8th Annual Summer Symposium-Role
of Reaction Rates
Significance of Rates and Equilibria in Electroanalytical Chemistry JOSEPH JORDAN Department o f Chemistry, Pennsylvania State University, University Park, Pa.
where E denotes the potential of the indicator electrode, Ellz the half-current (or half-wave) potential, i the current a t potential E,and i, and i, the anodic and cathodic limiting currents, respectively.
On the basis of Nernstian equilibria and kinetics, reversibility and irreversibility of simple electrode reactions are interpreted in terms of two competing rate processes-viz., mass transfer and electron transfer. The corresponding relationships are illustrated by experiments of hydrodynamic voltammetry where currentvoltage curves are determined in solutions which are allowed to stream with varying flow velocities past a stationary indicator electrode. By adjusting the rate of flow, the shape of the current-voltage wave of ferricyanide can be changed from reversible to irreversible over an appreciable range of potentials. A generalized wave equation is derived for current voltage curves obtained by hydrodynamic voltammetry, which can be made use of for the determination of the rate of electron transfer. Because the latter is a monotonic exponential function of the potential, control by mass transfer prevails whenever a true limiting current is attained, irrespective of the apparent reversibility or irreversibility of the ascending portion of the wave. Since the rate of diffusive and/or convective transport is in all known instances proportional to the bulk concentration of the electroactive species, limiting currents of this type may always be applied with Confidence to quantitative anal?sis.
I
N THIS day and age, a ghost called irreversibility is haunting electroanalytical chemists. Irreversibility appears to be responsible for a good deal of lack of confidence in the analytical utilization of certain electrode processes. The t x o methods primarily affected are potentiometry and voltammetry. Notwithstanding the very successful application of fiome procedures based on admittedly irreversible electrode reaction-e.g., potentiometric titrations with permanganate-the loose common usage of the adjective “irreversible” implies “nnt well understood.”
POTENTIAL OF INDICATOR VERSUS REFERENCE
Figure 1. Idealized composite current-voltage curves (Illustrating criteria of reversibility for electrode reaction: O x ne Red) I. Residual current 11. O x = 10 Red 111. O x = Red IV. 1 O O x = Red A , R , C. Potentiometric zero current potentials
+
where both the oxidized and the reduced form of the electroactivc species are soluble, reversibility is considered to prevail whenevey the zero current potential, E,=”, varies in accordance with th(7 equation:
E+o = Eo
+ ( R T I n F ) In
(UOx/aRed)
(2)
a R e d denote the activity of the oxidized and of the reduced form, respectively. In voltammetry, reversibility or iwrversibility is est,nhlisheti by considering the ent,ire current-voltage curve and taking into account concentration polarization. The equation of a reversible current,-voltage wave corresponding to Reaction 1 has the form:
+
~
CRITERIA OF REVERSIBILITY
The criterion by which electrode processes are classified into reversible and irreversible is the Nernst equation. I n potentiometry, the zero current potential is compared with the theoretical equilibrium potential. Thus, in an “electrode equilibrium” of the type Os ne Red (1)
ELECTRODE, VOLT ELECTRODE
The correlation between the potentiometric and voltammetric tests for reversibility is illustrated in Figure 1. I n the figure, the potentiometric criterion represents a single point in the currentvoltage wave. The potentiometric zero-current value is rigorously significant only when the corresponding residual current is negligibly small, as was first pointed out by Kolthoff and Orlemann ( 7 ) . Consequently, a voltammetric wave-equation represents generally a more reliable criterion of reversibility than the classical potentiometric test.
aox and
PHEWOMEVOLOGIC4L IhTERPRETATION OF REVERSIBILITY
For an insight into the physical significance of reversibility, it is necessary to consider that even the simplest electrode reaction involves two kinds of competing rate processes: I. lllass transfer processes-i.e., the transport of one form of the electroactive species from the hulk of the solution to the sur-
1708
V O L U M E 27, NO. 11, N O V E M B E R 1 9 5 5
1709
face of the indicator electrode (and the analogous transport of the other form in the opposite direction). 11. Electron transfer between the electrode on the one hand, and the oxidized and reduced forms of the electroactive species on the other.
If electron transfer is rapid compared to mass transfer, the concentrations at. the electrode surface of the oxidized and reduced forms of the electroactive spec-ics at. any given potential adjust instantaneously to the ratio of values corresponding to Serristiau equilibrium. This yields a current-volt,age wave of the shape shown in Figure 2, curve 1, which has been calculated 011 the h s i s of Equat,ion 3, aysuming that the bulk concentration of the oxidized species is finite and that of the reduced form equal to zero. If the rat,e of electron t,ransfer is conipara1)lr to (or slower than) the rate of mass transfer, the adjustmeiit, to the equilibrium ratio of the electrode surface concentrations (ailti the corresponding current) on the ascending portioii of the wave lags behind the change in potential and a drawn out, irreversi!ile wave (Figure 2, curve 11)is obtained, everything else being equal. I n the case of multistage electron transfer processes, and in situations complicated by cheniical (nonelectrode) reac tions, ot.lier factors must he taken in regard besides I and 11, The treatment in this paper, however, is confined to the type of reaction which consists of a single electron t.ransfer pro( arid does riot, pertain to c:italytic waves, etc. QC..khTlTATI\ E ASPECTS O F ELECTRON TRANSFER ANI) 1 1 4 S S TRANSFER I S YOLTAXIMETRY
-1 unified theory of current-voltage waves at the dropping mercury electrode, covering currents controlled by diffusion and by the rate of electron t,ransfer, was developed by Delahay ( 1 ) . A generalized equation for current-voltage curves obtained in streaniiiig solutions a t indicator electrodes of constant area is nom presented. Its deriwtion \!-as inspired by unpublished work carried out in 1954 ut the University of Minnesota (6). Conrider Reaction 1, involving one oxidized and one reduced forin of the electroactive slxvies, b0t.h soluble. Let Cox and C ' I + ~ 0;
(t[(cd
=
0
(-1)
Mass transfer affects the oxidized species, which is consumed a t the electrode interface, as well as reduced species which is formed concomitantly. The rate of mass transfer depends on its mechanism, which may be diffusion ( 9 ) , forced convection ( b ) , or both ( 2 , 10, 16). I n all known instances, however, the rate of mass transfer is proportional to the difference between the bulk c'oncentration and the concentration a t the electrode surface. G e n e d l y , the rate of mass transfer can be expressed as follows: I'or the oxidizctl P p e t k
IU~,~(C OC'~L)
=
- / ? M , R ~ =~
Foi. t hr, r~cducotlPpecies.
mRed(c'iFd
-
...
(9)
C R ~=~ ) (10)
mRedCied
denotes mass transport coefficients expressed in centiniet)ers per second. I n streaming solutions a stead)- state is assumed to prevail (2). Consequently: 7n
122 = K R r d
-
!?O.Y
= 1?nt,Ox
=
- RH.Red
(11)
Substituting in Equation 11 values from Equations 5 , 8, !I> and 10 and eliminating the Co - 8 , the following expression ip obtained:
where K
=
kk = exp [ ( E kRed
In the limiting current ( i l ) region Cox >> Cx,; it follows from Equation? 5, 9 and 11 that,:
EO)~LF/RTI
(12)
and, themlore,
(13)
which substituted in Equation 11 yields: j
=
_
_
)llU\ AO, f H c d
~
+
I1
~-
~
x
'1%
TRRed
r""K
+1
( 14)
fox
Ihp~esmon14 represents a generalizrd wave equation R hich takes into account control bv both m a v transfer and electron transfm
Under these initial conditions, cathodic current-voltage waves of the type shown in Figure 2 are obtained. The current density (corrected for residual current) at, a given potential represents a nieasure of t8herate, Kr, of the over-all net electrode process:
u-here i is expressed in amperes a ~ i dd denotes the area of t h e iiitlir:Ltor electrode expressed in square cent'imeters. The net electron transfer involves a forward (reduction) and a backward (oxidat,ion) process. The corresponding rat'es are expressed in accordance with Kimball's treatment of the Eyring theory (3,4) applied to electrode reactions, for which experimental verificat,ion is available in t,hP literature ( 2 , 15). Forward.
Ox
+ ne
~3aciiW:lI'ci. Ked
+.
-+
ox
S e t electron transfer.
R F . ~R: I + ~ = ~k~,daOo.; ,~ xhere k ~ =~ kO exp [a(E" - E)nF//L'?']
d
((j)
+
lie: - K O x = kox&d; where kcjh = k" c'sp [ ( I - m ) ( E - E0)nF/f27'] ( i )
l < t + e c ~- K o ,
=
+ 0.4
k ~ + , , a $ - koxa",d = - koh.fRrd('kd ( 8 )
POTENTIAL
h d , f O x ( %
The synihols kn.d and ko. denote first-order rate cons'tants referred to unit electrode area and exprcssed in centimeters per second; X.0 is called the specific rat,e constant at the standard potential where kox = h C i = k". OL is a transfer coefficient defined by Kimhall (S, 4 ) ; d and Co indicat'e activities and concentrations a t the elect,rode surface, respectively, and f denotes activity coefficients.
t0.3
OF
f0.2
C0.1
INDICATOR ELECTRODE, VERSUS S S C E
VOLT
Figure 2. Current-voltage curves of M ferricyanide in 1M potassium chloride I.
Calculated reversible wave, assuming A = 1.25 X 1 0 - 2 s g . c m . , mor = 2.89 X 10-1 Irreversible wave obtained i n hydrod>namic voltammetry cell a t flow velocity a t 136 cm. per sec. EF. Formal potential of ferricyanide-ferrocyanide couple i n 1.W KCI
11.
ANALYTICAL CHEMISTRY
1710
and is, as such, analogous to expressions derived by various authors, pertaining to somewhat different experimental situations ( 2 ) . The significance of Equation 14 becomes apparent if we assume that mox = m R e d [a reasonable assumption when convection controlled mass transfer prevails ( 6 ) , or when the diffusion coefficients of the oxidized and reduced form are equal] and that fRed = f o x = 1. Considering, under these simplifying assumptions, the point on the w v e which corresponds to the standard potential where K = 1 and k ~ = ~ dkO, the following expression is obtained:
\\ I
'POLIETHVLENE CELL ELLCTRDDL
[ , 'AzHP
Obviously the current is controlled by the rate of mass transfer when
7
moJ2.
..~. - I
( 0 . 0 2 TO 20 RP-SEC3 TURNTABLE
I --GLARS
MOTOR
I500 RPY
Figure
If this condition is satisfied, the curren t-voltage wave is reversible. Conversely, if kO nzox/2 t h e current is controlled by the rate of electron transfer and the wave is irreversible. Similar considerations hold for the entire ascending portion of the wave. On the limiting current region, however, control by mass transfer prevails in all instances. Control by electron transfer could not account for a limiting current region, because both k R e d and kox are monotonic exponential functions of the applied potential (Equations 6 and 7 ) . Equation 13 is valid whenever a true limiting current is attained. The latter is always proportional to the bulk concentration of the electroactive species. Therefore, voltammetric limiting currents may be used with confidence for quantitative determinations, even when the ascending part of the wave is irreversible.
3.
Diagram
of
experimental setup voltammetry cell
of
hydrodynamic
