7th Annual Summer Symposium-Developments in Titrimetry
Relations between Voltam metry and Potentiometric and Amperometric Titrations I. M. KOLTHOFF School o f Chemistry, University o f Minnesota, Minneapolis, M i n n .
ifter polarization and depolarization have been defined, potentiometric and aniperometric titrations are discussed on the basis of roltammetrj. Potentiometric titrations at zero current with a specific indicator electrode can be carried out with an outside reference electrode (classical method) or with an attackable inside reference electrode-e.g., tungsten in oxidation-reduction titrations. Potentiometric titrations at constant current can be carried out with one or two indicator electrodes. The titration of copper(1I) with Versene at a constant cathodic current is discussed and explained. \ large jump in potential occurs just before the end point. A mercury electrode is advantageous in the titration of many metal ions. Various other sjstems are discussed. The titration lines with two indicator electrodes are interpreted on the basis of the voltammetric behavior of the sjstem titrated and the reagent. iniperometric titrations can be carried out with one or two indicator electrodes. Changes of current in the neighborhood of the end point are calculated.
S
I N C E 1951, when the aut,hor presented the essentials in this paper in a seminar, various classes of titration methods have been studied syst,ematically in this country and abroad. Ileilley, Cooke, and Furman (9,15) were the first to discuss the relation between voltammetry and potentiometric titrations a t constant current a t two identical indicator electrodes, a type of titration introduced by Willard and Fenwick ( 1 8 ) in 1922. Gauguin, Charlot, and their coworkers (1-3, 1 0 ) in Paris and Duyckaerts ( 7 ) in LiEge have contributed by studying the relations between current and potential in partially reversible systems. I n this reqiect Duyckaerts’ recent review paper ( 7 ) is especially recommended. Aiclassical contribution to the theory of aniperometric titr:ttions with two indicator electrodes has been made recently by Bradbury ( 2 ) . The present paper deals with reversible or entirely irreversible s>.stems. The theory of partly reversible systems is knovm. However, such systems in general do not lend themselves readily to :I quantitative experinient,al treat,ment (although easily to a qualitntive interpretation), because current-voltage curves of suc.h systems a t solid electrodes are difficult t o reproduce and oftcii depend upon the pretreatment of the electrode and the time of contact betneen electrode and solution. For a systematic discussion of the interrelation between potentiometric titrations a t constant current and also a t zero current with an attackable electrode and amperometric titrations with two indicator electrodes on the one hand and voltammetry on the other hand, it is desirable to classify potentiometric and nniperonietric titrations. It is also desirable to be consistent in thc terminology used, especially for the expression “polarized” electrodes. This notation is often used in potentiometric titrations a t constant current, when the electrode(s) are polarized only before the end point and depolarized after the end point, or are polarized only a t the end point. The best definition of polarization and depolarization appears to be a combination of t h a t given by Kortum and Bockris ( 1 4 ) niid by Kolthoff and Lingane ( 1 3 ) . iln electrode is polarized
when it adopts a potential imprebwd upon it nith little 0 1 no change of the current ( I S ) , and it is ideally depolarized when upon passage of current the potential does not deviate from its reversible value ( 1 4 ) This reversible value refers to activities of the electroactive species a t the surface of the electrode. I n this respect the notation “concentration polarization” is a misnomer and “concentration overpotential” is a better term. The so-called overpotential 01, better, “activation potential” (voltage) is a quantitative measure of the degree of polarization of the electrode. According to the above definition, a platinum electrode in a solution of thallour thallium remains polarized until thallium is deposited on the electrode. Cntil this potential is attained, there is no (reversible) electrode reaction. After some thallium has been plated out, the electrode becomes depolarized, the activation potential being zeio. I n the electrooxidation of hydroxyl ions or water a t a noble metal, the electrode is polarized, even though the current changes noticeably upon a change of the imposed potential, because there is a marked activation potential all along this anodic current-voltage curve. The notation “bimetallic electrode system” is not used in this paper, because practically all electrolytic cells are bimetallice.g., mercury in the saturated calomel electrode (S.C.E.); silver in the silver-silver chloride electrode Typical notations for various types of titrations have been proposed in the recent literature or are proposed in the present paper. I t m s desirable to gather a considerable amount of euperimental information in connection with a quantitative interpretation of the various types of potentiometric and amperometric titrations. Eutensive, although not exhaustive, experimental studies have been made by E. R. Sightingale in this laboratory, the results of which will be described in separate papers. C LA S SI FIC 4TIOK OF POTENTIOMETRIC TITR 4TIOYS
At Zero Current. ENATTACKABLE ELECTRODE.The classical method with specific or unattackable indicator electrode 4 ith an outside reference electrode. The various methods for finding the end point and experimental variations in the system used are fully discussed in the literature. ATTACK.4BLE ELECTRODE.Lhe of an attackable electrode as inside reference electrode in oxidation-reduction titrations lvvith a noble metal as indicator electrode. The first metal couple used in this method is pslladium-platinum in the titration of ferrous iron with dichroniate by Hostetter and Roberts (11). Willard and Fenwick ( 1 8 ) and Van Kame and Fenwick ( 1 7 ) made an extensive study of the use of various attackable electrodes in ouidatioii-reducton titrations. The tungsten-platinum couple was found particularly useful. The behatrior of the tungsten electrode in this type of titration is accounted for on the basis of its voltammetric characteristics. At Constant Current. OSE INDICATOR ELECTRODE (us. reference electrode). This method % a s introduced by Dutoit and von Weisse (6). Although not explicitly stated in their extensive papers, the idea underlying their method is to make the potential of less noble metals more reproducible by depositing continuously fresh layers of metal upon a noble metal electrode. They state (6, page 579) that copper, lead, nickel, zinc, and cadmium concentrations (activities) can be measured in this way. This is a gross ex-
1685
1686
A N A L Y T I C A L CHEMISTRY
aggeration. They used only the “po1arized”cathode (actually depolarized in the presence of excess metal ion a t constant current) in the titration of cupric copper with ferrocyanide and sulfide. For several reasons, the method as proposed is very impracticable. On the basis of the voltamnietric behavior of copper i t is shown below that the principle introduced (but not understood) by Dutoit can be made the basis of some potentiometric titrations (see also S, 10). The application of the method is limited except with a mercury cathode. Even when it can be applied, i t is simpler and more accurate to make a n amperometric titration nit’h one indicator electrode. I n the titration a t constant current with one indicator electrode the large break in potential is found just before or after the equivalence point. Two (IDENTICAL) I N D I C A T O R ELECTRODES. The method was introduced by Willard and Fenwick (18), who passed a small, but constant, current through the system. I n a most instructive paper Van Name and Fenwick ( 1 7 ) gave the values of the potentials of the cathode and of the anode separately during the titration of various oxidation-reduction systems. From these data valuable information can be derived concerning the degree of polarization of both electrodes during the entire titration. A qualitative interpretation of the potential break a t the end point on the basis of voltammetry was first given by Reilley, Coolie, and Furman (15). When the voltammetric characteristics of the electrodes used and the system titrated and the reagent are known, the titration line can be predicted quantitatively.
the proper pH. The titration is carried out at a constant cathodic current in the absence of oxygen, and the solution, or the electrode, or both are well stirred, I n the medium used the cupric copper is reduced to the metal. Cnder the conditions of stirring and a t the particular temperature it is assumed that the diffusion current of l O - 4 N ( 5 X 1 O - b J I ) cupric copper is 10 pa. By vsrying the size of the rltctrode this “concentration sensitivity” can he varied widely. In t’he present titration, which is carried out a t 2 bn., we start with 0.05.U cupric copper and assume, for the sake of convenience, that the volume rcmains unchanged during the titration. We further assume that the copper-copper(I1) potentin1 is rever,Qit)leundcr tlic cJxpc.ri:ntnt,al tonditions.
CLASSIFICATION OF AMPEROMETRIC TITRATIONS
Figure 1. Current-\-oltage Curves of Copper(I1) at Rotated Platinum Electrode
One Indicator Electrode. I n this method the current ie measured during the titration a t a constant applied voltage, at which the substance titrated, or the reagent or both yield a limiting (usually diffusion) current. This type of titration with the dropping mercury or a rotated solid electrode is fully discussed by Kolthoff and Lingane ( I S ) . Two Indicator Electrodes. This method was introduced by Foulk and Bawden (8) under the name of “dead-stop end point.” \Then a small and constant electromotive force is applied between the two electrodes in a cell containing a completely reversible system, like iodine-iodide which is being titrated with thiosulfate, current flows until the end point. At and after the end point the current is zero or close to zero. The reverse of a dead-stop end point (might be called “kick off”) is found in the reverse titration. When both the system titrated and the reagent are reversible oxidation-reduction systems, the current is zero or close to zero only a t the end point. The various types of curve? on the basis of voltammetry are discussed qualitatively by Delahay ( 5 ) and more in detail by Duyckaerts ( 7 ) . For a reversible system the current changes linearly with the concent,ration near the end point, as postulated by Bradbury ( 2 ) and sho\vn experimentally by Stone and Scholten (16). The end point, therefore, can be found graphically as in a n amperonietric titration with one indicator elect~rode,or the galvanometer can be used as a visual indicator, the end point being taken when the current becomes zero. In an amperometric titration with one indicator electrode there is a (slight) loss of the clectroactive species when current passes through the system. This can be avoided by using a tap key and having current pass only during its measurement on the microammeter or galvanometer. I n a titration with two identical indicator c1t.ctrodes there is no loss of electroactive species upon p current, because in a reversible system the amount of the oxidized form reduced a t the cathode is equal to that formed by oxidation of the reduced form at’ the anode. POTENTIOMETRIC TITRATIONS AT CON STANT CURREXT
One Indicator Electrode (us. standard reference electrode).
METALI o x REDUCED TO METAL,REAGENT NOT REDUCED..4n example is the potentiometric titration at a platinum or copper electrode of cupric copper with ethylenediamine tetraaretate a t
I
I
I
t
PCTENT~CL
sr..~
1. 0.05M 2 . 0.005M 3. 0.0005.1.1 4. 5 x 10-5.11 3. 5 x 10-6.v 6 . 0.05M Cu(I1)-Versene in supporting electrolyte
Figure 1 gives idealized currenbvoltage curves of solutions of cupric copper at varying concentrations. Curve 6 is the residual current of cupric Versenate in the medium, which is about identical to that of the medium TI ithout the copper compound. With a current of 2 Ma. the potential ( L S . S.C.E.)in the copper Versenate solution is found a t A. ;\fter 90% reaction of the copper with rragent, the potential has changed only 30 mv. since the start (from E to D in Figure 1). Between 90 and 99% i t changes another 30 inv., and again 30 mv. between 99 and 99.9%) as in an ordinary potentiometric titration. When 99.9% of the copper has been changed into the complex, the concentration of the residual copper is 5 X 10-631 under the chosen experimental conditions. This solution yields a diffusion current of 10 Fa. Between 99 9 and 100% the potential c h n g e s from about B to A . The break is large and occurs just before the equivalence point. The error is negligibly small. The magnitude of the break in potential near the end point is not determined by the stability constant of the complex formed. In titrations of very dilute copper solutions the error may be appreciablr. Whrn the concentration of the cupric in the bulk of the solution becomes very small, the depletion of the ion around the electrode a t a given current must be considered in a more (,\act calculation of the electrode potential. The sensitivity, k , of the electrode is calculated from the dlffusion current at a given copper concentration. In thc present example a current of 10-6 ampeie is assumed for a 10-4N solution. 1 = ll( - cO) where c is bulk concentration and eo is concentration a t the electrode. Tf7hen c is expressed in equivalents per liter, k in this example is equal to 0.1 ampere per equivalent per liter. When the cupric concentration is very small, eo is no longer equal to c. For example, if Cu++ = 5 X 10-5LV,e o = 3 X 10-sN = 1.5 X 10-6AlI, and the potential of the copper electrode is 71- = 71-n 0.03 log
+
V O L U M E 26, NO. 11, N O V E M B E R 1 9 5 4 1.5 X 1 0 - 5 (in a medium of constant ionic strength concentration is written instead of activity). I n this way the potential at small copper concentrations greater than 2 X 10-sN is calculated with :i current of 2 pa. When the concentration becomes smaller than 2 X the potential shifts to a value close to A in Figure 1 (Yee curve 5 for 10-6A' copper in Figure 1). \Vhen 98% of the cyuivalmt amount of reagent has been added, in the titration of 0.001.j7 copper (no change in volume) the concentration of the copper has become 2 X l O - 5 S . Cpon further addition of reagent tlie potential jumps from about B to 9.I n ot8herwords, the end ~ioiiitoc'curs about 2% earl,v. The error becomes greater with increasing dilution of the copper and can he made smaller by using :In electrode with greater sensitivity. For example, if in the :ihove example a n electrode with ten times greater sensitivity (1, = I ) had been used, the end point would appear 0.2% illstead of 27, early. For the titration of very dilute solutions it is therefure desirable to use a n electrode with as high a sensitivit pr~rticable. As will be reported in a subsequent ptiper with Xightingale, :i ]:)rye break in potential is found a t the end point in the potention~etrictitration of copper(I1) with the ethylenediamine tetrawetat'e by the method described above. This break is much greater :it a mercury than a t a platinuiii electrode. OXIDIZED AND REDUCED FoRY O F REVERSIBLES Y S T E l I SOLUBLE IN WATER.We consider the case that the system titrated : i l i t l rcagmt both give reversihlc potentials at, small currents.
1687 of the ferrous iron is oxidized, the ferrous concentration is 5 X lO-5.lf, and the current-voltage curve is t h a t given on line 4. Between 99.9 and 100% the potential jumps to t h a t of a 0.0531 cerous solution a t a current of 2 pa. (line 7 ) . When the titration is carried out at zero current, the potential jumps to the equivalence potential and with the addition of a slight excess of ceric to the corresponding ceric-cerous potential. The plot of potential against volume of reagent is practically the same when the CUIrent is 2 pa. as when the current is zero. However, the largest break in potential occurs between 99.9 and 100% when i, = 2 pa., while i t occurs exactly a t 100% when i = 0. Evidently there is no advantage in carrying out the titration a t a n anodic current of 2 pa. There is a disadvantage when very dilute ferrous solutions are titrated, because the end point is found early. Using a n electrode with a sensitivity of 0.1 i t is calculated t h a t in the titration of 0.001.11 ferrous iron (no volume change) the break in potential occurs about 2% before the equivalence point. If the titration were carried out a t a constant cathodic current of 2 pa. (i is +2 in Figure 2), the large jump in potential ~ ~ o u be l d found between 100 and 100.1% in the titration of 0.05Jf ferrous iron and a t about 102% in the titration of 0.001Jf ferrous iron. \\-e consider now the case that the system titrated is reversible, and the reagent is irreversible, or vice versa. This case is very similar to the titration of cupric copper with Versene. When the syPteni titrated is reversible and the reagent system is irreversible or vice vcrsa, a much larger potential break may be found in the titration a t constant current than a t zero current. T h e potential of the thiosulfate-tetrathionate system is irreversible, while that of iodide-iodine is reversible, I n an air-saturated solution the thiosulfate potential corresponds to t h a t of the oxygen potential a t air prcbsure (and therefore depends upon pH). I n the absence of oxygen the potential is more negative but fluctuating and irreproducible. Thus at and after the end point the potential in the titration of a n iodine-iodide solution with thiosulfate is ill determined. If the titration is carried out v-ith a small cathodic current, the potential at and after the end point beconies well determined.
Figure 2. Idealized Current-Voltage Curves of Iron(II1)-Iron(1I) and Cerium(1V)-Cerium(II1) Explanation of curves in text
K i t h ordinary reversible oxidation-reduction systems the titration at a constant current us. a depolarized reference electrode has no advantages over the classical method. Consider, for example, the titration of ferrous iron with ceric cerium and assunit' t h a t the ferrous-ferric and ceric-cerous systems are reversible :tt the rotated platinum electrode (which is far from being the c u e ) . I n Figure 2 some idealized current-voltage curves at a rotating platinum electrode of the system iron(II1)-iron(I1) are drawn schematically. Again i t is assumed t h a t a 10-4M solution of ftarric. or ferrous ion yields a diffusion current of 10 pa. Line 1 is part of the current-voltage curve 2's. S.C.E. near zero current, potential of a solution 0.0551 in ferric and 0.05M in ferrous in the proper supporting electrolyte. Line 2 is t h a t of 0.05N iron(II1)0.005.11iron(III), line 3 t h a t of 0.05M iron(III)-O.O005M iron(II), liiie 1 tlint of 0.05M iron(III~-0.00005Jf iron(II), line 5 that of 0.05.lI iron(III)-0.00001J1 iron(I1). Line 6 is the idealized cwrrent-voltage curve of 0.05M cerous-5 X 10-6M ceric cerium. Tlie dottrd line corresponds to a n anodic current of 2 pa. during t h r titration. Without volume change a 0.05M ferrous solution is titrated with ceric cerium a t a constant anodic current of 2 pa. (1' = -2 in Figure 2). During most of the titration the potential of the electrode is pr:~cticallyequal to t h a t when the current is zero. When 99.9%
Figure 3. Current-Voltage Curves of Iodide-Iodine with Large Excess of Iodide 1 2. 3 4
Large concentration of 13lo-4.V iodine 2 X IO-5.V iodine No free iodine
A sketch is given in Figure 3. When the concentration of iodine becomes less than corresponds to a diffusion current of 2 pa., the potential jumps from A t o B in the presence of oxygen and to C in the absence of oxygen. The end point occurs before the equivalence point, the error being negligible in the titration of relatively concentrated iodine solutions, but, depending on the sensitivity of the electrode, i t may become pronounced in the titration of very dilute solutions.
1688
ANALYTICAL CHEMISTRY
It is easily shown that the break in potential would be much smaller if t,he above titration were carried out a t constant anodic instead of a t cathodic current. The anode pot,ential cannot become more negat'ive t h m -4 (Figure 3). This conclusion is illust'rated in Figure 4. Titration at Constant Small Current with Two Identical Electrodes. We consider the titration of ferrous iron with ceric cerium, assuming that both systems are reversible. When there is a reasonable amount of ferric ion formed during the titration of 0.1.U ferrous with 0.1X ceric ion, the system is well buffered (poised) until close to the end point. If the current ampere, and the resistance of the cell is kept constant a t 2 X is 500 ohms, the difference in potential E b e k e e n the cathode and the anode is iR = 0.001 volt. Very close to the end point t,he concentration overpotential a t t'he anode and cathode must be considered again. At the cathode: i = k ( [ F e + + ] ' - [FeT+])-k = 0.1; [ F e + + ] = 5 X then [Fe++]' e.g., i = 2 X = 7 X 10-5. The correct,ion for the concentration overpot,ential is small in this example. At' the equivalence point the solution is 0.05Jf in ferric and 0.05JI in cerous. At the cathode ferric iron is reduced, while there is oxidation of cerous at, the anode; hence the e.m.f. between t,he tn-o electrodes is large (see Figure 2 ) . K i t h a slight excess of ceric ion the situat'ion is the same as before the end point; the system is depolarized and the e.m.f. drops to almost zero.
Table I.
\-slues of Potential
5 X 10-*2f. Potentials and e.m.f. near equivalence point in titration of € e t + with C e T + + . i = 2 X 10-6 ampere: k = 0.1) i = 0 Reagent One Indikator Two Indicator Electrodes Added, Ea. R Electrode, T Toathode Tanode E , rolt 99.90 0.94 0.931 0 954 0 023 99.9G 0.95 0.946 Indet. -1.20 -0.25 99.98 0.96 0.954 1.23 0.278 100.00 1.105 0 964 1.246 0.282 100.02 1.25 0 982 1.257 0.275 100.10 1.27 1 237 1.279 0.022 (FeiT+
=
REAGENT ADDED IN EQUIVALENT PER CENT NEAR EQUIVALENT POINT
Figure 5. Potentionietric Titration of 0.1,M Iron(I1) with 0.LM Cerium(1V) 1. i = O 2. i a n o d i c = 2 X 10-0 pa. (one indicator electrode) 3. i = 2 X 10-6 pa. (two indicator electrodes, E on right side 4. 0.00lM Fey'ivith C e +of+ +ordinate) T , i = 2 X 10-6 (twoindicator electrodes.
,
'91
m,
7c5
'4
l03lNZ
sc.u-
CLI
Figure 4. Titration of Thiosulfate with Iodine (1 7) i. 1. 2. 3.
4.
i = 0 in curx.e 1. 25 X 10-6 ~ a in. ciii'ves 2, 3. and 4 One indicator electrode T anode T cathode E (two indicator electrodes)
I n Table I some values of the potential, rrof the indicator electrodes ( k = 0.1) are given near the equivalence point when z = 0 and i = 2 pa. The potentials Tcathode and Tanode are also equal to those calculated for a titration with one indicator electrode with i =2 X ampere. The e.m.f. E betm een two identical indicator electrodps ( k == 0.1) is equal t o the difference in anode and cathode potential. Some data are plotted in Figure 5 . T h e maximum in the e.m.f. is acute and occurs in a region slightly before and after the equivalence point. If 0.OOl.U ferrous (no volume change during titration) mould be titrated under the same conditions, this maximum would not be acute and would extend from a few per cent before the equivalence point t o a few per cent after the equivalence point. As an illustration, a few calculated figures near the equivalence point are presented in TabIe I1 and given graphically in Figure 5.
Lower ordinate refers t o this curve)
;\t 2 % before the end point the e.ni.f. jumps from almost 0 to 0.47 volt and a t 2y0 after the equivalence point it drops from this value to almost, 0. The mid-poiiit of the maximum potent,ial range corresponds t,o the end point. Of course, this range beconies shorter with larger electrodes (of greater sensitivity) but more pxtended with larger currents. For example, if k were equal to 1, the high value of t,he e.m.f. would be found from between 0.2% before and 0.270 after the end point. A411the calculations have been made assuming t h a t bot'h systems are ideally reversible. This is certainly not true for the cerous-ceric system, when the cerous concentration is very small. Under such conditions the anodic cerous wave almost coincides n-ith the oxygen evolution wave observed in the medium alone. -11~0,the ferric-ferrous syst,em is not idenllj- reversible and the current-voltage curves deviate considerably from the theoretical curves, especially a t small concentrations of ferrous or ferric ion. The pretreatment of the electrodes has a relatively large effect 011 the shape and Iocation of thp current-voltage curves, as mill be shon-n in a subsequent paper. R e consider now the case that the system titrated is reversible, and the reagent system is irreversible, or vice versa. IT-hen one oxidation-reduction is completely reversible while t,he rmgent system is irreve or vice versa), the e.m.f. remains large after the end point. The titration a t conat,ant current can he carried out with one indicat,or electrode (cathode in the present example) or t w o indicator electrodes. With both systems the break in potential of the cathode and in e.m.f. bet-xeen tv-o indicator electrodes a t the end point is of the same order of niagnitude. This is illustrated (Figure 4) b)' measurements by Van S a m e and Fenwick ( 1 7 ) of the cathode and anode potential? separately and of the e.ni.f. E between the tn-o indicator electrodes in the titration of thiosulfate with iodine a t constant current. As can be shown easily (see Figure 3), the change in anode potential a t the end point is relat,ively small because the anode
V O L U M E 26, N O . 11, N O V E M B E R 1 9 5 4 responds only t o changes in iodine concentration around the electrode. Even when there is no free iodine left in the solution, some is formed a t the anode when current flows. The break in potential a t the anode decreases and a t the cathode increases n i t h increasing current during the titration. This is to be anticipated, because a t the end point the concentration of iodine a t the anode increases a i t h increasing current. The reduction of oxygen (or of hydrogen ions in the absence of oxygen, Figure 3) is irreversible and the cathode potential a t the end point becomes more negative with increasing current. The change in e.m.f. between the two electrodes nil1 be greater in the absence than in the presence of oxygen (Figure 3).
Table 11.
Calculated Potential
(Two identical electrodes; i = 2 X 10-6: k = 0.1. Titration of 0 O O l X Fe +; no volume change) +
Reagent Added, Eq. R
iic
97
0.84
98
99 100E.Pt 101
0.844
0.84 0.862
0.87.5
XI1
0.88 (-1.2-1.3) 1.31 1.33 1.36
E 0.04
(-0.4) 0.46
0.47 0.47.5 n 17
POTENTIOMETRIC TITRATIONS AT ZERO CURRENT WITH ATTACKABLE (TURGSTEN) ELECTRODE AS INTERNAL REFERENCE ELECTRODE
Current voltage curves observed at’ a tungsten electrode account for the behavior of this electrode as internal reference electrode in oxidation-reduction titrations. I n 0.1~1-perchloric acid tungsten gives a drawn-out anodic wave, starting at about 0 volt (S.C.E.), and a t more negative potentials a t higher pH. A film, probably of WOS, is formed on the electrode upon passage of an anodic current. The cathodic current a t potentials more negative than 0 volt in 0.1S perchloric acid remains small until hydrogen evolution starts (see “residual current” H M G D in Figure 6). Qualitatively the current-voltage curve is comparable t o those observed a t other att,ackable electrodes, like silver and mercury, in inert supporting electrolytes. From the shape and location of the anodic wave it, can be predicted that tungsten will be attacked by oxidizing agents, such as iodine or ferric iron or ceric cerium. I~xperimentallyNightingale in this laboratory found the rate of reduction of ferric ion by tungsten to be independent of the ferric concentration and to decrease slightly with decreasing osidation potential (increasing ferrous iron concentration) of the solution. The effect of the reaction is negligible in a titration of relatively concentrated solutions of iron(I1) [with cerium(1V) or chromium(T’I)], but it is appreciable in the tit,ration of very dilute (of t h r order of lOP4Jf) solutions. For t,his and other reasons, t h r tungsten-platinum couple is not, suitable for the potentiometric titration of very dilute solutions. The behavior of the tungsten-platinum couple in the titration of ferrous iron with ceric is accounted for qualitatively by the current-voltage curves and the potential of the tungsten electrode in solutions containing ferrous and ferric (with cerous) or cerous and ceric (with ferric) ions. Line .4BC in Figure 6 is an idealized current-voltage curve observed at the rotated platinum electrode in a 0.1S hydrochloric acid solution xhich is 0.01M in ferric and 0,OLlf in ferrous iron. Point B corresponds t o the oxidation potential of the system where t,he cathodic current is equal t o the anodic current. Even if the ferric-ferrous system were ideally reversible at a tungsten electrode, the current would not be equal t o zero a t a potential B, but equal to “the residual current,” B D (Figure 6 ) . At the tungFtni electrode the oxidation potential of the iron s)-stem is found where the cathodic reduction current of ferric iron is equal t o the anodic current, observed a t the tungsten electrode in the absence
1689 of iron. This is the case a t E , where the cathodic current, EF, is equal to the anodic current, EG. From Figure 6 it is seen t h a t the potentials a t B and E differ only very slightly. Euperimentally in the iron solution under consideration the potential a t a rotated tungsten electrode was found to be only 1 mv. more negative than a t the rotated platinum electrode. Generally, the potentials measured a t tungsten and platinum in solutions containing relatively large concentrations of ferric iron are of the same order of magnitude. This is of importance in the interpretation of the titration line of ferrous iron with an oxidizing agent like ceric cerium using the platinum-tungsten couple. K h e n the concentration of ferric iron becomes very small, the potential of the tungsten electrode is much more negative than of the platinum electrode. At the platinum electrode the potential of a solution 0.0001M in both ferric and ferrous in 0.LY hydrochloric arid is about the same as that of the more concentrated solution ( B , Figure 6). Line KBI gives the current-voltage curve of this dilute iron solution a t the rotated platinum electrode. At the tungsten electrode the cathodic current, K L , is equal to the anodic current, L M , a t a potential, L, nhich is much more negative than B. This interpretation of the difference in potential a t the platinum and tungsten electrodes is naturally of a qualitative naturr only, a quantitative interpretation being dependent on the size and rate of rotation of the electrodes and the shape of the anodic current-voltage curve of tungsten in the presence of iron. Experimentally in the solution 10-4.1f in both ferric and ferrous the potential a t the rotated tungsten electrode was found 0.24 volt more negative than a t the platinum electrode. Evidently the difference in potential betn-een the two electrodes will be much greater in a solution 10-*11f in ceric cerium (plus some cerous) than in the dilute iron solution, because a t the tungsten electrode a value of the order of E to L n-ill he found.
Figure 6. Current-\-oltage Curves at Platinum Electrode Residual current and potential a t tungsten electrode H M G D . Current-voltage c u r r e in 0.l.W perchloric acid a t W A B C . Current-voltage curve a t Pt of solution 0.01.bl in F e + + +and 0.0141 in F e r ? K B I . As .4BC but 10-4 in Fe’++ and 10-4 I\. in F e + +
When ceric cerium is added to a solution of 10-2-1f ferric iron, xhich is free of ferrous, the potential a t the tungsten electrode will be more positive than E and it will increase gradually with increasing escess of ceric cerium ( M G D , Figure 6, becomes fairly flat at, potentials more positive than D ) . Thus the tungsten electrode does not exhibit a break in potential at t,he equivalence point in the titration of ferrous iron with ceric cerium a8 the platinum electrode does. The largest difference in potential between the two electrodes is found after the equivalence point, and it decreases slightly with increasing excess of ceric cerium.
ANALYTICAL CHEMISTRY
1690 T h i s interpretation accounts for the type of titration curves observed by Willard and Fenwick (18), using the platinum-tungsten couple in the potentiometric titration of ferrous iron with dichromate or permanganate. A more quantitative account of the behavior of a tungsten electrode will he given in :I subsequent paper with Xightingale.
Near the end point the line (region 3) becomes straight as in a n amperometric titration with one indicator electrode at constarit applied e.m.f. After the end point the current remains zero, since the system S20s---S40s-- is irreversible. The end point can he found graphically or by titration t o zero current.
AMPERO\IETRIC TITRATIONS
Two Identical Indicator Electrodes. Consider a n oxidationreduction system like iodine-iodide or bromine-bromide which gives reversible waves at a well-stirred platinum electrode. When the system is well poised, the coricentration overpotential can be neglected when two electrodes are placed in the solution and a small e.m.f. E is applied. -4s long as the system can 1)e considered ideally depolarized (region 1 in Figure i ) ,E = iR---e.g., E = 0.020; R = 200; i = 100 pa. We now remove one of the constituents--e,g., iodine by titration x i t h thiosulfate (the thiosu1i:Lte-tetrathionate system is completely irrevervible :it a p1:itinum electrode). When t,he concentration of iodine gets smaller, the concentration overpotential a t both electrodes can no longer be neglected (region 2 in Figure 7 ) .
E - iK
=
T o
-
Xe
r-----
/'
I
/
T u
-
(2I
Xc
Writing for convenience the iodide-iodine system :is :I one-electron transfer sj-stem, we have: X~ =
C
+ RTF In ~~
co
(3,
-5 c1-
c: is concentration of iodine a t anode,. c: that :it c:itliode.
For
the current me have the rehtioii
i
=
k(?: -
(,
,
=
k/p
-
co,
(.X
The vttlues of k are the sttine for identical electrodrs of the enme size and a completely revwsible system. From 1:quation 5 :
Figure 7 .
E
c.g., c,?
=
0.020
=
1.2i
x
IO-3-v;
c:
=
2.i3
x
10-3.\-
It is of interest to consider the relation between the current and the ditfusion current near the end point as a function of t,he applied e.m.f., neglecting again the iR drop and assuming perfect revrrsihility. This relation is given in the last rolunin of Tahli. 111. Khile the current is 0.2id at an applied e.ni.f. of 0.010 volt ancl 0.4id a t 0.020 volt), it increases t o practically j d at 0.100 volt or greater. The advantage of an applied e.m.f. of ronsiderahly mow than 0.1 volt is that the value of iR remains negligibly sm:d OVCI' a larger region t.han when the applied e.m.f. is small and thus the titration line remains straight over a longer dist'ince than a t small applied e.ni.f. The larger currerit sensitivity at a giver1 concentration of iodine a t higher applied e.m.f. is of little prartical consequence, hecause the sensitivity Can he varied a t will b y varying the size of the electrodrs and to a minor extent by varying the speed and kind of st,irring.
0.050 0.100
0.200 0.500
or =
L'c
kc(1 - k ' ) = Zd(l - k ' ,
Volt
-0
012
-0 036 -n.o93 -0.182 -0.48
Volt
illd
0.008 0.014 0.017 0.018 0.02
0 37 0.74 O.O(i 1.0 1.0
(9 I
and =
=
0.002.V; c,"
E , Volt 0 020
i
Amperometric Titration Lines Using Two Indicator Electrodes
Table 111. rC and A. and i / i d in Titration of Iodine (Large Excess of Iodide) with Thiosulfate in Region 3 (Near End Point) 1(2c - cp,/c; = 10Eia.Q6o= A . T~ - li represents difference hetween cathode potential li a n d oxidation potential li(i = 0 ) 1 ilc(-=) m i - II),
From Equations 3, 4, and G :
cp
I
I
I
EX\LIPIX.
=
REGION 2
F . ,e
(1)
Kear the end point (region 3) Z Rusually becomes negligibly sniiill as compared to X. - re( a is anode; c is cathode). If necessary, a correction for iR can he n u d e by a series of nppro\;im:ttions. Keglecting I R
E
I
REO'OY I
(10)
Therefore, i varies in proportion to c and, neglecting volume changes, according to a straight line. T h e characteristics of the line giving the change of 1 from the point where the system is well poised (region 1) to the end point is represented in Figure 7 .
From the concentrations of iodine a t the cathode and a t the anode the cathode and anode potentials can be calculated. The concentration overpotential a t the cathode is greater than a t the anode and increases with increasing applied e.m.f. When 7 has become practically equal to Z d , the iodine concentration at the cathode approaches zero The difference between the oxidation potential of the system ( i = 0 ) and the cathode potential increases
V O L U M E 26, NO. 11, N O V E M B E R 1 9 5 4
1691
\\-ith the applied e.m.f. and becomes virtually equal t o this e.m.f. a t large E applied, as can be seen from the figures in Table 111. In Table I11 the calculated relative cathode and anode potentials are reported, the difference between the two, of course, al\vays being equal to the applied e.m.f. It is a simple matter t o calculate the cathode and anode potentials during the amperometric titrat,ion. Construction of a potentiometric titration curve :tt constant applied e.m.f. (16) reveals that such a pot,entiometric titration has no advantage over the classical type of potentiomet,ric titration or t,hat a t constant current. In the coulometric titration of iodide in 2-b- hydrochloric acid in the presence of bromide n-ith electrolytically generated bromine Wooster et al. (19) determined the end point amperometrically by nieasuring the current between two platinum electrodes. Tiiw (12) carried out the amperometric titration with brominr. iis rcagent. I n the beginning of t.he titration iodide is oxidizd to iodine and the current increases t o a maximum. Thereafter the iodine is being osidized to iodine monobromide and the currrnt decreases:
I? + Br,-
+ Br-
$
21Br2-
Thc systrms I, -+ IBrs- and Ur3- + Br2 give composite ( 1 2 ) :iiid prol~alilyreversible waves. At the equivalence point the current has :L minimum value, but is not zero because of dissociation of 1Hr.- into iodine a,nd bromine (19). Otherwise the titration linr bcxfore the end point is similar to that in Figure 7. Since the systcm becomes depolarized again after t h r end point, the titration line will be the approximate mirror image of t h a t I)c.fore the end point (see dotted line in Figure 7 ) . .klthough the systems ferrous-ferric and especially cerous-ceric are not perfrctly reversible, a similar t,ype of titration line is approached in the titration of ferrous iron wit,h ceric sulfate. T h e incomplete rclversibility of the ceric-cerous system is evident from the plots in Figure 2 in the paper by Stone and Scholt,en ( 1 6 ) . I n the titration of ferrous iron with ceric sulfate a t a n applied e.m.f. of 100 mv. the current, is large and increases according to a straight line after t,he end point. However, in the titration of ferrocyanide with ceric sulfate a t a n applied e.m.f. of 50 mv. the current remains practically zero after tthe end point. I n the calculation of the titration line obtained r\-it,h systems which are not ideally reversible, the activation e.m.f. E,,* must be considered (see Bradbury, 2 ) . Seglect,ing the iR drop \\-e have
E
- Eact=
7ra
-
7rc
(11)
Assuming that the sum of the activation potentials (equal t o E,,,t) is constant a t small currents, the effective e.m.f. is equal to the 'tpplied minus the activation e.m.f. Therefore, in the above calculations Equation 11 is used instead of 2. From data listed for various systems by Stone and Scholten ( 2 6 ) on the e.m P. to be :ipplied in this type of amperometric titration, i t can be inferred thixt the iodine-iodide couple is the only ideally reversible couple under the experimental conditions. T h e ferrous-ferric, titanoust i t:inic, and ferro-ferricyanide couples have activation potentials
which are smaller than those for ceric. cerous, permanganate (acid)-manganese( 11), and vanadate-vanadyl. Exact information on these potentials can be derived from current-voltage curves at rotated electrodcs. CONCLUSIONS
I n conclusion a brief comparison might be made bet\\-ecn nmperometric titrations with one and with two indicator electrodes. With reversible systems the titration lines near the end point in both methods become ident,ical, if the applied e.m.f. (tn-o indicator electrodes) is large enough to yield id. In many instances the equipment required is much simpler in amperometric titrations with one indicator electrode. For example, in the titration of iodine with thiosulfate the indicator electrode is simply connected with the saturated calomel electrode without applying any e.m.f. I n other titrations it is usually not necessary to apply a n e.m.f. when a suitable reference electrode, is available. As far as simplicity, accuracy, speed, and sensitivity are concerned, amperometric titrations (with one or two indicator electrodes), in general, have great advantages over potentiometric titrations. This is especially true in dealing Lvith extremely dilute solutions (10-4LV or less), which can be titrated much more accurat,ely and precisely by thc nmperometric than hy t'he potentiometric technique, especially when one indicattorelectrode is used and the titration is carried out at a suitable potential. With relatively strongly irrcversible systems the amperometric titration with one electrodr is superior to the two-electrodr systems. REFERESCES
(1) Bertin, C . , Anal. C h i m dcta, 5, 1 (1951). (2) Bradbury, .J. H., Trans. Faraday Soc., 49, 304 (1953). (3) Charlot, G., ,4nal. Chim. Acfa., 7 , 408 (1952); 8, 65 (1953). (4) Coursier, ,J., Ibid., 7, 77 (1952); 10, 182, 265 (1954). (5) Delahay, P., Ihid., 1, 19 (194T); 4, 035 (1950); 6, 542 (1952). (6) Dutoit, P., and von ll'eisse, G.. .T. c h i m . p h y s . , 578, 008, 630 (1911). ( 7 ) Duyc.kaerts, G., Anal. Chivi. Acta, 5 , 233 (1951); 8, 57 (1053); ITid, chin,. belge, 18, 795 (1953). (8) l..oulk, C'. JT., and Bawden. A . T.. -1..4m. Chem. Soc., 48, 2045 (1926). (9) Furinan. S . H., SAL. CHEJI.,26, 84 (1954). (IO) Gaiiguin, It., Anal. C h i m . Acta, 5 , 200 (1951); 7, 172. 360, 408 (1952), (11)
Hostetter, ,J. C., and Robert., TI. S..J . .4m.C h e m . Soc., 41, 1343
(12) (13)
Kies, H. L., .471a2. C h i m . d c f a , 6, 190 (1952). Kolthoff, I. AI., and Lingp.7.2, J. J.. "Polarography," 2nd ed., Sew York, 1nterscien.e Publishers. 1952. Kortiim. G., and Bockris, .J. O ' l I . , "Textbook of Electrocheniis" p. 39, Kern Tork, Elsevier Publishing Co.. 1951. Reilley, C. N., Cooke, \I7. D . , and Furman, S . H.. A x . 4 ~ . CHEM.,23, 1223 (1951). Stone, K. G., and Srholren, FI. C . . I b i d . , 24, 071 (1952). Tan Same, R. G., and I'cur\-ick. I:., J . A m . C h e m . Soc., 47, 19
(1919).
(14) (15)
(IO) (17)
(1925).
(15) Willard, H. H., and I.'Ls;lwick,F., I h i d . , 44, 2504, 251(3 (1922). (19) Wooster, W.S., FarriJi-ton. P. S., and Swift, E. H., A r a ~CHEM., . 21, 1457 (1949).
RECEIVED for rev:c\v July 3 , 1934. Accepted September 27, 1954.
