Toward the Reactivity Prediction: Outersphere Electroreduction of

J. Phys. Chem. C 2009, 113, 2881–2890

2881

Toward the Reactivity Prediction: Outersphere Electroreduction of Transition-Metal Ammine Complexes Renat R. Nazmutdinov,† Maria Yu. Rusanova,‡ David VanderPorten,‡ Galina A. Tsirlina,*,§ and W. Ronald Fawcett‡ Kazan State Technological UniVersity, Kazan 420015, Republic Tatarstan, Russian Federation, Department of Chemistry, UniVersity of California, DaVis, California 95616, USA, and Department of Electrochemistry, Moscow State UniVersity, Moscow 119899, Russian Federation ReceiVed: September 6, 2008; ReVised Manuscript ReceiVed: December 6, 2008

The electrochemical reactivity of Co(III), Cr(III), and Ru(III) ammine complexes is discussed within the framework of a quantum chemical approach. The electronic structure of oxidized and reduced complex forms is addressed at the density functional theory level; intramolecular and solvent contributions to the Franck-Condon barrier, as well as electronic transmission coefficients are computed. Standard electrode potentials are estimated for certain redox pairs. The charge distribution in the reactants is employed to calculate electrostatic interactions with a mercury electrode surface and to make double layer corrections more realistic as compared to point/spherical reactant models. The model predictions are employed to explain the rate constant and transfer coefficient. Also discussed are trends in the available experimental data, including new original results on a comparative study of Co(III) and Cr(III) hexaammine complexes. Introduction The prediction of the rate constant of a chemical process remains a central problem of chemical kinetics. Earlier attempts at such predictions have been based on examination of some sequences of relative reactants and correlations with certain independently measured values. One of the most systematic analyses of this sort was reported by Vlcˇek1 for electron transfer (ET) reactions. At that stage, the configurational effects were addressed on the basis of spectroscopic data. Usually, the majority of reactant features were assumed to be fixed, whereas only one or two were varied. Comparison with experiment was limited to considering a single factor or a very few that affect the rate constant. Although the uncertainty in the electrostatic correction for reactants with an inhomogeneous atomic charge distribution had been already mentioned at that time,2 a quantitative description remained unavailable. This article is an attempt (similar to that made by Weaver et al.3,4) to consider the data corrected for double layer effects and to get a self-consistent description of corrected key kinetic parameters, namely the rate constant and transfer coefficient R, at least semiqualitatively. Our analysis rests on the Marcus theory (ET activation barrier) and a modern quantum mechanical theory (pre-exponential factor).5,6 We employ a quantum chemical approach to estimate the intramolecular reorganization energy and transmission coefficients for a number of inert transition-metal ammine complexes being good model reactants. We restrict ourselves to reduction from aqueous solutions on a mercury electrode, despite the availability of precise data for solid metals as well;3,4,7-15 the reason is the possible specific electrode-reactant interaction, which follows from earlier data treatment. On the other hand, there is strong evidence for the * To whom correspondence should be addressed. Tel. +7(495)9391321. Fax +7(495)9328846. E-mail: [email protected]. † Kazan State Technological University. ‡ University of California, Davis. § Moscow State University.

outersphere reduction of all complex species under discussion just on mercury. Our goal is to understand the nature of the anomalously high values of the transfer coefficient R and to decrease the uncertainty of the rate constant interpretation (bearing in mind that standard potentials of the redox pairs are strictly unknown). To attain this goal, we also need to decrease the uncertainty of R, an important parameter of the elementary act, being typically determined from the corrected Tafel plot (cTp). We demonstrate that significant deviations of a so-called corrected R from 0.5 at low overvoltages can result from under- or overestimated double layer corrections. Thus, one should examine various possible configurations of the reaction layer to avoid misleading conclusions. To address the last point more precisely, we apply the original concept of a “molecular Frumkin correction”,14-19 which recasts an idea qualitatively formulated earlier.2 We are rather skeptical regarding the use of simplified double layer corrections without a special molecular level analysis of the experimental data. To demonstrate the importance of this point, the original data for electroreduction of Co(III) and Cr(III) hexaammine complexes at mercury are presented below. Data treatment under various assumptions on the structure of the reaction layer is reduced to construction of cTp with variation of effective parameters in a wide range. When comparing the literature data on the reduction of various amminecomplexes on mercury (Table 1), one sees a satisfactory mutual agreement of the rough data for solutions of certain composition (observed rate constants, kf) and a poorer agreement of the rate constants resulting after the double layer correction (kcorr), if the latter corresponds to different concentrations and/or nature of the supporting electrolyte. As the transfer coefficient is also sensitive to the double layer correction, there is no chance to extract a true value of this important parameter without more thorough analysis. The transfer coefficient, in turn, plays a crucial role in extrapolation of the rate constant data to certain potentials, a procedure widely used in various comparative studies.

10.1021/jp807926t CCC: $40.75  2009 American Chemical Society Published on Web 01/26/2009

2882 J. Phys. Chem. C, Vol. 113, No. 7, 2009

Nazmutdinov et al.

TABLE 1: Kinetic Parameters Reported Earlier for the Reduction of Certain Metal Ammine Complexes on a Mercury Electrode; All Potentials Are Presented in sce Scale reactant

solution

-4

-1

[Co(NH3)6]3+

0.04 M La(ClO4)3 0.003 M HClO4 0.1 M NaClO4

kcorr ) 4 × 10 cm s (-1V) kf ) 2 × 10-2cm s-1 (-1 V) kf ) 5.1 × 10-4 cm s-1 (-0.3 V)

[Co(NH3)6]3+ [Co(NH3)6]3+ [Co(NH3)5F2+

0.1 M KPF6 0.1 M KPF6 0.1 M NaClO4

kf ) 2.7 × 10-4 cm s-1 (-0.3 V) kf ) 5 × 10-6 cm s-1 (-0.175 V) kf ) 6.5 × 10-5 cm s-1 (-0.3 V)

[Co(NH3)5F]2+

0.1 M KPF6

kf ) 4.5 × 10-4 cm s-1 (-0.3 V)

[Co(NH3)5F]2+ [Co(NH3)5F]2+

0.2 M.NaF 0.2 M.NaF

[Ru(NH3)6]3+ [Ru(NH3)6]3+

0.1 M KPF6 0.1 M KPF6

kf ) 8 × 10-6 cm s-1 (-0.2 V) kf ) 1 × 10-2 cm s-1 (-0.47 V) 2.8 × 10-5 cm s-1 (-0.27 V) kcorr )1 × 10-2 cm s-1 (-0.47 V) 3.3 × 10-4 cm s-1 (-0.27 V) kf ) 0.3 cm s-1 (-0.175 V) kcorr ) 2. cm s-1 (-0.18 V)

3+

[Cr(NH3)6]

R from cTpsa

reaction rate constants (k)/ corresponding potential

ref.

0.75

7

0.5 0.5

10, 11

0.5 0.5 0.5 0.5 0.75 0.69

0.6

10, 11 9 10, 11 10, 11 11 10

9 3

a In all of these studies, the effective charge of the reactant was assumed to have an integer charge number at the double layer (ionic association was never considered), and the potential drop across the double layer was calculated according to the Gouy-Chapman theory.

We pay special attention to the problem of the distance of closest approach as this parameter affects considerably the rate constant values. Unfortunately, for the usual electrode reactions this distance can be neither fixed, nor determined directly, so that we have the only possibility to compare several versions of model calculations performed for various distances and orientations. We try to improve this situation by looking for a self-consistency between the computational results and the experimental data. Another reason to quantify the transfer coefficients arises from our recent generalization of its dependence on the mechanism of the electrode reaction.22 Experimental Methods A conventional three-electrode glass cell with glass ground joints was used for all of the experiments. The working electrode was a hanging drop mercury electrode (Controlled Growth Mercury Electrode manufactured by BAS). The saturated calomel reference electrode and platinum counter electrode completed the three-electrode setup. The cleanliness of the supporting electrolyte solutions was verified by cyclic voltammetry. To minimize contamination of the working solutions with chloride ions and decrease the value of the liquid junction potential, the reference electrode was connected to the cell through a Luggin capillary filled with a 0.5 M NaClO4 solution. All solutions were prepared with Nanopure water with a minimum resistivity of 18 MΩ/cm (Barnstead). Glassware was cleaned in boiling 50% nitric acid or in a mixture of concentrated nitric and hydrochloric acids (1:3) and washed with Nanopure water before each set of experiments. Perchloric acid, (Aldrich), sodium perchlorate (GFS Chemicals) were of the best quality available. If it was necessary, the salt was additionally dried. The reactant Co(NH3)6(ClO4)3 was prepared from the corresponding chloride by precipitation with saturated sodium perchlorate and recrystallization from water. The reactant Cr(NH3)6(NO3)3 was synthesized by the reaction between unhydrous CrCl3 and liquid ammonia in the presence of metal Na and Fe(NH4)2(SO4)2. The working solutions were thoroughly deaerated with highpurity argon moisturized before coming to the cell by bubbling through Nanopure water. For the cell used, which had a volume of 50 mL, the deaeration time was 45-60 min. During the

experiments, an argon atmosphere was maintained above the working solution. Cyclic voltammograms (CVs) were obtained by using a Princeton Applied Research (PAR) 173 potentiostat with a PAR 175 universal programmer. The data were collected in digitized form by using the PAR 175 and a computer with a GPIB standard interface. The irreversible one-electron reduction of [Cr(NH3)6]3+ and [Co(NH3)6]3+ ions in acidic (10-3 M HClO4) solutions of 0.03 M NaClO4 was studied by cyclic voltammetry at a scan rate of 50 mV s-1. It was also assumed that specific adsorption of ClO4- ion is negligibly small in the studied potential range. Double layer capacity data were taken from the literature.23 The potential of zero charge (Ez) was determined from the minimum current observed on a CV obtained in a 10-4 M solution of HClO4. The value of Ez is in good agreement with literature data.24 All the potentials are corrected for liquid junction potentials using the Henderson equation and are given on the saturated calomel electrode scale. The reduction current was also corrected for the charging current. The rate constants were calculated for a totally irreversible reduction process from the experimental CV (corrected for the charging current), using the expression

[

ln kf ) -ln

]

Ilim - I(t) + ln DA1⁄ 2 i(t)

(1)

where I(t) is the semi-integral of the measured current i(t), Ilim is the limiting current, and DA is the diffusion coefficient of the reacting species calculated as follows

DA1⁄ 2 )

Ilim . FAcA

(2)

Here, A is the effective surface area of the working electrode and cA is the reactant concentration.24 The semi-integration procedure was performed as described in the literature.24-26 Freshly prepared working solutions of Cr(NH3)6(NO3)3 (10-4 M) and Co(NH3)6(ClO4)3 (5 × 10-4 M) were used in all experiments. Between the experiments, the working solutions were kept in the dark at a constant temperature of 25 °C. The experiments were carried out at a constant temperature of 25 °C. The effective surface area of the hanging drop mercury

Transition-Metal Ammine Complexes

J. Phys. Chem. C, Vol. 113, No. 7, 2009 2883

TABLE 2: Me-N Bond Length, r(Me-N), Vibration Frequency, ν, and Intramolecular Reorganization Energy, λin, Calculated for Certain Amminocomplexes in Oxidized and Reduced Statesa redox pairs

state

ground-state spin

Ox

1/2

[Ru(NH3)6]3+/2+ Red [Cr(NH3)6]3+/2+

Ox

0 3/2

Red

2

Ox

0

[Co(NH3)6]3+/2+

Red Ox

3/2 0

[Co(NH3)5F]2+/+ Red

3/2

ν/cm-1

f

r(Me-N) /nm 0.2195÷0.2201 (0.2104)34 0.2128÷0.2217 (0.212÷0.214)34 0.2152÷0.216 {0.2104÷0.2107}g 0.222 (eq)c; 0.262 (ax) {0.219÷0.221}(eq)g {0.2474÷0.2487}(ax)g 0.2039÷0.2045 (0.1972÷0.1992)37 (0.1908÷0.1977)38 (0.203)39d {0.2034÷0.2045)g 0.2255÷0.2268 (0.216);40 (0.222)41d {0.226÷0.2272)g 0.2023(ax) 0.2003÷0.2005(eq) (0.1817)b 0.2195 (ax) 0.223÷0.2269 (eq) (0.193)b

403 (500)35

e

λin/eV 0.05

373 380 (465);35 (470)36

0.02 0.93 {0.41}g

316 (eq); 147 (ax)

0.87 {1.32}g

413 (494)35 (486÷490)39

307 (357);35 (327)42

1.13 {1.17}g

1.22 {1.22}g

420

1.37

309

1.21

a

Experimental data obtained for crystal salts using XRD, IR, and Raman spectroscopy are given in parentheses, with corresponding references. For Co(III/II) and Cr(III/II), both gas-phase and (in braces) solvent-affected values are listed. b The Co-F bond length. c eq and ax note equatorial and axial coordination, respectively. d results of DFT calculations. e The averaged intramolecular reorganization energy λ˜ in can Ox Red Ox Red 2 Ox Red be calculated as follows:43 λ˜ in ) 4λin λin in ref 22. f Frequency of a normal vibration λin /(λin + λin ) ; λin and λin correspond to a λin and b leading to the intramolecular reorganization at ET. g The geometry of complex was fully optimized by using the PCM.

electrode was kept constant (0.024 ( 0.001 cm2). All the potentials are given in the SCE scale. Computational Details The calculations were performed at the density functional theory (DFT) level with the hybrid functional B3LYP27-30 as implemented in the Gaussian 03 program suite.31 The standard 6-311++g(d, p) basis set31 augmented by diffusive and polarization functions was used to describe the electrons in the Co, Cr, N, F, and H atoms. A basis set of DZ quality32 was employed to describe the valence electrons of Ru atom, whereas the effect of inner electrons was included in the relativistic effective core potential developed by Hay and Wadt.32 The geometry of the complexes in oxidized and reduced states was fully optimized without symmetry restrictions. The unrestricted DFT formalism was employed to describe the open-shell systems. The vibration frequencies were calculated as well. The solvation effects in the bulk of aqueous solutions were addressed by using the polarized continuum model (PCM).31 Although such calculations were performed mostly at the fixed optimized geometry of complex forms found previously for the gas phase, the geometry of some complexes was newly optimized; a value of 78 being assumed for the dielectric constant of liquid water. The atomic charges were computed within the framework of the ChelpG scheme,33 which yields the best fit of the molecular potential (metallic radii values were employed in these calculations for the Ru, Co, and Cr atoms). Results and Discussion Molecular Features of the Reactants and Products. The low-spin electronic configuration was found for the groundstate of the [Co(NH3)6]3+, [Co(NH3)5F]2+, and [Ru(NH3)6]3+

complexes, whereas [Cr(NH3)6]3+ has a value of 3/2 for the total spin number (Table 2). In the reduced state, the electrons form a high-spin structure for [Co(NH3)6]2+, [Co(NH3)5F]+, and [Cr(NH3)6]2+; the ground-state for the [Ru(NH3)6]2+ complex is singlet. Six N atoms from the nearest coordination shell form an octahedron for [Co(NH3)6]3+ and [Cr(NH3)6]3+; for [Ru(NH3)6]3+ the octahedron structure reveals a very slight distortion due to the Jahn-Teller effect. As can be seen from Table 2, the computed Co-N and Ru-N bond lengths are in good agreement with experimental data. The calculated intramolecular energy (λin) values are also summarized in Table 2 (pertinent details of calculations in ref 44). The smallest λin values (which are comparable with kT in order of magnitude) were found for the [Ru(NH3)6]3+/2+ redox pair as the electron is transferred to the bonding molecular orbital of the complex. On the contrary, electron transfer to the antibonding orbitals of [Co(NH3)6]3+, [Co(NH3)5F]2+, and [Cr(NH3)6]3+ entails a significant reorganization of the nearest coordination sheath (a lengthening of the Co-N bonds). For [Co(NH3)5F]3+/2+ and [Cr(NH3)6]3+/2+, an attempt was made to address solvent effect (in terms of the PCM) on the equilibrium geometry of species and, therefore, on their intramolecular reorganization. Our calculations predict no any noticeable change either for the geometry of [Co(NH3)5F]3+/2+, or for the corresponding λin values when going from gas phase to solution. At the same time, for [Cr(NH3)6]3+/2+ redox couple the solvent influence is found to be significant (Table 2). The solvent weakens the Jahn-Teller distortions of [Cr(NH3)6]2+, although they still remain pronounced. The most important prediction relates to the character of intramolecular reorganization. As can bee seen from Table 2, the solvent environment results in a considerable asymmetry of λin values. Moreover,


Nazmutdinov et al.

Figure 1. Optimized structure of [Cr(NH3)4]2+; r(Cr-N) ) 0.2167 nm; ∠(N-Cr-N) ) 175 grad.

the averaged43 λin value for [Cr(NH3)6]3+/2+ calculated for gas phase (0.9 eV) exceeds only slightly the value obtained for this redox couple taking into account the solvent effect (0.7 eV). Note that the reduction of the ammine complexes does not lead to a noticeable change of the high frequency (quantum) degrees of freedom in the inner coordination shell, namely, the N-H bond lengths and the valence angles ∠H-N-H in the ammonia molecules. The vibration frequencies of the Me-N bond computed for oxidized and reduced forms of the ammine complexes (Table 2) are close to the region for which a classical approximation (frequently used at calculations of the ET activation energy6) is still valid. Therefore, the innersphere reorganization of these redox pairs does contribute to the Franck-Condon barrier. Note that the calculated frequency values are lower as compared with experimental values presented in Table 2 (and also considered in ref 3). This discrepancy might be attributed to the fact that the experimental data have been obtained for crystalline salts, whereas the model calculations of frequencies presume a gas phase. Because the Jahn-Teller distortions for [Cr(NH3)6]2+ are especially strong (Table 2), one should not ignore the possible formation of a tetracoordinated Cr(II) ammine complex (Figure 1):

[Cr(NH3)6]2+(solv) ) [Cr(NH3)4]2+(solv) + 2NH3(solv) (3) The symmetry of starting geometry of [Cr(NH3)4]2+ is close to tetrahedron. Because of the Jahn-Teller effect, this structure, however, is not stable and its optimized geometry was found to form a slightly distorted planar-square complex. According to our estimations, the free energy of the process (eq 3) amounts to -0.5 eV (the entropy term plays a key role), which makes it feasible in electrolyte solutions. Exploring the rate of reaction (eq 3) remains out of the scope of our studies. The λin values given in Table 2 are valid if we separate the ET elementary act and the ligand detachment treating the latter as a subsequent chemical step. Because the product of [Co(NH3)5F]2+ reduction is apparently the most labile species among the other reduced ammine complexes, one cannot exclude the following substitution process:

[Co(NH3)5F]+(solv) + H2O(solv) ) [Co(NH3)4H2OF]+(solv) + NH3(solv) (4) +

In [Co(NH3)4H2OF] , the water molecule resides in trans position relative to the F atom. The model estimations predict

a rather small negative value (ca. -0.06 eV) for the reaction free energy (the key contribution also comes from the entropy term), and this process can be neglected in further modeling except in estimating standard electrode potentials. Using the computed hydration energy values of the complex species (on the basis of PCM), we have estimated the effective radii (reff) of the reactants: namely 0.32 nm for [Co(NH3)5F]2+ and 0.36 nm for the other ammine complexes, which exceed significantly the Me-N bond lengths. In the absence of specific adsorption, the reff values are assumed to be of value for estimating the distances of electrode-reactant closest approach. We have calculated the solvent contribution (λs) to the total reorganization energy using the model of a conducting sphere in the vicinity of the inner double layer part (interlayer) developed by Kharkatz et al.45 with correction for quantum modes of solvent by a factor of 0.8. For a reasonable configuration of the reaction layer, the λs values do not exceed 0.4 to 0.6 eV (reff ) 0.36 nm) and 0.6 to 0.7 eV (reff ) 0.32 nm) at the interlayer thickness ranging from 0.3 to 0.4 nm. These values are significantly lower as compared to previously estimated from usual Marcus formula (ca. 0.9 eV).3 They are believed to be more close because they reflect the real physical features of spatially inhomogeneous reaction layer. Mercury Electrode-Reactant Electronic Overlap. An important parameter that allows one to asses the adiabaticity of electron transfer in condensed media is the Landau-Zener (LZ) factor, 2πγe. This quantity for the case of heterogeneous ET (assuming linear response theory for the region of low overvoltages) can be defined in the following way:5,6

γe ≈ F(εF)kT

|Vif|2 pωeff

π (λs + λ˜ in)kT

(5)

where F(εF) is the density of electronic states at the Fermi level of a metal electrode, and ωeff is the effective polarization frequency of liquid water (∼1013 c-1, see in refs 5 and 6). The key quantity in eq 5 governing the order of Landau-Zener factor is the resonance integral Vif, which depends crucially on the composition and symmetry of the acceptor orbital of the complex reactant. We estimated this quantity on the basis of perturbation theory using the results of DFT calculations and the jellium model for a uncharged mercury surface (pertinent computational details and discussions of the model assumptions in refs 46-48). Then the electronic transmission coefficient (κ) can be calculated as follows5

κ ≈ 1 - exp(-2πγe)

(6)

The electronic transmission coefficient values (κ) were computed for various values of the central atom-mercury electrode surface distance (x); three different orientations of the reactants are shown in Figure 2. It is convenient to fit the results of calculations in the exponential decay form:

κ(x) ) κ0 exp(-βx)

(7)

where x is computed from the metal electrode edge (also see the discussion in ref 47). The κ0 and β values are presented in Table 3; the electronic transmission coefficients calculated for several electrode-reactant distances and three orientations are given in Table 4. It can be argued that for all ammine complexes residing outside the compact layer ET proceeds in diabatic limit, which originates mostly from a strong localization of the molecular acceptor orbitals on the central atoms (RMe values in Table 3). For [Co(NH3)6]3+ and [Cr(NH3)6]3+ complex forms, the electrode-



Figure 2. Orientations of the complex reactants relative the electrode surface (which is assumed to be on the bottom) considered when computing the electronic transmission coefficient; [Ru(NH3)6]3+/2+, [Cr(NH3)6]3+/2+, [Co(NH3)6]3+/2+ - a; [Co(NH3)5F]3+/2+ - b.

TABLE 3: Parameters of Eq 7, K0, β, and Contribution of the Central Atom of Complex (Me) to the Molecular Acceptor Orbital (rMe) redox pairs orientation apex

ridge

model parameters RMea κ0

0.35

β/ nm-1

0.2

κ0 β/ nm-1

face

κ0 β/ nm-1

a

[Ru(NH3)6]3+/2 0.95

[Cr(NH3)6]3+/2+ 0.66 23 0.206

570

40

0.227

0.209

440

47

0.227

0.215

[Co(NH3)6]3+/2+ 0.5 8.7 0.213 7.1 0.225 10 0.218

[Co(NH3)5F]2+/+ 0.53 380 (F) 300 (N) 0.304 (F) 0.212 (N) 6.2 (F) 1.1 (N) 0.204 (F) 0.179 (N) 4.9 (F) 11 (N) 0.209 (F) 0.202 (N)

RMe ≈ ΣiMeci2/Σjcj2, where ci(j) are contributions to the molecular orbital.

reactant orbital overlap depends slightly on their orientation, whereas the orientation of [Ru(NH3)6]3+ and [Co(NH3)5F]2+ noticeably affects the electronic transmission coefficients. The κ values were found to increase in the series [Co(NH3)6]3+ < [Cr(NH3)6]3+ < [Co(NH3)5F]2+ ≈ [Ru(NH3)6]3+. This model prediction can be qualitatively understood bearing in mind that the radius of the valence d-orbitals of the transition metals increases in the same sequence (namely by a factor of 2 going from Co to Ru). Previous Data on the Rate Constants in View of the New Computational Results. A challenging problem of the electrochemistry of metal ammine complexes is the fact that the standard potentials (E0) of some redox pairs are unknown. This

was the reason for the approximations and additional assumptions complicating the last analysis of the problem by Weaver.4 Ru(III) and Co(III) Hexaammine Complexes. Tabulated values are available for cobalt and ruthenium systems (0.108 and 0.10 V in nhe scale, respectively49). The former is known to be less perfect but under the rough approximation we can consider [Co(NH3)6]3+ and [Ru(NH3)6]3+ reductions at a certain potential as taking place at the same overvoltage and without a noticeable difference in double layer effects. The effect of inner sphere reorganization can be readily appraised (Table 2) and is responsible for the difference in the rate constants of ca. 5 orders of magnitude. Such a difference agrees with available experimental values (observed at an overvoltage close to zero,9 Table


Nazmutdinov et al.

TABLE 4: Electronic Transmission Coefficient Values Computed for Certain Ammino Complexes vs the Cental Atom-Mercury Electrode Distance (x) at Three Different Orientations complex reactant 3+

[Ru(NH3)6]

[Cr(NH3)6]3+ [Co(NH3)6]3+ [Co(NH3)5F]2+

orientation apex ridge face apex ridge face apex ridge face apex(F) apex(N) ridge(F) ridge(N) face(F) face(N)

x ) 0.4 nm -4

1.2 10 6.5 10-2 5. 10-2 6.1 10-3 9.4 10-3 8.7 10-3 1.7 10-3 8.8 10-4 1.6 10-3 2. 10-3 6.2 10-2 1.8 10-3 8.5 10-4 1.1 10-3 3.4 10-3

x ) 0.5 nm -5

1.6 10 6.7 10-3 5.2 10-3 7.7 10-4 1.2 10-3 1. 10-3 2.1 10-4 9.2 10-5 1.8 10-4 9.5 10-5 7.5 10-3 2.3 10-4 1.4 10-4 1.4 10-4 4.5 10-4

x ) 0.6 nm -6

2.2 10 6.9 10-4 5.4 10-4 1. 10-4 1.4 10-4 1.2 10-4 2.5 10-5 9.7 10-6 2.1 10-5 4.6 10-6 9.10-4 3.10-5 2.4 10-5 1.8 10-5 6.10-5

x ) 0.7 nm 2.9 10-7 7.2 10-5 5.5 10-5 1.3 10-5 1.8 10-5 1.4 10-5 2.9 10-6 1. 10-6 2.4 10-6 2.2 10-7 1.1 10-4 3.9 10-6 4.10-6 2.2 10-6 8. 10-6

TABLE 5: Kinetic Data for the Reduction of [Cr(NH3)6]3+ and [Co(NH3)6]3+ on Hg Treated by Means of Construction of Classical cTp reactant

solution

[Cr(NH3)6]3+ [Co(NH3)6]3+

0.03 M NaClO4 + 0.001 M HClO4 0.3 M NaClO4 + 0.001 M HClO4

[Co(NH3)6]3+

0.03 M NaClO4 + 0.001 M HClO4

1). At the same time, one could expect an additional difference up to 2 orders of magnitude stemming from the electronic transmission coefficient (see the most reactive ridge and face orientations of [Ru(NH3)6]3+ in Table 4). There are, however, at least two possible reasons which could diminish the preexponetial effect mentioned above: (i) predominating ET at the significantly less reactive apex orientation of Ru(III) complex and (ii) the closer approach of [Co(NH3)6]3+ to the electrode surface. Both situations probably occur when some additional specific interaction of the ammine complexes with the mercury surface is assumed. This should depend on the nature of central atom. Being still open, this challenging question stimulates further modeling the metal ammine complexes in the reaction layer. It should be stressed that the observed discrepancy is even stronger, if the standard potential for [Co(NH3)6]3+/2+ is less positive, as it was stated in ref 4. Co(III) Hexaammine and Pentaammine Fluoride Complexes. The comparison of the rate constants for other reactants under discussion looks even more complicated, as there are no reliable data for their standard potentials. Despite currently existing approaches to predict the E0 values, they still cannot be obtained with a good accuracy. At the same time, they are not available from experiment for certain systems for various reasons. We have estimated standard potentials using the scheme applied earlier to cyanide cobalt complexes50 and found E0 ) +0.13 V nhe for [Co(NH3)6]3+/2+, which is in surprisingly good agreement with the tabulated value.48 The same approach yields a value of -0.38 V nhe for [Co(NH3)5F]2+/+ redox pair (or -0.32 V nhe, when reaction (eq 4) is considered as well). From first glance, both values look too negative, as some authors assumed more positive potentials (on the nhe scale). One can argue from this analysis, however, that the replacement of one ammine ligand with fluoride entails a decrease in the equilibrium potential. This fact together with the higher innersphere reorganization energy (Table 2) is expected to induce a slower reduction of mixed-ligand species, in agreement with the data for perchlorate solutions10,11 (Table 1). At the same time, our calculations predict the higher transmission coefficients for the

reaction rate constants, k (the range of potential) kf ) 1.3 × 10-3 cm s-1 (-0.8V) kcorr ) 6.7 × 10-3 cm s-1 (-0.15V) kf ) 1.8 × 10-4 cm s-1 (-15V)

R from cTps 0.77 0.45 0.40

apex orientation of [Co(NH3)5F]2+ as compared with any orientation of [Co(NH3)6]3+. This difference might be additionally increased by the closer approach of the former reactant to the electrode surface (judging from the effective radii, the total difference in the corrected rate constants induced by the preexponential term is expected to be roughly 2 orders of magnitude). The electrostatic repulsion of [Co(NH3)5F]2+ should be less than for [Co(NH3)6]3+ because of their difference in ionic charges. To address this point more quantitatively, we estimated the effective charges from internal charge distributions in oxidized and reduced species using the approach described in refs 19-21. We considered the apex(F) orientation as evidently the most favorable for positively charged surface and assumed that the effective spheres touch the outer Helmholtz plane. The difference between the effective charges calculated for [Co(NH3)5F]2+ (+1.76) and [Co(NH3)6]3+(+2.71) is very close to the difference in their integer charges. The same conclusion remains true for the reduced forms (+0.84 and +2.03, respectively).52 Finally, we have to deal with a complex compensation effect: the pre-exponential term and double layer effect favor the faster reduction of substituted species, whereas the innersphere contribution and standard potential difference provide an opposite effect. Finally, if the latter effect overcompensates the influence of other factors by 2 orders of magnitude, one may conclude that the standard potential of the [Co(NH3)5F]2+/+ couple is essentially more negative than that of the hexaammine Co(III) couple. Cr(III) and Co(III) Hexaammine Complexes. The problem of the unknown standard potential for [Cr(NH3)6]3+/2+ appears to be even more serious when we compare the rate constants of cobalt and chromium hexaammine complexes, as there are no any data related to any fixed potential. From the model calculations, we obtained E0 ) -0.5 V nhe for [Cr(NH3)6]3+/2+, and E0 ≈ 0 V nhe for the reaction

[Cr(NH3)6]3+ + e ) [Cr(NH3)4]2+ + 2 NH3 3+

(8)

As the reduction of [Cr(NH3)6] starts at several hundred mVs more negative potentials than the latter value, it is hardly



Figure 3. Corrected Tafel plots constructed under various assumptions about the effective reactant charge (indicated at the plot) for [Co(NH3)6]3+ (a) and [Cr(NH3)6]3+ (b). The data for 0.001 M HClO4 + 0.03 M NaClO4.

reasonable to consider dissociative redox equilibria (eq 8). Assuming a value of -0.5 V nhe for the standard potential, the data7 (Table 1) correspond to overvoltage above 200 mV, and there are no similar data for comparison with [Co(NH3)6]3+. The values presented in Tables 2 and 4 demonstrate that the difference in key parameters (intramolecular reorganization energy and transmission coefficient) does not differ drastically, predicting a higher rate constant of about 1 order of magnitude for the chromium complex at a fixed overvoltage. We report below some original data for these two reactants, to have a more robust basis for comparison. Transfer Coefficients as Related to Innersphere Reorganization. The transfer coefficient is less sensitive to the choice of approximate standard potential values as compared to rate constant. Hence, to verify any model of elementary act one can get more certain result when analyzing differential kinetic parameters. Reexamination of various possible by-side reasons related to the double layer correction should precede; however,

the model treatment of anomalously high transfer coefficient observed for Cr(III) complex. Below, we attempt to do this in a more convincing way (with allowance for possible molecular level effects), using the original data for Co(III) and Cr(III) hexaammine complexes. We do not aim at any exact value; the main goal is to estimate the degree of uncertainty induced by unknown location of the species having inhomogeneous internal charge distribution. Experimental Data treatment: Various Types of cTp. To construct any cTp, the experimental dependence of the rate constant kf on the electrode potential is usually plotted as (lnkf + zAfφd) against (φm - φd), where zA is the charge of the reacting species, f ) F/RT (F, R, and T have their usual meaning), φd is the potential at the oHp, and φm is the rational potential (φm ) E - Ez where E is the experimental potential and Ez is the zero charge potential). We employ exactly the same approach, although sometimes use the effective values instead of the


Nazmutdinov et al.

Figure 4. Corrected Tafel plots for [Co(NH3)6]3+ (a) and [Cr(NH3)6]3+ (b), constructed under various assumptions about the decreased effective potential Nφd, of the outer Helmholz plane, N being 0.7 or 0.90 (indicated in the plots). The reactant’s charge is assumed to be +2. The data for 0.001 M HClO4 + 0.03 M NaClO4.

integer reactant charge and/or φd (ref 17 for more detailed explanations). Table 5 collects the values obtained on the basis of usual cTp procedure and assuming more realistic integer charges +2 for highly charged species (most probably associated with cations, see discussion in refs 16, 18). There is no serious contradiction with the values presented in Table 1, if one takes into account the difference in solution composition. However, the dependence of transfer coefficient on supporting electrolyte concentration indicates that the result is approximate. In what follows, we try to find whether some version of the double layer correction can decrease the difference of transfer coefficients for the cobalt and chromium complexes. Performing simple tests, we have examined how strong should be the deviation of an effective charge from an integer value to get a transfer coefficient equal to 0.5 from the experimental data. The effective charge was varied in a wide interval assuming both internal charge distribution and ion pairing effects. For

[Co(NH3)6]3+, the decrease in the effective charge changes the slope significantly (part a of Figure 3), and if this charge is below +2, the values of the transfer coefficient close to 0.5 can be basically obtained. This is not the case for [Cr(NH3)6]3+ (part b of Figure 3), for which the transfer coefficient remains close to unity for a wide range of effective charges. To investigate what effective charges can be expected and how these values can depend on the electrode-reactant distance and electrode potential, we computed the values for various distances between the reactant edge and the outer Helmholtz plane. When this distance increased, the effective charges slightly decreased (by about ca. 0.1 per 0.1 nm) but the difference of effective charges for oxidized and reduced forms always remained the same (close to unity). This means that we can exclude any serious distortions of cTps induced by the arbitrary choice of the distance of approach. Another possibility to correct the transfer coefficients is to take into account the difference of molecular work terms for



reactant and product when constructing the abscissa axis of the corrected Tafel plots. If the reactants are rather large and do not penetrate into the inner layer, this value is always lower as compared to φd. After we have chosen two values as realistic ones (Figure 4), a possible decrease of φd was found to induce only minor changes in the transfer coefficient. Furthermore, the difference for cobalt and chromium complexes still remained considerable. All combined versions (with simultaneous application of effective charge and effective φd) lead to the same conclusion. Basically, it is possible to consider the specific work term, which depends (linear, in the first approximation) on the electrode potential. We are aware of the fact that such a hypothesis looks purely phenomenologically without model calculations. However, the data under consideration correspond to rather narrow potential ranges, and it is hardly possible to expect the predominating role of potential-dependent work term. Nevertheless, the analysis performed above reduces the number of factors responsible for the behavior of transfer coefficient. The effective charges applied in this analysis include all possible versions of double layer corrections. Our treatment of polarization curves for cobalt ammine complex clearly demonstrates that ca. 0.1 deviation of transfer coefficient from 0.5 can easily result from some uncertainty of double layer correction. As the Ru(III) complex undergoes the reduction in the same region of electrode charges, one can extend this conclusion to ruthenium redox system as well, and do not consider R ∼ 0.6 as the fact requiring more deep and detailed explanations. At the same time, Cr(III) complex deviation from 0.5 is much higher than 0.1. We can maintain that the true transfer coefficient (as well as the apparent value) for [Cr(NH3)6]3+ is anomalously high and calls for some explanation. Possible Interpretation. It was already mentioned above that considering the effect of solvent environment leads to a significant asymmetry of the intramolecular reorganization of Ox Red ) 0.41 eV, λin ) 1.32 eV). [Cr(NH3)6]3+/2+ redox system (λin This may, in turn, entail increasing the asymmetry coefficient at the vicinity of pzc. According to a general equation22

1 R) ∂Ui ⁄ ∂r 1+ |∂Uf ⁄ ∂r|

(9)

where Ui(f)(r) are potentials along intramolecular coordinate r for reactant and product; the derivatives are taken at the saddle point of the reaction free energy surface. Ox Red /λin , the Since at small overvoltages ∂Ui/∂r/|∂Uf/∂r| ≈ λin transfer coefficient for Cr(III) ammine complex amounts to ca. 0.76, which may explain the experimental data just within the accuracy available for the cTp slope. Conclusions We have explored the intramolecular reorganization and electronic transmission coefficient in the electroreduction of inert Co(III), Cr(III), and Ru(III) ammine complexes on a mercury electrode using a combined computational approach. The estimates of the standard electrode potentials of certain redox pairs have been performed as well, which facilitates the treatment of the experimental kinetic data. Because the level of the model calculations does not allow one to attain a reasonable quantitative accuracy in the prediction of the rate constant values ab initio, emphasis was put on elucidating some effects of a qualitative nature. We paid special attention to the

dependence of electronic transmission coefficient and the reorganization of inner sheath on the nature of central atom and the orientation of the complex reactant at the electrode surface, as well as on the problem of stability of certain labile product forms in aqueous solution. The discussion of computational results is closely intertwined with attempts to interpret available experimental data (including original data). Despite the apparent qualitative character of the main conclusions, we believe that our analysis is useful to gain a new molecular level insight into the electrochemistry of transition-metal ammine complexes. It would be tempting in the future to increase the accuracy of model calculations of the reactant-electrode coupling to get more reliable estimates of electronic transmission coefficient. Another important and challenging problem is the estimation of specific work terms for complex reactants in terms of a microscopic approach. In other words, the role of water molecules on the electrode surface, the descreteness-of-charge, and the microscopic details of the double layer potential profile should be properly addressed. The latter problem looks exceedingly complicated and still remains a bottle-neck of various quantum chemical approaches developed recently to describe the heterogeneous ET reactions.43 Nevertheless, in the near future a considerable computational effort will be made to advance in exactly this field. From the experimental side, the kinetic data for wider potential ranges are of great interest to compare rate constants and transfer coefficients at different electrode charges and overvoltages keeping the same molecular features of the reacting species. In this case, the procedure of cTp construction should be modified and even becomes more perfect, despite higher technical complexity.51 Acknowledgment. We thank M. Probst for the help with quantum chemical calculations and D.V. Glukhov for useful discussions. This study was supported in part by the RFBR (project 08-03-00769-a). References and Notes (1) Vlcˇek, A. A. Discuss. Faraday Soc. 1958, 26, 164. (2) Vlcˇek, A. A. Nature 1963, 197, 786. (3) Hupp, J. T.; Liu, H. Y.; Farmer, J. K.; Gennett, T.; Weaver, M. J. J. Electroanal. Chem. 1984, 168, 313. (4) Weaver, M. J. J. Electroanal. Chem. 2001, 498, 105. (5) Kuznetsov, A. M.; Ulstrup, J. Electron Transfer in Chemistry and Biology; Wiley: New York, 1999. (6) Kuznetsov, A. M. Charge Transfer in Physics, Chemistry and Biology. Mechanisms of Elementary Processes and Introduction to the Theory; Gordon and Breach Science Publishers: Berkshire, 1995. (7) Liu, H. Y.; Hupp, J. T.; Weaver, M. J. J. Electroanal. Chem. 1984, 179, 219. (8) Barr, S. W.; Guyler, K. L.; Weaver, M. J. J. Electroanal. Chem. 1980, 111, 41. (9) Gennett, T.; Weaver, M. J. Anal. Chem. 1984, 56, 1444. (10) Fawcett, W. R.; Solomon, P. H. J. Electroanal. Chem. 1988, 251, 183. (11) Satterberg, T. L.; Weaver, M. J. J. Phys. Chem. 1978, 82, 1784. (12) Hamelin, A.; Weaver, M. J. J. Electroanal. Chem. 1986, 209, 109. (13) Hromadova´, M.; Fawcett, W. R. J. Phys. Chem. A 2000, 104, 4356. (14) Hromadova´, M.; Fawcett, W. R. J. Phys. Chem. B 2004, 108, 3277. (15) Muzikar, M.; Fawcett, W. R. Anal. Chem. 2004, 76, 3607. (16) Fawcett, W. R.; Hromadova, M.; Tsirlina, G. A.; Nazmutdinov, R. R. J. Electroanal. Chem. 2001, 498, 93. (17) Tsirlina, G. A.; Petrii, O. A.; Nazmutdinov, R. R.; Glukhov, D. V. Russ. J. Electrochem. 2002, 38, 132. (18) Rusanova, M. Yu.; Tsirlina, G. A.; Nazmutdinov, R. R.; Fawcett, W. R. J. Phys. Chem. A 2005, 109, 1348. (19) Fawcett, W. R. Cond. Mat. Phys. 2005, 8, 413. (20) Fawcett, W. R.; Hromadova, M. J. Solid State Electrochem. 2008, 12, 347. (21) Fawcett, W. R.; Chavis, G. J.; Hromadova, M. Electrochim. Acta 2008, 53, 6787.

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Nazmutdinov et al. (34) Stynes, H. C.; Ibers, J. A. Inorg. Chem. 1971, 10, 2304. (35) Schmidt, K. H.; Mu¨ller, A. Inorg. Chem. 1975, 14, 2183. (36) Blyholder, G.; Vergez, S. J. Phys. Chem. 1963, 67, 2149. (37) Schroeder, D. R.; Jacobson, R. A. Inorg. Chem. 1973, 12, 211. (38) Meek, D. W.; Ibers, J. A. Inorg. Chem. 1970, 9, 465. (39) Long, T. V.; Herlinger, A. W.; Epstein, E. F.; Bernal, I. Inorg. Chem. 1970, 9, 459. (40) Herlinger, A. W.; Brown, J. N.; Dwyer, M. A.; Pavkovic, S. F. Inorg. Chem. 1981, 20, 2366. (41) Endres, R. G.; LaBute, M. X.; Cox, D. L. J. Chem. Phys. 2003, 118, 8706. (42) Sacconi, L.; Sabatini, A.; Gans, P. Inorg. Chem. 1964, 3, 1773. (43) Nazmutdinov, R. R. Russ. J. Electrochem. 2002, 38, 131. (44) Nazmutdinov, R. R.; Tsirlina, G. A.; Kharkats, Yu. I.; Petrii, O. A.; Probst, M. J. Phys. Chem. B 1998, 102, 677. (45) Kharkats, Yu. I.; Nielsen, H.; Ulstrup, J. J. Electroanal. Chem. 1984, 169, 47. (46) Nazmutdinov, R. R.; Glukhov, D. V.; Tsirlina, G. A.; Petrii, O. A. Russ. J. Eletrochem. 2003, 39, 97. (47) Nazmutdinov, R. R.; Kuznetsov, A. M.; Schmickler, W. Chem. Phys. 2005, 310, 257. (48) Nazmutdinov, R. R.; Glukhov, D. V.; Petrii, O. A.; Tsirlina, G. A.; Botukhova, G. N. J. Electroanal. Chem. 2003, 552, 261. (49) CRC handbook of Chemistry and Physics, 84th Edition; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 2004. (50) Kotov, V. Yu.; Nazmutdinov, R. R.; Botukhova, G. N.; Petrii, O. A.; Tsirlina, G. A. MendeleeV Commun. 2004, 14, 113. (51) Zagrebin, P. A.; Tsirlina, G. A.; Nazmutdinov, R. R.; Petrii, O. A.; Probst, M. J. Solid State Electrochem. 2006, 10, 157. (52) The effective charges for cobalt, chromium, and ruthenium hexaammine complexes (the latter two are not reported) are always very close if one and the same location relative to the outer Helmholtz plane is assumed.

JP807926T

Toward the Reactivity Prediction: Outersphere Electroreduction of

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