AC Research
Articles Anal. Chem. 1996, 68, 581-584
Diffusion Currents in Concentrated Redox Solutions Scott C. Paulson, Nathan D. Okerlund, and Henry S. White*
Department of Chemistry, University of Utah, Salt Lake City, Utah 84112
Diffusion-limited currents, ilim, corresponding to the 1 e- reductions of nitrobenzene (NB) and benzophenone (BP) at Pt microdisk electrodes are reported for CH3CN solutions containing 0.01 e [NB] e 9.1 M and 0.01 e [BP] e 4.0 M. Values of ilim exhibit a highly nonlinear dependence on redox concentration, obtaining maximum values in solutions containing either ∼3 M NB or ∼2 M BP (i.e., above these intermediate concentrations, ilim decreases with increasing redox concentration). The data are analyzed within the framework of a redox species/ solvent interdiffusion model, employing the CullinanVignes equation to estimate concentration-dependent binary mutual diffusion coefficients for the NB/CH3CN and BP/CH3CN systems. An analytical equation is derived that allows ilim values for a microdisk to be computed using the viscosities and hard-sphere molecular radii of the solvent and redox species in their pure state. Predicted values of ilim are found to be in reasonable agreement with voltammetric currents measured for NB and BP reduction. Diffusion of an electroactive species under the influence of a chemical potential gradient involves the interdiffusion of the redox species, solvent molecules, and supporting electrolyte ions.1 Despite the complexities of this transport process, simplified expressions for diffusion-limited rates based on a single empirically determined diffusion constant are frequently, and justifiably, employed in investigations of dilute redox solutions. In concentrated redox solutions, molecular diffusivities are anticipated to show a significant dependence on the concentrations of various solution constituents.1 This expectation is based on the kinetic theory of transport processes in liquids, which assumes that diffusion occurs by thermally activated displacements of individual (1) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960. 0003-2700/96/0368-0581$12.00/0
© 1996 American Chemical Society
species over molecular dimensions.1,2 Because this activation energy will be largely determined by molecular interactions and structural parameters, the diffusion velocity of a molecule will be greatly influenced by the solution composition. In previous investigations, we have observed behavior consistent with a significant decrease in the diffusion coefficient of the redox species when its concentration is increased above ∼50 mM. Specifically, voltammetric limiting currents, ilim, measured using microdisk electrodes for the reductions of small organic molecules, e.g., cyanopyridine3 and nitrobenzene,4 are found to exhibit a highly nonlinear dependence on redox concentration, Credox. At very high redox concentrations, ilim is observed to decrease with increasing Credox. In contrast, a linear ilim vs Credox relationship is predicted if the diffusion coefficient of the redox species is independent of solution composition.5 In the current report, we present a simple analytical expression for the diffusion-limited current at a microdisk electrode that accounts for the concentration dependence of the interdiffusion of solvent and redox species. The key component of our analysis is the computation of the concentration-dependent binary diffusion coefficient for liquids containing comparable mole fractions of solvent and redox species. Our preliminary results indicate that ilim can be predicted within a factor of ∼2 for all redox concentrations accessible in electrochemical investigations. EXPERIMENTAL SECTION Benzophenone (Janssen Chimica, 99%) and acetonitrile (Fisher Scientific, HPLC grade) were used as received. Nitrobenzene (Aldrich, 99+%) was stored over molecular sieves. Tetra-nbutylammonium hexafluorophosphate [(n-butyl)4NPF6; Aldrich, 98% or Sigma] was recrystallized twice from absolute ethanol and vacuum dried at 150 °C. (2) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes; McGrawHill: New York, 1941. (3) Morris, R. B.; Fischer, K. F.; White, H. S. J. Phys. Chem. 1988, 92, 5306. (4) Malmsten, R. A.; Smith, C. P.; White, H. S. J. Electroanal. Chem. 1986, 215, 223. (5) Saito, Y. Rev. Polarogr. 1968, 15, 177.
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Pt microdisk electrodes were constructed by sealing a nominally 12.5-µm-radius Pt wire in soft glass and grinding one end to expose a microdisk surface. The electrode was polished using successively finer grits of emory paper and, as a final step, with 1.5-µm alumina powder on a cloth pad. Precise values of electrode radii were calculated from limiting current values for the oxidation of 2 mM ferrocene in CH3CN containing 0.2 M (n-butyl)4NPF6. The literature value6 for the diffusivity of ferrocene in CH3CN (2.4 × 10-5 cm2/s) was assumed in this calibration procedure. Electrochemical measurements were made using a conventional three-electrode cell equipped with a Pt wire counter electrode and a Ag wire quasi-reference electrode. Voltammetric curves were obtained using either a Princeton Applied Research Corp. Model 173 potentiostat and Model 175 Universal Programmer or a Bioanalytical Systems Model CV-27. Solutions were purged with N2. Absolute solution viscosities were determined using size 25 and 50 calibrated Cannon-Fenske Routine (capillary) viscometers and measured solution densities. Measurements were performed at room temperature (21.5 ( 0.5 °C). RESULTS AND DISCUSSION The diffusion-limited current for the reduction of a neutral species at a microdisk electrode is given by6
ilim ) 4nFDC*ro
(1)
where F is the Faraday constant, ro is the electrode radius, C* is the concentration of the redox species in bulk solution, n is the number of electrons transferred per molecule, and D is the diffusion coefficient of the electroactive species. For a dilute solution of a redox species, D can be assumed to have a constant value throughout the solution (i.e., the value of D for redox molecules in the diffusion layer region is the same as in the bulk solution). However, the reduction of a neutral redox species that is present in the bulk solution at molar concentrations is likely to alter the physical properties of the diffusion layer, since the electrochemical process will necessarily generate molar quantities of the corresponding anion near the surface. Consequently, it is anticipated that the near-surface diffusivity will be smaller than in the bulk solution and that this spatial dependence may result in a decrease in ilim. However, a simple analysis of this problem demonstrates that ilim will never be less than a factor of ∼2 from the value predicted using eq 1, regardless of how much D decreases within the depletion layer.3 Thus, in employing eq 1, we assume a constant value of D equal to the bulk solution diffusivity. Since the mole fraction of supporting electrolyte is generally small relative to either the mole fraction of solvent or redox species, the bulk phase of the solution can be modeled as a binary liquid, consisting only of redox and solvent molecules. The binary mutual diffusion coefficient, Dij, for such a system is given by
(
Dij ) (Doij)xj(Doji)xi 1 +
)
∂ ln γi ∂ ln xi
(2)
where Dijo is the diffusion coefficient for a dilute solution of i in j, Djio is the diffusion coefficient for a dilute solution of j in i, γi is the activity coefficient of i, and xj and xi are the mole fractions of (6) Kuwana, T.; Bublitz, D. E.; Hoh, D. E. J. Am. Chem. Soc. 1960, 82, 5811.
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species j and i, respectively. Equation 2 was originally derived by Cullinan using the Eyring theory of absolute reaction rates.7 In a companion paper, Vignes tested this relationship using diffusivity data for ∼25 binary organic solutions and found excellent agreement between theory and experiment for both ideal solutions (∂ ln γi/∂ ln xi ) 0, e.g., heptane/hexadecane) and nonideal solutions (e.g., acetone/benzene and ethanol/chloroform).8 Substituting the Cullinan-Vignes equation into eq 1 yields the diffusion-limited current at a microdisk electrode (where the subscripts i and j refer to the redox species and solvent, respectively). Equation 3 is applicable to both dilute and con-
(
ilim ) 4nFC*ro(Doij)xj(Doji)xi 1 +
)
∂ ln γi ∂ ln xi
(3)
centrated solutions, and, thus, represents a more general expression than eq 1 for the diffusion current at a microdisk. In the limit of an infinitely dilute solution (i.e., xi f 0), eq 3 reduces to eq 1 as expected. In most instances encountered in electrochemistry, one or both of the infinite dilution binary diffusivities (Dijo and Djio ) will not be known a priori. For instance, for the reduction of nitrobenzene (NB) in acetonitrile (CH3CN), the diffusivity Dijo may be readily obtained from measurement of ilim for a dilute solution of NB in CH3CN. However the corresponding measurement of Djio can not be made since the redox species is reduced more easily than the solvent. In such cases, the infinite dilution binary diffusivities may be written in terms of measurable physical parameters using the Stokes-Einstein equation1
Doij )
kT 6πriηj
(4)
where ri is the hard-sphere radius of the redox molecule and ηj is the absolute viscosity of the pure solvent. Transposing the subscripts i and j in eq 4 yields Djio in terms of the radius of the solvent molecule, rj, and the viscosity of pure redox species, ηi. Combining the Stokes-Einstein expression for Dijo and Djio with eq 3 yields the limiting diffusion current
ilim )
(
)
2nFC*kTro ∂ ln γi (ηirj)-xi(ηjri)-xj 1 + 3π ∂ ln xi
(5)
If the pure redox species is a liquid at room temperature then ηi and ηj may be measured by straightforward viscometric measurements. Molecular radii may be estimated from liquid densities, molecular models, or crystallographic data. The activity correction term (∂ ln γi/∂ ln xi) for a binary solution is generally obtained from partial pressure data. In general, neglecting this term leads to an overestimation of the binary diffusion coefficient by a factor of 1.2-2 at intermediate solution compositions.8 However, since ∂ ln γi/∂ ln xi approaches zero as xi f 0 or 1, the (7) Cullinan, H. T. Ind. Eng. Chem. Fundam. 1966, 5, 281. (8) Vignes, A. Ind. Eng. Chem. Fundam. 1966, 5, 189.
error incurred in ignoring activity effects vanishes in the limits of dilute and highly concentrated redox solutions.9 Equation 5 provides a remarkably simple means of computing diffusion-limited currents over the entire range of solution compositions encountered in electrochemistry (0 < xi e 1). Neglecting the activity correction, a more compact form of this expression can be written as
ilim )
2nFC*kTro 3πηavrav
(6)
where ηav represents the geometrically weighted average value of the viscosity of the solvent/redox pair,10
ηav ) (ηixiηjxj)
(7)
rav ) (rjxirixj)
(8)
and rav is defined as
The accuracy of eq 6 (as well as the more exact form, eq 5) is limited, in part, by the approximate nature of the Stokes-Einstein equation for predicting the binary diffusivities at infinite dilution (see Appendix). More accurate semiempirical correlations for computing Dijo and Djio written in terms of the molar volumes of the pure solvent and redox species may be used in place of eq 4. However, the error resulting from using the Stokes-Einstein equation will generally be smaller than the error resulting from neglecting activity effects and/or the variation of the diffusivity within the diffusion layer. Equation 6 was tested by measuring ilim values for the 1 - ereduction of NB in CH3CN. Steady-state voltammetric curves (Figure 1) were obtained using a 13.2-µm-radius Pt disk (calibrated) in CH3CN/0.2 M (n-butyl)4NPF6 solutions containing NB at concentrations ranging from 0.01 to 9.1 M. These solutions may be approximated as binary solutions, since the mole fraction of (n-butyl)4NPF6 is less than 0.019. The viscosities of CH3CN and NB solutions containing 0.2 M (n-butyl)4NPF6 were measured to be 0.394 and 2.377 cP, respectively. These latter values were used to calculate ηav (eq 7) for each mixed CH3CN/NB solution used in the voltammetric experiments. The effective hard-sphere radius for CH3CN was estimated as 2.4 Å from self-diffusion coefficients reported for CH3CN11 and the Stokes-Einstein equation for self-diffusion.12 A value of 2.0 Å for the effective hardsphere radius of NB was computed (equations 1 and 4) from ilim values measured in dilute NB solutions (e0.1 M). The solvent (9) For a binary solution at constant temperature and pressure, the GibbsDuhem equation yields ∂ ln γi/∂ ln xi ) ∂ ln γj/γ ln xj. After substitution of this relationship, and applying Raoult’s law for xi f 1 and xj f 1, the activity correction term in eq 5 vanishes at both high and low redox concentrations. (10) The measured viscosity (ηmix) of the NB/CH3CN (0-9.1 M NB) solutions was found to be closely approximated by the geometrically weighted average defined by eq 7), i.e., ηmix ∼ ηav. Thus, it is possible to use measured values of ηmix in eq 6 for each solution, rather than computing ηav based on the viscosities of pure solvent and redox species. However, this procedure does not appear to offer any advantage. (11) (a) Kunz, W.; Calmettes, P.; Bellissent-Funel, M.-C. J. Chem. Phys. 1993, 99, 2079. (b) Kovacs, H.; Kowalewski, J.; Maliniak, A.; Stilbs, P. J. Phys. Chem. 1989, 93, 962. (c) Easteal, A. J. Aust. J. Chem. 1980, 33, 1667. (12) The Stokes-Einstein equation for the self-diffusion of species i is given by Dii ) kT/4πriηi.
Figure 1. Voltammetric response of a 13.2-µm-radius Pt disk in CH3CN/0.2 M (n-butyl4N)PF6 solutions containing 0.7, 3.0, and 8.0 M nitrobenzene. Scan rate, 10 mV/s.
Figure 2. Experimental (points) and theoretical (solid line) diffusionlimited currents, ilim (eq 6), as a function of nitrobenzene concentration. The solid line is obtained using the parameters rNB ) 2.0 Å, rCH3CN ) 2.4 Å, ηNB ) 2.377 cP, and ηCH3CN ) 0.394 cP. The dashed line is the current predicted by eq 1 using the dilute solution diffusivity of NB.
and redox species radii were then used to calculate rav ( eq 8) for each mixed CH3CN/NB solution. Figure 2 shows experimental and theoretical values (eq 6) of ilim for the CH3CN/NB system. For NB concentrations up to ∼1 M, the agreement between experiment and theory is good. At higher concentrations, both the observed and predicted values of ilim decrease as the redox concentration increases. However, in this range, the theoretical values of ilim computed using eq 6 are ∼2 times larger than the corresponding measured values. This difference is most likely to result from ignoring the decrease in diffusivity near the electrode surface, an effect that should become more pronounced at higher redox concentrations.3 Indeed, as noted above, experimental ilim values are anticipated to be too small by a factor of ∼2 in comparison to values computed using eq 6. Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
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Figure 3. Experimental (points) and theoretical (solid line) diffusionlimited currents, ilim (eq 6), as a function of benzophenone concentration. The solid line is obtained using the parameters rBP ) 2.3 Å, rCH3CN ) 2.4 Å, ηBP(4.0 M) ) 2.374 cP, and ηCH3CN ) 0.394 cP. The dashed lines show the effect of varying the solvent radius by factors of 0.5 and 2×.
Thus, taking into account this limitation of the theory, eq 6 predicts ilim values over the entire range of assessable redox concentrations with reasonable accuracy. In many instances, the pure form of the redox species is a room-temperature solid, and thus, ηi and ηav cannot be obtained. However, eq 6 may still be employed by defining ηi as the measured viscosity of the most concentrated solution employed in the investigation. For example, we have used eq 6 to predict ilim values for the 1 - e- reduction of benzophenone (BP) in CH3CN/0.12 M (n-butyl)4NPF6 solutions containing 0.01-4.0 M BP. The viscosity of the 4.0 M BP solution was measured to be 2.374 cP. Using the values for the viscosity and radius of CH3CN given above, and rBP ) 2.3 Å (obtained as previously described for NB), we have computed ilim values for each CH3CN/BP solution. Figure 3 shows that the experimental and theoretical values are again in reasonable agreement. Values of ilim predicted by eq 6 are quite sensitive to the solvent molecular radius. For instance, the dashed lines in Figure 3 show the expected dependence of ilim on redox concentration when the radius of CH3CN is assumed to be 0.5 and 2 times as large as the value computed based on the self-diffusion coefficient. These curves illustrate the importance of interpreting diffusion-limited voltammetric currents in terms of an interdiffusion model. The analysis presented above does not take into account the electron flux resulting from homogenous electron-transfer reactions (i.e., “electron-hopping”) that are coupled to physical diffusion and migration of the redox reactant and product. For instance, electron conduction has been shown to represent ∼50% of the total current in solutions containing molar concentrations of NB.13 Thus, we anticipate that this additional flux would tend to increase ilim at high NB concentrations to values above those predicted by eq 6. However, the complex dependencies of the mutual binary diffusion coefficient (eq 2) and the electron flux contribution (see ref 14) on redox concentration prevent a straightforward quantitative analysis of this problem. (13) Norton, J. D.; Anderson, S. A.; White, H. S. J. Phys. Chem. 1992, 96, 3. (14) Saveant, J. M. J. Phys. Chem. 1988, 92, 4526.
584 Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
Figure 4. Plot of D vs η-1 for ferrocene in mixed CH3CN/NB solutions.
In conclusion, the solvent/redox species interdiffusion model provides a method for computing diffusion currents in concentrated redox solution. We have shown that eq 6 captures the essential features of the nonlinear dependence of ilim on redox concentration. Quantitative agreement between theory and experiment is expected by using data obtained at lower electrolyte concentrations, correcting Dij for activity effects and electron diffusion, and accounting for the variation of molecular diffusivity across the depletion layer. ACKNOWLEDGMENT This work was funded by the National Science Foundation/ Electric Power Research Institute and the Office of Naval Research. APPENDIX The Stokes-Einstein equation is not strictly applicable when the dimensions of the solute molecule are comparable to that of the solvent molecule. However, we find that plots of D vs. η-1 for small redox molecules in CH3CN/NB mixtures are linear. For instance, Figure 4 shows values of D for ferrocene (Fc) obtained from voltammeric currents in mixed CH3CN/NB solutions containing 15 mM Fc, plotted as a function of η-1. The slope of the resulting straight line is 9.56 × 10-8 cm3/g, ∼50% higher than that predicted by eq 4 assuming a hard-sphere radius of 3.4 Å (the latter value estimated from the crystallographic structure of ferrocene). A similar straight-line plot was obtained for BP (0.2 M) in CH3CN solutions, where η was varied by adjusting the concentration of (n-butyl)4NPF6. Again, the slope of the line yielded an effective radius for BP that was ∼30% smaller than the anticipated value based on molecular structure.
Received for review September 25, 1995. November 17, 1995.X
Accepted
AC950958Q X
Abstract published in Advance ACS Abstracts, January 1, 1996.
