Abstract
With the help of a new theory using the notion of path in a graph, a physical interpretation and meaning can be given to electroanalytical measurements, without recourse to mathematical treatments. The frequency dependence of impedances measured by ac techniques, or the scan rate dependence of current vs. potential characterizations in large signal techniques (cyclic voltammetry), can be interpreted through this approach as a determination of the proportion of conserved energy vs. total energy involved in a mechanism. This total energy includes the energy which is lost by dissipation, i.e., converted into heat. This new way of thinking sheds an interesting light on the difference between normal and anomalous or fractal diffusion and provides to the analyst a simple and immediate tool for interpreting experimental data.
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Abbreviations
- A:
-
Initial species of the reaction
- A*:
-
Initial species of the reaction in the bulk or in its initial state
- A°:
-
Initial species of the reaction on the charge transfer site
- A:
-
Area (vector) [m2]
- \( \widehat{A} \) :
-
Area (operator) [m2]
- B:
-
Product species of the reaction
- B*:
-
Product species of the reaction in the bulk or in its initial state
- B°:
-
Product species of the reaction on the charge transfer site
- C:
-
Differential electric capacitance [F]
- \( \widehat{C} \) :
-
Integral electric capacitance (operator) [F]
- \( {\widehat{\text{C}}_{{\rm n} }} \) :
-
Integral physical chemical capacitance (operator) [mol2 J−1]
- \( {\widehat{\text{C}}_q} \) :
-
Integral generalized capacitance (operator) [q2 J−1] (q stands for the unit of the basic quantity (e.g., mol, C, m, m3, etc.) that defines the energy variety.)
- c :
-
Volumic concentration of substance [mol m−3]
- c* :
-
Bulk or initial volumic concentration of substance [mol m−3]
- \( c_{\text{A}}^0 \) :
-
Volumic concentration of species A on the charge transfer site [mol m−3]
- \( c_{\text{B}}^0 \) :
-
Volumic concentration of species B on the charge transfer site [mol m−3]
- D :
-
Diffusivity (diffusion coefficient) [m2 s−1]
- d:
-
Differential operator
- E :
-
Electrode potential [V]
- E i :
-
Initial value of the electrode potential [V]
- F :
-
Faraday constant [≈9.64853 × 104 C mol−1]
- e− :
-
Electron
- e q :
-
Effort (generalized potential) [J q−1]1
- f q :
-
Flow (generalized current) [q s−1]1
- G :
-
Electric conductance [S]
- \( \widehat{\text{G}} \) :
-
Integral electric conductance (operator) [S]
- \( {\widehat{\text{G}}_{{\rm n} }} \) :
-
Integral physical chemical conductance (operator) [mol2 J−1 s−1]
- \( {\widehat{\text{G}}_q} \) :
-
Integral generalized conductance (operator) [q2 J−1 s−1]1
- I :
-
Electric current [A]
- \( \widetilde{{\rm I} } \) :
-
Fourier transform of the electric current [A]
- I v0 :
-
Scaling electric current [A]
- ℑ:
-
Substance flow (mass flux) (I Fraktur) [mol s−1]
- i :
-
Imaginary number (i 2 = −1)
- J :
-
Substance flow density (mass flux density) (vector) [mol m−2 s−1]
- k A :
-
Forward heterogeneous rate constant (vector) [m s−1]
- k B :
-
Backward heterogeneous rate constant (vector) [m s−1]
- \( \mathbf{k}_\mathbf{CT}^{0} \) :
-
Intrinsic heterogeneous rate constant (vector) [m s−1]
- L :
-
Electric inductance [H]
- \( \widehat{\text{L}} \) :
-
Integral electric inductance (operator) [H]
- \( {\widehat{\text{L}}_q} \) :
-
Integral generalized inductance (operator) [J q−2 s2]1
- ℓ:
-
Displacement, characteristic distance, thickness (vector) [m]
- M :
-
Inertial mass [kg]
- \( \widehat{\user2{m}} \) :
-
Mass transfer rate (operator) [m s−1]
- n :
-
Substance amount [mol]
- \( {{\rm n}^{{\rlap{--}{0}}}} \) :
-
Reference substance amount [=1 mol]
- n e :
-
Stoichiometric number of electrons [–]
- \( \widehat{\text{O}} \) :
-
Any operator
- p :
-
Degree of derivation or mass transfer mode [–]
- p q :
-
Impulse (generalized quantity of movement) [J q−1 s]1
- Q :
-
Electric charge [C]
- q :
-
Basic quantity [q]1
- R :
-
Gas constant [≈8.3144 J mol−1 K−1]
- R :
-
Electric resistance [S]
- \( \widehat{\text{R}} \) :
-
Integral electric resistance (operator) [S]
- \( {\widehat{\text{R}}_q} \) :
-
Integral generalized resistance (operator) [Js q−2]1
- r :
-
Space vector [m]
- s :
-
Sink of substance [mol s−1 m−3]
- T :
-
Temperature [K]
- t :
-
Time [s]
- u :
-
Convective rate (vector) [m s−1]
- V :
-
Electric potential [V]
- V :
-
Volume [m3]
- \( \widetilde{{\rm V} } \) :
-
Fourier transform of the electric potential [V]
- v :
-
Sweep rate [V s−1]
- v 0 :
-
Scaling sweep rate [V s−1]
- ×:
-
Composition of operators by commutative multiplication (in a Formal Graph)
- Y :
-
Differential electric admittance [S]
- \( \widehat{\text{Y}} \) :
-
Integral electric admittance (operator) [S]
- \( \widetilde{Y} \) :
-
Fourier transform of the integral electric admittance [S]
- \( {\widehat{\text{Y}}_{{\rm n} }} \) :
-
Integral physical chemical admittance (operator) [mol2 J−1 s−1]
- Z :
-
Differential electric impedance [S]
- \( \widehat{\text{Z}} \) :
-
Integral electric impedance (operator) [Ω]
- \( \widetilde{Z} \) :
-
Fourier transform of the integral electric impedance [Ω]
- z :
-
Charge number of an ion [–]
- \( {\Phi _{B}} \) :
-
Induction flux [Wb]
- Γ:
-
Euler or Gamma function [–]
- \( \lambda \) :
-
Integration variable
- μ :
-
Chemical potential [J mol−1]
- \( {\mu^{{\rlap{--}{0}}}} \) :
-
Reference chemical potential [J mol−1]
- \( \overline \mu \) :
-
Electrochemical potential [J mol−1]
- μ V :
-
Translated chemical potential from electrodynamics [J mol−1]
- ν :
-
Frequency [Hz]
- π:
-
Archimedes constant [≈3.14159]
- τ :
-
Relaxation or characteristic time [s]
- τ Cq :
-
Capacitive time constant [s]
- τ Lq :
-
Inductive time constant [s]
- ω :
-
Pulsation or angular velocity [rad s−1]
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Vieil, E. Simple and direct interpretation of phase angles or derivation degrees in term of energy conservation vs. dissipation with Formal Graphs. J Solid State Electrochem 15, 955–969 (2011). https://doi.org/10.1007/s10008-011-1308-9
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DOI: https://doi.org/10.1007/s10008-011-1308-9