- Email: [email protected]
Kinetics of dissociative electron transfer for homogeneous and heterogeneous reductive bond-breaking reactions
Journal of Molecular Structure (Theochem) 532 (2000) 87±93
www.elsevier.nl/locate/theochem
Kinetics of dissociative electron transfer for homogeneous and heterogeneous reductive bond-breaking reactions Zheng Yu Zhou a,b,*, Yu Mei Xing a b
a Department of Chemistry, Qufu Normal University, Shandong, Qufu 273165, People's Republic of China State Key Laboratory of Crystal Materials, Shandong University, Shandong, Jinan 250100, People's Republic of China
Received 5 January 2000; received in revised form 16 March 2000
Abstract The kinetics of dissociative electron transfer reactions of butyl halides by outer-sphere heterogeneous (glassy carbon electrodes) or by homogeneous (aromatic anion radicals) reagents is investigated in aprotic solvents. The potential energies of the reactants and products are described by the Morse and the exponential curves as functions of the C±X distance. Both the homogeneous and the heterogeneous reactions are consistent with an outer-sphere adiabatic electron-exchange process. The activation free energy in both cases is suitable for a quadratic driving force free energy relationship and varies with the electrode potential. The standard activation free energy contains two contributions. There is a good agreement between theory and experiment in the heterogeneous case, while the good agreement is only for the tertiary halides in the homogeneous case, which is due to the stability of tertiary radical with sterical effect. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Dissociative electron transfer; Heterogeneous and homogeneous reaction; Butyl halides
1. Introduction Electron transfer reactions are among the simplest of solution phase reactions and have been the subject of extensive theoretical and experimental studies [1± 11]. These reactions include two categories: one involves no breaking or formation of chemical bonds, but rather only changes in the bond lengths and angles and changes in the molecular orientation of the solvent molecules. The other is very often accompanied by the breaking of an existing bond and/or the formation of a new bond [4±11]. In the case where a radical and an anionic fragment are formed upon electron transfer to a neutral molecule, an important question is whether electron transfer and * Corresponding author. Department of Chemistry, Qufu Normal University, Shandong, Qufu 273165, People's Republic of China.
bond breaking are concerted (reaction (1)) or occur successively (reactions (2) and (3)) [5]. RX 1 e2 ! Rz 1 X2
1
RX 1 e2 ! RXz2
2
RXz2 ! Rz 1 X2
3
The theory for the concerted mechanism has been described by Saveant [12], who was convinced of the adiabatic aspects of this reaction. More recently a corresponding nonadiabatic treatment has been proposed [13]. In the classical limit, Saveant used the Marcus theory to describe the solvent polarization and the Morse-like curves for the reactants' and products' molecular potential. The treatment of solvent reorganization in the Marcus± Hush fashion leads to a quadratic activation force
0166-1280/00/$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0166-128 0(00)00504-2
88
Z.Y. Zhou, Y.M. Xing / Journal of Molecular Structure (Theochem) 532 (2000) 87±93
2. Theoretical model A few classical semi-molecular and molecular models of electron transfer reactions involving bond-breaking reactions are the following: RX 1 e2 ! Rz 1 X2
Electrochemical
RX 1 Dz2 ! Rz 1 X2 1 D
Fig. 1.
relationship [1±4]: #
DG wR 1
DG#0
DG0 2 wR 1 wP 11 4DG#0
!2 4
for both homogeneous and heterogeneous (electrochemical) electron transfers (DG #, free energy of activation for the forward reaction; DG 0, standard free energy of the reaction, 2DG 0, a measure of the driving force of the reaction; wR and wP, work required to bring the reactants and products, from in®nite separation to the reacting distance). The standard free energy of activation or the intrinsic barrier free energy, DG#0 ; i.e. the free energy of activation at zero driving force, appears as the sum of two factors, i.e. an internal reorganization (bond lengths and angles) factor and a solvent reorganization factor. The external reorganization factor was obtained from Marcus± Hush solvent reorganization factor. The internal reorganization factor is approximately equal to one-fourth of the dissociation energy of the broken bond. Saveant described a simple model of the kinetics of dissociative electron transfer in polar solvents, showing the validity of the quadratic activation-driving force free energy relationship depicted by Eq. (4) and relating the standard activation free energy to the bond dissociation energy as well as to the solvent reorganization. In this paper, we discuss the case where electron transfer to an organic molecule is accompanied by the breaking of a bond and use the model given by Saveant to test the agreement of the theoretical calculation with experiments of the kinetic parameters.
Homogeneous
RX stands for alkyl halides or any molecule, X designing the ªleaving groupº, i.e. the group that leaves carrying on an electron pair and R the ªremaining groupº, i.e. the group remaining with an unpaired electron. D/D z2 is a chemically stable redox couple reacting in an outer-sphere manner. The potential energy of the reactants was assumed to depend upon the R±X distance according to the Morse curve, UR DRX 1 2 2 exp 2by2 ; where y rR±X 2 r0;R±X and r0 is the equilibrium bond length of the bond C±X; DRX is the dissociation energy of the R±X bond; b y 0 2p 2 m=DRX with y 0 representing the vibration frequency of the RX bond and m representing the reduced mass of the C and X atoms. The potential energy of the products was modeled by an exponential function, U P DRX exp 22by: The potential energy is zero at the minimum of the Morse and of the exponential curve, as shown in Fig. 1. The free energy of activation shown below was thus obtained in terms of the potential energy. 3. Results and discussion 3.1. Electrochemical reduction of alkyl butyl halides In the electrochemical case, the intrinsic barrier is given by # DRX 1 lhet DG0;het 0 =4
5
with a Marcus-like estimate of solvent reorganization 2 [1,2] lhet 0 e0 1=Dop 2 1=Ds 1=4a1 and with a 2 Hush-like estimation [14] lhet 0 e0 1=Dop 2 1=Ds 1=2a1 (Dop and Ds are the optical and static dielectric constants of the solvent, respectively, a is the radius of the outer-sphere equivalent to the reactant and products and e0 is the charge of the electron), which looks more consistent with the experimental
Z.Y. Zhou, Y.M. Xing / Journal of Molecular Structure (Theochem) 532 (2000) 87±93
89
Table 1 Kinetics of the electrochemical reduction of butyl iodides and bromides (in DMF, at a glassy carbon electrode at 108C; scan rate v 0:1 V=s; the values in the parentheses when DG04 ± 0 DG04 ; standard Gibbs free energy of the reaction RX 1 RzDMF , RXDMF 1 Rzg ), D 1025 cm2 s21 : All the potential are in V vs. SCE, all the energies are in eV). Comparison between the prediction for the theory of dissociative electron transfer and the experimental data Compd.
0 a z 1X2 2ERX=R Z el (cm s 21) Ê) aRX,aX,a(A b lhet 0 DRX b 2Ep 2f r DG#0 theor DG#0 exp DG#el theor DG#el exp DG#21 theor a (theor) a (pred) a (exp) a b
BuI
BuBr
n-BuI
sec-BuI
t-BuI
n-BuBr
sec-BuBr
t-BuBr
1.075(1.209)
0.946(1.080) 4:5 £ 104 3.58,3.05,3.50 0.902 2.28(2.49) 2.05 0.084 0.796(0.848) 0.808(0.760) 0.368(0.463) 0.379 1.388(1.349) 0.340(0.369) 0.342(0.357) 0.33
0.826(0.960)
1.109(1.230)
0.929(1.050)
3.62,3.06,3.53 0.892 2.20(2.31) 1.91 0.081 0.773(0.801) 0.802(0.751) 0.353(0.425) 0.379 1.356(1.294) 0.337(0.364) 0.343(0.355) 0.32
3.50,2.82,3.37 0.935 3.00(3.12) 2.85 0.102 0.984(1.014) 1.043(1.000) 0.335(0.397) 0.385 1.974(1.915) 0.292(0.313) 0.303(0.310) 0.25
1.049(1.190) 5:1 £ 104 3.50,2.82,3.37 0.935 2.99(3.07) 2.63 0.097 0.982(1.001) 0.987(0.936) 0.380(0.442) 0.385 1.864(1.785) 0.311(0.332) 0.312(0.321) 0.25
3.56,3.04,3.48 0.905 2.56(2.67) 2.33 0.09 0.866(0.894) 0.864(0.817) 0.381(0.453) 0.380 1.546(1.484) 0.332(0.354) 0.331(0.341) 0.30
3.54,2.83,3.40 0.927 2.87(2.99) 2.51 0.090 0.949(0.979) 0.993(0.949) 0.350(0.414) 0.388 1.841(1.784) 0.303(0.325) 0.312(0.319) 0.20
See Refs. [16±18] (from thermodynamic data compilations [18] using a previously described method [16,17]). From literature thermochemical data in Ref. [13].
results than Marcus' approximation in the present case. Thus the ensuing variations of lhet 0 with 1=a approximates to lhet 0 3:15=a A:
6
Previous estimations [12] have used the arithmetic mean of the halogen radius, aX, and of the radius of the sphere equivalent to the whole molecule, aRX, which was derived from the molar mass, M, and the density, r , aRX 3M=4NA r1=3 : The following equation: a
aX 2aRX 2 aX aRX
7
probably gives a more realistic estimation and was used in the following. thus obtained and the Using the values of lhet 0 values of the bond dissociation energy from the literature [13], we obtained the theoretical values of the intrinsic barrier, i.e. DG#0 theor; in terms of Eq. (5), listed in Table 1. At a given sweep rate, the transfer coef®cient (symmetry factor), a , can be considered as approximately constant along the cyclic voltammetric wave.
Such a simpli®cation is perfectly legitimate for each scan rate individually, since the potential difference between the base and the peak height of each individual voltammogram is small. Thus, DG#el ; the activation free energy of the forward electron transfer at the peak potential Ep is DG#el
" !1=2 # RT RT RT el ln Z 2 0:78 a p F nD F F
8
where D is the diffusion coef®cient of RX, n is the scan rate and Z el is the collisional frequency, Z el RT=2pM1=2 (M is the molar mass, for the butyl iodides and butyl iodides and butyl bromides and Z el was taken as equal to 4:5 £ 104 and 5:1 £ 104 cm s21 : The transfer coef®cient a (exp) is
a
1:85 RT £ Ep=2 2 Ep F
9
The experimental values thus obtained are compared with the theoretical values, i.e. from
90
Z.Y. Zhou, Y.M. Xing / Journal of Molecular Structure (Theochem) 532 (2000) 87±93
in a linear way:
Eq. (10) DG#el
" #2 0 z 1X2 2 fr Ep 2 ERX=R DRX 1 lhet 0 11F 4 DRX 1 lhet 0 10
(expressing the potential in volts and the free energies in eV) where f r is the potential difference between the reaction site and the solution. The reaction site is usually assumed to be located in the outer Helmoltz plane [14] and f r is obtained from previous measurements carried out in the same solvent (DMF) [15]. Given a scan rate and a speci®c temperature, a ®xed activation free energy is seen at the peak. Thus, increasing the scan rate induces a negative shift of the peak potential and hence an increase of the driving force offered to the reaction. At a ®xed scan rate, decreasing the temperature induces the following negative shift of the peak potential, hence offering more driving force to the reaction. The activation free energy of the backward reaction thus derived from the similar equation: DG#21
" #2 0 z 1X2 2 fr Ep 2 ERX=R DRX 1 lhet 0 12F 4 DRX 1 lhet 0 11
For obtaining DG#0 exp; we obtained from Eq. (13) according to Eq. (10) DG#0 exp { Ep 2 E0 2 fr 2 2DG#el 2 2 Ep 2 E0 2 fr 2 1=2 2 Ep 2 E0 2 fr 2 2DG#el }=4
12
The results are displayed in Table 1. We see that there is satisfactory agreement between the prediction of the theory and the experimental data when DG04 0: The agreement is not so good as the second estimation when DG04 ± 0: The comparison among the experimental values of the transfer coef®cient a (exp), the predicted a (pred) and the theoretical a (theor) values also testi®es the dissociative electron transfer theory. If the activationdriving force relationship is quadratic as depicted in Eq. (4), when wR wP 0; the transfer coef®cient a is expected to decrease upon raising the driving force
2DG# 1 DG0 1 1 ap 2 4DG#0 2DG0 1 E 2 E0 2 fr 11F P 2 4DG#0
! ! 13
This indicates that a p is related to the driving force (2DG 0) and the intrinsic barrier free energy DG#0 : Its value lies between 0 and 0.5. When 2DG 0 and DG#0 are high, a p is closer to 0.5 and vice versa. a (pred) is the value obtained from Eq. (13), taking DG#0 exp; whereas a (theor) is obtained from Eq. (13) using DG#0 theor Table 1 explains that, although the experimental values of a p are somewhat smaller than the predicted ones, the reason that the transfer coef®cient is distinctly below 0.5 appears as a consequence of the fact that the reduction potential is much negative relative to the standard potential. The potential-dependent heterogeneous rate constant k, according to Marcus theory of outersphere electron transfer, is related to the activation free energy, DG #, by the following k Z el exp 2DG#el =RT: In fact, the identi®cation of the pre-exponential factor with the collision frequency is certainly a crude approximation, but to an extent, it is valuable to show the variable tendency of the rate constant. From Table 1, we can see the theoretical values of DG#el in the ®rst case are in better agreement with those of the experimental results than in the second case. The activation free energy of the backward reaction, DG#21 ; is much larger than that of the forward reaction, DG#el : This shows that it is impossible for the backward reaction to take place. 3.2. Homogeneous reduction of alkyl butyl halides by aromatic anion radicals (D z2) These reactions can be thought of as a sequence of elementary steps: the diffusion of the two reagents to form the precursor complex, then the activation of the precursor complex to yield the successor complex, at last the separation of this latter into the products, according to Eq. (14) RX 1 Dz2 , RXuDz2 , RXz2 uD , Rz 1 X2 1 D 14
1:39 £ 104 2:79 £ 104
0.798 0.671
8:50 £ 103 8:73 £ 105
DG#0 theor DG#0 exp
khom(theor) khom(exp)
d
c
b
a
0.753 0.735
0.439 0.322
DG #(theor) DG #(exp)
Temperature, 208C. The mediator is anthracene. The mediator is 4-cyanopyridine. The mediator is diphenyl.
0.412 0.395
0.946 1.730 c 0.631 2.38 0.784
1.075 1.900 b 0.633 2.56 0.825
2ERX=Rz1 X2 2ED/D z2 lhom 0 DRX 2DG 0
s-BuI
n-BuI a
Compd.
4:54 £ 105 2:17 £ 105
0.707 0.726
0.327 0.345
0.826 1.730 c 0.628 2.20 0.904
t-BuI
67.95 3:06 £ 102
0.914 0.874
0.561 0.523
1.109 1.900 b 0.656 3.00 0.766
n-BuBr a
86.41 5:69 £ 102
0.912 0.863
0.536 0.490
1.049 1.900 b 0.656 2.99 0.851
s-BuBr
1:79 £ 103 1:52 £ 103
0.885 0.885
0.462 0.466
0.929 1.900 b 0.653 2.87 0.971
t-BuBr
6:24 £ 102 1:89 £ 103
1.044 1.014
0.505 0.477
1.267 2.540 d 0.674 3.50 1.273
n-BuCl a
5:69 £ 102 1:95 £ 103
1.031 0.998
0.490 0.460
1.258 2.540 d 0.672 3.45 1.282
s-BuCl
4:24 £ 103 6:13 £ 103
1.022 1.012
0.441 0.432
1.138 2.540 d 0.668 3.42 1.402
t-BuCl
Table 2 Kinetics of homogeneous reduction of butyl halides by aromatic anion radicals in DMF. Comparison between the experimental data (temperature 108C unless otherwise stated) and Ê , lhom the prediction of the theory (all standard potentials in V vs. SCE, all free energies in eV, all radii in A in eV) 0
Z.Y. Zhou, Y.M. Xing / Journal of Molecular Structure (Theochem) 532 (2000) 87±93 91
92
Z.Y. Zhou, Y.M. Xing / Journal of Molecular Structure (Theochem) 532 (2000) 87±93
The same strategy as in the electrochemical case was followed here. For the cross-exchange reaction between RX and D z2 [2] 1 1 1 1 1 2 lhom e 2 2 2 15 0 DOP DS 2a1 2a2 d
case of the secondary and the primary halides, even though the difference remains small (about 50 meV at maximum except n-BuI and sec-BuI). The homogenous rate constant can thus be derived from
where a1 and a2 are the radii of the spheres equivalent to the RX and D molecules, respectively, and d is the distance of their centers in the precursor complex. The theoretical value of DG#0;hom is thus obtained from Eq. (16)
20
DG#0;hom DRX 1 lhom 0 =4
16
is estimated from Eq. (15) and DRX is where lhom 0 obtained from the gas-phase thermochemical data as in the electrochemical case. The theoretical prediction of activation free energy is thus obtained from !2 DG0 # # 17 DGhom DG0;hom 1 1 4DG#0;hom where the driving force of the reaction is obtained from 0 0 z 1X2 2 ED=Dz2 DG0 2nF ERX=R
18
0 z2 is the standard potential of the mediator couple, ED=D 0 z 1X2 is the standard potential of reaction (1), n ERX=R 1: Whereas the experimental value of DG#0;hom is then derived from Eq. (4) (with wR wP 0 by means of
DG#0;hom { DG0 2 2DG# 2 2 DG0 2 1=2 2 DG0 2 2DG# }=4
19
the experimental activation free energy DG # is obtained from the rate constant k. All the results are shown in Table 2. It however distinctly appears that, given the halogen and the nature of the reacting carbon, the agreement is better for the tertiary halides than for the secondary and the primary halides. From the mere fact that the tertiary alkyl radicals are more stable than the secondary and the primary alkyl radicals, DRX increases in this series and E 0 becomes less and less negative. DG#0 theor becomes larger and larger from the tertiary to the secondary and the primary, but the trend is opposite for DG#0 exp: Thus, taking the theory±experiment agreement as good for the tertiary halides, it can be seen that the theory overestimates DG#0 and DG # in the
k Z hom exp 2DG#hom =RT
where Z hom a1 1 a2 2 8pRT=m1=2 (m is the reduced molar mass of the two reactants; a1 and a2 are the outer-sphere equivalent radii of the two reactants). For the whole series of homogeneous mediators Z hom was found to be equal to 3 £ 1011 M21 s21 with a maximal deviation of ^10%, i.e. perfectly negligible in testing the theory. From the data of homogeneous rate constant, it is obvious that the agreement of the theory with the experiment is much better for the tertiary halides than for the secondary and the primary halides and the rate constant of the ®rst is larger than those of the last two. These observations suggest that in the absence of strong steric hindrance, the tertiary halides are easier to break the C±X bond and to inform the tertiary radicals. This also suggests the fact that the tertiary radical is more stable than the secondary and the primary radicals. 4. Conclusions The dissociative electron transfer reaction has two mechanisms. In the cases where concerted mechanism prevails, the main parameter governing the reduction of the alkyl halides is the dissociation energy of the C±X bond undergoing cleavage in the reaction. Based on a Morse curve description of the potential energy surfaces for bond breaking, it leads, for both homogeneous and heterogeneous processes, to a quadratic activation-driving force free energy relationship with the standard activation free energy being the sum of two contributions. One is the bond breaking that appears as being equal to one-fourth of the bond dissociation energy, the other is the solvent reorganization that can be estimated on the basis of the Marcus±Hush dielectric continuum model for solvation. In the electrochemical case, there is a good agreement between the theoretical and experimental values of the intrinsic barrier for all the butyl iodides and bromides. In the homogeneous case, there is also a
Z.Y. Zhou, Y.M. Xing / Journal of Molecular Structure (Theochem) 532 (2000) 87±93
good agreement between theoretical predictions and experiment in the case of tertiary butyl. The homogeneous reduction of tertiary butyl halide is faster than those of the secondary and the primary. This shows that the tertiary butyl radical is more stable than the secondary and the primary. Acknowledgements This work is supported by the Natural Science Foundation of Shandong Province and National Natural Science Foundation of China. References [1] R.A. Marcus, J. Chem. Phys. 24 (1956) 4966. [2] R.A. Marcus, J. Chem. Phys. 43 (1965) 679. [3] R.A. Marcus, in: P.A. Rock (Ed.), Theory and Applications of Electron Transfers at Electrodes and in Solution. Special Topics in Electrochemistry, Elsevier, New York, 1977, pp. 161±179.
93
[4] R.A. Marcus, J. Phys. Chem. A 101 (1997) 4072. [5] C.P. Andrieux, A.L. Gorande, J.M. Saveant, J. Am. Chem. Soc. 114 (1992) 6892. [6] J.M. Saveant, in: P.S. Mariano (Ed.), Dissociative electron transfer, Advances in Electron Transfer Chemistry, vol. 4, JAI Press, New York, 1994, pp. 53±116. [7] S. Antonello, M. Musumeci, D.D.M. Wayner, F. Maran, J. Am. Chem. Soc. 119 (1997) 9541. [8] M.S. Workentin, F. Maran, D.D.M. Wayner, J. Am. Soc. 117 (1995) 2120. [9] E.D. German, A.M. Kuznetsov, J. Phys. Chem. 99 (1995) 9095. [10] D.A. Pratt, J.S. Wright, K.U. Ingold, J. Am. Chem. Soc. 121 (1999) 4877. [11] C. Costentin, P. Hapiot, M. Medebielle, J.M. Saveant, J. Am. Chem. Soc. 121 (1999) 4451. [12] J.M. Saveant, J. Am. Chem. Soc. 109 (1987) 6788. [13] S.W. Benson, Thermodynamical Kinetics, 2nd ed., Wiley, New York, 1976. [14] P. Delahay, Double Layer and Electrode Kinetics, Wiley, New York, 1965. [15] H. Kojima, A. Bard, J. Am. Chem. Soc. 97 (1975) 6317. [16] N.S.Z. Hush, Elektrochemie 61 (1957) 734. [17] L. Eberson, Acta Chem. Scand. B 36 (1982) 533. [18] C.P. Andrieux, I. Gallardo, J.M. Saveant, K.B. Su, J. Am. Chem. Soc. 108 (1986) 638.
www.elsevier.nl/locate/theochem
Kinetics of dissociative electron transfer for homogeneous and heterogeneous reductive bond-breaking reactions Zheng Yu Zhou a,b,*, Yu Mei Xing a b
a Department of Chemistry, Qufu Normal University, Shandong, Qufu 273165, People's Republic of China State Key Laboratory of Crystal Materials, Shandong University, Shandong, Jinan 250100, People's Republic of China
Received 5 January 2000; received in revised form 16 March 2000
Abstract The kinetics of dissociative electron transfer reactions of butyl halides by outer-sphere heterogeneous (glassy carbon electrodes) or by homogeneous (aromatic anion radicals) reagents is investigated in aprotic solvents. The potential energies of the reactants and products are described by the Morse and the exponential curves as functions of the C±X distance. Both the homogeneous and the heterogeneous reactions are consistent with an outer-sphere adiabatic electron-exchange process. The activation free energy in both cases is suitable for a quadratic driving force free energy relationship and varies with the electrode potential. The standard activation free energy contains two contributions. There is a good agreement between theory and experiment in the heterogeneous case, while the good agreement is only for the tertiary halides in the homogeneous case, which is due to the stability of tertiary radical with sterical effect. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Dissociative electron transfer; Heterogeneous and homogeneous reaction; Butyl halides
1. Introduction Electron transfer reactions are among the simplest of solution phase reactions and have been the subject of extensive theoretical and experimental studies [1± 11]. These reactions include two categories: one involves no breaking or formation of chemical bonds, but rather only changes in the bond lengths and angles and changes in the molecular orientation of the solvent molecules. The other is very often accompanied by the breaking of an existing bond and/or the formation of a new bond [4±11]. In the case where a radical and an anionic fragment are formed upon electron transfer to a neutral molecule, an important question is whether electron transfer and * Corresponding author. Department of Chemistry, Qufu Normal University, Shandong, Qufu 273165, People's Republic of China.
bond breaking are concerted (reaction (1)) or occur successively (reactions (2) and (3)) [5]. RX 1 e2 ! Rz 1 X2
1
RX 1 e2 ! RXz2
2
RXz2 ! Rz 1 X2
3
The theory for the concerted mechanism has been described by Saveant [12], who was convinced of the adiabatic aspects of this reaction. More recently a corresponding nonadiabatic treatment has been proposed [13]. In the classical limit, Saveant used the Marcus theory to describe the solvent polarization and the Morse-like curves for the reactants' and products' molecular potential. The treatment of solvent reorganization in the Marcus± Hush fashion leads to a quadratic activation force
0166-1280/00/$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0166-128 0(00)00504-2
88
Z.Y. Zhou, Y.M. Xing / Journal of Molecular Structure (Theochem) 532 (2000) 87±93
2. Theoretical model A few classical semi-molecular and molecular models of electron transfer reactions involving bond-breaking reactions are the following: RX 1 e2 ! Rz 1 X2
Electrochemical
RX 1 Dz2 ! Rz 1 X2 1 D
Fig. 1.
relationship [1±4]: #
DG wR 1
DG#0
DG0 2 wR 1 wP 11 4DG#0
!2 4
for both homogeneous and heterogeneous (electrochemical) electron transfers (DG #, free energy of activation for the forward reaction; DG 0, standard free energy of the reaction, 2DG 0, a measure of the driving force of the reaction; wR and wP, work required to bring the reactants and products, from in®nite separation to the reacting distance). The standard free energy of activation or the intrinsic barrier free energy, DG#0 ; i.e. the free energy of activation at zero driving force, appears as the sum of two factors, i.e. an internal reorganization (bond lengths and angles) factor and a solvent reorganization factor. The external reorganization factor was obtained from Marcus± Hush solvent reorganization factor. The internal reorganization factor is approximately equal to one-fourth of the dissociation energy of the broken bond. Saveant described a simple model of the kinetics of dissociative electron transfer in polar solvents, showing the validity of the quadratic activation-driving force free energy relationship depicted by Eq. (4) and relating the standard activation free energy to the bond dissociation energy as well as to the solvent reorganization. In this paper, we discuss the case where electron transfer to an organic molecule is accompanied by the breaking of a bond and use the model given by Saveant to test the agreement of the theoretical calculation with experiments of the kinetic parameters.
Homogeneous
RX stands for alkyl halides or any molecule, X designing the ªleaving groupº, i.e. the group that leaves carrying on an electron pair and R the ªremaining groupº, i.e. the group remaining with an unpaired electron. D/D z2 is a chemically stable redox couple reacting in an outer-sphere manner. The potential energy of the reactants was assumed to depend upon the R±X distance according to the Morse curve, UR DRX 1 2 2 exp 2by2 ; where y rR±X 2 r0;R±X and r0 is the equilibrium bond length of the bond C±X; DRX is the dissociation energy of the R±X bond; b y 0 2p 2 m=DRX with y 0 representing the vibration frequency of the RX bond and m representing the reduced mass of the C and X atoms. The potential energy of the products was modeled by an exponential function, U P DRX exp 22by: The potential energy is zero at the minimum of the Morse and of the exponential curve, as shown in Fig. 1. The free energy of activation shown below was thus obtained in terms of the potential energy. 3. Results and discussion 3.1. Electrochemical reduction of alkyl butyl halides In the electrochemical case, the intrinsic barrier is given by # DRX 1 lhet DG0;het 0 =4
5
with a Marcus-like estimate of solvent reorganization 2 [1,2] lhet 0 e0 1=Dop 2 1=Ds 1=4a1 and with a 2 Hush-like estimation [14] lhet 0 e0 1=Dop 2 1=Ds 1=2a1 (Dop and Ds are the optical and static dielectric constants of the solvent, respectively, a is the radius of the outer-sphere equivalent to the reactant and products and e0 is the charge of the electron), which looks more consistent with the experimental
Z.Y. Zhou, Y.M. Xing / Journal of Molecular Structure (Theochem) 532 (2000) 87±93
89
Table 1 Kinetics of the electrochemical reduction of butyl iodides and bromides (in DMF, at a glassy carbon electrode at 108C; scan rate v 0:1 V=s; the values in the parentheses when DG04 ± 0 DG04 ; standard Gibbs free energy of the reaction RX 1 RzDMF , RXDMF 1 Rzg ), D 1025 cm2 s21 : All the potential are in V vs. SCE, all the energies are in eV). Comparison between the prediction for the theory of dissociative electron transfer and the experimental data Compd.
0 a z 1X2 2ERX=R Z el (cm s 21) Ê) aRX,aX,a(A b lhet 0 DRX b 2Ep 2f r DG#0 theor DG#0 exp DG#el theor DG#el exp DG#21 theor a (theor) a (pred) a (exp) a b
BuI
BuBr
n-BuI
sec-BuI
t-BuI
n-BuBr
sec-BuBr
t-BuBr
1.075(1.209)
0.946(1.080) 4:5 £ 104 3.58,3.05,3.50 0.902 2.28(2.49) 2.05 0.084 0.796(0.848) 0.808(0.760) 0.368(0.463) 0.379 1.388(1.349) 0.340(0.369) 0.342(0.357) 0.33
0.826(0.960)
1.109(1.230)
0.929(1.050)
3.62,3.06,3.53 0.892 2.20(2.31) 1.91 0.081 0.773(0.801) 0.802(0.751) 0.353(0.425) 0.379 1.356(1.294) 0.337(0.364) 0.343(0.355) 0.32
3.50,2.82,3.37 0.935 3.00(3.12) 2.85 0.102 0.984(1.014) 1.043(1.000) 0.335(0.397) 0.385 1.974(1.915) 0.292(0.313) 0.303(0.310) 0.25
1.049(1.190) 5:1 £ 104 3.50,2.82,3.37 0.935 2.99(3.07) 2.63 0.097 0.982(1.001) 0.987(0.936) 0.380(0.442) 0.385 1.864(1.785) 0.311(0.332) 0.312(0.321) 0.25
3.56,3.04,3.48 0.905 2.56(2.67) 2.33 0.09 0.866(0.894) 0.864(0.817) 0.381(0.453) 0.380 1.546(1.484) 0.332(0.354) 0.331(0.341) 0.30
3.54,2.83,3.40 0.927 2.87(2.99) 2.51 0.090 0.949(0.979) 0.993(0.949) 0.350(0.414) 0.388 1.841(1.784) 0.303(0.325) 0.312(0.319) 0.20
See Refs. [16±18] (from thermodynamic data compilations [18] using a previously described method [16,17]). From literature thermochemical data in Ref. [13].
results than Marcus' approximation in the present case. Thus the ensuing variations of lhet 0 with 1=a approximates to lhet 0 3:15=a A:
6
Previous estimations [12] have used the arithmetic mean of the halogen radius, aX, and of the radius of the sphere equivalent to the whole molecule, aRX, which was derived from the molar mass, M, and the density, r , aRX 3M=4NA r1=3 : The following equation: a
aX 2aRX 2 aX aRX
7
probably gives a more realistic estimation and was used in the following. thus obtained and the Using the values of lhet 0 values of the bond dissociation energy from the literature [13], we obtained the theoretical values of the intrinsic barrier, i.e. DG#0 theor; in terms of Eq. (5), listed in Table 1. At a given sweep rate, the transfer coef®cient (symmetry factor), a , can be considered as approximately constant along the cyclic voltammetric wave.
Such a simpli®cation is perfectly legitimate for each scan rate individually, since the potential difference between the base and the peak height of each individual voltammogram is small. Thus, DG#el ; the activation free energy of the forward electron transfer at the peak potential Ep is DG#el
" !1=2 # RT RT RT el ln Z 2 0:78 a p F nD F F
8
where D is the diffusion coef®cient of RX, n is the scan rate and Z el is the collisional frequency, Z el RT=2pM1=2 (M is the molar mass, for the butyl iodides and butyl iodides and butyl bromides and Z el was taken as equal to 4:5 £ 104 and 5:1 £ 104 cm s21 : The transfer coef®cient a (exp) is
a
1:85 RT £ Ep=2 2 Ep F
9
The experimental values thus obtained are compared with the theoretical values, i.e. from
90
Z.Y. Zhou, Y.M. Xing / Journal of Molecular Structure (Theochem) 532 (2000) 87±93
in a linear way:
Eq. (10) DG#el
" #2 0 z 1X2 2 fr Ep 2 ERX=R DRX 1 lhet 0 11F 4 DRX 1 lhet 0 10
(expressing the potential in volts and the free energies in eV) where f r is the potential difference between the reaction site and the solution. The reaction site is usually assumed to be located in the outer Helmoltz plane [14] and f r is obtained from previous measurements carried out in the same solvent (DMF) [15]. Given a scan rate and a speci®c temperature, a ®xed activation free energy is seen at the peak. Thus, increasing the scan rate induces a negative shift of the peak potential and hence an increase of the driving force offered to the reaction. At a ®xed scan rate, decreasing the temperature induces the following negative shift of the peak potential, hence offering more driving force to the reaction. The activation free energy of the backward reaction thus derived from the similar equation: DG#21
" #2 0 z 1X2 2 fr Ep 2 ERX=R DRX 1 lhet 0 12F 4 DRX 1 lhet 0 11
For obtaining DG#0 exp; we obtained from Eq. (13) according to Eq. (10) DG#0 exp { Ep 2 E0 2 fr 2 2DG#el 2 2 Ep 2 E0 2 fr 2 1=2 2 Ep 2 E0 2 fr 2 2DG#el }=4
12
The results are displayed in Table 1. We see that there is satisfactory agreement between the prediction of the theory and the experimental data when DG04 0: The agreement is not so good as the second estimation when DG04 ± 0: The comparison among the experimental values of the transfer coef®cient a (exp), the predicted a (pred) and the theoretical a (theor) values also testi®es the dissociative electron transfer theory. If the activationdriving force relationship is quadratic as depicted in Eq. (4), when wR wP 0; the transfer coef®cient a is expected to decrease upon raising the driving force
2DG# 1 DG0 1 1 ap 2 4DG#0 2DG0 1 E 2 E0 2 fr 11F P 2 4DG#0
! ! 13
This indicates that a p is related to the driving force (2DG 0) and the intrinsic barrier free energy DG#0 : Its value lies between 0 and 0.5. When 2DG 0 and DG#0 are high, a p is closer to 0.5 and vice versa. a (pred) is the value obtained from Eq. (13), taking DG#0 exp; whereas a (theor) is obtained from Eq. (13) using DG#0 theor Table 1 explains that, although the experimental values of a p are somewhat smaller than the predicted ones, the reason that the transfer coef®cient is distinctly below 0.5 appears as a consequence of the fact that the reduction potential is much negative relative to the standard potential. The potential-dependent heterogeneous rate constant k, according to Marcus theory of outersphere electron transfer, is related to the activation free energy, DG #, by the following k Z el exp 2DG#el =RT: In fact, the identi®cation of the pre-exponential factor with the collision frequency is certainly a crude approximation, but to an extent, it is valuable to show the variable tendency of the rate constant. From Table 1, we can see the theoretical values of DG#el in the ®rst case are in better agreement with those of the experimental results than in the second case. The activation free energy of the backward reaction, DG#21 ; is much larger than that of the forward reaction, DG#el : This shows that it is impossible for the backward reaction to take place. 3.2. Homogeneous reduction of alkyl butyl halides by aromatic anion radicals (D z2) These reactions can be thought of as a sequence of elementary steps: the diffusion of the two reagents to form the precursor complex, then the activation of the precursor complex to yield the successor complex, at last the separation of this latter into the products, according to Eq. (14) RX 1 Dz2 , RXuDz2 , RXz2 uD , Rz 1 X2 1 D 14
1:39 £ 104 2:79 £ 104
0.798 0.671
8:50 £ 103 8:73 £ 105
DG#0 theor DG#0 exp
khom(theor) khom(exp)
d
c
b
a
0.753 0.735
0.439 0.322
DG #(theor) DG #(exp)
Temperature, 208C. The mediator is anthracene. The mediator is 4-cyanopyridine. The mediator is diphenyl.
0.412 0.395
0.946 1.730 c 0.631 2.38 0.784
1.075 1.900 b 0.633 2.56 0.825
2ERX=Rz1 X2 2ED/D z2 lhom 0 DRX 2DG 0
s-BuI
n-BuI a
Compd.
4:54 £ 105 2:17 £ 105
0.707 0.726
0.327 0.345
0.826 1.730 c 0.628 2.20 0.904
t-BuI
67.95 3:06 £ 102
0.914 0.874
0.561 0.523
1.109 1.900 b 0.656 3.00 0.766
n-BuBr a
86.41 5:69 £ 102
0.912 0.863
0.536 0.490
1.049 1.900 b 0.656 2.99 0.851
s-BuBr
1:79 £ 103 1:52 £ 103
0.885 0.885
0.462 0.466
0.929 1.900 b 0.653 2.87 0.971
t-BuBr
6:24 £ 102 1:89 £ 103
1.044 1.014
0.505 0.477
1.267 2.540 d 0.674 3.50 1.273
n-BuCl a
5:69 £ 102 1:95 £ 103
1.031 0.998
0.490 0.460
1.258 2.540 d 0.672 3.45 1.282
s-BuCl
4:24 £ 103 6:13 £ 103
1.022 1.012
0.441 0.432
1.138 2.540 d 0.668 3.42 1.402
t-BuCl
Table 2 Kinetics of homogeneous reduction of butyl halides by aromatic anion radicals in DMF. Comparison between the experimental data (temperature 108C unless otherwise stated) and Ê , lhom the prediction of the theory (all standard potentials in V vs. SCE, all free energies in eV, all radii in A in eV) 0
Z.Y. Zhou, Y.M. Xing / Journal of Molecular Structure (Theochem) 532 (2000) 87±93 91
92
Z.Y. Zhou, Y.M. Xing / Journal of Molecular Structure (Theochem) 532 (2000) 87±93
The same strategy as in the electrochemical case was followed here. For the cross-exchange reaction between RX and D z2 [2] 1 1 1 1 1 2 lhom e 2 2 2 15 0 DOP DS 2a1 2a2 d
case of the secondary and the primary halides, even though the difference remains small (about 50 meV at maximum except n-BuI and sec-BuI). The homogenous rate constant can thus be derived from
where a1 and a2 are the radii of the spheres equivalent to the RX and D molecules, respectively, and d is the distance of their centers in the precursor complex. The theoretical value of DG#0;hom is thus obtained from Eq. (16)
20
DG#0;hom DRX 1 lhom 0 =4
16
is estimated from Eq. (15) and DRX is where lhom 0 obtained from the gas-phase thermochemical data as in the electrochemical case. The theoretical prediction of activation free energy is thus obtained from !2 DG0 # # 17 DGhom DG0;hom 1 1 4DG#0;hom where the driving force of the reaction is obtained from 0 0 z 1X2 2 ED=Dz2 DG0 2nF ERX=R
18
0 z2 is the standard potential of the mediator couple, ED=D 0 z 1X2 is the standard potential of reaction (1), n ERX=R 1: Whereas the experimental value of DG#0;hom is then derived from Eq. (4) (with wR wP 0 by means of
DG#0;hom { DG0 2 2DG# 2 2 DG0 2 1=2 2 DG0 2 2DG# }=4
19
the experimental activation free energy DG # is obtained from the rate constant k. All the results are shown in Table 2. It however distinctly appears that, given the halogen and the nature of the reacting carbon, the agreement is better for the tertiary halides than for the secondary and the primary halides. From the mere fact that the tertiary alkyl radicals are more stable than the secondary and the primary alkyl radicals, DRX increases in this series and E 0 becomes less and less negative. DG#0 theor becomes larger and larger from the tertiary to the secondary and the primary, but the trend is opposite for DG#0 exp: Thus, taking the theory±experiment agreement as good for the tertiary halides, it can be seen that the theory overestimates DG#0 and DG # in the
k Z hom exp 2DG#hom =RT
where Z hom a1 1 a2 2 8pRT=m1=2 (m is the reduced molar mass of the two reactants; a1 and a2 are the outer-sphere equivalent radii of the two reactants). For the whole series of homogeneous mediators Z hom was found to be equal to 3 £ 1011 M21 s21 with a maximal deviation of ^10%, i.e. perfectly negligible in testing the theory. From the data of homogeneous rate constant, it is obvious that the agreement of the theory with the experiment is much better for the tertiary halides than for the secondary and the primary halides and the rate constant of the ®rst is larger than those of the last two. These observations suggest that in the absence of strong steric hindrance, the tertiary halides are easier to break the C±X bond and to inform the tertiary radicals. This also suggests the fact that the tertiary radical is more stable than the secondary and the primary radicals. 4. Conclusions The dissociative electron transfer reaction has two mechanisms. In the cases where concerted mechanism prevails, the main parameter governing the reduction of the alkyl halides is the dissociation energy of the C±X bond undergoing cleavage in the reaction. Based on a Morse curve description of the potential energy surfaces for bond breaking, it leads, for both homogeneous and heterogeneous processes, to a quadratic activation-driving force free energy relationship with the standard activation free energy being the sum of two contributions. One is the bond breaking that appears as being equal to one-fourth of the bond dissociation energy, the other is the solvent reorganization that can be estimated on the basis of the Marcus±Hush dielectric continuum model for solvation. In the electrochemical case, there is a good agreement between the theoretical and experimental values of the intrinsic barrier for all the butyl iodides and bromides. In the homogeneous case, there is also a
Z.Y. Zhou, Y.M. Xing / Journal of Molecular Structure (Theochem) 532 (2000) 87±93
good agreement between theoretical predictions and experiment in the case of tertiary butyl. The homogeneous reduction of tertiary butyl halide is faster than those of the secondary and the primary. This shows that the tertiary butyl radical is more stable than the secondary and the primary. Acknowledgements This work is supported by the Natural Science Foundation of Shandong Province and National Natural Science Foundation of China. References [1] R.A. Marcus, J. Chem. Phys. 24 (1956) 4966. [2] R.A. Marcus, J. Chem. Phys. 43 (1965) 679. [3] R.A. Marcus, in: P.A. Rock (Ed.), Theory and Applications of Electron Transfers at Electrodes and in Solution. Special Topics in Electrochemistry, Elsevier, New York, 1977, pp. 161±179.
93
[4] R.A. Marcus, J. Phys. Chem. A 101 (1997) 4072. [5] C.P. Andrieux, A.L. Gorande, J.M. Saveant, J. Am. Chem. Soc. 114 (1992) 6892. [6] J.M. Saveant, in: P.S. Mariano (Ed.), Dissociative electron transfer, Advances in Electron Transfer Chemistry, vol. 4, JAI Press, New York, 1994, pp. 53±116. [7] S. Antonello, M. Musumeci, D.D.M. Wayner, F. Maran, J. Am. Chem. Soc. 119 (1997) 9541. [8] M.S. Workentin, F. Maran, D.D.M. Wayner, J. Am. Soc. 117 (1995) 2120. [9] E.D. German, A.M. Kuznetsov, J. Phys. Chem. 99 (1995) 9095. [10] D.A. Pratt, J.S. Wright, K.U. Ingold, J. Am. Chem. Soc. 121 (1999) 4877. [11] C. Costentin, P. Hapiot, M. Medebielle, J.M. Saveant, J. Am. Chem. Soc. 121 (1999) 4451. [12] J.M. Saveant, J. Am. Chem. Soc. 109 (1987) 6788. [13] S.W. Benson, Thermodynamical Kinetics, 2nd ed., Wiley, New York, 1976. [14] P. Delahay, Double Layer and Electrode Kinetics, Wiley, New York, 1965. [15] H. Kojima, A. Bard, J. Am. Chem. Soc. 97 (1975) 6317. [16] N.S.Z. Hush, Elektrochemie 61 (1957) 734. [17] L. Eberson, Acta Chem. Scand. B 36 (1982) 533. [18] C.P. Andrieux, I. Gallardo, J.M. Saveant, K.B. Su, J. Am. Chem. Soc. 108 (1986) 638.