AIterna ti ng Current Pola rogra phy Improved Experimental Arrangement, Examination of Theory, and Study of Cadmium(II) Reduction. HENRY H. BAUER and PHILIP J. ELVING University of Michigan, Ann Arbor, Mich.
b The technique of alternating current polarography employing an alternating potential of millivoltage amplitude has important potentialities for analysis and the study of electrode processes. The fundamental theoretical and experimental factors involved in the technique are critically reviewed. Certain innovations in handling the experimental and calculated data have been introduced to facilitate use of the technique. An experimental procedure is described, which overcomes the greatest uncertainty previously connected with the use of alternating current polarography-viz., the making of adequate corrections for the effect of the series resistances in the circuit. The data obtained on the reduction a t the dropping mercury electrode of cadmium(ll) in hydrochloric acid solution have been used to test the validity of the theoretical treatment of alternating current polarographic phenomena, These data yielded a value of the heterogeneous rate constant for the reaction specified of 0.165 cm. sec.-l The experimental and mathematical procedures are described in detail to facilitate use of the technique.
T
HE USE of alternating potentials
under polarographic conditions was first introduced in attempts to develop more rapid methods of polarographic analysis and better ways of determining the half-wave potential, Ellz; later, it was extended to the study of more fundamental electrode phenomena (2, 3, ?‘, 10,16, 22, 34, 26). The techniques, developed by research along these lines, may be usefully, if arbitrarily, divided into two groups: The first involves the use of alternating potentials of amplitudes as large as the whole potential range under investigation-e.g,, 1 volt or more-the second involves the use of much smaller amplitudes, of the ordei of millivolts or tens of millivolts. The technique of “alternating current polarography” falls into the second category. It aims a t investigating electrode pr’ocesses by the use of as small an alternating potential as is feasible, superposed on the steady polarizing potential; the alternating current pro334
ANALYTICAL CHEMISTRY
duced under these conditions is measured and recorded as a function of the applied steady potential, with the resulting graphs termed “alternating current polarograms” (a.c. polarograms). The main characteristic of an alternating current polarogram is the feature of observing, when a reversible redox process is being studied, a maximum alternating current a t steady potentials corresponding to E I , ~in the conventional (direct current) polarogram (Figure 1). The literature on alternating current polarography has been reviewed through 1958 (8); since then, Miller (26) has described the adaptation of a Sargent Model XXI polarograph for use as a recording alternating current polarograph.
Potential
--t
Figure 1. Relation between direct current and alternating current polarograms
- - - -. D.C. polarogram
-.
A.C. polarogram
Almost all the studies conducted to date by the technique of alternating current polarography have used a sinusoidal alternating potential. Barker and Jenkins ( I ) , however, used a superposed alternating potential of squaren-ave shape. It is convenient to reserve the term “alternating current polarography” to refer to the use of sinusoidal alternating potentials, and t o use “square-wave polarography” to describe the latter method; this terminology has become general usage in recent years. Square-wave polarography under the proper conditions has advantages over alternating current polarography in that the capacitative current admitted by the double layer is relatively less in comparison to the faradaic current.
This fact permits analytical application to lower concentrations of reducible species (of the order of 10-7M). However, the apparatus required for squarewave polarography is of greater complexity than that used in alternating current polarography. illternating current polarography can be used t o test the reversibility of a redox process, and to measure the heterogeneous rate constant of the electrochemical process (6) and the transfer coefficient (14). As an analytical tool, it has certain advantages over conventional polarography-in rapidity (usually oxygen need not be expelled from the cell), in the separation of waves occurring a t potentials close to one another, and in the determination of a small concentration of a substance in the presence of a large amount of a more easily reduced substance. The technique can also be used to study adsorption processes a t the dropping mercury electrode with nonreducible surface-active substances (13) as well as with reducible ones (4,6). The present paper is the first result of an investigation whose objective is a critical study of alternating current polarography from the viewpoint of its application to studying the nature of organic electrode processes as they are affected by experimental conditions, and t o organic analysis. I n view of the relative unavailability of the journals in which the work of Breyer and his collaborators on alternating current polarography has been published, the fundamental experimental and theoretical factors involved in the technique have been critically reviewed. An apparatus and experimental procedure for alternating current polarography are described which are felt to be more reliable than any previously published, because the series circuit resistances have been carefully niinimized and corrected for. The results thus obtained inspire much more confidence than previous ones; consequently, they are used to test the validity of the theoretical treatment of alternating current polarographic phenomena. As part of this program, the heterogeneous rate constant for
the reduction of cadmium(I1) to Cd(Hg) in hydrochloric acid solution was measured; the methods of handling the data are described in detail t o facilitate use of the technique by others.
1-
L
o
v
6 7
A T
c
Q-
-
D +b
D
---_
E@$(G
~ _ _ _ _
Figure 2. Schematic circuii for alternating current polarography
BASIS OF ALTERNATING CURRENT POLAROGRAPHY
Generally, apparatus for alternatiug current polarography consists of a conventional polarographic circuit, plus provisions for applying a knonn alternating potential of low amplitude across the interface and for measuring the alternating current produced (Figure 2). The function of such an arrangement can best be seen by considering the electrical nature of the polarographic cell in such a circuit, as represented in Figure 3, where Ro represents resistances in the circuit outside the cell; R.tf, resistance of the dropping mercury electrode; Rs, resistance of the solution; Cx,the capacity of the double layer a t the dropping electrode; and Z, the faradaic impedance, which describes the behavior of a redox system and cannot be satisfsctorily espresscd in terms of conventional circuit elements (19). I n the following discussion, the total of RM and Rs is referred to as the (total) series resistance, R x ; CH is termed the capacitative impedance and the current flowing through it, the capacitative current; the current through Z is the faradaic current. I n using alternating current polarography to investigate redox reactionsLe., the behavior of the equivalent element, 2-effects due to the other elements in the circuit must be eliminated or corrected for. I n practice, measurements are made of the quantities RT’, the total effective impedance of the cell in the presence of the redox system, and Rc‘, the total effective impedance of the cell in the absence of the redox system; both impedances are measured under the influence of the small alternating potential, AV’, operative across the cell a t the steady direct current potential, E,, where the maximum altern lting current flow in the presence of the redox system. From these measured quantities, the faradaic impedance, 2, and the alternating potential, AVO, operative across the interface-Le., across 2-have to be calculated. Effect of Series Resistances. I n conventional polarography, the presence of series resistances is not so disturbing as in alternating current polarography. I n the case of a reversible redox process, the resistance simulated by the reaction, dE/di, is a t a minimum a t where the slope of the current-potential curve is greatest; dE/di, which can be calculated from
D.C. potentiometer Source of alternating potential Monitoring [alternating potential) voltmeter (input voltmeter) D. Shunting capacitors E. Polarographic cell (pool anode) F. Reference cell G. Potential-dropping resistor H. Amp Iifier K. Output vacuum-tube voltmeter I . D.C. galvanometer A. B. C.
I
Gel I [ potentlal=&’)
*
Figure 3. Electrical representation of polarographic cell and circuit Ro.
Resistance of circuit elements outside cell R>w. Resistance of capillary electrode R s . Resistance o f solution Z. Faradaic impedance CH. Capacity of double l a y e r
the familiar Heyrovsk9-IlkoviE equation
tive and will make the slope of the wave around Eliz less by lo%, compared to the theoretical values. hIoreover, only the actual potential values across the interface and, hence, the shape and position of the curve are affected by series resistances; the magnitude of the diffusion current is not altered. In alternating current polarography the situation is different. For a reversible redox system, a t concentrations of the order of millimolar and a t frequencies greater than a few hundred cycles, the impedance of the interface becomes less than 100 ohms; a t frequencies of 10 kc. and higher, the impedance may be less than 20 ohms. Since the alternating potential is applied across the total impedance of the circuit, only a fraction of it will be operative across the interface: this fraction is responsible for the obserred alternating current flow. Thus, it is necessary to measure not only the alternating current, but also the alternating potential actually operative across the inteifacial region. The resistance of the capillary dropping mercury electrode can vary from values of the order of 10 ohms to upwards of 100 ohms. A platinum wire sealed into the capillary near the tip, to make contact with the mercury column, decreases the effective resistance to a very useful extent (23). Calculation of Alternating Potential A vacuum-tube across Interface. voltmeter can be used t o measure the alternating pohential, AV’, across the cell. The alternating potential operative across the interface, is given by
(4)
is given by
Determination of the series resistance,
Rx, is subsequently described. and at, E112has the value
The minus sign has, of course, no significance for the present purpose. For a case where n = 2, T = 300°, and i d = 30 Fa., Equation 3 gives a value of about 1700 ohms for the simulated resistance of the redox reaction. Under normal polarographic conditions, the value will not be much less than this, as polarographic currents are rarely greater than 30 pa. and generally are much less Series resistances in the circuit will not invalidate the results, if these resistances are small or negligible compared to 1700 ohms, as they usually are. A total series resistance of 170 ohms, for example, will make the apparent value of El,, only about 2.6 mv. mole nega-
Measurement of Alternating Current. Instead of calibrating the vacuum-tube voltmeter-amplifier arrangement (K-H of Figure 2) to determine absolute currents, it is preferable to substitute for the polarographic cell after each experiment a suitable variable resistance and to determine the resistance which produces the same meter deflection (16); in other words, to measure the impedance of the cell rather than the current. Fundamental Theoretical Equations. The essential approach t o a n understanding of the factors involved in alternating current polarography has been through the formulation of equations relating the faradaic alternating current, and the frequency and amplitude of the alternating potential to the other parameters in the system. On this basis, Breyer and coworkers have derived two equations expressing the faradaic alternating current. VOL. 30, NO. 3, MARCH 1958
335
The first equation was derived (12) for a reversible redox process, small amplitude of the alternating potential, and frequencies sufficiently low that the process can follow the alternating field; the following equation is obtained when the peak current used in Equation 40 of the original reference ( I $ ) is converted to the r.m.s. current: (Ai),,
=
:.In.*.
nFACB(wD)l/a tanh ( n 3 ) (5)
2RT
2312
where (Ai)z8, r,m,,, is the root-meansquare value of the faradaic alternating current flowing a t the summit potential E,, A is the area of the dropping electrode (at maximum age of the drop), CB the bulk concentration of the electroactive species, o the angular frequency of the alternating potential (radians per second or 27rj where f is the frequency in cycles per second), and n, F, R, T, and D have their usual significance; AVOhas been defined The second equation (6) expresses the variation of the alternating current with changing frequency of the alternating potential: (Ai).&,
Equations 8 and 9, which can be esperimentally examined and whose effects can be used to evaluate the relative merits of the two equations: the proportionality of the current to tanh (nF AV0/2RT) or to AVO (the difference between these functions increases markedly with increasing AVO) (Table I); and the correctness of Equation 9 in predicting the frequency dependence of the current.
Table
I.
Mv. 1 2 5 10 15 20 25 30
Variation of Function tanh (nF L ~ V C / ~ Rwith T ) ~AVO 1 nFAV0 tanh nFAVo ~
2 RT 0.078 0.156
0.388 ~.~ 0.742 1.05 1.31 1.50 1.65 Based on n = 2 and T =
tanh 2RT 0.078 0,078 0.077 0.0742
0.070
0.0655 0.060 0.055 298’ I(.
-
I rn,B.
n2F2AC~AV0 1
1
1
li
2
[a- ( 2 k , 2 ~ D ) 1 -l’ 2 a]
2512R T
(6)
where k , is the heterogenmus rate constant of the electrochemical process, given by (7)
k , = (wAtfD/8)’’2
where w.,, is the angular frequency a t which the faradaic alternating current is a maximum ( w M = 2.rrJ,u). Equations 5 and 6 can be written more conveniently in the forms (Ai)E,,
-
r.m.a.
Applied
4RT
x x x.
- - -.
alternating
Experimental points Interpolated capacity current due to double layer Meter deflection proportional to alternating current
-
n2F2AAV&E (TD)”’ 2 2 1’2
[:t
(ff..u)”*
+
(9)
6 1
The treatment leading t o Equation 9 is rather more approximate than that used in deriving Equation 8, because the problem is complicated in the former case by taking into account the effect of the heterogeneous rate constant, assumed in the latter derivation to be very large. There is an obvious discrepancy between these tm-o equations in respect to the dependence of the current on the amplitude of the alternating potential, though in the limiting case of vanishingly small amplitudes, where the equations are most rigorously applicable, this discrepancy disappears. There are two factors implicit in
336
F. ->
potential
Figure 4. Typical current polarogram
M. r rn s.
DC
ANALYTICAL CHEMISTRY
FUNDAMENTAL EXPERIMENTAL APPROACH
Plotting of the Alternating Current Polarogram. The oscillator ( B ,Figure 2 ) is set to the desired frequency; the steady (direct current) potential is set a t a suitable value; the alternating potential control is set to give the desired potential amplitude (at maximum drop age) across the cell; and the output vacuum-tube voltmeter deflection is recorded (again a t maximum drop age). This sequence of steps is repeated a t suitable intervals in the range of steady potential under investigation. The whole procedure can then be repeated at, other frequencies and amplitudes as desired. Characteristically, a t potentials where no redox process occurs, the alternating current is the same whether or not a depolarizer is present in the solution
(Figure 4). I n the potential range where the redox process occurs, the current is greater than in absence of the depolarizer, reaching a maximum a t the summit potential, E,; it is the faradaic alternating current a t E. which is of most interest. Consequently, the meter deflections of primary interest in Figure 4 are M r and M e , where M T represents the total alternating current flowing a t E, in presence of a depolarizer and Mc in its absence. Mc is obtained by interpolating (linear interpolation is usually adequate) readings taken a t potentials sufficiently removed from those potentials where the redox reaction occurs; M c represents the current admitted capacitatively a t the potential, E,, by the double layer in absence of a redox process. To obtain data for calculating the currents, the polarographic cell is replaced by a variable resistance. Vsing zero applied direct current potential and the same alternating potential as was used in obtaining the polarogram, the resistance values, RT’ and Rc’, necessary to produce meter deflections, MT and 3fc, respectively, are determined. Correction for Series Resistances. There are two ways of correcting the measured values, RT’ and Rc’, for the effect of the series resistance, R,. The first, which is most convenient, and is the one used in the present work, is based on the following considerations : The capacity of the double layer in the absence of a redox process is independent of the frequency of the alternating potentia1 (18). Capacitative impedance is inversely proportional to both frequency and the capacity itself, Rc
l/fC
(10)
where R c represents the impedance of the double layer and C, its capacity. Since C is independent of frequency, the product jRcis independent of frequency; this fact enables calculation of Rx by measuring Rc’ over a wide range of frequencies: RT
=
RT’ - Rx
(114
- Rx
(Ilb)
R c = Rc‘
and f R c = f(Rc’
- Rx)
(1lc)
If the measured quantity Rc’ is 505 o h m a t 200 c.p.s. and 15 ohms a t 10 kc., then, from the frequency independence of f R c and Equation l l c , 200 (503 - R x ) = 10,000 (15 - R x ) ( 1 2 4 and R X = 5ohms (12b) In practice, of course, to allow for experimental error in the measurements, more than tn.0 values of Rc’ are used in these calculations.
gives the value of Rx. The values of RT’must then be corrected vectorially, after measurement of the phase angles involved (15). It is doubtful whether this way of making the correction will be better in practice than the first, although in theory it is more rigorous. The cumulative effect of the experimental errors, especially in the measurement of the phase angles, would cancel the better accuracy which could be theoretically obtained. Calculation of Faradaic Impedance. The faradaic impedance, %, must be obtained from RT and Rc by a vectorial calculation, because the capaci-
a cos+
Figure 5. Phase relations of alternating current AVO. Alternating potential ocross interface a = AVo/Rc. Capacitative current b = A V ~ / R T . Total alternating current c = AVa/Z. Faradaic current
duced sDecies). Eauation 13 can be more convenikntly writtin aa cot e’
=
where F Consequently,
=fM/f
sin 8’ = 1/(2
+F
- 2F19*’*
and COS
e‘ = (F”’ - l ) / ( 2
+F
- 2F”’)’’’
From Figure 5, b2 = az cosz .$
+ (c + a sin +)* = u2 + c2 + 2ac sin 4
or
1/Rr2 = 1/Rc2
L
-1
~ 1 ’ 2
+ l/Za + (2 sin 9)/RcZ (19)
From Equation 13a sin =
+
+
sin e‘ cos ( ~ / 4 ) cos e‘ sin ( ~ / 4 ) (20a) ~1:2/21’2(2+ F - 2F1/2)1/2 (20b) =
Combining Equations 19 and 20
I
Figure 6.
G, Figure 2. Resistance bank RI to RIZ Continuously variable amplitude control afforded b y RIS to R1.
500 ohms
R E . 100 ohms
Rls.
Rz. RI.
4 5 0 ohms 400 ohms 3 0 0 ohms 2 5 0 ohms 2 0 0 ohms 150 ohms
Rg.
75 ohms 5 0 ohms
Rib.
R4.
Rs. Rs. R7.
Rlo.
RII.
RIZ. R13. R14.
This method of correcting for the series resistances is only an approxiniation; strictly, a vectorial method should be used. The value of Rx calculated by the procedure given does not represent the actual resistance, but rather the average correction for the series resistance in the direction of the c,Lpacitative impedance. I n applying the same correction to RT’, an error is introduced due to the phase difference hetwcen RT’ and Rc’. The latter error nil1 be small, as the correction becomes appreciable only a t higher freytiencies, when RT’ is not much smaller t h i n &’-the capzcitative and the total currents are nearly equal and the faradaic current small by comparison, the phase angle betn een the capacitative and the total currents is then small. In the second correction method, the actual value of the series resistance, Rr, can be determined by measuring Rc’ a t sufficiently high frequencies, n here the capacitatii e iinpednnce becomes small (Equation 10) nit11 respect to the series resistance; extrapolation of Rc‘ to infinite frequency
+F
- 4F”’)
1
Calculation of 2 by Equation 21 requires knowledge of F , and hence of f~ (Equation 15), as well as of RT and Rc, I n the present study, calculations r e r e made for several assumed values of fM, selected from the range of frequencies indicated by the experimental results (see subsequent discussion).
Ri7
9 ohms 9 0 ohms 150 ohms SP, 12T (shorting) ,
R17
Si.
2 5 ohms 10 ohms 100 ohms pot. 1 ohm
(4
[m - 2 ( 2 + F - 2F1’3 z z ]
Polarizing circuit details
Sz. SP, ST Ss. DP, 3T
tative current through the double layer is known to be almost exactly r / 2 out of phase with the alternating potential, while the faradaic current and, hence, the total current have different phase angles, $I and y, respectively (Figure 5) (17). The angle y can be determined experimentally, but not too accurately, by observations of the Lissajous patterns produced on an oscillograph screen by inipressing the alternating current and the superposed alternating potential on the plates of the oscillograph; the procedure is laborious because the phase relations change during the life of the drop (1.4). The angle 4 can be calculated from the theoretically derived expression (6) cot e’ = 2k,z - 1
where
+ ?i/4
+
=
e’
2
=
@/&)I
113) (13%)
and (13b)
(D is assumed equal for oxidized and re-
EXPERIMENTAL
Electrical Apparatus. The polarizing circuit used is shown in Figures 2 and 6. A Fisher Elecdropode was used as the source of the steady direct current polarizing potential and as the direct current galvanometer. The variable resistance inserted in place of the cell to determine the simulated resistances, Rr‘ and Rc‘, was a Leeds & Northrup No. 4750 decade resistance box. The oscillator used to provide the alternating potential was a HewlettPackard Model 200D, feeding into the circuit through a continuously variable amplitude control (Figure 6). This is helpful, because, for a given gross superposed alternating potential, the potential operative across the cell (and even more so across the interface) changes markedly in magnitude T\ ith different concentrations of the reactive species, with the magnitude of the applied direct current potential, and with the frequency of the alternating potential. The significance of the alternating current polarogram (Figure 7 ) is enhanced by keeping the alternating potential across the interface constant a t each value of the steady applied potential.
vot.
30, NO. 3, MARCH 1 9 5 8
337
In practice, the former potential is kept relatively constant by keeping the alternating potential across the cell constant. For example, the impedance of the interface may be 300 ohms at potentials just outside the region where the redox process occurs, but only 60 ohms neai Elil ithis is roughly the case for 5 X lO-4-V cadmium chloride in 0.5N hydrochloric acid at 1000 C.P.S. and an alternating potential of a few millivolts amplitude). If the impedance in the remainder of the circuit totals only 30 ohms and a fixed alternating potential of 5 mv. is superposed, then a t potentials where no redox process occurs, the alternating potential operative across the interface will be 4.54 mv. (assuming the impedance vectors to be parallel), while near El 2 the effective potential mill be only 3.33 mv. As a consequence, the peak on the alternating current polarogram will not be as pronounced as it would he if the alternating potential across the cell were kept a t the same level a t all values of the steady potential (Figure 7 ) . The direct current potential source, galvanometer, reference electrode, and salt bridge 1%-ereshuntpd by 100-pf. capacitors. A pool anode was used so that the reference electrode could be thus shunted ( I I ) . The purpose of such shunting is to decrease the alternating current resistance of the circuit elements (Figure 3, EO),making it unnecessary to change the setting of the alternating potential control as much as would otherwise be required to keep the alternating potential across the cell constant a t each value of the steady potential. The amplifier (Figure 8) was constructed following the circuit of Breyer and cou-orkers (9) with a number of changes, the most important of which was the direct coupling of the amplifier to the vacuum-tube voltmeter, omitting the output transformer previously used. The sensitivity of the instrument is such that the extra gain resulting from the
and the bias a t about 12 volts, resulting in an instrument current of about 50 ma. This arrangement considerably improved the stability of the amplifier and voltmeter, and decreased the stray pickup. A cathode-ray oscilloscope was incorporated into the circuit, taking its input from the plate lead of the last amplifier stage; this provides a continuous check on the wave form of the signals, ensuring that measurements are actually being made of a sinusoidal current. By connecting the cathode-ray oscilloscope to the amplifier via a suitable frequency filter rather than directly, it should be possible to measure the transfer coefficient of electrode
use of the output transformer is superfluous. I n addition, complications due to the necessity of matching the impedances in the primary and secondary circuits are avoided. The output vacuum-tube voltmeter (Figure 9) differs from that of Breyer and coworkers (9) in the provision of a four-position switch with shunts for the meter, giving an "off" position and three "on" positions of different sensitivities for more convenient operation. Instead of the po\.i.er-pack described by Breyer and coworkers (9),a HewlettPackard RIodel 712A pover supply, fed from the lines through a Sola No. 30864 constant-voltage transformer, lyas used. The plate voltages were set a t 400 volts
CIZ
*
13
IOVTVM
c
CI
c7+
I
I
T
rzo I
*
4 B-
BA
Figure 8. 1 0 0 kohms 2 5 0 kohms R3. 5 kohms R4. 100 kohms R6. 500 kohms Re. 2 . 5 kohms Rj. 2 megohms R8-R1j. 50 kohms RIB. 200 kohms R1g. 2 kohms RZO. 1 kohm
R1.
Rz.
High gain amplifier (H, Figure 2) 300 100 1 20
kohms kohms megohm kohms 5 0 0 kohms 5 K ( 1 0 watts) 1 0 kohms 2 5 0 kohms 10 kohms 1 0 0 kohms pot. 1 megohm
c1.
0.01 pf.
c12.
Cz.
0 . 0 0 2 5 pf. 8 . 0 pf." 2 5 . 0 pf." 0 . 2 5 pf. 2 5 pf. 0.01 pf. 0 . 0 2 pf.
ci3.
C3.
C.i. C5. Cs.
c7.
c*. Cg.
cm. c11.
8' pf. 0. 1 pf. 8" pf.
Electrolytic condenser.
Applied steady potential
-
Figure 7. Schematic alternating current pola rog ra ms demonstrating importance of series resistance effects I.
Taken with fixed alternating potential AV II. Token with alternating potentiol kept at AV' across cell at all values of E 111. Ideal curve, for olternoting potential AVc across interface a t every value of the potentiol E
338
ANALYTICAL CHEMISTRY
Rt. 2 megohms Rs. 100 kohmr R,. 100 kohmr R4. 2 megohms Rr. 5 kohmr Re. 5 kohmr pot.
R 7 . 5 kohms RE. 1 0 megohms Rg. Rle.
R11.
c1.
500 kohms 1-ma. shunt 1 0-ma. shunt 0.1 pf.
Cz. 0 . 5 pf. 6H6 Vr. 6 V 6 VJ. 6 V 6 MI. 0-100ma. Si. DP, 4T Vi.
o. 1
pf.
0.1 pf. Clr. 25" pf. CIS. 0.5 pf. C16. 0.1 pf, Vi. 6SJ7 Vz, 6SJ7 Vs. 6 V 6 SI. SP, 12T (shorting switch)
processes by the method of van Cakenberghe (14). The input vacuum-tube voltmeter, reading between the pool anode and the dropping mercury electrode, was a Heathkit Model AV-3 voltmeter, which has a maximum sensitivity of 10 mv. r.ni.s. full-scale. This instrument was calibrated against a Hewlett-Packard lIodP1 400C voltmeter. Dropping Mercury Electrode. The capillary used (Corning marine barometer tubing) was 12.5 cm. long and had a bulb (approximate internal diameter of 7 mm.) blown near its lower end, into which a platinum mire n as sealed. T h e impedance between the wire and a mercury pool into which the tip was immersed was about 3 ohms. The rate of flow of the mercury was measured, as an average over the useful life of the capillary, by allowing the latter to drop into distilled water, with the same head of mercury as in the actual experiments (68.8 em.), during those periods when measurements were not being made. Over a period of 141 hours, m averaged 0.44 mg. per second. The drop time, t, measured a t E , (about -0.66 volt us. S.C.E.), changed slowly over this week, being 10 seconds when first used and 12 seconds just before the capillary stopped dropping. The area of the drop, A , at maximum age, calculated for t = 11 seconds, is 2.43 X sq. em. Assuming the same value for m, but values for t of 10 and 12 seconds, respectively, the possible error in area is less than zk4'%. Subsequent measurement of a similar capillary showed the product of m and t to be constant within 4% for an 8% change in t with time. The low value of m and the high value of t appear to be characteristic for a capillary with a bulb near the tip. The measured t includes a n apparent quiescent period of 4 to 5 seconds, probably due t o retraction of the mercury up into the capillary just after the fall of each drop. The slom, progressive change in t may be due to accumulation of solute inside the capillary near the tip. No correction for the quiescent period is necessary when calculating the drop area a t maximum age from the measured values of m and t , as this area is proportional to (mt)2'3 and any correction for the quiescent period would involve multiplying m and dividing t by the same factor. Polarographic Cell. T h e H-cell used (21) contained a saturated calomel half-cell in the reference electrode arm, an agar plug in the bridge, and a mercury pool in the test solution arm. The latter pool as connected to the mercury pool of thp wturated calomel electrode via a shunting capacitor (Figure 2). Test Solution. A solution 5 . 5 5 X 10-4A11in cadmium chloride and 0.5JP in hydrochloric acid mas used to test the thcory and apparatus. The temperature
Table II. Calculation of Series Resistance Correction R J ~ , Values off& ( X for R X in Ohms Assumed Equal to C.P.S. Ohms 0 5 6'/4 7'/2 8'/s 8'/r 10 400 802 3.21 3.19 3.19 3.18 3.18 3.18 3.17 700 420 2.94 2.91 2.90 2.89 2.89 2.88 2 . 8 i 1,000 265 2.65 2.60 2.59 2.58 2.58 2.57 2.55 1,500 185 2 . i 8 2.71 2.69 2.67 2.66 2.65 2.63 2,000 136 2.72 2.62 2.60 2.57 2.56 2.55 2.52 3,000 90 2.70 2.55 2 52 2 48 2 46 2 44 2 40 4 000 72.7 2.91 2.71 2 66 2 61 2 59 2 56 2 51 6; 000 53.6 3.22 2.92 2.85 2 . j i 2.?4 2.70 2 . 6 2 8 > 000 40.1 3.21 2.81 2.71 2.61 2.56 2.51 2.41 10,000 34.4 3.44 2.94 2.82 2.69 2.63 2.57 2.44 14,000 27.5 3.85 3.15 2.98 2.80 2.72 2.63 2.45 Mean, R 3.06 2.83 2.77 2.72 2.69 2.66 2.60 Standard deviation, u 0.37 0.22 0.20 0.19 0.20 0.20 0 . 2 3 Percentage error, 100 c/R 12.1 7.9 7.2 7.0 7.4 7.5 8.8
f,
Table 111.
Dependence of Faradaic Current on Amplitude of Alternating Potential"
Tanh AVO, Current,b pa. Mv. (nFAVol2RT) 1.62 7.53 0.126 3.14 14.7 0.243 4.48 22.1 0.346 7.25 29.5 0.55 15.1 60.4 1.06 21.9 82.3 1.39 35.3 98.9 1.76 Mean Frequency = 100 c.p.5. Current = AVo/Z.
Current AVO 4.65 4.68 4.94 4.06 4.00
3.75 2.80 4.13
Dev. from Current Dev. from, Mean, tanh Mean, % (nFAVd2RT) % 112.6 59.6 f1.7 $13.3 60.5 f3.Z +19.6 63.9 $9.0 - 1.7 53.6 -8.5 - 3.1 57.0 -2.7 - 9.2 59.2 11.0 -32.2 56.2 +4.1 zk13.1 58.6 zk4.3
0
varied between 25.4" and 23 7" C., because the thermostat had to be off while readings were being taken t o avoid interference from stray pickup produced by the stirrer, circulating pump, and the relay of the thermostat The temperature coefficient of the faradaic alternating current is known to be less than 1% per degree a t these temperatures (10). Operating Procedure. The power supply, oscilloscope, and oscillator were switched on and allowed to 15-arm up for 30 minutes. Meanwhile, t h e test solution was placed in the polarographic cell and deoxygenated by purging with oxygen-free nitrogen; the direct current potpntiometer was standardized; the thermostat was switched off; the oscillator was set to give the desired frequency; the voltagedropping reqistor and gain controls were set to suitable values; and the meter of the output \ acuum-tube voltmeter TVas zeroed. The direct current potentiometer was then set to the desired value and the alternating potential control set to give the desired reading on the input vacuum-tube voltmeter a t the maximum age of the mercury drop. The deflection of the output vacuum-tube voltmeter m s recorded. These steps lvere repeated over tlie range of direct current potentials to be investigated. The drop time was measured a t E,. The lead3 to the polarographic cell &-ere then replaced by leads to tlie resistance box and the resistance i-alues corresponding to the meter deflections, J f T and .Vc, as revealed by inspection or plotting of the output vacuum-tube voltmeter reading 0.9. direct current potential, w r e determined.
DATA AND DISCUSSION
The results obtained in the study of the cadmium(I1) reaction are given in Tables I1 to IV. I n Table 11, the quantity Rc',measured as a function of frequency, has been used to calculate fRc for different assumed values of the series resistance, Rx. Graphical inspection indicates that the condition that fRc should be independent of frequency, is best satisfied by a value of R x of about 8 ohms, which figure has been used for R x in all subsequent calculations. The significance of the data in Tables I11 and IV is discussed on the basis of examining the theoretically postulated relations of the variations of faradaic alternating current to the frequency and the amplitude of the alternating potential. Dependence of Faradaic Current on Amplitude of Alternating Potential. T h e two main difficulties in obtaining accurate values of the faradaic current, both of which become increasingly disturbing a t higher frequencies, are t h e corrections for series resistances and for phase differences. Because of t h e uncertainty unavoidably connected with these corrections, the problem is best attacked experimentally by working at comparatively low frequencies, where the capacitative current is only a small fraction of the total current. This is permissible, as it may reasonably be assumed that the nature of the dependence of the faradaic current on the amplitude of the alterVOL. 30, NO. 3, MARCH 1958
* 339
nating potential does not change with frequency of the alternating potential. Values of the faradaic current, obtained a t a frequency of 100 c.P.s., for different amplitudes of the alternating potential, have been corrected for phase differences on the basis of Equation 21 (Table 111). F in the equation is based on the value of j M found in the range of 4 to 7 kc.; the correction a t 100 C.P.S. is the same for all values of jM in the latter frequency range. The calculation of fJW is discussed in the next subsection. The nature of the current dependence is resolved by comparing the ratios of current t o AVO and current to tanh (nF AV0/2RT) at different amplitudes. The values of the former ratio progressively decrease with increasing AVO, whereas the values of the latter ratio seem t o be randomly distributed about their mean. In other words, the latter ratio is apparently independent of amplitude, while the former is not, showing that the faradaic alternating current is linearly proportional to tanh (nF AV0/2RT) and not t o AVO itself. Consequently, the equation for the faradaic alternating current should be written not as in Equation 9. but as
..
(22) Frequency Dependence of Alternating Current. Because of the rather unwieldy dependence of the current on the amplitude of the potential, it is conA7enient t o express the results in a manner which is independent of amplitude, so that results a t a given frequency but a t different amplitudes may be averaged. At the same time, this approach expresses the results in terms of a parameter characteristic of
the system being studied, no matter what the magnitude of the amplitude of the alternating potential used (provided it is not greater than tens of millivolts, the normal range for alternating current polarography). This is conveniently done by multiplying each current value by the factor, (nF AVo/2RT)/AVO, tanh (nF AVO/ 2RT), to give a quantity J F which is independent of amplitude: Jp
(Ai) Ea, = AVO, r.m.,.
tanh nFAVo/2RT (nFAVd2RT)
r.m.B.
-
nFAV&RT Z tanh (nFAV&RT)
where AVO, r m . s . is the root-meansquare value of the amplitude of the alternating potential. It is also convenient to define a parameter J , given by
J is independent of amplitude and frequency of the superposed alternating potential. The test of whether Equation 22 correctly represents the frequency dependence and magnitude of the alternating current is that the esperimental values of J should be independent of frequency and equal in magnitude to the term n2F2ACo( ~ T D ) ~ / * / ~which R T , contains no adjustable parameters. For the reduction of cadmium, a value of J = 0.336 pa. per mv., can be calculated for the present expcrimental situation, where n = 2, F = 96,500 coulombs, A = 0.0243 sq. em., C B = 5.55 X 10-7 mole per cc., D = 7 X 10-6 sq. cni. per second, R =
Table IV.
8.3 joules per mole-degree, and T = 298' (mean). The validity of the correction made for the phase difference between the faradaic and the capacitative current cannot with certainty be established by comparing the constancy of the experimental values of J when correction is made with that u~hen correction is neglected (Table IV), as the magnitude of the deviation of individual values from the mean is of the same order m ith and without this correction. The results calculated by making the correction fit the theory slightly better. In any case, the correction for phase differences as made is a n implicit part of the theoretical treatment, and so must be used in testing the theory. Determination of the most probable value of jawfrom the experimental results is best carried out by trial and error-i,e,, by finding which value best satisfies the condition that J should be constant over the whole frequency range. This is a more reliable procedure than taking jM as that frequency where the experimental value of J p is a maximum. I n the present case, for instance, the experimental values of J p are reproducible to about 5 or lo%, and thus the values given in Table IV should be used only as an indication that f i l lies in the range of 4 to 10 kc. By trial and error, ho.lvever, it is seen that a value of jM equal to 5 kc. makes the fit to the theoretical expression more satisfactory than values of 4, 6, or 7 kc. This most likely value for fdtl in the present case should not be taken to be accurate to better than perhaps 20%. The frequency of 5 kc. corresponds t o a value of k , for the cadmium(I1) t o Cd(Hg) reduction of 1.65 X 10-1 em. see.-' (accurate to perhaps 10%). The results are consistent with the theory, since J is constant t o within
Frequency Dependence of Faradaic Current
7000 C.P.S.
f,
C.P.S. 50
JF 3.61 - 4.50 6.29 8.85 11.0 13.2 14.5 16.3 17.0 20.3 20.2 20.4 16.7 10.8
100 . ~ 200 400 700 1,000 1, 500 2,000 3,000 4.000 6 ;000 8,000 10,000 14,000 Mean, ? Standard deviation. u % deviatkn, 100 u/x
340
Uncorrected for Phase Differences 0.459 0.462 0.469 0.465 0,396 0.401 0.396 o.39i 0.374 0,378 0.360 0.367 0.354 0.348 0.333 0.342 0.316 0.310 0.294 0.304 0.307 0.296 0.301 0.298 0.268 0.265 0.276 0.268 0.27i 0.261 0.261 0.269 0.268 0.250 0.235 0.230 0.291 0.320 0.268 0.258 ~. 0.325 0.291 0.263 0.247 0.338 0.296 0.265 0.246 0.280 0.219 0.204 0.247 0.186 0.147 0.148 0,165 0.314 0.294 0.292 0.303 ~
0.064 20
ANALYTICAL CHEMISTRY
0.072 24
0.081 28
0,085 29
JF 3.64 4.56 6.39 8.95
11.0
13.2 15.1 16.8 18.4 23.4 25.9 25.9 26.8 21.5
J 0.462 0.396 0.366 0.337 0 294 0.296 0.287 0.286 0.291 0.370 0.416 0.430 0.450 0.372 0.361 ~
0.064 18
JF 3.64 4.53 6.41 8.95 11.1 13.2 15.0 16.7 18.0 22.5 24.7 24.6 24.5 20.1
J 0.466 0.394 0.374 0.343 0.306 0.298 0.278 0.275 0.264 0.322 0.356 0.357 0.362 0.308 0.337 0.055 16
JF 3.65 4.56 6.42 9.0 11.2 13.2 14.9 16.8 17.5 21.9 23.9 23.4 24.4 19.1
J 0.471 0.402
JP 3.64 4.60 6.41 8.93 10.8 13.3 14.8 16.4 18.7 21.6 23.1 21.3 18.9
0.381 0.354 0.316 0.301 0.273 0.269 0.242 0.289 0.311 0.304 0.319 0.260 18.1 0.321 0.062 19
J 0.473 9.410 0.386 0.358 0.310 0.310 0.274 0.262 0.253 0.275 0.282 0.257 0.231 0.248 0.309 0.072 23
a standard deviation of 1670 over the frequency range of 50 to 14,000 c.p.s., and of magnitude 0.337 pa. per mv. with a probable error of not more than compared to the theoretical value (Equation 25) of 0.336. The theoretical value itself is not reliable to better than 5yo because of the uncertainty in II) and t , and a possible error in the value of the diffusion coefficient. This agreement between theoretical and observed values is more satisfactory than that observed in the study (6) where the frequency dependence was first considered. I n that paper the theoretical values of the current, and not the observed values, were corrected for the residual series resistances in the circuit (110 ohms) and for phase differences: the reported values of .f.lf (450 c.P.s.) and. hence, of k , (0.5 X 10-I em. see.-’) were therefore too lo^, as the apparent value of .f,,f decreases with increasing series resistances (20). The latter fact was realized by the authors ( G ) , d i o indicated that only relatiye signific:ince could be ascribed to the reported value of f w and k,. As the observed values in the present stud>- were corrected for the effects of series resistances and phase differences, tlic values of jlIand k , can be regarded as indicative of the absolute, rather than merely relative, magnitudes. Conclusions. T h e results obtained with cadmiuni(I1) in 0.5M hydrochloric acid are consistent u-itli Equntion 2 2 . The procedure evolved appears to ovmxmie most of the significant un-
certainties previously connected with the use of alternating current polarography. It may now be justifiable to regard the values of k, obtained by this technique as having absolute, rather than relative significance. The experimental error (10 to 15%) connected with the present results might be decreased by use of a low-resistance capillary of more stable characteristics, more precise temperature control, and use of more concentrated solutions to give a higher ratio of Rc/Rr, thus allowing results to be obtained a t even higher frequencies. Kork currently in progress is directed a t modification of the apparatus t o permit determination of transfer coefficients by the method suggested by van Cakenberghe ( l e ) , and to the application of the technique for investigating the reversibility of organic electrode reactions vith varying euperimental conditions. ACKNOWLEDGMENT
The authors wish to thank the Atomic Energy Commission, n-hich helped support the work described, and Carl Miller, mho constructed the electronic equipment. LITERATURE CITED
(1) Barker, G. C., Jenkins, I. L., Analyst 77, 685 (1952).
12) Boeke. J.. Suchtelen. H. van. Philim Tech. R e v . 4. 213 ’11939). ‘ (3) Boeke, J., S&ht&n, -H. van, Z. Elektrochem. 45, 753 (1939). (4) Breyer, B., Bauer, H. H.. Australian J . Chenz. 9, 437 (1956). \
,
(5) Breyer, B., Bauer, H. H., Hacobian, S . , Zbid., 7, 305 (1954). (6) Zbzd., 8 , 322 (1955). (7) Breyer, B., Gutmann, F., Trans. Faraday SOC.42, 650 (1946). (8) Breyer, @,., Gutmann, F., Bauer, H. H., Osterr. Chemiker-Ztg. 57, 6 i (1956). (9) Breyer, B., Gutmann, F., Hacobian, S., Australian J . Chem. 6 , 188 (1953).
(10) Breyer, B., Gutmann, F., Hacobian, S., dustralzan J . S a . Research, Ser. A, 3 , 567 (1950). (11) Ibzd., Ser. A, 4, 595 (1951). (12) Breyer, B., Hacobian, S., Bustralian J . Chem. 7, 225 (1954).
(13) Breyer, B., Hacobian, S., Azcstralian J: Scz. Research, A5, 500 (1952). (14) Cakenberghe, J. van, Bull. SOC. chim. Belges 60, 3 (1951). (15) Delahay, P., “Sew Instrumental
Methods in Electrochemistry,” Interscience, Sew York, 1954. (16) Delahay, P., Rec. trav. c h m 67, 165 (1948). (17) Drlahay, P., Adams, T J., J . Am. Chern. SOC.74, 5740 (1952). (18) Graharne, D. C., Zbid., 68, 301 (1946). (19) Grahame, D. C., J . Electrochem. SOC. 99. C370 11962). (20) Iial~anasundaram,A,, Proc. Indian =Icad. Sei. A33, 316 (1951). (21) Komyathy, J. C., llalloy, F., Elving, P. J., ASAL.CHEX.24, 431 (1952). Elving, P. J., Chun. (22) Loveland, J. IT,> Revs. 51. 67 11952). (23) Loveland, ’J, IT.>Elving, P. J., J. Phys. Chon. 5 6 , 250 (1952). Sichols, S.,Trans. (24) Matheson, L. -4.. Electrochenz. Soc. 73, 193 (1938). (25) Miller, D. AI., Can. J . Chern. 34, 942 (1956). (26) RIuller, R. H., Garman, R. L., Droz,
11.E , Petras, J., ISD.ENG.CHEM., AXAL.ED 10, 339 (1938).
RECEIVED for review July 22, 1957 cepted Sovember 27, 1967.
,4c-
AIternating Current Polarography Determination of Transfer Coefficient of Electrochemical Processes HENRY H.
BAUER and PHILIP J. ELVING
Universify of Michigan, Ann Arbor, Mich.
b The apparatus and technique described for use in sinusoidal wave alternating current polarography have been modified to permit determination o f the transfer coefficients for electrochemical reactions. The procedure i s a modification of that proposed by van Cakenberghe, which i s based on measurement of the even harmonics of the alternating current. The improved method i s simple and rapid, and permits determination o f the transfer coefficient to a precision o f about i0.01 or =k 0.02. The apparatus, operation, and critical features are described in some detail. The procedure was used to measure the transfer coefficients for
the reduction of cadmium(l1) a t the dropping mercury electrode in various media. A significant difference was observed between the values in potassium chloride and sodium sulfate solutions; the value in hydrochloric acid solution i s midway between the other two. N o significant changes in the transfer coefficient were apparent on varying the concentrations o f cadmium(11) or of background electrolyte, temperature, or amplitude and frequency o f the superposed alternating potential.
V
(6) has shown mathematically, from the theory of absolute reaction rates, that the alterAN C a K E N B E R G H E
nating current, produced under polarographic conditions as a result of a superposed alternating potential of small amplitude, should contain harmonics of the fundamental frequency. Furthermore, the even harmonics should disappear a t the point of symmetry of the polarographic curve-Le., a t the symmetry potential. For a perfectly reversible reaction, the latter would occur at the half-wave potential, E1 2 ; for other than a perfectly reversible reaction, the point of disappearance of the even harmonics would be a t some other potential, E z . The potential difference between Ell?and Ez can be used to culculate the ratio of Co’CR where Co and VOL. 30, NO. 3, MARCH 1958
341
