An insight into thermodynamics of adsorptive removal of fluoride by calcined Ca–Al–(NO₃) layered double hydroxide

Abstract

The conventional method for the estimation of ΔG⁰ through the van't Hoff equation was critically analysed. The development of an analytical framework for the determination of the thermodynamic parameters for the adsorption process had been attempted in this work. A calcined Ca–Al–(NO₃) layered double hydroxide was employed for the adsorptive removal of fluoride. The effect of the temperature on the adsorption process was assessed through thermodynamic and statistical methods. The adsorption equilibrium was described by Freundlich and linear isotherms. The isotherm constants were used in the van't Hoff equation by the proposed method. The thermodynamic assessment revealed that the adsorption reaction is endothermic and spontaneous. Furthermore, the ΔH⁰, E_a, E_ad, and ΔH_x values confirmed that the adsorption process is in between physisorption and chemisorption in nature. The interactive effect of the temperature with other important process parameters, i.e., pH, initial fluoride concentration and adsorbent dose, was evaluated through response surface methodology. The solution pH had very little effect on the adsorption. On the contrary, the adsorbent dose and initial concentration influenced the adsorption process significantly. The interactive effect of temperature was prominent at a lower adsorbent dose and a higher initial concentration. The highest adsorption capacity, obtained in the RSM study, was 59.60 mg g⁻¹. However, the highest K_f value from Freundlich isotherm was 8.48 (mg g⁻¹) (L mg⁻¹)^1/n at 50 °C.

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Introduction

Fluorine is the 13^th most abundant element in the earth crust¹ and is an essential micronutrient for human beings, helping in the formation of bone and dental enamel.² Fluorine occurs in the form of fluoride in ground water. Fluoride is required at a low concentration for preventing tooth decay, although it is detrimental at higher concentrations as it initiates fluorosis in different proportions.^3,4 As fluorosis spread over more than 35 nations,⁵ fluoride turn out to be an important contaminant in drinking water. The defluoridation of drinking water has enormous significance in the arena of research as well as for the drinking water industry/organizations. Adsorption is found to be a cheap and feasible technology in this regard.^6,7 Extensive research on the adsorptive removal of fluoride gave rise to a number of natural and synthetic adsorbents, wherein layered double hydroxides acquired significant attention.^2,8–10

Layered double hydroxides (LDHs), commonly known as hydrotalcite (HTlc)-like compounds are basically a class of synthetic or natural anionic clay. The layered structure of LDH materials is similar to brucite [Mg(OH)₂]. Two or more kinds of metallic cations present in the main layers and anionic species present in the hydrated interlayer. In this material, some divalent ions are replaced by trivalent ions resulting in a net positive surface charge, which is further counterbalanced by intercalated exchangeable anions. LDHs have the general chemical formula [M(II)_1−xM(III)_x(OH)₂]^x+[(Aⁿ⁻)_x/nyH₂O], where M(II) and M(III) represent bivalent metal ions and trivalent metal ions, respectively. A denotes an interlamellar anion with valency n- and x is the molar ratio, i.e., the ratio of the trivalent metal to the total metal ions (ranging between 0.2 and 0.4).^11,12 The net positive surface charge and availability of the exchangeable anions in the hydrated interlayer creates the high anion trapping capacity of LDHs.¹³ A wide application of LDHs is documented in various fields of chemistry, engineering, materials and bio-sciences, i.e., magnetic materials,¹⁴ fluorescence and luminescence,^15–18 nanocomposite hydrogels,¹⁹ polymer thermal stabilizers,²⁰ UV-shielding composites,²¹ biomedical,²² environmental engineering,²³ photochemistry,²⁴ catalytic processes²⁵ etc. In the field of water and wastewater treatment, LDHs are used as pollutant scavengers for various anions and oxyanions.^13,26 The defluoridation of water was performed with various LDHs,^10,27–29 of which the Mg–Al LDH was preferred by many researchers.^30–33 The calcination of LDHs expels anionic species and intercalated water resulting in the formation of metal oxides. The calcined LDH may reconstruct its layered structure by trapping target anions from an aqueous anionic solution.¹² A number of literatures established the enhancement of the defluoridation capacity of LDHs upon calcination.^27,34 Given the affinity of calcium and aluminum towards fluoride and their wide application in the field of water treatment, a calcined Ca–Al–(NO₃) LDH has been considered for the adsorption of fluoride in the present study.

The adsorption of anions by a LDH is largely dependent on the factors related to the adsorption process. The temperature of the adsorption process controls the thermodynamics of the adsorption. The study of the effect of temperature on the adsorption is essential for understanding and controlling the adsorption reaction. One of the most important parameters for adsorption thermodynamics is the Gibbs free surface energy change (ΔG⁰), which indicates the spontaneity of a reaction. Although experimental methods for the determination of the enthalpy of adsorption have been reported using calorimetric measurements,³⁵ in a number of literatures the determination of thermodynamic parameters is conducted from isotherm experiments performed at different temperatures. The ΔG⁰ value was attained mostly from the van't Hoff equation, where the equilibrium constant is correlated with ΔG⁰. However, crucial issues such as the correct estimation of the equilibrium constant and the non-dimensionality of the van't Hoff constant were not adequately addressed so far.^36–40 Furthermore, the change in the standard entropy and enthalpy would also be erroneously appraised without the correct assessment of the thermodynamic equilibrium constant (K_eq). Correct methodology for the estimation of the van't Hoff constant is unequivocally a gap in this field.

The assessment of the combined influence of temperature with other important process parameters is necessary to understand the adsorption thermodynamics. Multivariate optimization techniques are of pivotal importance for assessing the optimal conditions of the adsorption and the interactive effects of the influencing parameters on the adsorption. The ‘one factor at a time’ approach was commonly performed for the optimization, although the interactive effect of the influencing factors and overall optima of the system were rarely achieved. A recent trend of using multivariate optimization became popular in this field to overcome this drawback. Among the different techniques of multivariate optimization, response surface methodology (RSM) is one of the efficient tools.⁴¹ The main advantages of RSM lie in the systematic design of experiments, application of statistical and mathematical models and achieving accuracy in the prediction model with the minimum number of experiments. RSM has been efficiently utilized to model the adsorption process, addressing the effect of the interactive parameters.^42–49 To the best of the knowledge of the authors, the use of RSM for addressing the adsorptive removal of fluoride by LDH materials has been rarely conducted. RSM can be applied for the determination of the interactive effect of the parameters influencing the defluoridation by LDH materials as well as appraising the overall optima of the adsorption process.

In this paper, an analytical method for the estimation of the thermodynamic equilibrium constant has been introduced. The effect of the temperature on the adsorption process in the defluoridation by calcined Ca–Al–(NO₃) LDH was investigated through the thermodynamic and statistical approach. The Gibbs free surface energy change was determined from the van't Hoff equation through the proposed methodology. The thermodynamic parameters, e.g., activation energy, isosteric heat of adsorption, change in standard entropy and enthalpy etc. were also addressed to describe the adsorption process. A four factor face centered central composite design (FCCD) coupled with the response surface method was attempted to appraise the combined influence of the temperature and the other parameters, viz., pH, initial fluoride concentration and adsorbent dose.

Experimental section

Materials

All chemicals used in the study, e.g., sodium fluoride (NaF), calcium nitrate tetrahydrate [Ca(NO₃)₂·4H₂O], aluminum nitrate nonahydrate [Al(NO₃)₃·9H₂O], sodium hydroxide (NaOH) and nitric acid (HNO₃) were purchased from Merck, India. The chemicals were of analytical reagent grade and were used without further purification. Deionized (DI) water was used for the material preparation and the sorption experimentation process.

Synthesis of calcined Ca–Al–(NO₃) LDH and its characterization

The preparation and subsequent characterization of the material are described elsewhere.⁵⁰ Typically, the synthesis of the Ca–Al–(NO₃) LDH was conducted through a co-precipitation method. The precursor salts of 0.04 M Ca(NO₃)₂·4H₂O and 0.02 M Al(NO₃)₃·9H₂O were dissolved in DI water. An alkali solution of 4 M NaOH was prepared in DI water. The salt solution and alkali solution were together added drop-wise at a constant pH of 11 with continuous stirring. A nitrogen atmosphere was used to avoid carbonate intercalation. The resulting solution was thermally treated in an aging process and dried in a vacuum oven to obtain the pristine LDH. The LDH was calcined at 500 °C and manually ground to a powder form to use for further experiments. The characterization of the adsorbent was performed with XRD, FTIR, EDX, SEM and N₂ adsorption desorption analyses and was reported elsewhere.⁵⁰

Sorption experiments

A series of batch adsorption experiments were conducted in a BOD incubator shaker at controlled temperatures and agitation rate. The working volume of the fluoride solutions was taken as 100 mL in a 300 mL polypropylene bottle. The adsorption isotherms were obtained at various temperatures (10 °C to 50 °C) by varying the adsorbent dose at a range of 0.05 to 3 g L⁻¹ for initial concentration of 10 mg L⁻¹. As per the preliminary experiments, the influencing parameters selected for the RSM study were temperature, pH, initial concentration and adsorbent dose. The sorption experiments for the RSM study were conducted under the designed values of these parameters. An Orion Star A214 benchtop pH/ISE meter with an Orion 9609BNWP fluoride electrode (Thermo Scientific, USA) and TISAB III buffer were used to measure the fluoride concentration. The adsorption capacity q_t (mg g⁻¹) (the amount of fluoride adsorbed per unit mass of LDH at any time t) was calculated from eqn (1) as:

q_t = (C₀ − C_t) × V/M (1)

where C₀ (mg L⁻¹) and C_t (mg L⁻¹) are the initial concentration and the concentration of fluoride at any time t, respectively. M is the mass (g) of the adsorbent, and V is the volume (L) of the solution.

Experimental design

RSM is a combination of statistical and mathematical tools and has become an established technique for the design of experiments (DoE). The efficiency of a DoE is represented as the ratio of the number of coefficients present in the model equation to the number of experiments required.⁵¹ The central composite design, Box–Behnken design (BBD), Doehlert matrix, 3³ factorial design etc. are different RSM methods. Amongst these, BBD has the highest efficiency and 3³ factorial design has the highest accuracy.⁵¹ FCCD has become a very popular technique as the accuracy of the method is comparable to 3³ factorial design, however, the efficiency of FCCD is close to BBD. A four factor three level FCCD was employed to design the experiments and the optimization was performed through the generation of response surfaces. The temperature, pH, initial fluoride concentration and adsorbent dose were considered as the factors and the adsorption capacity was optimized as the response. The coded values of variables (X_i) were computed using eqn (2) for the design of experiments as follows:⁴¹where Z_i and Z₀ represent the actual value at the i^th level and the central point, respectively. ΔZ is the difference between the maximum or the minimum value and the value in the central point, and β is the major coded limit value for each variable.

The actual and coded values of independent variables are presented in Table 1. FCCD consists of a total number of 30 sets of experiments, out of which there are sixteen factorial, eight axial and six centre points (2⁴ + 2 × 4 + 6). The experimental design was computed by Design Expert 8.0.7.1 (Stat-Ease, Inc. USA) and is represented in Table 2. A second order polynomial model [eqn (3)], appropriate for a 4-factor, 3-level design,⁵² was used to establish a numerical relationship between the dependent and independent variables as follows:

where Y is the response; X_i and X_j are the coded variables; β₀ is a constant coefficient; β_i, β_ii and β_ij are the coefficients of the linear, quadratic and second-order interactions, respectively; k is the number of studied factors; and e is the error. Design Expert 8.0.7.1 (Stat-Ease, Inc. USA) was used to perform the RSM analysis of the experimental data. The significance of the independent variables and their individual and interactive effect on the adsorption capacity was identified and was expressed by P-values and F-values.

Table 1 Experimental range and levels of factors used in RSM

Factors	Level
	−1	0	+1	Coded value
Temperature (A)	10	30	50	Actual value
pH (B)	6	8	10
Initial fluoride concentration (C)	5	20	35
Adsorbent dose (D)	0.5	1.0	1.5

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Table 2 Design matrix and experimental results of FCCD

Run order	(A) Temperature (°C)	(B) pH	(C) Initial concentration (mg L^−1)	(D) Adsorbent dose (g L^−1)	Adsorption capacity (mg g^−1)
1	30	8	20	1	16.62
2	30	8	20	1.5	12.01
3	50	8	20	1	17.65
4	50	6	5	1.5	3.29
5	30	10	20	1	16.86
6	50	6	5	0.5	7.62
7	10	6	35	1.5	20.59
8	30	8	20	0.5	30.28
9	30	8	20	1	16.94
10	10	6	5	1.5	3.15
11	30	8	35	1	30.31
12	50	6	35	1.5	21.61
13	30	8	20	1	15.92
14	10	10	5	1.5	3.17
15	10	10	35	1.5	20.68
16	30	8	20	1	15.91
17	50	10	5	0.5	7.8
18	50	10	5	1.5	3.29
19	30	6	20	1	16.64
20	30	8	5	1	4.4
21	10	10	5	0.5	5.9
22	10	10	35	0.5	55.44
23	50	6	35	0.5	59.3
24	50	10	35	0.5	59.6
25	10	6	35	0.5	54.98
26	10	6	5	0.5	5.76
27	10	8	20	1	15.84
28	30	8	20	1	17.31
29	50	10	35	1.5	21.73
30	30	8	20	1	16.14

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Results and discussion

Adsorption isotherm

The adsorption of fluoride on the calcined Ca–Al–(NO₃) LDH was represented by the adsorption isotherms for various temperatures, viz., 10 °C, 20 °C, 30 °C, 40 °C and 50 °C. The isotherm studies were conducted for an initial fluoride concentration of 10 mg L⁻¹ with a varied adsorbent dose from 0.05 to 3 g L⁻¹. The Langmuir, Freundlich, linear and Dubinin–Raduskevich adsorption models were attempted in this study. In view of the inherent bias of the linearized forms of these models, a non-linear fit was performed using MATLAB, 2010a (The MathWorks, INC.). The general forms of the Langmuir, Freundlich, Dubinin–Raduskevich and linear models are represented in eqn (4)–(6) and (8), respectively, as follows:where q_e represents the amount of the adsorbate adsorbed per unit weight of the adsorbent at the equilibrium (mg g⁻¹), C_e is the concentration of the solute at the equilibrium (mg L⁻¹), and q_max and b are the Langmuir constants. q_max is represented as the maximum adsorption capacity (mg g⁻¹) and b is related to the binding energy or the affinity parameter of the adsorption system.

q_e = K_fC_e^1/n (5)

where K_f and n are the Freundlich constants and indicate the relative adsorption capacity and adsorption intensity, respectively.

q_e = Q_m exp(−K_adε²) (6)

where Q_m and K_ad are the maximum adsorption capacity and a constant related to the adsorption energy, respectively. The parameter ε is the Polanyi potential and is represented by eqn (7) as follows:where R is the universal gas constant and T is the absolute temperature in K.

q_e = K_cC_e (8)

where K_c is the isotherm constant. The isotherm curves are plotted in Fig. 1. The non-linearized plot represents the better applicability of the Freundlich isotherm. The linear isotherm was also fitting for this experimental data.

Fig. 1 Adsorption isotherms at (a) 10 °C, (b) 20 °C, (c) 30 °C, (d) 40 °C and (e) 50 °C.

Adsorption thermodynamics

Adsorption is a temperature dependent phenomena and the assessment of the thermodynamic parameters delineates the feasibility of the adsorption process. The spontaneity of a system is defined by evaluating ΔG⁰. At equilibrium conditions, ΔG⁰ is defined as follows:

ΔG⁰ = −RT ln K_eq (9)

where K_eq is the thermodynamic equilibrium constant. Substituting eqn (9) into the Gibbs Helmholz equation [eqn (10)], the van't Hoff equation is obtained [eqn (11)] as follows:where ΔH⁰ is the change in the standard enthalpy. Integrating eqn (11) for T at a constant pressure, the following equation is attained:

−RT ln K_eq = ΔH⁰ − TRY (12)

where Y is the integration constant. The relationship between the standard Gibbs free surface energy change, standard enthalpy change and standard entropy change (ΔS⁰) is as follows:

ΔG⁰ = ΔH⁰ − TΔS⁰ (13)

From eqn (9), (12) and (13) the following equation is attained:

If ln K_eq is plotted against 1/T (van't Hoff plot), multiplying the slope and intercept of the plot by R, ΔH⁰ and ΔS⁰ is obtained. The equilibrium constant should be dimensionless in accordance with the dimensionality of eqn (9) and (14).

Existing approaches for the determination of K_eq

A critical literature review revealed that the determination of K_eq was performed by various approaches.^{8,31,32,37,53–57} In this regard, the use of b as K_eq was found in a number of literatures.^{8,36,39,54,58} Besides, the use of the isotherm constant from other adsorption isotherm models, i.e., Freundlich, Frumkin, Flory–Huggins model etc., was also practiced.^40,57,59 The equilibrium constant was also derived by adopting the distribution coefficient (K_d) in terms of q_e/C_e,^29,60,61 the equilibrium constant (K_e) in terms of C_s/C_e (C_s is the concentration of solute adsorbed at equilibrium in mol L⁻¹ or mg L⁻¹),^56,62,63 or other theoretical approaches.^64,65 The key issue in dealing with the isotherm constant is the consideration of the dimensionality. The various approaches for the determination of the equilibrium constant from the isotherm study and other theoretical approaches are mostly incapable of explaining the ambiguity of the dimensions of the constant. Furthermore, the different methods estimated the thermodynamic parameters in various ways, which unequivocally depicts the non-uniformity and ambiguity associated with this field. Many researchers proposed a number of modifications to existing methodologies to rectify the error associated with the different approaches.

Milonjić has proposed an equation for an equilibrium constant given in L mol⁻¹ as follows:⁶⁶

ΔG⁰ = −RT ln(55.5 × K_eq) (15)

This multiplication was used to nullify the unit of K_eq (L mol⁻¹) with 55.5 mol of water per litre of aqueous solution and if K_eq is represented in L g⁻¹, a factor of 1000 can be used, as the unit weight of water is 1000 g L⁻¹. Albadarin et al. had adopted the same concept and developed a similar equation for the Langmuir isotherm constant as follows:³⁸

ΔG⁰ = −RT ln(55.5 × b) (16)

However, Albadarin et al. used the unit b in L mg⁻¹.³⁸ Dawood and Sen attained a positive value of ΔG⁰ by employing K_d, taken as q_e/C_e with a unit of L g⁻¹ in the van't Hoff equation and commented on the non-spontaneity of the process.³⁷ Canzano et al. commented on the work in terms of the erroneous estimation of the value of K_d without considering the non-dimensionality.⁶⁷ Accordingly, Dawood and Sen corrected the value of K_d by multiplying by 1000 (considering the density of water as 1000 g L⁻¹) and achieved a negative value of ΔG⁰ after the rectification, demonstrating the spontaneity of the adsorption reaction.⁶⁸ Biggar and Cheung, Khan and Singh proposed the method for the estimation of K_eq considering the effect of activity.^64,65 The K_eq value was determined from the intercept of the plot for ln(C_s/C_e) versus C_s, where C_s and C_e are the concentration of the adsorbate at the surface and the suspension in the equilibrium. Many researchers followed this method.^55,60,69 Sawalha et al. found K_eq from the above mentioned methods without considering the non-dimensionality.⁶⁰ Milonjić analysed the work of Sawalha et al. and showed that the value of ΔG⁰ was wrongly calculated.^60,66

The above mentioned approaches and the proposed modifications thereof, for the determination of the equilibrium constant, have several shortcomings due to the inadequate theoretical background or improper applications of the methods. The method of multiplying K_eq by mol L⁻¹ (or g L⁻¹) of water is not applicable everywhere. In the existing literature, the use of K_d, K_e or the constant from the method proposed by Khan and Singh did not consider the equilibrium relationship between q_e and C_e represented by the appropriate isotherm model.⁶⁵ Subsequently, the non-linear relationship of q_e and C_e may lead to a poor fitting of the experimental data and improper estimation of the thermodynamic parameters.

Proposed approach for the determination of K_eq

In view of the drawbacks discussed above, a theoretical analysis is essential for the correct estimation of K_eq. The equilibrium relation of q_e and C_e shall be confirmed in order to adopt a suitable approach. In most of the literature, the equilibrium relationship is described either by the Langmuir or the Freundlich isotherm model. The linear isotherm may also be applicable for a lower range of equilibrium concentration. An adsorption equation can be represented as follows:⁷⁰

A(aq) + M(s) ⇔ M–A(ad) (17)

where A is the adsorbate and M is the vacant site of adsorbent. M–A represents the adsorbed sites on the adsorbent. The equilibrium constant, K_eq can be defined as follows:

The molar concentration of the adsorbate and the adsorbent should be represented in terms of activity and is as follows:

where a_A, a_M, and a_M–A represent the activities of A, M and M–A, respectively. The Langmuir isotherm constant, b in mol L⁻¹, is numerically equal to the equilibrium constant for the non-ionic or the dilute solution of the ionic substances. However, a suitable modification of b, represented with other units, should be performed in order to adopt it as the equilibrium constant. The influence of the activity coefficient shall also be accounted for a concentrated solution of the ionic solutes. The stated method can only be used if the Langmuir isotherm defines the adsorption process. However, a number of existing literatures also showed the applicability of the Freundlich and other isotherm models. The determination of the thermodynamic constants from the Langmuir isotherm shall not be performed in those cases. Two widely used approaches, the representation of K_eq as K_d or K_e, adopted in a diversified manner in the literature have to be investigated as mentioned above.

K_e and K_d may be represented by eqn (20) and (21) as follows:

Now, considering the activity of the solid adsorbent a_M is one in eqn (19), K_eq can be represented as follows:

However, the activity will represent the molar concentration for the non-ionic and dilute solution of electrolytes. Hence, eqn (22) can be re-written as follows:

C_MA is equal to the concentration in the adsorbed phase at equilibrium which is equal to C_s in mass per mass units. In the existing literature, C_s is mostly defined in mol L⁻¹ (or mg L⁻¹). It is customary to represent C_s as the solid phase concentration of the solute in a mass to mass ratio. As a consequence, C_s can be represented as mol g⁻¹ (or mg g⁻¹). C_e is represented in mol L⁻¹ (or mg L⁻¹). By representing the concentration in this manner one may get

K_e = K_d (24)

The two entities, K_e and K_d have units of L g⁻¹. However, the equilibrium constant K_eq is unitless. If C_e is converted to mol (or mg) of the solute per gram of the solvent, the dimensions of K_e or K_d can be omitted. Considering the mass of water per litre is M_w(T), in g L⁻¹ at a temperature T, K_e or K_d can be written as follows:

From eqn (25), one can postulate eqn (26) as follows:

K_eq ≈ K_eM_w(T) = K_dM_w(T) (26)

Eqn (26) is applicable if the equilibrium is defined by the linear isotherm. If the equilibrium is defined by the Freundlich isotherm model with C_e and q_e in mg L⁻¹ and mg g⁻¹, respectively, the following equation may be developed by analysing the dimension of K_d and K_f:

However, calculating the value of K_eq from the slope of the plot of q_e (or C_s) versus C_e may not be wise unless the linear isotherm describes the adsorption equilibrium. Similarly, the method proposed by Khan and Singh did not account for the influence of the equilibrium relationship between C_s and C_e.⁶⁵ In the present paper, K_eq is computed from the slope of the linear and Freundlich isotherm plots (R² = 0.91 to 0.97) for different temperatures and modified through eqn (26) and (27), respectively. The van't Hoff plot (Fig. 2) of K_eq versus 1/T provided ΔH⁰ and ΔS⁰ as the slope and intercept of the graph multiplied by R. ΔG⁰ was obtained from eqn (13) for different temperatures.

Fig. 2 van't Hoff plot using isotherm constants from (a) the Freundlich model and (b) the linear model.

The positive value of ΔH⁰ indicated that the adsorption reaction is endothermic and the increase in temperature favoured the adsorption process.⁷¹ Furthermore, the value of ΔH⁰ ranges between 2.1 and 20.9 kJ mol⁻¹ and between 80 and 200 kJ mol⁻¹ in case of physisorption and chemisorption, respectively.⁷² In the present study, the value of ΔH⁰ (Table 3) showed the nature of adsorption was in between physisorption and chemisorption. The positive value of ΔS⁰ (Table 3) reflected an affinity of the calcined LDH for the fluoride ion along with an increase in randomness at the interface during the sorption process and an increase in the degree of freedom of the adsorbed species.^58,72 The negative value of ΔG⁰ confirmed that the adsorption process is spontaneous and thermodynamically feasible (Table 3). The decrease in ΔG⁰ with the increase in the temperature unequivocally confirmed an enhancement of the adsorption with the increase in the temperature. This is plausible due to the increase in the mobility of the fluoride ions in aqueous solution and the increase of the affinity of the fluoride ions for the calcined LDH with the increase in temperature.⁷²

Table 3 Thermodynamic parameters from isotherm constant

Temperature (°C)	Freundlich isotherm			Linear isotherm
	ΔG^0 (kJ mol^−1)	ΔH^0 (kJ mol^−1)	ΔS^0 (kJ mol^−1 K^−1)	ΔG^0 (kJ mol^−1)	ΔH^0 (kJ mol^−1)	ΔS^0 (kJ mol^−1 K^−1)
10	−18.07	25.33	0.15	−17.78	27.38	0.16
20	−19.60			−19.38
30	−21.13			−20.98
40	−22.67			−22.57
50	−24.20			−24.17

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Mean free surface energy

The Freundlich or Langmuir isotherm equations do not account the influence of temperature in the isotherm relationship. However, the D–R isotherm parameters correlate the adsorption equilibrium with the thermodynamic parameters. The adsorption constant K_ad is related to adsorption energy. The D–R isotherm was computed from the adsorption data to obtain the values of K_ad at different temperatures (Fig. 1). The mean free surface energy (E_ad) is the free energy change when one mole of an ion is transferred from a solution to the surface of the adsorbent and is calculated by the following equation:³⁸

A value of E_ad that is less than 8 kJ mol⁻¹ represents physical adsorption. However, the value of E_ad in ion exchange reactions ranges between 8 and 16 kJ mol⁻¹.⁷³ The calculated values of E_ad, represented in Table 4, may be ascribed to the fact that physisorption is predominant in the present study.

Table 4 Energy parameter from thermodynamic analyses

Temperature (°C)	Ead (kJ mol^−1)	Ea (kJ mol^−1)	S*
10	0.40	53.18	0.28
20	0.42
30	0.47
40	0.53
50	0.55

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Activation energy

The effect of temperature on fluoride adsorption was further assessed by evaluating the activation energy from a modified Arrhenius type equation as follows:⁷⁴where S* is the sticking probability and depends on the temperature of the system. E_a is the activation energy. The surface coverage (θ) is estimated by the following equation:⁷⁴

S* and E_a were calculated from the plot of (1 − θ) versus 1/T through linear regression (Fig. 3). S* is defined as a function of the adsorption system depicting the potential of an adsorbate to remain on the adsorbent. A value of S* between 0 and 1 favours sorption (Table 4). A positive value of E_a indicates an endothermic reaction. The value of E_a ranges between 5 and 40 kJ mol⁻¹ for physisorption and between 40 and 800 kJ mol⁻¹ for chemisorption.⁵⁵ The E_a value (Table 4) indicates that the adsorption process is between physical and chemical adsorption.

Fig. 3 Variation of surface coverage with temperature for determination of the activation energy.

Isosteric heat of adsorption

Isosteric heat of adsorption is an important thermodynamic parameter depicting the influence of heat on the adsorption process. The standard enthalpy of the adsorption at a constant amount of the adsorbate adsorbed is defined as the isosteric heat of adsorption. It is an indicator of the performance of an adsorptive separation process and surface energetic heterogeneity. The isosteric heat of adsorption at a constant surface coverage is obtained from the Clausius–Clapeyron equation as follows:where ΔH_x is the isosteric heat of adsorption. Furthermore, assuming ΔH_x is independent of the temperature, the following equation may be obtained upon integration:where C is integration constant. ΔH_x can be attained from the slope of the isosteres at different surface loading values (mg g⁻¹) in the plot of ln C_e versus 1/T (Fig. 4). The isosteres for the different equilibrium concentrations at a constant amount sorbed was obtained from the adsorption isotherm data at different temperatures.

Fig. 4 Plots of ln C_e for fluoride adsorption at constant amounts adsorbed as a function of 1/T.

The nature of the adsorption may be appraised from the value of ΔH_x. The value of ΔH_x is less than 80 kJ mol⁻¹ for physisorption. However, a value of ΔH_x between 80 and 400 kJ mol⁻¹ may be found for a chemisorption reaction. In this study, the value of ΔH_x (Fig. 4) is attributed to the occurrence of physisorption. Furthermore, the degree of heterogeneity of the adsorbent may be confirmed from the variation of ΔH_x with the surface loading. A constant value of ΔH_x, however, may be attained in the case of a homogeneous surface.^72,75 The moderate variation of ΔH_x (Fig. 4) confirmed a less heterogeneous adsorbent surface in the present study.

Empirical modeling through RSM

Optimization of fluoride removal. The empirical relationship between the fluoride adsorption capacity and the adsorption process parameter was established through RSM. A FCCD was employed to perform the experimental design and the polynomial model equation was developed by regression analysis. The actual adsorption capacity in accordance to the experimental run varied from 3.15 to 59.60 mg g⁻¹ (Table 2). The substantial variation of the adsorption capacity may be attributed to the fact that the process parameters had a potential influence in the experimental domain. The quadratic model equation (eqn (3)) for the adsorption capacity with the coded values of the independent variables (i.e. temperature, pH, initial fluoride concentration and adsorbent dose) was attained as follows:

q = +16.54 + 0.91A + 0.09B + 16.66C − 9.84D − 0.01AB + 0.41AC − 0.62AD + 0.04BC − 0.05BD − 8.16CD + 0.14A² + 0.15B² + 0.75C² + 4.54D² (33)

ANOVA was perform for the regression model [eqn (33)], which exhibited its adequacy with a considerably higher F value. A low F value for ‘lack of fit’ [Table S1 in ESI†] also demonstrated the significance of the model. It was revealed from ANOVA that the model terms with the pH showed less significance. A stepwise model reduction was performed to eliminate the insignificant model terms. Accordingly, the reduced model is given in eqn (34) as follows:

q = +16.55 + 0.91A + 16.66C − 9.84D + 0.41AC − 0.62AD − 8.16CD + 0.19A² + 0.80C² + 4.59D² (34)

The equation of the reduced model was evaluated by Fisher's F value, P value, lack of fit, sequential model sum of squares (Table S2 in ESI†) and coefficient of determination (R² = 0.99). The model parameters were improved after reduction (Table S2 in ESI†). The validity of the fitted quadratic model was further evaluated by graphically analysing the residual. The plot of the predicted versus actual values exhibited a reasonable agreement between the predicted values from the model equation and the experimental results (Fig. S1 in ESI†). The studentized residual was plotted against the predicted value and confirmed the adequacy of the model (Fig. S2 in ESI†). The normal distribution of the error was exhibited from the presence of scatter points in close proximity to the straight line in the normal probability plot for the studentized residual (Fig. S3 in ESI†). The random distribution of the residual without a definite pattern was depicted in the graphical analyses.

Influence of parameters. The influence of the independent variables on the fluoride adsorption capacity was assessed through the coefficient of the model equation [eqn (33)] and the perturbation curve (Fig. 5). The three dimensional surface plot provided the interactive effects of the process parameters and the overall optima of the system (Fig. 6). The adsorption capacity was mostly influenced by the initial fluoride concentration and the adsorbent dose compared to the other two parameters. The pH had the most insignificant effect and least interaction with the other three parameters. A prominent synergistic and antagonistic effect was exhibited for the initial fluoride concentration and the adsorbent dose, respectively. However, the temperature had a comparatively lower synergistic effect in the tested range. The interaction between the initial concentration and the adsorbent dose had pronounced significance whereas the combined influences of the temperature with these two parameters was comparatively less significant (Fig. 6). The effect of the temperature is more prominent at a higher initial concentration and a lower adsorbent dose. The increase in the adsorption capacity with the increase in the initial concentration at a lower adsorbent dose was greater compared to the same at a higher adsorbent dose (Fig. 6c). The greater amount of adsorbate available at the active sites of the adsorbent at a higher initial fluoride concentration facilitated the increasing trend of the adsorption capacity. The possibility of contact between the adsorbate and the adsorbent was enhanced at a higher concentration gradient due to the diffusion and mass transfer rates as well as faster transport.^46,76 The decreasing trend of the adsorption capacity with the adsorbent dose may be ascribed to the fact that the availability of more active sites may reduce the amount of the utilization of the adsorbent, depending on the concentration of the adsorbate.⁴⁵

Fig. 5 Perturbation curve ((A) temperature, (B) pH, (C) initial concentration, (D) adsorbent dose).

Fig. 6 Three-dimensional response surface for the interactive effects of (a) temperature and initial concentration, (b) temperature and adsorbent dose and (c) initial concentration and adsorbent dose.

Conclusions

A critical review for the estimation of the thermodynamic equilibrium constant has been performed, indicating discrepancy in the existing literature. An analytical approach has been proposed for the appropriate estimation of the equilibrium constant. In the present study, the thermodynamic aspects of the adsorption of fluoride onto the calcined Ca–Al–(NO₃) LDH were delineated. Analysing the isotherm data at different temperatures, it was revealed that the Freundlich and linear isotherm models were appropriate for this study. The estimation of K_eq was conducted with the proposed method. ΔG⁰ was obtained by incorporating K_eq into the van't Hoff equation from both the Freundlich and linear isotherm data. The value of ΔG⁰ was comparable using both the isotherm constants after the proposed modification. The negative ΔG⁰ confirmed the spontaneity of the reaction. ΔH⁰, ΔS⁰, E_a, E_ad and ΔH_x were also calculated. The obtained values of the thermodynamic constant may be ascribed to the thermodynamic feasibility of the adsorption reaction. Furthermore, the thermodynamic constant confirmed that the type of sorption reaction is in between physisorption and chemisorption and is endothermic in nature. The RSM study exhibited the prominent interactive effect of the temperature at a lower adsorbent dose and a higher initial concentration. The pH of the adsorption process had an insignificant influence. However, the influence of the adsorbent dose and initial fluoride concentration has a predominant effect on the adsorption capacity. The present method for the determination of the thermodynamic parameters from the van't Hoff equation will be instrumental in appropriately delineating the adsorption process from isotherm constants.

List of abbreviations

ANOVA	Analysis of variance
ΔG^0	Gibbs free surface energy change
ΔH^0	Change in standard enthalpy
ΔHx	Isosteric heat of adsorption
ΔS^0	Change in standard entropy
Ea	Activation energy
Ead	Mean free surface energy
FCCD	Face centered central composite design
LDH	Layered double hydroxide
Keq	Thermodynamic equilibrium constant
Ke	Equilibrium constant from Cs/Ce
Kd	Distribution coefficient
RSM	Response surface methodology
R	Universal gas constant
T	Temperature

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5ra20538g

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