Langmuirisotherm

The Langmuir isotherm is a model of adsorption that describes monolayer adsorption on a homogeneous surface with a finite number of identical sites. Proposed by Irving Langmuir in 1918, it assumes that all adsorption sites are equivalent, that adsorbed molecules do not interact laterally, and that the adsorption energy is constant and independent of surface coverage. These assumptions lead to a simple relationship between the amount adsorbed and the driving force of adsorption.

For gas-phase adsorption, let p be the pressure. The fraction θ of occupied sites is θ = (K p)/(1 + K p), where K is the Langmuir constant reflecting the affinity between adsorbate and surface. The amount adsorbed q is q = q_max θ, with q_max the maximum monolayer capacity. This gives the standard form q = q_max K p /(1 + K p). A common linear form is p/q = (1/(K q_max)) + (p/q_max), which facilitates data fitting. For adsorption from solutions, the equation has a similar form with concentration C replacing p: q = q_max (K C)/(1 + K C).

Applications and limitations: the Langmuir isotherm is widely used to fit adsorption data to estimate q_max and the affinity constant K, and to compare adsorbents. It is most appropriate for relatively homogeneous surfaces and conditions where monolayer adsorption dominates. Deviations arise from surface heterogeneity, lateral interactions between adsorbates, multilayer adsorption, or changes in adsorption energy with coverage. Temperature dependence of K provides thermodynamic insights, often explored via van’t Hoff analysis.