3.2.2 Langmuir isotherm
The Langmuir isotherm is one of the adsorption models that describe physisorption of neutral molecules onto adsorption sites [38], [39]. Based on the Langmuir model assumption, adsorption takes place on homogeneously distributed adsorbent sites by monolayer adsorption.
The nonlinearized Langmuir equation is expressed as:
\begin{equation} q_{\text{Be}}=\frac{b{q_{\max}C}_{B_{e}}}{1+bC_{B_{e}}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ (3.2)\nonumber \\ \end{equation}
where \(q_{\text{Be}}\) (mg/g) is the amount of TCO adsorbed by AgNO3 at equilibrium, \(q_{\max}\) (mg/g) represents the maximum adsorption capacity, \(C_{B_{e}}\) (g/dm3) is the equilibrium concentration of the adsorbate in solution, and \(b\)(dm3/ mg) is a constant that amongst others is related to the heat of adsorption.
Moreover, Eq. 3.2 can be expressed as Eq. 3.3.
\begin{equation} \frac{C_{B_{e}}}{q_{\text{Be}}}=\frac{C_{B_{e}}}{q_{\max}}+\frac{1}{q_{\max}b}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3.3)\nonumber \\ \end{equation}
Plotting \(\frac{C_{B_{e}}}{q_{\text{Be}}}\) against \(C_{B_{e}}\), allows the calculation of the parameters, \(b\) and \(q_{\max}\).