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Fit adsorption isotherms to data

Notes:

Isotherm models

The isotherm is the analytical relation between the solute concentration in the solid, or solid load, \(q\), and its concentration in the fluid phase, \(C\), at constant temperature.

Linear

\[q = H C\]

where \(H\) is the linear equilibrium constant.

Langmuir

\[q = \frac{a C}{1 + b C}\]

where \(a = Q b\), \(Q\) is the adsorbent capacity and \(b\) is the adsorption equilibrium constant.

Freundlich

\[q = k C^{1/n}\]

where \(k\) and \(n\) are Freundlich isotherm constants for a given adsorbate and adsorbent at a given temperature.

Linear-Langmuir

\[q = m C + \frac{a C}{1 + b C}\]

which adds a linear contribution, \(m\), term to the Langmuir model.

Bi-Langmuir

\[q = \frac{a_1 C}{1 + b_1 C} + \frac{a_2 C}{1 + b_2 C}\]

where two Langmuir contributions are considered representing two adsorption sites.

Langmuir-Freundlich

\[q = \frac{a C^n}{1 + b C^n}\]

a combination of the Langmuir and Freundlich isotherms.

Fitting performance

The absolute average relative deviation (\( AARD \)), determination coefficient (\(R^2\)), and adjusted determination (\(R_{adj}^2\)) were used to evaluated the quality of fit of the isotherm models. These are defined below, where \( y_i^{exp} \) is the experimental value, \(y_i^{calc}\) is the calculated or expected value, \( \bar{y}_i \) is the mean of the experimental values, \( NDP \) is the number of data points, and \( p \) is the number of model parameters.

Absolute Average Relative Deviation

\[AARD = \frac{100}{NDP} \sum_{i=1}^{NDP} \frac{|y_i^{calc}-y_i^{exp}|}{y_i^{exp}}\]

Determination coefficient

\[R^2 = 1 - \frac{ \sum_{i} (y_i^{exp}-y_i^{calc})^2 }{ \sum_{i} (y_i^{exp}-\bar{y}_i)^2 }\]

Adjusted determination coefficient

\[R_{adj}^2 = 1 - (1-R^2) \frac{NDP-1}{NDP-p-1}\]

References:

J.P.S. Aniceto, C.M. Silva. Preparative Chromatography: Batch and Continuous, in: Analytical Separation Science, Vol 4 (2015).

A. Seidel-Morgenstern, M. Schulte, A. Epping. Fundamentals and General Terminology, in: Preparative Chromatography, 2nd Edition (2012).