Langmuir Adsorption Isotherm
Langmuir isotherm: theta = (K*c)/(1 + K*c).
Model the fraction of surface covered by adsorbed molecules at equilibrium concentration.
The Formula
For gas-phase adsorption: θ = (K · P) / (1 + K · P)
The Langmuir adsorption isotherm describes how the fraction of a surface covered by adsorbed molecules depends on the concentration (or partial pressure) of those molecules in the surrounding medium, at constant temperature. Developed by American chemist and physicist Irving Langmuir in 1916, it was one of the first quantitative models of surface adsorption and earned Langmuir the Nobel Prize in Chemistry in 1932. The model assumes that adsorption occurs on a fixed number of equivalent sites, each capable of holding only one molecule, and that there is no interaction between adsorbed molecules.
In the formula, θ (theta) is the fractional surface coverage — ranging from 0 (empty surface) to 1 (fully saturated monolayer). K is the Langmuir adsorption constant, which reflects the affinity of the adsorbate for the surface: a larger K means the surface binds molecules more strongly. The variable c is the concentration of the adsorbate in solution (or P is its partial pressure in the gas phase).
At low concentrations, when K·c is much less than 1, the denominator approaches 1 and the equation simplifies to θ ≈ K·c — a linear relationship known as Henry's law adsorption. At high concentrations, when K·c is much greater than 1, θ approaches 1 — the surface becomes saturated and adding more adsorbate has no further effect. This saturation behavior is a key feature that distinguishes the Langmuir isotherm from simpler linear models.
The Langmuir model is widely used in heterogeneous catalysis, drug delivery, water treatment, and the characterization of porous materials. Its linearized form — c/θ = 1/K + c — allows the adsorption constant K and maximum coverage to be determined experimentally from a plot of c/θ versus c.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| θ | Fractional surface coverage — fraction of sites occupied | dimensionless (0 to 1) |
| K | Langmuir adsorption constant — surface affinity for the adsorbate | L/mol or 1/Pa |
| c | Concentration of adsorbate in solution | mol/L |
| P | Partial pressure of adsorbate gas (gas-phase form) | Pa or atm |
Example 1
A catalyst surface has a Langmuir constant K = 50 L/mol. What fraction of the surface is covered when the adsorbate concentration is 0.02 mol/L?
K · c = 50 × 0.02 = 1.0
θ = 1.0 / (1 + 1.0) = 1.0 / 2.0 = 0.50
θ = 0.50 — exactly half the surface sites are occupied
Example 2
For the same catalyst (K = 50 L/mol), what coverage is achieved at a much higher concentration of 0.2 mol/L?
K · c = 50 × 0.2 = 10.0
θ = 10.0 / (1 + 10.0) = 10.0 / 11.0 = 0.909
θ = 0.91 — the surface is nearly saturated, confirming the monolayer limit
When to Use It
The Langmuir isotherm is appropriate when:
- Modeling gas adsorption onto solid catalyst surfaces in heterogeneous catalysis
- Analyzing contaminant removal from water using activated carbon or ion exchange resins
- Characterizing surface area and porosity of porous materials such as zeolites
- Describing protein or drug molecule binding to cell surface receptors in biochemistry
- Designing biosensors where analyte molecules adsorb onto a functionalized surface
- Interpreting BET surface area measurements, which extend the Langmuir model to multilayers