Table Of ContentADVANCES IN
POLAROGRAPHY
PROCEEDINGS OF THE SECOND
INTERNATIONAL CONGRESS
HELD AT CAMBRIDGE
1959
IN THREE VOLUMES
EDITED BY
IAN S. LONGMUIR
Institute of Diseases of the Chest, London
Volume 2
SYMPOSIUM PUBLICATIONS DIVISION
PERGAMON PRESS
OXFORD · LONDON · NEW YORK · PARIS
1960
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THE DIFFUSION EQUATION IN D.C.
POLAROGRAPHY
I. CURRENT-TIME CURVES WITHOUT
DEPLETION EFFECT
By J. M. Los and D. W. MURRAY*
Department of Chemistry, University of New Brunswick,
Frederic ton, N.B., Canada
THE MODEL of an expanding spherical electrode has been the basis of all
theoretical derivations of the limiting diffusion current in d.c. polarography
since 1950(1_6). In every case, the expansion of surface area was considered
to be proportional to (mt)$, where m, the rate of flow of mercury, was assumed
to be independent of the time t. The methods applied by Matsuda< 5> and
Koutecky^) seem to be the most rigorous. Since Matsuda's final equation
has coefficients which are still somewhat approximate, we shall use Koutecky's
equation. The instantaneous current, i , as given by Koutecky's equation is:
t
it = AnDimmC{\ + BDhm-m + CDm-m + ...) (1)
•j.1 4 nf\n cm2 coulomb
withal =709^ .
|cquivalcnt
B= 39 mg* cm-1.
C = 150mgf cm-2.
In this paper, we will express i in microamperes, D in cm2 sec1, m in
t
mg sec-1, t in sec and C in millimoles litre -1.
The above model of an expanding spherical electrode is certainly an
idealisation of conditions prevailing in polarography. The following effects
should also be considered:
(i) Depletion or impoverishing effect—This effect was first discussed by
Airey and Smales<7). When a drop dislodges from the capillary electrode,
it leaves part of its depleted diffusion layer at the orifice of the capillary, so
that succeeding drops grow in a depleted (or impoverished) depolariser solu
tion. This leads to smaller currents than correspond with equation (1),
especially at the earlier stages of drop growth. Hans, Henne and Meurer* 8)
eliminated this effect in their study of instantaneous current-time curves,
by investigating " first " drops (i.e. preceding drops were more or less ideally
* Holder of a National Research Council of Canada Studentship.
408
I. CURRENT-TIME CURVES WITHOUT DEPLETION EFFECT 409
polarised and the depolarising potential was applied by a fast-action switch
at the inception of the " first " drop). Comparison with " second " and, in
general, " nth " drops clearly showed that this effect can be considerable.
Very recently, Markowitz and Elving*9) have attached a semi-empirical factor
to equation (1), in order to account for depletion. This procedure is hardly
permissible, since it ascribes all deviations from " ideal " behaviour to
depletion.
(ii) Back pressure effect—The counter pressure, due to interfacial tension
σ*10), is inversely proportional to R, the radius of the electrode and is there
fore a function of time. This causes m to be dependent on time as well.
A question thus arises as to the significance of m in equation (1). To the few
authors who have considered this point at all·8' n>, it seems to be a foregone
conclusion that the instantaneous rate of flow should be substituted (however,
see the third paper in this series where the inconsistency of m will be con
sidered right from the beginning in the derivation of the diffusion equation).
(iii) Shielding effect—Matsuda*5* has given an approximate correction for
the shielding by the bottom of a cylindrical capillary. His final equation is
of the same form as equation (1), with B = 23-5 and 0 — 62-9 (instead of
B = 36-3 and C = 343, which followed from Matsuda's treatment without
regard to shielding). This implies that the shielding increases with time.
Some authors*8· 12> seem to disagree: apparently they deem it possible that
the shielding effect becomes zero at very long times. Lingane* 11* has con
sidered that a part of the spherical electrode is inactive due to the connection
of the drop with the mercury inside the capillary. This lumen correction
is very small.
(iv) Convection effect—Pretty well all current-time curves reported in the
literature show that the current at longer times increases more rapidly than
predicted by equation (1). Those of Hans, Henne and Meurer* 8) exhibit the
same trend. Rather than ascribing this to a decrease in the shielding effect,
this effect may well be due to increased convection in the diffusion layer as
the latter grows thicker*11) (also cf. Smith*13), who found proportionality
with Ü for extremely long drop times).
Other effects have been considered, such as deviation of the electrode
surface from spherical shape*8' 14) and " Anbaueffekt "*15), but these effects
seem to be negligible under the conditions of polarography.
A quantitative test of equation (1) requires an accurate knowledge of the
diffusion coefficient involved. Very few diffusion coefficients are known with
an accuracy of 1% or better for depolariser concentrations around 10~ 3 molar
in indifferent electrolyte concentrations of 0-1-1 molar. MacDonald and
Wetmore*16) have used precise values of D from conductivity measurements
for Cu(II) in various concentrations of sulphuric acid, but nothing pertinent
with respect to equation (1) evolved from their work, since the depletion
B 2
410 j. M. LOS and D. w. MURRAY
effect was not excluded. Hans, Henne and Meurer* 8*, in their measurements
with 10-3ifCd(II) as depolariser in 01M KC1, 0-01% in gelatine, used
D = 7-3 X 10-6cm2sec-1, which is close to D = 7-17 x 10-ecm2sec-1, as
independently determined by the Cottrell method of linear diffusion < 17>.
Accurate values of " tracer " diffusion coefficients, suitable for the purpose
of testing equation (1), were obtained by Wang and by Wang and Polestra* 18)
for Pb(II), Zn(II) and T1(I) ions in various concentrations of electrolytes at
25 i 0-01 °C. Most of these determinations were done without gelatine.
The measurements presented in this paper have been planned in agreement
with the conditions of Wang's experiments, except that we have used 0-01 %
gelatine in most cases (see below). Our experimental conditions are listed
in Table 1 and they are such that no complexation of the ions to the gelatine
needs to be feared. Tanford<19> has shown that ions of Pb(II), Cd(II), Zn(II)
and Cu(II) form complexes with the imidazo] group of serum albumin; only
Cu(II) and Pb(II) seem to be weakly bonded to carboxylate groups in alka
line solution. Below the isoelectric point of the protein the complexed metal
ions become rapidly replaced by protons and at pH = 1, none of these metal
ions is complexed. In the case of Pb(II), we have assumed that the same is
true for gelatine (see Table 1). For Zn(II) in ammonia buffer, Wang found
that addition of gelatine does not affect the value of D. Apparently, the
Zn(II) is completely complexed with the large excess of ammonia; hardly at
all with the amino groups of the small concentration of gelatine. Since
Tanford also has shown that T1(I) ion does not complex with protein at
any pH, Wang and Polestra's D value can be used for the same solution
with gelatine.
TABLE 1
Concentra Gelatine D x 105, cm2 sec-1
Supporting
Depolariser tion concentra
electrolyte
(m moles/1.) tion {%) tracer polarographic
Pb(II) 1011 0-1MKC1 + 001 0-963 ± 0-956 ±0-014
01MHC1 0011
T1(I) 2032 0-2MKC1 001 1-79 ± 1-739 ±0022
002
Zn(II) 2018 1-0MKC1 + 0-818 ± 0-791 ±0014
00005 M HC1 0-008
Zn(II) 1-454 1-0MNH C1± 001 1-020 ± 1-053 ±0-013
4
IOMNH3 0015
The small change in viscosity of the solution, caused by 0Ό1 % of gelatine,
does not seem to affect the diffusion coefficient to a measurable extent( 20>.
The amount of gelatine added to the solution (0-01%) was considered
as an optimum amount. Maxima of the first or second kind* 21» 22> are often
I. CURRENT-TIME CURVES WITHOUT DEPLETION EFFECT 411
suppressed by the traces of organic material present in the solution anyway.
However, in the case of an insufficient concentration of maximum suppressor,
the current during the early stages of drop life may well be high, since then
the rate of growth of the drop's surface is greatest, and adsorption equili
brium with the maximum suppressor is not immediately established. Also
for the investigation of the effect of back pressure in the subsequent papers
of this series, it is of great importance to have equilibrium in the double
layer of the electrode right from the beginning, so as to ascertain constancy
of interfacial tension.
For the depolarisers and under the conditions listed in Table 1, we have
measured oscillographic current-time curves of first drops. Also, peak
currents, obtained for first drops with a fast recording polarograph, could be
compared with the end points of the corresponding current-time curves.
In order to compare the shape of our current-time curves with those of Hans,
Henne and Meurer<8> (see above) we have applied the approximate way of
correcting for back pressure, as was done by these authors, which implies
that the m in equation (1) was tentatively taken to be the instantaneous
rate of flow.
EXPERIMENTAL
A. Apparatus
First drops were obtained by the technique of " artificial " drop-time
control, so that the current pulse which actuated the drop-detaching device,
simultaneously operated an automatic switch which applied the required
potential to the drop at the beginning of its growth. In Fig. 1 this " control "
circuit is distinguished from the " measuring " circuit.
The measuring circuit consists of:
The dropping mercury electrode (DME).
A Leeds and Northrup, Type E, Electrochemograph. This polarograph
is equipped with a " Speedomax " pen recorder provided with four degrees
of damping; damping " 0 " has been exclusively used here (1 sec balancing
time across 10-in. chart paper).
A Du Mont, Type 304-A, Cathode-ray oscillograph, equipped with a
Du Mont oscillograph-record camera, Type 296 (35 mm film).
A decade resistance box, R (0-11,111 Ω) to supply an iR drop of about
30 m volt for the oscillographic measurements.
A BZ-2RS micro switch (Minneapolis-Honeywell), S , to short the just
m
mentioned iR drop momentarily, at definite time intervals, in order to mark
the time (zero-current) axis on the oscillograms. This very useful device was
first introduced by Lingane(n>. The switch was operated by a constant-
speed motor (speeds, at any setting used, were constant to better than
412 J. M. LÖS and D. W. MURRAY
0-1%). A soldered cam on the motor's chuck closed the micro switch once
during each revolution.
A 2 V storage cell powering a slide wire (SW) for applying the desired
potential to the DME and for calibration of the vertical deflection of the
oscillograph.
Tandem Drop Control
Recycling Device
Timer r
Electrochemo-
graph,
Type E
; i J ~m
Measuring
Circuit K JJ Constant
Control ^ —S SSppeeeedd MM otor
Circuit Shield
FIG. 1. Circuit diagram.
A Leeds & Northrup, Type K2, potentiometer to measure the potential
applied by SW.
A double pole, double throw switch, S , to change from the oscillographic
3
to the polarographic circuit and vice versa.
The secondary circuit of the Bll AXA sequence relay (see below), which
acts as the automatic switch.
I. CURRENT-TIME CURVES WITHOUT DEPLETION EFFECT 413
A ΙΟΟ,ΟΟΟΩ (+0-1%) precision resistor, R .
c
Switch $2, which in position (a) allows measurement of currents affected
by depletion; in position (b) allows measurement of currents not affected
by depletion; in position (c) permits calibration of the measuring devices
by means of the iR drop across R , with the secondary circuit of the sequence
c
relay permanently open.
The control circuit consists of:
The Tandem Recycling Timer, Type A (equipped with timing elements
ET 5S and ET 15S; Industrial Timer Corporation). This dual timer is
capable of producing two successive time intervals in each cycle, each timing
element continually actuating the other. For the elements mentioned, these
intervals could be varied from 0 to 5 sec and from 0 to 15 sec, respectively.
The second element was always set for the desired drop time, the
first one for the pulse needed to actuate both the sequence and the striker
relays (see below). Best results were obtained with this pulse time set at
0-05 sec. The reproducibility of a dual cycle was found to be better than
0-01 sec.
The " Standard Sequence " Relay Switch (Bll AXA relay; Struthers
Dunn). This automatic switch is of the " roll-over " type and has double
pole, double throw, cam-operated contacts (in the present application only
one pole was used). The contacts remain alternately open or closed after
each momentary impulse of 0-05 sec from the timer to the relay coil. This
coil was operated at 30 V d.c. For the operation of the switch, Si must be
closed, while S2 should be in the neutral position (b). If $1 is open with S2
at (b), the measuring circuit would be either permanently open or perman
ently closed, depending on the position of the relay switch at the instant
Si was opened.
The solenoid-operated striker for drop-time control. This device was
adapted from a General Electric " Instantaneous Overcurrent Relay "
(12PAC 11 B19), by removing the secondary circuit parts, leaving the coil
encased in the metal shield, the soft iron core and the striker to which it is
attached. A light spring was then connected between the striker and the
relay case, to return the iron core to its equilibrium position after each
activation. This modified relay was bolted firmly to the upright wooden
superstructure on the laboratory bench, in such a position that when the
relay was not energised, the striker was about 5 mm from the channelled
aluminium support (supplied with the Electrochemograph) (see Fig. 2). The
solenoid coil was originally designed to operate at 12 V a.c, but since it had
to be energised by the same 0-05 sec pulse as the sequence relay coil (operating
at 30 V d.c), the arrangement as indicated in Fig. 1, using a heavy duty
potential divider, was found most expedient (operating current was 6 A at
25 V d.c).
414 j. M. LOS and D. W. MURRAY
z—support
bracket
channeled
I—aluminum
support
&
coil
overcurrent
relay
Heyrovsky cell
FIG. 2. Drop control device.
The dropping electrode assembly, also shown in Fig. 2, was rigidly mounted
on a securely braced framework (" flexiframe "), so that when the striker
hit the side of the aluminium support, no visible vibration occurred and no
stirring of the solution was noticed. This method of drop detachment has
certain features in common with the method used by Wâhlin and Bresle* 23).
These authors do not seem to use as forceful a stroke as we apply. With the
present method, drop times as small as 5% of the natural drop time could
easily be achieved without noticeable distortion of the polarographic waves.
In Fig. 3 can be seen an example of such a polarogram.
I. CURRENT-TIME CURVES WITHOUT DEPLETION EFFECT 415
B. Materials (see Table 1)
Depolariser salts: ZnS0 .7H 0 (B. & A. Reagent); Pb(N0 ) (Fisher
4 2 3 2
Reagent); TI2SO4 (Fisher C.P.). The water of hydration and absorbed
moisture were determined in each case and corrections were applied to the
weights. In the case of ZnSC>4. 7H2O a gravimetric determination of Zn was
also made.
Supporting electrolytes: KC1 (B. & A. Reagent); NH4CI (Merck Reagent);
HC1 and NH3 (Nichols C.P.), diluted with distilled water and standardised;
gelatine was B. & A. powder.
Oxygen-free nitrogen, saturated with water vapour at 25 °C was used to
remove dissolved oxygen. For Zn(II) in ammonia buffer a 800 ml saturator
containing 1M NH and 1 M NH C1 was used to saturate the nitrogen with
3 4
ammonia. The ammonia of the cell solution was titrated before and after
each set of measurements and it was shown that the ammonia concentration
could be kept constant in this manner within 1 % for several hours.
The mercury used had been washed in nitric acid (1:3) and distilled
water, and was triple distilled under vacuum.
C. Measurements
For each of the solutions of Table 1 several polarographic and oscillographic
current measurements were made with a few cylindrical capillaries and at
various heights of the mercury reservoir (i.e. various natural drop times).
Ci. Polarographic Measurements
In agreement with the results of other investigators (cf. réf. 24), the pen
speed of the Speedomax recorder at damping 0 was considered to be suffi
ciently fast accurately to record the current at the end of drop life, i.e. the
maximum or " peak " current, provided the actual drop time is greater
than 2-5 sec. In order to record these maximum instantaneous limiting
currents, for each solution, head of mercury and capillary, the following
polarographic measurements were made:
(a) A complete current-voltage wave with $2 in position (a). This is
needed in order to judge the plateau of the limiting current. The
waves are subject to the depletion effect (Fig. 3a).
(b) The maximum current (i ) of first drops at a fixed potential, at least
max
0*1 V within the potential range of the limiting current. Switch $2
was in position (b), while the sequence relay switch and Si were open
for 5-10 drops. This is sufficient to insure the absence of depletion
