Table 2. Isothermal Models Used for Modeling of CO₂ Adsorption on Shale^{57,60,75,76,80,92,93}.

isotherm model	description	linear	nonlinear	limitations
Langmuir	Describes adsorption of gas molecules on solid surfaces. It assumes that adsorption takes place at homogeneous sites.			Presupposes monolayer adsorption with no interactions between adsorbed molecules on a homogeneous surface. It is not applicable to heterogeneous surfaces or multilayer adsorption. At high pressures, where multilayer adsorption becomes substantial, the model breaks down.
Freundlich	Describes solutes interacting with a solid surface, particularly the adsorption process. It is one of the oldest and simplest models to explain the adsorption characteristics. The model assumes that the surface is heterogeneous and heat of adsorption is distributed unevenly.			A theoretically unfounded empirical model that suggests it is unable to forecast the adsorption mechanism. It is less useful at larger concentrations since it does not explain saturation, just suggesting adsorption on heterogeneous surfaces.
D-R	Utilized in the investigation of vapor and gas adsorption on uneven surfaces. In contrast to more basic models such as Freundlich or Langmuir isotherms, the D-R model takes into consideration both the porosity and heterogeneity of the adsorbent material.			Assumes a heterogeneous surface with a Gaussian energy distribution, which is more appropriate for subcritical vapor adsorption. Particularly at higher pressures, it might not be able to forecast the adsorption process with complete accuracy for all types of adsorbates.
D-A	Describes an adsorption process where a substance adheres to the surface of a solid. The D-A model is especially relevant for microporous materials, where it can accurately predict the adsorption behavior within the pores.			Expansion of the D-R isotherm that adds a new parameter in an effort to overcome some of its shortcomings. Still, it makes an assumption about a certain kind of energy distribution, thus it might not apply to other adsorbent–adsorbate systems, especially those with intricate interactions.
Sip	Hybrid of Freundlich and Langmuir models that is intended to accurately represent adsorption processes across a broad concentration range. It assumes that the surface is energetically heterogeneous and the adsorption process is localized (i.e., there is no adsorbate transmigration in the plane of the surface).			Hybrid of the Freundlich and Langmuir isotherms that is intended to forecast heterogeneous adsorption at both high and low concentrations. For some systems, its complexity and requirement for several parameters may make it less useful.
Toth	An extension of the Langmuir that assumes a surface with identical sites, and each can hold only one adsorbate molecule. The Toth model, however, modifies this by incorporating a distribution function for the energy of adsorption sites, which accounts for the heterogeneity of real surfaces.			Specially designed to provide a more flexible fit than the Langmuir model for adsorption on heterogeneous surfaces. Its presumption that the adsorption energy will drop smoothly and continuously might not apply to all heterogeneous surfaces, though.
BET	Extends the Langmuir theory to multilayer adsorption with the assumption that the adsorbate molecules form a multilayer on the adsorbent surface. In this first layer of adsorbate it behaves similarly to the Langmuir model, adhering directly to the solid surface. Additional adsorbate layers can then form on top of the first. This model is applied where the adsorption process is physisorption, involving relatively weak van der Waals forces.			Presumes multilayer adsorption, mostly useful in a narrow range of relative pressures. At very low or high relative pressures, where monolayer or capillary condensation events predominate, respectively, it is unable to anticipate adsorption with any degree of accuracy.
OK	Based on a well-developed lattice theory, the Ono-Kondo model describes adsorption behavior in both monolayer and multilayer formats. It describes adsorption behavior based on the physical properties and accessible characterization of the adsorbent and makes predictions under a variety of circumstances			Less useful for heterogeneous surfaces and complicated interaction situations due to its assumptions about homogeneous adsorption sites and simpler interactions. Its practicality in real-world scenarios is further impacted by the difficulty of parameter estimation.