Operations, Logistics & Engineering
Sips (Langmuir-Freundlich) & Redlich-Peterson 3-Parameter Adsorption Isotherm Formula
Bridge heterogeneous Freundlich low-concentration behavior and finite Langmuir monolayer saturation using Robert Sips’s (1948) isotherm q_e = q_max*(K_S*C_e)^n / (1 + (K_S*C_e)^n) and the Redlich-Peterson (1959) hybrid equation.
Why 3-parameter Sips and Redlich-Peterson isotherms fix Freundlich’s infinite-loading flaw!
In activated-carbon water treatment, PFAS/heavy-metal sorption, and gas separation, the 2-parameter Freundlich isotherm q_e = K_F * C_e^n captures surface energetic heterogeneity accurately at trace concentrations—but unphysically diverges to infinity as equilibrium concentration C_e rises! Conversely, Langmuir saturates cleanly at q_max but assumes every surface site is identical!
Robert Sips (1948, Journal of Chemical Physics) unified both limits in the Langmuir-Freundlich (Sips) equation q_e = q_max * (K_S * C_e)^n_S / (1 + (K_S * C_e)^n_S): at low C_e it reduces to Freundlich, at high C_e it plateaus at monolayer capacity q_max, and when heterogeneity index n_S = 1 it collapses identically to Langmuir! Complementarily, Otto Redlich & D. L. Peterson (1959) formulated q_e = (K_RP * C_e) / (1 + a_RP * C_e^g) (0 < g <= 1), which obeys Henry’s linear law K_RP * C_e at infinite dilution (unlike Sips for n_S != 1) while transitioning to Freundlich at high loading!
| 3-Parameter Hybrid Isotherm | Exact Algebraic Equation | Limiting Physical Regimes |
|---|---|---|
| Sips (Langmuir-Freundlich, 1948) | q_max * (K_S * C_e)^n_S / (1 + (K_S * C_e)^n_S) | Low C_e -> Freundlich; High C_e -> q_max; n_S=1 -> Langmuir |
| Redlich-Peterson (1959) | (K_RP * C_e) / (1 + a_RP * C_e^g) | Low C_e -> Henry’s Law; g=1 -> Langmuir; g=0 -> Linear |
Sips (Langmuir-Freundlich) Equilibrium Uptake q_e (mg/g)
Evaluates q_e (mg/g) from monolayer capacity q_max (B2), Sips affinity K_S (B3, L/mg), heterogeneity exponent n_S (B4), and equilibrium concentration C_e (B5, mg/L).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 < 0), "Invalid Sips Isotherm Inputs", IF(B5 = 0, 0, ROUND(B2 * ((B3 * B5) ^ B4) / (1 + ((B3 * B5) ^ B4)), 4)))Redlich-Peterson Hybrid Uptake q_e,RP = (K_RP * C_e) / (1 + a_RP * C_e^g)
Evaluates Redlich-Peterson equilibrium sorption from K_RP (B6), a_RP (B7), exponent g in (0, 1] (B8), and C_e (B5).
=IF(OR(B5 < 0, B6 <= 0, B7 < 0, B8 <= 0, B8 > 1), "Invalid Redlich-Peterson Inputs", IF(B5 = 0, 0, ROUND((B6 * B5) / (1 + B7 * (B5 ^ B8)), 4)))Try the example
Use the original values or change them to test your understanding.