Operations, Logistics & Engineering
Toth (1971) 3-Parameter Adsorption Isotherm Formula: Henry’s Law & Monolayer Limit
Model sub-monolayer gas adsorption (CO2 capture, CH4 shale/coalbed storage, VOC zeolites) using József Tóth’s (1971) thermodynamically consistent 3-parameter isotherm q = q_max * b * P / (1 + (b * P)^t)^(1/t).
Why József Tóth’s (1971) isotherm beats Sips at ultra-low pressure: it satisfies BOTH Henry’s Law AND finite saturation!
In gas-separation PSA units (CO2/N2 on zeolites and MOFs) and coalbed/shale methane storage, thermodynamic rigor requires TWO simultaneous boundary conditions: (1) as pressure P -> 0, uptake dq/dP must approach a finite nonzero Henry’s Law constant K_H = q_max * b (the Sips isotherm violates this with an infinite slope whenever n_S < 1!), AND (2) as P -> infinity, uptake must plateau at a finite saturation capacity q_max (Freundlich and Redlich-Peterson for g < 1 violate this!)!
Hungarian physical chemist József Tóth (1971, Acta Chimica Academiae Scientiarum Hungaricae) solved both requirements simultaneously with q(P) = (q_max * b * P) / (1 + (b * P)^t)^(1 / t), where b (bar^-1) is the affinity constant and 0 < t <= 1 characterizes asymmetrically widened adsorption-energy heterogeneity (when t = 1, Tóth reduces identically to Langmuir q_max * b * P / (1 + b * P))! At q_max = 6.50 mmol/g, b = 0.80 bar^-1, t = 0.60, and P = 5.0 bar, q = 4.4446 mmol/g with exact zero-pressure Henry slope K_H = q_max * b = 5.20 mmol/(g*bar)!
| Heterogeneity Exponent t (q_max = 6.5 mmol/g, b = 0.8 bar^-1, P = 5 bar) | Tóth Uptake q(P) (mmol/g) | Zero-Pressure Henry Constant K_H = q_max * b |
|---|---|---|
| t = 1.00 (Homogeneous Langmuir Surface) | 5.2000 mmol/g (80.0% Saturation) | 5.2000 mmol/(g*bar) |
| t = 0.60 (Moderately Heterogeneous Zeolite/MOF) | 4.4446 mmol/g (68.4% Saturation) | 5.2000 mmol/(g*bar) — Finite Henry Slope Preserved! |
| t = 0.35 (Strongly Heterogeneous Activated Carbon) | 3.2982 mmol/g | 5.2000 mmol/(g*bar) |
Tóth (1971) Equilibrium Gas Uptake q(P) = (q_max * b * P) / (1 + (b * P)^t)^(1 / t)
Evaluates equilibrium loading q (mmol/g) from monolayer capacity q_max (B2), affinity b (B3, bar^-1), Tóth heterogeneity exponent t in (0, 1] (B4), and gas pressure P >= 0 (B5, bar).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B4 > 1, B5 < 0), "Invalid Toth Isotherm Parameters (Requires 0 < t <= 1)", IF(B5 = 0, 0, ROUND((B2 * B3 * B5) / ((1 + ((B3 * B5) ^ B4)) ^ (1 / B4)), 4)))Thermodynamic Henry’s Law Slope K_H = lim_{P->0} (q / P) = q_max * b (mmol/(g*bar))
Computes the exact low-pressure Henry’s Law partition slope required for PSA blowdown design.
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Capacity or Affinity", ROUND(B2 * B3, 4))Try the example
Use the original values or change them to test your understanding.