ANALYTICAL CHEMISTRY Campanile, V. A , . h d l e y , J. H., Peters, E. D., Agaaai, E. J.. and Brooks, F. R., ASIL. CHEY.,23, 1421 (1951). Corwin, A. H., RIicrochemical Section, 94th Meeting, AM.CHEM. SOC., Rochester, N. Y., Sept. 9, 1937. Deinum, H. W.,and Schouten, A , , Anal. Chim. Acta, 4, 286 (1950). Dexheimer, L., 2. ami. Chem., 58, 13 (1919). Dombrowski, A , , Mikrochemie, 28, 125, 136 (1940). Dundy and Stehr, ANAL.CHEM.,23,1408 (1951). Elving, P. J., and McElroy, IT. R., ISD. EXG.CHEM.,ANAL. ED.,13,660 (1941). Fischer, F. o.,- ~ N A L .CHEhf., 21, 827 (1949). . 216, 74 (1933). Friedrich, iZ., 2. p h ~ s i o l Chem., Gouverneur, P., Shell, Amsterdam, private communication. Gouverneur, P., Schreuders, LI. A, and Degens, P. N., Jr., -Anal. Chim. Acta, 5, 293 (1951). Crote, W., and Krekeler, H., Angew. Chem., 46, 106 (1933). Gysel, H., H e h . Chim.Acta, 22, 1088 (1939). Harris, C. C., Smith, D. hI., and Mitchell, J., Jr., ANAL.CHEM., 22, 1297 (1950). Holowchak, H., and Wear, G. E. C., Ibid., 23, 1405 (1951). and Rees. TT. J.. Trans. Ceram. Soc. (Enol.). Hubbard. D. UT.. 28,277 (1929). Huffman, E. 15’. D., ANAL.CHEM.,23, 531 (1951). (23) IIPY,R., and Riley, H. L., J . Chem. Soc. London, 150, 1362 i1948). (24) Iiigram, G., Mikrochemie, 36/37, 690 (1951). ( 2 5 ) himescu, J., and Popescu, B., 2.anal. Chem., 128, 185 (1948). (26) Jones, W. H., AKAL.CHEM.,23,532 (1951). ANAL.CHEM.,19, 925 (1947). . (27) Kimten, W., (28) Ibid.,22,358 (1950). (29) Kirsten, W., Mikrochemie, 34, 149 (1949). (30) Ibid., p. 151. (31) Ibid., 35, 174 (1950). , (32) Ibid., p. 217. (33) Ibid., 36/37, 609 (1951). (34) Kirsten, W., SvenslzKevi. Tidskr., 57, 69 (1945). (35) Ihid,, 58, 265 (1946). (36) Ibid., “Determination of Halogens.” (37) Ibid., “Dumas Determination of Nitrogen.” (38) Kirsten, W., and A41perowicc,I., Mikrochemie, 34, 234 (1952).
(39) Kirsten, IT., and \~-allberg-Olausson,B., ANAL.CHEM.,23, 927 (1951). (40) Kuck, J., Ibi:;, 23, 531 (1951). (41) Lindner, J., Mikromassanalytische Bestimmung des Kohlenstoffes und Wasserstoffes mit grundlegender Behandlung der Fehlerquellen in der Elementaranalyse,” Berlin, Edit Chemie, 1935. (42) Maylott, A. O., and Lewis, J. B., ANAL.CHEM.,22, 1051 (1950). (43) Muller, Ernst, and Willenberg, H. B., J . prakt. C h a . , 99, 34-44 (1919). (44) Siederl, 3. B., and TVithman, B., Mikrochemie, 11,274 (1932). (45) Ogg, C. L., Willits, C. O., and Cooper, F. J., ANAL.CHEM.,20, 83 (1948). (46) Pieters, H. A. J., Chem. Weekblad, 43, 455 (1947). (47) Pregl, F., and Roth, H., “Quantitative organische Mikroanalvse.” Wien. Surineer-Verlae. 1947. (48) Rhead: Th. F. E., a-nd YVheeler, k.V., J . Chem SOC.London, 97, 2178 (1910). (49) Ibid., 99, 1140 (1911). (50) Royer, G. L., Norton, B. R., and Sundberg, 0. E., IWD.EKG. CHEM.,ANAL.ED., 12, 688 (1940). (51) Schoberl, A,, Angew. Chem., 50, 334 (1937). (52) Schutze, M., 2.anal. Chem., 118, 245 (1939). (53) Steyermark, 9 1 , Bass, E., and Littman, B., ANAL.CHEM.,20, 587 (1948). (54) Unteraaucher, J., Ber., 73, 391 (1940). (55) Unterzaucher, J., Chem. Ing. Tech., 22, 39 (1950). (56) Ibid., p. 128. (57) Unteraaucher, J., Mikrochemie, 36/37, 706 (1951). (58) Walton, W.,McCulloch, F. TI’.. and Smith, W. H., J . Reseaich NatatE. Bur. Standard, 40, 493 (1947). (59) White, L. h l . , and Kilpatrick, A I , D., AXIL. CHEM.,22, 1049 (1950). (60) White, L. hI., and Secor, G. E., Ibid., 22, 1047 (1950). (61) Zimmermann, W., 2.anal. Chem., 118,258 (1939). (62) Zinnecke, F., Ibid., 132, 175 (1951). RECEIVED for review M a y 19, 1952. Accepted October 31, 1052. Presented before the Division of Analytical Chemistry, Symposium on Microchemistry, a t t h e 120th Lileeting of t h e . ~ ~ I E R I c A NC H E M I C A L S O C I E T Y , S e w York,
N. Y.
Principles of Highfrequency Titrimetry CHARLES N. REILLEYl AND W. H. MCCURDY, JR. Princeton University, Princeton, N . J .
S T I L recently, the majority of attention concerning highfrequency titrations has centered around instrument development and, as yet, the theory underlying these measurements has not been explained in detail for conducting solutions. Since this work was completed, articles by Hall (29) and by Blaedel and coworkers (10) have shed further light on this subject. T h e present study is the result of a n attempt to phrase in a comprehensive manner the relationship of the high-frequency propertie8 of a cell containing a solution t o the low-frequency conductance of the solution and the dielectric constant of the solvent. It is believed t h a t this explanation will prove advantageous in the clarification, development, and application of high-frequency methods to new situations. The various instruments t h a t have appeared so far may be classified into three main groups depending upon the electrical proprrty measured. These measurements are a function of both the solution and its container. The first group contains instruments whose response depends upon measurements of the high-frequency conductance or loss. Examples of this type measure circuit factors such as plate, grid, or cathode voltages or currents in a tuned circuit where measurements take place a t the resonant frequency (1-5, 12-22, 26, 27, 83-37). Thermal effects ( 2 7 ) have also heen employed to measure loss a t high frequencies. T h e second group consists of those instruments based upon measurement of capacitance changes. Examples of this group are the beat-frequency type (6-9, 28, 41-43) and the frequency discriminator type (Sargent Model V) (39, 40). 1
Present address, University of S o r t h Carolina, Chapel Hill, Y . C.
The third type includes those which give a measurement dependent upon both the capacitance and the high-frequency conductance. Instruments which measure a voltage, or current influenced by a tuned circuit (containing the solution and cell) Tvhich is not operated a t its resonant frequency, are examples of this type (1, 5, 32-37). Thus the mode of instrument operation can convert a group one type response into a group three type response. Some instruments give measurements nearly proportional to admittance and could be classified in this group under those conditions (12-16, 19, 26, 28). Results typical of a group three device can be qualitatively interpreted from a knowledge of the first and second type responses. Some instruments are capable of giving separate readings for capacitance and conductance simultaneously and therefore belong t o both groups one and two (30, 31). The Beckman instrument is capable of operating under any of the three conditions-either measuring the first and second type responses simultaneously or obtaining a mixture response of the third type. APPARATUS
The apparatus employed in this study was chosen because measurements of parallel capacitance and high-frequency conductance can he obtained independently of other circuit components. The instrumentation is essentially identical to t h a t described by Hall and Gibson (31). A General Radio Type 684-A signal generator, a General Radio Type 821-A Twin-T impedance bridge, and a Hallicrafters SX-25 receiver comprise the essential parts. I n addition, a Triumph Illode]-841 oscilloscope was used as a null detector for visual observation of the tone produced by the beat-frequency
a7
V O L U M E 25, NO. 1, I A N U A R Y 1 9 5 3
This study was undertaken to extend and unify the concepts for interpretation of high-frequency titrimetric methods. Instruments used in highfrequency titrimetry are classified into three groups according to the measured electrical property of the cell and its solution. The electrical behavior of the cell and its contents is quantitatively described, using an equivalent circuit analogy. Each factor influencing these electrical properties is studied in detail by comparison of derived equations and experimental data. Effects studied include frequency, geometric factors, such as thickness of cell walls, area, and separation of electrodes, chemical factors, such as degree of ionization, the chemical reaction, ionic concentration, and solvent differences. Transfer plots, obtained from experimental data,
oscillator in the receiver. All low-frequency conductance measurements were made with a Serfass RClI-15 conductivity bridge (Arthur H. Thomas Co.). This instrument was equipped with a dipping-type conductivity cell having a cell constant of 0.473 reciprocal cm. a t 20" C. All cells were constructed in this laboratoiy from 100-ml. borosilicate glass beakers or 50-mm. borosilicate glass tubing. Fine tinned copper wire (0.16 mm.) of 10-em. length was soldered to tin-foil plates, which were then glued to the beaker walls with Duco cement in the desired position. Ten different cells were fabricated having ditTerent wall thickness, plate geometry, plate separation, or plate area, The significant cell types are shown in Figure 1. Exceedingly thin cell walls were obtained by mounting the plates inside the beaker and coating them with Beckman Dessicote 01 airplane dope as shown in the case of beaker D in Figure 1. Several cell holder arrangements were studied to minimize stray conductance losses. The optimum conditions involved suspension of the unshielded beaker above the bridge by its lip from a Lucite sheet l/g inch thick, having a hole of the beaker diameter and of suitable shape to allow proper seating of the cell. Shielding the cell produced only slight changes in the high-frequency conductance readings and so was omitted for simplicity of operation. IVith this arrangement the beaker position was reproducible with rrspect to the bridge plug sockets and no difficulty was experienced in obtaining precise measurements. Although temperature affects the accuracy of the measurements to some extent, no special precautions were taken. EXPERIMEYTAL
Thv experimental procedure vas uniform throughout the progress of this study, as comparisons between various effects were desired. The bridge, xvith cell disconnected, was balanced initially and this value was employed as a reference check point for all SUC(seeding rradings. Any drift in frequency was corrected by rebalancing the bridge a t the initial capacitance and zero highfrequency conductance settings. In all cases the volume of sohtion employed was such that the liquid level was a t least 1 cm. above the cell plates. Studies of high-frequency conductance, G,, or capacitance, C,,
A
B Figure 1.
C
D
Cell Designs
E
are found to agree with the shape expected theoretically. By use of these transfer plots all apparent anomalies in the shape of the highfrequency titration curves are correlated with the shape of the corresponding low-frequency conductance titration curves. Because the titration curve shape is determined quantitatively by the transfer curve, undesirable end-point characteristics may be avoided by proper dilution, addition of foreign electrolyte, change of frequency, or use of a cell with the proper geometrical characteristics. The interpretation of high-frequency titration curves is facilitated by this analogy which focuses attention on underlying variables affecting the results. This viewpoint should prove valuable in extending highfrequency methods to new analytical problems.
versus low-frequency conductivity xere carried out by adding a concentrated electrolyte solution dropwise t o distilled water. After suitable stirring, the low-frequency conductivity was measured by dipping the conductivity cell into the beaker cell. The conductivity cell was then removed, and the Twin-T bridge rebalanced. Values of AC, were obtained by subtracting the capacitance readings from the C, readings for distilled water in the particular cell. I n regions near maximum values of G, and rapidly changing values of AC,, dilute electrolyte was added in order that points could be taken a t more frequent intervals. With cell A1 curves were obtained a t 3 Rfc. using aqueous solutions of aluminum chloride, acetic acid, and potassium chloride and potassium chloride in methanol (Figures 3, A , and 5, A ) and using aqueous potassium chloride solutions (0.0001 t o 1.0 N ) a t 1, 3, 10, 20, and 30 Mc. as shown in Figures 3, B , and 5, B.
A large number of titrations of 0.01 A' hydrochloric acid with 0.01 N sodium hydroxide were performed in which G,, C,, the volume of titrant, and low-frequency conductivity were recorded. From a knowledge of the change in conductivity due to changes in concentration of hydrochloric acid and sodium chloride during a titration with sodium hydroxide it was possible t o select appropriate initial amounts of 0.01 N acid so this conductivity change would occur within a desired range. I n this way the various types of high-frequency conductance and capacitance titration curves were obtained at 3 and 10 Mc. using beaker (A)1. Several of these titration wrves are presented in Figure 4 and Figurr ij and related through a transfer plot to the lowfrequency vonductivity curves obtained for the same titration. Because the inert electrolyte has a significant effect on the shape of the high-frequency titration curves, several additional experiments were undertaken. The conductivities of 0.0001, 0.001, 0.01, 0.10, and 1.0 LV potassium chloride solutions were measured, and thus it as possible to select the proper amount of potassium chloride electrolyte so that the conductivity change produced by titrating a small amount of 0.01 N hydrochloric acid would o ( w r in a desired rpgion. A sample result is illustrated in Figure3 4 and 6. High-frrqucncj- titration cell studies at 3 and 10 Mc. were carried out in a fashion similar to experiments already discussed. High-frequencv conductance and eapacitance measurements were made on the cell empty, filled with distilled water, and filled with mercury or 1.0 AT potassium chloride. I n addition, enough readings w r e taken on potassium chloride solutions of increasing conductivity to locate the point of most rapid change in the capacitance and the maximum value of G , for the given cell. The fa'ctors investigated included cell wall thickness, plate geometry, plate area, and plate separation. -4 summary of results appears in Table I. Experimentation with cells of verr thin walls was limited by the amount of high-frequency conductance that could be halmced
ANALYTICAL CHEMISTRY
88 Table I. Cell
Type
Study of Cell Factors at 3 Mc. for Aqueous Potassium Chloride Solutions
Wall Thickness
Plate Area
Mm. 1,s
Sq. cm. 30.4:12.6
Plate Separation
Cm.
Gp
Peak
pmhos 1.0 50.0 Geometry 3.3 37.0 (A)21.5 9.O:g.O 3.7 34 2 @)I 1.5 9.0:9.0 5.0 31.2 (C)il.5 9.0:9.0 \Tali Thicknrss 1.O:l.O 5.0 2.05 (C)z 1 . 5 1.O:l.O 3.5 34 8 (C)3 0 . 2 5 5.0 232 (D)i 0,075" 1 . O : l . O Plate Area 1.O:l.O 3.0 2.05 4.0:4.0 5 0 11 5 (C)11.5 9.O:Q.O 5.0 31.2 Plate Separation 5.0 11.5 (C)r1.5 4.0:4.0 1.5 8.3 (E)11,5 4.0:4.0 a Plates coated with airplane dope.
(AI1
{${: ?:
KO
Peak pmhos/ cm.
ACp
A CP Peak
ppfd.
ppfd.
160
5.15
2.57
192 203 190
4.10 3.83 3.45
2.05 1.91 1.70
148 273 1420
0 24 3.6 31.5
0.12 1.9 15.7
148 161 190
0.24 1.32 3.43
0.12 0.65 1.70
161 153
1.32 1.05
0.63 0.50
can be represented by a qimple. rquivalent circuit of lumped values in a manner that gives perfect agreement between theory and practice. However, the equivalent circuit represents a logical analogy and gives a good approximation to experimental measurements. The series circuit is the more convenient form to use when a resonant circuit is employed having an inductor in series with the cell composed of the solution and its container. The Clapp oscillator uses a resonant circuit of this type. In practice a variable condenser is placed across the cell instead of in series and this complicates the exact interpretation, although a qualitative picture will still hold. The parallel equivalent circuit is most useful for tuned circuits where the coil is placed in parallel with the ccll, since a t resonance the equivalent capacitative and resistive values can operste independently of each other, as related by the simplified parallel equivalent circuit equations. Equations for the parallel and series equivalent circuits are derived in terms of the fundamental equivalent circuit; the effects of cell design, solution properties, and frequrncy are interpreted in terms of thcw ecju:itions. SIMPLIFIED EQLIVALEYT PARALLEL CIRCUIT
by the Twin-T bridge. In many cases where very thin walls were employed, the G , response was much larger than the range of the bridge, so that only a qualitative examination of cell characteristics could be made simply.
The parallrl equivalciit circuit, Equation 1, is derived by a stand'rrd method ( 2 3 ) through summation of admittances, rationalizing into r(srL1 :tnd imaginary terms.
USE O F EQUIVALENT CIRCUIT O F CELL AND SOLUTIOA
An equivalent circuit which expresses the parameters of the various measurements has been found t o be a very convenient and fruitful method for interpreting results and predicting ways of changing these parameters so as to obtain superior response curves. Blake (18) was the first to use an equivalent circuit, by which he explained a maximum in his cell response curve. Recently, use has been made of the equivalent circuit shown in Figure 2, A (10, 11, 29, 40), but no detailed quantitative studies have been published on its application in evaluating the factors involved in high-frequenry titrations of electrolytic solutions.
(A)
1 T
-T
A. B. C.
=
G,
+ jB,
(2)
I' = net admittance of circuit G, = conductance term (or 1 /Rp,the parallel resistance), real part of admittance B, = susceptance term (equal t o wC,), imaginary part of admittance w = 2Hj where f is frequency R = actual resistance offered by solution (equal t o I l k where k is actual low-frequency conductance of solution) C1 = capacitance due t o walls of container C2 = capacitance due to solution j = operator ( - 1 ) 1 / 2 Each term is evaluated individually to interpret the effects of thr cell, polution, and frequency parameters.
cs
Qcz
Figure 2.
Y where
Equivalent Circuit of Cell and Solution Fundamental equivalent circuit Simplified series equivalent circuit Simplified parallel equivalent circuit
T h e equivalent circuit of the cell may be simplified into either a series (Figure 2, B ) or parallel (Figure 2, C) equivalent circuit. C1 is the capacitance due to the dielectric properties of the container walls, CZ is the capacitance due to the dielectric pfoperties of the solution, and R is the resistance of the solution. T h e resistive component of the container walls is assumed t o be BO high as t o be a negligible contribution to the over-all response. No distributed system, such as the container and its solution,
HIGH-FREQUENCY CONDUCTAYCE TERM
(3) I n this discussion vmktions of G, and k are described, as the instrument employed in this study gave answers in these terms. Some circuits are based on measurements proportional to R,, which is inversely proportional t o G,. From Equation 3 , one can see that as Ci increases, the actual value of G , increases, and furthermore when C1 approaches infinite capacitance, the high-frequency conductance approaches the low-frequency conductance as a limit. Equation 3 states that G , is a unique funct,ion of k , the low-frequency conductivity. That the highfrequency conductance is actually independent of the nature of the electrolyte and dependent only on the low-frequency conductivity is shown in Figure 3, A , with curves for potassium chloride, acetic acid, and aluminum chloride in water. These data confirm recent st.atements by others (10, $9). From Equation 3, one can see that G, decreases as frequency decreases and becomes zero a t zero frequency. K h e n k approaches very small or very large values, G , approaches zero, so that there is a peak for a given frequency. The theoretical curve shown in Figure 3, A , was calculated for 3 hIc., using Equation 3. The necessary constants for this calculation mere obtained from two experimental measurements on cell (A)i. These measurements of C, and C1 are the capacitance values received with the cell filled with pure m t e r and filled
89
V O L U M E 25, NO. 1, J A N U A R Y 1 9 5 3
proportional t o the frequency a t the peak. Figure 3, B , shows the experimental verification of these conclusions. For a thick container wall where C1 approaches a negligibly small value in comparison n-ith Cz, Equation 4 becomes
with mercury or 1 S potassium chloride, respectively. Using cell (A)1, values of 22.9 and 28.0 Ipfd. were obtained for C, and C1. Czwas calculated from the relationship C, = CICz/(Cl+C,). Corresponding k values n-ere determined for a number of specific conductivities from k = Koa/l. The cell constant (a/2) needed for this calculation was evaluated by means of the equation ( a l l ) = 3.6 X 1012rCz/D.
GP
80
00
I 5 I
4") II
z
=
1.8 X 10°K~
(6)
D
since k = Koa/l and CZ = Da/3.6 X 1012rl. Thus the frequency for a peak under these conditions is inI dependent of other cell factors and approaches a value which ran be determined by the dielectric constant, D, of the solvent 4 ACETIC ACID and the low-frequency specific conductivity, KD,of the solution. Equation 6 gives the same relationship for the peak as determined by the empirical methods (6, $ 7 ) and that derived by Debye and Falkenhagen (24, 25). Table I gives results illustrating the shift in kVeak IT-hen the cell walls are of such a thickness that C1 is not negligible compared to Cz. Thus it is seen t h a t
I
m 0 c m
fpeak
CURVE FOR WATER
10
0
-
