Advisory Panel Jonathan W. Amy Richard A. Durst G. Phillip Hicks
INSTRUMENTATION Donald R. Johnson Charles E. Klopfenstein Marvin Margoshes
Harry L. Pardue Howard J. Sloane Ralph E. Thiers
Instruments for Rate Determinations HOWARD V. MALMSTADT, COLLENE J. DELANEY, and EMlL A. CORDOS School of Chemical Sciences University of Illinois a t Urbana-Champaign Urbana, Ill. 61801 Widespread commercial introduction of inexpensive general-purpose minicomputers and hardware interfaces is greatly changing the design concepts of chemical reaction-rate instruments. Circuits, integration systems, variable-time methods, hybrid analog-digital systems, and software systems for reaction-rate determinations are presented and utilization T of the rate of change of one physical quantity with respect to another are HE DETERMISATIOK
.
frequently encountered in analytical methods. For example, one type of automatic titrator (1) electronically generates a signal proportional to the rate of change of transducer output ( E )us. volume ( V )of titrant and utilizes the first derivative signal (dE/dV) to control the rate of titrant delivery. Another type of automatic titrator electronically computes a second derivative signal (dZE/dVZ) during the course of the titration. This signal is ideally suited for the automatic termination of a titration a t the endpoint, because it crosses the zero reference a t the inflection point of the titration curve independently of the absolute magnitude of the transducer output ( 2 , s ) . A series of regular and derivative titration curves is shown in Figure l a . Recently, there has been increased interest in derivative spectrophotometry ( 4 , 5 ) ’ where the rate of change of absorbance ( A ) or emission (8‘) with respect to wavelength (A) is determined and recorded vs. wavelength. The dA/dX derivative signal (Figure l b ) is especially useful in locating overlapping absorption bands (5, 6). The second derivative signal ( d 2 A / d A 2 ) , Figure IC, is said to provide improved sensitivity in the determination of absorbing gases in the ultraviolet regions (4). The second derivative emission signal (d2F/ dX2),Figure Id, has been used to provide increased sensitivity for trace constituents that have emission lines super-
imposed on strong background signals ( 7 )* Of all the analytical examples where rate information is utilized, none has had so much attention focused on the various instrumental methods of encoding rate data as have the so-called kinetic or reaction-rate methods of analysis (8). I n these methods the time rate of change of some property P that is proportional to the concentration of one of the reaction products is measured, as illustrated in Figures l e and If. For most practical procedures the desired rate information is obtained shortly after the start of the reaction. The rate measurement system must be sensitive and capable of determining average initial rate amidst considerable background noise. After the first automated reactionrate methods and instrumentation for the rapid determination of glucose were developed in our laboratories about a dozen years ago (&la), many further investigations were initiated-both of specific chemical reactions and of the instrumentation that would provide sensitive and selective quantitative rate procedures. It was apparent from the start of these studies that more instrumentation developments were essential if rate methods were to be widely accepted as routine quantitative chemical methods. Also, of course, reliable automated instrumentation was important for more rapid investigation of chemical reactions that might be used for quantitative rate procedures. For over a decade there have been
many published improvements from several laboratories for all major sections of reaction-rate instrumentation for quantitative analyses. Many of these developments, including computer-controlled instruments with automated sample and reagent handling systems, are described in this month’s Report for Analytical Chemists, page 26 A (IS). I n this report the principles and characteristics of several electronic devices for rate determinations are presented which provide excellent noise immunity. Although the methods presented here were specifically developed for chemical reaction-rate procedures, several are generally applicable mhenever the direct readout of the rate of change of some parameter with respect to another is required. Classical Circuits for Rate Determinations
A differentiator circuit that provides a n output proportional to the rate of change of a n input signal can be easily assembled from a single operational amplifier (OA), an input capacitor, and a feedback resistor (14). However, relatively small noise components on the input signal can result in a high noise level a t the output that often obscures the desired rate information. Even when the time constant of the differentiator is increased, the noise is usually prohibitive for chemical reaction-rate procedures (16). An all-electronic rate comparison technique has been successfully used for some reaction-rate procedures. An
ANALYTICAL CHEMISTRY, VOL. 44, NO. 12, OCTOBER 1972
79A
Instrumentation
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methods can also be considered as general categories for the classification of circuits for rate determinations (8, 17, 18). Pardue (I?‘) and Ingle and Crouch (18) have indicated that the choice of method might be dependent on considerations other than noise, but in any event, it is important to modify the basic techniques to eliminate the noise problems illustrated in Figure 2. I n making a fixed-time measurement (Figure 2a), the noise component causes uncertainty in determining the value of P I and PZand, therefore, in the measured value, AP. The values determined for P I and P2 can fall anywhere within the dashed lines. The same considerations apply to the variable-time measurement shown in Figure 2b. I n addition to the uncertainty caused by the noise, an even greater error can be introduced by random spikes that can cause false triggering of the measurement system. For example, if the spike shown in Figure 2b is interpreted by the measurement system as a n indication that the preselected value of PZhas been reached, then the Gme interval (tz‘ - tl) is erroneously taken as the measured value, At. Greater noise immunity has been obtained for all three general methods by using hardware and software averaging and smoothing techniques (1925). Because space restrictions prevent a detailed discussion of all three categories, it was considered of greater value and interest to show how only one general method, the fixed-time method, can be implemented to provide high noise immunity with analog and digital hardware or with software procedures. The variable-time and derivative methods are briefly outlined and referenced. Integration (Fixed-Time) Technique for Reaction-Rate Determinations
TIME
(f )
Figure 1. Analytical methods by use of rate determinations. (a) Titration curves illustrating first and second derivative curves. (b) Benzene absorption curves recorded by minicomputer-controlled spectrophotometer (6). (c) Atomic absorption line showing first and second derivative curves. (d) Flame emission spectra of barium in presence of calcium (7). (e) and (f) Reaction-rate curves
OA integrator is used to generate a reference rate curve that is continuously compared by an OA comparator and maintained equal to the unknown input rate curve (15, 16). Although the performance of the comparison system is superior to the simple OA differentiators, it does not give sufficient averaging of noise for many typical chemical reaction-rate measurements, especially where high sensitivity is required. Both the simple OA differentiator and the rate comparison circuits can have fast response to provide output data proportional t o the “in80A
stantaneous” rate. Instruments capable of providing more or less instantaneous rate data for reaction-rate methods have been classified as derivative methods (8,17,18). An early approach to combat noise on rate signals mas to use one of the two-point methods as illustrated in Figure 2. The change in monitored parameter, AP (fixed-time), is measured over a preselected time interval, or the change in time, At, is measured over a preselected change in monitored parameter, A P (variable time). The so-called fixed-time and variable-time
ANALYTICAL CHEMISTRY, VOL. 44, NO. 12, OCTOBER 1972
Greater noise immunity has been obtained by applying an integration fixed-time technique rather than the two-point fixed-time method (16). The method involves the integration of two segments of the rate curve and subsequent subtraction of the resultant areas. As shown in Figure 3, the integration is made over two equal time increments, At, which are sequential, and the difference, A A , between the resultant areas A I and A2 is the parallelogram A B C D . The area of the parallelogram is given by: A A = (At)a
(1)
The slope is defined as: S = tana =
a
--
At
(2)
Substitution of Equation 1 into Equation 2 provides an expression relating
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(b) Figure 2. Illustration of noise susceptibility in (a) fixed-time measurement technique and (b) variable-time measurement technique
Figure 3. Principle of integration method for reaction-rate determinations. (a) Expanded section of typical rate curve illustrating integration and subtraction of two sequential areas AI and A2. (b) Expanded section of typical rate curve illustrating integration and subtraction of two areas AI and A2 separated by a meatl surement delay interval t2
-
slope to the difference in area: (3) Since At is a constant the slope is directly proportional to the measured difference, AA. I n Figure 3b the two time increments are not consecutive but are separated by a time interval t 3 - tz. I n this case Equation 3 can be rewritten as:
Since (tS - t2) is also kept constant, the slope remains directly proportional to the measured area. The total time for the measurement is 2 At (t3 - t2).
+
Rate Determinations with Analog Integration System
The integration and substraction operations illustrated in Figure 3 can
be implemented by use of the relatively simple analog circuit shown in Figure 4a. This circuit uses two operational amplifiers (Oh's) which are tied to one another and to a signal modifier by a switch network. During the first integration period, the signal is directed through switches 1 and 2 to the integrator (0x2). If the signal modifier output is as illustrated in Figure 4b, then line BC represents the signal applied to the integrator which charges capacitor C, causiug the output of OA2 to rise to a voltage VI as shown in Figure 4c. During the second integration period the signal from the modifier is first directed to the gain-of-one inverter (OAl) by switch 1 and then to the integrator by switch 3. The voltage represented by line DE is applied to the integrator input which discharges capacitor C, and the integrator output decreases in magnitude by the amount
Tiz. The voltage difference, A V , which is read out a t the end of the measurement period, is proportional to the difference in area, AA, and is therefore proportional to the initial slope according to Equation 3. The switches (Sl-S4) are activated with pulses generated by a logic circuit. Thus, upon receipt of a suitable trigger pulse, the logic circuit controls the start of the sequence of delay time, the length of the integration periods, the subtraction, and the readout of the result. The integration period is selected to be as long as feasible to average the noise most effectively. The shortest practical integration period is governed by the switch times and OA response and a t present is about 0.1 msec. During the integration period, the output voltage, eo, of the integrator changes as the square of the time, t, so that eo = Stz/(E3C). Thus, the rising and falling portions of the in-
ANALYTICAL CHEMISTRY, VOL. 44, NO. 12, OCTOBER 1972
81 A
Instrumentation I
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Figure 5. Oscilloscope traces of integrator input and output voltages for three successive measurement cycles. Integrator input (a) and integrator output (b) without curve-following suppression. Integrator input (c) and output (d) with curve-following suppression
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Table I. Automatic Rate Measurements with Long Integration Time and Signals Typical of Slow Reactions input rate, rnV/sec
Digital readout,a rnV
Proportionality constant
2.50 3.60 5.30 7.90 10.80
111.8 163.4 239.7 357.3 485.6
0.0223 0.0220 0.0221 0.0221 0.0222
4 Averages of 10 results; RZ = 450 kilohms; integration time: ment time: 30 sec.
Re1 SD,
%
0.24 0.24 0.33 0.15 0.09 10 sec; premeasure-
Table II. Automatic Rate Measurements with Short Integration Times and Signals Typical of Fast Reactions Input rate, mV/sec
Digital readout,a mV
180 385 590 795
81.8 175.3 268.8 362.8
Proportionality constant
2.20 2.19 2.19 2.19
a Averages of 10 results; integration time: msec; R Z = 15 kilohms.
Rei SD,
%
0.24 0.20 0.14 0.11
50 rnsec; premeasurernent time:
100
Table 111. Automatic Results for Glucose Direct concn readout5
l a ken
5.0 10.0 15.1 20.1
5.0 10.0 15.0 20.0
Glucose concn in fig/rni Rei error, %
0.0
... +0.7 +0.5
Rei SD, %
1.6 1.0 0.7 0.8
a Averages of five results; 10.0 pglml standard used to set readout; integration time: 10 sec; premeasurement time: 30 sec.
82A
ANALYTICAL CHEMISTRY, VOL. 44, NO. 12, OCTOBER 1972
tegrator output illustrated in Figure 4 should exhibit a curvature. This curvature is most pronounced when the integration operation is initiated a t a time close to the moment when a slope is generated ( t is small). I n the case of the utilization of a long premeasurement delay interval (t is large), the curvature is quite small. This effect canbe seen inthe photograph inFigure 5 . The rate meter described above is capable of measuring slopes over the range 0.45 mV/sec to 450 V/sec. I t has been evaluated with synthetic signals from a ramp generator simulating both s l o ~and fast reactions. The results given in Tables I arid I1 show from the proportionality constant that the output voltage is a linear function of the input slope, and that in both cases the relative standard deviations are typically about 0.2%. Tables I11 and IV show results obtained with the rate meter for the enzymatic determination of glucose with the H202-13- method (24) and for the determination of phosphate n-it'li the 123IP.1 met'hod ( 2 5 ) . Relative errors and standard deviations )!-ere about 1% for glucose determinations a t the gg/n-il level and were less bhan 1% for phosphate determinations a t the pg/ml level. Modified Analog Integration System
Complex chemical systems often exhibit rate curves that require the rate measurement, to be made on a segment of the curve where the absolut'e value of the input signal to the rate meter is high. The measurement of AV (Figure 4c) thus requires the determination of a small difference between two relatively large values, which can result in a decrease in the accuracy of the rate measurement. A similar situation occurs when successive measurements are made along the same rate curve, even with a simple chemical system, since the absolute value of the input signal to
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HEATH
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.
Figure 6. Block diagram of digital integration system for rate determinations
1 TRANSDUGER ANDSlGNAL ,MODIFIER \
Table IV. Automatic Reaction-Rate Results for Phosphate P, r g / m l
mV
Re1 SD,a %
0.50 1.00 1.25 1.50 1.75 2.00 2.25 2.50b 3.00 3.50 4.00 4.50 5.00
54 103 125 149 172 200 226 251 302 352 401 451 501
1.01 0.79 -0.85 1.05 0.97 0.54
a The
0.40 0.42 0.79 0.95 0.67 0.44 0.21
% relative standard deviation be-
tween averages of 10 results. * T h e 2.50 pg/ml standard was used t o set t h e readout for direct digital concentration data.
84A
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the rate meter increases after each measurement cycle. This cnse is demonstrated in Figures 5a and 5b, which are oscilloscope traces of the integrator input and output voltages obtained ivlien three successive measurements were made on n 1.0 V s e c ramp signal. Whereas tlie slope remains constant for each measurement cycle, voltages T71 and Ti? increase with each cycle. S o t only is the accuracy of tlie measurement of AV decreased, but, t'lie integrator might limit' during tlie measurenient. To prevent this problem, the basic arialog integration system lias been redesigned (26) to provide a curve-followiiig suppression circuit which a l l o ~ s more accurate measurement of rate over a wider dynamic range. -\lso, provision has been made for automatic recycling of the measurement operations
Av readout,
t
FROM GHEMlCAL REAC T l ON
for multiple integrations along successive portions of the same rate curve and printout of the result after each measurement cycle. .\t tlie beginning of each measurement cycle, tlie input signal is sampled and held by the curve-following suppression circuit. This voltage is directed to tlie integration and subtraction circuit during the measurement cycle in such a way that it is always in opposition to the signal which is integrated. Thus, for each measurenient period the initial input signal to the integrator is essentially zero, as shoivn in Figure 5c. The effect of the curve-folloiving suppression circuit on the readout for the multiple measurement operation can readily be seen from the oscilloscope trace for the integrator output voltage shown in Figure 5d. Now, regardless of the cycle, AT7 is a relatively large difference measured between two voltage levels which do not vary from cycle t o cycle. The result.. tabulated in Table V demonstrate tlie improvement in accuracy and precision obtained during multiple measurement operation by use of the modified analog system.
I
Digital Integration System
Rather than performing the integration and subtraction operations with a n analog circuit, these operations can be implemented with a digital circuit ( 2 7 ) . A simplified block diagram of such a rate iiistrumeiit is s h o ~ nin Figure 6. The input signal to tlie rate meter is applied to a voltage-to-frequency converter which generates a pulse train. The frequency of tlie pulses is proportional to tlie input voltage. The pulse train is then applied to a n up-down counter d i i c h is set to count up to AT1 during the first integration period arid set to count down in magnitude by S?during tlie second integration period. The resultant output of tlie up-down counter, the number S ? - S I , is decoded and read out. The relative standard deviations reported for measurement of synthetic slopes with this rate meter were about 0.27,. These results are similar to those obtained with the basic analog rate meter described above. The primary reason is that although tlie integration and subtraction operations are
Table V. Automatic Reaction-Rate Results for Successive Integrations with and Without Offset Suppression Measurement condition"
First cycle with curve-following suppression First cycle without curve-following suppression Sixth cycle with curve-following suppression Sixth cycle without curve-following suppression
Re1 error,c
Re I SD,
%
%
581.1
...
0.05
582.5
0.24
0.21
582.1
0.17
0.15
566.9
2.45
0.39
Digital readout,b mV
integration t i m e = 100 rnsec: integrator t i m e constant = 0 I n p u t rate = 1 V/sec; 20 rnsec. b Averages for 10 results. c Based on first-cycle measurement with curvefollowing suppression.
ANALYTICAL CHEMISTRY, VOL. 44, NO. 12, OCTOBER 1972
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.
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READER SERVICE CARD
ANALYTICAL CHEMISTRY, VOL. 44, NO. 12, OCTOBER 1972
85A
Instrumentation
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Minicomputer Implementation of Integration Technique
The integration and subtraction operations can be performed by use of a minicomputer-interfaced software approach (25). The block diagram shown in Figure 7 illustrates that, as with the digital integration system, the analog signal obtained from the signal modifier system is fed to a n analog-to-digital (-1-to-D) converter, and the digitized data are operated on. The output of the A-to-D converter is first fed to a 86A
2
22 42
TA ’
DATA
performed with a digital circuit, the limiting operation is the analog-todigital conversion. The same operation is the limiting factor when using the basic analog rate meter in conjunction with a digital readout device. The accuracy cannot be expected to change appreciably, since the primary difference is when the analog-to-digital operation is carried out. A truly digital rate measuring system would be one in which the monitored signal was obtained in a digital form. For example, in making spectrophotometric rate measurements, rather than using currentto-voltage and voltage-to-frequency converters, a pulse amplifier and signal level discriminator could be used in conjunction with the photomultiplier tube as a photon counting system.
21 41
POlNTS
register where a preselected number of A-to-D conversions (e.g., 128) is added and averaged. Each resultant average is stored as a data point. The data points are obtained a t a constant rate; the numbers o n the data point axes in Figure i are proportional with time. hfter a predetermined number of data points has accumulated (e.g., 121), the first S points (e.g., 0-20) are integrated, and this value is subtracted from the computed integral of the neyt S points (e.g., 20-40). The resulting number, Ro, which is proportional to the integral of the change in signal, is stored. The integration and subtraction processes are repeated by use of the same number of data points, with the S segments now taken bet\veen 1 to 21 and 21 to 41, and the resulting number, R1, is stored. This process of sliding the integration intervals pointby-point is repeated until all data points have been included in the integration process. The stored R values are averaged, and the resultant value is printed out on the teletype. This approach exhibits escellent noise rejection, since the data have undergone smoothing three times (23). Variable-Time Methods for ReactionRate Determinations
The variable-time method has been applied by use of one fundamental ap-
ANALYTICAL CHEMISTRY, VOL. 44, NO. 12, OCTOBER 1972
proach-the two-point technique. The method has been implemented in a variety of ways (16, 10-12, 28-30) by use of both hardware and softxare systems. The primary operations required in applying the variable-time method are measurement of the time interval, At, required for a preselected change in monitored parameter, AP, and computation of the reciprocal of the time interval, 1/At. Analog Circuits for Variable-Time Method
l n analog variable-time rate meter has been developed by Stehl et al. (28) which measures the interval At, computes log At, and subsequently differentiates the espressioii to provide a readout voltage directly proportional to l/At. -4similar approach has been reported by James and Pardue (29). h computation circuit integrates the signal from a voltage interval detector for the measurement interval At. .1 log computation and exponential computation are then performed sequentially on the output of the integrator. The resultant of these operations is a voltage level which is directly proportional to l / A t . The rate meter, when evaluated by a colorimetric procedure for the determination of alkaline phosphatase activity,
Instrumentation
FROM GHEMlCA L REACTION
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Figure 8. Block diagram of hybrid anavariable@-digital time system for rate determinations
1
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nr gave results with relat'ive standard deviations of about' 1%. Hybrid Analog-Digital Systems
One hybrid analog-digital approach which implements these operations (SO) is outlined in Figure 8. The signal from the reaction monitor, after it is transduced and appropriately modified, is directed to a comparator and logic network. This circuit starts and stops a timing circuit !Then the transduced signal reaches two preselected voltage levels. The timing circuit' is a part of a time interval-to-voltage converter which produces a n output voltage, V , proportional to the input time int,erval, A t . This output voltage is directed to a voltage-to-frequency converter which generates a pulse train whose frequency is proportional to the input)voltage and, therefore, to the time interval, At. By measuring the period of this pulse train, the reciprocal of the time interval, l l a t , is obtained. The digital meter used as the readout device is operated in its multiple period averaging mode to provide a more accurate result. The output of the digital average period meter can then be fed to a computer or printed out. K h e n this measurement system was evaluated by using synthetic slopes which varied from 5 mV/sec to 2 V/eec, the rate measurements were obtained n-ith rela-
VOLTAGE TO FREQUENCY GONVE RTER
fovzk'n
t
t tive standard deviations which ranged from 0.15 to 1.8%. A similar, yet complementary, hardware approach has been taken (20) by substituting a time-to-period converter for the time interval-to-voltage and voltage-to-frequency converters. I n this system, it is the frequency which is measuEed to obtain a readout proportional to the reciprocal of the time interval, 1,'At. The time-to-period conversion is performed by using a n entirely digital approach. The system also provides the possibility of noise averaging a t either end of the measurement interval. This digital system yielded results with about 0.2% relative standard deviation when synthetic slopes were used as the input signals. Software System
X software approach based on the variable-time method for on-line processing of reaction-rate data has been used (21). The transduced signal is directed to a n analog-to-digital converter (aDC), and the digitized signal is entered in computer memory where it is stored as a function of time. The acquired data points are processed in two parallel branches by the variable-time method and by a pseudofixed time method which provides some versatility to the system. To ensure that the available range of the ADC is
not exceeded, a preliminary rate measurement is made by the variable-time program. The time constant of a digital filter is also adjusted on the basis of such preliminary rate measurements, allowing a n analysis of rates over a wide dynamic range without making any changes in hardware. After the preliminary computations and adjustments are completed, a series of independent measurements are made for each reaction. The results are averaged and printed out along with individual results on the teletype. Data conversions (e.g., transmittance to absorbance) are performed either prior to or after the data acquisition process. The computer-assisted system has been evaluated by using two chemical systems. The results for the determination of alkaline phosphatase activity indicate a relative standard deviation of about O.3y0 and linearity (rate vs. activity) of 1%. Results from the catalytic determination of osmium by use of the Ce(1V)-=ls(III) reaction system gave relative standard deviations of about 5r0 a t 10-1131. Computer implementation of Derivative Method
-4 software approach based on the derivative method has been reported by Willis et al. (62). -4s with the software system used for variable-time measure-
ANALYTICAL CHEMISTRY, VOL. 44, NO. 12, OCTOBER 1972
*
87A
Instrumentation
ments, the transduced signal is directed to a n analog-to-digital converter (ADC), and the digitized signal is entered in computer memory where it is stored as a function of time. The acquired data points are averaged, and then a smoothing routine and computation of slope based on the methods of Savitsky and Golay (31) are performed. Several kinetic parameters are either displayed on an oscilloscope, or a permanent record is obtained via teletype printout or punched paper tape. These parameters include the rate curve (-4vs. t ) and the apparent first-order rate constant (d In ( A , - A / & ) . General Considerations of Methods for Rate Determinations
-4lthough the rate measurement methods in principle differ, there is a definite relationship between them. For example, in considering the integration technique (Figure 9a), the measurement interval is composed of the two fixed At integration periods. If the integration periods are made small, and the measurement interval is held constant, the result is the two-point fixed-time technique (Figure 9b). If the measurement interval is made small (Figure Sc), then the result is the derivative method. This conclusion can be supported by the fact that any practical instrunientation system used requires that the measurement be made over a finite time interval as determined by the time constant of the instrument. However, in practice it becomes necessary to consider the derivative method apart from the two-point methods, because the resultant of t'he measurement is a n indication of the instantaneous rate, and the instrumental implementations are different. Considerations in Selecting Method of Rate Computation
The factors that influelice the noise immunity of the techniques are integration time, measurement interval, noise amplitude, and the period of the noise. For the integration technique, the most
important factor is the relationship between the integration time and the period of the noise. A good noise average, and consequently a high noise immunity, are obtained when the integration time is much larger than the period of the noise. Giren the same rate curve, no noise averaging, is accomplished in applying the two-point fixed-time technique in the absence of software averaging and smoothing methods. I n this case, the factor that has the greatest influence on the measurement error is the amplitude of the noise. I n the case of the derivative method, the measurement interval is much smaller than the period of most noise. Consequently, the measured value is the summation of the slope of the monitored signal and the slope of the noise. The slope of the noise can be either positive or negative and can be much larger than the slope of the monitored signal during bhe measurement interval. If the rate inetruments are to be applied to the determination of chemical reaction rates, then additional factors must be considered (27, 18). This is a result' of the fact that the rate information is obtained on a segment' of the reaction-rate curve which does not coincide Ivith time t = 0. HoIyever, the ratio A W A t must be related to the tangent a t time t = 0 by use of the basic kinetic equations relating concentratioii and time. Hence, the relationship between initial concentration and the measured ratio A P i At depends upon the reaction order, reaction mechanism, the kinetic role of sought-for species (substrate, catalyst), and the transfer function of the transducer used t'o monitor the reaction. I n discussing these factors, the kinetic equation in rvhich they are included must be considered, and an explicit relationship between initial coilcentration Ca and the remaining parameters must be obtained. Generally, the relatioiiship between the initial coilcentration of sought-for-species Co, the measured parameter P ,and time t can be expressed asCo = j ( P ) ! p ( t ) .
I n choosing between the variabletime and fixed-time methods, one has to consider the complexity of functions j ( P ) and +(t). If f ( P ) can be expressed in the simple form, f ( P ) = a . AP, where a is a proportionality coefficient, then a fixed-time method is advantageous since $ ( t ) is kept constant, and the readout value, A P , is linearly related to CO. The variable-time method is preferred when $ ( t ) can be reduced to +(t) = b . At, where b is a proportionality coefficient. I n this case the numerator, j ( P ) , is held constant, and Co is proportional to l / A t . When the two functions are complex, the problem is solved by making several approsimations unt'il one of the functions can be expressed as a first-degree equation. For first or pseudo first-order kinetics, the fixed-time method yields the most accurate results (I?', 18). For catalytic and enzymatic methods, a variable-time procedure could be the most appropriate method, although the rate equations must be individually considered for each case. Conclusions
The widespread commercial introduction of inexpensive general-purpose minicomputers and hardiyare interfaces is greatly changing the design concepts of chemical reaction-rate instruments. Certainly, the specific-purpose electronic rate instruments will be useful and more economical in certain specialized applications. However, when the minicomputer is a part of a versatile reaction-rate system and is interfaced and programmed for control of instrument settings; sample and reagent handling operations; and smoothing, manipulation, and coilversion of rate data, then it can provide economy and elegaiice in design of the total package and in operation. A block diagram of a complete system is shown in this month's Report for Alnalytical Chemists (13). .After the rate curve is acquired under optimum computer-controlled conditions and is stored in memory, then the data can be smoothed, converted, and read out in any desired
/4 Figure 9. Illustration of relationship between (a) integration technique, (b) twopoint fixed-time technique, and (c) derivative method
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ANALYTICAL CHEMISTRY, VOL. 44, NO. 12, OCTOBER 1972
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form. The absorbance and derivative curves of Figure l b were obtained with such a minicomputer-interfaced system
(6)’ References
(1) H. V. Malmstadt, Rec. Chem. Progr.,
17. l(1956). (2) H. V. Malmstadt a n d , E . R. Fett, Anal. Chem., 26, 1348 (1954). (3) H. V. Malmstadt and C. B. Roberts, ibid., 27, 741 (1955). (4)D. T. Williams and R. N. Hager, Jr., Appl. Opt., 9, 1597 (1970). f5) F: Goiurn, D. Paine, and L. Zoeller, ibid., 11, 93 (1972). (6) R. Timmer and H. V. Malmstadt, submitted for publication, Anal. Chem. (1972). (7) N. Snelleman, T. C. Rains, K. W. Yee, H. D. Cook, and 0. Menis, Anal. Chem., 42, 394 (1970). (8) H. V. Malmstadt, C. Delaney, and E. Cordos, Critical Rev. Anal. Chem., 2, 559 (1972). (9) H. V. Nalmstadt and G. D. Hicks, Anal. Chem., 32, 394 (1960). (10) H. V. Malmstadt and H. L. Pardue, ibid.,33, 1040 (1961). (11) H. V. Nalmstadt and T. P. Hadjiioanou, ibid., 34, 455 (1962). (12) H. V. Malmstadt and H. L. Pardue, CZin. Chem., 8 , 606 (1962). (13) H. V. Malmstadt, C. J. Delaney, and E. A. Cordos, Anal. Chem., 44, 26A (1972). (14 H. V. Malmstadt and C. G. Enke, “Digital Electronics for Scientists,” w. A. Benjamin, Xenlo Park, Calif., 1969. (15) H. V. Malmstadt and S. R. Crouch, J. Chem. Educ., 43, 340 (1966). (16) H. L. Pardue, C. S. Frings, and C. J. Delaney, Anal. Chem., 37, 1426 (1965). (17) H. L. Pardue in “Advances in Analytical Chemistry and Instrumentation,” Vol 7, pp 141-207, C. N. Reilley and F. W. AlcLafferty, Eds., Interscience, New York, N.Y., 1969. (18) J. D. Ingle, Jr., and S. R. Crouch, Anal. Chem., 43, 697 (1971). (19) E. A. Cordos, S. R. Crouch, and H. V. Malmstadt, ibzd., 40, 1812 (1968). (20) R. A. Parker, H. L. Pardue, and B. G. Willis, ibid., 42, 56 (1970). (21) G. E. James and H. L. Pardue, ibid., 41, 1618 (1969). (22) B. G. Willis, J. A. Bittekofer, H. Ls Pardue, and D. W. Margerum, ibid., 421 1340 (1970). (23) E. S. Iracki and H: V. Malmstadt, submitted for publication, Anal. Chem. (1972). (24) H. V. Malmstadt and S. I. Hadjiioanou, Anal. Chem., 34, 452 i 1962). A. C . Javier, S. R. Crouch, and H. V. Malmstadt, ibid., 41, 239 (1969). (26) C. J. Delaney and H. V. Malmstadt, unpublished report, 1971. (27) J . D. Ingle, Jr.,?and S. R. Crouch, Anal. Chern., 42, 1Oaa (1970). (28) R. H. Stehl, D. W. &largerum, and J. J. Latterell, ibid., 39, 1346 (1967). (29) G. E. James and H. L. Pardue, ibid., 40, 796 (1968). (30) S. R. Crouch, ibid., 41, 880 (1969). (31) A. Savitzky and AI. J. E. Golay, ibid., 36, 1627 (1964).
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