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Electrode kinetics at open circuit at the streaming mercury electrode
JOURNAL
OF ELECTROANALYTICAL
ELECTRODE MERCURY I.
KINETICS ELECTRODE
CHEMISTRY
AT
OPEN
7:
CIRCUIT
- AT
THE
STREAMING
THEORY
PAVL
DELAHAY*
CQU~PS Chemical Labomtovy. Lorrisimla (Received March 16th. 1965)
State
U~zitxusity.
Batott
Rouge,
Louisiana
70803
(U.S._4
_)
Consider an electrode which exhibits the equilibrium potential 2% corresponding to the charge-transfer reaction 0 + ~ze=R for given activities of 0 and R. If the area, -4, of the electrolyte-electrode interface varies, the double-layer capacity, which is proportional to A, varies_ There results a double-layer charging current (capacity current) which, at open circuit, must be entirely compensated by the faradaic current for the reaction 0 +ne=R_ This reaction proceeds with an overvoltage, 1;1,which can be measured with an instrument of very high input resistance (electrometer amplifier) _ The current -potential characteristic for the reaction 0 + Tze= R can thus be determined at open circuit by varying the capacity current density. Current densities as high as 0.0~ A cm-2 are possible. Until this work was completed, we knew of only two related investigations, (i) the measurement of double-layer capacities with a vibrating mercury-electrolyte interface (\vATAEi_4BE1**) and (ii) oxygen analysis from the shift of the open-circuit potential of a dropping mercury electrode (LAITINEN, HIGUCHI AND CZUHA~). The earlier paper of ROSEXBURG _~XD STEGEMXN~ on the variation of potential of the streaming mercury electrode at open circuit should also be mentioned, but the interpretation is not correct and involves no consideration of electrode kinetics. After completion of this work we became aware of the work of SPEARS on the potential variations of a mercury-electrolyte interface of varying area. This author used a funnel of suitable profile, which was independently studied in this laboratory by COLE~, and examined the variations of potential as the level of the mercury-electrolyte interfacevaried. He considered the ideal polarized electrode and the case of an electrode with a faradaic process, namely the reduction of oxygen. The idea of compensation of the charging current by the faradaic current was clearly stated and examined by SPEAR. The essential ideas involved in the present work were outlined in a preliminary communications, and an equation for the shift of potential for the streaming mercury electrode was reported *** . Details on theory are given here, and experimental results are discussed in a subsequent paper 7_ Extension to the expanding mercury drop electrode is made elsewhere*_ Work on the “funnel electrode”, which_was mentioned in l x&3ress after August 1963: Department of Chemistry, New York University, \‘Vashingtdn Square. New York, N-Y., rooo3_ ** See previous references in this review. -*** It should benoted that the charging current for a streaming mercur-v eiectro-de is caused by continuous removal of the double layer alongitith mercury_ There is no v&iation of th&Iectrode.s.rea_
I’.
2
DELAHAY
the prelixninary communication6, was not completed because of difficulties caused by- erratic electrode response_ Likewise, studies on a \-ibrating mercury-electrolyte interface9 were discontinued because of difficulties in the analysis of experimental results. GENERAL
EQUATION
The capacity current, idz, for an electrode of \rarying area, A, is obtained by differentiating, with respect to time t, the charge on the electrode c(E -E,)_-2, where c is the double-layer integral capacity, E the potential (IUPAC-Stockholm convention), and E, the point of zero charge_ The capacity current is
If this quantity is positive, a positive charge must be supplied to the electrode, and a net cathodic reaction must occur on the electrode at open circuit_ Conversely, a net anodic reaction occurs at open circuit when the above quantity is negative. We adopt for the faradaic current density, If, the polarographic convention -according to which If 2 o for a cathodic or anodic process, respectively. Thus, one has in general at open circuit (E-E,)
A&g
-~-GA
dE dt
+ c(E-Ez)
$$
-
IfA=
(1)
The current density, If, is a function of E and of kinetic parameters for the density and the transfer coefficient_ In electrode reaction, e.g., the exchange-current principle, eqn. (I) can be solved for E as a function of time, and the shift of potential from its equilibrium value can be correlated to the kinetic parameters for the chargetransfer process_ CAPACITY
CURRENT
FOR
THE
STREAMING
MERCURY
ELECTRODE
We consider a streaming mercury electrode of the Heyrovsky type* which is characterized by a length. Z, and a rate of fIow of mercury, m. The radius, Y. of the m%ercu+ column in contact with the solution is supposed to be constant_ The capacity cur&& at constant potential can be computed immediately if one assumes that the -mercury-jet car&es with it the double layer. -This assumption is essentially valid, as _shoti by measurements -of the charging current4-7, and the capacity current is thus directly related to the Gate-of renewal of the- mercury-solution interface. At -cor&ant E ___-
where 6 is the gensity of &rercury. The term, z mfu 6, in eqn. (2) is the rate of renewal lof the me&ry-electrolyte interface, i+_,, the-area 2 m rZ divided-by the time required ‘~o~:l-eieinent-ofre~. to-tiavel the length of the column. This time is. Z/U, v being the _f._ See
ref. ro for a rev&w_
ELECTRODE
KINETICSATTHE
Hg ELECTRODE.
STREAMING
3
I
of mercury (v = wz/_z ~2 a) _ For example, idrz 3-10~~ A for E -E, = I V and the data: c=zo PF cm-“, Y = 0.005 cm, wz= r g sec- 1. The corresponding capacity density is approximately 0.02 A cm-2 for Z=O.~ cm, i.e., a relatively high
veIocit_v
typical current value.
occurs, idt=o, and E = Ez according behavior of the streaming mercury
If no electrode reaction corresponds to the well-known determination of E,. FARADAIC
CURRENT
FOR
THE
STREAhIIXG
MERCURY
to eqn. (2). This electrode in the
ELECTRODE
We consider a simple electrode process 0 + 72 e = R involving soluble we consider mass transfer_ The faradaic current density, If, is then
1-f
species
and
(3)
=
where 31 is the overvoltage ; LU, the transfer coefficient ; Ia 0 the apparent (not corrected for double-layer effects) exchange-current density ; I@ and 1_ad the cathodic and anodic diffusion-current densities, respecti\Tely; and F, R and T have their usual meaning_ Then Ifso for qp7o. 1~ d > o and I_*d > o. Equation (3) is derived by combining the usual I vs. r characteristic with the following relationships
co
If
ICd ego==--
CR
If
CR0
IAd
----=I+-
(4)
where the C’s are the concentrations at the electrode surface (no double-layer correction) and the CO’s are the bulk concentrations. Equations (4) follow directly from the Nemst model of the diffusion layer. They are not rigorous but will suffice for our purpose_ The current densities, lcd and IAd, vary along the mercury column but are readily derived by assuming,as KORYTA 11 did, that semi-infinite linear diffusion is thesole modeofmasstransfer and that the solution adjacenttothe mercury column moves Thus
with
the same
Id =
nFD%Co TWcs-
velocity as mercury*_
where D is the diffusion coefficient of the substance being reduced or oxidized electrode_ It is seen from eqn. (5) that Id varies along the mercury column inversely proportional** to ZS_The faradaic current, +, is directly obtained by nation of eqns. (3) and (5) and subsequent integration.
at the and is combi-
12andthedetaiiedstudiesofW~~v~~~x~P~~~~13 * The earlierpaperof RIUS,L~OPISALNDPOLO onuelocit~~distributionandothertopicsmayalsobecolzsulted-ReferenceisalsomadetoS~~~~~~'~ work'-'. ** Thediffusioncurrent,~~hichiSobtainedbyintegrationofIdfrom Z = o to I, isproportionalto Et See K~RYTA~~_ J. EZectvoanaZ. Chewz.. IO (1965) 1-7
I?. DELAH_-IY
4 Equation
(5) applies
to a process
with
diffusion
from
solution
toward
the elec-
trode. This equation can also be applied to the opposite case of diffusion from inside the mercury column toward the surface (anodic oxidation of amalgam) _ The approsimation should be about the same as for eqn. (5) _ It was indeed assumed in the apphcation of the equation for the flux for semi-infinite linear diffusion in the derivation of eqn. (5) that the diffusion-layer thickness is very small in comparison with the radius of the mercury STEADY-STATE
column. OVERVOLTAGE
AT
OPES
CIRCUIT
C;ine7aZ case By- equating the capacity obtains for the steady state* 2 IzFrG
c
A=---
ms PZF
J+ (>Cnt
I
-
;ln
exp C i c.o~D&
current
(=+A)]
eqn.
(2)
to
the
faradaic
= G(q)
exp +
of
[
one
(6)
yzaTfF ?I]I cI?oa2+
current
o a
1
+
(rut) -
PI
There,_c is taken at the potential, E, corresponding to the particuIar value Z/uzz at which the electrode operates. It is seen that G(r/) is proportional to (G/nz)s and to the fun&on 1 _ (I/%) In (I +A) (Fig. I).
ELECTRODE
KJXETICS
AT THE
STREAXLNG
When A-+-co. I - (I/A) In (I 4” A>+I, then proportional to ~~~~~~~~ according to
~~ELECTRODE.
I
5
and there is pure diffusion control_ 6(q) is
provided the integraf capacity c is independent considered. When d-m, x - (x/&l In (I + 31)-G/2, controL Equation (6) then reduces to
of potential in the interval being and there is pure charge transfer
E-E,
with
When A-XKJ (pure diffusion controf), - X/Q is proportional to (Z/m)* provided c is constant.
When &SO (pure charge transfer
ELECTRODE
KINETICS
AT THE
STREAMINIG
Hg
ELECTRODE.
1
7
REFERE5XES r A. 2
3 4 5 6 7 S g IO
II 12 13 x4
Sot.. RIO (1963) 72. AND H_ CZURA, J_ z-l,% cfitwz. SOC., 70 (IQ.@) 56r. J. E. ROSENBERG AND G. STEGEXXN. J_ Phys. Chem., 30 (IgzG) 1306~ C. D. SPEAR, T~cmsieszt Electrode Potentit& of .Merczrvy, dissertation, Tinfversity of ‘Utah, Ig6o; microfilm Nit 60-3379. Universit_); ;Ilicrofilms, Inc., Ann Arbor, Xichigan. D. COLE, unpublished investigation. P. B)ELAHAY, J. P,@s_ Chem., 68 (1964) 98x. v. s. S~IxIv_~sax, G. TORSI AND P. DELAHAY, J_ ~&xt~i?a22at. C&2?%, in press. D. COLE, P. DELAWAY XND E. !%USBIELLES, Col1ectiot.z Czech- Cfrem_ Co~lz~rruyr..submitted. R. DE I;EVIE, V. S. SRIXWASAN AXD P. DEL_AHAx-, unpublished investigation. X'oi. 2, edited by P. ZUMAX, I. Jr_ ~oLTHOFF AXD Y. OKIXAKA, Pro,oress ifz ~oL'n~ogr+z~, Interscience-Wiley, Xew York, 1962, pp. 367-372_ J_ Koxzlr~_4, CoEIectrozr Czeck. Cliema. Covvzmtrvt.. x9 (1954) 433A. -VS. J_ ~xwIS AND s. POLO. Anabs ReaE Sot. Es@m. Fis. y Qziinz,, 433 (xg_tg) 1039. J_ R. WE_A~ER ASD R. T?‘. P.~RRY, J. An2. Clsens. SOG., 76 (Ig%) 6253; 7s (I9561 554~ R. A. SLOTTER, liiweti~ Shdies at the Stveamixg Mercury Elecfmde, dissertation, University of Michigan, xg6o; microfilm Mic Go-2055, Universit>- Microfilms, Inc., Ann Srbor, Xichigan. XVATASABE, J. Elecfrwc?m~~. H2. X. LZII-~IXEN. -I-.HIGUCI~~
J_ ElecfuouvtaE.
Chew.,
IO (1965)
r-7
OF ELECTROANALYTICAL
ELECTRODE MERCURY I.
KINETICS ELECTRODE
CHEMISTRY
AT
OPEN
7:
CIRCUIT
- AT
THE
STREAMING
THEORY
PAVL
DELAHAY*
CQU~PS Chemical Labomtovy. Lorrisimla (Received March 16th. 1965)
State
U~zitxusity.
Batott
Rouge,
Louisiana
70803
(U.S._4
_)
Consider an electrode which exhibits the equilibrium potential 2% corresponding to the charge-transfer reaction 0 + ~ze=R for given activities of 0 and R. If the area, -4, of the electrolyte-electrode interface varies, the double-layer capacity, which is proportional to A, varies_ There results a double-layer charging current (capacity current) which, at open circuit, must be entirely compensated by the faradaic current for the reaction 0 +ne=R_ This reaction proceeds with an overvoltage, 1;1,which can be measured with an instrument of very high input resistance (electrometer amplifier) _ The current -potential characteristic for the reaction 0 + Tze= R can thus be determined at open circuit by varying the capacity current density. Current densities as high as 0.0~ A cm-2 are possible. Until this work was completed, we knew of only two related investigations, (i) the measurement of double-layer capacities with a vibrating mercury-electrolyte interface (\vATAEi_4BE1**) and (ii) oxygen analysis from the shift of the open-circuit potential of a dropping mercury electrode (LAITINEN, HIGUCHI AND CZUHA~). The earlier paper of ROSEXBURG _~XD STEGEMXN~ on the variation of potential of the streaming mercury electrode at open circuit should also be mentioned, but the interpretation is not correct and involves no consideration of electrode kinetics. After completion of this work we became aware of the work of SPEARS on the potential variations of a mercury-electrolyte interface of varying area. This author used a funnel of suitable profile, which was independently studied in this laboratory by COLE~, and examined the variations of potential as the level of the mercury-electrolyte interfacevaried. He considered the ideal polarized electrode and the case of an electrode with a faradaic process, namely the reduction of oxygen. The idea of compensation of the charging current by the faradaic current was clearly stated and examined by SPEAR. The essential ideas involved in the present work were outlined in a preliminary communications, and an equation for the shift of potential for the streaming mercury electrode was reported *** . Details on theory are given here, and experimental results are discussed in a subsequent paper 7_ Extension to the expanding mercury drop electrode is made elsewhere*_ Work on the “funnel electrode”, which_was mentioned in l x&3ress after August 1963: Department of Chemistry, New York University, \‘Vashingtdn Square. New York, N-Y., rooo3_ ** See previous references in this review. -*** It should benoted that the charging current for a streaming mercur-v eiectro-de is caused by continuous removal of the double layer alongitith mercury_ There is no v&iation of th&Iectrode.s.rea_
I’.
2
DELAHAY
the prelixninary communication6, was not completed because of difficulties caused by- erratic electrode response_ Likewise, studies on a \-ibrating mercury-electrolyte interface9 were discontinued because of difficulties in the analysis of experimental results. GENERAL
EQUATION
The capacity current, idz, for an electrode of \rarying area, A, is obtained by differentiating, with respect to time t, the charge on the electrode c(E -E,)_-2, where c is the double-layer integral capacity, E the potential (IUPAC-Stockholm convention), and E, the point of zero charge_ The capacity current is
If this quantity is positive, a positive charge must be supplied to the electrode, and a net cathodic reaction must occur on the electrode at open circuit_ Conversely, a net anodic reaction occurs at open circuit when the above quantity is negative. We adopt for the faradaic current density, If, the polarographic convention -according to which If 2 o for a cathodic or anodic process, respectively. Thus, one has in general at open circuit (E-E,)
A&g
-~-GA
dE dt
+ c(E-Ez)
$$
-
IfA=
(1)
The current density, If, is a function of E and of kinetic parameters for the density and the transfer coefficient_ In electrode reaction, e.g., the exchange-current principle, eqn. (I) can be solved for E as a function of time, and the shift of potential from its equilibrium value can be correlated to the kinetic parameters for the chargetransfer process_ CAPACITY
CURRENT
FOR
THE
STREAMING
MERCURY
ELECTRODE
We consider a streaming mercury electrode of the Heyrovsky type* which is characterized by a length. Z, and a rate of fIow of mercury, m. The radius, Y. of the m%ercu+ column in contact with the solution is supposed to be constant_ The capacity cur&& at constant potential can be computed immediately if one assumes that the -mercury-jet car&es with it the double layer. -This assumption is essentially valid, as _shoti by measurements -of the charging current4-7, and the capacity current is thus directly related to the Gate-of renewal of the- mercury-solution interface. At -cor&ant E ___-
where 6 is the gensity of &rercury. The term, z mfu 6, in eqn. (2) is the rate of renewal lof the me&ry-electrolyte interface, i+_,, the-area 2 m rZ divided-by the time required ‘~o~:l-eieinent-ofre~. to-tiavel the length of the column. This time is. Z/U, v being the _f._ See
ref. ro for a rev&w_
ELECTRODE
KINETICSATTHE
Hg ELECTRODE.
STREAMING
3
I
of mercury (v = wz/_z ~2 a) _ For example, idrz 3-10~~ A for E -E, = I V and the data: c=zo PF cm-“, Y = 0.005 cm, wz= r g sec- 1. The corresponding capacity density is approximately 0.02 A cm-2 for Z=O.~ cm, i.e., a relatively high
veIocit_v
typical current value.
occurs, idt=o, and E = Ez according behavior of the streaming mercury
If no electrode reaction corresponds to the well-known determination of E,. FARADAIC
CURRENT
FOR
THE
STREAhIIXG
MERCURY
to eqn. (2). This electrode in the
ELECTRODE
We consider a simple electrode process 0 + 72 e = R involving soluble we consider mass transfer_ The faradaic current density, If, is then
1-f
species
and
(3)
=
where 31 is the overvoltage ; LU, the transfer coefficient ; Ia 0 the apparent (not corrected for double-layer effects) exchange-current density ; I@ and 1_ad the cathodic and anodic diffusion-current densities, respecti\Tely; and F, R and T have their usual meaning_ Then Ifso for qp7o. 1~ d > o and I_*d > o. Equation (3) is derived by combining the usual I vs. r characteristic with the following relationships
co
If
ICd ego==--
CR
If
CR0
IAd
----=I+-
(4)
where the C’s are the concentrations at the electrode surface (no double-layer correction) and the CO’s are the bulk concentrations. Equations (4) follow directly from the Nemst model of the diffusion layer. They are not rigorous but will suffice for our purpose_ The current densities, lcd and IAd, vary along the mercury column but are readily derived by assuming,as KORYTA 11 did, that semi-infinite linear diffusion is thesole modeofmasstransfer and that the solution adjacenttothe mercury column moves Thus
with
the same
Id =
nFD%Co TWcs-
velocity as mercury*_
where D is the diffusion coefficient of the substance being reduced or oxidized electrode_ It is seen from eqn. (5) that Id varies along the mercury column inversely proportional** to ZS_The faradaic current, +, is directly obtained by nation of eqns. (3) and (5) and subsequent integration.
at the and is combi-
12andthedetaiiedstudiesofW~~v~~~x~P~~~~13 * The earlierpaperof RIUS,L~OPISALNDPOLO onuelocit~~distributionandothertopicsmayalsobecolzsulted-ReferenceisalsomadetoS~~~~~~'~ work'-'. ** Thediffusioncurrent,~~hichiSobtainedbyintegrationofIdfrom Z = o to I, isproportionalto Et See K~RYTA~~_ J. EZectvoanaZ. Chewz.. IO (1965) 1-7
I?. DELAH_-IY
4 Equation
(5) applies
to a process
with
diffusion
from
solution
toward
the elec-
trode. This equation can also be applied to the opposite case of diffusion from inside the mercury column toward the surface (anodic oxidation of amalgam) _ The approsimation should be about the same as for eqn. (5) _ It was indeed assumed in the apphcation of the equation for the flux for semi-infinite linear diffusion in the derivation of eqn. (5) that the diffusion-layer thickness is very small in comparison with the radius of the mercury STEADY-STATE
column. OVERVOLTAGE
AT
OPES
CIRCUIT
C;ine7aZ case By- equating the capacity obtains for the steady state* 2 IzFrG
c
A=---
ms PZF
J+ (>Cnt
I
-
;ln
exp C i c.o~D&
current
(=+A)]
eqn.
(2)
to
the
faradaic
= G(q)
exp +
of
[
one
(6)
yzaTfF ?I]I cI?oa2+
current
o a
1
+
(rut) -
PI
There,_c is taken at the potential, E, corresponding to the particuIar value Z/uzz at which the electrode operates. It is seen that G(r/) is proportional to (G/nz)s and to the fun&on 1 _ (I/%) In (I +A) (Fig. I).
ELECTRODE
KJXETICS
AT THE
STREAXLNG
When A-+-co. I - (I/A) In (I 4” A>+I, then proportional to ~~~~~~~~ according to
~~ELECTRODE.
I
5
and there is pure diffusion control_ 6(q) is
provided the integraf capacity c is independent considered. When d-m, x - (x/&l In (I + 31)-G/2, controL Equation (6) then reduces to
of potential in the interval being and there is pure charge transfer
E-E,
with
When A-XKJ (pure diffusion controf), - X/Q is proportional to (Z/m)* provided c is constant.
When &SO (pure charge transfer
ELECTRODE
KINETICS
AT THE
STREAMINIG
Hg
ELECTRODE.
1
7
REFERE5XES r A. 2
3 4 5 6 7 S g IO
II 12 13 x4
Sot.. RIO (1963) 72. AND H_ CZURA, J_ z-l,% cfitwz. SOC., 70 (IQ.@) 56r. J. E. ROSENBERG AND G. STEGEXXN. J_ Phys. Chem., 30 (IgzG) 1306~ C. D. SPEAR, T~cmsieszt Electrode Potentit& of .Merczrvy, dissertation, Tinfversity of ‘Utah, Ig6o; microfilm Nit 60-3379. Universit_); ;Ilicrofilms, Inc., Ann Arbor, Xichigan. D. COLE, unpublished investigation. P. B)ELAHAY, J. P,@s_ Chem., 68 (1964) 98x. v. s. S~IxIv_~sax, G. TORSI AND P. DELAHAY, J_ ~&xt~i?a22at. C&2?%, in press. D. COLE, P. DELAWAY XND E. !%USBIELLES, Col1ectiot.z Czech- Cfrem_ Co~lz~rruyr..submitted. R. DE I;EVIE, V. S. SRIXWASAN AXD P. DEL_AHAx-, unpublished investigation. X'oi. 2, edited by P. ZUMAX, I. Jr_ ~oLTHOFF AXD Y. OKIXAKA, Pro,oress ifz ~oL'n~ogr+z~, Interscience-Wiley, Xew York, 1962, pp. 367-372_ J_ Koxzlr~_4, CoEIectrozr Czeck. Cliema. Covvzmtrvt.. x9 (1954) 433A. -VS. J_ ~xwIS AND s. POLO. Anabs ReaE Sot. Es@m. Fis. y Qziinz,, 433 (xg_tg) 1039. J_ R. WE_A~ER ASD R. T?‘. P.~RRY, J. An2. Clsens. SOG., 76 (Ig%) 6253; 7s (I9561 554~ R. A. SLOTTER, liiweti~ Shdies at the Stveamixg Mercury Elecfmde, dissertation, University of Michigan, xg6o; microfilm Mic Go-2055, Universit>- Microfilms, Inc., Ann Srbor, Xichigan. XVATASABE, J. Elecfrwc?m~~. H2. X. LZII-~IXEN. -I-.HIGUCI~~
J_ ElecfuouvtaE.
Chew.,
IO (1965)
r-7