## 5 Kinetics  
The process is describing, by definition, the attachment of charged species (like the toxic metal ions) from a solution to a coexisting biosolid surface. Kinetics may be controlled by several independent processes that can act in series or in parallel. These sorption (a general term) processes fall in one of the following general categories: (i) bulk diffusion; (ii) external mass transfer (film diffusion); (iii) chemical reaction (chemisorption) and (iv) intraparticle diffusion. Kinetic analyses not only allow estimation of sorption rates but also lead to suitable rate expressions characteristic of possible reaction mechanisms [91]. Many studies engaged so far to examine sorption phenomena involved analysis of batch experiments where data were sampled at even time intervals over the entire course of the process. As a result, fast changing kinetic data characteristic of the phenomena just after the onset of sorption could not be accurately depicted in an adequately short time scale. The model selection criteria proposed by [PERSON] _et al._ concerning sorption of pollutants in aqueous systems were used herein, as a guideline [92]. For equilibrium and kinetic modeling, usually an overall mass balance of the sorbate across the biosorbent surface is initially written. The used example below deals with an investigation of the removal and depletion of cadmium and chromates from aqueous solutions by biosorption on _Aeromonas cavia_; this microorganism is often present in groundwater and general in aquatic environments. From the data, evidence was provided that the examined system was a complex process. More than one sorption models were often reported to describe correctly a case study [93]. From the chemical reaction category (chemisorption), the best fit for the data sets of this study is achieved by 2 nd order-type chemical reactions [94]. The solution of the standard 2 nd order reaction based on a constant stoichiometry of one metal ion per binding site is:  
\[C_{i}=\frac{C_{0}}{1-\left(\frac{C_{0}}{C_{\epsilon}}\right)\mathrm{exp} \left(-k_{i}C_{\epsilon}t\right)} \tag{12}\]  
where \(k_{1}\) is the reaction rate constant [\(\mathrm{L}\)\(\times\) (mg\({}^{-1}\) of metal) \(\times\) min\({}^{-1}\)]; \(C\) is the metal bulk concentration (mg\(\cdot\)L\({}^{-1}\)). Subscripts 0 and \(t\) denote conditions at the beginning and any other instant (time, \(t\)) of the process, respectively; and the subscript \(e\) denotes equilibrium conditions. This adsorption model has been very effective in describing the kinetics of adsorption of gases on solids. Nevertheless, when the data were plotted as a figure, it was shown that Equation (12) clearly fails to capture the steep concentration gradient of the early removal stage. This was a direct indication that adsorption on solids from a liquid phase is a different process than adsorption from a gas phase where traditionally the remaining bulk concentration dictates the kinetics. If the rate of sorption depends not on bulk concentration but on uptake by the sorbent this can be described by the so-called Ritchie 2 nd order equation according to which one metal ion occupies two binding sites [95]:  
\[q_{t}=q_{e}\left(1-\left(\frac{1}{1+k_{2}t}\right)\right) \tag{13}\]  
where \(q\) is the specific metal uptake (mg of metal per g of sorbent) and \(k_{2}\) is the reaction rate constant (min\({}^{-1}\)). When in the above treatment it is not necessarily \(q_{e}\) to dictate the sorbate uptake then a pseudo 2 nd order rate expression is more appropriate:  
\[\frac{t}{q_{t}}=\frac{1}{{\rm k}_{m}{q_{m}}^{2}}+\left(\frac{1}{q_{m}}\right)t \tag{14}\]  
where \(k_{m}\) is the reaction rate constant [g of sorbent \(\times\) (mg\({}^{-1}\) of metal) \(\times\) min\({}^{-1}\)] and \(q_{m}\) is a numerically determined parameter which under ideal 2 nd order rate control corresponds to \(q_{e}\).