# 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 average absolute 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.

##### Average Absolute Relative Deviation

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

##### Determination coefficient

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

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