Chemically modified Retama raetam biomass as a new adsorbent for Pb(II) ions from aqueous solution: non-linear regression, kinetics and thermodynamics

2.2.1 Preparation of STRR bio-adsorbent:

Three levels of succinic acid treated STRR with various carboxyl content were prepared by keeping other reaction conditions constant and varying the amount of succinic acid, as shown in Table 1. The reaction was carried out as follows: to a 400-ml beaker containing 4-g ground RR particles, certain weight of succinic acid (dissolved in the least amount of distilled water) was added under continuous mixing with a mechanical stirrer until a homogenous paste was obtained. The paste was then transferred to a Pyrex Petri dish and dried at 100°C for 1 h in an air-circulated oven. The dried material was thermally treated at 140°C for 2 h. The thermally treated sample was then cooled to room temperature and then ground. Soluble by-products and unreacted succinic acid were removed by extraction in a Soxhlet extractor for 3 h with a water/ethanol (20:80) mixture. The final purified material was dried for 5 h at 60°C. The adsorbent samples were characterized by scan electron microscopy (SEM) and estimation of carboxyl content.

2.2.2 Adsorption studies:

A known volume (100 ml) of a Pb(II) ions solution with a concentration in the range of 100–1000 mg/l was placed in a 125-ml Erlenmeyer flask. An accurately weighed sample of STRR adsorbent (0.05 g) with a particle size in the range of 50–150 μm was then added to the solution. A series of such flasks was prepared; the pH values of the contents were adjusted by the addition of 0.1 M HNO₃ or 0.1 M NaOH and then shaken at a constant speed of 150 rpm in a shaking water bath at 30°C for a known time length. At the end of the agitation time, the metal ion solutions were separated by filtration. Blank experiments were carried out simultaneously without the addition of the STRR adsorbent. The extent of metal ion adsorption onto adsorbent was calculated mathematically by measuring the metal ion concentration before and after the adsorption through direct titration against the standard EDTA solution. The amount of Pb(II) ions adsorbed on STRR at equilibrium, q_e (mg/g) and percent removal of lead were calculated according to the following relationships:

where C_o and C_e are the initial and final concentrations of Pb(II) ions (mg/l), V is the volume of Pb(II) ions (l), W is the weight of STRR adsorbent (g). All experiments were carried out in duplicate, and the mean values of q_e were reported.

2.3.1 The sum of absolute error (EABS):

2.3.2 Marquardt’s percent standard deviation (MPSD):

2.3.3 Hybrid fractional error function (HYBRID):

2.3.4 The sum of the squares of the error (ERRSQ):

2.3.5 Average percentage error (APE):

2.3.6 Average relative error (ARE):

2.4.1 Fourier transform infrared spectroscopy (FTIR):

The FTIR spectroscopy was used to determine the vibrational frequencies of the functional groups in the adsorbents. The infrared (IR) spectra were recorded on a Perkin-Elmer Spectrum 1000 spectrophotometer over 4000–400 cm^-1 using KBr disk technique.

2.4.2 Scanning electron microscopy (SEM):

To carry out the SEM analysis, the sample was first mounted on a standard microscope stub and coated with a thin layer of gold using a Polaron Diode Sputter unit. The analysis was performed using a JEOL JSM 840 scanning electron microscope. The SEM was used to identify the surface quality and morphology of RR and STRR before and after Pb(II) adsorption.

2.4.3 Carboxyl content:

The carboxy group contents of the modified and unmodified adsorbents were estimated according to the method reported [15] as follows: an accurate amount (0.2 g) of adsorbent was introduced into a 125-ml Erlenmeyer flask containing 50 ml of NaOH solution (0.03 N), a corresponding blank experiment was carried out without the addition of the adsorbent sample. The flasks were left overnight at room temperature, after which the contents were titrated with standard HCl solution using phenolphthalein as the indicator. The carboxy group content of the modified starch was calculated as follows:

where V_o is the volume of HCl (ml) consumed in the blank experiment, V₁ is the volume of HCl (ml) consumed by the adsorbent sample, N is the normality of the standard HCl solution, and W is the weight of adsorbent (g).

3.1.1 Fourier transform infrared spectroscopy (FTIR) studies: FTIR spectra

The FTIR spectra of RR, STRRR and Pb(II) ions loaded STRR are illustrated in Figure 1A, B and C, respectively. The difference in the fingerprint regions of the spectra is clear and confirms the modification and adsorption processes.

Figure 1:

FTIR spectra of RR (A), STRR (B) and Pb(II) ions loaded STRR (C).

Figure 1A shows a broadband at ≈3420 cm^-1, characteristic of the hydroxyl group of RR cellulose. The bands at 2928.54 cm^-1 in IR spectra of RR biomass may be due to the C-H stretching vibrations. The band appearing at 1744.03 cm^-1 is attributed to the formation of oxygen functional groups like a highly conjugated C=O stretching in carboxylic groups.

Figure 1B shows the appearance of new peak at 1618.95 cm^-1 characteristic of the carboxylic acid ester due to the formation of STRR after thermal treatment with succinic acid.

The appearance of new peaks at 1433.38, 2027.07, 2052.42, 2077.75, 2103.11, 2128.48 and 2153.72 cm^-1 could be attributed to the chemical modification of RR to from STRR. The bands in Figure 1A and B are in accordance with data obtained in similar work [18].

Figure 1C exhibits broadening at 3428.46 cm^-1 compared with the spectra of RR, and this can be attributed to the adsorption of Pb(II) ions onto the adsorbent (STRR). Figure 1C also exhibits a small shift in the absorbance peaks in Pb(II) loaded STRR compared with that in RR and STRR (Figure 1A and B). The peak at 1618.95 cm^-1 was shifted to 1615.75 cm^-1. The peak observed at 1316.23 cm^-1 was shifted to 1315.58 cm^-1. The peak at 1153.41 cm^-1 was shifted to 1158.2 cm^-1. It should also be noted that the FTIR results did not provide any quantitative analysis as well as the information about the level of affinity to metal of the functional groups presented in the adsorbents. They only presented the possibility of the coupling between the metal species and the functional group of the adsorbents.

3.1.2 Scanning electron microscopy (SEM) studies

The surface morphologies of the RR, succinic acid treated STRR and Pb(II) ions-loaded STRR are shown in Figure 2A, B and C, respectively. SEM micrograph of RR and STRR (Figure 2A and B) reveals the irregular nature of biomass particles, which is rough and heterogeneous with considerable amount of voids and lot of ups and downs. The adsorption of Pb(II) ions by STRR is demonstrated by the change in morphology in the surface of adsorbent (Figure 2C). After Pb(II) ions adsorption, the tubes appear to be prominently swollen as Pb(II) ions enter the pores of the STRR (Figure 2C). This observation indicates that Pb(II) ions are adsorbed to the functional groups present inside the internal wall of the STRR structure. Based on the surface morphology results of STRR, it is suggested that STRR can be used as adsorbent for liquid-solid adsorption processes.

Figure 2:

SEM of RR (A), STRR (B) and Pb(II) ions loaded STRR (C).

3.2.1 Effect of pH of adsorbate

Figure 3 shows the adsorption capacity of STRR towards Pb(II) ions as a function of the pH value of the adsorbate at fixed adsorbent concentration, fixed agitation time and fixed adsorbate concentration at 30°C. It is clear from the figure that the adsorption capacity of STRR particles towards Pb(II) ions increased as the pH value of the solution increased within the range studied from 2 to 4.5. The adsorption capacity towards Pb(II) ions was a minimum under highly acidic conditions (pH=2). This was because the binding sites for metal ions onto the adsorbent were mainly occupied by H⁺ ions, which restricted the approach of Pb(II) ions cations as a result of repulsive forces. However, as the pH value of the solution increased, the number of associated H₃O⁺ ions diminished, exposing an increasing proportion of the negatively charged adsorbent surface and thereby allowing an increasing number of positively charged Pb(II) ions to be adsorbed [13]. At pH values above 4.5, no further adsorption occurred due to the precipitation of Pb(II) ions as Pb(OH)₂ [13].

Figure 3:

Effect of pH on adsorption capacity of Pb(II) ions onto STRR at 30°C.

3.2.2 Effect of adsorbent dose

The dependence of the adsorbent dose on both the adsorption capacity and the percentage removal of Pb(II) ions onto STRR was studied at pH 4.5, employing adsorbent doses within the range of 0.5–5 g/l at 30°C (Figure 4). It is apparent from the corresponding data depicted in Figure 4 that the percentage removal of Pb(II) ions increased with the increasing adsorbent dosage, which may be due to the increase in the surface area of the adsorbent and the availability of more adsorption sites. On the other hand, the adsorption capacity (q_e) or the amount of Pb(II) ions adsorbed per unit mass of adsorbent decreased with increasing adsorbent dosage. This behavior may be due mainly to the overlapping of the adsorption sites as a result of the overcrowding of adsorbent particles in the system and also to the competition among Pb(II) ions for surface sites [19].

Figure 4:

Effect of adsorbent concentration on both adsorption capacity and % removal of Pb(II) ions onto STRR at 30°C.

3.2.3 Effect of contact time and adsorbate concentration

Figure 5 shows the effect of contact time and adsorbate concentrations (204 and 398 mg/l) on the adsorption capacity of STRR towards Pb(II) ions. It will be seen from the figure that the adsorption capacity (mg/g) increased with the increase in contact time and concentration of adsorbate but remained virtually constant after equilibrium had been attained. This equilibrium time increased with the increase in the concentration of adsorbate in the system. Thus, equilibrium times of 30 and 45 min were observed for initial adsorbate concentrations of 204 and 398 mg/l, respectively. It is clear that the adsorption capacity of STRR towards Pb(II) ions depended on the adsorbate concentration, with the contact time required for Pb(II) ions removal being quite small. This latter observation is an important parameter for the possible application of STRR for economical wastewater treatment plant applications.

Figure 5:

Effect of adsorbent concentration on both adsorption capacity and % removal of Pb(II) ions onto STRR at 30°C.

3.2.3.1 Adsorption kinetics

Kinetic analysis is required to get an insight on the rate of adsorption and the rate-limiting step of the transport mechanism, which are primarily used in the modeling and design of the process. The kinetic models were pseudo-first-order, pseudo-second-order, intra-particle diffusion, Bangham’s equations and Elovich model, which are expressed as follows:

The pseudo-first-order model [20] is represented as:

(10)log(qe-qt)=logqe-k⋅t2.303. (10)

The pseudo-second-order model [21] is represented as:

(11)tqt=1(k2⋅qe)2+tqe. (11)

The intra-particle diffusion model [22] can be expressed by the following equation:

(12)qt=kP⋅t+12C (12)

where k is the pseudo-first-order rate constant (mg/g/min); k₂ is the pseudo-second-order rate constant (g/mg/min); k_P is the intra-particle diffusion rate constant (mg/g min);C is the thickness of the boundary layer (mm); and q_e is the amount of adsorbate adsorbed at equilibrium (mg/g), while q_t is the amount of adsorbate adsorbed at time t (mg/g) before reaching equilibrium.

Bangham’s equation [23] was employed for applicability of adsorption of Pb(II) onto STRR. Bangham’s equation was used to evaluate whether the adsorption is pore-diffusion controlled.

(13)loglog(CoCo-q.m)=log(kom2.303V)+αlogt (13)

where C_o is the initial concentration of the adsorbate (mg/l), V is the volume of the adsorbate (ml), m is the weight of adsorbent (g/l), q (mg/g) is the amount of adsorbate retained at time t, and α (<1) and k are the constants.

The Elovich model equation [24] is generally expressed as:

(14)qt=1βln(αβ)+1βln(t) (14)

where α (mg/g/min) is the initial adsorption rate and the parameter β (g/mg l) is related to the extent of surface coverage and activation energy for chemisorption.

Pseudo-first-order model rendered the rate of occupation of the adsorption sites to be proportional to the number of unoccupied sites. A pseudo-first-order kinetic process (Eq. 10) is usually considered physical adsorption, and the whole process is diffusion controlled. The slopes and intercepts of plots of log (q_e-q_t) vs. t for the concentrations of 204 and 398 mg/l were used to determine the pseudo-first-order constant and equilibrium adsorption density, q_e. However, q_e of experimental data deviated considerably from the theoretical data from the pseudo-first-order model (Table 2). A comparison of the results with the correlation coefficients is shown in Table 1. The correlation coefficients for the pseudo-first-order kinetic model obtained for concentrations of 204 and 398 mg/l were low (Figure 6). This suggests that this adsorption of Pb(II) ions onto STRR is not acceptable for pseudo-first-order reaction.

Table 2

Kinetic parameters for adsorption of Pb(II) ions onto STRR at 30°C.

Models	Parameters	Values
		(204 mg/l)	(398 mg/l)
Pseudo-first order	k1	0.0013818	0.0013818
	R2	0.673	0.8411
	qe (exp.)	121.3856	132.7655
	qe (calc.)	–	–
Pseudo-second order	K2	0.001036849	0.001110208
	qe (exp.)	121.3856	132.7655
	qe (calc.)	114.9425287	136.9863014
	R2	0.9876	0.9989
Bangham’s equation	Ko	23.69321163	19.20984823
	α	0.1997	0.1283
	R2	0.8954	0.9773
Intra-particle diffusion	KP	7.3638	4.1825
	C	50.746	86.999
	R2	0.9718	0.9797
Elovich equation	α	121.7031047	1593.614413
	β	0.062861453	0.07384979
	R2	0.8676	0.9785

Figure 6:

Pseudo-first-order reaction of Pb(II) ions onto STRR at 30°C.

Pseudo-second-order kinetic model (Eq. 11) is assumed as the chemical reaction mechanisms [25], and the biosorption rate is controlled by chemical biosorption through sharing or exchange of electrons between the adsorbate and adsorbent. Therefore, the adsorption behavior belonging to the pseudo-second-order kinetic model is a chemical process. The pseudo-first-order and pseudo-second-order kinetic models cannot describe the diffusion mechanism, thus kinetic results need to be evaluated by using the intra-particle diffusion model, as given in Eq. 11.

The slopes and intercepts of plots t/q_t vs. t were used to calculate the pseudo-second-order rate constants k₂ and q_e. The straight lines in plots of t/q vs. tFigure 7 show good agreement of experimental data with the pseudo-second-order kinetic model for concentrations of 204 and 398 mg/l. Table 2 lists the computed results obtained from the pseudo-second-order kinetic model. The correlation coefficients for pseudo-second-order kinetic model obtained were >0.98 for the two concentrations used. The calculated q_e values also agree very well with the experimental data (Table 2). These indicate that the adsorption of Pb(II) ions onto STRR is acceptable to the second-order kinetic model.

Figure 7:

Pseudo-second-order reaction of Pb(II) ions onto STRR at 30°C.

The slopes and intercepts of plots q_t vs. t^1/2 were used to calculate the intra-particle diffusion rate constant k_P. The straight lines in plots of q_t vs. t^1/2 (Figure 8) show good agreement of experimental data with the intra-particle diffusion model for concentrations of 204 and 398 mg/l.

Figure 8:

Intra-particle diffusion of Pb(II) ions onto STRR.

Such plots may present a multilinearity [26, 27], indicating that two or more steps take place. The first portion is the external surface adsorption or instantaneous adsorption stage. The second portion is the gradual adsorption stage, where intra-particle diffusion is rate-controlled. The third portion is the final equilibrium stage, where intra-particle diffusion starts to slow down due to extremely low adsorbate concentrations in the solution. Figure 9 shows a plot of the linearized form of the intra-particle diffusion model for concentrations of 204 and 398 mg/l. Furthermore, such deviation of straight line from the origin (Figure 9) indicates that the pore diffusion is not the rate-controlling step [28]. The values of k_P (mg/g/min) obtained from the slope of the straight line (Figure 9) are listed in Table 2. The value of R² for the plot is listed also in Table 2. The values of intercept C (Table 2) give an idea about the boundary layer thickness, i.e. the larger the intercept, the greater the boundary layer effect [29]. This value indicates that the adsorption of Pb(II) ions onto STRR is acceptable for intra-particle diffusion mechanism.

Figure 9:

Test intra-particle diffusion of Pb(II) ions onto STRR.

The plots of loglog(CoCo-q⋅m) against log t in Figure 10 for the concentrations of 204 and 398 mg/l were found to be linear for each concentration with good correlation coefficient (≥0.90) for two concentrations, indicating that kinetics confirmed Bangham’s equation, and therefore, the adsorption of Pb(II) ions onto STRR was pore diffusion controlled.

Figure 10:

Bangham’s model of Pb(II) ions onto STRR.

If Pb(II) ions adsorption onto STRR fits the Elovih model, a plot of q_t vs. ln(t) should yield a linear relationship with a slope of (1/β) and an intercept of (1/β) ln(αβ). The straight lines in plots of q_t vs. ln(t) (Figure 11) show good agreement of experimental data with the Elovich model for concentrations of 204 and 398 mg/l (R²>0.90). This suggests that this adsorption system is acceptable for this system.

Figure 11:

Elovich model of Pb(II) ions onto STRR.

3.2.4 Effect of temperature on adsorption of Pb(II) onto STRR

The effect of temperature on the adsorption capacity of Pb(II) ions onto STRR. was studied at pH 4.5, employing adsorbent dose of 0.3 g/l under different temperatures at 30, 40 and 50°C, as shown in Figure 12. The data from this figure indicate that the adsorption capacity increases with the increase in temperature from 30 to 50°C. This increase may be attributed to the development of new active sites at high temperature or to the increase of the chance of metal ions diffusion into the adsorbent. The observed data also suggest that STRR can be considered as an efficient adsorbent for Pb(II) ions from aqueous solution.

Figure 12:

Effect of duration temperature on adsorption capacity of Pb(II) ions onto STRR.

3.2.4.1 Thermodynamic parameters

The original concepts of thermodynamics assume that in an isolated system, where energy cannot be gained or lost, the entropy change is the driving force. In environmental engineering practice, both energy and entropy factors must be taken into account in determining what processes will occur spontaneously.

Thermodynamic parameters such as the standard free energy (ΔG^o), the enthalpy change (ΔH^o) and the entropy change (ΔS^o) have been calculated for the present system using the following equations [30]:

(15)Ko=CsolidCliquid (15)

(16)ΔG°=-RTlnko (16)

(17)ΔGo=ΔHo-TΔSo (17)

(18)lnKo=ΔSoR-ΔHoRT (18)

where K_o is equilibrium constant, C_solid is solid phase concentration at equilibrium (mg/l), C_liquid is liquid phase concentration at equilibrium (mg/l), T is absolute temperature in Kelvin, and R is a gas constant. Free energy values are obtained from the above equations at all temperatures; enthalpy and entropy value is calculated from the slope and intercept of plot lnK against 1/T, as shown in Figure 13 and Table 3. The negative values of ΔG^o in the range of 30–50°C (303–323 K) indicate the spontaneous nature of the adsorption of Pb(II) ions onto STRR. The positive values of ΔH^o in the range of 30–50°C listed in Table 3 suggest that the adsorption process was endothermic over this temperature range. The positive values of ΔS^o in the range of 40–50°C show the increase in randomness at the solid/solution interface during the adsorption of Pb(II) ions onto STRR.

Figure 13:

Van’t Hoff plot of the effect of temperature on adsorption of Pb(II) ions onto STRR.

Table 3

Thermodynamic parameters for the adsorption of Pb(II) ions onto STRR at different temperatures.

Temperature (K)	ΔGo (KJ/mol)	ΔHo (KJ/mol)	ΔSo (KJ/mol K)
303	-40.03223
313	-32.6628
323	-31.3329	173.42	0.4425

3.2.5 Effect of adsorbate concentration (adsorption isotherm)

Adsorption isotherm describes how adsorbates interact with adsorbents and are critical in optimizing the use of adsorbents. The amount of adsorbate per unit mass of adsorbent at equilibrium, q_e (mg/g), and the adsorbate equilibrium concentration, C_e (mg/l), allows plotting the adsorption isotherm, q_e vs. C_e (Figure 14), at 30°C and mathematical models can be used to describe and characterize the adsorption process. The most common isotherms for describing solid-liquid sorption systems are Langmuir, Freundlich, Temkin, Dubinin and Halsey (two-parameter isotherms); Redlich-Peterson, Toth, Sips, Khan and Hill, Radke-Praunsnitz and Freudlich-Langmuir (three-parameter isotherms); Baudo and Fritz-Schlunder (four-parameter isotherms); and Fritz-Schlunder (five-parameter isotherm). Therefore, in order to investigate the adsorption capacity of Pb(II) ions onto STRR, the experimental data were fitted to these equilibrium models.

Figure 14:

Equilibrium adsorption isotherm of Pb(II) ions onto STRR at 30°C.

3.3.1 The Langmuir isotherm

The Langmuir equation [31] was applied for adsorption equilibrium of Pb(II) ions by STRR. The assumption of this model is based on the maximum adsorption corresponding to a saturated monolayer of adsorbate molecules on the adsorbent surface and the energy of adsorption is constant; as well, there is no transmigration of adsorbate in the plane surface. The non-linear form of the Langmuir isotherm is given by the following equation:

where a_L is the Langmuir isotherm constant (l/mg), K_L is the Langmuir constant (l/g), and a_L/K_L represents the adsorption capacity, Q_max.

3.3.2 The Freundlich isotherm

The Freundlich model [32] is a special case applied to non-ideal adsorption on heterogeneous surfaces and also to multilayer adsorption, suggesting that binding sites are not equivalent and/or independent. This model is described by Eq. 20 as follows:

where q_e is the equilibrium concentration of Pb(II) ions on STRR adsorbent (mg/g), C_e is the equilibrium concentration of Pb(II) ions (mg/l), and K_F (mg/g) and n are the Freundlich constants characteristic of the system, indicators of adsorption capacity and adsorption intensity, respectively.

3.3.3 Tempkin isotherm

The Temkin isotherm [33] has been used in the following form:

where R is the universal gas constant (8.31441 J/mol/K), T is the absolute temperature (K), A_T is the Temkin isotherm constant (g/mg), and b_T is the Temkin constant.

3.3.4 Dubinin-Radushkevich isotherm

This isotherm [34] is generally expressed as follows:

where q_D and β_D are Dubinin-Radushkevich constants.

3.3.5 Halsey isotherm

The Halsey [35] adsorption isotherm can be given as:

where k_H and n are the Halsey isotherm constant and exponent, respectively. This equation is suitable for multilayer adsorption, and the fitting of the experimental data to this equation attests to the heteroporous nature of the adsorbent.

3.4.1 Redlich-Peterson isotherm

The Redlich-Peterson model is used as a compromise between Langmuir and Freundlich models, which can be written as [36]:

where q_e is the amount of mercury adsorbed (mg/g) at equilibrium, C_e (mg/l) is the concentration of adsorbate at equilibrium, A (l/g) and B are the Redlich constants, and g is an exponent that lies between 1 and 0.

3.4.2 Toth isotherm

Toth isotherm model [37] is useful in describing heterogeneous adsorption systems, which satisfy both low- and high-end boundary of the concentration [38]. It can be represented by the following equation:

where K_T, a_T and t are the Toth isotherm constants.

3.4.3 Sips isotherm

Sips isotherm [39] is a combined form of Langmuir and Freundlich expressions. Sips model can be represented by the following equation:

where k_s is the Sips model isotherm constant (l/g), a_s the Sips model constant (l/mg), and B_s is the Sips model exponent.

3.4.4 Khan isotherm

The Khan isotherm [40] is a generalized model suggested for the pure solutions; b_K and a_K are devoted to the model constant and model exponent. The Khan isotherm model can be represented by the following:

Where b_k is the Khan model constant, a_k is the Khan model exponent, and q_K is the maximum uptake (mg/g).

3.4.5 Radke-Prausnitz isotherm

The Radke-Prausnitz [41] isotherm can be represented as follows:

where α_R and r_R are Radke-Prausnitz model constants, and β_R is the Radke-Prausnitz model exponent.

3.4.6 Langmuir-Freundlich isotherm

The Langmuir-Freundlich equation is given by [39] the following:

where q_mLF is the Langmuir-Freundlich maximum adsorption capacity (mg/g), k_LF is the equilibrium constant for a heterogeneous solid, and mLF is the heterogeneity parameter lies between 0 and 1.

3.4.7 Hill isotherm

The Hill model [42] assumes that adsorption process as a cooperative phenomenon, with the ligand binding ability at one site on the macromolecule, may influence the different binding sites on the same macromolecule. It is described by the following equation:

where k_D, n_H and s_H are constants for the Hill model.

3.5.1 Baudo isotherm

The Baudo isotherm [43] can be written in the following form:

where q_e is the adsorbed amount at equilibrium (mg/g), C_e is the equilibrium concentration of adsorbate (mg/l), q_mo is the Baudo maximum adsorption capacity (mg/g), b_o is the equilibrium constant, and x and y are the Baudo parameters .

3.5.2 Fritz-Schlunder isotherm

Another four-parameter equation of the Langmuir-Freundlich type was developed empirically by Fritz and Schlunder [44]. It is expressed by the equation:

where q_e is the adsorbed amount at equilibrium (mg/g), C_e is the equilibrium concentration of adsorbate (mg/l), A and B are the Fritz-Schlunder parameters, and α and β are the Fritz-Schlunder equation exponents. At high liquid-phase concentrations of the adsorbate, the last equation reduces to the Freundlich equation.

3.6.1 Fritz-Schlunder isotherm

Fritz and Schlunder [44] have proposed a five-parameter empirical expression that can represent a broad field of equilibrium data as follows:

where q_e is the adsorbed amount at equilibrium (mg/g), C_e is the equilibrium concentration of adsorbate (mg/l), q_mFS is the Fritz-Schlunder maximum adsorption capacity (mg/g), and K₁, k₂, m₁ and m₂ are Fritz-Schlunder parameters.

The comparison of the adsorption capacity of various adsorbents reported in the literature for the removal of Pb(II) ions is shown in Table 4 [45–52]. It was found from this table that ZscL had a good affinity for removal of lead compared with other adsorbents.

The comparison between the experimental data and the data obtained from two-, three-, four- and five-parameter models are shown in Figures 15–18; also, the constants and error analysis of two-, three-, four- and five-parameter models are given in Tables 5–8.

Figure 15:

Comparison between experimental isotherm data and two parameter isotherm models at 30°C.

Figure 16:

Comparison between experimental isotherm data and three parameter isotherm models at 30°C.

Figure 17:

Comparison between experimental isotherm data and four parameter isotherm models at 30°C.

Figure 18:

Comparison between experimental isotherm data and five parameters model at 30°C.