to have variances greater than predicted from Poisson statistics (68,82). Since the contribution of non-thermionic dark current pulses to the total dark current pulse rate is usually small at room temperature, little error will result if it is assumed that all dark current pulses follow Poisson statistics. Therefore, the variance in the observed numerical readout for dark current pulses, 00 , is given by Equation A.8. Od D. Signal-to-noise Ratio Expressions The S/N for a light intensity measurement using photon counting in- volves making two measurements: the combined photoelectron and dark cur- rent pulse rate and the dark current pulse rate alone. The S/N is thus the number of signal counts over the square root Of the sum of variances of the two measurements as shown in Equation A.13. N - NO N ot d op = (A.13) (02 + 02 )1/2 (dfi + 202 )1/2 Not Nod Op NOd .5... N 141 where the total observed pulse count, Not’ and the total variance, 02 Not are given by Not = N0d + N0p (A.14) 2 _ 2 2 ON —.ON +-0N (A.15) ot Op od It has been assumed that in the absence of photoelectron pulses, overlap of dark current pulses is negligible. In the presence of photo- electron pulses, dark current pulses may overlap with photoelectron pulses i :_o§ + OS . Usually for 0t Od op is much greater than R0d at photoelectron pulse so that in general NO N0d + NOp and o t.i a fairly fast system, R 0P rates where overlap is significant so that Equations A.l4 and A.15 are assumed to be valid. In some spectrometric applicatons, photoelectron pulses from back- ground sources may be part of the total observed pulse count, Not’ These background pulses experience the same statistics and discriminator and dead time loss effects as the signal pulses. For such cases, it is necessary to add another variance term, which is twice the variance in the background counts, under the square root in the denominator of Equation A.13. Since for these cases the signal is only a fraction Of the total photoelectron pulse rate, the linear range for the signal counts is reduced by the presence of background counts. The S/N expression for each type Of counting circuit is derived by substituting the appropriate relationship for Nop’ ofi , and as into op Od Equation A.13. Using Equations A.7 and A.8 in Equation A.13, the S/N expression under conditions of negligible pulse overlap becomes l/2 (A R t) 34 ”P * (A.16) R A R d l 2 [1+213tc+,;9 R—a—H/ ap 1 ap 142 At high light levels where the dark current variance term is negligible, the S/N increases with (A R t)1/2. ' m 1 ap At low l1ght levels where Rad - R atc/Rap * (Rad/Rap) decrease rapidly as Rap increases. The enhancement of the S/N by dark current discrimination is also evident from Equation A.16. In- aP the S/N increases faster than Ra9/2 because the ratios (R ) and creasing the discriminator level slightly above zero should improve the * S/N since the ratio Ad/A] decreases rapidly if Rad is comprised mostly Of small pulses from down the dynode chain. Increasing the discriminator level further should lead to decreased S/N since the S/N is proportional A 4/2 if Kg-is small or constant. This effect should be most noticeable l at low light levels. to A If photon counting is used in spectrometric applications where light levels are moderately high, the effects Of dead time loss and gain must be critically considered. Minimum dead time should be a primary criterion for choosing or constructing a photon counting system since the linear range of operation increases directly with the reciprocal Of the dead time. For the purposes Of comparing the advantages and disadvantages of the three limiting types of counting circuits, assume that 1 equals 10'7 sec for all three systems and the dark pulse rate is negligible. For type Ib and II counting circuits, the maximum photoelectron pulse rate is 104/A] sec“1 for 0.1 % linearity. Note that discrimination increases the maximum input photoelectron pulse rate but does not affect the maximum Observed pulse rate. For a type Ia system, Equation A.5 is valid to 0.1% 5 1 up to a maximum pulse rate Rap of 2.6 x 10 sec' , although the Observed number of counts Nop is not necessarily linearly related to Rap‘ If will be linearly dead time compensation is employed (A2 = 2A1), NOp 143 related to Rap up to 2.6 x 105 sec']. To see how the compensation can increase linearity, consider type Ib and II photon counting systems where such compensation is impossible. If compensation occurs at a discriminator level where A1 = 0.45, for example, type Ib and II counting circuits will 4 sec—1. exhibit linearity within 0.1% up to Rap = 2.2 x 10 Thus dead time compensation has extended the linear range by over an order of mag— nitude. This effect was used to advantage by Ash and Piepmeier (74) for molecular absorption spectroscopy where the light levels are normally moderately high. It should be noted that although discrimination extends the linearity for type Ia systems, as in any Of the types of counting circuits, the S/N is decreased by discrimination (by a factor of 0.67 in the above example). Also for type Ia circuits, use of higher discrimina- tor levels reduces the effective pulse width because Of the pulse shape. Under conditions where Ra r is greater than 0.1, type Ib and II cir- P cuits have the advantage since an exact equation exists to correct the observed pulse rate to the true pulse rate. Of course, this would be very tedious to do manually, but possibly attractive if an on-line com- puter is available. It is interesting to look at the S/N for cases in which pulse overlap is serious. Table XVI compares the noise, signal, and S/N of type Ib and II counting circuits to the noise, signal, and S/N expected if no pulse overlap is present. To expedite calculation, it is assumed that 7 the dark current is negligible. A1 = 1, T = 10' sec and t = 1 sec. In the absence of pulse overlap the S/N is given by S/N = N 1/2 = A OD 1Ra t (A.17) P The exact S/N expressions for type Ib and II counting circuits are given by Equations A.18 and A.l9. 144 . x . . mod n sq : no“ mu m no“ x 0H m noH x e~.~ nofl x m~.~ noH x c~.~ no” x no.~ «ed a no.~ «OH x co.“ . x o .4 . I No” x mm a NoH nN u an. x no u x I x o . I «3 ma n NS 3 n «3 x 3 m 10H x co.” o” x oo.~ o“ x oo.H e N N Ama.< dewumavmVAmH.< cauumacmVAna.< sewuoaqmv HH dash pH «use a.o.m.z z\m mgwaugwu mcwpcaoo copes; Lou ovumm mmvozuop-_m=mvm ucm wmwoz .pmcmwm 6o comwgmqeou no” A NM.« «on x 68.6 mofi x od.n no” u ~N.~ nofi u ao.~ mod n 3~.N X . I o. I o No“ no a No“ a. a no” oo H x . l . x . «0“ ha 0 Noa on o NoH so A I . I . X . Noa NH n ~oH N" n «ca 3H m x . x . a . Ho” on o Hod aa a ~oa oo H “HH.< co«ddacmv Am.< cofiudaamv A“.< couadaemv HH daze 8H «use q.o.a.a no. 70 I omwoz 0m x Aq.< aquudawmv An.< cowuaawuv An.< cemumawmv oo.n mm.n mo.o mm.o “a Oahu aa~u0>o awash x . won me n oOa u no.n mad I no.o x I men on a x . cod O¢ a x . noa aa a a“ «aha nod com 00H no~ now «on I.0 oz. 9 A z no z I anemwm .H>x m4m 10'3) were presented in Appendix A. C. Noise Expressions The noise sources considered here are those instrumental sources which often limit measurement precision. A modern spectrometric system is considered which, after a suitable warming time, has excellent stabili- ty over the measurement time and is free from undirectional drifts in the radiation source, the photomultiplier transducer, the amplifier and the readout device. Thus the errors considered are random and independent so that the total variance is the sum Of the variances from each noise source. 1. Total Variance For a DC measurement, the final variances of the signal voltage, 02 , the background voltage, 02 , and the dark current voltage, 02 , Eos Eob E0d are the sums of individual variance terms and are shown in Equations 8.10- 8.12. 152 All variance terms in Equations 8.10-8.12 are defined in Table XVII. 2 2 2 2 2 2 o = (o ) + (o ) + (O ) + (O ) + O (8.10) 05 EOS q EOS sec EOS f EOS pm amp-rd 2 2 2 2 2 2 o = (o ) + (0 ) + (G ) + (O ) + 0 (8.11) Eob E0b q E0b sec E0b f E0b pm amp-rd 2 _ 2 2 2 2 CEOd - (CEOd)q + (OEOd)SeC + (OEOd)pm + Camp-rd (8.12) where Gimp_rd = 03 + Gimp + Cid = amplifier-readout variance, V2 Equations B.lO-B.12 are complete expressions assuming separate measurements are made of E05, Eob’ and Eod' If a comb1ned measurement is made, as of dark current plus background, only one amplifier-readout variance term need be included for the measurement. For photon counting the total variances in the signal counts, 02 N . the background counts, 030b, and the dark counts, ofiod’ are given by 05 Equations 8.13-8.15, where all terms are defined in Table XVII. Usos = (000$)q + (0,305)f + Usrd + (000$)ex . (B 13) O[gob : (Ofiob)q + (ofiob)f + Usrd + (Oigob)ex (B 14) A more detailed discussion of the individual variance terms in the preceeding equations follows below. 2. Quantum Noise The variance in the number of counts due to quantum noise in photon counting systems, Equations 8.19-8.21, was discussed in Appendix A. For a DC system, the corresponding variance in the signal voltage due to quantum noise in the signal photoanodic current, (0% )q is found by os 153 AON.mv unmmp< u noz u mmwo: Esucmzc ou msu noz CA mucmwcm> u A Nov . mm A mo mo a moz AmA mv p m < u z u mmwo: Ezucmsc ow was 2 CA mucmwcm> u A Nov mmmACOAmcmEAv .va An um>wmumc CAmm ucmccsu m>wuummmm u > .1 AmA.mV AUMP» + dpmweveqmmdm u no a vow N> .mmwo: Ezpcmsc ow man m :A mucmwcm> u A Nov < .E\nmw n ucmccsu OwnocumOOpoza uczocmxumn m>Agumwmm u new 0 m ANA.mV eqmmn AANEN u eqmmn _meN u no N> .mmwo: Ezpcmzc ou man now CA mucmwcm> u UA way .Aomv emp_we Amen 30_ um mpaswm a An umuAEAA m? cva2ucmn mmwoc mmocz muA>mu usonmmg com .um¢\A u p we?“ acmemcgmmms saw; muw>mu “soummc mcwpmcmmpcw so» .uN\A u N: .Emumxm “soummL-LmAewAQEm on“ mo spuwzucmn pcmAm>A3cm mmAOc u e< < .cmuw u ucwcgzu Ownogum00poca Amcmwm m>wpumwmm u mu_ _ < .EC\mmA u acmccau ownogumOOpoca Amcmwm n mu_ m u Ao_.mv teem d_a EN H 040mm Pea EN u teemmaAAEN u N N N . 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mmwoz mmmoxm UL mocooo cw mocowoo> u mo H La ULO N H waULD N L.oeo + m.oco u ooooomg co mocowgm> N No ”mocowgo> poooomm 158 multiplying Equation 8.19 by (meRf/t)2 and setting A] = l, which yields CS 2 . 2 . (02 ) = (me) e1 = mee1as EOS q t t (8.36) Equation 8.36 is applicable in cases where DC charge integration or digital integration is used if the integrating or measurement time is much longer than the time constant of the amplifier-readout system. If an analog readout device such as a meter or recorder is used, or a digital readout device with a measurement or sampling time much less than the time constant of the amplifier-readout system, Equation 8.36 can be written in terms of the noise bandwidth of the amplifier-readout system as shown in Equation 8.16 (the Schottky formula (90)). For the remainder of this paper, all variance terms for a DC measurement system will be written in terms of the noise bandwidth of the amplifier-readout system since this is the usual convention. The variance in the background output voltage due to quantum noise is derived from Equation 8.20 in a similar manner as above. For 0C measurements, the expression for the variance of the dark current due to quantum noise is more complex because different components of the dark current receive different average gains. Equation 8.18 takes this into account by defining a different gain, y, for izd. Because leakage current is not composed of pulses, it is assumed not to contri- bute to the dark current quantum noise. 3. Secondary Emission Noise The statistical nature of secondary emission gives rise to a variance in the gain and produces the pulse height distribution. For DC measure- ment, this adds another source of variance. The factor a in Equations 8.22, 8.24, and 8.25 accounts for this variance in gain and is found from 159 Equation.8.23 assuming a photomultiplier with equal gain dynode stages described by Polya statistics (59,91). In Equation 8.25, it is assumed * that 1ad experiences the same relative variance in gain as i s’ iab’ and a Iatc' 4. Signal and Background Radiation Flicker Noise Non-ideal variations in the incident photon arrival rate (variations above quantum noise) can arise from many causes depending on the specific spectrometric application. In molecular absorption and fluorescence spectrometry, variations of the average arrival rate of photons can be caused by variations in the primary light source intensity from changes in source temperature, power supply voltage and current fluctuations, arc wander, and filament motion. In flame atomic emission spectrometry, fluctuations in the average arrival rate of photons can be caused by fluctuations in flame background, fluctuations in the solution aspiration rate, etc. In flame atomic absorption and fluorescence spectrometry fluctuations can occur in the primary light source, in scattered light and in the flame background. Empirical flicker factors, a and ;,are introduced in Equations 8.26-8.29, to account for all fluctuations of either the signal intensity or the background intensity which occur prior to the radiation striking the photocathode. Depending on the source of the signal and background radiation, these may be simple or may be com- posed Of several factors (52,86-89). Although the treatment here assumes independent signal and background flicker noise sources, as is the case with flame background flicker and hollow cathode flicker in atomic absorption, there may be dependent flicker noises, as occurs in molecular luminescence spectrometry (86). In this latter case the rms noise will include a cross term whose magnitude depends 160 on the degree of correlation of the two noise sources. In some types of spectrometry where background radiation is not a significant factor, the signal radiation flicker factor can be readily measured. For example, in molecular absorption spectrometry radiation flicker is usually due to fluctuations in the tungsten or deuterium lamp intensity. The flicker factor can be measured by removing the monochro- mator and monitoring the source directly with a vacuum or solid state photodiode. Under conditions of high photon flux, the relative contri- butions of shot noise and other variances should be small compared to source flicker, providing an accurate estimate of g. In other types of spectrometry the flicker factors are more difficult to measure. In atomic absorption, for example, the contribution of the hollow cathode lamp or continuum source to the total radiation flicker can be measured in a manner similar to that for a molecular absorption source. However, the contributions of flame flicker and background instability can only be measured under conditions identical to the analysis since these factors depend on the wavelength and spectral bandpass settings of the monochro- mator. It is therefore difficult to Obtain an accurate measurement Of the total signal and background flicker because any measurement will include contributions from photomultiplier shot noise. 5. Photomultiplier Flicker Noise Noise generated in the photomultiplier in excess Of quantum and secondary emission noise will be designated as photomultiplier flicker noise. Variations in the work function of the photocathode or dynodes or in the photomultiplier gain due to power supply fluctuations are possible causes of photomultiplier flicker noise. Flicker noise with a l/f dependence due to work function fluctuations is predicted to be l6l proportional to the photocathodic current squared (92). It has been shown to be negligible above 0.1 Hz for most photomultipliers (70) and will be assumed negligible here. Equations 8.30-8.32 can be used to estimate the variance in the signal, background, and dark current output voltages due to photomultiplier flicker only if the predominant source is supply voltage fluctuations (93). 6. Johnson Noise The variance in the output voltage due to Johnson noise is given by the Nyquist formula (90) in Equation 8.33 where it is assumed that Rf is a high quality resistor with negligible current noise. 7. Amplifier Noise In the DC mode, the 0A current-to-voltage converter contributes another source of noise to the system. Equation 8.34 is only an approxi— mation (94) since amplifier noise is not white. Because the noise per unit bandwidth is not constant, the noise contribution from the amplifier must be calculated or measured at the particular noise bandwidth used for the measurement. 8. Readout Variance The readout device itself may add variance in the overall measure- ment. This variance can be attributed either to noise in the circuits Of the readout device as in the amplifiers of an oscilloscope or servo re- corder, or to the resolution or readability of the readout display as: indicated in Equation 8.35. The latter source Of variance is not noise in the sense of electrical noise, but does contribute to the total variance and limit the maXimum obtainable S/N. Readout variance was discussed in detail in Chapter V. 162 9. Other Noise Sources 19_99_Measurements If the signal from the current-to-voltage converter, Eot’ is further modified before reaching the readout device, additional variance terms must be added to the total variance to take into account noise from voltage and logarithmic amplifiers and noise present on top of any suppression signals used in scale expansion. 10. Excess Noise If the photon counting circuitry is performing perfectly, the total observed variance in the number of signal pulses counted is due to quantum noise, signal radiation flicker noise, and readout variance. However, noise and fluctuations in the counting circuitry can affect the number of pulses passed by the discriminator and hence add an additional variance. This excess variance is designated as (ofi )ex' Similar terms 05 are denoted for variance in the background and dark current count readout due to excess noise. Pulses arising from Johnson or amplifier noise might be counted as true pulses and add to the dark current pulse rate. These spurious pulses are assumed to have been eliminated so that excess noise arises only from fluctuations in the photomultiplier gain, in the amplifier or discriminator gain, and in the discriminator level. If these fluctuations are small, 2 (egos)ex is proportional to Nos‘ The fluctuations due to excess noise are characteristic Of l/f noise so that their relative contributions to the variance are greater for longer measurement“times. The magnitude of the excess noise should be small for a stable system and dependent on the discriminator level. At low discrim- inator levels, excess noise variance is small, but is a maximum at dis- criminator levels where the rate of change Of the observed count rate 163 with respect to the discriminator level (dRos/dAl) is at a maximum. 0. Signal-to-Noise Ratio Expressions l. 99_Measurement §ystem In the absence of background radiation and dark current,the S/N would be the ratio of the desired signal voltage to the square root of its variance. When dark current and background are present, however, two measurements are needed to extract the desired signal voltage EOS from the total readout voltage Eot' First, E0 is measured and then the t sum of the background and dark current output voltages (Eob + Eod) is measured and subtracted from Eot’ TherefOre, the measured S/N, (S/N)dc’ is given by E0t - (E 2 (O + o + O Eot EOb E (S/N)dc (8.37) + 20 - E0d amp-rd 2 2 (o + 20 E05 EOb where 2 2 Q N total variance 1n Eot’ V 2 2 2 2 0E + O + 0E Oamp-rd ot - 2 E od 05 ob Equation 8.37 is arranged so that 2 amplifier-readout variance terms appear in the final expression, one for the measurement of E0t and one for the combined measurement of E0b + Eod‘ The subtraction indicated in Equation 8.37 may be performed electronically by suppressing out the sum of the background and dark current voltages. However, Equation 8.37 is equally valid for manual or electronic subtraction since in this case voltage suppression is a measurement. 8y arranging the variance terms in Equation 8.37 into groups which show similar dependencies on the signal photocathodic current ics’ 164 Equation 8.38 results. E _ os (S/N)dC (62 + b2 + C2)]/2 (8.38) where 2 2 2 2 a = (0 ) +(O ) + (O ) (8.39) EOS q EOS sec EOS pm 62 = (OE )f (8.40) 05 2 2 2 2 C = 2(0 + o - o ) (8.41) EOb E0d amp-rd The terms in a2 are proportional to ics’ those in b2 are proportional to 165’ while c2 terms are independent of ics' If in a specific appli- cation individual variances other than those considered are present or if cross variance terms exist, this useful classification should still be valid. For instance, if there is an interaction between background and signal flicker, the cross terms will be directly prOportional to iCS and additive to the above expression for a2. For many spectrometric applications simplifications can be made in some of the variance terms. As discussed in Chapter V for molecular absorption spectrometry, background radiation is usually negligible and c2 would consist only of dark current and amplifier-readout variance. In other applications a knowledge of the predominate variance terms will allow considerable simplification of Equation 8.38. In most analytical spectrometric situations light levels are high enough that Johnson noise will be negligible as discussed in Appendix C. By selection of a high quality, low-noise operational amplifier, the contribution of amplifier noise can be made negligible with respect to other noise sources. For -1/2 -1/2 instance fer a good 0A, en = 1 AV Hz and in = l pA Hz , making the variance calculated from Equation 8.34 negligible in most cases. 165 Also for a well regulated photomultiplier power supply, photomultiplier flicker noise will be small. Thus the total variance, and hence the S/N, is usually determined by shot noise from dark current, background and signal, signal and background radiation flicker noise, and readout noise. With these simplifying assumptions the group variance terms become 2 2 2 2 . a = (O ) + (O ) = Zmei R Af(1 + 0t) (8.42) EOS q EOS sec as f 2 _ 2 _ . 2 b - (0E0 )f — (giast) (8.43) s 2 _ 2 2 2 2 2 2 C - 2L(oEob)q + (0E0b)sec + (OEob)f + (OE ) + (OE ) + 0rd] 0d SEC . . 2 . 2 2 4me(1ab + 16d)RfAf(l + a) + 2(g1abe) + 20rd (3.44) To obtain Equation 8.44, the assumption is made that y = m in Equations 8.18 and 8.25, which for most cases is an over estimate of the shot noise * of 1ad' Substitution of the above group variance terms into Equation 8.38 and division of all terms by me yields (S/Nldc = .CS 2 2 2 2 2(81::) 2eAf(1 + o)(iCS + 21Gb + Zicd) + g ics + 22 icb + 20rd mzRi where iCd = —%g-= effective cathodic dark current, A If the readout variance is negligible, Equation 8.45 indicates that (S/N)dc is basically independent of feedback resistante R and photomultiplier f gain m, although there is a slight dependence Of o, icd’ i , and icb on cs m. Figures 12 and 13 show plots of log(S/N)dC vs. log iCS for different values of background and dark photocathodic current, and signal and I.” 166 Figure 12 Effect of Signal Shot and Flicker Noise On the Signal- TO-Noise Ratio. (a) Signal shot noise limited (b) 6 =10’4 (c) 5 = 10-3 (d) a = 10'2 167 4.. / / 3— / / / / / / 2— / / k/ |)— o l I I 1 l 16 15 I4 13 12 -Logi¢, Figure 12 Figure 13 168 Effect of Dark and Background Current Shot Noise and Back- ground Flicker Noise on the Signal-TO-Noise Ratio. (a) Signal shot noise limited (b) icd + icb = 10'16 A, c = 0 (c) icd + icb = 10’15 A, c = 0 (d) 1cd + 1cb = 10’14 A, c - 0 (e) icd + icb + 10’]3 A, c = 0 I ..I OI (f) 1.cb - Log (S/Nldc 169 Figure 13 / / 4 — / 13 r- 1’ / 2 1- / o/ b c I d e f o I 4 L I I I l 16 l5 I4 13 12 ll 10 9 8 -Log in 170 background flicker factors. In constructing the plots, it was assumed that the noise bandwidth Af = 1 Hz, a = 0.275, and did/mzRi is negligible. The dotted line in both figures indicates the signal shot noise limit (a2 >> 02 + c2 for all 1C5). Figure 12 illustrates the effect of signal radiation flicker and hence the magnitude of b2 or g on the S/N. Figure 13 indicates the effect of the magnitude of c2 variances (dark current and background shot noise and background radiation flicker noise) on (S/N)dc for different values Of icd’ icb’ and g. The figures clearly illustrate the 3 limiting regions where the 1/2 1 (S/N)dc 1s dependent on ‘cs’ cs , and independent of iCS and indicate the importance of reducing the signal flicker factor 5 for high photo- currents and icd’ icb’ and g for low photocurrents. The two figures can be used in common spectrometric applications to estimate the S/N expected and the dependence of the S/N on iCS since order of magnitude values of the group variance terms can be derived from a few experimental parameters. For molecular absorption spectrometry iCb is negligible and (S/N)dc depends only on ics’ g, and icd' The relatively high amounts of background radiation in many analytical flame spectrometric applications make it necessary to consider in addition iCb and C. 2. Photon Counting System The S/N expressions for photon counting are similar to those for DC measurements, but less complicated because the number of variance terms is smaller. The S/N for a photon counting measurement, (S/N)p, including dark current and background is given by Equation 8.46 - 8.46 (S/N) = Not (Nob + Nod) = Nos ( ) p [ofi + 00 + as )1/2 [03 + 203 + 203 - 303 ]1/2 ot ob od 05 ob od rd 171 where N0t = NOS + N0b + N d = total observed count 2 _ 2 2 2 2 _ - ON OH + ON + ON - 20N — total observed variance 1n N0t Equation 8.46 can be rewritten by grouping the variance terms according to their dependence on N05 as N (S/N)p = [x2 + ygs+ 2211/2 (8.47) where y2 = (000$)f + (000$)ex 22 2|:Ofiob + Osod - CJfird] The term x2 is proportional to Nos’ those in y2 are proportional to N35, while 22 terms are independent of "05' As was true with the DC measurement case, practical considerations can lead to the simplification of Equation 8.47. For example, the read- out variance of 1 count is negligible to 1% if more than 100 counts are accumulated. Also, if nearly all pulses are counted so that Al is near unity, the number of pulses passed by the discriminator is essentially independent of gain and discriminator level fluctuations, allowing the excess noise terms to be eliminated. Under these conditions, the grqu variance terms become 2 2 x =(O )=AR t (8.48) NOS q 1 as 2 _ 2 _ 2 y (0” )f - (A1Rast5) (8.49) )q] = 2t[A1Rab + A R + A t 1 atc dRad] 2 (8.50) + 2(A1Rabtg) Substituting the simplified group variance terms in Equation 8.47 yields t][A R t + 2t(A 'k 1 as T A R + AdRad) lRab l atc 2 2 -1/2 + 2(A1Rabtg) ] s N = A R ( / )p I 1 as (B 5]) + (AlRastg) Plots of log (S/N)p vs. log NOS or log iCS under equivalent noise bandwidth conditions would be quite similar in shape to those shown in Figures 12 and 13 for the DC system. However, Equation 8.51 holds only for count rates which are low enough that pulse pileup is negligible. Hence curves for photon counting would extend only to a photocathodic current Of 10']3 A, which is the upper limit for even a very fast count- ing system. Because of the necessity of Operating at relatively low light levels where pulse overlap can be neglected, flicker in the signal and background will rarely become dominant noise sources in photon count- ing. E. Comparison of Photon Counting and DC Measurement Techniques The main advantages of photon counting usually mentioned are higher signal-tO-noise ratio, discrimination against dark current not origina- ting at the photocathode, direct processing Of discrete spectral infor- mation, elimination of reading error, and system stability against drift. The merits of these advantages compared to DC measurement are discussed below. Photon counting has been shown to be most advantageous in com- parison to DC measurements under low light level conditions where the S/N is near unity or less (68). Since the S/N's in most common analytical spectrometric applications are much greater than unity, the advantages 173 Of photon counting are not to be expected to be as great as for low light level measurements. 1. S/N Comparison To compare theoretical S/N's of photon counting with DC techniques, equivalent spectrometric system parameters up to the output Of the photo- multiplier tube are assumed so that ias = meRaS. Some confusion exists in the literature (67) in correlating the counting time, t, in photon counting to the noise equivalent bandwidth, Af, in a DC system. For equivalent conditions, Af must equal t/2. Of course if an integrating procedure is used in the DC case, the integration time should equal the counting time for equivalent noise bandwidths. If the readout variance, Cid, in the DC system is assumed negligible and l/2t is substituted for Af in the DC system, the ratio of Equation 8.45 to Equation 8.51 can be taken to indicate the improvement in S/N expected when using photon counting. (S/N) _____J1 ,=. (S/N) dc (8.52) -1 . . . 2.2 2.2 1/2 [AlRast] [ et (1+ a)(1cs + 21cd + 21Gb) + E 1CS + 221Gb 24 . * 2 1cs A1Rast + 2t(AlRab + AlRatc + AdRad) + (AlRastg) + 2(AlRath) Using the relationship ics = eRas and assuming that A] = Ad = l and iCd = eRad’ Equation 8.52 can be reduced to 2 2 2 2 (S/N) _ [(Ras + 2Rab + 2Rad)(l + a) + g Ras + 2g RabJI/Z (B 53) T 2 2 2 2 ' (S/N)dc (Ras + 2Rab + 2Rad) + g Ras + 2; Rab Equation 8.53 gives a good indication Of the S/N improvement to be expected from using photon counting. If flicker noises in the signal or 174 background predominate, the first terms in the numerator and denominator of Equation 8.53 can be dropped, and the S/N's of the two techniques are equivalent. As was previously mentioned, this is not likely to occur because pulse pileup limits the maximum light level which can be measured in photon counting. At the other extreme, if the flicker noises are negligible, the last two terms in both the numerator and denominator can be drapped, which reveals a net advantage in S/N of /T§§ for photon counting under shot noise limited conditions. Since a usually ranges from 0.1 to 0.5 for typical values of dynode gain 6, the S/N for photon counting is better by a factor of 1.05 to 1.22 under these conditions. 2. Discrimination Against Dark Current Results of previous studies concerning the relative merits of using discrimination against dark current as a means to improve the S/N in photon counting systems (68,69) are inconclusive. Clearly the effect of discrimination against dark current on the S/N depends on the relative magnitude of the dark current compared to the photocurrent and also on the sources of dark current. For a DC measurement, if ias + iab is much greater than iad’ the shot noise of the dark current makes little contri- bution to the total shot noise. Thus, even if photon counting allowed discrimination against all dark current, the improvement in S/N would be negligible. The relative magnitudes of the different kinds of dark current, which are dependent on the size and nature of the photocathodic surface, are important because pulse discrimination will only improve (S/N)p as Al is decreased if the dark current pulse rate decreases much more rapidly than the photoelectron rate. Since Ratc has the same discriminator coefficient 'k as Ras’ if Ratc >> Rad’ little improvement in S/N will result with pulse 175 * discrimination. If Rad > R then pulse discrimination will signifi- —- atc’ cantly reduce R:d’ since Ad decreases more rapidly than A1 if R:d is composed of smaller pulses from down the dynode chain. Note that discrimination reduces the S/N by the discriminator coef- ficient A1 in the absence of dark current pulses, or if all the dark current pulses originate from cathodic thermal emission. Thus, under these conditions, the S/N advantage of photon counting is lost when A1 §_(l + 0)-]. Because of this, a small amount of discrimination may improve the S/N if Rad is significantly reduced. However, too much discrimination will reduce ROS significantly and reduce the S/N. In a DC measurement dark current pulses which originate from down the dynode chain are inherently discriminated against since they receive an average gain which is lower than m, the full average gain. Hence, if the major contributor to igd is thermionic emission from down the dynode chain, rather than cold field emission or radioactivity, the contribution of izd to the total dark current shot noise in a DC measurement is in- herently small. In photon counting, all pulses no matter what their origin are weighted equally in the absence of pulse height discrimination. Because of this the S/N improvement for photon counting with no pulse height discrimination may be less than predicted by the /T35 factor. Thus, a small amount of discrimination is usually necessary to bring the level above the amplifier noise and to reduce the observed rate of small dark current pulses. If such factors as cosmic rays or after pulses are a large contributor to 12d and igd is of comparable magnitude to iatc’ pulse counting would be advantageous, since the large pulses are counted only once or can be eliminated by upper level discrimination. Considering that for most analytical applications of spectrometry, l76 ias + iab is considerably larger than the total dark current and that for commonly used photomultipliers, such as the RCA lP28, thermionic dark current from the photocathode i contributes approximately 90% atc of the dark current at room temperature (95), pulse discrimination in photon counting appears to offer little S/N advantage for most analytical spectrometric applications. 3. Reading Error and Direct Digital Processing If the readout device for the DC measurement system utilizes scale expansion and high resolution digital readout, the readout variance can often be made negligibly small. Under these conditions, the absence of reading error in photon counting presents no real advantage, although there is certainly an advantage over the low resolution meter or recorder readouts which are still common in analytical spectrometers. The direct digital processing of data in photon counting is prac- tically, as well as philosophically, an advantage. Although the DC measurement system is assumed to have digital readout, a number of data domain conversions (96) are necessary to obtain the final numerial read- out. Therefore conversion errors, non—linearities and the resolution of the A/D converter or the readout device can limit the ultimate measure- ment precision, while photon counting measurements show high linearity in the absence of pulse overlap. 4. Measurement System Stability Probably the greatest advantage of photon counting is the inherent system stability, which allows precise, long time integration to reduce the bandwidth. To increase the S/N by a factor of l0, the counting time must be increased by a factor of 100. Increasing the integrating time or reducing the bandwidth in a DC measurement system can also increase l77 the S/N. However, drift and l/f noise in the gain of the photomultiplier, in the amplifier, and in the readout device or A/D converter can become limiting for small bandwidths. Thus many of the variance terms that were dropped to obtain Equation 8.45 may be important at small bandwidths and should be included. Robben((67) has found that his photon counting system was a factor of 5 more stable than a DC system for long integration times (greater than 10 seconds). For photon counting, especially at low discriminator levels, drifts in the pulse heights due to photomultiplier or amplifier gain drifts have little effect on the readout. If dead time compensation is utilized to extend the linearity in photon counting, A1 is approximately 0.5, and the system will be more susceptible to drift. In many routine analytical spectrometric applications, when S/N's are reasonably high, the increased stability of photon counting is not a significant advantage. However, in some applications where low intensi- ties are being measured near the limit of detection, the ability to in- crease the measurement time without drift may be of importance. 5. g§g_gf_Modulation Techniques In many analytical applications modulation techniques are used for various reasons including S/N improvement. In flame atomic absorption and fluorescence, modulation of the source is often necessary to reduce the signal from flame emission. Modulation also allows the use of ac amplification so that amplifier drifts become less important. Modulation techniques can also be used with photon counting by the use of up-down counters synchronized to the modulation frequency (97). Again, however, the unmodulated background radiation must not be so intense that pulse overlap occurs, or non-linearity will result. Nhen modulation techniques are utilized many of the same S/N l78 considerations previously discussed are valid, although modulation may suppress certain low frequency noise components. If a modulation system is used which has the same noise equivalent bandwidth as a conventional DC or photon counting system, noise whose power spectrum is white, such as Johnson and shot noise, will not be suppressed. Flicker noise which show greater amplitudes at low frequency can be suppressed by the appro- priate modulation technique. In flame spectrometry, mechanical light chapping, pulsing of hollow cathodes, selective modulation of specific absorption lines, and modulation of the aerosol stream (98) have been utilized. Such techniques can be used with either analog detection or synchronous photon counting, and the same conclusions discussed above as to the relative S/N's should hold, although photon counting should be of advantage for the detection of weak sources, where the long term stability allms very long measurement times. In molecular absorption spectrometry double beam ac systems are often used to suppress low frequency source fluctuations and can likewise be used with synchronous photon counting. F. Conclusions Under equivalent conditions, photon counting systems compared to DC measurement systems can provide the advantages of 5 - 22% higher S/N under shot noise limited conditions, greater stability, and better linearity if pulse overlap is negligible. The significance of these advantages depends on the magnitude of the S/N. For applications where the DC S/N is inherently high with normally used noise bandwidths (i,§,, l Hz), the above advantages are not very significant. However, under conditions where the S/N is small, the 5 - 22% S/N advantage of photon counting can be highly significant, and the stability advantage can be used effectively to improve the S/N by increasing the counting time. l79 Whether photon counting will show an improvement over DC techniques in a given spectrometric measurement depends to a large extent on the type of spectroscopy (absorption, emission, or luminescence), the magnitude of the signal, the magnitude of the background, and the resolution re- quired. Since the S/N in a given application depends upon many experi- mental and spectral parameters whose influence may differ from one appli- cation to another, it is difficult to generalize. The spectral experi- mental conditions necessary to Optimize the S/N should be the same for both detection systems. If the S/N is optimized (55) within the resolu- tion requirements of the particular application, the choice of the bet- ter detection system is relatively simple. If the photoelectron pulse rate is in the region where pulse overlap is negligible [(Ras + Rab)r < l0'3], photon counting provides superior characteristics. Under con- ditions of significant pulse overlap, non-linearity makes photon counting undesirable unless mathematical correction can be used. Once S/N opti- mization has been carried out, the photocathodic signal current should not be reduced (i,e,, by decreasing the signal radiation intensity) to bring the pulse rate into the linear range of photon counting. Such practice is unsound because the reduction in S/N due to the reduction of iCS will offset any advantages of photon counting. From the above considerations photon counting appears to be most useful in analytical applications where signal and background radiation intensities are low and signal-to-noise ratios are small. Such appli- cation may include non-flame atomic fluorescence spectrosc0py and molecu- lar luminescence techniques. For non-flame methods where peak responses are obtained, the integrating and fast response characteristics of photon counting may be advantageous. In flame emission spectrometry, the l80 presence of background may prevent the use of photon counting techniques under conditions of Optimum S/N. In atomic and molecular absorption spectrometry light levels are often high enough under optimum conditions that photon counting is not the advantageous detection technique. How- ever, for certain atomic and molecular absorption applications where very high resolution is desirable, such as atomic absorption with a continuum source and scanning molecular absorption, the low light levels encountered may make photon counting the more attractive measurement system. APPENDIX C. SIGNAL-TO-NOISE RATIO COMPARISON OF PHOTOMULTIPLIERS AND PHOTOTUBES A. Introduction The question of whether to use a photomultiplier tube or a vacuum photodiode in a given spectroscopy application has been asked many times and conflicting answers have resulted (99). This question is important because these two transducers are used almost exclusively for measure- ments involving ultraviolet and visible radiation. For many years, it has been concluded that the process of internal amplification in photo- multiplier tubes gives better S/N characteristics than could be obtained by using a single stage photodiode in conjunction with a high-gain amp- lifier-readout system (100). Recently, however, the opposite conclusion was reached and experimental data presented to show greater S/N's for the photodiode-amplifier-readout system than for the corresponding system using the photomultiplier tube (12). This apparent reversal was justi- fied as being caused by the develOpment of solid-state, low-noise amplifiers. There are many factors which must be considered in choosing the optimum transducer-amplifier system for a given spectrometric application. Dynamic range, stability, linearity, and response speed must all be con- sidered in addition to the inherent S/N characteristics. In this appendix, only the inherent S/N characteristics of photomultipliers and photodiodes are compared. 181 182 8. Signal-to-Noise Ratio Expressions To compare the inherent S/N characteristics of the two transducers, the generalized single beam system described in Appendix 8 is considered. The DC S/N for the photomultiplier, (S/N)PMT’ is given by Equation 8.38 in Appendix 8. Substituting the unsimplified group variance terms (Equations 8.39-8.41) into Equation 8.38 and assuming that background radiation is absent (i = i E = E , and 0E =(D and that the read— cp cs’ op 05 ob out variance is negligible yields ( ) zfluo 2 2 = (E ) /[(O ) + (O ) PMT op PMT EOp q E0p sec op pm 05 C.1) 2 2 2 2 1/2 ( d)q + (OEOd)sec + (OEOd)pm + 0J + OampJ + (..E o where (Eop)PMT is the signal output voltage for the photomultiplier system. Note that actually only c2/2 is used to obtain Equation C.1 since we are concerned with the inherent S/N of the transducer-amplifier system. This is equivalent to neglecting the variance in making a separate measurement of Eod' Substituting into Equation 6.1 the exact expressions for the individual variance terms found in Table XVII and assuming y = m yields _ - 2 2 . - [icmef][2em Rf(1 2 0amp ( ) )2 :zun PMT Cp + lcd)(Af(1 + a) + x) + (gicmef (C.2) 1/2 + + 4kTRfAf]- The corresponding DC S/N for the vacuum photodiode-amplifier system, (S/N)PD, is derived from Equation C.2 by noting that m = l, a = x = 0 as shown in Equation C.3 * i R (A) = 2 .* .* .Sp f 2 2 + f 1/2 (C'3) PD [2eRfAf(iCp + lcd) + (gichf) + Camp 4kTRfA ] 183 .* where (Eop)PD = 1Cp Rf is the signal output voltage for the photodiode system and the superscript asterisk denotes parameters for the photodiode system. It is assmed that the same 0A current-to-voltage converter, readout device, and feedback resistor are used for both transducers. Further voltage amplification may be necessary in the photodiode system to obtain the comparable output signal voltage as the photomultiplier system. noise in the photodiode system. The additional amplifier is assumed to contribute negligible In many cases, it is realistic to assume that only Johnson and shot nOise contribute to the total rms noise. and C.3 become mR i For these cases, Equations C.2 (fil = -20 f CD -19 2 2 1/2 (C°4) PMT [1.6 x 10 Rf + 6.4 x10 Rfm 1ct] .* (81 = -20 RfICp -19 2 * 172 (C 5) PD [1.6 x 10 Rf + 3.2 x 10 Rfict] where 1Ct = 1Cp + icd’ the appropriate fundamental constants are utilized. and it is assumed that Af = 1 Hz, a = 1.00 (worst case), and T = 293°K. Further simplification is possible under certain experimental con- ditions, In the case of the photomultiplier, the first term in the denominator of Equation C.4 is negligible to 1% if RfmziCt > 2.6 V. Con- sidering that typical values of m and Rf the Johnson noise term is negligible if iCt > 2.6 x 10 6 and 1 M9, respectively. -18 are 10 Aor fiat) 2.6x 10"12 A. This allows reduction of Equation C.4 to i (§) = 1.3 x 109-—Jfll—— 3 1.3 x 1091 1/2 (C.6) N . 1/2 cp PMT 1Ct under conditions where the dark current is negligible with respect to the 184 photocurrent. Thus for most practical purposes, the photomultiplier is shot noise limited and the S/N depends on the square root of the photo- cathodic current. Likewise, Equation C.5 can be simplified under certain limiting con- * ditions. The second term in Equation C.5 is negligible to 1% if RfiCt = (Eop)PD < 500 uV which yields s_ = 9 .* 1/2 (N)PD 7.9 x 10 leRf (C.7) Thus for even a high resistance value such as 100 M9, the photodiode is -12 Johson noise limited if 1: < 5 x 10 A. P At the other extreme, if R > 5 V, then Equation C.5 can be .* f1ct reduced with 1% accuracy to i * (§) = 1.8 x 109 ——;99———-% 1.8 x 109(1 )”2 (C.8) ct * and the photodiode is shot noise limited. If 500 uV < Rfict < 5 V, then Equation C.5 must be used to calculate (S/N)PD. The only valid means of comparing the two transducers is under equivalent conditions. Thus it is assumed that the radiant power impingent on the photocathode surfaces, the collection efficiencies, and the sensitivities and areas of the photocathode surfaces are equal so that iCp = izp. Figure 14 presents plots of log S/N versus icp’ for the photomultiplier and photodiode. Since in the Johnson noise limit, the S/N of the photodiode depends on the magnitude of the feedback or load resistor, plots are shown for three typical Rf values. The data were calculated from Equations C.4 and C.5 where it is assumed that dark current is negligible. It is clear from Figure 14 that the photodiode and photomultiplier are applicable in different regions of photocathodic 185 14 13 ' 12 Figure 14. 11109 8 7 6 5 4 -Log(kp) S/N As a Function of Photocathodic Current x o 13 D Photomultiplier Photodiode, Rf = 1 M9 Photodiode, Rf = 10 M9 Photodiode, Rf = 100 M9 186 current. The plot of log (S/N)PMT is extended only to iCp = 10‘8 A because the corresponding photoanodic current would be in the milliampere range; the maximum output current for most photomultipliers. Acutally, most photomultipliers should be operated with photoanodic currents much less than one milliampere for linear quantitative results. The crossover point can be found by dividing Equation C.4 by Equation C.5 and simplify- ing which yields (S/N)PMT = m 1 + 20 Rfict 1/2 (c 9) (S/N)PD 1 + 40 Rfmzict By setting Equation C.9 equal to one and solving for Rfict’ it is found that if RfiCt < 50 mV, then (S/N)PMT > (S/N)PD. It is interesting to note that when Equation C.9 is exactly one, the Johnson noise and shot noise in the photodiode system are equal. C. Discussion By comparing Equations C.4 and C.5, or by observing Figure 14, a difference can be seen between internal amplification in the photomulti- plier and the use of external amplification with the photodiode. In a photomultiplier, only the quantum noise is amplified with the signal, while with a photodiode both the quantum and Johnson noise receive the same amplification as the signal. Any subsequent voltage amplification after the 0A or load resistor to increase the signal level cannot improve the S/N in either transducer system. The basic S/N is determined only by the signal and rms noise voltage across the feedback or load resistor. The equations presented indicate that the S/N of a photomultiplier or photodiode amplifier-readout system depends on the:magnitude of the feedback or load resistor, the gain of the photomultiplier, and the 187 photocathodic current, which is directly proportional to the radiant power impingent on the photocathode. The criteria for choosing between a photomultiplier and photodiode on the basis of S/N theory are estab- lished by Equation C.9. If the signal voltage output of the photodiode system, (Eop)PD = Rdict’ is greater than 50 mV or the output of the photomultiplier system, (Eop)PMT = meict, is greater than 50m mV, then the inherent S/N of the photodiode is higher than that of the photo- multiplier. Since some approximations were used to obtain these criteria, the value of 50 mV is not exact, but can be used as a guide for choosing between the two transducers. Since usually m = 105 - 107 and Rf = l - 100 M , the photocathodic current (i,g,, incident light level), which is the most variable parameter, determines the S/N. For a typical feedback or load resistor of 10 M9, the photocathodic current msut be greater than 5 x 10'9 A or 3 x 1010 photoelectrons/sec for the photodiode to be superior. A more exact expression for the crossover point can be obtained by inclusion of variance terms dropped to obtain Equations C.4 and C.5. Thus, Equations C.2 and C.3 can be used to calculate a crossover point with the same procedure used to obtain Equation C.9. However, it is easily shown that the basic criteria remains unchanged and the crossover point is usually within a factor of two or three of (Eop)PD = 50 mV. The magnitude of the photocathodic current depends on the spectro-: scopy application and the particular chemical system studied. For either transducer system, the S/N is improved by increasing icp’ unless the signal radiation flicker limit is reached, so that the photocathodic current should be maximized by using intense light sources, spectral bandpasses as large as can be tolerated, photocathodic surfaces with high 188 sensitivity in the wavelength region of interest, and efficient optical designs. The type of light source and the allowable spectral bandpass are extremely dependent on the particular application. Applying the criteria given above to analytical spectrometric methods, leads to the conclusion that in all but a few cases, light in- tensities are in the range where the photomultiplier has the better S/N characteristics. For example, in UV-visible recording spectrophotometry, and molecular fluorescence and phosphorescence spectrometry, the low intensities of sources and the small spectral bandpasses required usually cause the photocathodic current to be below 10'9 A where the photomulti- plier has the better characteristics. Also in flame atomic absorption, emission and fluorescence spectrometry, the photocathodic currents are quite small and the photomultiplier will have the higher S/N. Only for certain applications of quantitative single beam absorption spectrometry is it feasible to obtain light levels where the photodiode shows better characteristics. For example, with some chemical systems large absorption bandwidths make it possible to use intense light sources with filter photometers or monochromators with large slit widths. For chemical systems involving narrow absorption bandwidths, where deviations from Beer's law due to non-monochromatic radiation are serious, the much smaller spectral bandwidths required may reduce the light intensity to the point where, by the criteria established above, the photomultiplier will yield the higher S/N. Also, the use of high intensity sources and large slitwidths may result in heating of the sample and possibly decom- position. The incident light intensity must also be such that the photo- cathodic current is in the linear range of the transducer. For cases in which molecular absorption spectrosc0py is used to 189 measure initial rates of reactions, large spectral bandwidths and thus large photocathodic currents often make the photodiode the better trans- ducer. Large spectral bandwidths can often be used in kinetic methods because only small differences in absorbance are measured so that the error due to non-monochromatic radiation is small. Weichselbaum et. al. (12) have indicated that in such a system the S/N of the photodiode is l to 2 orders of magnitude greater than a corresponding photomultiplier. However, in comparing the transducers, the incident light intensity was adjusted for equal anode currents. Thus the result is not surprising since the photocathodic current was high enough that both transducers were shot noise limited (Equations C.6 and C.8). Since the photocathodic current of the photodiode was adjusted to be 102-104 higher to obtain equal anode currents, the S/N is expected to be one or two orders of magnitude higher. It should be noted that choosing the gain of the photomultiplier by varying the photomultiplier supply voltage involves a compromise. High gains are advantageous because the statistical factor a, decreases with the gain, and at higher gains, fluctuations in the photomultiplier supply voltage have a smaller effect on the gain. However, if the gain is too high, regenerative effects occur and the S/N expressions developed are no longer valid. Thus, if in a particular application the voltage out- put of the photomultiplier system is out of the range accepted by the readout device, the magnitude of the feedback resistor should be reduced rather than reducing the radiant intensity or the photomultiplier gain. This is true for the photomultiplier at high light levels where Equation C.6 is valid because the S/N is independent of Rf but will decrease if the radiant intensity is reduced or ifcxls increased by reducing the photomultiplier gain. m1|1111111111|111111111111111111111111111158 31293 0306] 9351