Operations, Logistics & Engineering
Koros-Nowak & Madon-Boudart Test Formula: Catalytic Turnover Frequency & Transport Disguise
Perform the definitive experimental test for heat- and mass-transfer artifacts in heterogeneous catalysis (Koros & Nowak, 1967; Madon & Boudart, 1982) by computing log-log slope d(LN R_vol) / d(LN Site_Density) across diluted active-metal loadings (Slope = 1.0 intrinsic vs. 0.5 pore-diffused vs. 0.0 film-limited!).
Why Madon & Boudart’s log-log slope test works even when you don’t know pore diffusivity D_eff or thermal conductivity!
Calculated Weisz-Prater and Mears criteria require estimating tortuous effective pore diffusivity D_eff—which can be uncertain by an order of magnitude in bimodal or liquid-filled catalyst pores! Robert M. Koros & Edwin J. Nowak (1967, Chemical Engineering Science) and Rostam J. Madon & Michel Boudart (1982, I&EC Fundamentals) devised the gold-standard experimental diagnostic requiring ZERO transport-property estimates: synthesize 2 or 3 catalysts with identical support pore structure and metal nanoparticle size, varying ONLY volumetric active-site concentration L_v (mol_sites / cm^3_pellet)!
Plot LN(Volumetric Rate R_v) against LN(Active-Site Concentration L_v): in the true intrinsic kinetic regime, Turnover Frequency TOF = R_v / L_v is strictly invariant, yielding log-log slope = 1.00! Under strong internal Knudsen/bulk pore diffusion (where R_v scales with SQRT(k_v) ~ L_v^0.5), the log-log slope drops to 0.50! Under complete external stagnant-film control, R_v is independent of catalyst loading (slope = 0.00); while an exothermic intraparticle hot spot yields slope > 1.00!
| Madon-Boudart Log-Log Slope d(LN R_v) / d(LN L_v) | Turnover Frequency TOF Behavior | Catalytic Regime Diagnosis |
|---|---|---|
| Slope ≈ 1.00 (0.95 to 1.05) | Constant TOF Across Loadings! | True Intrinsic Kinetics Confirmed (Artifact-Free) |
| Slope ≈ 0.50 (0.45 to 0.65) | TOF Drops as L_v^-0.5 | Strong Internal Pore-Diffusion Disguise (Crush Pellets!) |
| Slope ≈ 0.00 (< 0.15) | TOF Drops as L_v^-1.0 | External Film Mass-Transfer Control (Raise Flow/RPM!) |
| Slope > 1.10 | TOF Rises with Loading! | Exothermic Intraparticle Hot-Spot Falsification! |
Madon-Boudart Log-Log Diagnostic Slope = SLOPE(LN(R_v), LN(Site_Density))
Computes the OLS slope across helper log-rate column D2:D4 and log-site-density column C2:C4.
=IF(OR(MIN(A2:A4) <= 0, MIN(B2:B4) <= 0, VARP(A2:A4) = 0), "Requires Distinct Positive Site Loadings & Rates", ROUND(SLOPE(D2:D4, C2:C4), 4))Koros-Nowak / Madon-Boudart Transport-Disguise Regime Classifier
Diagnoses Intrinsic Kinetics (0.95..1.05), Internal Pore Diffusion, External Film Control, or Exothermic Hot-Spot Artifacts.
=IF(OR(MIN(A2:A4) <= 0, MIN(B2:B4) <= 0, VARP(A2:A4) = 0), "Invalid Inputs", IF(AND(SLOPE(D2:D4, C2:C4) >= 0.95, SLOPE(D2:D4, C2:C4) <= 1.05), "Intrinsic Kinetics Confirmed (Slope ≈ 1.00, Invariant TOF)", IF(SLOPE(D2:D4, C2:C4) > 1.05, "Thermal Hot-Spot Artifact (Slope > 1.05)", IF(SLOPE(D2:D4, C2:C4) >= 0.35, "Internal Pore-Diffusion Disguise (Slope ≈ 0.5)", "Severe External Film Mass-Transfer Limit (Slope -> 0)"))))Try the example
Use the original values or change them to test your understanding.