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Method for the evaluation of the reorganization energy of electron transfer reactions in water–methanol mixtures
Chemical Physics Letters 407 (2005) 342–346 www.elsevier.com/locate/cplett
Method for the evaluation of the reorganization energy of electron transfer reactions in water–methanol mixtures F. Pe´rez, M. Herna´ndez, R. Prado-Gotor, T. Lopes-Costa, P. Lo´pez-Cornejo
*
Departamento de Quı´mica Fı´sica, Facultad de Quı´mica, Universidad de Sevilla c/ Profesor Garcı´a Gonza´lez s/n. 41012 Sevilla, Spain Received 24 February 2005; in final form 15 March 2005 Available online 14 April 2005
Abstract The kinetics of the electron transfer reactions between [Co(C2O4)3]3 and different ruthenium complexes [Ru(NH3)5L]2+ (L = pyridine, pyrazine and 4-cyanopyridine) have been studied in water–methanol mixtures. A method for the evaluation of the reorganization energy of these reactions, based on comparison of their rate constants, is proposed. The results showed a decrease of the rate constants and a decrease of the reorganization energies when the dielectric constant of the medium decreases. This behaviour is according to that obtained from the use of classical continuum methods. The slight differences found between the distinct methods could be related to the existence of specific effects of the medium. 2005 Elsevier B.V. All rights reserved.
1. Introduction Study of the medium effects on reactivity of electron transfer processes has been of great interest to our group for the last decade [1–5]. Solvent effects in water–cosolvent mixtures are more difficult to explain than those in pure solvents because the reactivity can be dependent on preferential solvation phenomena [6]. However, this kind of mixing medium is interesting because of its relation to several areas of chemistry and biology. An electron transfer process occurs in, at least, the following steps: k1
A þ D A=D k 1
k et
A=D A =Dþ k2
A =Dþ A þ Dþ k 2
*
This scheme corresponds to the simplest case in this type of reaction. Situations more complicated are also possible [7]. The formation of the precursor complex, A/D, happens in the first step. In the second step, the activation of the precursor complex, the electron transfer process and the formation of the successor complex take place. If the reaction is accomplished by a major decrease in the free energy, the contribution from the reverse processes of the second and third steps can be ignored. On the other hand, if the forward and reverse processes in step 1 are diffusion controlled, that is they are rapid, it can easily be shown that
ð1Þ k¼ ð2Þ ð3Þ
Corresponding author. Fax: +34 954557174. E-mail addresses: [email protected], [email protected] (P. Lo´pez-Cornejo).
0009-2614/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.03.105
k et K IP ½X ; 1 þ K IP ½X
ð4Þ
where KIP is the equilibrium constant corresponding to the formation of the precursor complex (KIP = k1/k1). Eq. (4) assumes that the concentration of the reactants (acceptor or donor) is in excess with respect to the other, in such a way that X represents the reactant in excess and [X] its concentration. Besides, if KIP[X] << 1:
F. Pe´rez et al. / Chemical Physics Letters 407 (2005) 342–346
k ¼ k et K IP :
ð5Þ
On the other hand, we can write the following expression of ket given by the classical electron transfer theory [8]: 6¼ =RT
k et ¼ jet mn eDG
;
Absorbance changes were monitored at 407 nm for [Ru(NH3)5py]2+ and at 489 nm for [Ru(NH3)5 (4-CNpy)]2+ (these wavelengths correspond to the maximum absorbance values for the different ruthenium complexes).
ð6Þ
where jel and mn are the electronic transmission coefficient and the nuclear frequency factor, respectively [9]. DG6¼, the Gibbs free energy of activation from the electron transfer process, is given by: 2 k þ DG00 6¼ ; ð7Þ DG ¼ 4k k being the (free) energy of reorganization for the electron transfer reaction. DG0 0 is related to the free energy of the process 2, DG0, in the previous scheme, through: DG00 ¼ DG0 þ wp wr ;
343
ð8Þ
where wp is the work of forming the successor complex from the separate products and wr the work needed to form the precursor complex from the separate reactants. The free energy DG0 is a thermodynamic parameter which can be obtained from experimental data (measurements of standard formal redox potentials): DG0 ¼ nF E00acceptor E00 ð9Þ donor : Taking into account all these relations between the different parameters, the experimental results on k are 0 due to the influence of the solvent on DG0 and/or k. It is extremely difficult to calculate KIP, wp, wr and k in water–organic cosolvent mixtures and, therefore, to obtain information about such an influence. Recently, we have published a Paper [10] in which we show a method to obtain relative reorganization energy values in restricted geometry media without the necessity of known KIP, wp and wr. We apply this same method in this work to evaluate the effects on the water– methanol mixtures and, accordingly, to be able to explain the kinetic behaviour of the electron transfer processes in these complicated reaction media. The reactions studied here are RuðNH3 Þ5 L2þ þ 3 CoðC2 O4 Þ3 ; L being pyridine (py) and 4-cyanopyridine (4-CNp).
2. Experimental 2.1. Material and kinetic measurements The preparation of the reactants, [Ru(NH3)5L] (ClO4)3 (L = py and 4-CNpy) and Na3 [Co(C2O4)3], as well as the way of performing kinetic experiments have been described in a previous work [11–13] ([Ru(NH3)5L]2+ = 8 · 105 mol dm3 and [Co(C2O4)3]3 = 1 · 103 mol dm3).
2.2. Electrochemistry The standard formal redox potentials of the [Ru(NH3)5 py]3+/2+ (py = pyridine) pair in the different reaction media were obtained by cyclic voltammetry technique. The equipment, procedure and electrodes have been previously described [14]. The working conditions (pH and ionic strength) were the same as those in the kinetic experiments. A carbon working electrode, a saturated calomel electrode, as reference, and a platinum electrode, as auxiliary electrode, were utilized. Sweep rates used were between 0.1 and 0.5 V s1. Liquid junction potentials were not corrected using the redox potentials in the different solvent mixtures due to the type of calculation made (Section 4).
3. Results Table 1 contains the kinetic data corresponding to the three electron transfer reactions. The data corresponding to [Ru(NH3)5pz]2+ in Table 1 were taken from [13]. Table 2 contains the standard formal redox potentials of the [Ru(NH3)5 pz]3+/2+ couple obtained from experimental measurements.
4. Discussion In agreement with the classical Marcus–Hush Treatment for electron transfer reactions, variations in the second order rate constant, obtained as the ratio k ¼ k obs =½CoðC2 O4 Þ3 3 (kobs being the observed rate constant, that is, the pseudofirst order rate constant), are due to changes in both reorganization free energy, k, 0 and the free energy DG0 (see Eqs. (5)–(7)). In order to have the variations of k when the reaction medium changes, it is necessary, first of all, to separate KIP and ket. In conventional reaction media, there are several procedures to accomplish this separation [15]. In fact, in favourable cases KIP can be calculated through an Eigen–Fuoss approach [16]. This procedure, based on a simple continuum model, seems to be satisfactory enough for pure solvents [16,17]. However, the application of this model to mixed solvents, notwithstanding, used in several Letters [6,9a,18], has been criticised and still remains an open question [19].
F. Pe´rez et al. / Chemical Physics Letters 407 (2005) 342–346
344
Table 1 2+ Observed rate constants (ki; i = py, pz or 4-CNpy) and calculated rate constant ðk calc + i Þ from fit of Eq. (10) for the reaction [Ru(NH3)5L] [Co(C2O4)3]3 (L = pyridine, pyrazine and 4-cyanopyridine) in water–methanol mixtures X
kpy (s1)
1 k calc py ðs Þ
a
0 0.034 0.064 0.126 0.203 0.248 0.362
80.2 76.7 66.4 54.4 47.2 42.9 35.4
82.8 72.9 66.2 55.7 46.8 42.9 35.8
2.50 2.31 2.14 1.81 1.63 1.54 1.43
a
kpz (s1)
1 k calc pz ðs Þ
k4-CNpy (s1)
1 k calc 4-CNpy ðs Þ
2.53 2.29 2.12 1.86 1.65 1.55 1.38
0.76 0.68 0.64 0.54 0.46 0.41 0.38
0.77 0.68 0.62 0.54 0.46 0.43 0.37
Data taken from Ref. [13].
Table 2 Redox potentials (E0 0 ) vs. NHE at 298.2 K for the couple [Ru(NH3)5pz]3+/2+ in water–methanol mixtures x
0
0.034
0.064
0.126
0.203
0.248
0.362
E0 0 (mV)
524
538
542
545
546
547
547
Our approach, in order to obtain the variations of k in water–methanol mixtures, is based on the study of the kinetics of three closely related reactions: in the present case, the oxidation of the three ruthenium complexes, [Ru(NH3)5L]2+ (L = py, pz and 4-CNpy) by [Co(C2O4)3]3. This approach relies on the assumption that the interaction of these reductant complexes with the oxidant and the reaction medium will be the same. The values of the observed rate constants for the different studied processes are represent versus the relative mole fraction of the organic solvent x (x = xc/xw; xc and xw being the mole fraction of methanol and water, respectively). These values are fitted to the equation: k obs ¼
k w þ Kk c x : 1 þ Kx
On the other hand, the idea of the same interactions with the oxidant receives support from the fact that, in the oxidation of this kind of ruthenium complexes with anionic oxidants, the union of the reactants happens by the ammonia side of the complexes because this is favourable for the formation of hydrogen bonds, which reinforces the electrostatic interactions between the [Co(C2O4)3]3 and the ruthenium complexes [18,20]. According to this, it seems reasonable to suppose that the preexponential term in Eq. (6) is the same for the three reactions studied here. Under these circumstances it is easy to show, from Eqs. (5)–(9), that the reorganization energy is given by [21]: k¼
ðE2 D3 E3 D2 ÞF 2 ; D3 K 2 D2 K 3
where F is the Faraday constant and Di ¼ ðE00 ÞRui ðE00 ÞRu1 2
2
Ei ¼ ðE00 ÞRui ðE00 ÞRu1 K i ¼ 4RT ln k 1 =k i
ð10Þ
This equation implies that on the average a fraction 1 of the solvation shells of the reactant (actually of 1þKx the encounter complex) is preferentially constituted by kw water and contributes with 1þKx to the observed rate conK stant of the process 2. A second fraction 1þKx of the solvent shell is composed of methanol and contributes with Kk c x to the observed rate constant of the studied pro1þKx cesses. Parameter K gives an idea of the relative affinity of methanol and water for the encounter complex, that is, about the interaction of the encounter complex with the medium. Eq. (10) fits well the values of kobs with similar values of parameter K (4.3 for py, 4.2 for pz and 4.2 for 4-CNpy). The values of the observed rate constants obtained from the fit are collected on Table 1. The fact that the values of K are practically the same is proof that the interaction between the different ruthenium complexes and the medium is the same.
ð11Þ
ð12Þ
1ðpyÞ; i ¼ 2ðpzÞ and 3ð4 CNpyÞ: Parameters appearing in Eq. (11) can be calculated from data appearing in Table 1 (we have used in this procedure the values of kobs obtained from the fit of Eq. (10) to minimize errors in experimental data). In order to calculate the potential terms (Di, Ei) we considered that the differences in the redox potentials of the complexes are a constant, independently of the reaction media, which is consistent with the fact that all the complexes seem to have the same interaction with the medium. Thus, we take E0 0 [Ru(NH3)5pz]2+ E0 0 [Ru(NH3)5py]2+ = 0.175 V and E0 0 [Ru(NH3)5(4CNpy)]2+ E0 0 [Ru(NH3)5py]2+ = 0.224 V [22]. It is worth pointing out that the value of k calculated from Eq. (10), but not the variations in this parameter, are very susceptible to small changes in the values of the redox potentials used. Thus, we have calculated relative values of the reorganization energy, krel. The relative reorganization free energies are defined as the ratio between the reorganization energy at a given mole
F. Pe´rez et al. / Chemical Physics Letters 407 (2005) 342–346
complicated as water–methanol mixtures in which the use of a simple continuum model seems not to be satisfactory.
1 0.98
λrel
345
0.96
Acknowledgements
0.94
This work was financed by the D.G.I.C.Y.T. (BQU 2002-01063) and the Consejerı´a de Educacio´n y Ciencia de la Junta de Andalucı´a.
0.92 0.9 0
0.1
0.2
0.3
0.4
x Fig. 1. Plot of the values of the relative reorganization free energy (see text) versus the relative mole fraction x (x = xc/xw, see text).
fraction of the organic cosolvent and the reorganization energy at xc = 0. These relative values are given in Fig. 1. A decrease of krel is observed, and accordingly of k, when decreasing the mediumÕs dielectric constant. The reorganization energy is the sum of two contributions: the solvent reorganization energy, kout, and the intramolecular reorganization energy, kin. The latter is assumed to be constant when the medium changes while the former depends on it. From classical continuum models of solvent, kout can be written in function of the PerkarÕs factor, c, as follows [23] 1 1 kout ¼ F c c¼ 2 ; ð13Þ n D F being a geometry factor which depends on the acceptor and donor radii and on acceptor–donor distance. D represents the static dielectric constant of the solvent and n2 the optical dielectric constant of the medium. Changes in k with the medium are due to changes in the PekarÕs factor. In the case of water–methanol mixtures, this factor changes from a value of 0.5534 for water (D = 78.5) to a value of 0.5383 for x = 0.362 D¼60 (D = 60). The ratio ccD¼78:5 (= 0.973) is the variation that, according to the classical continuum model, one must find for krel. This value is quite similar to that obtained in this work of 0.92, as was expected. Also, it is similar to the value obtained by Pe´rez-Tejeda et al. [13] of 0.95 by using the classical continuum model and doing several assumptions. The slight differences between these methods is because Eq. (13) takes into account the medium as a continuum solvent when, in fact, it is not so. From this point of view, the differences could be related to the existence of specific effects of the medium that have an effect on the reaction rate. Therefore, the method used here seems to work well in order to get variations of the reorganization energy in water–methanol mixtures. In conclusion, we have shown a procedure to obtain variations of the reorganization energies in media so
References [1] (a) P. Perez-Tejeda, P. Lo´pez, M.L. Moya´, M. Domı´nguez, F. Sa´nchez, E. Carmona, P. Palma, New J. Chem. 20 (1996) 95; (b) P. Lo´pez, F. Sa´nchez, R. Jime´nez, P. Pe´rez-Tejeda, M.L. Moya´, Int. J. Chem. Kinet. 28 (1996) 57. [2] (a) P. Pe´rez-Tejeda, J. Benko, M. Lo´pez, M. Gala´n, P. Lo´pez, M. Domı´nguez, M.L. Moya´, F. Sa´nchez, J. Chem. Soc., Faraday Trans 92 (1996) 1155; (b) P. Lo´pez-Tejeda, R. Jime´nez-Sindreu, P. Lo´pez-Cornejo, F. Sa´nchez-Burgos, Curr. Top. Sol. Chem. 2 (1997) 49. [3] M.S. Matamoros-Fontenla, P. Lo´pez-Cornejo, P. Perez, R. Prado-Gotor, R. De laVega, M.L. Moya´, F. Sa´nchez, New J. Chem. (1998) 39. [4] (a) P. Neto-Ponce, F. Sa´nchez, F. Pe´rez, A. Garcı´a-Santana, P. Pe´rez-Tejeda, Langmuir 17 (2001) 980; (b) P. Pe´rez-Tejeda, P. Neto-Ponce, F. Sa´nchez, J. Chem Soc., Dalton Trans. (2001) 1686. [5] (a) R. Prado-Gotor, A. Ayala, A.B. Tejeda, M.B. Sua´rez, C. Mariscal, M.D. Sa´nchez, G. Hierro, A. Lama, A. Aldea, R. Jime´nez, Int. J. Chem. Kinet. 35 (2003) 367; (b) F. Muriel, R. Jime´nez, M. Lo´pez, R. Prado-Gotor, F. Sa´nchez, Chem. Phys. 298 (2004) 317. [6] F. Sa´nchez, A. Rodrı´guez, F. Muriel, J. Burgess, P. Lo´pezCornejo, Chem. Phys. 243 (1999) 159. [7] See for example: R.G. Wilkins, Kinetics and Mechanism of Reactions of Transition Metal Complexes, second edn., VCH, Weinheim, 1991. [8] R.A. Marcus, N. Sutin, Biochim. Biophys. Acta 811 (1985) 265. [9] (a) U. Fu¨lholz, A. Haim, Inorg. Chem. 26 (1987) 3243; (b) M. Weaver, Chem. Rev. 92 (1992) 463. [10] P. Lo´pez-Cornejo, M. Herna´ndez, P. Pe´rez-Tejeda, F. Pe´rez, R. Prado-Gotor, F. Sa´nchez, J. Phys. Chem. B 109 (2005) 1703. [11] C. Creutz, H. Taube, J. Am. Chem. Soc. 95 (1973) 1086. [12] R.D. Cannon, J. Stillman, Inorg. Chem. 14 (1975) 2207. [13] P. Pe´rez-Tejeda, F.-J. Franco, A. Sa´nchez, M. Morillo, C. Denk, F. Sa´nchez, Phys. Chem. Chem. Phys. 3 (2001) 1271. [14] E. Rolda´n, M. Domı´nguez, D. Gonza´lez-Arjona, Comput. Chem. 19 (1986) 87. [15] See for example R. Prado-Gotor, R. Jime´nez, P. Pe´rez-Tejeda, P. Lo´pez-Cornejo, M. Lo´pez-Lo´pez, A. Sa´nchez, F. Muriel-Delgado, F. Sa´nchez, Prog. React. Kinet. 25 (2000) 371, and references therein. [16] (a) M. Eigen, Z. Phys. Chem. (N.F.) 1 (1954) 176; (b) C.M. Fuoss, J. Am. Chem. Soc. 80 (1958) 5059. [17] J. Aqvist, T. Hannson, J. Phys. Chem. 100 (1996) 9512. [18] A.J. Miralles, R.E. Armstrong, A. Haim, J. Am. Chem. Soc. 99 (1977) 1416. [19] A. Chandra, B. Bagchi, J. Chem. Phys. 94 (1991) 8367. [20] C.D. Clark, M.Z. Hoffmann, Coord. Chem. Rev. 159 (1997) 359. [21] For the derivation of Eq. (11) see Ref. [11].
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F. Pe´rez et al. / Chemical Physics Letters 407 (2005) 342–346
[22] The redox potential of [Ru(NH3)5pz]2+ in water (I = 0.1 mol dm3) was obtained in this work as 0.524 V. The redox potentials of [Ru(NH3)5py]2+ and [Ru(NH3)54-CNPy]2+, in the same solvent, are 0.349 V (see Ref. [10]) and 0.574 V (see K.J.
Moore, L. Liangshiu, G.A. Mabbott, J.D. Petersen, Inorg. Chem. 22 (1983) 1108), respectively. [23] R.A. Marcus, Ann. Rev. Phys. Chem. 15 (1964) 155, and references therein.
Method for the evaluation of the reorganization energy of electron transfer reactions in water–methanol mixtures F. Pe´rez, M. Herna´ndez, R. Prado-Gotor, T. Lopes-Costa, P. Lo´pez-Cornejo
*
Departamento de Quı´mica Fı´sica, Facultad de Quı´mica, Universidad de Sevilla c/ Profesor Garcı´a Gonza´lez s/n. 41012 Sevilla, Spain Received 24 February 2005; in final form 15 March 2005 Available online 14 April 2005
Abstract The kinetics of the electron transfer reactions between [Co(C2O4)3]3 and different ruthenium complexes [Ru(NH3)5L]2+ (L = pyridine, pyrazine and 4-cyanopyridine) have been studied in water–methanol mixtures. A method for the evaluation of the reorganization energy of these reactions, based on comparison of their rate constants, is proposed. The results showed a decrease of the rate constants and a decrease of the reorganization energies when the dielectric constant of the medium decreases. This behaviour is according to that obtained from the use of classical continuum methods. The slight differences found between the distinct methods could be related to the existence of specific effects of the medium. 2005 Elsevier B.V. All rights reserved.
1. Introduction Study of the medium effects on reactivity of electron transfer processes has been of great interest to our group for the last decade [1–5]. Solvent effects in water–cosolvent mixtures are more difficult to explain than those in pure solvents because the reactivity can be dependent on preferential solvation phenomena [6]. However, this kind of mixing medium is interesting because of its relation to several areas of chemistry and biology. An electron transfer process occurs in, at least, the following steps: k1
A þ D A=D k 1
k et
A=D A =Dþ k2
A =Dþ A þ Dþ k 2
*
This scheme corresponds to the simplest case in this type of reaction. Situations more complicated are also possible [7]. The formation of the precursor complex, A/D, happens in the first step. In the second step, the activation of the precursor complex, the electron transfer process and the formation of the successor complex take place. If the reaction is accomplished by a major decrease in the free energy, the contribution from the reverse processes of the second and third steps can be ignored. On the other hand, if the forward and reverse processes in step 1 are diffusion controlled, that is they are rapid, it can easily be shown that
ð1Þ k¼ ð2Þ ð3Þ
Corresponding author. Fax: +34 954557174. E-mail addresses: [email protected], [email protected] (P. Lo´pez-Cornejo).
0009-2614/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.03.105
k et K IP ½X ; 1 þ K IP ½X
ð4Þ
where KIP is the equilibrium constant corresponding to the formation of the precursor complex (KIP = k1/k1). Eq. (4) assumes that the concentration of the reactants (acceptor or donor) is in excess with respect to the other, in such a way that X represents the reactant in excess and [X] its concentration. Besides, if KIP[X] << 1:
F. Pe´rez et al. / Chemical Physics Letters 407 (2005) 342–346
k ¼ k et K IP :
ð5Þ
On the other hand, we can write the following expression of ket given by the classical electron transfer theory [8]: 6¼ =RT
k et ¼ jet mn eDG
;
Absorbance changes were monitored at 407 nm for [Ru(NH3)5py]2+ and at 489 nm for [Ru(NH3)5 (4-CNpy)]2+ (these wavelengths correspond to the maximum absorbance values for the different ruthenium complexes).
ð6Þ
where jel and mn are the electronic transmission coefficient and the nuclear frequency factor, respectively [9]. DG6¼, the Gibbs free energy of activation from the electron transfer process, is given by: 2 k þ DG00 6¼ ; ð7Þ DG ¼ 4k k being the (free) energy of reorganization for the electron transfer reaction. DG0 0 is related to the free energy of the process 2, DG0, in the previous scheme, through: DG00 ¼ DG0 þ wp wr ;
343
ð8Þ
where wp is the work of forming the successor complex from the separate products and wr the work needed to form the precursor complex from the separate reactants. The free energy DG0 is a thermodynamic parameter which can be obtained from experimental data (measurements of standard formal redox potentials): DG0 ¼ nF E00acceptor E00 ð9Þ donor : Taking into account all these relations between the different parameters, the experimental results on k are 0 due to the influence of the solvent on DG0 and/or k. It is extremely difficult to calculate KIP, wp, wr and k in water–organic cosolvent mixtures and, therefore, to obtain information about such an influence. Recently, we have published a Paper [10] in which we show a method to obtain relative reorganization energy values in restricted geometry media without the necessity of known KIP, wp and wr. We apply this same method in this work to evaluate the effects on the water– methanol mixtures and, accordingly, to be able to explain the kinetic behaviour of the electron transfer processes in these complicated reaction media. The reactions studied here are RuðNH3 Þ5 L2þ þ 3 CoðC2 O4 Þ3 ; L being pyridine (py) and 4-cyanopyridine (4-CNp).
2. Experimental 2.1. Material and kinetic measurements The preparation of the reactants, [Ru(NH3)5L] (ClO4)3 (L = py and 4-CNpy) and Na3 [Co(C2O4)3], as well as the way of performing kinetic experiments have been described in a previous work [11–13] ([Ru(NH3)5L]2+ = 8 · 105 mol dm3 and [Co(C2O4)3]3 = 1 · 103 mol dm3).
2.2. Electrochemistry The standard formal redox potentials of the [Ru(NH3)5 py]3+/2+ (py = pyridine) pair in the different reaction media were obtained by cyclic voltammetry technique. The equipment, procedure and electrodes have been previously described [14]. The working conditions (pH and ionic strength) were the same as those in the kinetic experiments. A carbon working electrode, a saturated calomel electrode, as reference, and a platinum electrode, as auxiliary electrode, were utilized. Sweep rates used were between 0.1 and 0.5 V s1. Liquid junction potentials were not corrected using the redox potentials in the different solvent mixtures due to the type of calculation made (Section 4).
3. Results Table 1 contains the kinetic data corresponding to the three electron transfer reactions. The data corresponding to [Ru(NH3)5pz]2+ in Table 1 were taken from [13]. Table 2 contains the standard formal redox potentials of the [Ru(NH3)5 pz]3+/2+ couple obtained from experimental measurements.
4. Discussion In agreement with the classical Marcus–Hush Treatment for electron transfer reactions, variations in the second order rate constant, obtained as the ratio k ¼ k obs =½CoðC2 O4 Þ3 3 (kobs being the observed rate constant, that is, the pseudofirst order rate constant), are due to changes in both reorganization free energy, k, 0 and the free energy DG0 (see Eqs. (5)–(7)). In order to have the variations of k when the reaction medium changes, it is necessary, first of all, to separate KIP and ket. In conventional reaction media, there are several procedures to accomplish this separation [15]. In fact, in favourable cases KIP can be calculated through an Eigen–Fuoss approach [16]. This procedure, based on a simple continuum model, seems to be satisfactory enough for pure solvents [16,17]. However, the application of this model to mixed solvents, notwithstanding, used in several Letters [6,9a,18], has been criticised and still remains an open question [19].
F. Pe´rez et al. / Chemical Physics Letters 407 (2005) 342–346
344
Table 1 2+ Observed rate constants (ki; i = py, pz or 4-CNpy) and calculated rate constant ðk calc + i Þ from fit of Eq. (10) for the reaction [Ru(NH3)5L] [Co(C2O4)3]3 (L = pyridine, pyrazine and 4-cyanopyridine) in water–methanol mixtures X
kpy (s1)
1 k calc py ðs Þ
a
0 0.034 0.064 0.126 0.203 0.248 0.362
80.2 76.7 66.4 54.4 47.2 42.9 35.4
82.8 72.9 66.2 55.7 46.8 42.9 35.8
2.50 2.31 2.14 1.81 1.63 1.54 1.43
a
kpz (s1)
1 k calc pz ðs Þ
k4-CNpy (s1)
1 k calc 4-CNpy ðs Þ
2.53 2.29 2.12 1.86 1.65 1.55 1.38
0.76 0.68 0.64 0.54 0.46 0.41 0.38
0.77 0.68 0.62 0.54 0.46 0.43 0.37
Data taken from Ref. [13].
Table 2 Redox potentials (E0 0 ) vs. NHE at 298.2 K for the couple [Ru(NH3)5pz]3+/2+ in water–methanol mixtures x
0
0.034
0.064
0.126
0.203
0.248
0.362
E0 0 (mV)
524
538
542
545
546
547
547
Our approach, in order to obtain the variations of k in water–methanol mixtures, is based on the study of the kinetics of three closely related reactions: in the present case, the oxidation of the three ruthenium complexes, [Ru(NH3)5L]2+ (L = py, pz and 4-CNpy) by [Co(C2O4)3]3. This approach relies on the assumption that the interaction of these reductant complexes with the oxidant and the reaction medium will be the same. The values of the observed rate constants for the different studied processes are represent versus the relative mole fraction of the organic solvent x (x = xc/xw; xc and xw being the mole fraction of methanol and water, respectively). These values are fitted to the equation: k obs ¼
k w þ Kk c x : 1 þ Kx
On the other hand, the idea of the same interactions with the oxidant receives support from the fact that, in the oxidation of this kind of ruthenium complexes with anionic oxidants, the union of the reactants happens by the ammonia side of the complexes because this is favourable for the formation of hydrogen bonds, which reinforces the electrostatic interactions between the [Co(C2O4)3]3 and the ruthenium complexes [18,20]. According to this, it seems reasonable to suppose that the preexponential term in Eq. (6) is the same for the three reactions studied here. Under these circumstances it is easy to show, from Eqs. (5)–(9), that the reorganization energy is given by [21]: k¼
ðE2 D3 E3 D2 ÞF 2 ; D3 K 2 D2 K 3
where F is the Faraday constant and Di ¼ ðE00 ÞRui ðE00 ÞRu1 2
2
Ei ¼ ðE00 ÞRui ðE00 ÞRu1 K i ¼ 4RT ln k 1 =k i
ð10Þ
This equation implies that on the average a fraction 1 of the solvation shells of the reactant (actually of 1þKx the encounter complex) is preferentially constituted by kw water and contributes with 1þKx to the observed rate conK stant of the process 2. A second fraction 1þKx of the solvent shell is composed of methanol and contributes with Kk c x to the observed rate constant of the studied pro1þKx cesses. Parameter K gives an idea of the relative affinity of methanol and water for the encounter complex, that is, about the interaction of the encounter complex with the medium. Eq. (10) fits well the values of kobs with similar values of parameter K (4.3 for py, 4.2 for pz and 4.2 for 4-CNpy). The values of the observed rate constants obtained from the fit are collected on Table 1. The fact that the values of K are practically the same is proof that the interaction between the different ruthenium complexes and the medium is the same.
ð11Þ
ð12Þ
1ðpyÞ; i ¼ 2ðpzÞ and 3ð4 CNpyÞ: Parameters appearing in Eq. (11) can be calculated from data appearing in Table 1 (we have used in this procedure the values of kobs obtained from the fit of Eq. (10) to minimize errors in experimental data). In order to calculate the potential terms (Di, Ei) we considered that the differences in the redox potentials of the complexes are a constant, independently of the reaction media, which is consistent with the fact that all the complexes seem to have the same interaction with the medium. Thus, we take E0 0 [Ru(NH3)5pz]2+ E0 0 [Ru(NH3)5py]2+ = 0.175 V and E0 0 [Ru(NH3)5(4CNpy)]2+ E0 0 [Ru(NH3)5py]2+ = 0.224 V [22]. It is worth pointing out that the value of k calculated from Eq. (10), but not the variations in this parameter, are very susceptible to small changes in the values of the redox potentials used. Thus, we have calculated relative values of the reorganization energy, krel. The relative reorganization free energies are defined as the ratio between the reorganization energy at a given mole
F. Pe´rez et al. / Chemical Physics Letters 407 (2005) 342–346
complicated as water–methanol mixtures in which the use of a simple continuum model seems not to be satisfactory.
1 0.98
λrel
345
0.96
Acknowledgements
0.94
This work was financed by the D.G.I.C.Y.T. (BQU 2002-01063) and the Consejerı´a de Educacio´n y Ciencia de la Junta de Andalucı´a.
0.92 0.9 0
0.1
0.2
0.3
0.4
x Fig. 1. Plot of the values of the relative reorganization free energy (see text) versus the relative mole fraction x (x = xc/xw, see text).
fraction of the organic cosolvent and the reorganization energy at xc = 0. These relative values are given in Fig. 1. A decrease of krel is observed, and accordingly of k, when decreasing the mediumÕs dielectric constant. The reorganization energy is the sum of two contributions: the solvent reorganization energy, kout, and the intramolecular reorganization energy, kin. The latter is assumed to be constant when the medium changes while the former depends on it. From classical continuum models of solvent, kout can be written in function of the PerkarÕs factor, c, as follows [23] 1 1 kout ¼ F c c¼ 2 ; ð13Þ n D F being a geometry factor which depends on the acceptor and donor radii and on acceptor–donor distance. D represents the static dielectric constant of the solvent and n2 the optical dielectric constant of the medium. Changes in k with the medium are due to changes in the PekarÕs factor. In the case of water–methanol mixtures, this factor changes from a value of 0.5534 for water (D = 78.5) to a value of 0.5383 for x = 0.362 D¼60 (D = 60). The ratio ccD¼78:5 (= 0.973) is the variation that, according to the classical continuum model, one must find for krel. This value is quite similar to that obtained in this work of 0.92, as was expected. Also, it is similar to the value obtained by Pe´rez-Tejeda et al. [13] of 0.95 by using the classical continuum model and doing several assumptions. The slight differences between these methods is because Eq. (13) takes into account the medium as a continuum solvent when, in fact, it is not so. From this point of view, the differences could be related to the existence of specific effects of the medium that have an effect on the reaction rate. Therefore, the method used here seems to work well in order to get variations of the reorganization energy in water–methanol mixtures. In conclusion, we have shown a procedure to obtain variations of the reorganization energies in media so
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[22] The redox potential of [Ru(NH3)5pz]2+ in water (I = 0.1 mol dm3) was obtained in this work as 0.524 V. The redox potentials of [Ru(NH3)5py]2+ and [Ru(NH3)54-CNPy]2+, in the same solvent, are 0.349 V (see Ref. [10]) and 0.574 V (see K.J.
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