Adsorption Isotherms — Core Principles

Core Principles

Adsorption isotherms are graphical representations showing the relationship between the amount of adsorbate adsorbed per unit mass of adsorbent ($x/m$) and the equilibrium pressure (for gases) or concentration (for solutions) at a constant temperature. They are crucial for understanding the extent and mechanism of adsorption. The two main types are the Freundlich and Langmuir isotherms.

The Freundlich isotherm is an empirical model, expressed as $x/m = kP^{1/n}$ (for gases) or $x/m = kC^{1/n}$ (for solutions). It suggests multilayer adsorption and is valid over an intermediate range of pressures/concentrations but fails at very high pressures. Its linearized form is $log(x/m) = log(k) + (1/n)log(P)$.

The Langmuir isotherm is a theoretical model based on assumptions of monolayer adsorption on a homogeneous surface with no interaction between adsorbed molecules. Its equation is $x/m = \frac{(x/m)_{max} bP}{1 + bP}$.

It predicts a saturation limit and is often more accurate for chemisorption. Its linearized form is $rac{1}{x/m} = \frac{1}{(x/m)_{max} b} \frac{1}{P} + \frac{1}{(x/m)_{max}}$. Both isotherms provide constants that characterize the adsorption process, aiding in practical applications like catalysis and purification.

Important Differences

vs Langmuir Adsorption Isotherm

Aspect	This Topic	Langmuir Adsorption Isotherm
Nature of Model	Empirical (based on experimental observations)	Theoretical (based on specific assumptions)
Adsorption Type	Can describe multilayer adsorption	Assumes monolayer adsorption
Surface Homogeneity	Applicable to heterogeneous surfaces	Assumes a homogeneous surface
Intermolecular Interaction	Does not explicitly consider interactions	Assumes no interaction between adsorbed molecules
High Pressure Behavior	Fails at very high pressures (predicts indefinite increase)	Predicts saturation (monolayer capacity) at high pressures
Constants	$k$ and $n$ (empirical constants)	$(x/m)_{max}$ (monolayer capacity) and $b$ (affinity constant)
Mechanism Insight	Provides less insight into the mechanism	Provides more mechanistic insight (site-specific adsorption)

The Freundlich and Langmuir isotherms are two distinct models used to describe adsorption phenomena. Freundlich is an empirical model, derived from experimental data, and can account for multilayer adsorption on heterogeneous surfaces, but it fails to predict a saturation limit at very high pressures. In contrast, Langmuir is a theoretical model based on specific assumptions like monolayer adsorption on a homogeneous surface with no intermolecular interactions, and it accurately predicts saturation. While Freundlich is simpler and often fits data over a limited range, Langmuir offers deeper mechanistic insights into the adsorption process.