Heavy Metal Adsorption onto Kappaphycus sp. from Aqueous Solutions: The Use of Error Functions for Validation of Isotherm and Kinetics Models

Biosorption process is a promising technology for the removal of heavy metals from industrial wastes and effluents using low-cost and effective biosorbents. In the present study, adsorption of Pb2+, Cu2+, Fe2+, and Zn2+ onto dried biomass of red seaweed Kappaphycus sp. was investigated as a function of pH, contact time, initial metal ion concentration, and temperature. The experimental data were evaluated by four isotherm models (Langmuir, Freundlich, Temkin, and Dubinin-Radushkevich) and four kinetic models (pseudo-first-order, pseudo-second-order, Elovich, and intraparticle diffusion models). The adsorption process was feasible, spontaneous, and endothermic in nature. Functional groups in the biomass involved in metal adsorption process were revealed as carboxylic and sulfonic acids and sulfonate by Fourier transform infrared analysis. A total of nine error functions were applied to validate the models. We strongly suggest the analysis of error functions for validating adsorption isotherm and kinetic models using linear methods. The present work shows that the red seaweed Kappaphycus sp. can be used as a potentially low-cost biosorbent for the removal of heavy metal ions from aqueous solutions. Further study is warranted to evaluate its feasibility for the removal of heavy metals from the real environment.

constitute carrageenan that provides different binding sites (e.g., hydroxyl, carboxyl, amino, and sulfhydryl) responsible for the adsorption for heavy metals [9], they are the least focused group [13]. Therefore, further research studies are warranted on the selectivity of algal species [9].
The red seaweed Kappaphycus sp. is one of the most important commercial sources of kappa-carrageenan, which also has different medicinal and industrial applications [14]. Malaysia produced 331,490 tonnes of Kappaphycus sp., being 17.039% of the total world production in 2012 [15]. Recent studies suggest that both the living biomass and the waste biomass of Kappaphycus alvarezii are a good biosorbent for the removal of nutrients [13] and heavy metals from the aqueous environment [16,17]. However, there is no available literature report on the biosorption of heavy metals using the dry biomass of Kappaphycus sp. Hence, the present study was investigated to study the performance of the dried biomass of Kappaphycus sp. for the removal of Zn 2+ , Cu 2+ , Pb 2+ , and Fe 2+ from aqueous solutions in batch system at laboratory scale under different parameters like pH of solution, contact time, temperature, and initial metal ion concentrations. The Fourier transform infrared (FTIR) spectral analysis was made to identify the main functional groups involved in the biosorption process of those metal ions.
In general, mechanistic or empirical equations are used to express heavy metal adsorption capacities of different types of biosorbents using batch or column method [18]. Available literature reports confirm that nearly two dozens of empirical models involving 2, 3, 4, or even 5 parameters have been used to fit batch equilibrium isotherm curves to biosorbents [18,19]. Besides, kinetic models have been described by several authors elsewhere [20][21][22]. The equilibrium and kinetic models are often validated on the basis of coefficient of regression ( 2 ≥ 0.99) of the experimental data. In the present study, some error functions have been used for validating the experimental data along with an insight into the usual measures of model inferences.

Collection, Identification, and Preparation of the Biosorbent.
The red seaweed Kappaphycus sp. was collected from the Semporna coast of Sabah, Malaysia, in April 2013. The alga was washed for several times with running water and subsequently with deionized water to remove epiphytes and salts. The washed biomass was then dried in an oven at 60 ∘ C for 48 h until a constant weight was attained. The dried biomass was then crushed with an analytical mill, sieved (250 m size), and stored in polypropylene bottles until use.
The living biomass of the species was preliminarily identified following systemic morphological features [23]. It was then subjected to 28S DNA based molecular identification [24]. The species was identified as Kappaphycus sp. and the gene sequence of the nucleotide was submitted in the NCBI GenBank (accession number KM229320).

Chemicals and Reagents.
All the chemicals and reagents used in this study were of analytic reagent grade. The working solutions of different concentrations (10-200 mg L −1 ) of the heavy metals (Zn, Cu, Pb, and Fe) were prepared by diluting the stand solutions (1000 ± 2 mg L −1 ) of the metals (Merck, Germany) in double distilled deionized water. Different initial pH of the solutions was obtained by adding 0.1 N HCl (Sigma-Aldrich, USA) or 0.1 N NaOH (Merck, Germany).

Batch Biosorption Experiments.
All the experiments were conducted in a batch system using 150 mL Erlenmeyer flasks in a thermostatic shaker (25 ∘ C, 180 rpm), unless otherwise stated. Each flask was filled with 50 mL of solution and biosorbent as appropriate. The influence of several operational parameters on the biosorption characteristics of the metals such as pH of the aqueous solution (2-7), contact time (0−120 min), initial metal ion concentration (25−200 mg L −1 ), and temperature (25−50 ∘ C) were assessed using a constant biomass dosage (4 g L −1 ). Competitive adsorption of the four metal ions under mixed condition was also evaluated.
The adsorption studies were conducted with 50 mL of the metal solutions at an initial concentration of 10 mg L −1 .
For the kinetic studies sample solutions were withdrawn at regular intervals and the residual concentration of the heavy metals in the aqueous phase was analyzed after filtration as stated above.
The amount of the metal ions remaining in the solutions was measured by using Atomic Absorption Spectrometer (AAnalyst700, Perkin-Elmer, USA) after separation of the biosorbent by filtration through Whatman Filter number 1.
The amount of metal adsorbed per gram of the biosorbent at equilibrium, (mg g −1 ), was calculated from the difference of the metal concentration in the aqueous phase before and after biosorption as follows: where 0 and are the initial and equilibrium concentration of metal ions in the solution (mg/L), respectively, is the volume of metal solution (L), and is the mass of the dry biosorbent (g). The percentage of metal removal ( , %) from the solution was calculated as follows: Each experiment was done in triplicate and the data were expressed as the mean of the triplicate results. Statistical analyses were performed using Microsoft Office Excel 2007 (Microsoft Corp., USA).

Application of Adsorption Models.
In the present experiment, four two-parameter isotherm models: Langmuir, Freundlich, Temkin, and Dubinin-Radushkevich (D-R); four two-parameter kinetic models: pseudo-first-order (PFO), pseudo-second-order (PSO), Elovich, and intraparticle diffusion (IpD); and kinetic model were applied to describe the sorption behaviour of the adsorbent. The equations of these models are given in Table 1.

Error Function Analysis.
In order to evaluate the suitability of the equation to the experimental data error function is the best optimization procedure. Apart from the regression coefficient ( 2 ), nine error functions such as sum of square error (SSE), average relative error (ARE), hybrid functional error (HYBRID), sum of absolute error (EABS), Marquardt's percent standard deviation (MPSD), normalized standard deviation (Δ (%)), coefficient of determination ( 2 ), nonlinear chi-square test ( 2 ), and residual root mean square error (RMSE) were calculated to evaluate the best fit of the modeled equation to the experimental data. The equations of the error functions are given in Table 2.
2.6. FTIR Analysis. FTIR spectral analysis was carried out to determine the possible functional groups present in the dried biomass of Kappaphycus sp. Infrared spectra of the raw and metal-loaded biomass were obtained using a Fourier transform infrared (FTIR) spectrometer (Spectrum GX, Perkin-Elmer, USA).

Effect of Solution pH.
In the adsorption process of metal ions from aqueous solutions, pH of the solution plays an important role. It is apparent from the results represented graphically in Figure 1(a) that, with the increase in pH, the biosorption increased gradually. The maximum biosorption (54.13%, 81.84%, 84.17%, and 20.94% for Zn 2+ , Cu 2+ , Pb 2+ , and Fe 2+ , resp.) was observed at pH 5. At lower pH (2)(3)(4), biosorption of metal ions was inhibited greatly. This can be explained on the basis that cell wall of the Kappaphycus sp. contains various functional groups (as described in Section 3.9). The positively charged functional groups increase competition between protons and metal cations for binding active sites of biomass, resulting in decreasing the metal cations adsorption on the biomass surfaces [48]. At higher pH values (6)(7)(8), the biosorption efficiency of metal ions was significantly decreased (Figure 1(a)), which may be attributed to the formation of anionic hydroxide complexes that decrease the dissolved metal concentration in solution and their competition with the active sites [49]. Therefore, all the rest biosorption experiments were carried out at pH 5. Figure 1(b), biosorption of metal ions on the adsorbent increased with an increase in contact time and the equilibrium biosorption was attained within 90-120 min reflecting rapid initial biosorption. Maximum uptake of Cu 2+ , Pb 2+ , Fe 2+ , and Zn 2+ was reached up to 83.88, 85.89, 21.27, and 54.13%, respectively, within 90 min. A decrease in the biosorption was noticed during the subsequent time of incubation indicating the maximum adsorption level as a saturation point of biosorption. The rapid kinetic mechanism can be attributed to the formation of exterior surface complexes neglecting intraparticle diffusion, which is very advantageous in biotechnological processes for wastewater treatment [48]. In general, heavy metal biosorption efficiency of seaweeds attained a maximum level within 30 and 90 min [50]. Hence, a contact time of 120 min was selected for further experiments ensuring attainment of equilibrium conditions.

Effect of Contact Time. As shown in
,exp (mg g −1 ): value obtained from the batch experiment, ,calc (mg g −1 ): calculated value from the isotherm for corresponding ,exp , ,exp (mg g −1 ): mean of ,exp , : number of observations in the experimental isotherm, and : number of parameters in the respective model.

Effect of Temperature.
The solution temperature plays a vital role on the metal ions biosorption, which was found to increase with the increase of solution temperature. The rate of Zn 2+ , Cu 2+ , Pb 2+ , and Fe 2+ biosorption by the dried biomass of Kappaphycus sp. was rapid reaching a maximum of 61.74, 89.12, 86.68, and 40.06%, respectively, at 50 ∘ C. This phenomenon indicates that the biosorption process of the metal ions onto the biomass is endothermic. It can be attributed that, at the higher temperatures, the activation of the biosorbent surfaces is enlarged facilitating more active sites for biosorption of the metal ions. Moreover, an easy mobility and enhanced accessibility of metal ions from the bulk solution to the biomass active sites could also be the possible reason for the maximum biosorption of metal ions at higher temperatures [51].

Effect of Initial Metal Ion Concentration.
Biosorption capacity of the biomass was found to increase with increasing initial concentration of the metal ions. This phenomenon can be attributed to an increase in electrostatic interactions involving sites of progressively lower affinity for the metal ions up to the point of saturation [51,52]. It was further observed that the percentage removal of the metal ions decreased markedly from 77.52% to 30.58% for Zn 2+ , 87.52% to 36.09% for Cu 2+ , 87.12% to 40.04% for Pb 2+ , and 80.72% to 31.52% for Fe 2+ with an increase in the initial concentration of the metal ions from 25 to 200 mg L −1 . This might be due to the rapid saturation of all metal binding active sites of the biosorbent at a certain concentration of the metal ions [52,53] and an equilibrium state between adsorbate and biosorbent was attained.

Biosorption Isotherm Studies.
The equilibrium adsorption isotherms are essential data source to design, understand, and optimize the biosorption process. The data express the intrinsic properties of the biosorbent and interaction between adsorbate and adsorbent. The data can be used to compare the biosorptive capacities of the biosorbent for different pollutants.

Langmuir Isotherm Model.
The model isotherm parameters together with regression coefficient are represented in Table 3. As shown in Figure 2(a), approximately linear  (16.78), suggesting that Pb 2+ has a preferential uptake compared to the other metals, which can be attributed to its low tendency in forming strong complex [46]. Another reason might be attributed to carboxylate polysaccharides in seaweeds that show preferential binding of cations with large ionic radii [48]. However, the preferential sorption order of the metal ions in the present study can be explained by Pauling's electronegativity [54]: Pb 2+ (2.33) > Cu 2+ (1.190) > Fe 2+ (1.83) > Zn 2+ (1.65). This implies the fact that the higher the ion's electronegativity the higher the attraction for its electrons, and the attraction becomes stronger to the negative charge of the biomass ligands [55]. Furthermore, the separation factor, , for the metal ions (Table 3) falls within the range of 0 < < 1 suggesting that adsorption of the experimental ions is favourable at all the concentrations investigated [26]. Hence, the Kappaphycus sp. is a suitable biosorbent for the sorption of the experimental metal ions from aqueous solutions.
A comparative study on maximum heavy metal adsorption capacity of different low-cost adsorbents has been given in Table 4. The study shows that the dried biomass of Kappaphycus sp. is more promising than some other low-cost adsorbents for the removal of the metal ions. Figure 2(b); Freundlich constants and are represented in Table 3. The results suggest that the biosorption of Pb 2+ can be moderately described by the Freundlich model  (Table 3) indicate that the D-R model does not fit the experimental data well ( 2 ≤ 0.89) for describing the metal ions biosorption, suggesting the involvement of metal sorption mechanisms other than van der Waals force [56]. The mean free energy of biosorption, (0.297−0.489 kJ⋅mol −1 ), for the metal ions suggests that the sorption process is physisorption [57] and corroborative to the earlier reports in literature [46,48]. The positive values of indicate the endothermic nature of the sorption process [46]. Furthermore, the values of (<16 kJ mol −1 ) suggest that the mechanism of the ion exchange process is film-diffusion controlled [58].

Competitive Adsorption.
Competitive adsorption of the metal ions under quaternary system shows the adsorption preference of Pb 2+ > Cu 2+ > Fe 2+ > Zn 2+ with the rate of metal removal as 90.39, 90.00, 75.00, and 58.38%, respectively. The results suggest that the potentiality of the adsorbent in the quaternary system remains the same as that in the single metal system, which proves its unique adsorption quality.

Biosorption Kinetic Studies.
The kinetic data are essential to understand the rate and nature of adsorption onto the adsorbents. The data can be used to compare the kinetics of the biosorbent for different pollutants. Table 5 and Figure 3(a), the regression coefficient ( 2 ≥ 0.99) of the pseudo-first-order model suggests that the experimental data accurately support the PFO model to describe adsorption kinetics of the metal ions. But the differences between the experimental values, , were higher than the modelled values, . It refers to the fact that both the metal ions and adsorbent were involved in the adsorption process [52]. Therefore, it is suggested that the pseudo-first-order model is not suitable to explain the kinetic sorption of the experimental metal ions onto the dried biomass of Kappaphycus sp. over the range of experimental time and metal ion concentrations. Similar results have been reported for the sorption kinetic of different metal ions onto different adsorbents including seaweeds in the literature [51,52,59,60].

Pseudo-Second-Order Kinetic Model.
The values of the regression coefficient of the linearized PSO kinetic model as shown in Figure 3(b) were the highest ( 2 > 0.99) among the studied kinetic models, and the experimental values matched well with the calculated data (Table 5). Therefore, it can be suggested that the experimental data accurately support the best fit of the PSO model for the adsorption of the metal ions. Hence, chemisorption is the rate-limiting step which involves valence forces through the sharing or exchange of electrons between the metal ions and different functional groups in the sorbent [13,60].
The pseudo-second-order rate constant, 2 (g mg −1 min −1 ), was found in the range of 0.1874 to 0.9548, which supports that the metal ions uptake onto the sorbent from aqueous solution was more rapid and favourable. As shown in Figure 3(b), adsorption kinetic of the metal ions on Kappaphycus sp. occurred in two steps: a fast initial uptake rate, ℎ (0.17-3.82 mg g −1 min −1 ), in the first 30 min, where more than 85% of the total metal adsorption occurred, followed by a slower uptake rate leading to the equilibrium state (∼120 min). Similar observation was reported in literature [51,52]. This phenomenon supports that the diffusion is the rate-controlling step in the sorption process [60]. The halfadsorption time 1/2 (min) defined as the time required for the adsorption to take up half amount of the equilibrium metal ions was found within the range of 0.54 to 4.03 indicating high affinity between the adsorbate and adsorbent molecules [61].

Elovich Model.
The values of the regression coefficient ( 2 = 0.79-0.98) of the Elovich kinetic model (Table 5, Figure 3(c)) suggest that kinetic data did not follow the Elovich model. However, the higher values of the Elovich constants, (mg g −1 min −1 ) and (g mg −1 ), as shown in Table 5 are suggestive of an increased rate of chemisorption [13].

Intraparticle Diffusion Model.
The nonlinear regression data of versus 0.5 plots as shown in Figure 3(d) for different heavy metal ions suggests multilinearity (two phases in Pb 2+ adsorption and three phases in Cu 2+ , Fe 2+ , and Zn 2+ adsorption). The intraparticle diffusion rate constant ( ) as shown in Table 5 was obtained from the slope of the second linear portions of the plot of versus 0.5 for the metal ions. Apparently intraparticle diffusion plays a significant role in the adsorption of Pb 2+ , Cu 2+ , and Zn 2+ ( 2 ≥ 0.99) onto the dried biomass of Kappaphycus sp. suggesting the fact that there is a significant relationship between and 0.5 for the metal ions at the experimental conditions. However, versus 0.5 plots did not pass through the origin ( > 0) in any of the cases, suggesting that even though the adsorption process involved intraparticle diffusion, it was not the only rate-controlling step [13,62], and external mass transfer had also played an important role in the metal ions sorption by the dried biomass of Kappaphycus sp. [13].

Thermodynamic Studies.
The values of the thermodynamic parameters are shown in Table 6. The linearized Van't Hoff plots of ln ( eq ) versus 1/ are represented in Figure 4.
The negative values of Δ ∘ indicate that the thermodynamic process was spontaneous and feasible for all the tested metal ions [35]. Moreover, the increase in negative Δ ∘ values with an increase in temperature shows an increased feasibility of adsorption at higher temperature, which is corroborative to the earlier reports [48,52,59].
The positive values of enthalpy change (Δ ∘ ) suggest endothermic nature of the metal adsorption process [13,51,59,63,64]. In addition, the extent of enthalpy value gives indicative information on the type of biosorption, which can be either physical or chemical. The enthalpy change (Δ ∘ ) in the range of 2.1-20.9, 20.9-80.0, and 80.0-418.4 kJ mol −1 is indicative of physisorption, physisorption together with chemisorptions, and chemisorptions, respectively [48]. Based on the values of Δ ∘ , it can be presumed that the biosorption process took place physically for all the tested metal ions. This was also supported by D-R isotherm results with the (< 8 kJ mol −1 ) values of the metal ions (Table 3). Further, positive values of entropy change (Δ ∘ ) are suggestive of increased randomness at the solid-solution interface during   the biosorption process of the metal ions on the active sites of the biosorbent [59].

Error Function.
In the real-world, data samples from each experiment in a series of experiments differ due to measurement error affecting data precision. In order to ensure accurate measurement results, statistical error function is the measure to compensate data errors [65]. Hence, the isotherm and kinetic data were further analyzed using nine error functions in order to test the fitness of the models. Lower value of SSE, ARE, HYBRID, EABS, MPSD, 2 , Δ (%), and RMSE and higher value of 2 indicate the best fit of the model. The correlation of regression ( 2 ) for the adsorption isotherm models (Table 3) suggests that Pb 2+ , Fe 2+ , and Zn 2+ follow the Langmuir model while Cu 2+ follows the Temkin model accurately. The error functions of the isotherm data (Table 7) suggest that the Temkin model provides the best fit to the experimental data. Again, the correlation of regression ( 2 ) for the kinetic models (Table 5) shows that PSO is the best fit model. But the error functions of the kinetic data (Table 8) suggest that the best fit of the kinetic models is intraparticle diffusion. It is, therefore, strongly suggested that the regression coefficient ( 2 ) is not an appropriate method for comparing the best fitting of the isotherm and kinetic models; rather some forms of error analysis could be a better criterion for avoiding data errors.  [67], C=O stretching vibration of carboxylate groups [48,60], asymmetric stretching of -SO 3 − bonds in sulfonic acid [48], C=O stretching vibration of carboxylate groups [48,60], symmetric stretching of -SO 3 − bonds in sulfonic acid [48], C-O stretching vibration of carboxyl groups [60], S-O stretching [60], and S=O stretching bands of sulfonate groups [48], respectively.
After biosorption of Pb 2+ the peaks were shifted to 3355.9, 2917. 8   The significant change in the intensity of the peaks shows interaction between the metal ions and the functional groups. Because intensity depends on change in dipole moment and total number of functional groups present on biosorbent surface. Therefore, it can be concluded that the carboxylic, sulfonic acid, and sulfonate groups of Kappaphycus sp. dried biomass are involved in the biosorption of the metal ions.

Conclusion
In the present study, we examined adsorption of four heavy metal ions such as Pb 2+ , Cu 2+ , Fe 2+ , and Zn 2+ onto the dried biomass of the red seaweed Kappaphycus sp. from Malaysia.
The adsorption isotherm data for the metal ions fitted well with the Temkin model followed. Kinetic data for all the metal ions can be best described by the intraparticle diffusion model. Adsorption process was feasible, spontaneous, and endothermic in nature. We strongly suggest that analysis of error functions is a better criterion for validating isotherm and kinetic models in order to evaluate adsorptive behaviour of a typical adsorbent using linear method. Heavy metal adsorption process onto the dried biomass of Kappaphycus sp. was the complex one involving more than one mechanism. Both homogeneous and heterogeneous active sites were found to exist in the dried biomass. The FTIR study revealed the presence of carboxylic, sulfonic acid, and sulfonate groups in the cell wall matrix of the biomass that was involved in the adsorption of the metal ions. The dried biomass of Kappaphycus sp. may be used as a low-cost biosorbent for removal of heavy metal ions from aqueous solutions. Further study is warranted to evaluate the potentiality of the biosorbent for heavy metal removal from the real environment.

Conflict of Interests
The authors hereby declare no conflict of interests.