Langmuir Adsorption Calculator

The Langmuir adsorption isotherm describes how molecules from a solution or gas attach to a solid surface as concentration changes. It is the simplest quantitative model of adsorption, and despite its simple assumptions it works remarkably well for many real systems: gas binding to activated carbon, dye adsorption from wastewater, enzyme-substrate binding (where it becomes Michaelis-Menten kinetics), and antibody-antigen binding (where it underlies ELISA and surface-plasmon-resonance measurements).

The formula:

θ = (K · c) / (1 + K · c)

For gas-phase adsorption, c is replaced by partial pressure P:

θ = (K · P) / (1 + K · P)

Where θ is the fractional surface coverage (0 to 1, dimensionless), K is the Langmuir adsorption constant (units of L/mol or 1/Pa depending on phase), and c (or P) is the bulk-phase concentration or partial pressure.

Three regimes:

• Low concentration (Kc « 1): θ ≈ Kc, linear (Henry’s law). Adsorption proportional to concentration.
• Transition (Kc ≈ 1): θ ≈ 0.5, half-coverage. Adding more adsorbate has diminishing returns.
• High concentration (Kc » 1): θ → 1, saturation. The surface is full; extra adsorbate just stays in solution.

The half-saturation point c = 1/K is sometimes called the dissociation constant by analogy with enzyme kinetics. Larger K means stronger binding (lower c needed for half-saturation).

Langmuir’s four assumptions:

Irving Langmuir derived this in 1916 from kinetic theory. The model assumes:

1. A fixed number of adsorption sites, all equivalent (no heterogeneity).
2. Monolayer coverage only, no stacking or multilayers.
3. No interactions between adsorbed molecules (no cooperativity).
4. Equilibrium between adsorbed molecules and the bulk phase.

When these hold, the math works out exactly. Langmuir won the 1932 Nobel Prize in Chemistry largely for this and related surface-chemistry work.

Worked example, catalyst surface:

A heterogeneous catalyst surface has K = 50 L/mol for a target reactant. The reaction mixture contains 0.02 mol/L of the reactant.

Kc = 50 × 0.02 = 1.0

θ = 1.0 / (1 + 1.0) = 0.50 (50% coverage)

The catalyst is exactly at its half-saturation point. To approach full coverage (e.g., θ = 0.9), the reactant concentration would need to increase to:

0.9 = Kc / (1 + Kc) ⟹ Kc = 9 ⟹ c = 0.18 mol/L

So a 9× increase in concentration only doubles the coverage from 0.5 to 0.9 — the saturation nonlinearity is sharp.

Worked example, activated carbon for water purification:

Activated carbon binding atrazine (a pesticide, K ≈ 8,000 L/mol because the binding is strong) in tap water at trace concentration c = 1 × 10⁻⁵ mol/L (about 2 ppb):

Kc = 8,000 × 10⁻⁵ = 0.08 θ = 0.08 / 1.08 = 0.074 (7.4% coverage)

So a fresh activated-carbon filter is far from saturated and removes pesticide aggressively. As θ rises toward 1, removal becomes less efficient and the filter needs replacement. Filter capacity is the total amount of adsorbate that can be bound before breakthrough; calculating it requires multiplying maximum coverage by surface area per gram of carbon.

The Langmuir constant K:

K depends on the chemistry of both the surface and the adsorbate. Typical values:

For comparison, antibody-antigen binding constants are typically 10⁸-10¹⁰ L/mol — at the very strong end. Trace contaminants in water can be effectively removed by adsorbents with K well above 10³ L/mol because even minuscule c values produce non-trivial θ.

Linearization for fitting:

Experimental data is fit to Langmuir by rearranging to:

c/θ = 1/K + c

A plot of c/θ vs c gives a straight line with slope 1, intercept 1/K. This was the classic way to determine K before nonlinear regression became routine; many published Langmuir K values were extracted this way.

Beyond Langmuir:

When the four assumptions fail, other isotherms apply:

• Freundlich isotherm θ = K·c^(1/n): for heterogeneous surfaces with a range of binding affinities.
• BET isotherm: for multilayer adsorption (the basis of the BET nitrogen-adsorption method for measuring catalyst surface area; it extends Langmuir to multiple layers).
• Sips / Toth / Redlich-Peterson: hybrid forms that interpolate between Langmuir and Freundlich behavior.
• Hill equation: generalizes Langmuir with a cooperativity parameter n, fitting allosteric binding (oxygen to hemoglobin, etc.).

Connection to Michaelis-Menten:

Enzyme kinetics is mathematically identical to Langmuir adsorption with the enzyme-substrate binding playing the role of surface adsorption. The Michaelis-Menten equation V = Vmax·c/(Km + c) is just θ·Vmax with K = 1/Km. This is why graduate students who learn one then learn the other in another field often have a “wait, this is the same equation” moment.

Practical uses across fields:

• Catalysis design: predicting how reactant concentration affects reaction rate when the rate-limiting step is surface adsorption.
• Drug discovery: receptor-ligand binding studies fit Langmuir-style curves to get dissociation constants.
• Water treatment: activated-carbon and ion-exchange capacity calculations.
• Gas storage: characterizing porous materials like MOFs and zeolites for hydrogen, CO₂, or methane storage.
• Chromatography: retention times in HPLC follow Langmuir for many stationary phases.
• Sensor design: biosensors and chemical sensors use Langmuir to calibrate concentration measurements from surface signal.