Traditional electrochemistry, translated

If you arrive already fluent in electrochemistry, here is a Rosetta Stone: the familiar symbols and equations on the left, their species-voltage readings on the right.^[1] None of it is new physics — it is the same quantities re-expressed on one shared voltage axis (see what this is and isn't). Each entry points to the topic where the translation is worked out.

Potentials and levels

Traditional	In $V_i$ terms
electrochemical potential $\bar\mu_i$	$z_i F\,V_i$
Fermi level $E_{\mathrm{F}}$ ( $=\bar\mu_{\mathrm{e}^-}$ )	$-e\,V_{\mathrm{e}^-}$
(Galvani / inner) potential $\phi$	a stand-in for the $V^\circ_i$ ladder that no ion can measure ( $\phi$ under the microscope)
electrode potential $E$ (vs SHE)	$V_{\mathrm{e}^-}(\text{electrode}) - V^\circ_{\mathrm{e}^-}(\mathrm{SHE})$ (electrode potential)
standard electrode potential $E^\circ$	$V^\circ_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red}) - V^\circ_{\mathrm{e}^-}(\mathrm{SHE})$ (half-reactions)
redox (solution) potential $E_h$	$V_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red}) - V^\circ_{\mathrm{e}^-}(\mathrm{SHE})$
overpotential $\eta$	a drop in $V_{\mathrm{e}^-}$ : $V_{\mathrm{e}^-}(\text{electrode}) - V_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red})$ (or any $V_i$ step at a driven interface; kinetics)
cell voltage / EMF	$V_{\mathrm{e}^-}(\text{right}) - V_{\mathrm{e}^-}(\text{left})$ (reference electrodes & cells)
liquid junction potential	a step in the $V^\circ_{\mathrm{e}^-}$ ladder across the junction
Donnan potential	a step in the $V^\circ_i$ ladder at a fixed-charge medium's boundary (charge neutrality)
built-in potential	a step in the $V^\circ_i$ ladder across a junction at equilibrium (bipolar)

Concentrations and the Nernst equation

Traditional	In $V_i$ terms
activity $a_i$	defined by $V_i = V^\circ_i + \tfrac{RT}{z_i F}\ln a_i$ (solutions)
standard internal chemical potential $\mu^\circ_{\mathrm{int},i}$	$z_i F\,(V^\circ_i - \phi)$
Nernst, $E = E^\circ + \tfrac{RT}{zF}\ln\tfrac{a_{\mathrm{Ox}}}{a_{\mathrm{Red}}}$	the floating Nernst on $V_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red})$ (half-reactions)
mass-action / solubility constant $K$	a fixed $V_i - V_j$ gap (mass action)
$\mathrm{pH} = -\log_{10} a_{\mathrm{H}^+}$	$(V^\circ_{\mathrm{H}^+} - V_{\mathrm{H}^+})\big/(2.303\,RT/F)$

Transport and storage

Traditional	In $V_i$ terms
Nernst–Planck (drift + diffusion) flux	$J_i = -\sigma_i \nabla V_i$ (charge-current form; number flux $N_i = J_i/z_i F$ ) (transport)
ionic conductivity $\sigma_i$	$z_i^2 F^2 D_i\, c_i / RT$
"ohmic" electrolyte current $-\sigma\nabla\phi$	the uniform-concentration limit of $\textstyle\sum_i -\sigma_i\nabla V_i$
chemical capacitance	$z_i^2 F^2 c_i / RT$ (capacitance)

What does not translate cleanly

A few traditional quantities have no clean $V_i$ counterpart, and that is the point rather than a gap. The inner potential $\phi$ , single-ion activities $a_i$ , and single-ion activity coefficients $\gamma_i$ are each individually ambiguous; only charge-balanced combinations, and differences sampled within one solution, are well defined. The $V_i$ picture keeps exactly the unambiguous part and leaves the rest floating, which is why it never needs to commit to a value of $\phi$ . (See non-ideal solutions.)

(and if you're curious about this project, the next topic is about how this project came to be.)

The electron half of this dictionary has been done, and done well: Steve Byrnes's Translation guide for discussing electron energy concepts maps the chemist's, physicist's, and engineer's terms onto one another. This page is in the same spirit, with the ions brought along. ↩︎