High-Frequency Titration Theoretical and Practical Aspects JAMES L. HALL, West Virginia University, Morgantown, W . Va. change of the cell, but not to both. The conductance changes are generally significant at higher and lower concentrations than are the capacitance changes. The metal plate, the glass wall, and the solution of the type of titration cell in general use act as a series capacitor which determines, in part, the relation of the electrical properties of the solution to the response of the instrument. A proper understanding of the equivalent circuits of the cells and instruments should enable a calculation of the absolute values of conductances for solutions in high-frequency titration cells
If those interested in the high-frequency titration method are to devote their full efforts to the development of practical analytical applications, there must be a clear understanding of the relations among the properties of the solutions being studied, the electrical characteristics of the cells or titration vessels used, and the response J ielded by the various instruments. The instruments which have been described that depend upon the direct loading of a vacuumtube oscillator have generally been used in such a way as to respond primarily to either the effecthe capacitance change or the effective conductance
T
HE use of high-frequency oscillating electronic circuits for the determination of end points in volumetric analyses has attract.ed widespread attention during the past 5 years. In general, a glass vessel containing the solution under study is placed in such a position that it exerts a loading effect on some component of the oscillating caircuit, and a change in an clect,rical property of the circuit is measured as the composition of the Polution is changed. Several varieties of instruments have been described for this purpose. .I review of these was given by Blaedel and Malmstadt (,$j in 1950. Some addit,ional instruments are referred t o below. The interest in the use of high frequency is due in part t o the novelt,y of the method, and in part to the possibilities which it offers for making end-point determinations, in certain cases, more easily and more rapidly than by other methods. It has been shown that new and improved analytical procedures may be based upon the use of a high-frequency t,itration apparatus. Typical of the applications already described are nek analytical methods for thorium ( 6 ) , for chloride ( 5 ) , for beryllium ( 8 ) , and for calcium and magnesium ions in solution (17). Complexes of nickel(I1) ion and cyanide ion and of cobalt(I1) ion and cyanide ion in aqueous solution have been studied (11). The method has been adapted t o the determination of diffusion rat,es in solution ( I O ) , to the determination of reaction rates (f6), and to the detection of chromat,ographic zones (19). In spite of the widespread interest in the high-frequency method, there is still insufficient fundamental information upon the nat,ure of the memurementn being made and there is a general lack of understanding of the relation between the properties of the solution being studied and the response of the various instruments. The present article dewrihes and illustrates somc of these properties and relations. CELL CIIARACTEKISTICS
The over-all electrical properties of a glass or other insulatiug vessel containing a solution are distinctly different from the electrical properties of the solution itself. For the purposes of the present discussion a high-frequency “cell” is defined as a glass vessel containing a solution and having metal plates forming a capacitor attached to, or in the immediate proximity of, the outside of the glass vessel, and including the necessary leads and terminals. Except for the fact that the metal plates (electrodes) are outside the glass, this definition of “cell” is in general agreement with the use of “cell” in electrochemical and dielectric constant measurements. It also agrees with the stated or implied definition of high-frequency cells in several recent articles (14,
20, 2.2). In the following discubsion, \vhere only the glass tube :tnd its cont,ents are meant, the term “titration vessel” is used. .1 cell, as defined here, for use in high-frequency analysis, is a two-terminal circuit elenicnt which offers a definite impedance to an alternating current, which varies as the composition of t,he liquid or solution within the cell is varied. This impedance may be measured at, the terminals of the cell in terms of the equivalent series resistance and rvactancc or in terms of the parallel admittance components, suswptance and conductance. 4 1 high-frequency cells thus far described have a net capacit’ivc rcactance and so niay be descrihed in terms of series resistancc and series capacitive reactance or in terms of the parallel admittance coniponents, conductance and capacitive susceptance. Becausc, if the frequency is known, the capacitive reactance or susceptance may be converted to actual capacitance, we may commonly describe the properties of the cell in ternis of its effective over-all oonductance and its effective ovcr-all capacitance. S c v t ~ commercially l available high-frequency measuring instivnients may be used to determine directly the effective over-all capacitance and conductancc or resistance of a cell, and ,such instruments niay be used to folloTv changes in the over-all capacitance and conductance or resistancc as a titration or reaction is carried out within the cell. The Twin-T impedance-measuring circuit, ( H ) ,used as described by Hall and Gibson (14), is convenient, for these measurenicnts because t,he changes yiclded by many analytical react,ions in high-frequency cells are included in its direct-reading ranges. Deterniinations using t,he Twin-T circuit are made by a parallel substitution method which has thta advantage of yielding, as desired, ahsohite values of the effective over-all cell conductance and caparitance or absolute values of changes of effective over-all cell conduct~anceand capacitance. Data ok~tainedusing thifi circuit. noTv enable a simple explanation of the operation and result,s of most prcviounly described tl-pes of high-frequency analytical apparatus. CELL AND SOLUTION RELATIONSHIPS
I t has been shown experimentally that the effective over-all conductance and capacitance of an insulat.ed cell containing an electrolytic solution change with both concentration and frequency. Forman and Crisp ( 1 2 ) and Richards and Loomis (21) have determined the nature of the conductance change by observing the temperature rise of a solution placed in t’he field of a power oscillator. Forman and Crisp also observed the capacitance change by noting t8heshift of the frequency of the oscillator, Blake (8,9 ) has reported st,udies of the effective conductance as concentration was varied by measurement of the radio-
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V O L U M E 24, NO. 8, A U G U S T 1 9 5 2 frequency current transmitt.ed by a cell. hlore recently others have given further dat,aon this subject ( 3 , 2 2 ) . Figure 1 gives the over-all effective conductance and capacitance, a t 6 megacycles, of a cell containing hydrochloric acid, sodium hydroxide, or sodium chloride solutions. The data of this figure were obtained in this laboratory using the Twin-T impedance-measuring circuit, and are in t.erms of absolute values for the cell described by Hall and Gibson ( 1 4 ) . The units of conductance are micronihos. The zero on the conductance scale repreeents the balancc point on the bridge with the cell completely disconnected. With the empty cell connect,ed to the bridge, the conductance was 0.9 pmho, and when the cell was filled with distilled water the conductance rose t o 6.5 wmhos. These numbers are insignificant on the scale of Figure 1. The zero on the capacit,ancc scale also represents the balance point of the bridge n.ith the cell disconnected. The capacitancc of the empty cell including the leads and terminal st,rip wae 4.8 ppf. When t,hc cell was filled with distilled water the capacitance row to 13.4 ppf., ryhich is the starting point of t,hr curves shown. 6
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mum efiective conductance are all identical a t 37 rrf. There is R similar correspondence between conductance and capacitance values for the full range of all the curves. It therefore appears that if the actual conductances of the solutions were used as the abscissa, in place of concentration, the curves for the different electrolytes would be identical. The forms of the conductance curves are similar to curves obtained during the course of measu'rements of power transfer to fixed capacitancevariable resistance series circuits in elementary alternating current laboratory experiments. This all suggests that the data of Figure 1may be int'erpreted in terms of a variable resistance in series with a fixed capacitor with perhaps other k e d resistors and capacit,ors in the circuit,. The correspondence between the behavior of a high-freyueili.y cell, as the electrolyte concentration was varird, and a seiies variable resistor and fixed capacitor !vas pointed out by Blake (9) in 1945 and again in more dct,ail in 1946 (8). Blake s u h t i tuted for his high-frequency cell a Bakelite rod coated xvith graphite which was connected to a tuned rircuit by means of (-lips and leads containing fixed series capacitors. He then denionstrated that as the resistance was altered from zero, by moving the clips further apart along the rod, there occurred a cert:iin critical value of resistance a t \vhich the, pon-cr transfer to r.ho resktance pamed through 3 maximum. A more detailed approxiniatc equivalent circuit for the highfrequency ccll, based upon closer analysis of t8hedat,a of Figure 1 , is shown in Figure 2. Here Rl and C1 represent the resistance and capacitance which should be measured if the metal plates of the cell were in direct contact with the solution. C, represents a fixed capacitor which conb sists of the mrt,alplate, the dielectric, and t,he Figure 2. Equivalent Circuit of Titration Cell surface of the solut~ioriin contact with t.he glass. Actually the cell includes two such capacitors, hut since both are in series with the solutions one sufficesfor the equivalent circuit. Re represents the resistive component which results in power loss in the glass wall of the vessel and across the outside surface of the glass. The choice of reprmenting R, parallel to or in series with C2 is arbitrary. In cells where electrolyte concentration is less than about 0.1 M , Rz is usually negligible in comparison with R I ; its value varieH from a few tenths of an ohm to a few ohms, depending upon the design of the cell. Hence for such solutions, the equivalent circuit of Figure 2 is identical with that recently given elsewhere (3,22). This equivalent circuit does not include a term for t.he capacitance between the lead wires of the cell, and this must he c-orrected for before making cvdculations based upon the circuit,.
+G?!
The maxima of the effective cwiductance curves of Figure 1 may be simply explained in t.eriiis of t'lre circuit of Figure 2. Figure 1. Effective Over-all Conductance a n d Capacitance of a High-Frequency Titration Cell as a Function of Electrolyte Concentration Sevt.rd iiitercsting facts may be observed by inspection of Figurc 1. First, the effective ovei-all conductance curves go through t maximum as the concentration of the electrolyte is increascd from zero, while the capacitance curves show a continuous rise with an inflection. The maxima of the different effective conductance curves are at very nearly exactly the SaJlle ordinate and the concentrations a t which the maxima occur are simply related to the ordinary low-frequency conductance of the solutions; they occur at points of equal solution conductance. For each electrolyte the product of the concentration a t the maximum times its cquivnlent conductance (at 25" C.) a t that concentration equals 1.85, Furthermore, the effective capacitance values which correspond to the concentrations of the niaxi-
The resistance of a solution d e c r e e s as an ionic substance is added to water. Referring to Figure 2 and assuming that R? is negligible? let us consider the variation of power dissipated by this vircuit when an alternating current voltage is applied across the twniinals a - b and the value of Rl is decreased from infinity to zero. If Rl is infinite, there will be no current flowing through it, no power will be dissipated, and there will be no effective resistive component looking int,o a - b. If R1 is zero, there \vi11 be only capacitance in the circuit (still neglecting Rq) and again no power xvill be dissipated, and again there will be no effective resistive r o m p n e n t looking into a - b. At all intermediate values of R1, some power will be dissipated, and at some crit,ic*al value of til, de nding upon the reactances of C1 and CZ,the powt'r r b ie a maximum and hence there will be a maxidissipation w mum in the effective conductance component which can be mewured across a - b. This is in agreement with t,he observat,ions noted above in connection with Figure 1. T,?sing the circuit of Figure 2 as a basis of analysis, it is possible to calrulate the true v:ilutl~of R, and C,, the resistance and (.a-
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ANALYTICAL CHEMISTRY
pacitance which should be obtaiiied if the metal plates of the cell were direct,ly in contact with the eolution. First it is necessary t.o evaluate R2 and C Z . This may br done in several ways, but in this laboratory tentative calculations have been bayed on measurements of the cell filled Tvith rnei'cary or with the inside glass surface of the cell heavily coat,rd with a chemically deposited silver mirror. The silver mii,ror is bet.trr than mercury for evalu:+tiiig K?, but both silver and mrwury are equally valid foi, (.I.. Thcl silver mirror 01' nitxrcwry :tpprosiiiiatcly xhort-circuit,r K I and (,'I of Figure 2.
would be f'ollowd by an increase of cell conductance upon the addition of more sodium hydroxide. If we were to start with an 0.02 .FI solution of hydrochloric acid and add sodium hydroxide solution there would be an increase in the eft'ective over-all cell conductance from -100 for hydrochloric acid alone to 860 a t the equivalence point, followed by a decrease of mnductance as more wdiuin hydroxide was added.
r h t a of the hydrochloric, it&I I'UI'VI'S Figure 1 a t 0,00419 JI niay tw used as an example for the cd(ulation of Rl and Cl. .\t t,his concentration the effective conduotaricne of the cell, G, was S8Y pnihos and the effective capacitance was 35.9 ppf. By tncsitsuremnents using dummy leads the lead vapacitance was estiy a t e d to be 3.0 p p f . , and this value is suhtracted from the total j w t given to yield the net effec%ive capacitance of the cell, C, :I$ 32.9 ppf. These are the conduvtanrr and capacitance components a t the terminals u - b of Figure 2. The susceptance, B, is c*alculatedby the relation B = 2 4 ' , arid the absolute values of the equivalent series resistance, R,, and rrnctance, X c t , a,re then uilwlated by the equations: 0
(1
I
1
" t
I
ML. NaOH A D D E D
giving Rt equal to 381 ohms and X,r equal to 532 ohms. Folloning a similar calculat~ionfor the silvered vel1 which had an effective cqmcitance of 61.5 ppf. in parallel with an effective capacitance of 11.5 micromhos, series equivalents of 2.4 ohms' resistance : t i i d 453 ohms' capacitive reactance, are obtained. By subtractiiig the series equivalents of the silvered cell from the over-all wries equivalents, we obtain approximately 379 ohms' resistaiwe, R,, and 79 ohms' capacitive reactance, X,,, as the series equivalent of R1 and C, of Figure 2. Rl and C1 may then be evaluated by the equations:
1.
nud the relations Rl = 1/G1 and C1 = B 1 / ' 2 ~ / . In the example chosen, GI = 25.2 X lO-'mho and HI = 5.26 X lo-&mho, RI = :195 ohms, and C1= 14.0 ppf. These final values are only appioxiniate, a q the conductance of the water and certain other factors were neglected. In addition, the use of the silvered cell in this a ay may be subject t o furthei cmpirical correction. If the final value of the conductance ir divided by the product of the conrentration and the equivalent conductance a t that concentration, a constant, 0.00145, is obtained. This constant is maintained \Tithin 2 or 3% for all the data of all the electrolytes shown i n Figure 1. This indicates that variations of the equivalent rondu(*tanw at this frequency from the corresponding equivalent c*onductaIicesat low frequency are very bmall compared to the total vhanges illustrated in Figurcb 1. Further detailed data for several electiolj tes at several frtaquencies and analysis of the data in teims of the equivalent & w i t of Figure 2 are t o be submitted i n a separate article. FORMS OF TITRATLOY CURVES
The curves of Figure 1 may also be used t o explain why tht. tlirtwtion of the over-all effective conductance change of a cell reverses for some titrations the frequt.ric*v01 the initial concent i a t ion is varied. For example, if we neie t o start nith nn 0.005 '11 solution of hydrochloric acid and add sodium hydroxide solution at such concentration that the total volume vhange could be neglected, at the equivalence point where orill sodium and chloride ionn were in the solution there would have been a decrease in the effective over-all cell conductanre from ROO to 660 pmhos. This
Figure 3. Effect of Initial Concentration on Direction of Effective Conductance Change at Constant Frequency 1. 0.005 M HCl titrated with 0.05 M NPOH
2.
0.02 IW HCI titrated with 0.2 Vt YaOH
,, lheuc titratioris, plotted on the same coriductuiee scale as for Figure 1, are shown in Figure 3. The Hmall differences from the values just given are due to dilution with the addition of the sodium hydroxide solution. Since the concentrations corresponding to the conductance maxima of Figure 1increase with an increase of frequency, it may be deduced that similar reversals of ourve form, as in Figure 3, could occur at constant initial concentration with variation of frequency. A family of curves illustrating thiH reversal with frequency for the hydrochloric acidsodium hydroxide titration was given by Hall and Gibson (14 ) . T h o practicability of many end-point determinations by the highIrequency method should readily be predicted if curves of the t.yp of Figurc 1 w e w obtained for the reactants and produckQ. OSCILLATOR CHARACTER1 STICS
Several different kinds of apparatus are in use in which a titration vessel is pliwed in one component of a tuned circuit which is part, of a vacuuni tube osdlator, or i n which a titration cell is connected in parallel with t,he tuned circuit. Measurements are made of some change in t,he properties of the o~cillatingcircuit or of the curreiitn or voltages of the tube. Hence, in order tmo understand the mct,hod and its range of usefulness, it is necessary to consider the properties of tuned radio-frequency circuits and oscillators as well a8 the electrical characteristics of the cells previously considered. A brief description of some pertinent characteristics of radio-frequency oscillators is given below. Thin discussion is intended to illustrate certain principles and is not a completf: summary of all types of apparatus thus far described. Figure 4, a, shows achematically a simple tuned-plate vacuum tube oscillator. For this circuit the frequency of owillation may he shown to be (18)
cvhere J' is the frequency in cycles per sevond, and L is the induce of the circuit in henries. (' i R the net c,apacitance of the
aiiw
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V O L U M E 24, NO. 8, A U G U S T 1 9 5 2 rircuit in farads and inciludes the capacitanrc! of tht: lc.tttls, thv tube elements, and t'he ca,pacitance between the t,iirnr of t'he inductance coil. The resistanct, r p , is an apparent rivmigr plation(I,) as to whether thv c.ouplet1 part. I f , Xf-hich1s due to the effert. of the, solution, o I)(? iii series with the inductmcae coil t t s or whether it should he [:onsidered to bo in . Tf ronsitlerrd to he in parallel, Formula 5 thcrc: would he no tffert of the apparent r+ the, cell or vessel on the remiant frrqurJnc-y. ions hy Formula ., such an instrument could be 1ride1~useful wherever
