Adsorption isotherms are mathematical models that describe how molecules in a gas or liquid phase interact with surfaces. Two of the most common isotherm models are the Langmuir and Freundlich isotherms, which relate to Type I monolayer chemisorption. The Langmuir model is based on four key assumptions:
Adsorption cannot exceed monolayer coverage.
All surface sites are equivalent.
Molecules adsorb only at vacant sites.
There are no interactions between adsorbed molecules.
Consider the dynamic equilibrium between the molecules in the gas phase and those on the surface. The rate of change of surface coverage, dθ/dt, due to adsorption is proportional to the rate of collisions with the surface and, therefore to the partial pressure p of A and the number of vacant sites. The rate of change of fractional coverage due to desorption is proportional to the number of adsorbed species already present, which is equal to the number of occupied sites, Nθ. At equilibrium, there is no net change in θ, the Langmuir isotherm equation relates the surface coverage to pressure.
At low pressures, the surface coverage fraction is directly proportional to the partial pressure of gas molecules. The fractional coverage increases with increasing pressure and approaches 1 at very high pressure, making the reaction rate independent of the pressure.
The Langmuir-Hinshelwood isotherms are used when two different gas-phase species are adsorbed onto a solid surface. A modified form of the Langmuir isotherm describes dissociative adsorption, in which a diatomic molecule dissociates and each atom occupies a separate site. In this case, the adsorption rate is proportional to the pressure and to the square of the number of vacant sites. The desorption rate is proportional to the square of the number of sites occupied. Setting the rates equal to each other gives the Langmuir isotherm. While both models share the same underlying Langmuir assumptions, the stoichiometry of site occupation alters the pressure dependence. Non-dissociative adsorption requires only one site per molecule and, as a result, fills the surface more rapidly. Dissociative adsorption progresses more slowly, as each molecule requires a pair of adjacent vacant sites.
Another way to model the coverage versus the concentration of the adsorbing species is the Freundlich isotherm. The Freundlich equation is often applied to the adsorption of solutes from liquid solutions onto solids. This equation is usually plotted in terms of its logarithm and appears as a straight line. The Freundlich isotherm allows for several kinds of adsorption sites on the solid, each having a different heat of adsorption. While not valid at high pressures, it's often more accurate than the Langmuir isotherm for intermediate pressures.
Type I adsorption isotherms describe chemisorption, relating the amount of adsorbed molecules to pressure at a constant temperature.
According to the Langmuir isotherm for non-dissociative adsorption, the adsorbate forms a reversible monolayer on identical surface sites that do not interact with each other.
Consider gas A adsorbing reversibly on surface M, with adsorption governed by the rate constant ka and desorption by kd. The adsorption rate depends on pressure and the number of vacant sites, while the desorption rate depends on the number of occupied sites.
At equilibrium, equating these rates yields the Langmuir equation, linking fractional coverage to pressure through the parameter α— the ratio of the adsorption and desorption rate constants.
The modified Langmuir isotherm explains dissociative adsorption, where diatomic molecules dissociate into atoms and occupy separate sites on the adsorbent surface. At equilibrium, equating the adsorption and desorption rates and rearranging the expression gives the modified Langmuir equation.
In both models, fractional coverage increases linearly at low pressures and approaches saturation only at very high pressures, where most surface sites are occupied.
繁體中文
