# Modified Dual-Site Langmuir Adsorption Equilibrium Models from A GCMC Molecular Simulation
## 1 Introduction  
To date, with the development of adsorption theory and the improvement of adsorption separation technology, the pressure swing adsorption (PSA) technology has been widely used in petrochemical industry, biomedicine, energy, and environmental protection. In the modern industrial separation process, the PSA technology is used to separate and purify gases due to its low energy consumption, low cost, convenience, reliability, and environmental benignity [1; 2; 3; 4]. For example, the PSA process improved by [PERSON] et al. was a feasible process to obtain an almost complete separation of biogas components, and a recovery of more than 99% of methane was achieved [5]. The PSA technology alsoplays an important role in the field of liquid separation. The PSA process could be used to isolate individual bioactive compounds from complex plant extracts and recover solvents, which open new ways for the fractionation of various valuable natural compounds [6; 7]. In addition, PSA mathematical models were developed to assist with the process optimization, reducing the need for excessive experimentation [8]. The basic elements of design and application of the PSA process are adsorption isotherms at different temperatures for adsorbents. To date, adsorption equilibrium models that have been used to describe adsorption isotherms have mainly been empirical or semiempirical formulas, which have some disadvantages. Among these models, the dual-site Langmuir (DSL) adsorption equilibrium model is the mostly used [9; 10; 11; 12]. The DSL model is based on the Langmuir model, which assumes that there are two kinds of adsorption sites with different adsorption energies, and the adsorption behavior at each site is consistent with three assumptions of the Langmuir model: (1) the adsorption of gas molecules on a solid surface forms a monolayer; (2) the adsorption energy on the surface of an adsorbent is uniform and remains unchanged; and (3) there is no interaction between adsorbate molecules [13; 14; 15]. These assumptions make the DSL model suitable for most systems and applicable under different temperatures and pressures. For example, the adsorption module in Aspenplus, the chemical engineering simulation software, uses the DSL model to reflect temperature changes to correlate various adsorption isotherms [16; 17; 18]. However, the applicability of the DSL model remains limited. The results it provides on the regression correlation effect of the adsorption equilibrium data on materials with a complex pore structure are insufficiently accurate, and it cannot well describe the adsorption equilibrium characteristics of low-temperature and high-temperature ranges and the high-pressure ranges of each temperature range. The main reason for this is the underlying assumption that the adsorption energy on the surface of an adsorbent is uniform and constant [19; 20]. In fact, in the actual adsorption process, the characteristic change in adsorption energy occurs via a complex process that is related to the adsorption pressure. Under most conditions, it cannot be simply assumed that it is constant. To date, few studies have focused on the characteristics of adsorption energy with changes in pressure, and, at present, the research on the characteristics of adsorption energy at the micro level is insufficiently deep. Adsorption energy is a microscopic physical quantity, which needs to be obtained by molecular simulation. Molecular simulation is widely used to calculate the adsorption properties of various new materials, such as metal-organic frameworks (MOFs). MOFs with extremely large pore volumes and high surface areas are highly potential candidates as adsorbents on gas adsorption. For example, [PERSON] et al. examined upper bounds on CO\({}_{2}\) swing capacity existing via molecular simulation of a large collection of MOFs, which enabled the formulation of material selection criteria that allow for the efficient identification of materials with high CO\({}_{2}\) swing capacity [21].