Multi-Component Adsorption Isotherm Models

Key Insight

Multi-component systems are very common in industrial applications, involving complex mixtures found in fermentation broths or heterogeneous protein products with various isoforms. Predicting adsorption behavior in these systems from single-component data is often inaccurate, and experimental determination can be time-consuming and complicated. To address this, a pragmatic simplification groups compounds with lower affinity than the target product into one category and those with higher affinity into another, effectively reducing the system to a more manageable three-component description.

The multi-component Langmuir isotherm extends the single-component model, describing the competitive adsorption of multiple species for available sites. The general form is q_i = q_m K_i C_i / (1 + sum(K_j C_j)), where q_m, the maximum binding capacity, is assumed to be the same for all components for thermodynamic consistency. However, this model is sometimes applied with different q_m values for individual components, an assumption prone to error if actual binding capacities differ significantly. A key prediction of this model is a constant selectivity between components, independent of protein concentration, which can be used as a practical test for its applicability.

More advanced multi-component models account for specific protein characteristics. For instance, a modified multi-component isotherm model, developed by Gu et al., considers differences in binding capacity for individual proteins resulting from their varying sizes and associated size exclusion effects. This model incorporates 'discount factors' and allows for different concentrations of accessible surface-bound ligands for each component. Consequently, its selectivity is not constant but varies with solution composition, which can lead to phenomena like isotherm crossover and selectivity reversal. An example includes lysozyme (15 kilodaltons) displacing gamma-globulin (150 kilodaltons) on a cation exchanger due to its higher single-component binding capacity and gamma-globulin's size exclusion. The Steric Mass Action (SMA) model has also been extended for multi-component ion-exchange systems, predicting that when effective charges (z) differ among proteins, selectivity becomes dependent on both composition and salt concentration, with the selectivity for a protein with a higher effective charge decreasing as salt concentration increases.