404
Anal. Chem. 1900, 60, 484-488
Dr. E. Bourret for the Cu-doped GaAs samples. Registry No. Cu, 7440-50-8; GaAs, 1303-00-0.
LITERATURE CITED >
dv)-
2-
w -
f nw:
a: w -
I--
+ a0 v)
-
4
6
8
10
1
ENERGY i k e W
Flgure 3. X-ray spectra comparing the scattering distribution from GaAs relative to a hydrocarbon matrix. The two specira are normallzed
to an equivalent cross section for elastic and Inelastic Compton scattering. to arise in measurements involving tunable monochromatic synchrotron radiation where sensitivity enhancement for specific elements and discrimination against major matrix comnonents are desired.
ACKNOWLEDGMENT The authors thank W. Semles for his assistance in designing the experimental apparatus used in these investigations and
Sparks, C. J., Jr. I n Synchrotron Radletbn Research; Winlck, H., Donlach, S., Eds.; Plenum: New York, 1980 Chapter 14. Gilfrich, J. V.; Skelton, E. F.; Quadrl, S. B.; Kirkiand, J. P.; Nagel, D.J. Anal. Chern. 1083, 55, 187. Jones, K. W.; Gordon, B. M.; Hanson, A. L.; Hastings, J. 8.; Howeels, M. R.; Kraner, H. W. Nuel. Instrum. Methods Phys. Res., Sect. B 1884, 231, 225. Jaklevic, J. M.; Giauque, R. D.;Thompson, A. C. Nucl. Instrum. Methods Phys. Res ., Sect. B W85, 303. Giauque, R . D.;Jaklevic, J. M.; Thompson, A. C. Anal. Chern. 1986, 58, 940. Iida, A.; Goshi, Y. I n Advances in X-Ray Analysis; Barren, C. S., Predecki, P. K., Eds.; Plenum: New York, 1985; Vol. 28, pp 61-68. Sparks, C. J., Jr.; Raman, S.; Riccl. E.; Gentry, R. V.; Krause, M. 0. Phys. Rev. Lett. 1878, 40, 507. Goulding, F. S.; Jaklevic, J. M. Annu. Rev. Nucl. Sci. 1873, Vol. 2 3 ,
45.
Sparks, C. J. Phys. Rev. Lett. 1074, 3 3 , 262. . . Elsenberaer, P.; Platzman. P. M.; Wlnick. H. Phys. Rev. Lett. 1976, 3 6 , 623.(11) Eisenberger, P.; Platzman, P. M.; Wlnick, H. Phys. Rev. B : Solid State 1976, 13, 2377. (,2) Bannett, Y. B.; Freund 1. Phys. Rev. Lett. 1875, 34, 372.
RECEIVED for review August 20,1987. Accepted October 20, 1987. This work was S U P P O ~in part by the Director’s Office of Energy Research, Office of Health and Environmental Research, U.S. Department of Energy under Contract No. DE-AC03-76SF00098, and a t SSRL, partially by the Office of Basic Energy Sciences, U.S.DOE, and partially by the National Institutes of Health, Biotechnology Resource Program, Division of Research Resources.
Diffusional Microtitration: A Technique for Analyzing Ultramicrosamples Miklds Gratzl’ Institute for General and Analytical Chemistry, Technical University of Budapest, Gell6rt t6r 4, Budapest, H - 11 11 Hungary
THratlon Is one of the most preclse and rellable methods of quantitative analyds. Mlnknwn sample volumes that can be determined by exlstlng mechankal tltratlon devices are about 100 pL. I n this work ultramlcrosamples of micrditer and submicrollter range have been titrated by a method using only diffusion and caplllary forces, thus elimlnatlng the need of mechanical delivery and stlrrlng systems. The device for diffusional mlcrotltratlon Is simple and easy to operate. Acld/base, complexometrk, and precipltate tltrat4ons have been carried out partly with color lndlcators and visual detection and partly with an instrumental (potentiometric) detitrated was 50 pmol tector. The smaHest Bmown of sliver Ions In 0.1-pL droplets. Titration accuracy was Ilmlted by the accuracy of the sample size. I n addltlon to the llnear stationary dlffuslon used In this work, employment of equllibrlwn lnltlal condltlons and nonlinear reagent dellvery tor mlcrotltration Is also possible.
-
Present address: the University of Utah, Department of Materials Science and Engineering, Center for Sensor Technology, Salt Lake City, UT 84112.
One of the most precise and reliable methods of quantitative analysis of composite samples is an appropriate titration. With this technique, small portions of a reagent are added to a sample solution, while a selective and fast chemical reaction takes place with the component to be determined. If the reagent amount, being chemically equivalent to that of the analyzed component, can be indicated by any means of detection in the course of this process, then the analysis-tough in an indirect way-has been completed. With titrations absolute amounts instead of concentrations can be determined, in a remarkably reliable way. It is often necessary to titrate very small sample volumes, e.g. when monitoring organic microsyntheses ( l ) analyzing , binary (organic/water) microsamples (2),determining purity (3) or acid/base content ( 4 ) of precious radioactive preparations, analyzing fluids of small organisms (5), and determining proteins or nucleic acids involved in biological reactions that can be isolated often in only milligram amounts or less (61, etc. Equipment available for performing microtitrations (1-8) consists of more or less miniature versions of the ordinary devices used in macroscale titrations. Thus, very small vol-
0003-2700/88/0380-0484$01.50/00 1988 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 60,
umetric flasks, micropipets, piston burets with micrometer screws, and titration vessels with microstirrers are used. Stirring and reagent delivery are always done mechanically. A special stirring technique, e.g., makes use of swinging the titration vessel (5))while a recently developed delivery system (9) employs a vibrating glass needle to produce and “shoot” miniature titrant droplets into the sample. By use of these devices, many important determinations can be carried out in the micro range. They exhibit, however, certain drawbacks: the equipment is generally difficult to handle and time-consuming, as inner surfaces of burets and titration flasks must be recoated by silicone before each titration (to minimize capillary forces); the titration vessel may be filled with sample only by centrifugation (due to surface tension which represents an important constraint a t the sample sizes in question); mercury (pushing the reagent out of buret) can cause false delivery from the heat of hands; etc. The equipment is, in addition, rather expensive, as precise mechanical and electronic elements are required (7-9). Minimum volumes of solution that can be titrated by the existing devices are seldom less than 100 p L after the original sample has been prepared for reagent delivery (1-9). As an exception, an ion selective field effect transistor (ISFET) based coulometric setup (10)is reported to perform acid/base titrations using about l p L of solution. The sample, however, needs to be prepared by addition of a background electrolyte and a redox couple. Since it is hardly possible to accurately mix solutions of such a small size, the original sample volume must be significantly larger than that finally titrated. As the dimensions of the coulometric cell cannot be precisely measured, the method is not absolute (in contrast to coulometric titrations in the macro scale). Its applicability is also limited by the complexity of the device. Titration of a large variety of ultramicrosamplesin a simple way, would be useful-in addition to examples already mentioned (1-6)-e.g. in geochemistry (to determine preconcentrated rare elements) or even for space research (in analysis of Moon rocks). In this report a very simple solution to the problem of minimizing titration sample sizes down to the microliter and submicroliter range is presented. Experimental results of precipitate, complexometric, and acid/base titrations in this scale, with both visual and instrumental end point detection (the latter with automatic data handling) are also demonstrated. THEORETICAL CONSIDERATIONS Difficulties Arising at Titration of Ultramicrosamples. A substantial decrease of sample (and instrument) size relative to that used by available microtitration devices would be virtually impossible using the designs of present equipment (1-9), for the following reasons: (1)The mechanical method of reagent delivery and sample/reagent mixing would require a dramatic increase in technical difficulty and cost of mechanical components (coulometric titrations, a t which these problems partly vanish, are not treated here). (2) Due to the smaller volumes to be analyzed,surface tension and capillary forces would cause additional problems; “sticking” of sample to vessel walls and to buret tip would occur (even in spite of silicone coating), rendering quantitative weighing and complete stirring difficult. (3) In order for very small reagent additions to be made when titrating small samples, titrations would have to be performed with the reagent delivery tube immersed in the sample droplet. Then, with very small samples and slow rates of reagent delivery, diffusion between the sample droplet and buret tube becomes significant, leading to errors in quantity of titrant added. (This last problem is not encountered when minute reagent droplets are “shot” into the sample (9).)
NO. 5, MARCH 1,
1988
485
9
I
10
8
Flgure 1. Schematic diagram of the device for diffusional microtitration: (1) sample droplet; (2) reagent reservoir; (3) dmusional membrane made of agar gel; (4) membrane holder made of Teflon plate (thickness 0.4 mm); (5) Plexiglas body; (6) Plexiglas stopper, holding a Ag/AgCI reference electrode, when other kind of reference must be used, the stopper can be replaced by a liquld junction; (7) glass cap: (8) dlstilled water saturating the space above sample by water vapor or, in case of acid/base titrations, 0.1 M KOH solution for absorbing CO,; (9) bore of a diameter of about 0.5 mm; (10) Inner saturated space above sample droplet; (I) Indicator electrode of silver wire (optional); (R) Ag/AgCI reference electrode of the second kind (optional).
Diffusional Microtitration. In this work a solution to the above-mentioned difficulties is suggested, by using an approach different from the efforts summarized above. Namely, a significant decrease of titration scale can be achieved by employing capillary forces and diffusion to the advantage of the titration. Accordingly, mechanical (convective) stirring can be replaced by mixing by diffusion, as long as the sample droplet is sufficiently small and the titration itself is carried out slowly enough. Reagent delivery can also be accomplished by diffusion, presuming the former conditions (small sample, slow titration) are met. Capillary forces can be used for king the sample droplet to the diffusive reagent source if both are hydrophilic, thus assuring a reproducible cross sectional area for diffusive reagent delivery. Surface tension may separate the sample from the holder of the diffusive reagent source if the holder itself is hydrophobic. Cohesion forces of the same character may hold the droplet to be titrated together, thus rendering a titration vessel unnecessary. Finally, surface tension also minimizes free surface area of the sample droplet, thus minimizing evaporation, and diminishing also the effect of eventual undesired gas absorption (e.g. carbon dioxide in case of acid/base titrations), if the same conditions (sample droplet small enough) are assured. EXPERIMENTAL SECTION Apparatus. A simple device able to carry out diffusional microtitrations is presented in Figure 1. The sample droplet (1) is titrated by an appropriate reagent (2) diffusing through a membrane prepared of hydrophilic agar gel (3), surrounded by a membrane holder (4)made of a hydrophobic Teflon plate (thus decreasing dispersion of sample away from the membrane),the white color of which, in addition, facilitates visual detection of equivalence points with color indicators. The surface area of the membrane is completely covered by the sample. The integrity of the agar membrane and its adhesion to the holder are good
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 5, MARCH 1, 1988
enough at the required small sizes. Between the membrane holder and Plexiglas body (5)application of any adhesive is unnecessary, as both are hydrophobic. Thus,changing the diffusion membrane together with the membrane holder is easy to carry out. As the sample surface t o volume ratio increases, evaporation and gas absorption will take less time. The conical glass cap (7) decreases the rate of these processes, the inner space being saturated with distilled water (8) or-in the case of acid/base titrations-by a potassium hydroxide solution (8) t o extract carbon dioxide out of the air. A hole in the top (9) equilibrates pressure between the inner compartment (10) and the outside air. The membrane (3) is prepared as follows: the Teflon holder (diameter 12 mm, thickness 0.4 mm) with a hole in its center (4) is immersed for a short time (1-2 s) into a hot, fluid agar gel (lo%, made of purified agar-agar, Mallinkrodt). When the holder is taken out and cooled down, excess gel is cut off with a sharp scalpel; thus, diffusion membranes with sufficiently planar boundaries can be obtained (with diameters of 0.3-1.5 mm, depending on sample size). Hundreds of successive titrations may be performed on a single agar membrane over a period of several weeks. For sample application, microsyringes of appropriate volumes (Hamilton) were used. For indicating precipitate titrations with silver ions, silver wires of thicknesses of 0.1 or 0.04 mm (99.9%, Alfa) were used as indicator electrodes (I in Figure l),while a Ag/AgCl electrode was used for a reference electrode (R in Figure 1). To monitor cell potential, a digital pH-meter (Radelkis OP 208/1) connected via a BCD interface (HP 82941A) to a personal computer (HP 85A) was used. On-line control and evaluation (and all other calculations) were also done by the same computer. Reagents. Analytical grade reagents (Mallinkrodt,Merck and banal) were used. Visual indicators were prepared approximately 5-10 times more concentrated than as described in standard texts (8) to ensure their visibility in the small samples titrated here. Procedure. As stationary diffusion was employed in this work for reagent delivery (see next section), it was critical to apply the new sample fast and also reproduciblyjust after the equivalence point of the former sample has been detected. For this purpose, sample changes (sucking up of old sample by paper tissue, and applying the new one) were accomplished within about 2 s. RESULTS AND DISCUSSION Equilibrium Initial Conditions o r Stationary Diffusion. Diffusional microtitration of each sample must always be started at identical initial conditions, to ensure reproducibility of the diffusive reagent delivery. By use of the simple device displayed in Figure 1,this requirement can be fulfiied basically in two different ways: either one waits before each titration until reagent concentration in the diffusion membrane reestablishes homogeneity a t its original level, or each sample change must be done quickly, just after the equivalence point of titration of the previous sample has been detected. Results obtained by the first method, in which each titration starts a t the same equilibrium initial condition, will be reported elsewhere. In this paper the second method, using identical stationary conditions, will be presented. In this method the following further conditions must be fulfilled: (1)Diffusion of reagent has to be significantly slower within the diffusion membrane than in the sample or reagent reservoir. Thus, it can be ensured that mixing of reagent with sample, being not larger than several microliters, can be considered to be nearly homogeneous during the entire titration, due to diffusion processes within the droplet (see Figure 2A). (2) The concentration of reagent must be much higher than that of the sample. This prevents unreacted sample from escaping through the membrane; thus, sample loss via diffusion can be practically avoided. The first and second conditions together then assure a nearly linear concentration profile along the coordinate of
A
C
L X
0
B
Figure 2. Schematic diagram dlsplaying the geometry of diffusion in the microtitration device: (A) main directions of diffusive transport during titration (thin arrows, reagent; thick arrows, component titrated); (e) concentration profiles of reagent during titration at different time instants, (1) sample droplet; (2) reagent reservoir: (3) diffusion membrane; c,, reagent concentration in the reservoir; I,, thickness of the diffusion membrane; (e) profile corresponding to equilibrium initial conditions; (s)profile corresponding to stationary conditions (used in this work). The numbers at the other profiles correspond to different time instants (for profiles 1-8: t = 0.03, 0.15, 0.3, 0.6, 1.2, 2.5, 5, and 10 s, respectively). A numeric solution of Fick's second law for the given problem has been used for approximate simulation of reagent concentration (with I , = 0.4 mm and D , = 1.96 X cm2/s).
membrane thickness during operation at stationary conditions, as shown in Figure 2B. (3) Similar to ordinary macroscale titrations, the rate of the titration reaction, and that of equivalence point (or end point) detection, must be fast with respect to the time scale of the titration process. When all of these conditions are satisfied, the reagent amount, R, delivered by diffusion into the sample droplet during time, t, can be expressed in the case of stationary conditions (i.e. a t a constant concentration gradient within the membrane), according to Fick's first law, as follows:
R(t)=
D,dm2m, 41,
t
where D, is the diffusion coefficient of the reagent in the membrane, c, is the reagent concentration in the reservoir, and d, and 1, are diameter and thickness of the membrane. Acid/Base a n d Complexometric Titrations. Use of Calibration. Nitric acid can be properly titrated in ordinary amounts (not less than 1 Mmol in 5-10 mL of sample) with alkaline reagents, using a bromocresol purple/bromothymol blue acid/base color indicator mixture for detection of the equivalence point (8). In this work, with an agar gel diffusion membrane of diameter 1.0 mm and thickness of 0.4 mm an ultramicro amount of 150 nmol in droplets of 3 pL could be easily titrated. After the titration end point has been attained,
ANALYTICAL CHEMISTRY, VOL. 60, NO. 5, MARCH 1, 1988
487
Table I. Statistical Characteristics of Diffusional Microtitration
reagent titration type
concn,
detection
acid/base visual (BCP-BTB)* complexo- visual metric
precipi-
tate
(murexide)
kind
mol/L
KOH
0.01 0.1 EDTA 0.01 (pH ~ 1 0 ) 0.1 0.1 EDTA 0.01 (PH =5)
visual (xylenol orange) NaCl instrumental (potentiomet-
0.5 0.5
membrane diameter: mm
kind
2 x 10-4 (1-5) X lo-* NiSO, 2.5 X lo4 ( P -10) ~ (1-5)x 10-3 (1-5)x 10-3
0.3 1 1 1 1.5 1.1
HN03
0.5 0.5
AgN03
nmol
linearitv of calibration graph, r2
av std dev of det.'s nmol
1-10 5 2-a 3 7 2-6
0.2-2 50-250 0.5-2 3-15 7-35 2-6
0.993 0.991 0.996 0.999 0.995 0.973
f0.03 f4 10.02 f0.9 f3 10.2
2-5 2
10-25 2-10
0.985 0.992
10.3 f0.2
sample concn, volume, mol/L PL
Pb(N03)2 (pH ~ 2 ) ~ 5X (1-5)x 10-3
amount,
ric)d
Membrane thickness was always about 0.4 mm. Bromocresol purple-bromothymol blue indicator mixture. methylenetetramine for buffering. d Automatic control and evaluation were done by an HP85A personal computer. a fast sample change ensured a linear concentration profile of reagent within the membrane during the entire determination (apart from several initial titrations during which stationary diffusion developed). In this way, a potassium hydroxide reagent of 0.1 M provided end point times, teq,of 6.5 f 0.2 min. A similar time range (5.8 min) was calculated by eq 1where D, = 2.20 X loa cm2/s (referring to aqueous solution at room temperature (11))which indicates that the theoretical considerations outlined above are reasonable. However, as most diffusion coefficients in possible membrane materials are not very accurately known or not known at all (as in the actual case), instead of direct use of eq 1 for determinations, a calibration method is preferable. As a consequence of applying stationary diffusion in this work, a linear calibration curve can be established for the given experimental conditions and then used for determinations. In this way systematic errors of sample application, and those of visual end point indication are also minimized. With this calibration technique and visual detection, different sample ranges of nitric acid were determined. In addition to these acidlbase titrations, lead(I1) ions with xylenol orange, and nickel@) ions with murexide color indicators were successfully analyzed by using EDTA reagent in complexometric titrations (12) (Table I). Fulfillment of the Conditions for Proper Operation. Reagent diffusion was significantly slower within the membrane than in the reagent reservoir or sample droplet, due to the following facts: (1)An agar gel more concentrated than usual has been used, causing a corresponding decrease of the diffusion coefficients. In the case of nitric acid determination detailed above, e.g., a value of D, = 1.96 X cm2/s can be calculated with eq 1 from experimental t , versus the value of 2.20 X measured in water (11). (2) The geometry of diffusion within reagent reservoir and sample droplet was semispherical while within the membrane it was planar, the former enabling much larger material throughput than the latter, at the given small sizes (13) (see Figure 2A). Most remaining necessary conditions for proper operation, outlined above, were also fulfilled. The color of the diffusion membrane during determination corresponded to that of the overtitrated sample, indicating that only products of the titration reaction could leave the sample droplet via the membrane. According to visual observations with color indicators, homogeneity of the sample/reagent mixture can generally be ensured only to a certain extent. If calibration was not used, this lack of perfect homogeneity could cause end point errors which do not occur in macroscale titrations. Reproducibility of transport within the sample droplet, which
Contained hexa-
E ImVI
400
0
10
20
t [SI
Figure 3. Titration of a 0.1-pL droplet of 0.1 M AgNO, by 1 M NaCI. Membrane dimensions were d , = 0.3 mm and I , = 0.4 mm approximately. Dots represent individual digitized electrode potential data, while full line displays smoothed second derivative, calculated by the computer. Time of equivalence point, t,,, means time elapsed from titration start till appearance of the inflection point of potential-time curve.
is sufficient for proper operation when calibration is used, can always be perfectly realized. Precipitate Titrations with Instrumental End Point Detection. Titration of silver nitrate droplets by chloride ions has also been tested. Smoothed second derivatives of titration curves were calculated in real time, and the chemical equivalence point was determined by the zero point of the derivative (Figure 3). In other cases the time instant was indicated and alarmed by the computer when the end point potential has been reached. Statistics of some of these latter results are also included in Table I, while details of automatic data handling will be discussed elsewhere. It must be mentioned that eq 1 is not relevant for entire titration curves (such as that displayed by Figure 3) because the conditionsfor stationary diffusion cannot be ensured after the equivalence point has been reached and no sample change has been carried out. Thus, in such cases, even the starting conditions of a new titration are closer to those of the initial equilibrium reagent profile than to that of a stationary one. Since, as in a true titration, end point time should depend only on the absolute amount of the substance present in the droplet, calibrations can be done in at least two ways: as a function of either sample concentration (at fixed volume) or sample volume (at fixed concentration). Thus, the agreement of the results presented using both of these calibration methods lends support to the validity of the considerations outlined above. Statistics of the results (Table I) suggest that random errors are mostly due to syringes used for sample application (except
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 5, MARCH 1, 1988
when visual detection was involved where standard deviations are larger). For example, even 260 pmol of silver nitrate in droplets of 0.6 p L could be titrated with a standard deviation of about *7%, which corresponds to the approximate 0.03-0.05 p L uncertainty of the one microliter Hamilton syringe used here for sample application. Systematic errors (end point bias) are negligible even when visual detection is involved, due to calibration. Note, finally, the good quality of the titration curves of even smaller volumes (Figure 3). The smallest amount which could be successfully titrated was 50 pmol of silver ion in 0.1-pL droplets. A silver wire of a diameter of 0.04 mm was used for detection in this case.
CONCLUSIONS The reagent amounts that can be delivered by diffusion within several minutes into sample droplets of microliter and submicroliter range are sufficient to titrate the components to be determined in typical ultramicrosamples. Thus, the problem of minimizing size of titration assembly, and decreasing sample volumes to the mentioned scale, could be solved without any mechanical element, by use of capillary forces and diffusion: the same phenomena that impede the efforts of decreasing the scale of ordinary mechanical titrations below about 100 pL. As a consequenceof using diffusion-a highly reproducible natural transport process-for reagent delivery, diffusional microtitration might be as accurate and precise in the ultramicro range as ordinary titrations in macroscale. The limiting factor of accuracy in the experiments presented was due to the uncertainty i n sample size through addition by microsyringes. Ensuring identical initial conditions before each determination is, however, necessary for proper operation. In this report the use of a steady linear concentration profile within the diffusion membrane has been presented. In contrast to coulometric titrations where only a limited set of reagents can be generated (often accompanied by undesirable side products as gas bubbles), with diffusional microtitration most determinations of classical titrimetry are feasible in the microliter and submicroliter range. The as-
sembly introduced is very simple and inexpensive and provides fully automatic (diffusive) reagent delivery. It can be readily adjusted to the sample ranges to be analyzed by appropriate modifications of reagent concentration and membrane dimensions. Automatic real time evaluation with algorithmic end point indication may also be set up by an on-line personal computer, enabling serial microanalyses. Finally, it is interesting to note that-contrary to most other analytical techniques-diffusional microtitration can be characterized by a higher sample size limit of determination rather than a lower one. This maximum sample size is in the 10-pL range, since efficient mixing of the sample/reagent mixture must be ensured by diffusion. In the lower direction, accuracy of the sample size is practically the only limiting factor of further minimization.
ACKNOWLEDGMENT Valuable remarks of L. Ernyei and J. FucskB as well as technical assistance of J. Sziics are gratefully acknowledged. LITERATURE CITED (1) Belcher, R. Submicromehcds of Chemlcal Analysis; American Elsevler: New York, 1966. (2) Rogers, D. W.; Lllllan, D.; Chawla, I.D. Microchim. Acta 1988, 4 , 722-728. (3) Knobloch. V.; Mudrova, B. Microchim. Acta 1970, 2 , 235-239. (4) Ceska, M.; GrossmSller, F.; Slodln, A. V. I n t . J . A.m . i . Radiat. Isot. 1971, 22, 311-314. (5) Walshby, J. R. Anal. Chem. 1973, 4 5 , 2445-2446. (6) Spokane, A. 8.; Brill, R. V.; Gill, S. J. Anal. Biochem. 1980, 709, 449-453. (7) Tolg, G. Ukramicro fiemental Analysis ; Wiley Interscience: New York, 1970. (8) Kolthoff, I. M., Elvlng, Ph. J., Eds. Treatise on Analytical Chemistry; Why-Interscience: New York, 1975; Part I, Vol. 11; Part 11, Vol. 11. (9) Steel, A. W.; Hieftje, G. M. Anal. Chem. 1984, 56, 2884-2888. (IO) Van der Schoot, B.; Bergveld, P. Sens. Actuators 1985, 8 , 11-22. (11) Dobos, D. Nectrocbemicai Tables: Muszaki Konyvkladt? Budapest, 1965; p 99. (12) Flaschka, H. EDTA Titrations; Pergamon Press: London, 1959. (13) Carslaw, H.S.;Jaeger, J. C. Conduction of Heat in Solids; Clarendon Press: Oxford, 1959.
RECEIVED for review June 17, 1987. Accepted November 5 , 1987.
