​​Abstract​​

​​1. Introduction​​

​​2. Recent Advances in Adsorption Isotherm Theory​​

​​2.1. Refinement of Classical Models​​

• ​​Modified Langmuir Model​​: Incorporates factors like lateral interactions between adsorbed molecules, surface heterogeneity, and adsorbate size effects.
• ​​Freundlich Model Enhancements​​: Adjustments allow for better description of heterogeneous and multi-layer adsorption on nano-surfaces with non-uniform energy distribution.
• ​​Sips and Toth Models​​: Introduced to address heterogeneous surface adsorption with a more flexible mathematical form, combining features of both Langmuir and Freundlich models.

​​2.2. Emergence of Advanced and Hybrid Models​​

• ​​Redlich–Peterson Model​​: A hybrid of Langmuir and Freundlich, offering a more generalized form with an empirical parameter that improves fitting versatility.
• ​​Dubinin–Radushkevich (DR) and Dubinin–Astakhov Models​​: Useful for describing microporous adsorption, especially in nanoporous materials like activated carbons and MOFs.
• ​​Temkin Isotherm​​: Accounts for adsorbate-adsorbent interactions and heat of adsorption variation across the surface, valuable for nano-adsorbents with temperature-sensitive binding.
• ​​Quantum and Statistical Mechanical Models​​: Emerging computational models based on density functional theory (DFT) and molecular simulations provide molecular-level insights into adsorption mechanisms on nanostructured surfaces.

​​2.3. Multicomponent and Competitive Adsorption Models​​

• ​​Extended Langmuir and Ideal Adsorbed Solution Theory (IAST)​​
• ​​Competitive Freundlich Models​​
• ​​Statistical Thermodynamic Approaches​​

​​3. Practical Applications in Nanomaterials​​

​​3.1. Metal–Organic Frameworks (MOFs)​​

• Quantifying high-capacity adsorption of CO₂, CH₄, H₂, and volatile organic compounds (VOCs)
• Understanding the effect of pore geometry and functional groups on adsorption affinity
• Designing MOFs for selective gas separation and storage

​​3.2. Carbon-Based Nanomaterials (e.g., Graphene Oxide, Carbon Nanotubes)​​

• Heavy metal ion adsorption (e.g., Pb²⁺, Cd²⁺, As(V))
• Organic pollutant removal (dyes, pharmaceuticals, pesticides)
• Insights into π–π stacking, hydrophobic interactions, and electrostatic adsorption mechanisms

​​3.3. Nanostructured Metal Oxides (e.g., TiO₂, Fe₃O₄, ZnO, MnO₂)​​

• Adsorption of arsenic, fluoride, phosphates, and other inorganic contaminants
• Surface charge-dependent adsorption influenced by pH and oxidation state
• Synergistic effects in composite nanomaterials (e.g., GO-MOF or CNT-metal oxide hybrids)

​​3.4. Polymer-Based and Composite Nanomaterials​​

• Optimizing adsorption capacity and selectivity
• Understanding polymer–nanoparticle interactions and diffusion-controlled processes
• Designing smart adsorbents responsive to external stimuli (pH, temperature, light)

​​4. Future Perspectives​​

• ​​Machine Learning Integration​​: Using AI to predict adsorption behavior and optimize isotherm model parameters.
• ​​High-Throughput Experimental Data Analysis​​: Coupling automated adsorption experiments with advanced modeling for rapid material screening.
• ​​Molecular Modeling and Simulation​​: Combining DFT, Monte Carlo, and molecular dynamics (MD) simulations with empirical isotherm models for comprehensive understanding.
• ​​Sustainable and Green Adsorbents​​: Designing low-cost, biodegradable nanomaterials with predictable adsorption performance using refined isotherm frameworks.

​​5. Conclusion​​