## 2 Results and Discussion
### Influence of Contact Time on the Adsorption Process: Kinetic and Mass Transfer Studies
This difference around 10% between q\({}_{\mathrm{e-theor}}\) and q\({}_{\mathrm{e-exp}}\) values could be due to possible adsorption of stabilizing agents, e.g., polyols and salts, on the support surface that may prevent the adsorption of some lipase molecules. The theoretical q\({}_{\mathrm{e}}\) value obtained by fitting the [PERSON] kinetic model was also very similar to the q\({}_{\mathrm{e-exp}}\) value (q\({}_{\mathrm{e-theor}}\) = 28.5 mg\(\cdot\)g\({}^{-1}\) of support--Figure 1C); however, this kinetic adsorption model assumes chemisorption interactions [34]. Therefore, PFO was the model that better describes  
Figure 1: Non-linear adsorption kinetic models of (**A**) pseudo-first-order; (**B**) pseudo-second-order; and (**C**) Elovich plots for the adsorption of lipase Eversa® Transform 2.0 on Psty-DVB beads at 25 \({}^{\circ}\)C, 200 rpm, and pH 5.0 (buffer sodium acetate at low ionic strength—5 mmol\(\cdot\)L\({}^{-1}\)), using an initial protein loading of 40 mg\(\cdot\)g\({}^{-1}\) of support. Figure 2: SEM images of Psty-DVB beads at (**A**) 30\(\times\) and (**B**) 20,000\(\times\) magnification, and (**C**) immobilized lipase at 20,000\(\times\) magnification. the experimentally obtained data. This adsorption kinetic model has been widely applied for describing physisorption (reversible) processes [35; 36; 37; 38]. In this case, the reversible adsorption of the lipase occurs via preferential hydrophobic interactions (mechanism of interfacial activation) [9; 12; 19]. These results suggest a possible regeneration and recyclability of PSty-DVB beads by desorbing lipase molecules from its surface using several chemicals such as detergents for next lipase adsorption processes and, thus, new application in biotransformation reactions [39; 40]. This can be considered a crucial factor for its use as a support for preparing heterogeneous biocatalysts to be used in industrial processes. The adsorption process involves several mass transfer steps such as: (i) external mass transfer over the stagnant film formed between the liquid phase and external surface of the adsorbent (convective mass transfer); (ii) diffusional mass transfer inside the pores of the support; and (iii) interaction of the adsorbate, in this case lipase molecules, with the internal support surface via several mechanisms--physisorption and/or chemisorption [41; 42; 43]. An illustrative representation of mass transfer of lipase molecules from the solution to the support microenvironment (internal and/or external surfaces) is shown in Figure 3.  
Therefore, mass transfer studies were conducted to determine the limiting-step of the adsorption process. Firstly, it was necessary to determine some physical properties of the support and bulk solution (enzyme solution) for the calculation of mass transfer parameters at a liquid-solid interface, as shown in Table 2. Exceptionally for PSty-DVB, some properties were obtained from the supplier (Supelco(r) Analytical Products); however, other relevant properties for conducting the present study such as apparent specific mass (\(\rho_{\rm p}\)), porosity (\(\epsilon_{\rm p}\)), tortuosity (\(\tau\)) and external surface area (\(\rm S_{EXT}\)) were determined based on previous reports [44; 45]. According to Table 2, the calculated external surface of this support (\(\rm S_{EXT}\)) was 25 m\({}^{2}\cdot\)g\({}^{-1}\) that corresponds to only 5% of its overall surface area (500 m\({}^{2}\cdot\)g\({}^{-1}\)). This shows that the lipase is preferentially adsorbed on the internal support surface and, thus, internal mass transfer effects plays an important role in the adsorption process [43; 45], and, thus, the catalytic performance of the heterogeneous biocatalyst (immobilized lipase). In fact, SEM analysis illustrated in Figure 2B shows clearly that this support has a large internal surface required for preparing heterogeneous biocatalysts, which confirms this result of calculated \(\rm S_{EXT}\).