The three widely used isotherms Langmuir, Freundlich and Temkin were examined in an experiment using fluoride (F^{−}) ion adsorption on a geo-material (limonite) at four different temperatures by linear and non-linear models. Comparison of linear and non-linear regression models were given in selecting the optimum isotherm for the experimental results. The coefficient of determination, r^{2}, was used to select the best theoretical isotherm. The four Langmuir linear equations (1, 2, 3, and 4) are discussed. Langmuir isotherm parameters obtained from the four Langmuir linear equations using the linear model differed but they were the same when using the nonlinear model. Langmuir-2 isotherm is one of the linear forms, and it had the highest coefficient of determination (r^{2} = 0.99) compared to the other Langmuir linear equations (1, 3 and 4) in linear form, whereas, for non-linear, Langmuir-4 fitted best among all the isotherms because it had the highest coefficient of determination (r^{2} = 0.99). The results showed that the non-linear model may be a better way to obtain the parameters. In the present work, the thermodynamic parameters show that the absorption of fluoride onto limonite is both spontaneous (ΔG < 0) and endothermic (ΔH > 0). Scanning electron microscope and X-ray diffraction images also confirm the adsorption of F^{−} ion onto limonite. The isotherm and kinetic study reveals that limonite can be used as an adsorbent for fluoride removal. In future we can develop new technology for fluoride removal in large scale by using limonite which is cost-effective, eco-friendly and is easily available in the study area.

## INTRODUCTION

Equilibrium relationships between adsorbent and adsorbate are illustrated by the certain constants of adsorption isotherm, and values of these constants provide information about adsorbent capacity and affinity for some kind of adsorbate. At fixed temperatures, the isotherm equation explains the solid–liquid adsorption system. Isotherms can be obtained by examining adsorption in batch reactions. The linear regression model is used to determine the best-fitting isotherms. The quality of the fit of experimental results indicates whether the adsorption process is monolayer, multilayer, homogeneous or heterogeneous surface (Brdar *et al.* 2012). The linear least-square model with linearly transformed isotherm equations has been widely applied to confirm the experimental result and isotherms using coefficient of determination (Ho 2006). However, conversion of non-linear isotherm equations to linear forms unconditionally modify their error structure and may also violate the error variance and normality assumption of standard least squares (Brdar *et al.* 2012). For many years, several error analysis methods like coefficient of determination, the sum of the errors squared, the average relative error, the sum of the absolute errors and chi-squared test have been used to determine the best-fitting isotherm equation (Ho *et al.* 2002; Allen *et al.* 2003; Ho 2004).

In this study, adsorption capacity of the F^{−} ion onto a geo-material (limonite) was examined. The linear least-squares model and a non-linear model of three widely used isotherms, Freundlich (1906), Langmuir (1916) and Temkin & Pyzhev (1940), were compared in an experimental result of absorption of fluoride ion onto adsorbent. A trial-and-error procedure was used for the nonlinear model using the *solver* add-in with Microsoft Excel. The solver is a Microsoft Excel add-in program used for the non-linear model with adjustable parameters, through which some data can fit the model in the most effective way.

## MATERIALS AND METHODS

### Adsorbent preparation

In the present study the derived limonite was collected from the NMDC (National Mineral Development Corporation) Chemical Laboratory, BIOM (Bailladila Iron Ore Mines), Bacheli, Chhattisgarh (India). The physico-chemical characterization and composition of limonite is given in Table 1. The ore was divided into small pieces, then washed several times with water to remove the dirt and other materials attached to its surface. Final washing was done with double-distilled water, and the pieces were dried at 105–110 °C for 24 hours. The resulting product was cooled to room temperature and then screened through a 14-mesh sieve (>53); finally, the product was stored in vacuum desiccators until required (Chaudhari & Sasane 2014).

Composition/Properties | |
---|---|

FeO | 43.6% |

SiO_{2} | 26.5% |

Al_{2}O_{3} | 18.4% |

TiO_{2} | 10.1% |

P_{2}O_{5} | 0.56% |

Crystal system | Amorphous |

Color | Light yellow |

Hardness | 4.5–5.2 |

Density | 2.7–3.4 g/cm^{3} |

Specific gravity | 2.8–3.2 |

Composition/Properties | |
---|---|

FeO | 43.6% |

SiO_{2} | 26.5% |

Al_{2}O_{3} | 18.4% |

TiO_{2} | 10.1% |

P_{2}O_{5} | 0.56% |

Crystal system | Amorphous |

Color | Light yellow |

Hardness | 4.5–5.2 |

Density | 2.7–3.4 g/cm^{3} |

Specific gravity | 2.8–3.2 |

*Source:* Department of Geology, NMDC Ltd, BIOM, Bacheli, Chhattisgarh, India.

### Adsorption experiments

The adsorption isotherm and kinetic experiments were performed by a batch adsorption technique, followed by mixing 1.25 g (optimized amount) of adsorbent with 100 ml of sodium fluoride containing 3 mg/l as initial fluoride concentration. The mixture was agitated in a thermostatic shaker at a speed of 250 rpm at four different temperatures. The defluoridation studies were conducted for the optimization of various experimental conditions like contact time, initial F^{−} ion concentration, amount of adsorbent and particle size (Chen *et al.* 2010; Alagumuthu *et al.* 2012). The reagents used in the present study are of analytical grade. Standard solutions of F^{−} were prepared by the appropriate dilution of a stock standard solution (100 mg/l, Merck) of F^{−} with double-distilled water. F^{−} ion concentration was measured by a Nova 60 Spectroquant^{®} photometer. A LI-617 pH meter (Elico, Hyderabad, India) was used for pH measurements. Kinetic studies of adsorbent were carried out in a temperature controlled mechanical shaker (REMI RS-24BL, Rotator). The effect of different initial F^{−} ion concentrations 5.0, 10.0, 15.0, 20.0 , 25.0 and 30.0 mg/l at temperature 303, 313, 323 and 333 K were studied by keeping the mass of adsorbent as 1.25 g and volume of solution as 100 ml in pH 7 ± 0.3.

Initial pH adjustments were carried out by adding either a 0.1M H_{2}SO_{4} or 0.1M NaOH. After shaking the flasks for 3 hours, the reaction mixtures were filtered through filter paper (Whatman filter paper 40), and then the filtrates were analyzed for the remaining F^{−} ion concentration with a Nova 60 Spectroquant^{®} photometer.

## RESULTS AND DISCUSSION

### Isotherm study

^{2}, was used to test the best-fitting isotherm to the experimental result and, in addition, it also helped to determine the relationship between dependent and independent variables. Mathematically, it is given by the following equation: where

*Q*

_{m}is the equilibrium adsorption capacity obtained from the isotherm model,

*Q*

_{e}is the equilibrium capacity obtained from experiment, and is the average of

*Q*

_{e}.

### Linear regression model

*et al.*2002; Alagumuthu

*et al.*2012). Among the four Langmuir isotherms (1, 2, 3 and 4), Langmuir-1 and Langmuir-2 are frequently used. The experimental results are best fitted in the Langmuir-2 isotherm model because the deviations from the fitted equations is the minimum, resulting in the best error distribution (Karthikeyan

*et al.*2002; Ho

*et al.*2002). The four linear Langmuir equations with the experimental results for the adsorption of F

^{−}ion onto limonite at four different temperatures, 303, 313, 323 and 333 K, are shown in Figures 1–4. The values of the monolayer equilibrium adsorption capacity (Q

_{M}), and the Langmuir adsorption equilibrium constant, K

_{L}, are shown in Table 3 for the adsorption of F

^{−}ion onto limonite at 303, 313, 323 and 333 K. The coefficient of determination (r

^{2}), obtained from Langmuir-2 is the maximum among the four Langmuir isotherms. It gives strong positive evidence for the adsorption of F

^{−}ion onto limonite. The conversion of the non-linear Langmuir isotherm equation of linear forms absolutely amends the error structure. The linear analysis used for the calculation of Langmuir parameters through different linear forms of the Langmuir equation will be considerably changed. Among the four types of Langmuir isotherm equation, Langmuir-2 is the best fit to the experimental result as shown in Table 3.

Isotherm | Mathematical formula | Linear form | Plot |
---|---|---|---|

Langmuir-1 | Q_{e} = Q_{m}K_{L}C_{eq}/1 + K_{L}C_{eq} | C_{eq}/Q_{e} = C_{eq}/Q_{m} +1/K_{L}Q_{m} | C_{eq}/Q_{e} vs C_{eq} |

Langmuir-2 | 1/q_{e} = (1/K_{L}Q_{m})1/C_{eq} +1/Q_{m} | 1/Q_{e} vs 1/C_{eq} | |

Langmuir-3 | Q_{e} = Q_{m}−(1/K_{L}) Q_{e}/C_{eq} | Q_{e} vs Q_{e}/C_{eq} | |

Langmuir-4 | Q_{e}/C_{eq} = K_{L}Q_{m}−K_{L}Q_{e} | Q_{e}/C_{eq} vs Q_{e} | |

Freundlich | Q_{e} = K_{F} C_{eq}^{1/n} | log Q_{e} = logK_{F} + 1/n (log C_{eq}) | log Q_{e} vs log C_{eq} |

Temkin | Q_{e} = B_{T}ln(K_{T}C_{eq})/b | Q_{e} = B_{T}lnK_{T} + B_{T}lnC_{eq} | Q_{e} vs log C_{eq} |

Isotherm | Mathematical formula | Linear form | Plot |
---|---|---|---|

Langmuir-1 | Q_{e} = Q_{m}K_{L}C_{eq}/1 + K_{L}C_{eq} | C_{eq}/Q_{e} = C_{eq}/Q_{m} +1/K_{L}Q_{m} | C_{eq}/Q_{e} vs C_{eq} |

Langmuir-2 | 1/q_{e} = (1/K_{L}Q_{m})1/C_{eq} +1/Q_{m} | 1/Q_{e} vs 1/C_{eq} | |

Langmuir-3 | Q_{e} = Q_{m}−(1/K_{L}) Q_{e}/C_{eq} | Q_{e} vs Q_{e}/C_{eq} | |

Langmuir-4 | Q_{e}/C_{eq} = K_{L}Q_{m}−K_{L}Q_{e} | Q_{e}/C_{eq} vs Q_{e} | |

Freundlich | Q_{e} = K_{F} C_{eq}^{1/n} | log Q_{e} = logK_{F} + 1/n (log C_{eq}) | log Q_{e} vs log C_{eq} |

Temkin | Q_{e} = B_{T}ln(K_{T}C_{eq})/b | Q_{e} = B_{T}lnK_{T} + B_{T}lnC_{eq} | Q_{e} vs log C_{eq} |

C_{eq} = equilibrium concentration (mg/l), Q_{e} = amount of ion adsorbed (mg/g), equilibrium capacity obtained from the isotherm model = monolayer equilibrium adsorption capacity (mg/g), K_{L} = Langmuir adsorption equilibrium constant (l/mg), K_{F} = Freundlich adsorption capacity (mg/g)(l/mg)^{1/n}, n = adsorption intensity, B_{T} = Temkin isotherm constant (J/mol), K_{T} = Temkin isotherm equilibrium binding constant (l/mg).

Isotherm | Parameter | 303 (K) | 313 (K) | 323 (K) | 333 (K) |
---|---|---|---|---|---|

Langmuir-1 | Q_{m} (mg/g) | 8.403 | 8.703 | 9.140 | 10.802 |

K_{L} (l/mg) | 0.0348 | 0.0433 | 0.0546 | 0.047 | |

r^{2} | 0.952 | 0.956 | 0.9699 | 0.9586 | |

Langmuir-2 | Q_{m} (mg/g) | 6.402 | 6.570 | 6.720 | 6.860 |

K_{L} (l/mg) | 0.0352 | 0.0358 | 0.0366 | 0.0365 | |

r^{2} | 0.9879 | 0.9912 | 0.9932 | 0.9936 | |

ΔG (KJ/mol) | −6.00 | −6.21 | −6.41 | −6.61 | |

Langmuir-3 | Q_{m} (mg/g) | 6.185 | 6.445 | 6.724 | 7.064 |

K_{L} (l/mg) | 0.0363 | 0.0348 | 0.0333 | 0.0315 | |

r^{2} | 0.7549 | 0.7603 | 0.7630 | 0.7591 | |

Langmuir-4 | Q_{m} (mg/g) | 6.832 | 6.5261 | 6.1574 | 6.280 |

K_{L} (l/mg) | 0.0351 | 0.0441 | 0.0578 | 0.0617 | |

r^{2} | 0.8657 | 0.8952 | 0.8857 | 0.8222 | |

Freundlich | K_{F} [(mg/g) (l/mg)^{1/n}] | 0.2655 | 0.2758 | 0.2860 | 0.2861 |

1/n | 1.399 | 1.398 | 1.3956 | 1.3738 | |

r^{2} | 0.9102 | 0.9134 | 0.9132 | 0.9093 | |

Temkin | B_{T} (J/mol) | 1.247 | 1.298 | 1.354 | 1.430 |

K_{T} (l/mg) | 0.6856 | 0.6868 | 0.6852 | 0.6732 | |

r^{2} | 0.9099 | 0.9131 | 0.9126 | 0.9094 |

Isotherm | Parameter | 303 (K) | 313 (K) | 323 (K) | 333 (K) |
---|---|---|---|---|---|

Langmuir-1 | Q_{m} (mg/g) | 8.403 | 8.703 | 9.140 | 10.802 |

K_{L} (l/mg) | 0.0348 | 0.0433 | 0.0546 | 0.047 | |

r^{2} | 0.952 | 0.956 | 0.9699 | 0.9586 | |

Langmuir-2 | Q_{m} (mg/g) | 6.402 | 6.570 | 6.720 | 6.860 |

K_{L} (l/mg) | 0.0352 | 0.0358 | 0.0366 | 0.0365 | |

r^{2} | 0.9879 | 0.9912 | 0.9932 | 0.9936 | |

ΔG (KJ/mol) | −6.00 | −6.21 | −6.41 | −6.61 | |

Langmuir-3 | Q_{m} (mg/g) | 6.185 | 6.445 | 6.724 | 7.064 |

K_{L} (l/mg) | 0.0363 | 0.0348 | 0.0333 | 0.0315 | |

r^{2} | 0.7549 | 0.7603 | 0.7630 | 0.7591 | |

Langmuir-4 | Q_{m} (mg/g) | 6.832 | 6.5261 | 6.1574 | 6.280 |

K_{L} (l/mg) | 0.0351 | 0.0441 | 0.0578 | 0.0617 | |

r^{2} | 0.8657 | 0.8952 | 0.8857 | 0.8222 | |

Freundlich | K_{F} [(mg/g) (l/mg)^{1/n}] | 0.2655 | 0.2758 | 0.2860 | 0.2861 |

1/n | 1.399 | 1.398 | 1.3956 | 1.3738 | |

r^{2} | 0.9102 | 0.9134 | 0.9132 | 0.9093 | |

Temkin | B_{T} (J/mol) | 1.247 | 1.298 | 1.354 | 1.430 |

K_{T} (l/mg) | 0.6856 | 0.6868 | 0.6852 | 0.6732 | |

r^{2} | 0.9099 | 0.9131 | 0.9126 | 0.9094 |

_{T}and K

_{T}are calculated by the slope and intercept of the linear form of Equation (2). From Table 3 it is clear that the Temkin isotherm model gave the best fit after Langmuir-2 and Langmuir-1. The Temkin isotherm for the experimental results of the adsorption of F

^{−}ion onto limonite at four different temperatures (303, 313, 323 and 333 K) is shown in Figure 5.

^{−}ion. The same sets of experimental results have been used by plotting log (Q

_{e}) versus log (C

_{e}). The Freundlich isotherm for the experimental results of the adsorption of F

^{−}ion onto limonite at four different temperatures (303, 313, 323 and 333 K) is shown in Figure 6. The experimental results obtained from the linear form of Langmuir-2 and Langmuir-1 were more suitable than the Freundlich isotherm because of the higher value of the coefficient of determinations (Table 3). In contrast, the Freundlich isotherm was more suitable for the experimental results than was the Langmuir-3 and Langmuir-4. The Freundlich isotherm constant (K

_{F}), 1/n and the coefficients of determination are shown in Table 3. Figure 7 shows the comparison between the four types of Langmuir isotherm equation with respect to the experimental value at 303 K; it is clear that Langmuir-2 gave the best fit to the actual value.

^{−}ion onto limonite at a temperature of 313 K, the Langmuir model gave the best fit, because it is closer to the actual values (Figure 8).

### Non-linear method

_{L}and Q

_{m}was closer to those obtained in the non-linear model. Consequently, the Temkin isotherm was found to be a suitable model for this adsorption system.

Isotherm | Parameter | 303 (K) | 313 (K) | 323 (K) | 333 (K) |
---|---|---|---|---|---|

Langmuir-4 | Q_{m} (mg/g) | 6.265 | 6.149 | 6.505 | 6.562 |

K_{L} (l/mg) | 0.06183 | 0.05758 | 0.04410 | 0.03498 | |

r^{2} | 0.9999 | 0.9989 | 0.9979 | 0.9776 | |

Freundlich | K_{F} [(mg/g) (l/ mg)^{1/n}] | 6.325 | 5.776 | 6.075 | 6.475 |

1/n | 1.56 | 1.403 | 1.402 | 1.401 | |

r^{2} | 0.9999 | 0.9989 | 0.9978 | 0.9976 | |

Temkin | B_{T} (J/mol) | 1.3274 | 1.3224 | 1.3357 | 1.3486 |

K_{T} (l/mg) | 0.8254 | 0.8829 | 0.8847 | 0.8881 | |

r^{2} | 0.9906 | 0.9936 | 0.9942 | 0.9999 |

Isotherm | Parameter | 303 (K) | 313 (K) | 323 (K) | 333 (K) |
---|---|---|---|---|---|

Langmuir-4 | Q_{m} (mg/g) | 6.265 | 6.149 | 6.505 | 6.562 |

K_{L} (l/mg) | 0.06183 | 0.05758 | 0.04410 | 0.03498 | |

r^{2} | 0.9999 | 0.9989 | 0.9979 | 0.9776 | |

Freundlich | K_{F} [(mg/g) (l/ mg)^{1/n}] | 6.325 | 5.776 | 6.075 | 6.475 |

1/n | 1.56 | 1.403 | 1.402 | 1.401 | |

r^{2} | 0.9999 | 0.9989 | 0.9978 | 0.9976 | |

Temkin | B_{T} (J/mol) | 1.3274 | 1.3224 | 1.3357 | 1.3486 |

K_{T} (l/mg) | 0.8254 | 0.8829 | 0.8847 | 0.8881 | |

r^{2} | 0.9906 | 0.9936 | 0.9942 | 0.9999 |

After comparing the values in Tables 3 and 4 for the linear and non-linear method, it was found that Temkin and Freundlich isotherms are different models, but have a coefficient of determination that is similar for both methods at four different temperatures. It has been reported that it is inappropriate to use the coefficient of determination of a linear regression analysis for comparing the best-fitting solution of different isotherms (Allen *et al.* 2003). Linear regression has produced quite different outcomes as compared with non-linear regression. Among the three isotherms the Langmuir isotherm was considered to be a suitable model with a high coefficient of determination in the case of the non-linear method as compared with the linear method (Figure 9). Unlike the linear analysis, a different isotherm would significantly affect the r^{2} value and impact the final determination of parameters; however, use of the non-linear model would avoid such errors.

### Thermodynamic parameters

^{−}ion onto limonite was examined at four different temperatures of 303, 313, 323 and 333 K under the optimized condition and three thermodynamic parameters, free energy change (ΔG), enthalpy change (ΔH) and entropy change (ΔS) respectively. By using the experimental result all these thermodynamic parameters were calculated. Gibb's free energy change (ΔG) is the fundamental criterion of spontaneity. Reactions occur spontaneously at a given temperature if ΔG is a negative value. The thermodynamic parameters ΔG, ΔH and ΔS for the adsorption process are calculated using the following equations at four different temperatures: where R is the universal gas constant (8.314 J/(mol K)) and T is the temperature in K.

_{L}value (Table 3) of Langmuir-2 in Equation (3), a plot of ΔG versus temperature was obtained. The plot was found to be linear (Figure 10). The values of ΔH and ΔS were respectively determined from the slope and intercept of the plots (Equation (4)). The values of ΔG are shown in Table 3. Values of ΔH and ΔS for adsorption processes were calculated to be 147 J/mol and 20 J/(mol K) respectively. The negative value of ΔG confirms the feasibility of the process and the spontaneous nature of adsorption with a high preference for fluoride to adsorb onto limonite. The value of ΔH was positive, indicating that the adsorption is endothermic. The positive value of ΔS shows the increasing randomness at the solid–liquid interface during the adsorption of fluoride ions onto limonite.

### Instrumental analysis

^{−}ion. Figure 11(b) shows the image of treated limonite, quite different from the previous image. No pores seem clearer, and the shape and size of particles are very irregular. X-ray diffraction images of the untreated and fluoride-treated limonite material are given in Figure 12(a) and (b). Figure 12(b) shows appearance of new bands and disappearance of the old band, and also decrease in the percentage of transmittance indicates the adsorption of F

^{−}ion onto the limonite surface (Karthikeyan

*et al.*2002).

## CONCLUSION

It is not appropriate to use the coefficient of determination of the linear model for comparing the best-fitting isotherms. The nonlinear model is a better way to obtain the isotherm parameters because it had highest coefficient of determination. Langmuir-4 has the highest coefficient of determination compared with the other three Langmuir equations in non-linear form. The two Langmuir (1 and 2), Freundlich and Temkin isotherms had higher values of the coefficient of determination for the adsorption of F^{−} ions onto limonite in linear form. Thermodynamic parameters show that the adsorption of fluoride onto limonite is both spontaneous and endothermic. The isotherm and kinetic study reveals that limonite can be used as an adsorbent for fluoride removal. In future we can develop new technology for fluoride removal in large scale by using limonite, which is cost-effective, eco-friendly and easily available in the study area.