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Development of a Randles-Ševčík-like equation to predict the peak current of cyclic voltammetry for solid metal hexacyanoferrates

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Abstract

A mathematical model, derived from Fick’s second law for planar and spherical coordinates, is presented and solved to obtain novel equations that satisfactorily predict the charge transfer and peak current observed in cyclic voltammetry of solid hexacyanoferrates. For a planar geometry, two cases are considered. The first one takes into account the effect of the evolution of the activities of the oxidized and reduced phases of the hexacyanoferrates on the peak currents, while in the second one, the activities of both phases were considered as a constant. In the first case, an analysis of charges involved in the reaction is required to obtain the molar fractions of the oxidized and reduced forms of the hexacyanoferrates. In this case, the solution of the model is obtained numerically using the lines method. In the second case, the model is analytically solved obtaining a Randles-Ševčík-like equation. When spherical coordinates are considered, the activities of the solid phase are assumed to be constant and the model is analytically solved. In this way, other novel equations allowing the calculus of the electroactive electrode area and the diffusion coefficient of the alkali ions are presented. The advantages of an analytical expression for the peak current as a function of the square root of the potential scan rate, instead of a numerical solution, are analyzed. The validity of each model is proven by its comparison with experimental measurements for peak currents in a carbon paste electrode containing nickel hexacyanoferrate immersed in a 0.5M KNO3 solution.

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References

  1. Miller JS (2000) Inorg Chem 39:4392–4408

    Article  CAS  Google Scholar 

  2. Ohkoshi Shin-ichi. Abe Y, Fujishima A, Hashimoto K (1999) Phys Rev Lett 82:1285–1288

    Article  Google Scholar 

  3. Entley WR, Girolami GS (1994) Inorg Chem 33:5165–5166

    Article  CAS  Google Scholar 

  4. Carpenter MK, Conell RS (1990) J Electrochem Soc 137:2464–2467

    Article  CAS  Google Scholar 

  5. Duek EAR, De Paoli MA, Mastragostino M (1992) Adv Mater 4:287–291

    Article  CAS  Google Scholar 

  6. Itaya K, Uchida I, Toshima S (1983) J Phys Chem 87:105–112

    Article  CAS  Google Scholar 

  7. Kaneko M, Okada T (2008) Molecular catalyst for energy conversion. Springer

  8. Itaya K, Akahoshi H y, Toshima S (1982) J Am Chem Soc 104:4767–4772

    Article  CAS  Google Scholar 

  9. Zhao F, Zhang J, Hou X, Abe T, Kaneko M (1998) J Chem Soc Faraday Trans 94:277–281

    Article  CAS  Google Scholar 

  10. Neff VD (1985) J Electrochem Soc 132:1382–1384

    Article  CAS  Google Scholar 

  11. Paolella A, Faure C, Timoshevskii V, Marras S, Bertoni G, Guerfi A, Vijh A, Armand M, Zaghib K (2017) J Mater Chem 5:18919–18932

    Article  CAS  Google Scholar 

  12. Retter U, Widmann A, Siegler K, Kahlert H (2003) J Electroanal Chem 546:87–96

    Article  CAS  Google Scholar 

  13. Kahlert H, Retter U, Lohse H, Siegler K, Scholz F (1998) J Phys Chem B 102:8757–8765

    Article  CAS  Google Scholar 

  14. Heli H, Majdi S, Sattarahmady N (2010) Sens Actuators B Chem 145:185–193 

  15. Siang-Fu H, Lin-Chi C (2012) Sol Energy Mater Sol Cells 104:64–74

    Article  Google Scholar 

  16. Gholivand MB, Azadbakht A (2011) Electrochim Acta 56:10044–10054 

  17. Bard AJ, Faulkner LR (2002) Electrochemical methods fundamental and applications. Willey and sons, John

    Google Scholar 

  18. Randles JEB (1948) Trans Faraday Soc 44:327–338

    Article  CAS  Google Scholar 

  19. Ševčík A (1948) Collect. Czech Chem Commun 13:349–377

    Article  Google Scholar 

  20. Nicholson RS, Shain I (1964) Anal Chem 36:706–723

    Article  CAS  Google Scholar 

  21. Bárcena-Soto M, Scholz F (2002) J Electroanal Chem 521:183–189

    Article  Google Scholar 

  22. Orazem ME, Tribollet B (2008) Electrochemical impedance spectroscopy. The electrochemical Society Series, Willey

    Book  Google Scholar 

  23. Lovrić M, Scholz F (1997) J Solid State Electrochem 1:108–113

    Article  Google Scholar 

  24. Schröder U, Oldham KB, Mylan J, Mahon P, Scholz F (2000) J Solid State Electrochem 4:314–324

    Article  Google Scholar 

  25. Lovrić M, Hermes M, Scholz F (1998) J Solid State Electrochem 2:401–404

    Article  Google Scholar 

  26. Scholz F, Dostal A (1996) Angew Chem Int Ed in Engl 34:2685–2687 

  27. Ellis D, Eckhoff M, Neff VD (1981) J Phys Chem 85:1225–1231

    Article  CAS  Google Scholar 

  28. Gordon AR (1937) J Phys Chem 5:522–526

    Article  CAS  Google Scholar 

  29. Dawson HG (1898) Lond Math Soc s1-29:519–522

    Article  Google Scholar 

  30. Frankenthal RP, Shain I (1956) J Am Chem Soc 78:2969–2973

    Article  CAS  Google Scholar 

Download references

Acknowledgments

O. A. González-Meza is grateful to CONACyT for the financial support for his Ph. D. Studies (CVU: 665472).

The authors wish to dedicate this work and its results to Prof. Dr. Fritz Scholz as a deep and sincere acknowledgment to his valuable contribution to the field of the electrochemistry of metal hexacyanoferrates.

Glossary

Symbol

Meaning

Units

A

Area

cm2

a i

Activity of specie i

None

b

\( \frac{nF}{RT}v \)

s−1

C i

Concentration of species i

mol cm−3

\( {C}_{\mathrm{i}}^{\ast } \)

Bulk concentration of species i

mol cm−3

\( \overset{\sim }{C_{\mathrm{i}}}\left(x,s\right) \)

Laplace-transformed concentration

mol s cm−3

D i

Diffusion coefficient of species i

cm2 s−1

e−

Electron

None

E

Potential of an electrode versus a reference

V

E°

Standard potential of an electrode

V

E°′

Formal potential of an electrode

V

E°′′

Apparent formal potential of an electrode

V

E eq

Equilibrium potential of an electrode

V

E p

Peak potential

V

F

Faraday’s constant

C mol−1

γ i

Activity coefficient for species i

none

i 0

Straight line intercept in (33) \( \frac{nFA{D}_{{\mathrm{M}}^{+}}{C}_{{\mathrm{M}}^{+}}^{\ast }}{r_0} \)

A

I

Current

A

I p

Peak current

A

I pa

Anodic peak current

A

I pc

Cathodic peak current

A

\( \overset{\sim }{I(s)} \)

Laplace-transformed current

A s

m

Straight line slope in (33)

A s1/2 V−1/2

n

Stoichiometric number of electrons involved in the electrode reaction

None

(Qa)v

Total anodic charge, experimentally obtained, for a scan rate v

C

Qa(t)|v

Total anodic charge, experimentally obtained, for a scan rate v as a time function

 

(Qc)v

Total cathodic charge, experimentally obtained, for a scan rate v

C

Qc(t)|v

Cathodic charge, experimentally obtained, for a scan rate v as a time function

 

r

Radial distance from the center of the electrode

cm

r 0

Radius of an spherical electrode

cm

R

Gas constant

J mol−1 K−1

s

Laplace’s variable

s−1

t

Time

s

T

Absolute temperature

K

v

Linear potential scan rate

V s−1

w(r, t)

Auxiliary function

mol cm−2

x

Coordinate away from the electrode

cm

X i

Molar fraction of species i

None

Xi(t)

Molar fraction of species i as a time function

None

\( {X}_{\mathrm{i}}^{\mathrm{eq}} \)

Molar fraction of species i at equilibrium

None

Author information

Authors and Affiliations

  1. Departamento de Ingeniería Química, CUCEI, Universidad de Guadalajara, Guadalajara, Mexico

    O. A. González-Meza & E. R. Larios-Durán

  2. Departamento de Ciencias Básicas y Aplicadas, CUTonalá, Universidad de Guadalajara, Guadalajara, Mexico

    A. Gutiérrez-Becerra

  3. Departamento de Química, CUCEI, Universidad de Guadalajara, Guadalajara, Mexico

    N. Casillas, J. I. Escalante & M. Bárcena-Soto

Authors
  1. O. A. González-Meza
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  2. E. R. Larios-Durán
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  3. A. Gutiérrez-Becerra
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  4. N. Casillas
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  5. J. I. Escalante
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  6. M. Bárcena-Soto
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Corresponding author

Correspondence to M. Bárcena-Soto.

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González-Meza, O.A., Larios-Durán, E.R., Gutiérrez-Becerra, A. et al. Development of a Randles-Ševčík-like equation to predict the peak current of cyclic voltammetry for solid metal hexacyanoferrates. J Solid State Electrochem 23, 3123–3133 (2019). https://doi.org/10.1007/s10008-019-04410-6

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  • DOI: https://doi.org/10.1007/s10008-019-04410-6

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