## 2 Materials and Methods
### Cr (III) Adsorption at Equilibrium  
For the equilibrium studies of Cr (III) adsorption in WTOP, MTOP, and OPTmW, 0.5 g of bioadsorbent and 25 mL of Cr (III) solution were placed in a shaker (ZICHENG ZHWY-200D, Huizhou, China) with an orbital shaking of 200 rpm at 30 \({}^{\circ}\)C for 24 h of contact time. Varying concentrations were used, from 0 to 1000 ppm, at a pH of 0.91, 1.78 and 2.72 for 24 h. The pH selection of the test solutions was carried out based on the results of the predominant species diagram, obtained by simulation using the Hydra Medusa, San Jose, CA, USA)(r) program. During the development of the experiments, the pH of the solutions was monitored, and was kept stable by adding 0.05 M HCl.  
After this time, a sample aliquot was taken and centrifuged (HERMLE Z383K, Gosheim, Germany) at 4000 rpm for 10 min, subsequently, the absorbance was measured in the UV-Vis spectrum (Jenway 6705, Staffordshire, UK) [22; 23]. All experiments were carried out in triplicate and the data were reported as average values of the replications. The adsorption capacity (q) of Cr (III) was determined using Equation (1) [24]:  
\[\mathrm{q_{e}}=\frac{(C_{0}-C)V}{m} \tag{1}\]  
where C\({}_{0}\) and C correspond to the initial and equilibrium concentration (mg L\({}^{-1}\)), respectively, V is the volume of the solution (L) and m is the mass of the adsorbent (g). The most-used non-linear adsorption isotherm models are shown in Table 1. SigmaPlot 12(r) (Systat Software, San Jose, CA, USA) was used to obtain the parameters of each of the models.  
The DR isotherm is generally applied to express the adsorption mechanism with a Gaussian energy distribution on a heterogeneous surface. The mean energy of sorption, E, is calculated using Equation (2) [10,23]:  
\[\mathrm{E=}\frac{1}{\sqrt{2\mathrm{k_{DR}}}} \tag{2}\]  
The magnitude of E is useful for estimating the type of sorption reaction [10]. The Gibbs free energy (\(\mathrm{\Delta G}\)), which allows knowing the spontaneity of the adsorption process, was calculated using Equation (3); it should be noted that, in the present study, the only molecular weight taken into account was that of Cr (III), discarding the one of the water, since the dimensional analysis \(\mathrm{K_{L}}\) is in terms of \(\mathrm{L/mg}\) of Cr (III) [24,25]:  
\[\mathrm{\Delta G}\ =-\mathrm{RT\ln(55.5\ K_{L})} \tag{3}\]  
where \(\mathrm{K_{L}}\) is the [PERSON] model constant (\(\mathrm{L\ mol^{-1}}\)), R is the ideal gas constant (\(\mathrm{kJ\ mol^{-1}\ K^{-1}}\)), and T is the absolute temperature (K). The separation factor, \(\mathrm{R_{L}}\), which allowed predicting the affinity between adsorbent and adsorbate, was calculated using Equation (4) [24]:  
\[\mathrm{R_{L}=\frac{1}{1+K_{L}C_{0}}} \tag{4}\]  
From the proposed models, the one that fits the best needs to be chosen; for this, two different criteria were used: the first was the deterministic coefficient (\(\mathrm{R^{2}}\)), which must have a value close to 1 to consider a possible adjustment, and the second criterion is the normalized coefficient of determination, \(\mathrm{\Delta q}\), which was determined with the help of Equation (5) [22,26]: The choice of the model was made using the deterministic coefficient and the normalized standard deviation as selection criteria. The data processing was carried out with the Sigmaplot 12\({}^{\mathrm{\text{\textregistered}}}\) program (Systat Software, San Jose, CA, USA).  
\[\mathrm{\Delta q=}\sqrt{\frac{\left(\frac{q_{\mathrm{exp}}-q_{\mathrm{cal}}} {q_{\mathrm{exp}}}\right)^{2}}{\mathrm{N-1}}}\times 100 \tag{5}\]  
where N is the number of data points, \(q_{\mathrm{exp}}\) and q (mg g\({}^{-1}\)) are the values of the experimental adsorption capacity and that calculated using the model, respectively. Another parameter that was determined with the help of Equation (6) was the removal percentage, %\(\mathrm{R_{Cr}}\), [22,25]:  
\[\mathrm{\%_{o}R_{Cr}=\frac{(C_{0}-C)}{C_{0}}\times 100} \tag{6}\]