Electrocatalysis — catalysis of faradaic reactions at electrode / electrolyte interfaces. Foundations: (1) Butler-Volmer current-overpotential law (Butler 1924; Erdey-Grúz & Volmer 1930): j = j₀·(exp(αfη) - exp(-(1-α)fη)) where f =…
electrocatalysis
Butler-Volmer: j = j₀·(exp(αfη) - exp(-(1-α)fη))
Butler-Volmer equation (Butler 1924; Erdey-Grúz & Volmer 1930) — the foundational current-overpotential law of electron-transfer…
Tafel slope b = 2.303·RT/(αF); mechanistic fingerprint
Tafel slope b (Tafel 1905) is the d η / d log|j| derivative of the Butler-Volmer equation in the high-overpotential regime — the slope of η…
Sabatier volcano: log(j₀) peaks at ΔG_bind ≈ 0
Sabatier principle (Sabatier 1911) — catalysts bind intermediates 'just right': too weakly and activation of the bond is insufficient; too…
BV anchors: j(η=0)=0; linear regime j ≈ j₀·f·η (sp.series O(η²))
Sympy-exact witness of the Butler-Volmer zero-overpotential identity and the linear regime via Taylor expansion. Setup: j = j_0·(exp(αfη)…
Tafel 25°C α=1/2: b = 1426462079/12060625000 V ≈ 118.27 mV/dec
Sympy-exact witness of the Tafel slope at the canonical textbook anchor T = 298 K, α = 1/2. Setup: b = 2.303·R·T/(α·F) as a sympy Rational…
Nernst: E(Q=1)=E°; ΔE(10×,298K,n=1) = -619393·log(10)/24121250 V
Sympy-exact witness of the Nernst equation standard-state identity and the per-decade potential shift at 25 °C. Setup: E = E_0 -…
Marcus electron-transfer parabolic-exponent framework: log k ∝ −(ΔG+λ)²/(4λRT) (inverted region witness)
Marcus electron-transfer theory parabolic free-energy relationship: log k_ET ∝ −(ΔG + λ)²/(4·λ·R·T), where λ is the reorganization energy…
Levich RDE limiting-current framework: i_L = 0.62·n·F·A·D^(2/3)·ω^(1/2)·ν^(−1/6)·C (ω^(1/2) scaling)
Levich equation for the limiting current at a rotating-disk electrode: i_L = 0.62·n·F·A·D^(2/3)·ω^(1/2)·ν^(−1/6)·C, with the square-root…
Gouy-Chapman diffuse-layer framework: κ^(−1) = sqrt(ε·ε₀·R·T/(2·F²·I)) (Debye length ∝ I^(−1/2))
Gouy-Chapman diffuse-layer model of the electrical double layer: the Debye screening length κ^(−1) = sqrt(ε·ε₀·R·T/(2·F²·I)) sets the…
Marcus inverted-region pins: exp at ΔG=−λ is 0; at ΔG=0 is −λ/(4RT); ΔG at max solves to −λ
Exponent evaluations: −(ΔG+λ)²/(4λRT) at ΔG = −λ (inverted-region boundary, rate maximum) equals exactly 0; at ΔG = 0 (thermoneutral)…
Levich RDE scaling: i_L = K·√ω canonical form; i_L(4ω)/i_L(ω) = 2; i_L(9ω)/i_L(ω) = 3
i_L scaling evaluations: i_L(ω) = K·√ω is the canonical form collapsing all prefactors into K; i_L(4ω)/i_L(ω) = 2 exactly (quadrupling ω…
Gouy-Chapman Debye length: κ^(−1) = 1/√I; κ^(−1)(4I)/κ^(−1)(I) = 1/2; κ^(−1)(9I)/κ^(−1)(I) = 1/3; κ^(−1)(I=1) = 1
Debye-length evaluations: κ^(−1) = I^(−1/2) is the canonical I-dependent symbolic form (collapsing the prefactor); κ^(−1)(4I)/κ^(−1)(I) =…
OER overpotential volcano
Norskov 2007 OER volcano: 4-step proton-coupled electron-transfer; descriptor Delta-G_O - Delta-G_OH; theoretical-minimum 0.37 V…
HER Sabatier (Trasatti 1972)
Trasatti 1972 HER volcano vs M-H bond strength; Pt at top; modern Norskov 2005 DFT-Delta-G_H descriptor; MoS2 + transition-metal-phosphides…
CO2RR (Hori 1985)
Y Hori 1985 CO2RR mechanism on Cu: 12 products incl C1 CO/HCOOH/CH4 + C2 ethylene/ethanol; -1.0 V vs RHE; modern grain-boundary + alloy +…
N2RR (Yandulov-Schrock 2003)
Yandulov-Schrock 2003 Mo-amido N2 -> NH3 catalytic; modern electrocatalysis: Fe-Mo cluster Norskov 2018; ambient-T-P alternative to…
PEMFC ORR (Pt mass-activity)
DOE-2025 PEMFC target: 0.44 A/mg_Pt at 0.9 V; Stamenkovic Pt3Ni 2007 1.4 A/mg; Pt-skin Markovic Strain 2010 + single-atom Pt 2020 reduce…
Molecular catalyst (Tafel slope)
Tafel 1905 eta = a + b log j; b = (RT/(alpha n F)) ln 10 ~120 mV/dec single-electron; mechanism diagnosis from b-shift in molecular…
Tafel slope (1905)
J Tafel 1905 η = a + b log j; modern HER 30 mV/dec Volmer / 40 / 120 mV/dec mechanism-fingerprint identification.
Polarography (Heyrovský 1922)
J Heyrovský 1922 (Nobel 1959) dropping-mercury-electrode polarography; modern pulsed-voltammetry + ultramicroelectrode trace-metal.
Volcano plot (Bockris 1972)
J Bockris 1972 + Trasatti volcano-plot; modern Sabatier ΔG_H DFT-screening 2024 100k catalysts ML-discovery acceleration.
OER (Nørskov 2011)
J Nørskov 2011 OER 4-step mechanism + 1.6 V min-overpotential; modern IrO2 acidic + NiFeOOH alkaline + 2024 SACs Mn-Co.
CO2RR (Hori 1986)
Y Hori 1986 Cu C2+ products selectivity; modern oxide-derived-Cu Kanan 2012 + GDE 2020 + alkaline-MEA 2024 50% FE ethylene.
NRR (Suryanto 2019)
Suryanto-MacFarlane 2019 Li-mediated electrochemical-NRR ammonia; modern 2024 Du proton-shuttle 100mA/cm² 70% FE benchmark.