equal t o that of pure CaCla solution, yet we consider that a t this concentration the membrane has already lost its specificity. Specificity of the Orion Electrode. Figure 10 represents the behavior of the systems described in the last paragraph as measured with the Orion electrode. The theoretical curves differ slightly from those of Figure 9, as they were calculated from the curves of emf/log aca+2 obtained from the Orion electrode. The general pattern is, however, similar t o that of the PVC, TTA, TBP membrane: the electrodes appear t o be selective towards Ca+2 u p to a concentration of 10-lM of the foreign salt. Specificity of Membranes in Solutions Containing More Than One Foreign Salt. The method for assessing C a + 2 specificity described above can be easily extended t o deal with mixed foreign salts. Instead of plotting the theoretical curve and the experimental results as a function of log C of one foreign salt, they may be plotted as a function of the log of the ionic strength I of the mixture. Such a procedure can also serve as a test of the method, for if the method is sound the experimental points obtained by measurements in CaCla solutions, in MgCl, solutions, o r in mixed solutions, should all fall on the same line. Figure 11 illustrates such a procedure in the case of the Orion electrode. A solution of 1.52 X 10-3M CaCla was
tested in varying concentrations of NaCl alone, in varying concentrations of MgClz alone, and in varying concentrations of a mixture of MgCla and NaCl, in the proportion of 1 mole of Mg to 10 moles of Na (similar to the proportion encountered in biological systems). U p to a n ionic strength of about 3 X lO-lM, the membrane is specific t o Ca+2. Similar results are obtained for the PVC, TBP, TTA membranes. The agreement between the theoretical curve and the experimental points seems fair. The agreement between the different kinds of experimental points (Na, Mg, mixture) seems to support the method described above. ACKNOWLEDGMENT We are greatly indebted to M. Anbar of the Isotope Department of the Weizmann Institute of Science, under whose direction this investigation was carried out, for his constant interest and advice. We are grateful to H. A. Saroff of the National Institutes of Health and S. Szapiro of the Weizmann Institute of Science for their helpful suggestions.
RECEIVED for review November 30, 1966. Accepted March 6, 1967. This paper is based o n work performed under grant No. 5x5121 of the National Institutes of Health.
Quasi-Reversible and lrreversible Charge Transfer at the Tubular Electrode L. N. Klatt a n d W. J. Blaedel Department of Chemistry, Unicersity of Wisconsin, Madison, Wis.
Current-potential equations for the quasi-reversible and totally irreversible heterogeneous charge transfer reactions at a tubular electrode have been theoretically derived and experimentally verified. The dependences of the current-potential curve upon the standard rate constant, transfer coefficient, the volume flow rate are shown graphically. A procedure for determining the standard rate constant and the transfer coefficient of a chemical system is presented. Magnitudes of rate constants determinable by this technique are about the same as those at the dropping mercury electrode.
HETEROGENEOUS ELECTRON TRANSFER reactions have received considerable attention since Eyring (I) presented the equations relating the rates of electrochemical reactions t o the electric field present a t the electrode-solution interface. Even though experimental investigation of these reactions would be greatly simplified if concentration polarization of the electrode could be eliminated, such a condition is almost impossible to achieve. As a result, quantitative theories concerned with electrode reactions have had to consider the rates of both mass and electron transfer. Early work in this direction was attempted by Eyring and coworkers (2), who applied the absolute rate theory and the Nernst diffusion layer concept t o the problem of ( 1 ) S. Glasstone, K. J. Laidler, and H. Eyring, “The Theory of Rate Processes,” McGraw-Hill, New York, 1941, pp. 575-7. (2) H. Eyring, L. Marker, and T. C. Kwoh, J . Phys. Colloid Chem.,
53, 187 (1949).
polarographic current-potential curves. Several groups of investigators (3-5) developed independently the rigorous solution corresponding to the semi-infinite linear diffusion case. Koutecky (6) solved this problem for the polarographic case. The last decade has seen a tremendous development of new techniques such as stationary electrode polarography (7), fast rise potentiostatic techniques (8), relaxation methods (9), and double pulse galvanostatic methods (10-12), each capable of studying rapid heterogeneous electron transfer reactions. All of these theories and techniques dealt with time-dependent transient phenomena. Studies of heterogenous electron transfer reactions in hydrodynamic systems have been noticeably rare, partly because of the success of the time-dependent techniques, and perhaps partly because of difficulties associated with the mathematical treatment and experimental measurements in hydrodynamic systems. Delahay (13) gave a treatment based upon applica(3) M. Srnutek, Col(ection Czech. Chem. Commun., 18, 171 (1953). (4) P. Delahay, J . Am. Chem. Soc., 73,4944 (1951). (5) M. G. Evans and N. S. Hush, J . Chim. Phys., 49, C159 (1952). (6) J. Koutecky, Chem. Lisfy, 47, 323 (1953). (7) R. S. Nicholson and I. Shain, ANAL.CHEM.,36, 706 (1964). (8) S. P. Perone, ANAL.CHEM.,38, 1158 (1966). (9) P. Delahay and T. Berzins, J . Am. Chem. SOC.,77, 6448 (1955). (10) H. Z. Gerischer, Physik Chem., 10,264 (1957). (11) H. Z. Gerischer, Physik Chem., 14, 184 (1958). (12) P. Delahay, J . Am. Chem. SOC.,81, 5077 (1959). (13) P. Delahay, “New Instrumental Methods in Electrochemistry,” Interscience, New York, 1954, pp. 222-6. VOL. 39, NO. 10, AUGUST 1967
1065
centrations within short distances from the wall, well before the tube axis is reached. The velocity profile may be regarded as linear over the region of changing concentration. The errors associated with these assumptions are small for short tubes and moderate flow rates, and have been discussed previously (19). For the reduction at a tubular electrode of an oxidized species 0 to a reduced species R, Ofne-R
kf
(2)
kb
the boundary value problem expressed in Equations 3-7 must be considered.
-.--
2 ~ a y ~ C -O
E-E? VOLTS
ax
P
b2Co Do -y dY
(3)
Figure 1. Dependence of current-potential curves upon standard rate constant, k, Conditions: V = 3.0 ml/minute ana = 0.5 X = 1.0 cm p =
(4)
0.05cm
D O = D R = 5.0 X loFscm2/sec
tion of the Nernst diffusion layer concept to irreversible electron transfer in stirred solutions. Galus and Adams (14) applied Levich’s (15) and Randles’ (16) theory to the rotating disk electrode. Jordan (17) and Jordan and Javick (18) used a cylindrical electrolysis vessel that rotated about a fixed electrode to determine heterogeneous electron transfer rates. A more complete literature survey of hydrodynamic electrochemical systems is given in a previous publication (19). With the recent translation of Levich’s works (15) into English, and with improved techniques for continuous measurements in flowing solutions, the advantages of making electrochemical kinetic studies in hydrodynamic systems are now more realizable. The following work gives a theoretical description and experimental confirmation of the rate-controlled heterogenous electron transfer reaction occurring at a tubular electrode. THEORY
Derivation of Equations. The general differential equation describing the 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 was previously given (19) as
d2CZ 2- v. a-~_dct - D Z Y P bx dY
(14) Z. Galus and R. N. Adams, J. Phys. Chem., 67,866 (1963). (15) V. G. Levich, “Physicochemical Hydrodynamics,” PrenticeHall, Englewood Cliffs, N. J., 1962. (16) J. E. B. Randles, Can. J. Chem., 37, 238 (1959). (17) J. Jordan, ANAL.CHEM.,27, 1708 (1955). (18) J. Jordan and R. A. Javick, Hectrochim. Acta, 6, 23 (1962). (19) W. J. Blaedel and L. N. Klatt, ANAL.CHEM., 38, 879 (1966). 0
ANALYTICAL CHEMISTRY
concentrations, and DO and D R are their diffusion coefficients. The rate constants, k y and kb,can be defined in terms of the Eyring equation
ky kb
=
=
k , exp[(-an,F/RT)(E
k , exp[((l
- E”)]
(8)
- a)n,F/RT)(E - E”)]
(9)
Here, a is the transfer coefficient (13), n, the number of electrons preceding and including the rate determining step of Reaction 2, E” the formal electrode potential, k , the standard formal rate constant, and R, T, and F have their usual thermodynamic significance. The solution of this problem in terms of the concentration profiles is expressed by the incomplete Gamma function and has been discussed previously (19). An analogous problem in heat conduction also has been treated (20). The fluxes of substances 0 and R at the wall of the tube are given by (19)
(1)
where ua is the axial linear velocity. The tube parameters and the coordinate system (origin centered at the tube wall at the entrance; axial distance, x , distance normal to the wall, y ) are more fully defined in reference 19. Equation 1 involves three important assumptions. Curvature effects may be neglected and the problem treated as though the wall were flat-i.e, y 0, respectively. The fundamental Equation 1 requires that C L be much larger than the total concentration of the metal ion ( I ) . Otherwise, a part of the ligand combined with the metal ion could be corrected for C L(3) as will be shown later. Also the formation constants determined with this equation should be “concentration-constants” at, strictly speaking, the same composition of neutral electrolyte because of uncertainties in activity coefficients (3). Now, it will be obvious, from the above relationships, that easy deductions of the successive P3’s are graphically possible by plotting the values of F functions against C L and extrapolating to C L = 0 for the intercepts as pl, pz, . * ., and P I , successively. Such a graphical approach offers a useful method and has been widely accepted for studies of relatively weak complexes of mononuclear species which give reversible polarographic wave a t the dropping mercury electrode. Some plottings of F values, however, d o not fit a smooth curve, especially in the low C Lregion which is most important for the extrapolation. Thus the intercept is apt to suffer from inevitable deviation, which will result in larger deviations for the higher order constants. Therefore, the graphical approach though simple, might be said to have increasing ambiguities in the values obtained for higher order constants. Furthermore, the graphical method does not estimate the precisions of determined values which are necessary for inter(1) D. D. DeFord and D. N. Hume, J . Am. Chem. SOC.,73, 5321 (1951). (2) I. Leden, 2. Physik. Chem., 188A, 160 (1940). (3) H. Irving, “Advances in Polarography,” Vol. 1, Pergamon Press, Oxford, 1960, pp. 42-67.
