Sulu çözeltiden Pb2+ iyonlarının kızılçam kabuğu (Pinus brutia ten) üzerinde adsorpsiyonundan elde edilen kinetik ve denge verilerinin doğrusal olmayan regresyon kullanılarak ileri düzeyde yorumlanması: Microsoft Excel Çözücü ile Pratik Bir Yaklaşım

Year 2025, Volume: 7 Issue: 2, 108 - 131, 31.05.2025

Ali Gündoğdu , Volkan Numan Bulut

https://doi.org/10.51435/turkjac.1626570

Abstract

Bu çalışma, kinetik ve denge verilerini değerlendirmek için doğrusal olmayan regresyon analizi kullanarak sulu çözeltilerden çam kabuğuna (Pinus brutia Ten.) Pb(II) iyonlarının adsorpsiyonunu araştırır. Adsorpsiyon deneyleri bir dizi başlangıç ​​konsantrasyonu üzerinde yürütüldü ve denge verileri Langmuir, Freundlich, Temkin ve Dubinin-Radushkevich (D-R) dahil olmak üzere çeşitli izoterm modellerine ve Tóth, Sips, Redlich-Peterson (R-P) ve Brouers-Sotolongo (B-S) gibi gelişmiş üç parametreli modellere uygulandı. Kinetik veriler, sözde birinci mertebeden (PFO), sözde ikinci mertebeden (PSO), Elovich, Avrami ve B-S modelleri kullanılarak analiz edildi. Microsoft Excel Solver tarafından kolaylaştırılan doğrusal olmayan regresyon, model parametrelerini optimize etmek için kullanıldı ve uyum iyiliği SSE, ARE, HYBRID, MPSD ve MAE dahil olmak üzere birden fazla hata fonksiyonu aracılığıyla değerlendirildi. Sonuçlar, Brouers-Sotolongo (B-S) modelinin hem kinetik hem de izoterm verileri için en iyi uyumu sağladığını ve adsorbanın heterojen yüzey özelliklerini yansıttığını göstermektedir. Adsorpsiyon sürecinin, kinetik sabitler (αBS ve nBS) ve yarı reaksiyon süresi (τ1/2) ile kanıtlandığı üzere, fiziksel ve kimyasal etkileşimlerin bir kombinasyonunu içerdiği bulunmuştur. Denge modelleri arasında, özellikle B-S, Tóth ve Sips modelleri olmak üzere üç parametreli izotermler, iki parametreli modellere göre üstün performans göstererek, bu sistemdeki adsorpsiyon mekanizmalarının karmaşık doğasını vurgulamıştır.
Bu çalışma, çam kabuğunun ağır metal giderimi için düşük maliyetli ve çevre dostu bir adsorban olarak etkinliğini vurgulamakta ve adsorpsiyon çalışmalarında doğrusal olmayan regresyon ve gelişmiş hata analizinin faydasını göstermektedir. Bu yaklaşımın, model seçiminin hassasiyetini ve adsorpsiyon mekanizmalarının anlaşılmasını artırarak literatüre katkıda bulunacağı düşünülmektedir.

Keywords

Adsorpsiyon , Hata fonksiyonu , Excel Çözücü , İzoterm , Kinetik ve denge verileri , Doğrusal olmayan regresyon

Advanced interpretation of kinetics and equilibrium data obtained from adsorption of Pb2+ ions from aqueous solution onto pine bark (Pinus brutia Ten.) using nonlinear regression: A practical approach with Microsoft Excel Solver

Abstract

This study investigates the adsorption of Pb(II) ions from aqueous solutions onto pine bark (Pinus brutia Ten.) using nonlinear regression analysis to evaluate kinetic and equilibrium data. Adsorption experiments were conducted over a range of initial concentrations, and the equilibrium data were fitted to various isotherm models, including Langmuir, Freundlich, Temkin, and Dubinin-Radushkevich (D-R), as well as advanced three-parameter models like Tóth, Sips, Redlich-Peterson (R-P), and Brouers-Sotolongo (B-S). Kinetic data were analyzed using pseudo-first order (PFO), pseudo-second order (PSO), Elovich, Avrami, and B-S models. Nonlinear regression, facilitated by Microsoft Excel Solver, was used to optimize model parameters, and goodness-of-fit was assessed through multiple error functions, including SSE, ARE, HYBRID, MPSD, and MAE.
Results indicate that the Brouers-Sotolongo (B-S) model provided the best fit for both kinetic and isotherm data, reflecting the heterogeneous surface characteristics of the adsorbent. The adsorption process was found to involve a combination of physical and chemical interactions, as evidenced by the kinetic constants (αBS and nBS) and the half-reaction time (τ1/2). Among the equilibrium models, three-parameter isotherms, particularly the B-S, Tóth, and Sips models, showed superior performance over two-parameter models, highlighting the complex nature of adsorption mechanisms in this system.
This study underscores the efficacy of pine bark as a low-cost and eco-friendly adsorbent for heavy metal removal and demonstrates the utility of nonlinear regression and advanced error analysis in adsorption studies. This approach is thought to improve the precision of model selection and the understanding of adsorption mechanisms, contributing to the literature.

Keywords

Adsorption , Error function , Excel Solver , Isotherm , Kinetics and equilibrium data , Non-linear regression

References

• A.M. Badran, U. Utra, N.S. Yussof, M.J.K. Bashir, Advancements in adsorption techniques for sustainable water purification: A focus on lead removal, Separations 10(11), 2023, 565.
• N.e. Hira, S.S.M. Lock, N.F. Shoparwe, I.S.M Lock, L.G. Lim, C.L. Yiin, Y.H. Chan, M. Hassam, Review of Adsorption Studies for Contaminant Removal from Wastewater Using Molecular Simulation, Sustainability, 15(2), 2023, 1510.
• U.A. Edet, A.O. Ifelebuegu, Kinetics, isotherms, and thermodynamic modeling of the adsorption of phosphates from model wastewater using recycled brick waste, Processes, 8(6), 2020, 665.
• K.Y. Foo, B.H. Hameed, Insights into the modeling of adsorption isotherm systems, Chem Eng J, 156(1), 2010, 2–10.
• H.R. Ghaffari, H. Pasalari, A. Tajvar, K. Dindarloo, B. Goudarzi, V. Alipour, A. Ghanbarneajd, Linear and nonlinear two-parameter adsorption isotherm modeling: A case-study, Int J Eng Sci, 6(9), 2017, 1–11.
• L.S. Chan, W.H. Cheung, S.J. Allen, G. McKay, Error analysis of adsorption isotherm models for acid dyes onto bamboo derived activated carbon, Chinese J Chem Eng, 20(3), 2012, 535–542.
• K. Suwannahong, S. Wongcharee, T. Kreetachart, C. Sirilamduan, J. Rioyo, A. Wongphat, Evaluation of the Microsoft Excel Solver spreadsheet-based program for nonlinear expressions of adsorption isotherm models onto magnetic nanosorbent, Appl Sci, 11(16), 2021, 7432.
• E.A. Adekunbi, J.O. Babajide, H.O. Oloyede, J.S. Amoko, O.A. Obijole, I.A. Oke, Evaluation of Microsoft Excel Solver as a tool for adsorption kinetics determination, Ife J Sci, 21(3), 2019, 169–183.
• J. Sreńscek-Nazzal, U. Narkiewicz, A.W. Morawski, R.J. Wróbel, B. Michalkiewicz, Comparison of optimized isotherm models and error functions for carbon dioxide adsorption on activated carbon, J Chem Eng Data, 60(11), 2015, 3148−3158.
• Md A. Hossain, H.H. Ngo, W. Guo, Introductory of Microsoft Excel Solver function – spreadsheet method for isotherm and kinetics modelling of metals biosorption in water and wastewater, J Water Sustain, 3(4), 2013, 223–237.
• Y.S. Ho, G. McKay, Sorption of dye from aqueous solution by peat, Chem Eng J, 70(2), 1998, 115–124.
• Sahmoune Mohamed Nasser, Moussa Abbas, Mohamed Trari, Understanding the rate-limiting step adsorption kinetics onto biomaterials for mechanism adsorption control, Prog React Kinet Mec, 49, 2024, 1–26.
• B. Mehdinejadiani, S.M. Amininasab, L. Manhooei, Linear and non-linear methods for estimating isotherm parameters of nitrate adsorption, Water resources and wetlands, 4th International Conference Water resources and wetlands, 5-9 September 2018, Tulcea (Romania), p.312.
• G.W. Kajjumba, S. Emik, A. Öngen, H.K. Özcan, S. Aydın, Modelling of Adsorption Kinetic Processes—Errors, Theory and Application. In Advanced Sorption Process Applications. Edited by Serpil Edebali, Published: 05 November 2018, IntechOpen.
• J.-C. Liu, P. A. Monson, Monte Carlo simulation study of water adsorption in activated carbon, Ind Eng Chem Res, 45(16), 2006, 5649–5656.
• A. Wongphat, S. Wongcharee, N. Chaiduangsri, K. Suwannahong, T. Kreetachat, S. Imman, N. Suriyachai, S. Hongthong, P. Phadee, P. Thanarat, J. Rioyo, Using Excel Solver’s parameter function in predicting and interpretation for kinetic adsorption model via batch sorption: Selection and statistical analysis for basic dye removal onto a novel magnetic nanosorbent, Chem Eng, 8(3), 2024, 58.
• A. Gundogdu, D. Ozdes, C. Duran, V.N. Bulut, M. Soylak, H.B. Senturk, Biosorption of Pb(II) ions from aqueous solution by pine bark (Pinus brutia Ten.) Chem Eng J, 153(1-3), 2009, 62–69.
• S. Lagergren, About the theory of so-called adsorption of soluble substances. Kungliga Svenska Vetenskapsakademiens Handlingar, 24(4), 1898, 1–39.
• Y.S. Ho, Review of second-order models for adsorption systems. J Hazard Mater, 136(3), 2006, 681–689.
• L. Largitte, R. Pasquier, A review of the kinetics adsorption models and their application to the adsorption of lead by an activated carbon, Chem Eng Res Design, 109, 2016, 495–504.
• M. Avrami, Kinetics of phase change. I. General theory. J Chem Phys, 7(12), 1939, 1103–1112.
• R. George, S. Sugunan, Kinetics of adsorption of lipase onto different mesoporous materials: Evaluation of Avrami model and leaching studies, J Mol Catal B-Enzymatic, 2014, 105, 26–32.
• A.G. Marangoni, Kinetics of crystal growth using the Avrami model and the chemical potential approach. In: Kinetic analysis of food systems, 2017, Springer, Cham.
• F. Brouers, The fractal (BSf) kinetics equation and its approximations. J Mod Phys, 5(16), 2014, 1594–1601.
• F. Brouers, T.J. Al-Musawi, The use of the Brouers–Sotolongo fractal kinetic equation for the study of drug release. Adsorption 26, 2020, 843–853.
• S. Karoui, R.B. Arfi1, M.J. Fernández-Sanjurjo, A. Nuñez-Delgado, A. Ghorbal1, E. Álvarez-Rodríguez, Optimization of synergistic biosorption of oxytetracycline and cadmium from binary mixtures on reed-based beads: modeling study using Brouers-Sotolongo models, Environ Sci Pollut R, 28, 2021,46431–46447.
• A.M.B. Hamissa, F. Brouers, M.C. Ncibi, M. Seffen, Kinetic modeling study on methylene blue sorption onto agave americana fibers: Fractal kinetics and regeneration studies, Sep Sci Technol, 48(18), 2013, 2834-2842.
• I. Langmuir, The adsorption of gases on plane surfaces of glass, mica, and platinum, J Am Chem Soc, 40(9), 1918, 1361–1403.
• A. Saha, D. Bhaduri, A. Pipariya, R.K. Ghosh, Linear and nonlinear sorption modelling for adsorption of atrazine onto activated peanut husk, Environ Prog Sustain, 36(2), 2017, 348–358.
• H. Freundlich, Over the adsorption in solution. Journal of Physical Chemistry, 57, 1906, 385–471.
• M.J. Temkin and V. Pyzhev, Recent modifications to Langmuir isotherms, Acta Physiochim. USSR, 12 (1940) 217–222.
• M.M. Majd, V. Kordzadeh-Kermani, V. Ghalandari, A. Askari, M. Sillanpää, Adsorption isotherm models: A comprehensive and systematic review (2010−2020), Sci Tot Environ, 812, 2022, 151334.
• N.D. Hutson, R.T. Yang, Theoretical basis for the Dubinin-Radushkevitch (D-R) adsorption isotherm equation, Adsorption, 3, 1997, 189-195.
• R. Saadi, Z. Saadi, R. Fazaeli, N.E. Fard, Monolayer and multilayer adsorption isotherm models for sorption from aqueous media, Korean J. Chem. Eng., 32(5), 2015, 787–799.
• M.M. Dubinin, V.A. Astakhov, Development of the concepts of volume filling of micropores in the adsorption of gases and vapors by microporous adsorbents, Russ Chem Bull, 20, 1971, 3–7.
• G.J. Millar, S.J. Couperthwaite, M. de Bruyn, C.W. Leung, Ion exchange treatment of saline solutions using Lanxess S108H strong acid cation resin, Chem Eng J, 280, 2015, 525–535.
• V.J. Inglezakis, Solubility-normalized Dubinin–Astakhov adsorption isotherm for ion-exchange systems, Micropor Mesopor Mat, 103(1-3), 2007, 72–81.
• O. Redlich, D.L. Peterson, A useful adsorption isotherm, J Phys Chem, 63(6), 1959, 1024–1024.
• S. Kalam, S.A. Abu-Khamsin, M.S. Kamal, S. Patil, Surfactant adsorption isotherms: A review, ACS Omega, 6(48), 2021, 32342−32348.
• J. Toth, State equations of the solid gas interface layer, Acta Chem Acad Hung, 69 (1971) 311–317.
• L Bokányi, Some applications of Tóth-isotherm in mineral processing, XXVI International Mineral Processing Congress (IMPC) 2012 Proceedings, New Delhi, India, 24–28 September 2012.
• R. Ramadoss, D. Subramaniam, Removal of divalent nickel from aqueous solution using blue-green marine algae: adsorption modeling and applicability of various isotherm models, Sep Sci Technol, 54(6), 2019, 943–961.
• F. Brouers, O. Sotolongo, F. Marquez, J.P. Pirard, Microporous and heterogeneous surface adsorption isotherms arising from Levy distributions. Physica A, 349(1-2), 2005, 271–282.
• F. Gimbert, N. Morin-Crini, F. Renault, P.-M. Badot, G. Crini, Adsorption isotherm models for dye removal by cationized starch-based material in a single component system: Error analysis, J Hazard Mater, 157(1), 2008, 34–46.
• N. Sivarajasekar, R. Baskar, Adsorption of Basic Magenta II onto H2SO4 activated immature Gossypium hirsutum seeds: Kinetics, isotherms, mass transfer, thermodynamics and process design, Arab J Chem, 12(7), 2019, 1322–1337.
• C.J. Willmott, K. Matsuura, Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance, Clim Res, 30, 2005, 79–82.
• N.R. Draper, H. Smith, Applied Regression Analysis. 3th Edition, 1998, Wiley, New York.
• R.J. Hyndman, A.B. Koehler, Another look at measures of forecast accuracy, Int J Forecasting, 22(4), 2006, 679–688.
• W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical recipes: The art of scientific computing (3rd ed.), 2007, Cambridge University Press.