ACTIVITY COEFFICIENTS
Values of the molal activity coefficient of hydrochloric acid in these four methanol-water solvents have been calculated by Equation 2 and are given in Table V. The values of 6 were obtained from the slopes of the extrapolation lines whose intercepts were the standard potentials ,E”. The activity coefficients in 10 and 20% methanol are compared with those found by Harned and Thomas; the agreement is good except at O.lm in 20y0 methanol. Figure 1 is a plot of log -y& vs. m’l2 for 45 and 75y0methanol, the straight lines representing the limiting DebyeHuckel slopes. The activity coefficients
are much lower in 70 than in 45% methanol. Nevertheless, the experimental curves approach the theoretical straight lines from above in both instances. There is no evidence, therefore, of ion-pair formation even in 7oy0methanol.
Maclennan, W. H., J . Chem. SOC.1933,
p. 674.
(6) Feakins, D.. Watson. P.. Zbid.. 1963. (8) Gronwall, T. H., LaMer, V’ K.,Sandved, K., Physzk. 2. 29. 3.58 11428) (9) Harned, H. S., Th
LITERATURE CITED
(1) Albright, P. S.,Gosting, L. J., J . Am. Chem. SOC.68, 1061 (1946). (2) Austin, J. AI., Hunt, A. H., Johnson,
F. A., Parton, H. N., cited by Robinson, R. A., Stokes, R. H., “Electrolyte Solutions,” Butterworths, London, 1959. (3) Bates, R. G., Paabo, AI,, Robinson, R. A., J . Phys. Chem. 6 7 , 1833 (1963). (4)Bates, R. G., Rosenthal, D., Zbid., p. 1088. (5) Butler, J. A. V., Thomson, D. W.,
(1957).
(11) Nonhebel, G., Hartley, H., Phil.
Mag. 50, 729 (1925). (12) Oiwa, I. T., Sei. R e p k . Tohoku L’niv., Ser. 1 41, 47 (1957). (13) Paabo, AI., Robinson, R. A , , Bates, R. G., J . Chem. Eng. Data 9,374 (1964). (14) Schwabe, K., Ziegenbalg, S., 2. Elektrochem. 6 2 , 172 (1958). RECEIVED for review October 29, 1964.
Accepted January 27, 1965.
General Equation for Current-Potential Relationships at Rotating Disk Electrode ILANA FRIED and PHILIP J. ELVING Department o f Chemistry, The University of Michigan, Ann Arbor, Mich.
b The rotating disk electrode has been extensively investigated. The present paper discusses some of the existing theoretical treatment of the rotating disk electrode, evaluating its usefulness for actual electrochemical investigation, and presents the derivation of an equation which describes the complete current-potential curve obtained under standard voltammetric conditions: potential varying linearly with time and current varying as a function of potential. The treatment used assumes that the electrode reaction is reversible and controlled by the rate of mass transport and that an excess of nonelectroactive electrolyte is present. The treatment does not presuppose any particular mechanism for the electrode reaction.
T
HE rotating disk electrode has been the subject of many theoretical and experimental investigations. There are three reasons for the increasing current interest in this electrode type: the relative ease with which the electrode and the means of rotation are obtained; the availability of a rigorous hydrodynamical theory for this configuration; and the desirability of a well understood solid electrode for voltammetry and related techniques, especially for studying the electrochemical oxidation of organic compounds. The voltammetric theory of the rotated disk electrode essentially began with Levich (6), who proposed the following equation for the limiting current
464
ANALYTICAL CHEMISTRY
ili, = 0.62 nFAD1/I v - ’ / ~ w ~ C” / ~ (1) where ili, is the observed limiting current, n the number of electrons involved per molecule in the faradaic process, F the Faraday, A the area of the electrode, D the diffusion coefficient of the electroactive species, Y the kinematic viscosity, w the angular velocity of the disk, and C” the concentration of the electroactive species. The most important concept introduced by Levich is that of the hydrodynamic diffusion layer, 6. To solve the equation of diffusive convective mass transport
bt
(x,y, and z are the system’s coordinates; and vz, vy, and v,, the components of velocity of the solution in the respective directions) for the hydrodynamic conditions of the rotated disk, assuming that (bC/bt) = 0, Levich proved that the mathematical description of the rotating disk leads to results which are formally the same as if a layer of thickness 6 existed adjacent to the disk. The concentration gradient is limited to this layer. Outside this layer, the concentration is maintained constant by convection; inside the layer, diffusion gradually predominates over convection as the means of mass transport of the electroactive species as the surface of the disk is approached.
6 = 1.61 (D/V)’”( Y / U ) ~ ’ ~
(3)
6, as is evident from Equation 3, is independent of the concentration of the electroactive species in the solution and a t the electrode surface. Consequently, 6 does not change even if the concentration a t the electrode surface changes with time. By assuming (bC/bt) = 0 in Equation 2, Levich (6) found the current resulting from a faradaic process a t effectively constant current and constant potential conditions. The next step in the development of the theory of a rotated disk electrode is to let the current-or the potential-vary during the time of electrolysis. Siver (10) proposed equations for the current-time curve at constant potential for both reversible and irreversible electrode processes, Siver (11) also derived an equation for the potential of a rotating disk electrode as a function of time when the current is held constant. Bowers et al. (1) and Rosebrugh and Miller (9) solved the same boundary value problem, although they did not consider rotating disk electrodes. The resulting equation derived by Rosebrugh and Miller (9) is identical with the one derived by Siver (11). Bowers et al. (1) considered diffusion through a membrane, with the concentration on the side facing the solution held constant by stirring. The latter approach is applicable to the study of the rotating disk electrode as indicated by Buck and Keller ( 2 ) . -4 more detailed discussion of the equa-
tions for constant current is subsequently given. Objective of Present Study. The next step in evaluating a complete theory for voltammetry with the rotating disk electrode is to derive a current-potential equation for the actual conditions experienced in voltammetry, that is, potential varies linearly with time and current varies as a function of it. The derivation of this equation is the objective of the present study. BOUNDARY VALUE PROBLEM A N D ANALYSIS OF PAST W O R K
If it is assumed that the only concentration gradient near a rotating disk electrode is in the direction normal to the disk, the rate of change of concentration brought about by diffusion and convection is
bC
bC _ - D , - vb2C ,-
at
At time t
Co
bX
bX =
0, a t any x
=
Coo and CR = CR"
(4)
(5)
At the electrode surface
bC bx
i
- = -
bCo Do bX
nFAD
+ DR bCE bX --
=
0
[RTnF ( E - E " ) ]
Co = CRexp
-
=
Coo and C R =
C E O
bC - -- D -b2C bt 3x2
(7)
At the time t
=
0, a t any x
(9)
The three variables in this boundary value problem are C, E (the potential of the indicating electrode), and i (the current through the cell). One can assume either i or E constant and solve the problem for C, which will be a function of time and the constant quantity selected. When the proper expression is substituted in Equations 8 or 6, the final form of the solution is obtained, which gives the varying quantity as a function of time and the constant parameter. The various treatments which follow include the following assumptions. The reaction is reversible. No particular mechanism is presupposed. Roth the oxidized and the corresponding reduced species are soluble. The bulk concentrations of both species remain constant throughout the electrolysis. The current in the equations is the faradaic current-Le., observed current corrected for residual current. The potential is the true potential of the working electrode, corrected for iR drop. The same treatment is valid whether either the oxidized or the reduced species of the redox couple is considwed. The results depend on exactly where the distinction between the two species is introduced.
=
1-
4Dt
+
2m6 x X 2 4 6 2m6 x / erfc ~
+ (x) +
l
2m6
+
26 - x 2 6 t
4, X
('& 2 426E -
erfc
+
")-I
(14)
Substituting x = 0 in Equation 14 gives the concentration a t the electrode surface. The time elapsed until this concentration drops to zero is the transition time ( T ) , which cannot be explicitly evaluated from Equation 14. For fairly large values of t , when 6 -lo+ cm. the series converges rather slowly. When 6 + m , Equation 14 reduces to the equivalent expression for a quiet solution and stationary planar electrode. As subsequently shown, when t is small, for finite values of 6, the process is diffusion-, rather than convection-controlled; Equation 14 does not predict such behavior. Siver (11), using a different mathematical route, arrived a t the expression
Co(x,t) =
i
coo
iso
- DonFA -(60 - x ) t
zi
8
DonFA
1
2 (2m - 1)' X
cos
At x
+ 26 - x ) *
(2m6
[-
(8)
and in the bulk of the solution, when x m , at any time
Ca
The problem presented by Equations 4 to 9 cannot be solved analytically. Therefore, the following approximations bC are introduced: the term u,in bX Equation 4 is negligible, and all of the concentration change occurs within a layer of thickness 6 adjacent to the electrode surface. Although the first approximation may not seem to be completely justified, the error introduced is not great as shown by Hale (6),who solved the problem represented by Equations 4 to 9 numerically and compared his exact solution to the approximate one given by Siver (11). The second approxima tion is valid as evident from Levich's treatment, discussed previously. The form of the final equations obtained by solving the approximate boundary value problem depends on the exact mathematical route used. To illustrate this point and clarify the advantages and disadvantages of each form, the two existing treatments for constant current will be described. Roth treatments use the Laplace transformation method. This method is explained eleswhere (3, 7') After introducing the assumptions, the boundary value problem becomes as follows for constant current conditions. The differential equation is
(2m - 1 ) n x 260
6, a t any t
X
exp [ - (2m - 1)2n2Dot/46a2] (15) Substituting x = Oin Equation 15 gives and, a t the electrode surface,
Co(0, t) =
coo -
i60
~
DonFA
iso Derivation of the solution involves the following steps. The Laplace transformation is performed on the original differential equation to yield a simpler differential equation. The solution for the latter is obtained; the constants in the resulting equation are calculated using the specified boundary conditions. Introduction of these constants into the solution for the simpler differential equation produces a Laplace transform of the solution to the original differential equation. An inverse transform is then performed on the latter solution, resulting in the desired expression. Bowers et d.( I ) obtained the following equation for the concentration of electroactive material in the vicinity of the electrode. =
1
(2m - 1)2
X
- 1)2n2Dot/460z](16) When (DOt/602) < 0.1, Equation 16 exp [-(am
becomes Co(0, t ) =
2i coo - nFA
-
d&,
(I7)
which is the same result as that derived from Equation 14 for 6 + a, and is the equation applicable to the stationary electrode. However, when (Dot/6$) > 0.1, only the first term in the series is important and Equation 16 becomes C(0, t ) =
DonFA e [ l C(x,t)
zl
8
DonFA
t
coo - ~8 e x p ( - s t ) ] (18)
Thus, for a given value of a0, if t is short, (DOt/6O2) will be small and the same experimental results should be observed as with a stationary electrode. This is realized experimentally (4). There is another advantage to the form of expression as in Equation 16-i.e., VOL 37, NO. 4, APRIL 1965
0
465
having the term exp(-Dt/P) rather than exp(-6'/Dt). When C ( 0 ) = 0, t = 7 , which transition time can be evaluated explicitly.
( 8
The solution to Equation 26 has the form
C"
c = S
r 2 1 - C0"LI;~FA)
7 = - - l n460' T'DO
In order for 7 to exist and be positive, the argument under the In sign should be positive but smaller than 1. The values imposed on the current by this condition are 5.26 Co'DonFA
>i>
Co"DonFA
60
60
i > 0.606
4;CoDonFA/60
(21)
ala
B-
The initial condition is At t
=
0, C
=
a=
=
0, i
At x
= =
nFAD bc bx
I
(30)
i/n~~dDls
(31)
+
- 5
%FAdE[i exp (-26
-\/*)I (32)
i exp( -26
da)
%FA f i s [ 1 + e x p ( - 2 6 d s l D ) l (33)
Introducing Equations 32 and 33 into Equation 29 gives
C"
c=-+ S
i { -exp( -x dslD) exp [(z - 2 6) nFA fis X
%'%)I
+
+
nF cR= c0exp [= (EO- E ) ]
Substituting x = 0 in Equation 34 yields the Laplace transform of the equation for the concentration of electroactive species a t the surface of the rotating disk electrode. Rearrangement of the resulting expression yields
2-0
6 , C = Co
(25)
=
0, i = %FAD -
Atx
dx
2-0
C"
= 6,c = S
(27)
(28)
xhere c and i are the transforms of C and i, and s is the transform variable. 466
ANALYTICAL CHEMISTRY
+ at
(38)
Here Ei is the initial potential, and CY is the rate of change of the potential with time. To simplify the mathematics involved, it is helpful to assume that DO = DR and to introduce two notations:
4%tanh (6 45)
%F
a' = u
RT
The inverse transform of Equation 35 is C=C"-
The Nernst equation then becomes
CR = COee-a't (41) The equation for conservation of material ( 7 ) ,introducing the assumption DO = DR, is CO
Laplace transformations of Equations 24 and 25 are, respectively,
Ei
i
nFA
The Laplace transformation of Equation 22, introducing Equation 23, is
At x
=
(35) (24)
(37)
Under usual polarographic conditions, the potential varies linearly with time.
E
C"
(23)
The boundary conditions are At x
a =
c=--
C"
dslD)
The following expressions for a and p result from the combination of Equations 30 and 31:
8=
The next step in the development of a complete theory on voltammetry with a rotating disk electrode is the solution of the boundary value problem for the case where both potential and current change with time, which, as previously stated, is the objective of the present paper. The differential equation to be solved is Fick's second law (the assumptions and approximations involved have been stated).
-exp (-26
=
Another equation for a and P is derived by introducing Equation 27 into Equation 29.
The coefficient in Equation 20 is 1; that in Equation 21 is0.606 d ; = 1.07. DERIVATION OF EQUATIONS FOR CURRENTPOTENTIAL CURVE W H E N BOTH CURRENT A N D POTENTIAL VARY WITH TIME
(29)
Introduction of Equation 28 into Equation 29 yields the following r e lationship between a and 8.
(20)
An equation similar to the right side of Equation 20 was proposed by Buck and Keller ( 2 ) after observing that, for small currents, the transition time becomes immeasurable.
+ a exp (-x da)+ P exp (x d@)
the electrode reaction-Le., the oxidized form in a reduction reaction and the reduced form in an oxidation. If the integral is negative, C > C"and Equation 36 describes the concentration of the species formed during the reaction. The sign of the integral depends on thesign of the current-viz., positive for reduction and negative for oxidation processes-and the sign of the bracketed sum of Equation 36. T h e infinite series of the sum will always be negative because it starts with a negative term and each subsequent term is smaller than the preceding one. Therefore, the sign of the sum [I ( 2 X series)] depends on the actual value of the series, which, in turn, depends on the magnitude of the experimental variables 6, D , and t. Equation 36 is reduced to the equation for constant current by carrying out the indicated integration via the Laplace transformation. The Laplace transform of Equation 36 is identical with that of Equation 1 of reference ( I ) whenx = 0. To obtain an explicit expression for the current as a function of the potential, the Nernst equation is introduced to Equation 36.
+ CR = COO f CRo
(42)
Introduction of Equation 41 into 42 and rearrangement yields where 7 is the integration variable. Equation 36 gives the concentration of either the reduced or the oxidized species of the redox couple a t the surface of the electrode; the particular species to which it refers depends on the sign of the integral. If the latter is positive, C < C" and C'should be the concentration of the form consumed in
(43)
The current is best evaluated from Equation 36 via its Laplace transform, Equation 35 (8): i = n
~
(SCO
- co) ~ x
coth (6
m
-\/a) (44)
r t
The inverse transform of Equation 44 is
dCo/d7 can be evaluated from Equation
48 :
It is useful to change the place of ( t - r ) from the sum to the derivative of the concentration. This is permissible because of the properties of the Laplace transformation. Substituting Equation 46 into Equation 45 and changing the place of t as indicated results in the final equation :
r
Equation 47 represents the first theoretical description of the currentpotential curve for a rotating disk electrode when the potential varies linearly with time and the current varies &s a function of the potential, This equation is reduced to that for the stationary electrode case when 6 = ( 7 ) . The integral indicated in Equation 47 must be computed numerically. LITERATURE CITED
(1) Bowers, R. C., Ward, G., Wilson, C. AI., DeFord, D. I)., J . Phys. Chem. 6 5 , 672 (1961).
(2) Buck, R. P., Keller, H. E., ANAL. CHEM.35,400 (1963). (3) Churchill, R. V., "Operational Mathe-
matics," 2nd ed., p. 324, McGraw-Hill, New York, 1958. (4) Fried, I., Elving, P. J., unpublished data, 1964. (5) Hale, J. AI., J . Electroanal. Chem. 6 , 187 (1963). ( 6 ) Levich, V. G., "Physicochemical Hydrodynamics," p. 65, Prentice-Hall, Englewood Cliffs, N. J., 1962. (. 7,) Reinmuth, W. H., ANAL.CHEM.34. 1446 (1962): (8) Reinmuth. W. H., Columbia University, New York, H. Y., private communication. 1964. (9) Rosebrugh, T. R., LLIiller, IT.L., J . Phys. Chem 14, 816 (1910). (10) Siver, Yu. G., Russzan J . Phvs. Chem. 33, 533 (1959). (11) Ihzd., 34, 273 (1961)). RECEIVED for review- 3Iarch 15, 1963. Resubmitted March 26, 1964. Resubmitted December 31, 1964. Accepted February 10, 1965. The authors thank the U. S. Atomic Energy Commission, which helped support the work described.
Electrooxidation in Pyridine at Pyrolytic Graphite Electrode W. RICHARD TURNER and PHILIP J. ELVING The University of Michigan, Ann Arbor, Mich. The potential utility of pyridine as a solvent for the investigation of electrochemical oxidation was evaluated b y examining the available potential spans for typical background electrolytes and by a voltammographic investigation of a number of organic compounds. The pyrolytic graphite electrode was used. The potential range is limited b y oxidation of pyridine or of an impurity in it at about 1.4 volts vs. the silver-silver nitrate (1M in pyridine) reference electrode. The voltammetric behavior of the compounds studied indicates that the solvent-electrode combination examined should be of considerable value for studying electrolytic oxidation.
A
several investigations of electrochemical reduction in pyridine have been reported (6, 6, 16), there have been no reports of electrooxidations using only pyridine a s the solvent, although several investigators (10-12) have added small amounts of pyridine to acetonitrile and have used this medium for the electrooxidation of organic compounds. In the latter instances, pyridine has been observed to lower the potential a t which the organic compounds are oxidized. Being a strongly basic, aromatic solvent, which is apparently resistant to normal chemical oxidation, pyridine would seem to be an ideal medium for LTHOUGH
electrooxidation. Although it has a low dielectric constant (12.3), many inorganic salts are soluble in it and produce conducting solutions. For example, a 0.2M solution of lithium perchlorate, when placed in the cell used in the present investigation, produces a typical cell resistance of about 2000 ohms. Pyridine has a low proton availability so that reactions can be compared in the absence of protons and in the presence of a known available activity or, rather, concentration of protons (actually present as pyridinium ions). Because the oxidation of organic compounds frequently involves the removal of protons and/or the formation of carbonium ions, pyridine would greatly facilitate such reaction by acting as a profon acceptor and a carbonium ion stabilizer. Many pyridine adducts to carbonium ions are quite stable N-substituted pyridinium compounds. It is these two effects which contribute to the observed lowering of potential for oxidation in pyridine previously mentioned (10-12). In the present study, electrooxidation of pyridine solutions of a number of salts, which might be suitable for background electrolytes, was studied a t the pyrolytic graphite electrode; in addition, the electrolytic oxidation of a number of organic substances was investigated.
EXPERIMENTAL
Reagents. Analytical reagent grade pyridine (Mallinckrodt) was used without further purification as it showed the same voltammetric behavior as pyridine, which had been dried over barium oxide and fractionally distilled. Anhydrous lithium perchlorate (G. Frederick Smith) was dried at 200" C. Quinone (Eastman Yellow Label) was recrystallized from benzene after treatment with Xorit. Catechol, hydroquinone, quinhydrone, 2,2 - diphenyl - 1 - picrylhydrazyl (DPPH) , and the tetraalkylammonium salts were Eastman White Label grade; resorcinol and phenol were Mallinckrodt analytical reagent grade; and (ferrocene) dicyclopentadienyl - iron was Eastman practical grade. These were all used without further purification. The indicating electrode was machined from pyrolytic graphite purchased from the General Electric Co. A colorless, transparent, conducting gel, which is excellent for preparing salt bridges, was prepared by heating 5 grams of methyl cellulose (Dow Methai eel, 15-cp. viscosity) with 100 ml. of 0.5M lithium perchlorate in pyridine, and then cooling. Apparatus. The cell and electrode used in this investigation are shown in Figure 1; the cell was placed in a water bath, which was maintained a t 25.0" i 0.1" C. The pyrolytic graphite cylinder (4.0 mm. in diameter) was coated with polyethylene to insulate the sides from the solution; the ah VOL. 37, NO. 4, APRIL 1965
467
