- Email: [email protected]
Development of a generalized adsorption isotherm model at solid-liquid interface: A novel approach
Journal of Molecular Liquids 240 (2017) 21–24
Contents lists available at ScienceDirect
Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq
Short Communication
Development of a generalized adsorption isotherm model at solid-liquid interface: A novel approach Partha S. Ghosal, Ashok K. Gupta ⁎ Environmental Engineering Division, Department of Civil Engineering, Indian Institute of Technology, Kharagpur 721 302, India
a r t i c l e
i n f o
Article history: Received 20 January 2017 Received in revised form 11 April 2017 Accepted 10 May 2017 Available online 11 May 2017 Keywords: Langmuir adsorption isotherm Freundlich adsorption isotherm Generalized model Nonlinear regression
a b s t r a c t The adsorption at solid-liquid interface is mostly described through the Langmuir and Freundlich isotherm models. However, these models sometimes give an ambiguous estimation of the adsorption model parameters by ignoring the assumptions originally considered for the model development. In the present study, the generalized adsorption model was developed through analyzing the isotherm plot and isolating the adsorption zones. The initial linear zone was segregated and the combined Langmuir- Freundlich based adsorption model was applied for the nonlinear zone. The effect of other mechanisms apart from the adsorption was also addressed at the higher equilibrium concentrations. An adsorption system at solid-liquid interface was analyzed. The Langmuir model provides qmax, KL and R2 as 52.425 mg/g, 0.149 L/mg and 0.989, whereas, the proposed model provides the Ki, qmax, Kaeq, n1, Kom, n2 and R2 are 6.355 L/g, 40.437 mg/g, 0.090 L/mg, 1.430, 1.00 × 10−6, 0.5 and 0.997. The generalized model can be applied to demonstrate adsorption isotherm at the solid-liquid interface for other adsorption systems. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Environmental pollution has become an emergent by-product of the extensive development of the industrialization, exponential growth in population and urbanization. Concern about the detrimental effect of the various pollutants, environmental remediation drags attention of the global scientific community. Amongst, the treatment of water and wastewater as per the stringent water quality standards is one of the biggest challenges in this field. Given the development of the numerous traditional and advanced techniques, e.g., coagulation-precipitation, filtration, electro-coagulation, advanced oxidation, etc., the adsorption is still preferred for cost effectiveness, technical feasibility, ease of operation and diversified applications [1–9]. Eventually, the extensive research brought forth many natural and synthetic adsorbents along with the various facile synthesis processes. On the contrary, research on the process chemistry of the adsorption delineating the isotherm equilibrium has acquired comparatively lesser attention than to evolve a ‘novel adsorbent’ with ‘high adsorption capacity.’ The adsorption isotherm modeling is indispensable in delineating the equilibrium relationship between the solute in the solution and on the adsorbent surface. The isotherm parameters correlate the kinetic and thermodynamics of adsorption process. The quantitative and ⁎ Corresponding author. E-mail address: [email protected] (A.K. Gupta).
http://dx.doi.org/10.1016/j.molliq.2017.05.042 0167-7322/© 2017 Elsevier B.V. All rights reserved.
qualitative estimation of the efficiency of the adsorbent and the nature of the reaction at the interface is also enumerated from isotherm model. Langmuir (1916) and Freundlich (1906) had developed isotherm models based on some specific assumptions [1,10–12]. A literature survey was conducted on 100 papers, randomly selected, with different solute and adsorbent. The nature of adsorption isotherm was listed and presented in Table S1 in Supplementary Material (SM). It is ascribed from Table S1 that these two isotherm models were extensively used in representing equilibrium isotherm at solid-liquid interface. Successively, the model parameters are reported to represent the adsorption capacity of the adsorbent as well as the thermodynamics of adsorption [13]. The applications of Langmuir and Freundlich model are extensive irrespective of the assumption followed for the model development. The best fit model is selected through the coefficient of determination of mostly linear regression, which often lacks scientific sense [10,14]. Some other isotherm models have been developed from basic principal of Langmuir and Freundlich model or some other theoretical basis [1,11]. However, these models had not acquired considerable attention due to complexity and case-specificity. Some computational approach for modeling isotherm based on statistical or molecular dynamics also developed in this direction [15,16]. Nevertheless, the modeling of adsorption isotherm in the different physico-chemical system of adsorption process still demands extensive research in the field of physical chemistry. Given the lacuna in this direction, a generalized approach for modeling adsorption isotherm in solid-liquid interface
22
P.S. Ghosal, A.K. Gupta / Journal of Molecular Liquids 240 (2017) 21–24
based on Langmuir and Freundlich model and considering the effect of other influencing factors apart from adsorption has been demonstrated in the present study. 2. Theoretical background The Langmuir isotherm was primarily developed for the unimolecular adsorption on the gas-solid interface with the assumption of monolayer surface coverage, independent and homogeneous sorption surface and energy, mostly applicable for the chemisorption process. Freundlich isotherm evolved for the heterogeneous surface with non-uniform distribution of adsorption heat as well as not restricted to monolayer coverage. Generally, physical adsorption is described with this model. Nevertheless, the applicability of the two isotherm models is not restricted to the assumptions originally used. The Langmuir isotherm was theoretically derived from the concept of the surface coverage rate of adsorption and desorption reaction. The rate of surface coverage at equilibrium is as follows [12,17]: dθ ¼ K ads C e ð1−θÞ−K des C e θ dt ads
ð1Þ
where, θ is the fraction of the surface coverage represented as qe/ qmax; qe represents the amount of the adsorbate adsorbed per unit weight of the adsorbent at equilibrium (mol/g), Ce is the concentration of the solute at equilibrium mol/L and qmax is represented as the maximum monolayer adsorption capacity (mol/g). Kads and Kdes are the rate constant of adsorption and desorption, respectively. At equilibrium, the rate of surface coverage is zero, the Langmuir isotherm is as follows: qe ¼
qmax K L C e 1 þ KLCe
ð2Þ
where KL is related to the binding energy or affinity parameter of the adsorption system (L/mol). Although the Freundlich isotherm was obtained from the empirical relationship, some theoretical derivation of this isotherm can be interpreted as follows [18]: dC ¼ −K ads C ne 1 þ K des qne 2 dt ads
ð3Þ
dC q q ¼ −K ads C ne 1 1− −K m C ne 2 þ K des dt ads qmax qmax
ð5Þ
The third term of the right-hand side of the equation contributes the mechanism responsible for removal of sorbate other than sorption with an equilibrium constant Km. Considering the rate of change of the concentration is zero at equilibrium, Eq. (5) yields as follows: qe ¼
qmax K aeq C ne 1 þ K o C ne 2 1 þ K aeq C ne 1
ð6Þ
where Kaeq is the apparent equilibrium constant and is given by Kads/ Kdes and is equivalent to Langmuir constant KL. Ko is the constant from the contribution of other mechanisms. Now, the nonlinear equation can be represented as a polynomial of the nth order. The experimental data can be fitted with a polynomial by nonlinear regression as follows: qe ¼ K z þ K 1 C e þ K 2 C 2e þ ……… þ K n C ne
ð7Þ
Identifying the significant terms of the fitted polynomial, the slope of the curve has to be computed as follows: dqe ¼ K 1 þ 2K 2 C e þ 3K 3 C 2e þ ……… þ nK n C en−1 dC e
ð8Þ
Plotting dqe/dCe with Ce, the significant variation of slope of the isotherm curve is obtain which will help in the segregation of different zones of the plot, viz., initial linear zone, zone prevailing nonlinear adsorption and zone with multiple removal mechanisms. The Eq. (6) can be further modified with respect to the influence of the different zones as follows: qe ¼ f 1 ðC e Þ þ f 2 ðC e Þ þ f 3 ðC e Þ
ð9Þ
where
At equilibrium, the rate of change of concentration will be zero, which yields the Freundlich isotherm as follows: qe ¼ K F C ne
studies associated with linear regression may be associated with the wrong interpretation of isotherm due to the bias arises for linearization. The present approach stressed on analyzing the experimental graph of isotherm, investigating segment wise the isotherm relationship and model the adsorption isotherm judiciously. The model kinetics at equilibrium is as follows:
f 1 ðC e Þ ¼ K i C e ¼
ð4Þ
where KF is equal to Kads/Kdes and n is the ratio of n1/n2. KF and n are the Freundlich constants and indicate the relative adsorption capacity and adsorption intensity, respectively. Analyzing the assumptions of the isotherm models, it can be interpreted that the Freundlich or Langmuir isotherm may provide erroneous estimation of adsorption capacity depending on surface heterogeneity, nature of adsorption and type of isotherm curves. Moreover, the isotherm parameter from these models may vary depending on the experimental range for a particular type of isotherm curve [10,19,20]. One noticeable shortcoming of the Langmuir and Freundlich isotherm found in the existing literature is the large variation of isotherm parameters with varying initial concentration of the solute. A generalized observation is that the adsorption capacity is increased largely with the increase in initial concentration [10]. Furthermore, the adsorption isotherm experiments are usually performed by measuring the equilibrium concentration of solute at equilibrium in the liquid phase. At higher concentrations, some other mechanisms, such as precipitation may also play an important role, which cannot be differentiated from the adsorption. Eventually, over estimation of adsorption capacity may be reported from the adsorption isotherm model. In many adsorption
f 2 ðC e Þ ¼
K i Ce 0
C e bC L C e ≥C L
ð10Þ
0 qmax K aeq hC e −C L in1 q K ð C e −C L Þn1 aeq ¼ max n1 1 þ K aeq hC e −C L i 1 þ K aeq ðC e −C L Þn1
f 3 ðC e Þ ¼ K om hC e −C cr in2 ¼
0 K om ðC e −C cr Þn2
C e bC cr C e ≥C cr
C e bC L C e ≥C L
ð11Þ
ð12Þ
Accordingly, Eq. (9) can be written as follows: qe ¼ K i C e þ
qmax K aeq hC e −C L ine 1 þ K om hC e −C cr ine 2 1 þ K aeq hC e −C L ine 1
ð13Þ
where CL and Ccr are equilibrium concentrations limiting the initial linear zone and zone prevailing nonlinear adsorption, respectively and in mol/L (or mg/L). Ki is the constant for linear isotherm relationship and Kom is the constant for other mechanism modified from Eq. (6). 3. Methodology The data acquisition for isotherm relationship of experimental points was performed for the selected paper [10]. The relevant documents were opened with Adobe Acrobat 9 pro Extended. The isotherm
P.S. Ghosal, A.K. Gupta / Journal of Molecular Liquids 240 (2017) 21–24 Table 1 Isotherm parameter for Langmuir, and proposed model through nonlinear regression. Models
Parameters
Value
Freundlich
KF n R2 qmax KL R2 Ki qmax Kaeq n1 Kom n2 R2
10.626 0.446 0.941 52.425 0.149 0.989 6.355 40.437 0.090 1.431 1 × 10−6 0.500 0.997
Langmuir
Proposed model
plot was extracted with snipping tool and saved as a Potable Network Graphic (PNG) extension. Plot digitizer (v.2.6.6 for Windows) freeware program has been used to extract data from isotherm experiments, which were further used for the proposed methodology of the present work. MATLAB 7.10.0 (R2010a) (Mathworks, Inc.) was used for nonlinear regression of the proposed model. The experimental data of Ayoob & Gupta 2008, were further fitted with polynomial regression in MATLAB 7.10.0 and the generated large dataset has been chosen to demonstrate the proposed model. 4. Results and discussion The isotherm data has been fitted with the linear and polynomial model. A substantial improvement of fitting was observed by increasing the order of the equation upto the cubic model. The forth order was showing to have lesser improvement than the cubic model (Table S2 in SM). The regression equation of cubic model has been obtained as follows: f 1 ðqe Þ ¼ p0 þ p1 C e þ p2 C 2e þ p3 C 3e
ð14Þ
where, p0, p1, p2, and p3 are coefficient of the model term with values of 0.3678, 5.563, −0.2817 and 0.004898. The plot of predicted values, 1st derivative and 2nd derivative was presented in Fig. S1 in SM. The slope of the curve was reached towards zero at a value of Ce around 16 mg/L. A sudden increase in value has been shown thereafter (Fig. S1). The plot was inspected and the CL and Ccr were found to be 0.5 and 16 mg/L. The present model estimated the isotherm parameter for the linear zone upto 0.5 mg/L (approximately) [Eq. (10)] with a Ki value of 6.355. The remaining part of isotherm was nonlinear described with Eqs. (11) and (12). A unit step method was applied to fit the equation segregating it in different ranges. The fitting was performed by nonlinear least square method in Matlab (2010a). The proposed method was
23
employed segregating the zones in the adsorption curve. The estimated isotherm parameters and coefficient of determination (R2) of Freundlich model, Langmuir isotherm model and proposed model were depicted in Table 1. The proposed model showed a significant increase in R2 compared to Langmuir or Freundlich model. The adequacies of the models were further tested with the ANOVA of linear regression between predicted values from the models and the regressed dataset with the cubic polynomial (Table S3 of SM). The proposed model shows smaller error function and higher F values compared to the other models, which confirmed the applicability of the proposed model. The nonlinear fit of the Freundlich, Langmuir, and proposed model was presented along with the actual dataset and the dataset fitted from cubic polynomial (Fig. 1). The Freundlich model was found to be least fitted model. The Langmuir fitting shows an overestimation of experimental points at the end values due to the influence of very less values of Ce at the initial phase (Fig. 1). The proposed model exhibited the segregation of the other influencing factors and reduction in the maximum adsorption capacity (40.437 mg/g), which was found to be overestimated in Langmuir model (52.425 mg/g) (Table 1). A set of error functions, viz., The Sum of the square of the error(SSE), sum of absolute error (SAE), average relative error (ARE), hybrid fractional error function (HYBRID), Marquardt's percent standard deviation (MPSD) and chi-square test statistic were computed for the predicted values of the models and corresponding actual experimental values obtained by plot digitization (Table S4 in SM). The proposed model showed the least values of error functions and chi-square. The Freundlich model showed very high values of error functions. The values of SSE, SAE, ARE, HYBRID, MPSD and chi-square of Langmuir model are 79, 30, 19, 57, 17, and 28%, respectively higher than those of the proposed model. The applicability of the proposed model is unequivocally supported from the R2, error functions, chi-square and F statistics. 5. Conclusions In the present study, a generalized isotherm model in solid-liquid interface was developed. The proposed model is based on the concept of Langmuir and Freundlich isotherm model. The isotherm curve was fitted with polynomial model and the nature of the curve was analyzed. The adsorption isotherm plot is segregated with different zones, i.e., linear zone, nonlinear zone and the zone influenced by other mechanisms apart from adsorption. The isotherm curve from existing literature was analyzed and fitted with the present model. A high R2 value, F value and lesser values of different error functions were achieved in the proposed model compared to Langmuir and Freundlich model, which in turn exhibited the significant applicability of the model. The present study can be considered for modeling the adsorption isotherm for different concentrations of the solute in various solid-liquid adsorption systems, thereby carries tremendous significance in the field of physical chemistry.
Fig. 1. Isotherm plot for experimental data and different models.
24
P.S. Ghosal, A.K. Gupta / Journal of Molecular Liquids 240 (2017) 21–24
Appendix A. Supplementary data Supplementary data to this article can be found online at http://dx. doi.org/10.1016/j.molliq.2017.05.042. References [1] K.Y. Foo, B.H. Hameed, Insights into the modeling of adsorption isotherm systems, Chem. Eng. J. 156 (2010) 2–10. [2] S. Ayoob, A.K. Gupta, V.T. Bhat, A conceptual overview on sustainable Technologies for the defluoridation of drinking water, Crit. Rev. Environ. Sci. Technol. 38 (2008) 401–470. [3] A. Bhatnagar, E. Kumar, M. Sillanpää, Fluoride removal from water by adsorption-a review, Chem. Eng. J. 171 (2011) 811–840. [4] D. Ghosh, M.K. Sinha, M.K. Purkait, A comparative analysis of low-cost ceramic membrane preparation for effective fl uoride removal using hybrid technique, Desalination 327 (2013) 2–13. [5] D. Hou, J. Wang, C. Zhao, B. Wang, Z. Luan, X. Sun, Fluoride removal from brackish groundwater by direct contact membrane distillation, J. Environ. Sci. 22 (2010) 1860–1867. [6] M.A. Sandoval, R. Fuentes, J.L. Nava, I. Rodríguez, Fluoride removal from drinking water by electrocoagulation in a continuous filter press reactor coupled to a flocculator and clarifier, Sep. Purif. Technol. 134 (2014) 163–170. [7] P.S. Ghosal, A.K. Gupta, S. Ayoob, Effect of formation pH, molar ratio and calcination temperature on the synthesis of an anionic clay based adsorbent targeting defluoridation, Appl. Clay Sci. 116–117 (2015) 120–128. [8] P.S. Ghosal, A.K. Gupta, An insight into thermodynamics of adsorptive removal of fluoride by calcined Ca–Al–(NO3) layered double hydroxide, RSC Adv. 5 (2015) 105889–105900.
[9] P.S. Ghosal, A.K. Gupta, Enhanced efficiency of ANN using non-linear regression for modeling adsorptive removal of fluoride by calcined Ca-Al-(NO3)-LDH, J. Mol. Liq. 222 (2016) 564–570. [10] S. Ayoob, A.K. Gupta, Insights into isotherm making in the sorptive removal of fluoride from drinking water, J. Hazard. Mater. 152 (2008) 976–985. [11] I. Horsfall, M. Jr, Spiff, Equilibrium Sorption Study of AI3+, Co2+ and Ag+ in Aqueous solutions by fluted pumpkin (Telfairia occidentalis Hook F.) Waste Biomass, Acta Chim. Slov. 52 (2005) 174–181. [12] P. Atkins, J. de Paula, Atkins' Physical Chemistry, Eighth Oxford University Press, Oxford New York, 2006. [13] P.S. Ghosal, A.K. Gupta, Determination of thermodynamic parameters from Langmuir isotherm constant-revisited, J. Mol. Liq. 225 (2017) 137–146. [14] K.V. Kumar, S. Sivanesan, Comparison of Linear and Non-linear Method in Estimating the Sorption Isotherm Parameters for Safranin onto Activated Carbon, 2005. [15] R. Carr, J. Comer, M.D. Ginsberg, A. Aksimentiev, Microscopic perspective on the adsorption isotherm of a heterogeneous surface, J. Phys. Chem. Lett. 2 (2011) 1804–1807. [16] C.W. Davis, D.M. Di Toro, Modeling nonlinear adsorption to carbon with a single chemical parameter: a lognormal Langmuir isotherm, Environ. Sci. Technol. 49 (2015) 7810–7817. [17] S. Sohn, D. Kim, Modification of Langmuir isotherm in solution systems–definition and utilization of concentration dependent factor, Chemosphere 58 (2005) 115–123. [18] J. Skopp, Derivation of the Freundlich adsorption isotherm from kinetics, J. Chem. Educ. 86 (2009) 1341. [19] C.H. Giles, D. Smith, A. Huitson, A general treatment and classification of the solute adsorption isotherm. I. Theoretical, J. Colloid Interface Sci. 47 (1974) 755–765. [20] G. Limousin, J.-P. Gaudet, L. Charlet, S. Szenknect, V. Barthès, M. Krimissa, Sorption isotherms: a review on physical bases, modeling and measurement, Appl. Geochem. 22 (2007) 249–275.
Contents lists available at ScienceDirect
Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq
Short Communication
Development of a generalized adsorption isotherm model at solid-liquid interface: A novel approach Partha S. Ghosal, Ashok K. Gupta ⁎ Environmental Engineering Division, Department of Civil Engineering, Indian Institute of Technology, Kharagpur 721 302, India
a r t i c l e
i n f o
Article history: Received 20 January 2017 Received in revised form 11 April 2017 Accepted 10 May 2017 Available online 11 May 2017 Keywords: Langmuir adsorption isotherm Freundlich adsorption isotherm Generalized model Nonlinear regression
a b s t r a c t The adsorption at solid-liquid interface is mostly described through the Langmuir and Freundlich isotherm models. However, these models sometimes give an ambiguous estimation of the adsorption model parameters by ignoring the assumptions originally considered for the model development. In the present study, the generalized adsorption model was developed through analyzing the isotherm plot and isolating the adsorption zones. The initial linear zone was segregated and the combined Langmuir- Freundlich based adsorption model was applied for the nonlinear zone. The effect of other mechanisms apart from the adsorption was also addressed at the higher equilibrium concentrations. An adsorption system at solid-liquid interface was analyzed. The Langmuir model provides qmax, KL and R2 as 52.425 mg/g, 0.149 L/mg and 0.989, whereas, the proposed model provides the Ki, qmax, Kaeq, n1, Kom, n2 and R2 are 6.355 L/g, 40.437 mg/g, 0.090 L/mg, 1.430, 1.00 × 10−6, 0.5 and 0.997. The generalized model can be applied to demonstrate adsorption isotherm at the solid-liquid interface for other adsorption systems. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Environmental pollution has become an emergent by-product of the extensive development of the industrialization, exponential growth in population and urbanization. Concern about the detrimental effect of the various pollutants, environmental remediation drags attention of the global scientific community. Amongst, the treatment of water and wastewater as per the stringent water quality standards is one of the biggest challenges in this field. Given the development of the numerous traditional and advanced techniques, e.g., coagulation-precipitation, filtration, electro-coagulation, advanced oxidation, etc., the adsorption is still preferred for cost effectiveness, technical feasibility, ease of operation and diversified applications [1–9]. Eventually, the extensive research brought forth many natural and synthetic adsorbents along with the various facile synthesis processes. On the contrary, research on the process chemistry of the adsorption delineating the isotherm equilibrium has acquired comparatively lesser attention than to evolve a ‘novel adsorbent’ with ‘high adsorption capacity.’ The adsorption isotherm modeling is indispensable in delineating the equilibrium relationship between the solute in the solution and on the adsorbent surface. The isotherm parameters correlate the kinetic and thermodynamics of adsorption process. The quantitative and ⁎ Corresponding author. E-mail address: [email protected] (A.K. Gupta).
http://dx.doi.org/10.1016/j.molliq.2017.05.042 0167-7322/© 2017 Elsevier B.V. All rights reserved.
qualitative estimation of the efficiency of the adsorbent and the nature of the reaction at the interface is also enumerated from isotherm model. Langmuir (1916) and Freundlich (1906) had developed isotherm models based on some specific assumptions [1,10–12]. A literature survey was conducted on 100 papers, randomly selected, with different solute and adsorbent. The nature of adsorption isotherm was listed and presented in Table S1 in Supplementary Material (SM). It is ascribed from Table S1 that these two isotherm models were extensively used in representing equilibrium isotherm at solid-liquid interface. Successively, the model parameters are reported to represent the adsorption capacity of the adsorbent as well as the thermodynamics of adsorption [13]. The applications of Langmuir and Freundlich model are extensive irrespective of the assumption followed for the model development. The best fit model is selected through the coefficient of determination of mostly linear regression, which often lacks scientific sense [10,14]. Some other isotherm models have been developed from basic principal of Langmuir and Freundlich model or some other theoretical basis [1,11]. However, these models had not acquired considerable attention due to complexity and case-specificity. Some computational approach for modeling isotherm based on statistical or molecular dynamics also developed in this direction [15,16]. Nevertheless, the modeling of adsorption isotherm in the different physico-chemical system of adsorption process still demands extensive research in the field of physical chemistry. Given the lacuna in this direction, a generalized approach for modeling adsorption isotherm in solid-liquid interface
22
P.S. Ghosal, A.K. Gupta / Journal of Molecular Liquids 240 (2017) 21–24
based on Langmuir and Freundlich model and considering the effect of other influencing factors apart from adsorption has been demonstrated in the present study. 2. Theoretical background The Langmuir isotherm was primarily developed for the unimolecular adsorption on the gas-solid interface with the assumption of monolayer surface coverage, independent and homogeneous sorption surface and energy, mostly applicable for the chemisorption process. Freundlich isotherm evolved for the heterogeneous surface with non-uniform distribution of adsorption heat as well as not restricted to monolayer coverage. Generally, physical adsorption is described with this model. Nevertheless, the applicability of the two isotherm models is not restricted to the assumptions originally used. The Langmuir isotherm was theoretically derived from the concept of the surface coverage rate of adsorption and desorption reaction. The rate of surface coverage at equilibrium is as follows [12,17]: dθ ¼ K ads C e ð1−θÞ−K des C e θ dt ads
ð1Þ
where, θ is the fraction of the surface coverage represented as qe/ qmax; qe represents the amount of the adsorbate adsorbed per unit weight of the adsorbent at equilibrium (mol/g), Ce is the concentration of the solute at equilibrium mol/L and qmax is represented as the maximum monolayer adsorption capacity (mol/g). Kads and Kdes are the rate constant of adsorption and desorption, respectively. At equilibrium, the rate of surface coverage is zero, the Langmuir isotherm is as follows: qe ¼
qmax K L C e 1 þ KLCe
ð2Þ
where KL is related to the binding energy or affinity parameter of the adsorption system (L/mol). Although the Freundlich isotherm was obtained from the empirical relationship, some theoretical derivation of this isotherm can be interpreted as follows [18]: dC ¼ −K ads C ne 1 þ K des qne 2 dt ads
ð3Þ
dC q q ¼ −K ads C ne 1 1− −K m C ne 2 þ K des dt ads qmax qmax
ð5Þ
The third term of the right-hand side of the equation contributes the mechanism responsible for removal of sorbate other than sorption with an equilibrium constant Km. Considering the rate of change of the concentration is zero at equilibrium, Eq. (5) yields as follows: qe ¼
qmax K aeq C ne 1 þ K o C ne 2 1 þ K aeq C ne 1
ð6Þ
where Kaeq is the apparent equilibrium constant and is given by Kads/ Kdes and is equivalent to Langmuir constant KL. Ko is the constant from the contribution of other mechanisms. Now, the nonlinear equation can be represented as a polynomial of the nth order. The experimental data can be fitted with a polynomial by nonlinear regression as follows: qe ¼ K z þ K 1 C e þ K 2 C 2e þ ……… þ K n C ne
ð7Þ
Identifying the significant terms of the fitted polynomial, the slope of the curve has to be computed as follows: dqe ¼ K 1 þ 2K 2 C e þ 3K 3 C 2e þ ……… þ nK n C en−1 dC e
ð8Þ
Plotting dqe/dCe with Ce, the significant variation of slope of the isotherm curve is obtain which will help in the segregation of different zones of the plot, viz., initial linear zone, zone prevailing nonlinear adsorption and zone with multiple removal mechanisms. The Eq. (6) can be further modified with respect to the influence of the different zones as follows: qe ¼ f 1 ðC e Þ þ f 2 ðC e Þ þ f 3 ðC e Þ
ð9Þ
where
At equilibrium, the rate of change of concentration will be zero, which yields the Freundlich isotherm as follows: qe ¼ K F C ne
studies associated with linear regression may be associated with the wrong interpretation of isotherm due to the bias arises for linearization. The present approach stressed on analyzing the experimental graph of isotherm, investigating segment wise the isotherm relationship and model the adsorption isotherm judiciously. The model kinetics at equilibrium is as follows:
f 1 ðC e Þ ¼ K i C e ¼
ð4Þ
where KF is equal to Kads/Kdes and n is the ratio of n1/n2. KF and n are the Freundlich constants and indicate the relative adsorption capacity and adsorption intensity, respectively. Analyzing the assumptions of the isotherm models, it can be interpreted that the Freundlich or Langmuir isotherm may provide erroneous estimation of adsorption capacity depending on surface heterogeneity, nature of adsorption and type of isotherm curves. Moreover, the isotherm parameter from these models may vary depending on the experimental range for a particular type of isotherm curve [10,19,20]. One noticeable shortcoming of the Langmuir and Freundlich isotherm found in the existing literature is the large variation of isotherm parameters with varying initial concentration of the solute. A generalized observation is that the adsorption capacity is increased largely with the increase in initial concentration [10]. Furthermore, the adsorption isotherm experiments are usually performed by measuring the equilibrium concentration of solute at equilibrium in the liquid phase. At higher concentrations, some other mechanisms, such as precipitation may also play an important role, which cannot be differentiated from the adsorption. Eventually, over estimation of adsorption capacity may be reported from the adsorption isotherm model. In many adsorption
f 2 ðC e Þ ¼
K i Ce 0
C e bC L C e ≥C L
ð10Þ
0 qmax K aeq hC e −C L in1 q K ð C e −C L Þn1 aeq ¼ max n1 1 þ K aeq hC e −C L i 1 þ K aeq ðC e −C L Þn1
f 3 ðC e Þ ¼ K om hC e −C cr in2 ¼
0 K om ðC e −C cr Þn2
C e bC cr C e ≥C cr
C e bC L C e ≥C L
ð11Þ
ð12Þ
Accordingly, Eq. (9) can be written as follows: qe ¼ K i C e þ
qmax K aeq hC e −C L ine 1 þ K om hC e −C cr ine 2 1 þ K aeq hC e −C L ine 1
ð13Þ
where CL and Ccr are equilibrium concentrations limiting the initial linear zone and zone prevailing nonlinear adsorption, respectively and in mol/L (or mg/L). Ki is the constant for linear isotherm relationship and Kom is the constant for other mechanism modified from Eq. (6). 3. Methodology The data acquisition for isotherm relationship of experimental points was performed for the selected paper [10]. The relevant documents were opened with Adobe Acrobat 9 pro Extended. The isotherm
P.S. Ghosal, A.K. Gupta / Journal of Molecular Liquids 240 (2017) 21–24 Table 1 Isotherm parameter for Langmuir, and proposed model through nonlinear regression. Models
Parameters
Value
Freundlich
KF n R2 qmax KL R2 Ki qmax Kaeq n1 Kom n2 R2
10.626 0.446 0.941 52.425 0.149 0.989 6.355 40.437 0.090 1.431 1 × 10−6 0.500 0.997
Langmuir
Proposed model
plot was extracted with snipping tool and saved as a Potable Network Graphic (PNG) extension. Plot digitizer (v.2.6.6 for Windows) freeware program has been used to extract data from isotherm experiments, which were further used for the proposed methodology of the present work. MATLAB 7.10.0 (R2010a) (Mathworks, Inc.) was used for nonlinear regression of the proposed model. The experimental data of Ayoob & Gupta 2008, were further fitted with polynomial regression in MATLAB 7.10.0 and the generated large dataset has been chosen to demonstrate the proposed model. 4. Results and discussion The isotherm data has been fitted with the linear and polynomial model. A substantial improvement of fitting was observed by increasing the order of the equation upto the cubic model. The forth order was showing to have lesser improvement than the cubic model (Table S2 in SM). The regression equation of cubic model has been obtained as follows: f 1 ðqe Þ ¼ p0 þ p1 C e þ p2 C 2e þ p3 C 3e
ð14Þ
where, p0, p1, p2, and p3 are coefficient of the model term with values of 0.3678, 5.563, −0.2817 and 0.004898. The plot of predicted values, 1st derivative and 2nd derivative was presented in Fig. S1 in SM. The slope of the curve was reached towards zero at a value of Ce around 16 mg/L. A sudden increase in value has been shown thereafter (Fig. S1). The plot was inspected and the CL and Ccr were found to be 0.5 and 16 mg/L. The present model estimated the isotherm parameter for the linear zone upto 0.5 mg/L (approximately) [Eq. (10)] with a Ki value of 6.355. The remaining part of isotherm was nonlinear described with Eqs. (11) and (12). A unit step method was applied to fit the equation segregating it in different ranges. The fitting was performed by nonlinear least square method in Matlab (2010a). The proposed method was
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employed segregating the zones in the adsorption curve. The estimated isotherm parameters and coefficient of determination (R2) of Freundlich model, Langmuir isotherm model and proposed model were depicted in Table 1. The proposed model showed a significant increase in R2 compared to Langmuir or Freundlich model. The adequacies of the models were further tested with the ANOVA of linear regression between predicted values from the models and the regressed dataset with the cubic polynomial (Table S3 of SM). The proposed model shows smaller error function and higher F values compared to the other models, which confirmed the applicability of the proposed model. The nonlinear fit of the Freundlich, Langmuir, and proposed model was presented along with the actual dataset and the dataset fitted from cubic polynomial (Fig. 1). The Freundlich model was found to be least fitted model. The Langmuir fitting shows an overestimation of experimental points at the end values due to the influence of very less values of Ce at the initial phase (Fig. 1). The proposed model exhibited the segregation of the other influencing factors and reduction in the maximum adsorption capacity (40.437 mg/g), which was found to be overestimated in Langmuir model (52.425 mg/g) (Table 1). A set of error functions, viz., The Sum of the square of the error(SSE), sum of absolute error (SAE), average relative error (ARE), hybrid fractional error function (HYBRID), Marquardt's percent standard deviation (MPSD) and chi-square test statistic were computed for the predicted values of the models and corresponding actual experimental values obtained by plot digitization (Table S4 in SM). The proposed model showed the least values of error functions and chi-square. The Freundlich model showed very high values of error functions. The values of SSE, SAE, ARE, HYBRID, MPSD and chi-square of Langmuir model are 79, 30, 19, 57, 17, and 28%, respectively higher than those of the proposed model. The applicability of the proposed model is unequivocally supported from the R2, error functions, chi-square and F statistics. 5. Conclusions In the present study, a generalized isotherm model in solid-liquid interface was developed. The proposed model is based on the concept of Langmuir and Freundlich isotherm model. The isotherm curve was fitted with polynomial model and the nature of the curve was analyzed. The adsorption isotherm plot is segregated with different zones, i.e., linear zone, nonlinear zone and the zone influenced by other mechanisms apart from adsorption. The isotherm curve from existing literature was analyzed and fitted with the present model. A high R2 value, F value and lesser values of different error functions were achieved in the proposed model compared to Langmuir and Freundlich model, which in turn exhibited the significant applicability of the model. The present study can be considered for modeling the adsorption isotherm for different concentrations of the solute in various solid-liquid adsorption systems, thereby carries tremendous significance in the field of physical chemistry.
Fig. 1. Isotherm plot for experimental data and different models.
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P.S. Ghosal, A.K. Gupta / Journal of Molecular Liquids 240 (2017) 21–24
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