# Modeling of Hexavalent Chromium Removal with Hydrophobically Modified Cellulose Nanofibers

^{1}

^{2}

^{*}

## Abstract

**:**

^{−1}CNF), initial pH of the wastewater (3–9), initial chromium concentration (0.10–50 mg·L

^{−1}), and adsorbent dosage (250–1000 mg CNF·L

^{−1}). Furthermore, the corresponding adsorption mechanism was identified. Complete adsorption of hexavalent chromium was achieved with CNF hydrophobized with 1.5 mmol MTMS·g

^{−1}CNF with the faster adsorption kinetic, which proved the initial hypothesis that hydrophobic CNF improves the adsorption capacity of hydrophilic CNF. The optimal adsorption conditions were pH 3 and the adsorbent dosage was over 500 mg·L

^{−1}. The maximum removal was found for the initial concentrations of hexavalent chromium below 1 mg·L

^{−1}and a maximum adsorption capacity of 70.38 mg·g

^{−1}was achieved. The kinetic study revealed that pseudo-second order kinetics was the best fitting model at a low concentration while the intraparticle diffusion model fit better for higher concentrations, describing a multi-step mechanism of hexavalent chromium onto the adsorbent surface. The Freundlich isotherm was the best adjustment model.

## 1. Introduction

^{−1}[7].

^{2+}, Cu

^{2+}, and Zn

^{2+}(more than 95% removal at the optimum conditions) [14]. Furthermore, nanomaterials can be used as adsorbents such as the polyacrylonitrile nanofiber membrane modified with bovine serum albumin used to remove Ca

^{2+}from process streams, achieving removal efficiencies of about 62% [15]. Other membrane structure materials used as adsorbents are, for example, cellulose acetate-based membranes with glass nanoparticles for CO

_{2}separation [16]. The surfaces of the nanomaterial adsorbents are usually functionalized or chemically modified to improve Cr(VI) adsorption such as the attachment of active binding sites [17].

^{2+}, Cd

^{2+}, Ni

^{2+}, Pb

^{2+}, Zn

^{2+}, Fe

^{3+}, and Ag

^{+}adsorption, reached with none or low modifications. Compared to these cationic heavy metals, the adsorption capacity of CNC for anionic As(V) was reduced by an order of magnitude. The same trend was also observed by Liu et al. [26], who applied both untreated CNF and 2,2,6,6-tetramethylpiperidin-1-yl-oxyl (TEMPO)-mediated oxidized CNF to a list of 15 heavy metals, with hexavalent chromium the only anionic species. The authors found a maximum adsorption capacity of 87.5 mg·g

^{−1}using TEMPO-oxidized CNF-PAN membranes to adsorb Cr(VI), which was lower than the adsorption capacity reached when cationic Pb(II) was adsorbed.

## 2. Materials and Methods

#### 2.1. Materials

^{−1}as Cr(VI)) was used as a calibration standard for the spectrophotometric method. Analytical reagents for hexavalent chromium determination were purchased from Macherey Nagel (Dueren, Germany) following Standard Method 3500 Cr B. Poly(diallyldimethylammonium chloride) (PDADMAC) and polyethylenesulfonate (PesNA) solutions with a concentration of 0.00025 N were used as standard titration reagents during cationic and anionic demand determination.

#### 2.2. CNF Hydrogel Synthesis

#### 2.3. CNF Characterization

#### 2.4. Experimental Procedure: Batch Adsorption of Hexavalent Chromium Solution

^{−1}between 0.02 and 0.50 mg·L

^{−1}and the measurement was performed at the peak wavelength λ = 540 nm, as indicated in Standard Method 3500 Cr B [40]. Synthetic hexavalent chromium solution samples were diluted when the concentration exceeded the top of the calibration curve.

^{−1}. Hexavalent chromium concentration in the kinetic and isotherm studies varied from 0.1 to 50 mg·L

^{−1}. Hydrophobized CNF hydrogels were tested in a dosage range 250–1000 mg CNF·L

^{−1}. The dosage of the hydrophobic reagent MTMS during CNF hydrogel synthesis was varied from 0 to 5 mmol MTMS·g

^{−1}CNF hydrogel, according to the indications from Zhang et al. [30]. Each test was repeated three times.

#### 2.5. Isotherm and Kinetic Studies

_{0}and C

_{t}represent the initial and equilibrium concentrations of pollutants in solution, respectively (mg·L

^{−1}); V

_{water}is the volume (L) of the solution; m

_{ads}is the adsorbent mass (g CNF). The adsorbent mass (m

_{ads}) can be calculated by the following Equation (3), which relates to the volume dose of the adsorbent and the consistency measured as indicated before.

_{ads}is the volume of adsorbent and the consistency is the dried mass of adsorbent at 60 °C [42]. These compiled data were subsequently analyzed through different kinetic models.

## 3. Results and Discussion

#### 3.1. CNF Characterization

^{−1}. This content was higher than the values reported by Patiño-Masó et al. [48] (0.75 mmol COOH·g

^{−1}), Lu et al. [49] (0.73 mmol COOH·g

^{−1}) and Balea et al. [38] (0.50 and 0.25 mmol COOH·g

^{−1}) using bleached kraft eucalyptus pulp, bleached bagasse, recycled newspaper, and corrugated container as raw materials under equivalent NaClO dosages, respectively. These differences are related to the quantity of impurities and lignin present in the source of cellulose, which has a strong influence on the amount of carboxylic groups formed for a specific amount of NaClO [38]. Therefore, the high number of carboxylic groups that formed was due to the reduced amount of impurities and lignin in the bleached pine kraft pulp used in this case, which agrees with the good performance of the oxidized cellulose during the homogenization process to obtain the CNF, and explains the low number of homogenization cycles (three).

^{−1}. This value was higher than the one obtained by Balea et al. [38] for recycled fibers (200–600 µeq·g

^{−1}) and lower than those obtained by Patiño-Masó et al. [48] (1000 µeq·g

^{−1}) for bleached kraft pulp from eucalyptus under the same testing conditions. Cationic demand is indicative of the degree of defibrillation achieved by homogenization. As the cellulose surface is negatively charged, a higher amount of negatively charged groups in the suspension means that the specific surface of the cellulose material/nanomaterial was higher. Therefore, the cationic demand will be higher as the specific surface increases, which implies that a higher number of individual nanofibers was achieved during the homogenization process. The degree of nanofibrillation achieved was directly related to both the cellulose source and the cycles applied during the homogenization, which explains the higher value achieved regarding recycled fibers such as the one obtained with virgin fibers, as the amount of cellulose in virgin sources is higher than the cellulose in recycled ones.

#### 3.2. Kinetics of Cr(VI) Adsorption with Hydrophobic CNF

#### 3.2.1. Effect of MTMS Dosage

^{−1}cellulose). These adsorption batch experiments were evaluated at pH 3, 1000 mg·L

^{−1}of adsorbent dosage, and 0.1 mg·L

^{−1}of initial chromium concentration. Hexavalent chromium concentration in the water was measured along time, using the three different hydrophobic CNF used as adsorbents (Figure 1). The trends of chromium decreased for each dose of MTMS, indicating a variation in both the adsorption rate and the maximum adsorption capacity. Whereas the CNF hydrogel without the hydrophobic coating reached a maximum chromium removal of about 80%, after 75 h of contact time, both the MTMS doped CNF hydrogels showed a higher adsorption capacity as they adsorbed hexavalent chromium up to complete abatement, after 75 h of contact time. However, the lowest MTMS dosage applied to the CNF enhanced the adsorption rate (more than 90% of chromium was removed before 6 h of treatment) in comparison with the CNF without the MTMS coating and the CNF with the highest MTMS dosage. Other modified CNF (i.e., cationized [54], carboxylated [55,56], acid treated [57], and diethylenetriamine [58] modifications) achieved similar removal yields, over 90% of hexavalent chromium removal after 120 min of contact time using doses of the adsorbent between 0.3 and 3 g adsorbent·L

^{−1}at acidic pH values from 1 to 5.5.

^{−1}CNF when applying the 1.5 mmol MTMS·g

^{−1}CNF hydrogel (Figure 3), confirming the experimental results plotted in Figure 2. Non-modified CNF and hydrophobized CNF with 1.5 mmol MTMS·g

^{−1}worked similarly, both reaching high adsorption capacities and fast saturation, while hydrophobized CNF with 3 mmol MTMS·g

^{−1}showed slower adsorption, and saturation of the adsorbent was not found at 24 h of operation. The results of the adsorption capacity in the equilibrium and contact time to equilibrium were close to those found by other silanized cellulose applied by Jamroz et al. [33] to adsorb hexavalent chromium of 0.30 mg·g

^{−1}at 300 min, respectively. However, these authors applied hydrophobic cellulose to measure ultra-traces of Cr(VI) in water. They also found that the pseudo-second order kinetic was the best fitting model.

^{−1}. Compared to other silanized materials applied for hexavalent chromium removal, this amount of silanization agent was considerably lower. Around 40 mmol·g

^{−1}of silanization reagents ((3-aminopropyl)trimethoxysilane, [3-(2-aminoethylamino)propyl]trimethoxysilane, and 3-[2-(2-aminoethylamino)ethylamino]propyl-trimethoxysilane) were added to coat the graphene oxide to adsorb hexavalent chromium from water [32].

#### 3.2.2. Effect of pH

^{−1}of initial chromium concentration, and 1000 mg·L

^{−1}of CNF dosage. The trend of the Cr(VI) removal rate was similar under neutral (pH 7) and alkali (pH 9) conditions, while the adsorption was faster under acidic (pH 3) conditions, with a total removal of 80% after 1 h of operation (Figure 4). The high adsorption capacity found at pH 3 was associated with the equilibrium changes of hexavalent chromium under acidic conditions. While divalent chromate is the predominant specie when pH >6, the monovalent specie is mainly present between pH 2 and 4. Therefore, the amount of adsorbate, hexavalent chromium, is doubled at acid pH because only one active site is required per anion. A similar result of pH optimization can be found with independence of the kind of adsorbent, as indicated by Owlad et al. [9] and Saha and Orvig [61]. It can be concluded that the pH effect is related to the adsorbate ionic forms, being the effect on the adsorbent negligible. After these experiments, the pH 3 condition was selected for the rest of the optimization process.

^{−1}of the adsorption capacity and 80% of the maximum removal was reached in 1 h. The operation under neutral and alkaline media was similar in adsorption rate, but with a slightly lower adsorption capacity when operating at alkaline conditions.

#### 3.2.3. Effect of Adsorbent Dosage

^{−1}(Figure 7). The kinetic curves showed a clear tendency of increasing the adsorption rate as the dosage of CNF increased. Among the studied CNF dosages, the minimum one needed to obtain a 100% of adsorption of hexavalent chromium was 500 mg·L

^{−1}whereas the lowest adsorbent dosage studied showed a fast saturation of the CNF and a total removal below 10% (Figure 7). The highest dosage studied, 1000 mg·L

^{−1}, achieved the complete removal of hexavalent chromium after 25 h of contact time, the same as the dosage of 500 mg·L

^{−1}. However, the higher the dosage, the faster the adsorption, as expected. In terms of the adsorption capacity, the optimum value was found while applying 500 mg·L

^{−1}, reaching the largest efficiency in chromium removal per gram of CNF.

^{−1}with black wattle tannin-modified dialdehyde nanocellulose. Qiu et al. [62] and Huang et al. [58] achieved the complete depletion of hexavalent chromium employing 3000 mg·L

^{−1}of polyethylenimine facilitated ethyl cellulose and diethylenetriamine-modified hydroxypropyl methylcellulose, respectively. In the study developed by Singh et al. [63], a dosage of 5000 mg·L

^{−1}of aminated cellulose nanocrystals adsorbed 98.33% of hexavalent chromium. These results depict that hydrophobic CNF reached equivalent yields compared to other cellulosic materials, reducing the dosage by an order of magnitude.

^{−1}were necessary, respectively. Other typical materials used to adsorb hexavalent chromium are both commercial and mango kernel-synthetized activated carbons [67,68], and chitosan microparticles and nanoparticles [69], but they require larger adsorbent dosages than the hydrophobic CNF. The optimal dosage needed for the activated carbons was 2000 mg·L

^{−1}, while 800 mg·L

^{−1}was the minimum required dose of the chitosan microparticles and nanoparticles to reach hexavalent chromium removal.

^{−1}are shown in Figure 8. The optimal kinetic model corresponded to a pseudo-second order equation. All of the kinetic parameters obtained through the kinetic fittings from the dosage optimization experiments are shown in detail in Table A3. The results of the intraparticle diffusion model adjustments were similar for each dosage, corresponding to multi-step adsorption.

^{−1}when 500 mg CNF·L

^{−1}was dosed compared to 0.30 mg·g

^{−1}in the case of 1000 mg CNF·L

^{−1}. However, the pseudo-second order kinetic constant was fourteen-times higher in the case of the highest dosage. The contrast between reaching fast-equilibrium with high adsorbent dosages and increasing the adsorption efficiency can be observed through the use of other nanomaterial adsorbents such as Fe

_{2}O

_{3}nanoparticles [70]. This fact would suggest that the optimal dosage would depend on the objective between reaching the rapid total removal or maximizing the total capacity of the adsorbent.

#### 3.2.4. Effect of Initial Chromium Concentration

^{−1}and when the initial chromium concentration drastically increased to the range from 10 to 50 mg·L

^{−1}(Figure 10a,b, respectively). These kinetic experiments were developed by implementing the optimal tested MTMS dosage, the optimal tested pH, and the maximum adsorbent dosage (1000 mg CNF·L

^{−1}), thus prioritizing the process kinetic over the adsorption capacity. Furthermore, this CNF hydrogel dosage was selected to prevent the CNF from extremely fast saturation.

^{−1}(Figure 11). At this initial concentration, as it happened at 1 mg·L

^{−1}, the only model that allowed an adequate interpretation and simulation of the experimental results was the intraparticle diffusion model. Each kinetic parameter obtained by kinetic analysis for all the kinetic models can be checked in Table A4. The intraparticle diffusion model predicts a three-step adsorption mechanism including the first high adsorption rate step, a steady-state step, and the last slow adsorption step, confirming the trend seen in the experimental data. Evaluating the intraparticle adjustment of the first step in the experiments using an initial concentration of 5 mg·L

^{−1}and above, the intercept is extremely high, in some cases, close to the maximum adsorption capacity. This fact indicates a strong effect of the boundary layer, which means that external diffusion limitation will play a major role in the overall adsorption rate [59].

^{−1}). The equilibrium time of the hydrophobic CNF was similar to the one of another silanized cellulosic material applied by Jamroz et al. [33] (300 min) to detect ultra-trace concentrations of hexavalent chromium in water, suggesting that silanization processes lead to larger contact times than other kinds of adsorbents (Table 4) and could be associated with relevant mass transfer limitations due to the silane reaction with celluloses. The applied dosage was in the order of magnitude of other cellulosic adsorbents such as polypyrrole-bacterial CNF and polyaniline-functionalized CNF and was lower than that of activated carbons [72,73,74,75,76]. This comparison suggests that the hydrophobized CNF hydrogel is an efficient material for hexavalent chromium adsorption from wastewater compared to other adsorbents including other nanocellulosic materials.

#### 3.3. Isotherm Analysis

**Figure 13.**The isotherm experimental data and isotherm model adjustment of the Langmuir, Freundlich, Temkin, Dubinin–Raduskevich, and Sips equations.

**Table 4.**The results of the isotherm model adjustment to the adsorption equilibrium data of hexavalent chromium on a CNF hydrogel.

Model | Parameters | Values |
---|---|---|

Langmuir | Isotherm parameters | k_{L} [L·mg^{−1}] = 21.26q _{e} [mg·g^{−1}] = 0.3417R _{L} (C_{0} = 0.1 mg·L^{−1}) [-] = 0.9670R _{L} (C_{0} = 50 mg·L^{−1}) [-] = 5.53·10^{−2} |

Correlation parameters | R^{2} = 0.7420RSS = 2949.55 | |

Freundlich | Isotherm parameters | k_{F} [mg^{(1−1/n)}-L^{(1/n)}·g^{−1}] = 1.3914n _{F} [-] = 0.8404 |

Correlation parameters | R^{2} = 0.9902RSS = 108.01 | |

Dubinin–Raduskevich | Isotherm parameters | B_{DR} [mol^{2}·J^{−2}] = 9.93·10^{−8}q _{max} [mg·g^{−1}] = 27.72 |

Thermodynamic parameters | E_{DR} [J·mol^{−1}] = 2243.50 | |

Correlation parameters | R^{2} = 0.5754RSS = 2542.39 | |

Temkin | Isotherm parameters | B_{T} [J·mol^{−1}] = 12.83b _{T} [-J·mol^{−1}] = 188.08A _{T} [L·g^{−1}] = 1.3759 |

Correlation parameters | R^{2} = 0.7481RSS = 1415.93 | |

Sips | Isotherm parameters | n_{S} [-] = 1.2442k _{S} [L^{(1/nS)·}mol^{-(1/nS)}] = 6.16·10^{−2} |

Correlation parameters | R^{2} = 0.9023RSS = 1529.83 |

_{F}<1 indicates an unfavorable process as well as the small bond adsorbate–adsorbent compared to a favorable process [81]. A similar value of n

_{F}was reported by Dawodu et al. [82] for hexavalent chromium adsorption onto the seed coat biomass, showing a cooperative adsorption between Cr(VI) and adsorbent surface.

_{L}< 1, but close to the upper and lower limits of the interval, respectively. The variability of the separation factor indicates that while treating lower concentrations, the high value of R

_{L}implies a reduced affinity adsorbate–adsorbent. On the other hand, the treatment of concentrated solutions showed a reduction in the R

_{L}values close to 0, strengthening the chromium attachment onto the CNF surface [83].

^{−1}, similar to the typical values indicated for the physisorption of chromium (<8 kJ·mol

^{−1}) [84]. The value of the Temkin b

_{T}parameter, which is related to the heat of sorption, was 0.19 kJ·mol

^{−1}. Choudhary and Paul [85] indicated that values of b

_{T}below 8 kJ·mol

^{−1}revealed a weak interaction chromium-CNF surface. These low b

_{T}values are related to physisorption processes, where the values of adsorption enthalpy are in the order of physical processes such as intermolecular forces.

## 4. Conclusions

^{−1}as a hydrophobizing agent allowed for the increase in the Cr(VI) kinetic constant of adsorption k

_{2}by 84.97%. The pseudo-second order and intraparticle diffusion kinetic models were the best fitting models. These models revealed that both the sorption rate and hexavalent chromium diffusion played a major role as rate-limiting steps. The adsorption mechanism is ruled by the multi-step adsorption of hexavalent chromium on the CNF hydrogel dominated by internal diffusion at low concentrations and external diffusion with concentrations above 5 mg·L

^{−1}. The optimized conditions were found to be pH 3 and a dosage over 500 mg·L

^{−1}. More than 97% of hexavalent chromium removal was reached, treating concentrations below 1 mg·L

^{−1}and the maximum adsorption capacity of 70.38 mg·g

^{−1}was achieved at 50 mg·L

^{−1}. The isotherm analysis showed that Freundlich was the best fitting model, meaning that multilayer adsorption and the heterogeneous dispersion of surface energy is the main adsorption mechanism of hexavalent chromium onto the surface of hydrophobized CNF. The Freundlich unfavorable isotherm predicts a multilayer adsorption and a weak interaction between hexavalent chromium and CNF, associated with a physical sorption mechanism. The relatively low values of mean free energy of adsorption (2.24 kJ·mol

^{−1}) and heat of sorption (0.19 kJ·mol

^{−1}) calculated through the Dubinin–Raduskevich and Temkin models are also indicators of a physical sorption process. In general terms, the hydrophobized CNF hydrogel reached relatively high adsorption capacities compared to previously developed cellulose nanomaterials and activated carbons, and the complete removal of chromium could be found, even at low adsorbent dosages.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**The results of the kinetic equation adjustment to the adsorption experimental data of 0–3 mmol MTMS·g

^{−1}CNF.

Kinetic Model | 0 mmol MTMS·g^{−1} CNF | 1.5 mmol MTMS·g^{−1} CNF | 3 mmol MTMS·g^{−1} CNF | |
---|---|---|---|---|

Pseudo-first order | Kinetic parameters | k_{1} [h^{−1}] = 0.2378 | k_{1} [h^{−1}] = 8.63·10^{−2} | k_{1} [h^{−1}] = 9.16·10^{−2} |

Correlation parameters | R^{2} = 0.9653RSS = 7.11·10 ^{−3} | R^{2} = 0.7725RSS = 7.84·10 ^{−2} | R^{2} = 0.9674RSS = 9.38·10 ^{−3} | |

Pseudo-second order | Kinetic parameters | k_{2} [mg·g^{−1}-h^{−1}] = 3.7399q _{e} [mg·g^{−1}] = 0.2933 | k_{2} [mg·g^{−1}-h^{−1}] = 6.9175q _{e} [mg·g^{−1}] = 0.3058 | k_{2} [mg·g^{−1}-h^{−1}] = 1.4827q _{e} [mg·g^{−1}] = 0.2610 |

Correlation parameters | R^{2} = 0.9440RSS = 1.23·10 ^{−2} | R^{2} = 0.9661RSS = 5.96·10 ^{−3} | R^{2} = 0.9430RSS = 1.08·10 ^{−2} | |

Elovich | Kinetic parameters | α [h·mg·g^{−1}] = 0.8350β [g·mg ^{−1}] = 21.5517 | α [h·mg·g^{−1}] = 6.6752β [g·mg ^{−1}] = 28.1690 | α [h·mg·g^{−1}] = 1.6142β [g·mg ^{−1}] = 37.7358 |

Correlation parameters | R^{2} = 0.8109RSS = 1.23·10 ^{−2} | R^{2} = 0.8005RSS = 7.74·10 ^{−3} | R^{2} = 0.9540RSS = 3.90·10 ^{−3} | |

Intraparticle diffusion | Kinetic parameters: Step 1 | k_{i,1} [mg·g^{−1}·min^{−0.5}] = 0.1193C _{i,1} [mg·g^{−1}] = −1.62·10^{−2} | k_{i,1} [mg·g^{−1}·min^{−0.5}] = 0.1462C _{i,1} [mg·g^{−1}] = −2.30·10^{−3} | k_{i,1} [mg·g^{−1}·min^{−0.5}] = 0.1462C _{i,1} [mg·g^{−1}] = 7.00·10^{−4} |

Correlation parameters | R^{2} = 0.9709RSS = 1.23·10 ^{−4} | R^{2} = 0.9996RSS = 3.17·10 ^{−5} | R^{2} = 0.9918RSS = 5.06·10 ^{−5} | |

Kinetic parameters: Step 2 | k_{i,2} [mg·g^{−1}·min^{−0.5}] = 1·10^{−17}C _{i,1} [mg·g^{−1}] = 0.299 | k_{i,2} [mg·g^{−1}·min^{−0.5}] = 4.00·10^{−2}C _{i,2} [mg·g^{−1}] = 0.196 | k_{i,2} [mg·g^{−1}·min^{−0.5}] = 0.0239C _{i,1} [mg·g^{−1}] = 0.0658 | |

Correlation parameters | R^{2} = 0.5477RSS = 1.23·10 ^{−4} | R^{2} = 0.9996RSS = 4.25·10 ^{−8} | R^{2} = 0.9839RSS = 3.67·10 ^{−4} | |

Kinetic parameters: Step 3 | k_{i,3} [mg·g^{−1}·min^{−0.5}] = 2.10·10^{−3}C _{i,3} [mg·g^{−1}] = 0.2878 | |||

Correlation parameters | R^{2} = 0.8137RSS = 4.29·10 ^{−5} |

## Appendix B

**Table A2.**The results of the kinetic equation adjustment to the adsorption experimental data of pH 3–9.

Kinetic Model | pH 3 | pH 7 | pH 9 | |
---|---|---|---|---|

Pseudo-first order | Kinetic parameters | k_{1} [h^{−1}] = 5.55·10^{−2} | k_{1} [h^{−1}] = 6.09·10^{−2} | k_{1} [h^{−1}] = 0.2179 |

Correlation parameters | R^{2} = 0.6366RSS = 0.167 | R^{2} = 0.9219RSS = 7.25·10 ^{−3} | R^{2} = 0.9960RSS = 1.49·10 ^{−4} | |

Pseudo-second order | Kinetic parameters | k_{2} [mg·g^{−1}-h^{−1}] = 11.4405q _{e} [mg·g^{−1}] = 0.3050 | k_{2} [mg·g^{−1}-h^{−1}] = 0.9889q _{e} [mg·g^{−1}] = 0.1821 | k_{2} [mg·g^{−1}-h^{−1}] = 4.4596q _{e} [mg·g^{−1}] = 0.1139 |

Correlation parameters | R^{2} = 0.9980RSS = 3.29·10 ^{−4} | R^{2} = 0.9612RSS = 2.77·10 ^{−3} | R^{2} = 0.9801RSS = 8.14·10 ^{−4} | |

Elovich | Kinetic parameters | α [h·mg·g^{−1}] = 43.5512β [g·mg ^{−1}] = 33.7838 | α [h·mg·g^{−1}] = 0.3496β [g·mg ^{−1}] = 46.9484 | α [h·mg·g^{−1}] = 0.3013β [g·mg ^{−1}] = 66.2252 |

Correlation parameters | R^{2} = 0.8683RSS = 5.44·10 ^{−3} | R^{2} = 0.8989RSS = 6.16·10 ^{−3} | R^{2} = 0.9255RSS = 2.18·10 ^{−3} | |

Intraparticle diffusion | Kinetic parameters: Step 1 | k_{i,1} [mg·g^{−1}·min^{−0.5}] = 0.2346C _{i,1} [mg·g^{−1}] = 1.38·10^{−2} | k_{i,1} [mg·g^{−1}·min^{−0.5}] = 2.41·10^{−2}C _{i,1} [mg·g^{−1}] = 4.75·10^{−4} | k_{i,1}[mg·g^{−1}·min^{−0.5}] = 8.34·10^{−3}C _{i,1} [mg·g^{−1}] = 4.63·10^{−3} |

Correlation parameters | R^{2} = 0.9995RSS = 2.13·10 ^{−10} | R^{2} = 0.9841RSS = 5.36·10 ^{−5} | R^{2} = 0.9055RSS = 2.73·10 ^{−5} | |

Kinetic parameters: Step 2 | k_{i,2} [mg·g^{−1}·min^{−0.5}] = 3.01·10^{−2}C _{i,2} [mg·g^{−1}] = 0.2178 | k_{i,2} [mg·g^{−1}·min^{−0.5}] = 0.2399C _{i,2} [mg·g^{−1}] = −0.4314 | k_{i,2}[mg·g^{−1}·min^{−0.5}] = 4.82·10^{−2}C _{i,2} [mg·g^{−1}] = −3.30·10^{−2} | |

Correlation parameters | R^{2} = 0.9965RSS = 6.23·10 ^{−6} | R^{2} = 0.9999RSS = 2.13·10 ^{−8} | R^{2} = 0.9891RSS = 5.05·10 ^{−5} | |

Kinetic parameters: Step 3 | k_{i,3} [mg·g^{−1}·min^{−0.5}] = 2.10·10^{−3}C _{i,3} [mg·g^{−1}] = 0.288 | k_{i,3} [mg·g^{−1}·min^{−0.5}]= 7.05·10^{−3}C _{i,3} [mg·g^{−1}] = 0.1079 | k_{i,3}[mg·g^{−1}·min^{−0.5}] = 1.35·10^{−2}C _{i,3} [mg·g^{−1}] = 4.42·10^{−2} | |

Correlation parameters | R^{2} = 0.8137RSS = 4.29·10 ^{−5} | R^{2} = 0.9103RSS = 1.96·10 ^{−4} | R^{2} = 0.9999RSS = 6.74·10 ^{−9} | |

Kinetic parameters: Step 4 | k_{i,4} [mg·g^{−1}·min^{−0.5}] = 0C _{i,4} [mg·g^{−1}] = 0.1114 | |||

Correlation parameters | RSS = 1.98·10^{−6} |

## Appendix C

**Table A3.**The results of the kinetic equation adjustment to the adsorption experimental data of 250 to 1000 mg CNF·L

^{−1}.

Kinetic Model | 250 mg·L^{−1} | 500 mg·L^{−1} | 1000 mg·L^{−1} | |
---|---|---|---|---|

Pseudo-first order | Kinetic parameters | k_{1} [h^{−1}] = 4.49·10^{−2} | k_{1} [h^{−1}] = 5.56·10^{−2} | k_{1} [h^{−1}] = 6.69·10^{−2} |

Correlation parameters | R^{2} = 0.3713RSS = 3.31·10 ^{−3} | R^{2} = 0.9267RSS = 0.159 | R^{2} = 0.6605RSS = 0.152 | |

Pseudo-second order | Kinetic parameters | k_{2} [mg·g^{−1}-h^{−1}] = 3822.25q _{e} [mg·g^{−1}] = 2.76·10^{−2} | k_{2} [mg·g^{−1}-h^{−1}] = 1.1320q _{e} [mg·g^{−1}] = 0.5787 | k_{2} [mg·g^{−1}-h^{−1}] = 14.3041q _{e} [mg·g^{−1}] = 0.3047 |

Correlation parameters | R^{2} = 0.9256RSS = 1.55·10 ^{−4} | R^{2} = 0.9596RSS = 3.78·10 ^{−2} | R^{2} = 0.9957RSS = 9.31·10 ^{−4} | |

Elovich | Kinetic parameters | α [h·mg·g^{−1}] = 229.62β [g·mg ^{−1}] = 462.96 | α [h·mg·g^{−1}] = 3.2448β [g·mg ^{−1}] = 15.1976 | α [h·mg·g^{−1}] = 66.5435β [g·mg ^{−1}] = 35.9712 |

Correlation parameters | R^{2} = 0.6241RSS = 2.78·10 ^{−4} | R^{2} = 0.9246RSS = 4.49· 10 ^{−2} | R^{2} = 0.8683RSS = 5.34·10 ^{−3} | |

Intraparticle diffusion | Kinetic parameters: Step 1 | k_{i,1} [mg·g^{−1}·min^{−0.5}] = 0.1384C _{i,1} [mg·g^{−1}] = −1.23·10^{−2} | k_{i,1} [mg·g^{−1}·min^{−0.5}] = 0.1397C _{i,1} [mg·g^{−1}] = 5.45·10^{−2} | k_{i,1} [mg·g^{−1}·min^{−0.5}] = 0.2346C _{i,1} [mg·g^{−1}] = 1.38·10^{−2} |

Correlation parameters | R^{2} = 0.9631RSS = 1.51·10 ^{−4} | R^{2} = 0.9295RSS = 1.03·10 ^{−2} | R^{2} = 0.9995RSS = 2.13·10 ^{−10} | |

Kinetic parameters: Step 2 | k_{i,2} [mg·g^{−1}·min^{−0.5}] = 1·10·^{−17}C _{i,2} [mg·g^{−1}] = 2.76·10^{−2} | k_{i,2} [mg·g^{−1}·min^{−0.5}] = 7.03·10^{−2}C _{i,2} [mg·g^{−1}] = 0.2210 | k_{i,2}[mg·g^{−1}·min^{−0.5}] = 3.01·10^{−2}C _{i,2} [mg·g^{−1}] = 0.2178 | |

Correlation parameters | R^{2} = 0.5117RSS = 1.44·10 ^{−11} | R^{2} = 1.0000RSS = 4.19·10 ^{−7} | R^{2} = 0.9965RSS = 6.23·10 ^{−6} | |

Kinetic parameters: Step 3 | k_{i,3} [mg·g^{−1}·min^{−0.5}] = 0C _{i,3} [mg·g^{−1}] = 0.5658 | k_{i,3}[mg·g^{−1}·min^{−0.5}] = 2.10·10^{−3}C _{i,3} [mg·g^{−1}] = 0.2880 | ||

Correlation parameters | RSS = 1.73·10^{−9} | R^{2} = 0.8137RSS = 4.29·10 ^{−5} |

## Appendix D

**Table A4.**The results of the kinetic equation adjustment to the adsorption experimental data of the initial hexavalent chromium concentrations from 0.1 to 50 mg·L

^{−1}.

Kinetic Model | 0.1 mg·L^{−1} | 1 mg·L^{−1} | 5 mg·L^{−1} | |
---|---|---|---|---|

Pseudo-first order | Kinetic parameters | k_{1} [h^{−1}] = 8.00·10^{−2} | k_{1} [h^{−1}] = 4.76·10^{−2} | k_{1} [h^{−1}] = 1.5675 |

Correlation parameters | R^{2} = 0.8105RSS = 6.81·10 ^{−2} | R^{2} = 0.8880RSS = 1.1163 | R^{2} = 0.7040RSS = 35.56 | |

Pseudo-second order | Kinetic parameters | k_{2} [mg·g^{−1}-h^{−1}] = 4.9284q _{e} [mg·g^{−1}] = 0.3059 | k_{2} [mg·g^{−1}-h^{−1}] = 3.53·10^{−2}q _{e} [mg·g^{−1}] = 1.9361 | k_{2} [mg·g^{−1}-h^{−1}] = 2.99·10^{−2}q _{e} [mg·g^{−1}] = 6.9541 |

Correlation parameters | R^{2} = 0.9853RSS = 3.05·10 ^{−3} | R^{2} = 0.8975RSS = 0.7128 | R^{2} = 0.5884RSS = 54.66 | |

Elovich | Kinetic parameters | α [h·mg·g^{−1}] = 8.6542β [g·mg ^{−1}] = 31.257 | α [h·mg·g^{−1}] = 1.6140β [g·mg ^{−1}] = 4.7996 | α [h·mg·g^{−1}] = 1.20·10^{9}β [g·mg ^{−1}] = 4.7547 |

Correlation parameters | R^{2} = 0.8606RSS = 5.62·10 ^{−3} | R^{2} = 0.8760RSS = 0.5550 | R^{2} = 0.5875RSS = 3.5383 | |

Intraparticle diffusion | Kinetic parameters: Step 1 | k_{i,1} [mg·g^{−1}·min^{−0.5}] = 0.1099C _{i,1} [mg·g^{−1}] = 2.88·10^{−2} | k_{i,1} [mg·g^{−1}·min^{−0.5}] = 0.6325C _{i,1} [mg·g^{−1}] = −0.1448 | k_{i,1}[mg·g^{−1}·min^{−0.5}] = 0.3298C _{i,1} [mg·g^{−1}] = 4.4590 |

Correlation parameters | R^{2} = 0.9982RSS = 1.68·10 ^{−4} | R^{2} = 0.9271RSS = 3.91·10 ^{−2} | R^{2} = 0.8235RSS = 19.88 | |

Kinetic parameters: Step 2 | k_{i,2} [mg·g^{−1}·min^{−0.5}] = 6.00·10^{−3}C _{i,2} [mg·g^{−1}] = 0.2615 | k_{i,2} [mg·g^{−1}·min^{−0.5}] = 2.29·10^{−2}C _{i,2} [mg·g^{−1}] = 0.5194 | k_{i,2}[mg·g^{−1}·min^{−0.5}] = 0C _{i,2} [mg·g^{−1}]= 4.6922 | |

Correlation parameters | R^{2} = 0.9978RSS = 1.66·10 ^{−6} | R^{2} = 0.9690RSS = 2.79·10 ^{−4} | RSS = 9.68·10^{−9} | |

Kinetic parameters: Step 3 | k_{i,3} [mg·g^{−1}·min^{−0.5}] = 0C _{i,3} [mg·g^{−1}] = 0.3036 | k_{i,3} [mg·g^{−1}·min^{−0.5}]= 0.2121C _{i,3} [mg·g^{−1}] = −0.3775 | k_{i,3}[mg·g^{−1}·min^{−0.5}] = 0.617C _{i,3} [mg·g^{−1}] = 0.4177 | |

Correlation parameters | RSS = 1.68·10^{−4} | R^{2} = 0.9986RSS = 2.48·10^{−3} | R^{2} = 0.9999RSS = 7.46·10^{−8} | |

Kinetic model | 10 mg/L | 25 mg/L | 50 mg/L | |

Pseudo-first order | Kinetic parameters | k_{1} [h^{−1}] = 3.00·10^{−2} | k_{1} [h^{−1}] = 0.1664 | k_{1} [h^{−1}] = 3.86·10^{−2} |

Correlation parameters | R^{2} = 0.5969RSS = 307.88 | R^{2} = 0.5860RSS = 1592.74 | R^{2} = 0.8982RSS = 2433.18 | |

Pseudo-second order | Kinetic parameters | k_{2} [mg·g^{−1}-h^{−1}] = 5.40·10^{−2}q _{e} [mg·g^{−1}] = 12.115 | k_{2} [mg·g^{−1}-h^{−1}] = 8.70·10^{−3}q _{e} [mg·g^{−1}] = 34.51 | k_{2} [mg·g^{−1}-h^{−1}] = 6.16·10^{−3}q _{e} [mg·g^{−1}] = 70.92 |

Correlation parameters | R^{2} = 0.7242RSS = 121.03 | R^{2} = 0.6332RSS = 1286.27 | R^{2} = 0.9851RSS = 236.03 | |

Elovich | Kinetic parameters | α [h·mg·g^{−1}] = 3.82·10^{8}β [g·mg ^{−1}] = 2.2222 | α [h·mg·g^{−1}] = 7.29·10^{8}β [g·mg ^{−1}] = 0.8143 | α [h·mg·g^{−1}] = 153.74β [g·mg ^{−1}] = 0.1041 |

Correlation parameters | R^{2} = 0.9184RSS = 1.5841 | R^{2} = 0.7408RSS = 44.33 | R^{2} = 0.9934RSS = 166.49 | |

Intraparticle diffusion | Kinetic parameters: Step 1 | k_{i,1} [mg·g^{−1}·min^{−0.5}] = 1.3193C _{i,1} [mg·g^{−1}] = 8.1756 | k_{i,1} [mg·g^{−1}·min^{−0.5}] = 4.9473C _{i,1} [mg·g^{−1}] = 21.343 | k_{i,1}[mg·g^{−1}·min^{−0.5}] = 27.162C _{i,1} [mg·g^{−1}] = 4.5797 |

Correlation parameters | R^{2} = 0.8392RSS = 66.84 | R^{2} = 0.8504RSS = 455.52 | R^{2} = 0.9935RSS = 20.97 | |

Kinetic parameters: Step 2 | k_{i,2} [mg·g^{−1}·min^{−0.5}] = 0C _{i,2} [mg·g^{−1}] = 9.1085 | k_{i,2} [mg·g^{−1}·min^{−0.5}] = 0.1966C _{i,2} [mg·g^{−1}] = 24.783 | k_{i,2}[mg·g^{−1}·min^{−0.5}] = 8.0237C _{i,2} [mg·g^{−1}] = 24.416 | |

Correlation parameters | RSS = 2.08·10^{−9} | R^{2} = 0.9204RSS = 0.2037 | R^{2} = 0.9987RSS = 1.4264 | |

Kinetic parameters: Step 3 | k_{i,3} [mg·g^{−1}·min^{−0.5}] = 0.3399C _{i,3} [mg·g^{−1}] = 8.5209 | k_{i,3} [mg·g^{−1}·min^{−0.5}] = 2.2280C _{i,3} [mg·g^{−1}] = 10.785 | k_{i,3}[mg·g^{−1}·min^{−0.5}] = 1.2205C _{i,3} [mg·g^{−1}] = 56.515 | |

Correlation parameters | R^{2} = 0.9953RSS = 4.63·10 ^{−2} | R^{2} = 0.9999RSS = 2.05·10 ^{−7} | R^{2} = 0.9443RSS = 3.4399 |

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**Figure 1.**The evolution of the hexavalent chromium concentration [mg·L

^{−1}] at 0.1 mg·L

^{−1}of the initial chromium concentration, pH 3, and 1000 mg CNF·L

^{−1}of dosage during adsorption with different doses of MTMS in the CNF hydrogels.

**Figure 2.**The evolution of the kinetic adsorption experiment at 0.1 mg·L

^{−1}of the initial chromium concentration, pH 3, and 1000 mg CNF·L

^{−1}hydrophobized with 1.5 mmol MTMS·g

^{−1}cellulose and kinetic fitting of the pseudo-first, pseudo-second, Elovich, and intraparticle models.

**Figure 3.**The evolution of the kinetic adsorption experiment at the previously indicated conditions under different MTMS dosages and kinetic fitting of the pseudo-second order model.

**Figure 4.**The evolution of the hexavalent chromium concentration [mg·L

^{−1}] during adsorption with the CNF hydrogel at 0.1 mg·L

^{−1}of the initial chromium concentration, 1.5 mmol MTMS·g

^{−1}CNF applied during hydrophobization, and 1000 mg CNF·L

^{−1}of dosage under different pH conditions.

**Figure 5.**The evolution of the kinetic adsorption experiment at 0.1 mg·L

^{−1}of the initial chromium concentration, 1.5 mmol MTMS·g

^{−1}CNF applied during hydrophobization and 1000 mg CNF·L

^{−1}of dosage under pH 3 conditions and the kinetic fitting of the pseudo-first, pseudo-second, Elovich, and intraparticle models.

**Figure 6.**The evolution of the kinetic adsorption experiment at 0.1 mg·L

^{−1}of the initial chromium concentration, 1.5 mmol MTMS·g

^{−1}CNF applied during hydrophobization, and 1000 mg CNF·L

^{−1}of dosage under different pH and kinetic fitting of the pseudo-second order model.

**Figure 7.**The evolution of the hexavalent chromium concentration [mg·L

^{−1}] during adsorption with the CNF hydrogel at 0.1 mg·L

^{−1}of the chromium initial concentration, 1.5 mmol MTMS·g

^{−1}CNF, and pH 3 conditions under different adsorbent dosages.

**Figure 8.**The evolution of the kinetic adsorption experiment at 0.1 mg·L

^{−1}of the chromium initial concentration, 1.5 mmol MTMS·g

^{−1}CNF, and pH 3 conditions under 500 mg CNF·L

^{−1}of dosage and the kinetic fitting of the pseudo-first, pseudo-second, Elovich, and intraparticle models.

**Figure 9.**The evolution of the kinetic adsorption experiment at 0.1 mg·L

^{−1}of the chromium initial concentration, 1.5 mmol MTMS·g

^{−1}CNF, and pH 3 conditions under different adsorbent dosages and kinetic fitting of the pseudo-second order model.

**Figure 10.**(

**a**) The evolution of the hexavalent chromium concentration [mg·L

^{−1}] during the adsorption with CNF hydrogel at 1000 mg·L

^{−1}of dosage, 1.5 mmol MTMS·g

^{−1}CNF, and pH 3 conditions under 0.1 to 5 mg·L

^{−1}; (

**b**) 10 to 50 mg·L

^{−1}initial hexavalent chromium concentrations.

**Figure 11.**The evolution of the kinetic adsorption experiment of the CNF hydrogel at 1000 mg·L

^{−1}of dosage, 1.5 mmol MTMS·g

^{−1}CNF, and pH 3 conditions under 25 mg·L

^{−1}of the initial hexavalent chromium concentration and kinetic fitting of the pseudo-first, pseudo-second, Elovich, and intraparticle models.

**Figure 12.**(

**a**) The evolution of the kinetic adsorption experiment of the CNF hydrogel at 1000 mg·L

^{−1}of dosage, 1.5 mmol MTMS·g

^{−1}CNF, and pH 3 conditions under 0.1 to 5 mg·L

^{−1}; (

**b**) 10 to 50 mg·L

^{−1}of the initial hexavalent chromium concentration and kinetic fitting of the pseudo-second order model.

Model | Nonlinearized Equations | Linearization | Ref. | ||
---|---|---|---|---|---|

Pseudo-first order ^{1} | $\mathrm{q}={\mathrm{q}}_{\mathrm{e}}\xb7\left(1-{\mathrm{e}}^{-{\mathrm{k}}_{1}\xb7\mathrm{t}}\right)$ | (4) | $\mathrm{ln}\left({\mathrm{q}}_{\mathrm{e}}-\mathrm{q}\right)-\mathrm{ln}\left({\mathrm{q}}_{\mathrm{e}}\right)=-{\mathrm{k}}_{1}\xb7\mathrm{t}$ | (8) | [43] |

Pseudo-second order | $\mathrm{q}=\frac{{\mathrm{q}}_{\mathrm{e}}^{2}\xb7{\mathrm{k}}_{2}\xb7\mathrm{t}}{1+{\mathrm{q}}_{\mathrm{e}}\xb7{\mathrm{k}}_{2}\xb7\mathrm{t}}$ | (5) | $\frac{\mathrm{t}}{\mathrm{q}}=\left(\frac{1}{{\mathrm{q}}_{\mathrm{e}}^{2}\xb7{\mathrm{k}}_{2}}\right)+\left(\frac{1}{{\mathrm{q}}_{\mathrm{e}}}\right)\xb7\mathrm{t}$ | (9) | [43] |

$\frac{1}{\mathrm{q}}=\left(\frac{1}{{\mathrm{q}}_{\mathrm{e}}^{2}\xb7{\mathrm{k}}_{2}}\right)\xb7\frac{1}{\mathrm{t}}+\left(\frac{1}{{\mathrm{q}}_{\mathrm{e}}}\right)$ | (10) | [44] | |||

Elovich | $\mathrm{q}=\frac{1}{\mathsf{\beta}}\xb7\mathrm{ln}\left(\mathrm{t}\right)+\frac{1}{\mathsf{\beta}}\mathrm{ln}\left(\mathsf{\alpha}\xb7\mathsf{\beta}\right)$ | (6) | - | [45] | |

Weber and Morris (Intraparticle) | $\mathrm{q}={\mathrm{k}}_{\mathrm{i}}\xb7{\mathrm{t}}^{0.5}+\mathrm{C}$ | (7) | - | [46] |

^{1}The value of q

_{e}must be previously obtained by estimation or experimentally.

Model | Nonlinearized Equations | Linearization | Ref. | ||
---|---|---|---|---|---|

Langmuir | ${\mathrm{q}}_{\mathrm{e}}=\frac{{\mathrm{K}}_{\mathrm{L}}\xb7{\mathrm{q}}_{\mathrm{max}}\xb7{\mathrm{C}}_{\mathrm{e}}}{1+{\mathrm{K}}_{\mathrm{L}}\xb7{\mathrm{C}}_{\mathrm{e}}}$ | (11) | $\mathrm{Type}\mathrm{I}:\frac{{\mathrm{C}}_{\mathrm{e}}}{{\mathrm{q}}_{\mathrm{e}}}=\frac{1}{{\mathrm{K}}_{\mathrm{L}}\xb7{\mathrm{q}}_{\mathrm{max}}}+\left(\frac{1}{{\mathrm{q}}_{\mathrm{max}}}\right)\xb7{\mathrm{C}}_{\mathrm{e}}$ | (20) | [47] |

$\mathrm{Type}\mathrm{II}:\frac{1}{{\mathrm{q}}_{\mathrm{e}}}=\frac{1}{{\mathrm{q}}_{\mathrm{max}}}+\left(\frac{1}{{\mathrm{K}}_{\mathrm{L}}\xb7{\mathrm{q}}_{\mathrm{max}}}\right)\xb7\left(\frac{1}{{\mathrm{C}}_{\mathrm{e}}}\right)$ | (21) | ||||

$\mathrm{Type}\mathrm{III}:{\mathrm{q}}_{\mathrm{e}}=\left(-\frac{1}{{\mathrm{K}}_{\mathrm{L}}}\right)\xb7\left(\frac{{\mathrm{q}}_{\mathrm{e}}}{{\mathrm{C}}_{\mathrm{e}}}\right)+{\mathrm{q}}_{\mathrm{max}}$ | (22) | ||||

$\mathrm{Type}\mathrm{IV}:\frac{{\mathrm{q}}_{\mathrm{e}}}{{\mathrm{C}}_{\mathrm{e}}}=\left(-{\mathrm{K}}_{\mathrm{L}}\right)\xb7{\mathrm{q}}_{\mathrm{e}}+{\mathrm{K}}_{\mathrm{L}}\xb7{\mathrm{q}}_{\mathrm{max}}$ | (23) | ||||

${\mathrm{R}}_{\mathrm{L}}=\frac{1}{1+{\mathrm{K}}_{\mathrm{L}}\xb7{\mathrm{C}}_{0}}$ | (12) | $\mathrm{Type}\mathrm{V}:\frac{1}{{\mathrm{C}}_{\mathrm{e}}}=\left({\mathrm{K}}_{\mathrm{L}}\xb7{\mathrm{q}}_{\mathrm{max}}\right)\xb7\frac{1}{{\mathrm{q}}_{\mathrm{e}}}-{\mathrm{K}}_{\mathrm{L}}$ | (24) | ||

Freundlich | ${\mathrm{q}}_{\mathrm{e}}={\mathrm{K}}_{\mathrm{F}}\xb7{\mathrm{C}}_{\mathrm{e}}^{1/{\mathrm{n}}_{\mathrm{F}}}$ | (13) | $\mathrm{ln}\left({\mathrm{q}}_{\mathrm{e}}\right)=\frac{1}{{\mathrm{n}}_{\mathrm{F}}}\xb7\mathrm{ln}\left({\mathrm{C}}_{\mathrm{e}}\right)+\mathrm{ln}\left({\mathrm{K}}_{\mathrm{F}}\right)$ | (25) | |

Temkin | ${\mathrm{q}}_{\mathrm{e}}={\mathrm{B}}_{\mathrm{T}}\xb7\mathrm{ln}\left({\mathrm{A}}_{\mathrm{T}}\xb7{\mathrm{C}}_{\mathrm{e}}\right)$ | (14) | ${\mathrm{q}}_{\mathrm{e}}={\mathrm{B}}_{\mathrm{T}}\xb7\mathrm{ln}\left({\mathrm{A}}_{\mathrm{T}}\right)+{\mathrm{B}}_{\mathrm{T}}\xb7\mathrm{ln}\left({\mathrm{C}}_{\mathrm{e}}\right)$ | (26) | |

${\mathrm{B}}_{\mathrm{T}}=\frac{\mathrm{R}\xb7\mathrm{T}}{{\mathrm{b}}_{\mathrm{T}}}$ | (15) | ||||

Dubinin–Raduskevich | ${\mathrm{q}}_{\mathrm{e}}={\mathrm{q}}_{\mathrm{max}}\xb7\mathrm{exp}\left(-{\mathrm{B}}_{\mathrm{DR}}\xb7{\mathsf{\epsilon}}^{2}\right)$ | (16) | $\mathrm{ln}\left({\mathrm{q}}_{\mathrm{e}}\right)=\mathrm{ln}\left({\mathrm{q}}_{\mathrm{max}}\right)-{\mathrm{B}}_{\mathrm{DR}}\xb7{\mathsf{\epsilon}}^{2}$ | (27) | |

$\mathsf{\epsilon}=\mathrm{R}\xb7\mathrm{T}\xb7\mathrm{ln}\left(1+\frac{1}{{\mathrm{C}}_{\mathrm{e}}}\right)$ | (17) | ||||

$\mathrm{E}=\frac{1}{\sqrt{2\xb7{\mathrm{B}}_{\mathrm{DR}}}}$ | (18) | ||||

Sips ^{1} | ${\mathrm{q}}_{\mathrm{e}}=\frac{{\mathrm{K}}_{\mathrm{S}}\xb7{\mathrm{q}}_{\mathrm{max}}\xb7{\mathrm{C}}_{\mathrm{e}}^{1/{\mathrm{n}}_{\mathrm{S}}}}{1+{\mathrm{K}}_{\mathrm{S}}\xb7{\mathrm{C}}_{\mathrm{e}}^{1/{\mathrm{n}}_{\mathrm{S}}}}$ | (19) | $\mathrm{ln}\left(\frac{{\mathrm{q}}_{\mathrm{e}}}{{\mathrm{q}}_{\mathrm{max}}-{\mathrm{q}}_{\mathrm{e}}}\right)=\frac{1}{{\mathrm{n}}_{\mathrm{s}}}\xb7\mathrm{ln}\left({\mathrm{C}}_{\mathrm{e}}\right)+\mathrm{ln}\left({\mathrm{K}}_{\mathrm{S}}\right)$ | (28) |

^{1}The value of q

_{max}can be first estimated from the q

_{max}obtained from the Langmuir model as the first input for optimization using a calculation software.

**Table 3.**A comparison of the hexavalent chromium adsorption through different NC and activated carbon adsorbents.

Adsorbent | Contact Time [min] | Adsorbent Dosage [mg·L^{−1}] | Initial Cr(VI) Concentration [mg·L^{−1}] | pH | q_{max} [mg·g^{−1}] | Maximum Removal Yield [%] | Ref. |
---|---|---|---|---|---|---|---|

CNF from rice husk | 100 | 1500 | 30 | 6 | 3.76 | 92.99 | [57] |

Polypyrrole-bacterial CNF | 180 | 250 | 300 | 2 | 555.6 | 97.5 | [72] |

Thiol-modified CNF composite | 20 | 50 | 4 | 87.5 | 96 | [71] | |

Citric acid-incorporated CNF | 120 | 40 | 50 | 2 | 11 | 23 | [77] |

Amino-silanized cellulose membranes | 300 | 5000 | 50 | 4 | 34.7 | [33] | |

Polyaniline-functionalized CNC | 40 | 500 | 30 | 2.5 | 48.92 | 97.84 | [73] |

Microwave-assisted H_{3}PO_{4}/Fe-modified activated carbon | 200 | 1000 | 30 | 3 | 34.39 | 100 | [74] |

ZnCl_{2}-modified tamarind wood activated carbon | 70 | 3000 | 10 | 3 | 28.02 | 99 | [75] |

Acid-base surface modified activated carbon | 180 | 2000 | 50 | 13.89 | [76] | ||

Hydrophobized CNF Hydrogel (MTMS dosage = 1.5 mmol·g^{−1}) | 330 | 500 | 50 | 3 | 70.38 | >97.14 | This work |

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**MDPI and ACS Style**

Ojembarrena, F.d.B.; Sánchez-Salvador, J.L.; Mateo, S.; Balea, A.; Blanco, A.; Merayo, N.; Negro, C.
Modeling of Hexavalent Chromium Removal with Hydrophobically Modified Cellulose Nanofibers. *Polymers* **2022**, *14*, 3425.
https://doi.org/10.3390/polym14163425

**AMA Style**

Ojembarrena FdB, Sánchez-Salvador JL, Mateo S, Balea A, Blanco A, Merayo N, Negro C.
Modeling of Hexavalent Chromium Removal with Hydrophobically Modified Cellulose Nanofibers. *Polymers*. 2022; 14(16):3425.
https://doi.org/10.3390/polym14163425

**Chicago/Turabian Style**

Ojembarrena, Francisco de Borja, Jose Luis Sánchez-Salvador, Sergio Mateo, Ana Balea, Angeles Blanco, Noemí Merayo, and Carlos Negro.
2022. "Modeling of Hexavalent Chromium Removal with Hydrophobically Modified Cellulose Nanofibers" *Polymers* 14, no. 16: 3425.
https://doi.org/10.3390/polym14163425