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A model calculation of deaquation overvoltage
Electroanalytical Chemistry and lnterfacial Electrochemistry, 46 (1973) 181-183 © ElsevierSequoia S.A., Lausanne - Printed in The Netherlands
181
A MODEL CALCULATION OF DEAQUATION OVERVOLTAGE* H. STREHLOW and J. JEN Max-Planck-l~stitut fiir Biophysikalische Chemie, GSttingen (German Federal Republic) (Received 21st March 1973)
INTRODUCTION In a preceding paper 1 a mechanism was proposed for the reaction inner-sphere complex ~ outer-sphere complex for metal complexes with ionic ligands. An electrostatic model calculated explicitly for an aqueous s.olution of BeSO 4 is consistent with the experimental data 1"2. Figure 1 shows the model for the outer-sphere complex, the transition state and the inner-sphere complex of the B e S O 4 - H 2 0 system. The assumptions made are: Be 2÷ cations, water molecules, and sulphate ion are hard spheres of radii r + = 0 . 3 4 A, rw= 1.38 ~, and r =2.91 ~, respectively (the finite compressibilities of the species and their deviation from spherical shape are neglected). The respective polarizabilities are taken to be ct+ =0, ~ = 1.48 ~3, and 0¢_ = 5.8 /~a. The permanent dipole moment of the water molecule,/~0, is 1.85 Debye**.
outer sphere complex
tronsition state
inner
sphere" complex
Fig. 1. Complex formation (schematically).
The beryllium ion is tetrahedrally surrounded by four water molecules, thus forming the outer-sphere complex. A fourth water molecule on the opposite side completes the solvation shell of the anion. Formation of the inner-sphere Complex proceeds via the approach of the two ions, in which the water molecules assume positions that make the free enthalpy of the system a minimum for every given configuration while the ion-ion distance, r, changes. The electrostatic energy E was calculated as a function of the distance r, leading to the results indicated in Fig. 2. The calculated activation energies are similar to those experimentally observed. * Dedicated to Professor Wiktor Kemula on the occasion of his 70th birthday. ** 1 Debye= 10-18 e.s.u.~-3.336×.10-3° C m
182
H. STREHLOW, J. JEN
Energy/kcolmole -t - 700
- 710-
-
720-
I . . . . . . . J
4.5
r outer sphere
4.0
3.5 3.1 r inner sphere
Fig. 2. Calculated electrostatic energy for the BeSO 4 model. Parameters used: H20, r = 1.38 /L ~ = 1.48 ]k 3,/~= 1.85 D; Be 2+, r=0.34/~, ~ = 0 ; SO~-, r=2.91/~, ~=5.8 ]k. 1 ca1=4.184 J.
D E A Q U A T I O N AS THE M A I N CAUSE O F O V E R V O L T A G E
For small multiply charged cations the deaquation of the ion is likely to be the main cause of the considerable overvoltage required for electrolytic reduction in aqueous solution. By extending the model presented in ~the introduction this expectation may be put on a semiquantitative basis. Let us take the sulphate ion in Fig. 1 as being equivalent to an electrode. A change in the "electrode potential" may be effected by changing the charge of the "electrode" from z = 0 to z = - 5 . Some calculations on the same line as indicated above have been performed, the results of which are presented in Fig. 3. T h e curve with z = - 2 corresponds to Fig. 2 calculated for BeSO4. If these curves are interpreted as the potential curves at a spherical electrode at which Be 2+ ions are discharged, the ordinates at r=3.28 /~ (the distance r for the inner-sphere complex) correspond to the electrode potential E and the maximum for the curves corresponds to the activation energy EA. Since eA =
the transfer coefficient for the model may be evaluated. In Fig. 4 Ea and ( ! - ~) are plotted against - E . The zero point of E is taken to be at the ordinate with z = 0 . F r o m the ratio of charge to voltage we obtain a value for the electrode capacity of ~ 20 pF/cm 2 and a corresponding effective dielectric constant •eff 3.5 for the water layer separating the Be z + ion and the "electrode". The n~odel does not take into consideration other sources of overvoltage. It demonstrates that for small multiply charged ions deaquation is a strongly activated process which alone may account for the fact that these ions, for example Be 2+ or AP + (refs. 3 and 4 respectively), cannot be reduced in aqueous solution even at a mercury electrode with its high hydrogen overpotential. In a more sophisticated calculation for a Be(OHz),~ + ion discharging at a metal electrode, and considering image forces as well, a qualitatively similar result can be expected. ~
183
D E A Q U A T I O N OVERVOLTAGE
•
7.0
.20 ~
O,,Ss.°
Distance
from
"electrode"/
0 ,~
Fig. 3. Calculated electrostatic energies for Be 2 + ions at spherical electrodes with charge - z .
l EA/Volt
1-d~
1
0.7 03
.5 0.5
0.2~ 0
I
0.5
I
1
- E]Volt ~
Fig. 4. Deaquation overvoltage and transfer coefficient of Be 2 ÷ ions at model electrode.
SUMMARY
An estimation of the activation energy due to the deaquation of cations, preceding their reduction at an electrode, is presented. A correlation is drawn between this process and the rate of water exchange in the solvation sheath of the metal ion in homogeneous solution. REFERENCES 1 2 3 4
H. Strehlow and W. Knoche, Ber. Bunsenges. Phys. Chem., 73 (1969) 427. D. Hess, W. Knoche and C. A. Firth, Ber. Bunsenges. Phys. Chem., to be published. J. Heyrovsky and S. Berezicky, Coll. Czech. Chem. Commun., 1 (1929) 19. W. D. Treadwell and H. Stern, Helv. Chim. Acta, 7 (1924) 7.
181
A MODEL CALCULATION OF DEAQUATION OVERVOLTAGE* H. STREHLOW and J. JEN Max-Planck-l~stitut fiir Biophysikalische Chemie, GSttingen (German Federal Republic) (Received 21st March 1973)
INTRODUCTION In a preceding paper 1 a mechanism was proposed for the reaction inner-sphere complex ~ outer-sphere complex for metal complexes with ionic ligands. An electrostatic model calculated explicitly for an aqueous s.olution of BeSO 4 is consistent with the experimental data 1"2. Figure 1 shows the model for the outer-sphere complex, the transition state and the inner-sphere complex of the B e S O 4 - H 2 0 system. The assumptions made are: Be 2÷ cations, water molecules, and sulphate ion are hard spheres of radii r + = 0 . 3 4 A, rw= 1.38 ~, and r =2.91 ~, respectively (the finite compressibilities of the species and their deviation from spherical shape are neglected). The respective polarizabilities are taken to be ct+ =0, ~ = 1.48 ~3, and 0¢_ = 5.8 /~a. The permanent dipole moment of the water molecule,/~0, is 1.85 Debye**.
outer sphere complex
tronsition state
inner
sphere" complex
Fig. 1. Complex formation (schematically).
The beryllium ion is tetrahedrally surrounded by four water molecules, thus forming the outer-sphere complex. A fourth water molecule on the opposite side completes the solvation shell of the anion. Formation of the inner-sphere Complex proceeds via the approach of the two ions, in which the water molecules assume positions that make the free enthalpy of the system a minimum for every given configuration while the ion-ion distance, r, changes. The electrostatic energy E was calculated as a function of the distance r, leading to the results indicated in Fig. 2. The calculated activation energies are similar to those experimentally observed. * Dedicated to Professor Wiktor Kemula on the occasion of his 70th birthday. ** 1 Debye= 10-18 e.s.u.~-3.336×.10-3° C m
182
H. STREHLOW, J. JEN
Energy/kcolmole -t - 700
- 710-
-
720-
I . . . . . . . J
4.5
r outer sphere
4.0
3.5 3.1 r inner sphere
Fig. 2. Calculated electrostatic energy for the BeSO 4 model. Parameters used: H20, r = 1.38 /L ~ = 1.48 ]k 3,/~= 1.85 D; Be 2+, r=0.34/~, ~ = 0 ; SO~-, r=2.91/~, ~=5.8 ]k. 1 ca1=4.184 J.
D E A Q U A T I O N AS THE M A I N CAUSE O F O V E R V O L T A G E
For small multiply charged cations the deaquation of the ion is likely to be the main cause of the considerable overvoltage required for electrolytic reduction in aqueous solution. By extending the model presented in ~the introduction this expectation may be put on a semiquantitative basis. Let us take the sulphate ion in Fig. 1 as being equivalent to an electrode. A change in the "electrode potential" may be effected by changing the charge of the "electrode" from z = 0 to z = - 5 . Some calculations on the same line as indicated above have been performed, the results of which are presented in Fig. 3. T h e curve with z = - 2 corresponds to Fig. 2 calculated for BeSO4. If these curves are interpreted as the potential curves at a spherical electrode at which Be 2+ ions are discharged, the ordinates at r=3.28 /~ (the distance r for the inner-sphere complex) correspond to the electrode potential E and the maximum for the curves corresponds to the activation energy EA. Since eA =
the transfer coefficient for the model may be evaluated. In Fig. 4 Ea and ( ! - ~) are plotted against - E . The zero point of E is taken to be at the ordinate with z = 0 . F r o m the ratio of charge to voltage we obtain a value for the electrode capacity of ~ 20 pF/cm 2 and a corresponding effective dielectric constant •eff 3.5 for the water layer separating the Be z + ion and the "electrode". The n~odel does not take into consideration other sources of overvoltage. It demonstrates that for small multiply charged ions deaquation is a strongly activated process which alone may account for the fact that these ions, for example Be 2+ or AP + (refs. 3 and 4 respectively), cannot be reduced in aqueous solution even at a mercury electrode with its high hydrogen overpotential. In a more sophisticated calculation for a Be(OHz),~ + ion discharging at a metal electrode, and considering image forces as well, a qualitatively similar result can be expected. ~
183
D E A Q U A T I O N OVERVOLTAGE
•
7.0
.20 ~
O,,Ss.°
Distance
from
"electrode"/
0 ,~
Fig. 3. Calculated electrostatic energies for Be 2 + ions at spherical electrodes with charge - z .
l EA/Volt
1-d~
1
0.7 03
.5 0.5
0.2~ 0
I
0.5
I
1
- E]Volt ~
Fig. 4. Deaquation overvoltage and transfer coefficient of Be 2 ÷ ions at model electrode.
SUMMARY
An estimation of the activation energy due to the deaquation of cations, preceding their reduction at an electrode, is presented. A correlation is drawn between this process and the rate of water exchange in the solvation sheath of the metal ion in homogeneous solution. REFERENCES 1 2 3 4
H. Strehlow and W. Knoche, Ber. Bunsenges. Phys. Chem., 73 (1969) 427. D. Hess, W. Knoche and C. A. Firth, Ber. Bunsenges. Phys. Chem., to be published. J. Heyrovsky and S. Berezicky, Coll. Czech. Chem. Commun., 1 (1929) 19. W. D. Treadwell and H. Stern, Helv. Chim. Acta, 7 (1924) 7.