# Modeling of Experimental Adsorption Isotherm Data

^{1}

^{2}

## Abstract

**:**

^{2}) and standard errors (S.E.) for each parameter were used to evaluate the data. The modeling results showed that non-linear Langmuir model could fit the data better than others, with relatively higher r

^{2}values and smaller S.E. The linear Langmuir model had the highest value of r

^{2}, however, the maximum adsorption capacities estimated from linear Langmuir model were deviated from the experimental data.

## 1. Introduction

## 2. Experiments

#### 2.1. The Experimental Isotherm Data

_{0}= 0.5, 1.0, 5.0, 10.0, 25.0, and 55.0 mg P/L). As for the equilibrium experiments, about 0.05 g adsorbent was added into 100 mL phosphate solution in a polypropylene bottle with various initial concentrations. After being shaken at 35 ºC for 2 h, the solutions were removed by filtering thru 0.45 μm syringe nylon-membrane filters, and the equilibrium phosphate concentrations (C

_{e}, mg P/L) in the filtrates were analyzed by Autoanalyzer 3 (Bran and Luebbe Inc., Norderstedt, Germany). The amount of phosphate adsorbed at equilibrium (q

_{e}, mg P/g) was calculated by Equation (1):

_{0}and C

_{e}(mg P/L) are the initial and equilibrium phosphate concentrations, respectively, V (L) is the volume of the solution and m (g) is the mass of the adsorbent.

**Table 1.**Adsorption data of four adsorbents for phosphate removal in solution with different initial concentrations.

C_{0}\Samples | M1 | M2 | M3 | M4 | ||||
---|---|---|---|---|---|---|---|---|

C_{e} | q_{e} | C_{e} | q_{e} | C_{e} | q_{e} | C_{e} | q_{e} | |

0.5 | 0.49 | 0.021 | 0.061 | 0.93 | 0.037 | 0.98 | 0.052 | 0.95 |

1.0 | 0.78 | 0.39 | 0.023 | 1.91 | 0.013 | 1.92 | 0.017 | 1.94 |

5.0 | 3.67 | 1.91 | 0.35 | 8.56 | 0.036 | 9.26 | 0.034 | 9.26 |

10.0 | 9.03 | 3.88 | 2.45 | 17.18 | 0.15 | 19.49 | 0.33 | 18.97 |

25.0 | 23.98 | 4.82 | 16.43 | 19.74 | 5.74 | 42.22 | 4.12 | 44.95 |

55.0 | 53.25 | 5.01 | 44.40 | 20.91 | 29.54 | 44.30 | 26.09 | 52.14 |

#### 2.2. Non-linear Forms of the Isotherm Models

_{e}is the concentration of phosphate solution at equilibrium (mg P/L); q

_{e}is the corresponding adsorption capacity (mg P/g); q

_{m}(mg P/g) and K

_{L}(L/mg) are constants which are related to adsorption capacity and energy or net enthalpy of adsorption, respectively.

_{F}and n are the constants, which measure the adsorption capacity and intensity; respectively.

_{s}(mg P/g) is a constant in the Dubinin-Radushkevich isotherm model which are related to adsorption capacity; K

_{DR}(mol

^{2}/kJ

^{2}) is a constant in related to the mean free energy of adsorption; R (J/mol K) is the gas constant; and T (K) is the absolute temperature.

#### 2.3. Linear Forms of the Isotherm Models

_{e}/q

_{e}against C

_{e}it is possible to obtain the value of the Langmuir constant K

_{L}and q

_{m}; by plotting ln(q

_{e}) against ln(C

_{e}), the Freundlich constant of K

_{F}and n can be determined; and by plotting ln(q

_{e}) against ε

^{2}, the Dubinin-Radushkevich constants of K

_{D}and q

_{s}can be obtained.

Isotherm models | Linear form | |
---|---|---|

Langmuir | I | $\frac{{C}_{e}}{{q}_{e}}=\frac{1}{{q}_{m}{K}_{L}}+\frac{{C}_{e}}{{q}_{m}}$ |

II | $\frac{1}{{q}_{e}}=\left[\frac{1}{{q}_{m}{K}_{L}}\right]\frac{1}{{C}_{e}}+\frac{1}{{q}_{m}}$ | |

III | ${q}_{e}={q}_{\text{m}}-\left[\frac{1}{{K}_{L}}\right]\frac{{q}_{e}}{{C}_{e}}$ | |

IV | $\frac{{q}_{e}}{{C}_{e}}={K}_{L}{q}_{\text{m}}-{K}_{L}{q}_{e}$ | |

Freundlich | $\text{l}n{q}_{e}=\mathrm{ln}{K}_{F}+\frac{1}{n}\mathrm{ln}{C}_{e}$ | |

Dubinin-Radushkevich | $\mathrm{ln}{q}_{e}=\mathrm{ln}{q}_{s}-{K}_{D}{\epsilon}^{2}$ |

^{2}) was also used to determine the best-fitting isotherm to the experimental data, illustrated as Equation (6).

_{m}is the constant obtained from the isotherm model, q

_{e}is the equilibrium capacity obtained from experimental data, and $\overline{{q}_{e}}$ is the average of q

_{e}.

## 3. Results and Discussion

#### 3.1. Non-linear Fitting of the Isotherm Models

^{2}) and related standard errors (S.E.) for each parameter. In Table 3 and Figure 1, high r

^{2}are derived by fitting experimental data into the Langmuir isotherm model (r

^{2}> 0.973) and the Dubinin-Radushkevich isotherm model (r

^{2}> 0.946), as compared with the Freundlich isotherm model (r

^{2}> 0.858). Meanwhile, the values of S.E. for each parameter obtained in Freundlich isotherm model are correspondingly higher than that of the other two models. These suggest that both Langmuir isotherm model and the Dubinin-Radushkevich isotherm model can generate a satisfactory fit to the experimental data, while Freundlich isotherm model cannot. As shown, the values of maximum adsorption capacity determined using Langmuir model was 5.92, 20.82, 44.36, and 52.53 mg P/g for M1, M2, M3, and M4, respectively. These values are near the experimental adsorbed amounts and correspond closely to the adsorption isotherm plateau, which indicates that the modeling of Langmuir for the adsorption system is acceptable. Moreover, the adsorption feature of the experimental system might be caused by the monolayer adsorption. The values of theoretical monolayer saturation capacity in the Dubinin-Radushkevich model obtained using non-linear regression are all lower than the experimental amounts corresponding to the adsorption isotherm plateau, indicating that the modeling of Dubinin-Radushkevich for the adsorption system is unacceptable. Therefore, by comparison, the order of the isotherm best fits the four sets of experimental data in this study is Langmuir > Dubinin-Radushkevich > Freundlich.

**Figure 1.**Non-linear fitting of (

**a**) Langmuir, (

**b**) Freundlich, and (

**c**) Dubinin-Radushkevich isotherm models for the four adsorbents (M1, M2, M3 and M4).

**Table 3.**Langmuir, Freundlich and Dubinin-Radushkevich isotherm parameters obtained by nonlinear fitting for the four adsorbents (M1, M2, M3 and M4).

Model/parameters | M1 | M2 | M3 | M4 | |||||
---|---|---|---|---|---|---|---|---|---|

Value | S.E. | Value | S.E. | Value | S.E. | Value | S.E. | ||

Langmuir | q_{m} (mg P/g) | 5.92 | 0.51 | 20.82 | 0.62 | 44.36 | 2.38 | 52.53 | 3.24 |

b (L/mg P) | 0.152 | 0.045 | 1.890 | 0.310 | 4.470 | 1.154 | 1.682 | 0.518 | |

r^{2} | 0.973 | - | 0.993 | - | 0.979 | - | 0.978 | - | |

Freundlich | K_{F} | 1.18 | 0.45 | 9.91 | 1.96 | 21.61 | 4.17 | 22.88 | 4.32 |

n | 2.541 | 0.729 | 4.440 | 1.259 | 4.103 | 1.096 | 3.572 | 0.853 | |

r^{2} | 0.858 | - | 0.857 | - | 0.868 | - | 0.894 | - | |

Dubinin-Radushkevich | q_{s} (mg P/g) | 4.93 | 0.16 | 19.67 | 0.82 | 43.32 | 2.17 | 49.10 | 3.72 |

K_{D} (mol^{2}/kJ^{2}) | 2.79 | 0.37 | 0.074 | 0.014 | 0.032 | 0.005 | 0.080 | 0.024 | |

r^{2} | 0.989 | - | 0.977 | - | 0.975 | - | 0.946 | - |

#### 3.2. Linear Fitting of the Isotherm Models

^{2}) and related standard errors (S.E.) for each parameter. According to the r

^{2}and related S.E. for each parameter in Table 4, the Langmuir model fitted the experimental data best by linear analysis, while the Freundlich fitted worst. These results are in good agreement with the results indicated by nonlinear analysis. However, the r

^{2}(0.995, 0.999, 0.997 and 0.990 for M1, M2, M3, and M4, respectively) were all higher than the corresponding r

^{2}obtained by nonlinear analysis methods. Moreover, although the r

^{2}in Dubinin-Radushkevich model for M3 (0.980) is higher than that obtained by non-linear analysis (0.975), the r

^{2}for the other three adsorbents are lower. In particularly, the r

^{2}for M2 (0.688) is extremely lower than its corresponding r

^{2}by non-linear analysis (0.977). Meanwhile, all the S.E. values are much higher than those determined in non-linear analysis. These indicate that the linear fitting of experimental data into Dubinin-Radushkevich model may cause great fluctuation of r

^{2}, and the predicted parameters may induce deviation.

**Figure 2.**Linear fitting plots of (

**a**) Langmuir, (

**b**) Freundlich, and (

**c**) Dubinin-Radushkevich isotherm models for the four adsorbents (M1, M2, M3 and M4).

**Table 4.**Langmuir, Freundlich and Dubinin-Radushkevich isotherm parameters obtained by linear fitting for the four adsorbents (M1, M2, M3 and M4).

Model | M1 | M2 | M3 | M4 | |||||
---|---|---|---|---|---|---|---|---|---|

Value | S.E. | Value | S.E. | Value | S.E. | Value | S.E. | ||

Langmuir | q_{m} (mg P/g) | 6.29 | 0.19 | 21.18 | 0.26 | 45.17 | 1.11 | 54.23 | 2.59 |

b (L/mg P) | 0.135 | 0.018 | 1.352 | 0.442 | 1.815 | 1.01 | 0.985 | 0.524 | |

r^{2} | 0.995 | - | 0.999 | - | 0.997 | - | 0.990 | - | |

Freundlich | K_{F} | 0.67 | 0.14 | 6.95 | 1.87 | 15.97 | 7.15 | 16.38 | 7.19 |

n | 1.661 | 0.254 | 2.535 | 0.625 | 2.346 | 0.791 | 2.094 | 0.666 | |

r^{2} | 0.916 | - | 0.806 | - | 0.701 | - | 0.714 | - | |

Dubinin-Radushkevich | q_{s} (mg P/g) | 3.87 | 0.63 | 15.25 | 5.82 | 42.99 | 4.83 | 42.14 | 10.74 |

K_{D} (mol^{2}/kJ^{2}) | 0.52 | 0.06 | 0.032 | 0.009 | 0.026 | 0.002 | 0.028 | 0.005 | |

r^{2} | 0.946 | - | 0.688 | - | 0.980 | - | 0.900 | - |

#### 3.3. Comparison of Maximum Adsorption Capacities (q_{m})

_{m}) of the adsorbents, which were calculated by both the linear and nonlinear Langmuir model, with other adsorbents in literatures.

Adsorbents | Isotherm models | q_{m} (mg P/g) | D^{a} (mg P/g) | References |
---|---|---|---|---|

M1 | Nonlinear Langmuir | 5.92 | 0.91 | Present work |

Linear Langmuir | 6.29 | 1.28 | ||

M2 | Nonlinear Langmuir | 20.82 | 0.09 | Present work |

Linear Langmuir | 21.18 | 0.27 | ||

M3 | Nonlinear Langmuir | 44.36 | 0.06 | Present work |

Linear Langmuir | 45.17 | 0.87 | ||

M4 | Nonlinear Langmuir | 52.53 | 0.39 | Present work |

Linear Langmuir | 54.23 | 2.09 | ||

activated carbon fiber (ACF-NanoHFO) | Linear Langmuir | 12.86 | - | [3] |

Fe-Zr binary oxide | Nonlinear Langmuir | 13.65 | - | [4] |

aluminum pillared bentonites | Nonlinear Langmuir | 12.70 | - | [5] |

Fe(III)-coordinated amino-functionalized SBA-15 | Linear Langmuir | 20.70 | - | [9] |

^{a}means the differences between the maximum adsorption capacities derived from linear or non-linear models and the experimental data.

_{m}increases in the order M4 > M3 > M2 > M1, and the q

_{m}of M4 exhibits a superior phosphate adsorption capacity, when compared to other reported adsorbents. The values of q

_{m}predicted by the linear Langmuir model (6.29, 21.18, 45.17, and 54.23 mg P/g for M1, M2, M3, and M4) are all higher than that calculated from nonlinear Langmuir model, and beyond the adsorption isotherm plateau of the experimental data. Moreover, the differences (D

^{a}) between the q

_{m}derived from linear Langmuir model and the experimental data are higher than that between the nonlinear Langmuir model and the experimental data, shown in Table 5. This indicates that, as for this adsorption system, i.e., Fe(III)-coordinated amino-functionalized mesoporous silica materials for phosphate removal, the results derived from linear fitting of the isotherm models can cause discrepancy.

## 4. Conclusions

_{m}) increases in the order M4 > M3 > M2 > M1, and the q

_{m}of M4 exhibits a superior phosphate adsorption capacity than that of several other adsorbents in literatures. Although with higher r

^{2}values, the q

_{m}of the four adsorbents estimated from linear Langmuir model were all higher than that from nonlinear Langmuir model, which were all beyond the experimental data with higher differences (D

^{a}). Therefore, it can be concluded that, as for the adsorption system of Fe(III)-coordinated amino-functionalized mesoporous silica materials for phosphate removal, the non-linear isotherm models are more powerful and viable in modeling the adsorption isotherm data.

## Acknowledgements

## Conflicts of Interest

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Chen, X.
Modeling of Experimental Adsorption Isotherm Data. *Information* **2015**, *6*, 14-22.
https://doi.org/10.3390/info6010014

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Chen X.
Modeling of Experimental Adsorption Isotherm Data. *Information*. 2015; 6(1):14-22.
https://doi.org/10.3390/info6010014

**Chicago/Turabian Style**

Chen, Xunjun.
2015. "Modeling of Experimental Adsorption Isotherm Data" *Information* 6, no. 1: 14-22.
https://doi.org/10.3390/info6010014