Anal. Chem. 2000, 72, 4713-4720
Continuous On-Line True Titrations by Feedback-Based Flow Ratiometry. The Principle of Compensating Errors Hideji Tanaka,† Purnendu K. Dasgupta,* and Jimin Huang
Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas 79409-1061
We introduce a new concept for continuous on-line titrations based on feedback-controlled flow ratiometry and the principle of compensating errors. The system has been thoroughly tested by applying it to acid-base neutralization titrations with indicator-based end point detection. In a typical case, the total flow (FT, consisting of the sample and the titrant flows) is held constant while the titrant (e.g., a standard base containing an indicator) flow FB varies linearly in response to a controller output voltage. The sample (e.g., an acidic solution to be titrated) flow FA constitutes the makeup and thus also varies (FA ) FT - FB). The status of the indicator color in the mixed stream is monitored by an optical detector and used either for governing the controller output or for interpreting the results of the titration. Three methods (PID based control, fixed triangular wave control, and feedback-based triangular wave control implemented on a PC) were examined. In the last and the most successful approach, the titrant flow is initially ramped upward linearly. At the instant a change in the color is sensed by the detector, the titrant flow rate FH is higher than the true equivalence flow rate FE because of the lag time between the first compositional change and its detection. The sensing of the change in color causes the system output to immediately reverse its ramp direction such that the titrant flow now goes down linearly at the same rate. At the instant a change in color, in the opposite direction this time, is again sensed, the titrant flow rate FL is lower than FE by exactly the same amount that FH was higher than FE. This principle of compensating errors (FE ) (FH + FL)/2) allows true titrations with excellent reproducibility and speed (0.6% RSD at 3 s/titration and 0.2% RSD at 10 s/titration) and titrant volume consumption as little as 12 µL/titration and solves an old conceptual problem in flow based titrations. Titrimetry is one of the few classical analytical methods still in wide use, for the determination of both major and minor components (the latter most notably for the measurement of water by Karl Fisher titrations). Titrations are not limited to solutions. Indeed, the origin of titrimetry has been traced back to Geoffroy in 1729; he evaluated the quality of vinegar by noting the quantity * Corresponding author: (e-mail)
[email protected]. † Permanent address: Faculty of Pharmaceutical Sciences, Tokushima University, Shomachi, Tokushima 770-8505, Japan. 10.1021/ac000598t CCC: $19.00 Published on Web 08/25/2000
© 2000 American Chemical Society
of solid K2CO3 that could be added before effervescence ceased.1 The foundations of volumetric analysis, as it is presently known, were laid by Gay-Lussac between 1824 and 1832.1,2 Mohr is especially credited for popularizing volumetric analysis, through the 1855 publication of his classic treatise on titrimetry.3 Compared to competing techniques, titrimetry exhibits excellent precision, convenience, and affordability. But it is generally confined to a batch operation with slow throughput and requires significant amounts of sample and titrant. As long as the titrant concentration is exactly known and volumetric ware needs no further calibration, true titrations require no calibration curve. This can be important in situations where it is difficult to prepare a pure standard solution of the analyte being titrated. In routine analytical laboratories, automated titrators, often coupled to robotic workstations, have largely replaced manual titration. The hardware and algorithms for end point detection and the consequent feedback to control the rate of titrant addition have become increasingly sophisticated. In the research laboratory, titrators have been greatly miniaturized and automated4,5 and titrations have been demonstrated down to the femtoliter scale.6 The intrinsic batchwise nature of this measurement has, however, remained unaltered in such efforts. Blaedel, a pioneer in electronics, instrumentation, and automation, translated the paradigm of titrations from the volume to the flow domain more than three decades ago. The basic arrangement, shown in Figure 1a, involves two pumps. Typically, one pumps the sample at a constant rate while the other, a feedback based servo-controlled pump, delivers the titrant. The two streams are merged, and some desired property of the mixed stream (electrode potential in the original studies7,8) is measured by a suitable detector. In response to the detector output, the titrant pump changes speed and attempts to maintain the detector output at some constant level. This general system has been copied, reinvented, reconfigured, and (over)simplified numerous times since its original description. Linear titration plots have been (1) Ihde, A. J. The Development of Modern Chemistry; Harper and Row: New York, 1964; pp 288-289. (2) Tilden, W. A. Famous Chemists: The Men and Their Work; G. Routledge, and Sons: New York, 1930; pp 119-126. (3) Mohr, K. F. Lehrbuch der Chemish-Analytischen Titriemethode; F. Vieweg: Braunschweig, 1855. (4) Sweileh, J. A.; Dasgupta, P. K. Anal. Chim. Acta 1988, 214, 107-119. (5) Bartrolii, J.; Alerm, L. Anal. Lett. 1995, 28, 1483-1497. (6) Gratzl, M.; Yi, C. Anal. Chem. 1993, 65, 2085-2088. (7) Blaedel, W. J.; Laessig, R. H. Anal. Chem. 1964, 36, 1617. (8) Blaedel, W. J.; Laessig, R. H. Anal. Chem. 1965, 37, 332.
Analytical Chemistry, Vol. 72, No. 19, October 1, 2000 4713
Figure 1. (a) Blaedel-Laessig configuration; (b) present configuration; (c) arrangement to generate a programmed change in sample concentration. Flow diagrams: T, titrant; S, sample; W, waste; D, detector; MC, mixing coil; MR, mixing reactor; PS, dc power supply; LED, light-emitting diode; PD, photodiode; Amp, amplifier; Cont, controller.
shown to be possible9 using the McCallum-Midgely approach.10 Some three decades after the original work, manual rather than feedback-based control of the titrant pump is still being advocated, with finite pauses at discrete mixing ratios to allow the detector output to reach the equilibrium value reflective of that mixing ratio. This generates a complete multipoint titration curve.11 Such seemingly retrograde movements highlight the fact that the same problem faced by batch mode autotitrators reappears in the flow domain, perhaps in a more aggravated fashion. In the batch titration mode, following incremental titrant addition, a finite mixing and detection time is necessary before a detector reading is meaningful. In flow titrations, aside from the above factors, the time for the fluid transport to the detector is additionally necessary after a flow rate change. Altogether, these contribute to the overall lag time between the equivalence composition (or some other desired set point) being first reached at the sample-titrant confluence point and this actually being registered by the detector. Mixing is a more problematic issue in flow systems because it is difficult to incorporate active mixing (magnetic stirrers, etc.) without incurring additional mixing volume that increases the lag time further. Detector response time can also be factor. Optical detector response is rapid but potentiometric detection, especially with a glass electrode, can be slow. Because of the sigmoidal nature of most titration curves (near the end point, the change in the detector output is most rapid), the existence of a finite lag time is particularly important. Autotitrators solve this problem by reducing titrant delivery speed in proportion to the rate of change of the detector output, even performing discrete small incremental additions near the end point for highest accuracy. There is no barrier to implementing the same methodology in the flow domain, (9) Catalyud, J. M.; Falco´, P. C.; Albert, R. M. Analyst 1987, 112, 1063-1066. (10) McCallum, C.; Midgley, D. Anal. Chem. 1976, 48, 1232-1235. (11) Katsumata, H.; Teshima, N.; Kurihara, M.; Kawashima, T. Talanta 1999, 48, 135-142.
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to perform measurements with discrete changes in the flow ratio while waiting at each step for a steady-state reading11 and with progressively smaller changes as the end point is approached. Either this approach or Blaedel’s original concept (hold the system at the end point in a steady state under feedback control) has the virtue that they are true titrations. As long as the pumps are calibrated (in much the same way that volumetric ware used in conventional titrations have implicit calibrations), no further system calibration with samples of known concentration is needed. They have the considerable disadvantages of a substantial time (>5 min) and sample/titrant consumption per titration. For applications as on-line process titrators, the significant time involved can cause a real problem in streams with fast-changing compositions. Many ingenious and innovative approaches have emerged. The detector output has been used as the index to perform a halfinterval search in the sample/titrant ratio to attain the equivalence point with excellent precision in under 3 min/titration.12 With air bubbles at each end of a trial mixture to isolate it from the next mixture (monosegmented flow), the method could be used for even slow detectors without a major increase in the time spent.13 In such a binary approach to attain the end point, a maximum of n compositional trials are necessary for a desired accuracy of 1 part in 2n. Yarnitzky et al.14 described an approach with two variable-speed pumps, which alternate their roles as sample/titrant pumps with the help of two three-way valves. A titration is performed one way and then the pumps change their role to perform the titration again, and an average of the two compensates for the lag time. Further, the temporal flow profile for the pumps is exponential, rather than linear, to provide better accuracy. There are also some appealing techniques even though they are not true titrations in that calibration is required. Eichler15 was the first to advocate the merits of concentration gradients in the titrant, rather than a flow gradient;16 such gradients can be generated by exponential mixing15,17 or by electrochemical generation of the titrant.18 In more recent years, flow injection titrations have been introduced.19 In this approach, the width of an injected response peak at some set point, often the equivalence point, is measured and is linearly proportional to the logarithm of the injected concentration. Although gradient dilution chambers are advocated, they are not essential.20,21 While logarithmic response permits a large dynamic range,22 precision is limited. To get 1% accuracy, analysis time can be several minutes per sample and changes in sample viscosity can compromise accuracy. Instead, (12) Korn, M.; Gouveia, L. F. B. P.; de Oliveira, E.; Reis, B. F. Anal. Chim. Acta 1995, 313, 177-184. (13) Martelli, P. B.; Reis, B. F.; Korn, M.; Costa Lima, J. L. F. Anal. Chim. Acta 1999, 387, 165-173. (14) Yarnitzky, Klein, N.; Cohen, O. Talanta 1993, 40, 1937-1941. (15) Eichler, D. L. Advances in Automated Analysis, Technicon International Congress 1969; Vol. II. Industrial Analysis. Mediad Inc.: White Plains, NY, 1970; pp 51-59. (16) Lopez Garcia, I.; Vinas, P.; Campillo, N. Cordoba, M. Anal. Chim. Acta 1995, 308, 67-76. (17) Fleet, B.; Ho, A. Y. W. Anal. Chem. 1974, 46, 9-11. (18) Nagy, G.; Fehe´r, Zs.; To´th, K.; Pungor, E. Anal. Chim. Acta 1977, 91, 8796, 97-106. (19) Ramsing, A. U.; Ruzicka, J.; Hansen, E. H. Anal. Chim. Acta 1983, 129, 1-17. (20) Rhee, J. S.; Dasgupta, P. K. Mikrochim. Acta 1985, III, 49-64. (21) Rhee, J. S.; Dasgupta, P. K. Mikrochim. Acta 1985, III, 107-122. (22) Tyson, J. F. Analyst 1987, 112, 523-526.
using the configuration of Figure 1a, the time to reach a titration end point, as deduced from an indicator color change occurring as a result of a linear change in titrant flow rate, is linearly related to the sample concentration.23 This may be a more attractive approach in terms of time and precision. Further, for the purist, it is more likely to qualify as a titration.24 In the present study, we propose a new paradigm for flow titrations by feedback-based flow ratiometry in which the delay between the sample-titrant confluence point and the detector is made constant by modifying the Blaedel-Laessig configuration. The error due to lag time is continuously compensated for by averaging rapid backward and forward titrations. EXPERIMENTAL SECTION Flow System. The basic flow diagram of the present study is shown in Figure 1b. In the original Blaedel-Laessig configuration, variations in pump flow rates result in variations in total flow rate and a fixed hardware arrangement thus changes the lag time. The simple change to the configuration of Figure 1b results in a constant total flow rate FT where the titrant pump is adjusted to have a maximum flow rate equal to (or just below) FT. As such, the sample flow rate FA represents the difference between FT and the titrant flow rate FB. Since the mixed stream can go to 100% sample to 100% titrant, it is possible in principle to titrate a sample of any concentration with a titrant of any concentration. Obviously, to obtain good precision and accuracy, some judicious choice of titrant concentration will be more appropriate based on the sample concentration. In some experiments, we deliberately varied the sample concentration as a function of time using the arrangement shown in Figure 1c. A ramp generator (Tektronix FG 504), via a voltage-controlled pump, was used to vary the flow of water used to dilute a constant flow of a sample stream. Part of this mixed stream was aspirated by the sample aspiration line shown in Figure 1b while the rest was allowed to go to waste. Peristaltic pumps (10 stainless steel rollers, variable speed, Gilson Minipuls 2 or Rainin Rabbit-Plus/Dynamax) were used for the pumping needs. The pumps marked var in Figure 1 were externally voltage-controlled with 0-5-V dc analog input. For good mixing without significant residence time, single-bead string reactors (MR; 25 mm long, 0.81 mm i.d., average bead diameter 0.5 mm) were used on both the inlet and outlet of the final pump (Figure 1b). Residence time for mixing is less critical in Figure 1c, and a mixing coil MC (600 mm × 0.66 mm i.d.) sufficed for this purpose. Except as stated, 0.51-mm-i.d. Pharmed pump tubes were used throughout with the total flow rate FT held constant at 1.9 mL/min. The indicator was premixed in the titrant, and the indicator absorption was measured by a simple on-tube lightemitting diode (LED)-photodiode (PD)-based laboratory-made detector. The detector consisted of a 1/4-28 threaded male-male union with a center partition made for chromatography (P/N 39056, Dionex) with an LED emitting at 605 nm (P/N HAA5566X, Stanley Electric, Tokyo, Japan) on one side and a silicon photodiode (PD, P/N BPW 34, Siemens) on the other. A transparent tubing of FEP Teflon (0.8-mm i.d., 1.2-mm o.d.) passed between the LED and the PD in a perpendicular fashion (see Figure 7, ref 25). The PD output was converted to voltage by a current amplifier (23) Marcos, J.; Rı´os, A.; Valca´rcel, M. Anal. Chim. Acta 1992, 261, 489-494, 495-503. (24) Pardue, H. L.; Fields, B. Anal. Chim. Acta 1981, 124, 39-63, 65-79.
(Amp, model 427, Keithley, typically set at its minimum response time of 10 µs), and this signal, linearly related to optical transmittance, was processed by a controller (Cont). In the present study, two controllers were examined: a commercially available PIDtype process controller (Omega CN76160), and a PC-based system with a control algorithm described later. As an intermediate step, a function generator (Tektronix FG 504) was also used to ramp the flow of the titrant pump up and down, in a blind fashion. System output was acquired on a PC using a 12-bit data acquisition card (DAS-1601, Keithley/Metrabyte) using software written inhouse. In the case of the fully PC-based system, a PCMCIA card (PCM-DAS16D/12AO 12-bit A/D, D/A, Computerboards Inc., Middleboro, MA) housed in a Pentium II-class notebook computer (Latitude, Dell Computer Corp.) was used with software written in Visual BASIC. Reagents. Commercially available reagents of analytical reagent grade (indicator grade for indicators) were used without further purification. PRINCIPLES In the present configuration, at the equivalence point the following equation will hold:
CA(FT - FE) ) CBFE
(1)
where CA and CB represent the concentration of sample and titrant and FT is the invariant total flow rate of the system. The titrant flow rate FB is designated FE at the equivalence point. In the examples reported in this paper, generally the sample is an acid and the titrant is a standard base solution (already containing an indicator). The value of FB is linearly related to the output voltage from a controller (VC). Thus, eq 1 is rewritten as follows:
CA (FT - kVE) ) CBkVE
(2)
where k is a constant of proportionality, kVE is equal to FE, and VE is the value of VC at the equivalence point. Therefore, 1/VE is proportional to that of 1/CA in the configuration shown in Figure 1b; all other terms in the following equation being constant:
(VE)-1 ) (kCB/FT)(CA)-1 + k/FT
(3)
RESULTS AND DISCUSSION PID Control. Proportional integral derivative (PID) controllers are widely used for control of temperature, pressure, and other process parameters. Many controllers of this type are commercially available inexpensively. Compared to simple analog circuitry-based control utilized for the purpose more than 35 years ago,7 they provide substantially more built-in intelligence. Except for the configurational change (Figure 1a w Figure 1b), we therefore wanted to test the applicability of a PID controller in the Blaedel-Laessig concept. We chose a challenging task. The controller was asked to maintain the system at such a titrant flow rate that the mixed stream is exactly neutralized in a strong acidstrong base titration (titrant, 100 mM NaOH containing 0.2 mM bromthymol blue (BTB); sample, 50-200 mM HCl). There is no buffer capacity at the equivalence point except for that provided by the indicator, and a very slight deviation causes an indicator Analytical Chemistry, Vol. 72, No. 19, October 1, 2000
4715
Figure 2. Controller output voltage from a PID controller vs time. HCl concentration: (a) 50, (b) 100, (c) 150, and (d) 200 mM. PID values: 1, P ) 25, I ) 0.1, D ) OFF; 2, P ) 100, I ) 0.1, D ) 0.01.
color change. The detector output was premeasured with the mixed stream distinctly acidic (indicator completely yellow) and distinctly basic (indicator completely blue), and the controller was assigned the task of maintaining the detector output at the midpoint of these values. Under these conditions, if steady-state control can be achieved, VC will directly reflect analyte concentration (FB is linearly related to VC). Self-tuning abilities are provided in most modern microprocessor-based PID controllers to find optimum values of P (gain), I (bias), and D (time constant) values to maintain good control. For the present system, where the detector output is practically bistable with a sharp transition between two states, self-tuning was ineffective to maintain control. The controller output oscillated between its highest and lowest permitted values. Manual settings of the PID values led to better results; a typical run is shown in Figure 2. Some degree of control is possible with high P and low D values. With an increase of P value, oscillation became less significant, but it took a longer time for the system to stabilize at the set point. No significant improvements were observed by using greater time constants for the detector to reduce detector noise or by reducing lag time by placing the detector on the aspiration side of the final pump (Figure 1b). The degree of oscillation for the same sample concentration can be unpredictable (see, e.g., two runs for 2c, Figure 2). More importantly, if we take the midpoint of each oscillation as VE, the relative standard deviations would be significant. We concluded that PID control is not particularly well suited for maintaining a system at the equivalence point where the rate of change is very steep near the set point. Principle of Compensating Errors. We accept the premise that it is difficult for a control system to maintain the system at precise equivalence in a system that represents a steep titration curve. We can, nevertheless, still consider the merit of the information available when the titrant flow is scanned in the vicinity of the equivalence point, without an effort to keep the mixed effluent at equivalence. Consider a system in which the titrant flow is being ramped upward linearly. At the instant a change in the color is sensed by the detector, the titrant flow rate FH is higher than the true equivalence flow rate FE because of the lag time. Calibration of an entire system can include implicitly taking into account this lag time.23 FE can also be expressed as
FE ) kVE ) FH - rtlag
(4)
where r is the ramp rate (dFB/dt) and tlag, the lag time, is a combination of the physical transit time from the confluence point 4716
Analytical Chemistry, Vol. 72, No. 19, October 1, 2000
Figure 3. Fixed triangular wave control method: detector output Dout and controller output VC as a function of time.
and the detector response time. There may be nothing wrong with calibrating the systems with standards for subsequent assays except that it ceases to be a true titration; calibrations beyond flow rates are required. This may be a nontrivial issue; if a system is calibrated for use with one set point, it may have to be calibrated again for use with another set point, since for example, the precise response times of a glass electrode does vary depending on the pH regime. Consider now that the titrant flow FB is decreased from some high initial value downward at the same ramp rate r. At the instant a change in color, in the opposite direction this time, is again sensed, the titrant flow rate FL is lower than FE, in a mirror image fashion of the previous situation:
FE ) kVE ) FL + rtlag
(5)
We can then calculate FE from eqs 4 and 5 without the use of system calibrations and without knowing the specific values of r and tlag, thus permitting true titrations:
FE ) (FH + FL)/2
(6)
We refer to this as the principle of compensating errors. To monitor compositional changes in a stream accurately, it must be possible to carry out each forward and backward titration at a much faster rate than the rate at which the stream is changing composition. This concept was first tested with a function generator producing a triangular wave to control FB. Actual data output (titrant: as in Figure 2, sample 100 mM HCl) at a ramp rate of 100 mV s-1 is shown in Figure 3. In response to the triangular wave VC controlling the titrant flow, the detector output Dout basically executes a rectangular wave pattern. The yellow form of the indicator has practically no absorption at the monitoring wavelength. Thus, at the high end, Dout is flat. On the other hand, even after the indicator turns blue,, further increase of FB brings still more indicator in the system (since the indicator is incorporated in the titrant) and at the low end, Dout executes a shallow V, with the bottom of the V being approximately temporally coincident with the apex of VC, the difference being tlag. The effect of the scan rate r was examined at constant scan limits (1.0-4.5 V, 20-90% of maximum pump rate) using the same NaOH-HCl system as in the previous paragraph. Pump flow rates were calibrated so that the correct value of FE (or VE) will be known a priori. At low to moderate values of r (0.0302, 0.0607,
Figure 4. Fixed triangular wave control method: Detector output Dout vs controller output VC. HCl concentration: (a) 50 and (b) 100 mM.
and 0.1203 V/s), VE values obtained were in excellent agreement with the expected true value and exhibited good precision (RSD e1%). However, at high scan rates (0.24 V/s), the experimental VE values were inaccurate. Under these conditions and with the tlag in our specific system, VC reached its limits before a transition in Dout was detected. Since the number of titrations that can be conducted per unit time increases with increasing r, further experiments were conducted with r ) 0.100 V/s. HCl solutions (50, 75, 125, 150, and 200 mM) were measured as the sample with fixed scan limits of 1.0-5.0 V. Plotting the resulting data (1/VE vs 1/CHCl) exhibited good linearity (linear r2 0.9976). A plot of Dout as a function of VC is shown in Figure 4 for 10 titrations each of (a) 50 and (b) 100 mM HCl. It is interesting to note that although it is possible to visualize the conventional titration plot in Figure 4 by focusing on the ascending or descending halves individually, the overall image is most reminiscent of a cyclic voltammogram, the excitation and response functions being similar to the latter experiment. VE essentially represents the abscissa value corresponding to the center of mass of the parallelogram. Compared to extant literature methods, this approach is quite competitive (80 s/cycle, e1% precision), but Figure 4 also shows very clearly that the scheme results in large amounts of time being spent in a useless manner. For example, in Figure 4a, the system unnecessarily scans in the VC ) 3-5-V range and similarly it spends unnecessary time in the titration of Figure 4b in the VC ) 1-3-V range. Compensating Error Method Coupled to Feedback-Based Control of Titrant Flow. A more efficient method than the one above involves reversing of the ramp direction of the titrant pump once the equivalence point is crossed. The pump was endowed with such intelligence by using PC-based software that senses the detector output and thus changes the ramp direction as soon as some preset threshold in Dout is crossed. (Other standard end point functions based on the first or second derivatives of Dout can also be implemented as desired.) Both the principle and the results are illustrated in Figure 5 by using the same titrant and sample as in Figure 3. In contrast to the previous method, VC scan limits are not fixed but are in effect VH and VL and thus vary with the concentration of the analyte. As soon as Dout indicates an alkaline mixture (VH has been reached), VC is ramped downward. When Dout indicates an acidic mixture (VL has been reached), VC is immediately ramped upward. The resulting VC waveform has a constant frequency of (4tlag)-1 and displays an amplitude of 2rtlag.
Figure 5. Feedback-based triangular wave control method: detector output Dout vs controller output VC.
Note that these properties of the VC waveform are independent of the analyte concentration. These properties suggest the possibilities of diagnosing or compensating for flow inconstancies that have not been exploited in the present work. It is the dc component of VC that is related to the analyte concentration, and this dc bias moves up or down as the concentration of the analyte increases or decreases. One noteworthy item is that the detector set point (Dout,hi) at which VC begins a downward ramp does not have to be the same as the detector set point (Dout,lo) at which the ramp goes back upward. This is of practical importance since all real signals contain some noise. When these points are set identically, false triggering (premature ramp reversal in either direction) can and will occur. To avoid such problems, Dout,hi should differ from Dout,lo by at least 2 times the p-p detector noise. Because the transition is very steep, it makes no real difference in the ultimate results in VE. This method was then optimized, with the results as follows. Throughput Rate, Detector Position, and Mixing Means. In the above approach, it may be arguable as to what constitutes a complete titration. For a discrete sample, it is obvious that at least one VH and VL value is necessary to compute VE. However, when the system is being applied continuously to a flowing stream, the VE values will be computed by averaging the most recent VH or VL value with the immediately preceding VL or VH value. Since our intent is to apply the system in continuous-flow applications, we take this more liberal view of titration time requirement. Since the period of the VC waveform is directly dependent on tlag, it is essential to reduce tlag to improve throughput, but tlag cannot be reduced indefinitely without affecting the completeness of mixing and thus increasing detector noise and decreasing system reliability. These interrelated issues are therefore of critical importance. For these reasons, we extensively examined the effects of positioning the detector upstream or downstream of the final pump and of the type and the size of the reactor (knotted tubing, singlebead string reactor, and a porous frit-tee26) at various scan rates and sample (HCl) concentrations. One representative set of results is presented in Table 1. It was possible to reduce titration time to as little as 3.2 s with a small penalty to the precision in VE (RSD 0.56%) by locating the detector upstream of P1 pump. This loss of precision (although acceptable for many purposes) is attributable to increased detector noise since the active mixing provided by the pump tubing is no longer available. For the same reason (to (25) Dasgupta, P. K.; Bellamy, H. S.; Liu, H.; Lopez, J. L.; Loree, E. L.; Morris, K.; Petersen, K.; Mir, K. A. Talanta 1993, 40, 53-74. (26) Cassidy, R. M.; Elchuk, S.; Dasgupta, P. K. Anal. Chem. 1987, 59, 85-90.
Analytical Chemistry, Vol. 72, No. 19, October 1, 2000
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Table 1. Effect of Detector Position and Reactor Type Da
D
none
KTc
detector position reactor
reactor length, cm nae titration time,f s 10.4 % RSD of VE (n ) 20) 0.32
D KT
D KT
Ub
D two
SBSRd
15.0 30.0 60.0 2 × 2.5 12.6 15.5 20.1 10.2 0.46 0.40 0.35 0.22
U
SBSR SBSR 2.5 3.2 0.56
10.0 4.6 0.76
a Downstream of final pump. bUpstream of final pump. cKnotted tubing, 0.81 mm i.d. dSingle-bead string reactor. e Not applicable. f Titrant, 100 mM NaOH containing 0.2 mM BTB; sample, 100 mM HCl; r ) 0.100 V/s.
Table 2. Effect of Scan Ratea scan rate, V/s
VH
SDVHb
VL
SDVLb
VE
0.01 0.025 0.05 0.075 0.1 0.15 0.2
3.49 3.52 3.68 3.80 3.94 4.16 4.39
0.01 0.02 0.02 0.01 0.01 0.03 0.02
3.35 3.24 3.14 3.03 2.92 2.70 2.48
0.02 0.02 0.01 0.01 0.01 0.02 0.03
3.42 3.38 3.41 3.41 3.43 3.43 3.44
SDVEb RSDVE, % 0.01 0.01 0.01 0.01 0.01 0.02 0.02
0.40 0.43 0.35 0.26 0.23 0.48 0.44
titration time, s 14.3 11.6 10.8 10.35 10.2 9.8 9.6
aWherever applicable, the units are volts. b Standard deviations of preceding parameters.
improve pump-induced mixing), we chose a combination of high pump rotation rates and small-bore pump tubing, rather than largebore tubing and lower rotation rates, to be in our desired flow rate regime. A porous frit (pore size 90-130 µm)-tee, basically in the same design as a screen-tee reactor,25 proved to be incompatible with peristaltic pumps because of its substantial resistance to flow. We also attempted externally induced active mixing by connecting a microsized unbalanced motor (intended for use with vibrating pagers and cellular telephones), but the benefits were limited. For further work, a downstream detector position with a 2.5-cm-long SBSR installed on each side of the final pump (Figure 1b) was used for further work. Effect of Scan Rate. The effect of changing r was examined over the 0.010-0.200 V/s range with the same titrant-sample combination used in Table 1. Table 2 provides a summary of the results. In principle, the scan rate should not have a direct influence on the titration time. This is at least approximately true; a 20-fold increase in r results in only a 33% decrease of the titration time. The limited effect of r on the titration time that is observed is a practical consequence of a finite mixing and detection volume. The VE values were virtually constant irrespective of the scan rate, the observed range at different scan rates was within (0.7% of the mean, and this range included the independently determined true value. As may be intuitive, VH and VL increasingly diverge from VE as r increases, in accordance with eqs 4 and 5. A plot of VH vs r should thus have VE as the intercept and a slope equal to tlag. The data in Table 2 yield a value of VE ) 3.430 ( 0.011 V, tlag ) 4.87 ( 0.10 s, and a linear r2 value of 0.9979. Similarly, a plot of VL vs r exhibits a linear r2 value of 0.9979, an extrapolated VE of 3.366 ( 0.010 V and tlag ) 4.47 ( 0.09 s. This range of VE is almost the same as the range of VE values observed by numerical averaging of successive VH and VL values at individual scan rates, lending additional credibility to the theoretical basis of the scheme and the correspondence between theory and experiment. Although the scan rate was not a major factor in determining the titration time for constant or very slowly changing sample 4718 Analytical Chemistry, Vol. 72, No. 19, October 1, 2000
concentrations, a higher scan rate will reach VH or VL values more rapidly when the sample concentration changes quickly. However, a very fast scan may result in VC reaching the scan limits before a transition in Dout is detected (as was noted for function generatorcontrolled operation of the system without feedback, for r g 0.24 V/s, vide supra). In addition, a slower scan may improve the accuracy and precision in determining VE. In looking at the RSD values for VE, the RSD values increase at r > 0.100 V/s and also increase at r < 0.05 V/s. We concluded that the 0.05-0.100 V/s (corresponds to 1-2%/s of the maximum possible pumping rate) is the best range to operate in. The best precision that is observed here, ∼0.25%, is characteristic of the limits of peristaltic pumps for small volumetric aliquots, according to our independent experiments. Some 4 decades ago, Blaedel and Laessig7 reported a precision of 0.11% for peristaltic pumps using larger tubing and apparently significantly larger volumetric aliquots. It is obviously unclear as to what gains, if any, have been made in the intervening years and this highlights once again the need for high-precision liquid pumps that can pump in a voltage-controlled manner from an unconfined reservoir. Effect of Titrant Concentration. For virtually any process application, the sample consumption rate is not particularly important while minimizing the titrant consumption is highly desirable since this decreases the replacement frequency of a reagent that must be carefully made. We therefore examined the utility of more concentrated NaOH solutions as titrant, in conjunction with smaller diameter (0.25 and 0.44 mm) pump tubes in addition to the original 0.51-mm-diameter tube used in all the above experiments. Typical results are presented in Table 3. A combination of low CB and narrow tubing is insufficient to titrate the sample and was not tested. Likewise, a combination of high CB and wider bore tubes is impractical. In general, the absolute standard deviations decreased with an increase in NaOH concentration. However, not surprisingly, the RSD increased with increasing CB because VE decreased more significantly compared to SDVE. Nevertheless, the use of a small-diameter (0.25-mm i.d.) tube allowed the RSDVE for titrations using 500 and 1000 mM NaOH to be kept well below 0.5%. Similar results, not shown here, were obtained for a scan rate of 0.050 V/s. By using 1000 mM NaOH instead of 100 mM NaOH, the titrant consumption could be reduced to 18% of the original value. Note that, in the configuration of Figure 1b, the average volumetric consumption of the titrant (which we assume to be linearly related to FE) is linearly related to (1 + CB/CA)-1 (see eq 1). While large gains are made initially in FB consumption with increasing CB, there are diminishing returns on an absolute scale at higher and higher titrant concentrations. Nevertheless, it is remarkable that with a titrant concentration 25 times that of the sample, it is still possible to perform titrations with a precision only slightly over 1%, at only ∼10 s/titration and consuming 11.7 µL/titration. It is likely that it will be possible to further improve the precision at very high titrant/sample concentration ratios by using even smaller diameter pump tubes; these are commercially available. Continuous Titrations of a Stream of Changing Composition. A process stream in which the reciprocal of analyte (HCl) concentration changed linearly with time was created by the system depicted in Figure 1c. The dilution flow to a constant-flow
Table 3. Effect of Titrant (NaOH) Concentrationa tubing i.d., mm
nominal CB, M
VH
SDVHb
VL
SDVL
VE
SDVE
RSDVE, %
titratiion time, s
0.25 0.25 0.25 0.44 0.44 0.51 0.51 0.51
0.5 1.0 2.5 0.5 1.0 0.1 0.5 1.0
4.65 2.76 1.43 2.02 1.31 3.81 1.57 1.01
0.0148 0.0115 0.0187 0.0095 0.0071 0.0084 0.0130 0.0055
3.62 1.70 0.37 0.96 0.27 2.80 0.58 0.05
0.0170 0.0116 0.0134 0.0084 0.0091 0.0132 0.0112 0.0098
4.14 2.23 0.90 1.49 0.79 3.30 1.07 0.56
0.0112 0.0081 0.0115 0.0063 0.0057 0.0078 0.0085 0.0056
0.27 0.36 1.28 0.42 0.72 0.24 0.79 1.01
10.35 10.55 10.60 10.50 10.40 10.10 9.90 10.15
a
0.1 M HCl is sample, r ) 0.100 V/s. b Standard deviations of preceding parameters. Wherever applicable, the units are volts.
Table 4. Results for Different Acid-Base Neutralization Reactions sample HCl HCl CH3COOH CH3COOH H3PO4 H3PO4 H3PO4 H3PO4 NH3(aq) NH3(aq) a
titranta
indicator
scan rate, V/s
NaOH NaOH NaOH NaOH NaOH NaOH NaOH NaOH HCl HCl
BTBb
0.05 0.1 0.05 0.1 0.05 0.1 0.05 0.1 0.05 0.1
BTB TBc TB BCGd BCG TPe TP BCG BCG
measurement range, mM 50-200 50-200 25-200 25-200 12.5-200 12.5-200 12.5-125 12.5-100 25-200 25-200
no. of individual concns examined 6 6 7 7 9 9 9 9 7 7
linear r2 0.9989 0.9991 0.9994 0.9994 1.0000 0.9996 1.0000 1.0000 0.9999 0.9999
100 mM contg. 200 µM indicator. b Bromothymol Blue. c Thymol Blue. d Bromocresol Green. e Thymolphthalein.
Figure 6. Feedback-based triangular wave control method: continuous change in analyte (HCl) concentration: (a) function generator output FGout and controller output VC as a function of time; (b) reciprocal of controller output voltage VC that gives the equivalence point (1/VE).
stream of an acidic analyte was increased linearly with time by the slow triangular wave output (FGout) of a function generator. With a FGout cycle time of 93.67 min, the minimum and maximum analyte concentration were 50 and 180 mM, respectively. Since the entire range is spanned within a half cycle, this means the analyte concentration varied by almost a factor of 4 in a period of ∼45 min. In our judgment, this degree of change more than adequately represents the maximum change that occurs for a critical and major component in a real process stream. A total of 608 titrations were made within one cycle of the function generator, resulting in an average titration time requirement of
