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. 2025 Jun 5;15:19734. doi: 10.1038/s41598-025-04085-2

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© The Author(s) 2025

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 7 — (a) Nonlinear fitting plots of the Langmuir, Freundlich, Temkin, Redlich-Peterson, and Hill adsorption isotherm models of CR on Ce-BTC@MCC. (b) Nonlinear fitting plots of pseudo-first-order, pseudo-second order, and Elovich kinetic models for the adsorption of CR on Ce-BTC@MCC. (c) Intraparticle diffusion for the adsorption of CR dye on the Ce-BTC@MCC composite. (d) The thermodynamic parameters were calculated by plotting lnKe versus 1/T. The experimental conditions are similar to those in Fig. 6.