method reported here should not have application for the titration of bases in any electrolytic solvent in which the potentials for the oxidations of hydrogen, palladium, and the solvent bear the same relationship to each other that they do in water and methanol.
0
12
e e e e
ACKNOWLEDGMENT
0
The authors thank A. C. Makrides and D. N. Jewett and their colleagues for helpful discussions. -06
-04
-02
0
+02
to4
+06
108
t1.0
Figure 4. Current-voltage curves for pure Pd membrane in methanol, buffered with 0.1M each acetic acid-sodium acetate Electrode supplied with hydrogen
0 Hydrogen absent
with the maximum rate of change of pH with generation time ( t ) . This latter was located by the method of Gran (3) as the intersection of linear plots of At/ApH for short increments of 2 around the equivalence point. A summary of results is presented in Tables I and 11. The small but significantly positive titration errors shown in Tables I and I1 may be restated as negative deviations from 100% titration efficiency; failure to achieve 1 0 0 ~ ocurrent efficiency for hydrogen ion generation is the most
LITERATURE CITED
+I2
E , Volts us. S.C.E.
likely cause. The small anodic residual currents which flow in the absence of hydrogen (Figures 2 and 4) are of the same order of magnitude as the apparently inefficient portion of the generating current used in titrations. However, since the current-voltage performance of a given electrode changes with time in contact with solutions, there seems no exact way to correct for the residual current. Consequently, we have reported errors based on the total current. There is no a priori reason that the
(1) Blackburn,
T. R., Ph.D. thesis, Harvard University, Cambridge, &lass., 1962. (2) Cleary, H. J., Greene, N. D., Electrochim. Acta 10, 1107 (1965). (3) Gran, G., Acta Chem. Scand. 4 , 559 (1950). (4)Jewett, D. N., llakrides, A. C., Trans. Faraday soc. 61, 932 (1965). (5) Lingane, J. J., “Electroanalytical Chemistry,” 2nd ed., Interscience, New York, 1958. (6) Riddick, J. A,, Ann. X. Y . Acad. Scz. 92, 357 (1961); C . A . 56, 10884d (1962).
(7) Streuli, C. A., ANAL. CHEM.2 8 , 130 (1956). (81 Szebelledy, L., Somogyi, Z., 2. Anal. Chem. 112, 395 (1935). (9) Vasile, AI. J., Enke, C. G., J . Electrochem. SOC.112, 865 (1965). RECEIVEDfor review February 2, 1966.
Accepted April 11,1966. Work supported by National Science Foundation grant GP-3510 and by Wellesley College work scholarship to R. B. G.
Reversible Charge Transfer at the Tubular Platinum E ectrode W. J. BLAEDEL and L. N. KLATT Department of Chemistry, University o f Wisconsin, Madison, Wis.
b
Current-potential equations for the reversible charge transfer at the tubular platinum electrode (TPE) have been theoretically derived and experimentally verified with the ferricyanide-ferrocyanide system. At steady state, the dependences of the concentration profile upon distance along the tube and upon axial linear velocity are shown graphically. For flow rates of a few milliliters per minute in a 1 -cm. X 0.05-cm. diameter tube, the diffusion layer at steady state is comparable to that obtained in a potentiostatic experiment for times less than 1 second. For a given system, the half wave potential at the TPE differs slightly but definitely from that at the dropping mercury electrode.
H
electrochemical systems, where transport of the
YDRODYNAMIC
electroactive material occurs principally by convection, are of two general types: moving electrode systems, such as the rotating platinum wire electrode (14, I 7 ) , the rotating disk electrode (6,8), and the streaming mercury electrode ( 7 ) ; and moving solution systems, containing such electrodes as stationary wire electrodes (9, d 7 ) , a micro-bypass electrode ( d 6 ) , plane electrodes (19), spherical electrodes (IS, 28), a microconical electrode ( I d ) , and tubular electrodes (4, 5, IO). Much of the work on these systems has been empirical or has been based upon the Nernst diffusion layer concept, which assumes a linear concentration gradient. Only a few cases have been treated rigorously, usually in the diffusion-limited region. Hydrodynamic electrochemical systems have great analytical and practical capabilities; however, a more
extensive theory of the convective transport process and of the associated charge transfer is required to understand the fundamental processes occurring at hydrodynamic electrodes. The following work is concerned with a theoretical description and experimental confirmation of the reversible charge transfer at the tubular platinum electrode (TPE). THEORY
Derivation of Equations. T h e general equation for mass transfer of a chemical species, Ci, in a tube of circular cross section, with radius p and length X,and with a laminar velocity regime is given by Levich (20).
VOL. 38, NO. 7, JUNE 1966
879
= 5.306 X 105nD~r/3X2/3T/'1'3 .
This change of variable reduces Equations 6 and 7 to second-order differential equations which may be solved bv methods applicable to first-order differeritial equations, as shorrn in the appendix. The solution gives the explicit dependences of Co and CR upon v,, P , I,and y:
i
The fluxes of 0 and R to the wall of the tube may be found by differentiating Equations 12 and 13 with respect to y and evaluating a t y = 0.
from Equation 21 by substitution of the exponential form of 0 and rearranging.
cO*
1
- @R*
+ k0
(21)
When DO is in cm.2/second, X in em., and V in ~ m . ~ / s e c o n di is , in ma. The relation between the current and the potential may be obtained directly
Figure 1 . Coordinate system and tube parameters
The coordinate system and tube parameters are depicted in Figure 1. In the laminar flow regime, the velocity profile is of the form
where v, is the axial linear velocity. If the diffusion layer is very thin-Le., r p-the introduction of the variable y = p - r gives 2vay
v. = -
RT ide - i - In _ _ (22) nF i - ida
I n Equation 22, i d c and i d o are the limiting currents approached a t i g h l y cathodic potentials (0 approaches zero in Equation 21) and at highly anodic potentials (0 approaches infinity in Equation 21), respectively. The first two terms on the right side of Equation 22 correspond to the half-wave potential.
(3)
P
Equations 1 and 3 may be combined to give the equation for convective transport in a tube with laminar flow
% . bC 2 = D, b2C ?' P
(4)
ax
(15)
For the reversible reduction of an oxidized species 0 to a reduced species
R, 0+neeR
(5)
occurring a t a tubular electrode, the following boundary value problem must be considered.
Combining Equations 14 and 15 according to Equation 9 gives CR*
-
CR(y=O)= ( D O / D R ) ~ ~ ~ CO(~=O) ( D o / D R ) ~ ' ~ C O(16) *
which expresses the conservation of mass a t the electrode surface, and which relates the surface concentrations to the bulk concentrations. The current i due to Reaction 5 may be obtained from the flux of substance
sox (*)
0.
y = 0:
aco Do - +DE
i
nF -(E RT
- Eo)
(10)
Co and CR are the molar concentrations of substances 0 and R ; CO* and CR* are the bulk molar concentrations of substances 0 and R; DOand DR are the diffusion coefficients; n is the number of electrons; E o is the formal electrode potential; and R , T , and F have their usual thermodynamic significance. Equations 6-10 may be solved by defining a new variable q = (va/xP)"3Y
880
=
nF(2~p)
DO
bY
dx
(17)
Y-0
=
--
by
exp
+
E = E O - - RT 1n ( D O / D R ) ~ / ~ nP
ANALYTICAL CHEMISTRY
(11)
I n Equation 18, 0 is an abbreviation for the exponential term in Equation 10, and k is (D0/DR)2'3. For the purposes of measurement, it is convenient to use a volume flow rate V which may be expressed in terms of p and ua.
V
=
s,'
2arv,dr
v = irp2u,/2
(19)
p0)
Concentration Profiles. Equations 12 and 13 give the dependence of the concentrations of suhstances 0 and R upon the independent variables x and y, and upon t h e experimentally controllable parameters v , and p. Figure 2 indicates how the concentration profile (CO/Co* or CR/CO*us. y ) changes with distance (x em.) along the tube. Also given with each curve is the value of the parameter u,/xp (cm. second)-' from which the effect of u. and p upon the concentration profiles may be determined by simply evaluating this parameter for various values of va and p. Figure 2 shows that the thickness of the diffusion layer ranges from 0.002 to 0.006 em., or from 4 to 12% of the electrode radius. Examination of Equation 3 reveals that it is an excellent approximation for the velocity profile near the electrode surface, where molecular diffusion dominates relative to the linear velocity, and that the error of this approximation is under 5%, even for distances as far out as 0.006 cm. The times required to establish diffusion layers of comparable thickness in Dotentiostatic exDeriments are rather short, calculable as 0.1-1 second for DO = cm.2/second ( 7 ) . Thus. there may be some advantage in study: ing electrode reaction kinetics at steady .
The current may now be expressed in terms of volume flow rate.
I
SOLUTION OUTLET
(24)
CONTACT T O
Even if the diffusion coefficients of 0 and R differed by an order of magnitude, the half-wave potentials would differ by only about 10 mv. a t 25" C., which makes experimental verification of this difference quite difficult. EXPERIMENTAL
Figure 2.
Concentration profiles
The pair of numbers associated with each curve represents x (cm.) and the value of the parameter v , / x p (cm. second) -l. Profiles are calculated for Do = 1 0-6 cm.2/second, va = 10 cm./second, p = 0.05 cm., and C : =0
state with the TPE rather than with stationary electrodes in quiescent media, because of the experimental simplicity t h a t may be achieved. Variation of the Flux along the Electrode. Because t h e flux a t t h e surface of t h e electrode depends inversely upon t h e cube root of ;T, t h e flux does not change very rapidly with distance along t h e electrode. From Equation 14, with the same conditions as in Figure 2, and with Co* = 10-4Alf,the relative fluxes of substance 0 a t distances x of 0.1, 0.5, 1.0, and 2.0 em. are 1.24, 0.73, 0.58, and 0.46, respectively. According to Equations 14 and 15, the fluyes are undefined at T = 0, and this entire theory does not apply in the region near I = 0. T h e Half-Wave Potential. Unlike t h e thermodynamic standard potential, t h e half-wave potential is not a fundamental property of the cheniical system alone, b u t depends upon t h e electrode at which i t is measured. At the TPE, the half-wave potential depends upon ( D o / D R ) ~ whereas '~, at the dropping mercury electrode (DME) the half-wave potential depends upon ( D o / D R ) 1 ' 2(23). Thus, the half-wave potentials a t the TPE and D M E differ by :
Table I. Summary of Log ;-Log Plots at Various Potentials.
Pot,ent,ial 0,285 0 276 0,250
a
Slope 0.362 0.362 0.361 0.334 0.225 0.343 0.200 0.343 0.175 mean = 0.349 Data taken from Figure 4.
V
Apparatus. T h e current-potential measurements were made with a Sargent Model XXI polarograph, and all potential measurements were checked with a precision potentiometer (Rubicon Co., Philadelphia, Pa.). A peristaltic pump (IZIodel 000-1200, Harvard Apparatus Co., Dover, Mass.) powered by a regulated variable-speed motor (l/ls h.p., Type XSH34RH, S47 motor control; I3 and B Motor and Control Corp., 96 Spring St., S e w York, K. Y , ) was used to control the flow rate of the solutions. Figure 3 depicts the electrode assembly. The body consisted of two 2 x 11/4 1/2-inch Plexiglas blocks (Rohm and Haas Co., Philadelphia, Pa,), into which were milled 11/4 x 3/8 X 3/s-inch mirror-image channels. These channels formed the working and reference electrode compartments, when separated with an anion exchange membrane (1-104-EC Series, American Machine and Foundry Co., Springdale, Conn.). The platinum tube was cut from seamless platinum tubing (0.010-inch mall thickness), and the ends were finished squarely and smoothly. The platinum tube was then press-fitted into 0.003inch undersize holes in the Plexiglas and heated slight,ly, softening the Plesiglas and forming a tight seal. Electrical contact to the TPE was made by a soldered platinum wire before assembly. After assembly, the exposed portion of the T P E was completely encased in epoxy ceinent to provide mechanical strength. The reference electrode was an XgAgCl electrode of geometric area approximately 50 times greater than the T P E , prepared according to the procedure of Ives and J a m ( 1 1 ) . The reference elect'rode electrolyte mas 1-11 KCl saturated with silver chloride. The resistance of the cell, filled with Llf KCl, was 230 ohms, measured with a conductance bridge (Model RCJI 15B1, A. H.Thomas Co., Philadelphia, Pa.). Except for the ball joint, inlet and i.d. Tygon outlet tubes were 1/32-iii~h tubing (C. S. Stoneware Co., -tkron, Ohio), press fitted y i t h cyclohexanone into the Plexiglas. All solution channels upstream from the TPE were constructed from 2 mm. i.d. glass capillary tubing. Materials. All chemicals were of reagent grade quality, used without further purification. All solutions were prepared from triply distilled water, t h e second distillation being from alkaline permanganate. T h e solutions were deaerated, prior to and
+
