## 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

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