Competitive Langmuir Isotherm Equation
The Competitive Langmuir Isotherm extends the classical Langmuir adsorption model to systems containing multiple adsorbates competing for the same surface sites. In the context of POSEIDON, this model helps explain how different metal ions interact and compete for binding sites within the alginate–peptide matrix.
Our filter beads, immobilized with engineered phytochelatins and metallothioneins, present a limited number of adsorption sites. The ions Hg²⁺, Cr³⁺, Fe²⁺, and Al³⁺ compete for these active sites based on their affinity constants (KMi). The competitive model provides a quantitative description of this process, ensuring predictive insight into how selectivity and saturation behave under mixed-ion environments.
Each metal ion competes for a finite number of active sites on the adsorbent, governed by its individual Langmuir constant (KM).
Explains how phytochelatin–metallothionein immobilized beads exhibit preferential binding under mixed-ion environments.
The equation below represents the multicomponent or competitive Langmuir isotherm:
Here, Mi denotes the equilibrium concentration of metal ion i adsorbed per gram of adsorbent, while the denominator reflects competitive occupation across all available sites.
Variable Descriptions
- Mi: Adsorbed quantity of metal ion i at equilibrium (mg/g)
- Mmaxi: Maximum adsorption capacity (mg/g)
- KMi: Langmuir adsorption constant for ion i (L/mg)
- Ci: Equilibrium concentration of metal ion i remaining in solution (mg/L)
- Σj=1n (KMj · Cj): Competitive summation term accounting for all metal ions present
This form captures the saturation behavior observed experimentally, where ions with higher KM values dominate site occupancy until saturation, reducing the adsorption potential of other competing ions.
| Metal Ion | K_M (L/mg) | M_max (mg/g) | R² | Adsorption Strength |
|---|---|---|---|---|
| Hg²⁺ | 0.215 | 26.3 | 0.991 | High |
| Cr³⁺ | 0.188 | 21.4 | 0.982 | High |
| Fe²⁺ | 0.153 | 18.1 | 0.978 | Moderate |
| Al³⁺ | 0.122 | 14.7 | 0.964 | Low |
Experimental fitting of this equation provided metal-specific constants that correlate strongly with the observed order of binding affinity.
"Model Implications"
- The equation accurately describes ion exchange behavior in mixed-metal systems.
- It validates the observed preference order Hg²⁺ > Cr³⁺ > Fe²⁺ > Al³⁺.
- Predictive modeling using this equation assists in optimizing peptide composition for enhanced selectivity.
The sequence Hg²⁺ > Cr³⁺ > Fe²⁺ > Al³⁺ validates the competitive adsorption predicted by the model.
Enables rational tuning of peptide composition for selectivity and regeneration cycles in sustainable filters.
Summary and Relevance
In conclusion, the Competitive Langmuir Model allows a mechanistic understanding of adsorption behavior in POSEIDON’s biopolymer system. By combining theoretical and experimental validation, it provides a reliable framework for scaling up the filter performance to real-world contamination scenarios. This balance between adsorption efficiency and cross-ion selectivity underpins the sustainability and adaptability of our solution.