## 4 Conclusions  
Our analysis of the results of the phosphate sorption tests conducted on Rockfos(r) allowed the following conclusions to be drawn.  
1. The Langmuir isotherm, especially in the linearized form proposed by [PERSON], gave results that were the closest to those obtained from measurements. Linearization according to the Lineweaver-Burke suggestion ranked second. Only in some series did the calculations made according to the Scatchard and Eadie-Hofstee linearization show a good fit and a small error, but these results are too scanty to recommend use.
2. In assessing the degree of fit between the trend line and the survey results, it is insufficient to rely on the R\({}^{2}\) coefficient of determination since even a few outliers from the main set cause a sharp drop in the R\({}^{2}\) value. Additional error analyses, such as ESRQ, HYBRD, ARE, Fisher's test, and Chi-square test (\(\chi^{2}\)), are instrumental in better illustrating the degree of fit, since they are performed on the model in its general form. At the same time, R\({}^{2}\); refers to the linearized function.
3. The relationship between the amount of phosphate sorption and the granularity of the material analyzed was not demonstrated. The mass of adsorbed phosphates and the efficiency of their removal from the solution increased with the increase in the initial concentration of phosphates in the solution. The effect of temperature on phosphate removal efficiency was statistically significant.
4. The near-zero values of the separation coefficient R\({}_{\text{L}}\) prove that the sorption of phosphates on Rockfos(r) material is irreversible (chemsorption). That is, phosphates react with the active parts of the material. This conclusion was confirmed by qualitative changes in the solution, including increasing pH, hardness, and calcium concentrations with increasing sorbent weight. The pH can even exceed 10 at large weightings, and at that point, phosphate removal relies more on precipitation than adsorption.
5. The tested material can be used in practice in wastewater treatment, and maximum sorption capacities of 0.6-0.8 mgPO\({}_{4}\)\({}^{3-}\)/g of sorbent in the winter and 0.8-1.0 mgPO\({}_{4}\)\({}^{3-}\)/g in the summer can be assumed for designing the mass and adequate time of filter beds.  
Conceptualization, B.Z. and T.S.; methodology, B.Z. and T.S.; validation, T.S., M.M., K.J., and L.R.; formal analysis, K.J.; investigation, B.Z., T.S., M.M., and L.R.; data curation, B.Z., T.S., and L.R.; writing--original draft, B.Z., T.S. and M.M.; visualization, B.Z., T.S., M.M., and K.J.; supervision, T.S. and K.J.; project administration, T.S. All authors have read and agreed to the published version of the manuscript.  
This paper was written within the framework of a Ph.D. thesis prepared by [PERSON] from the project no DWD/5/0332/2021 financed by the Ministry of Education and Science (Poland) titled \"Development of guidelines for design and operation and implementation of a chemisorption modular phosphorus removal system in a selected wastewater treatment plant\".  
Not applicable.  
Data Availability StatementData are contained within the article.  
We acknowledge [PERSON] for her help in editing the English version of this article.  
Author [PERSON] was employed by the company Miroslaw Zalewski MONT-SAN. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.