Adsorption onto a surface saturates: once every site holds one molecule, no concentration increase can load more. The Langmuir isotherm is that saturation curve, and this Langmuir Isotherm Calculator evaluates it from the monolayer capacity qmax, the affinity constant KL and the equilibrium concentration Ce — returning the adsorbed amount q and the fractional surface coverage θ, for water treatment, catalysis and materials work.
Worked Example: qmax = 50 mg/g, KL = 0.1 L/mg, Ce = 20 mg/L
The adsorbed amount is q = qmax × KL × Ce / (1 + KL × Ce) = 50 × 0.1 × 20 / (1 + 2) = 100/3 = 33.33 mg/g, and the coverage θ = q/qmax = 0.6667 — two-thirds of all sites occupied. The landmark hiding in the numbers: coverage reaches exactly one-half when KL × Ce = 1, i.e. at Ce = 1/KL = 10 mg/L here, the isotherm’s analogue of Km in enzyme kinetics (the same hyperbola, wearing surface-science clothes). Below that concentration the curve is nearly proportional — q ≈ qmax × KL × Ce, the Henry regime, where doubling concentration doubles loading. Far above it, the curve pins against qmax and concentration increases buy almost nothing: at Ce = 90 (KL × Ce = 9) the coverage is 0.90, so nine times the half-saturation concentration still leaves a tenth of the surface empty.
The Model and Its Four Assumptions
Langmuir’s 1918 derivation balances adsorption and desorption rates on a surface of identical sites: adsorption proportional to concentration and to the empty fraction (1 − θ), desorption proportional to θ. Equilibrium sets them equal and θ = KL·Ce/(1 + KL·Ce) falls out. The assumptions are printed in the derivation: (1) one molecule per site — a monolayer, no stacking; (2) all sites energetically identical; (3) adsorbed molecules do not interact with each other; (4) dynamic equilibrium between surface and solution. Real surfaces violate all four to some degree, which is why the Freundlich isotherm (heterogeneous sites, no saturation) and BET (multilayer) exist as alternatives. The practical reading: fit Langmuir when your data visibly plateaus and the surface is reasonably uniform (activated carbons, crystalline adsorbents); when q keeps climbing without a ceiling or the fit’s residuals curve systematically, the model’s assumptions — not your arithmetic — are what failed, and qmax from a forced fit is a fiction with units.
Fitting KL and qmax From Data
Both constants come from equilibrium experiments: dose a known adsorbent mass into solutions of varying initial concentration, measure the equilibrium Ce, compute q = (C0 − Ce) × V/m for each bottle, then fit the hyperbola. Linearized forms (Ce/q vs Ce, or 1/q vs 1/Ce) remain common in the literature, but like all reciprocal transforms they weight the noisiest points most; nonlinear least squares on the raw (Ce, q) pairs is the defensible default, with linearized plots kept as visual diagnostics. Report both constants with their units every time — KL carries inverse-concentration units (L/mg here), so a KL quoted without them is meaningless, and qmax inherits the units of your q definition (mg/g, mmol/g, mol/m² are all in circulation). Temperature belongs in the report too: KL is an equilibrium constant and drifts with T, typically downward for exothermic adsorption, which is why a summer isotherm and a winter isotherm for the same carbon are genuinely different curves.
Common Mistakes
Unit mismatches: Ce in mg/L with KL in L/µg silently wrecks the product KL × Ce — it must be dimensionless. Reading qmax as a guaranteed capacity rather than a model parameter fitted over a measured range; extrapolation beyond your highest Ce is speculation. Confusing the separation factor RL = 1/(1 + KL × C0) with coverage: RL describes isotherm favorability at the initial concentration (0 < RL < 1 favorable, RL → 0 irreversible), a different line of the same algebra. And applying Langmuir to gas-phase data quoted in partial pressure without switching Ce to pressure — the formula is identical, but the constant becomes an inverse-pressure affinity.
The Separation Factor, Worked
The favorability index RL = 1/(1 + KL × C0) prices an isotherm before any experiment: with this page’s KL = 0.1 L/mg, RL is 0.667 at an initial concentration of 5 mg/L, 0.333 at 20 mg/L and 0.091 at 100 mg/L — all inside the favorable band (0, 1), sliding toward irreversible as concentration rises, because a fixed surface matters more against a richer solution. Designers screen adsorbents on exactly this number: an RL near 1 warns of a nearly linear, weak isotherm where most capacity sits out of reach at working concentrations, while an RL near 0 promises a surface that grabs hard and saturates early. Same constants as the main calculation — RL simply reads KL at the inlet concentration instead of the equilibrium one.
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Frequently Asked Questions
What is the Langmuir isotherm equation?
q = qmax × KL × Ce / (1 + KL × Ce), where q is the adsorbed amount at equilibrium concentration Ce, qmax the monolayer capacity and KL the affinity constant. Coverage is θ = q/qmax.
What does KL tell me?
Affinity strength. Half the surface is covered at Ce = 1/KL, so a large KL means saturation arrives at low concentrations — strong binding. KL must always carry its inverse-concentration units.
When does Langmuir fail?
When its assumptions fail: multilayer adsorption (use BET), strongly heterogeneous sites (Freundlich often fits better), or interacting adsorbates. A plateau in the data is the visual cue that Langmuir is appropriate.
What is the separation factor RL?
RL = 1/(1 + KL × C0) at the initial concentration C0. Between 0 and 1 the isotherm is favorable; RL = 0 is irreversible, RL = 1 linear, RL above 1 unfavorable.
Is qmax the amount adsorbed at my concentration?
No — qmax is the ceiling at infinite concentration. The amount at your Ce is q, always below qmax; in the worked example q = 33.33 of the 50 mg/g ceiling, θ = 0.6667.
Why does the same formula appear in enzyme kinetics?
Michaelis-Menten and Langmuir are the same rectangular hyperbola: occupancy of independent sites at equilibrium, whether the sites are surface positions or enzyme active sites. Km plays the role of 1/KL.