###### Abstract  
In this study, we aimed to develop a mathematical description of the process of phosphate sorption on Rockfos\({}^{\textcircledR}\) material using the Langmuir isotherm and determine the basic parameters for modeling this process. The Rockfos\({}^{\textcircledR}\) material was formed through the thermal treatment of opoka at 980 \({}^{\circ}\)C and is highly reactive due to its significant calcium and silicon compound content. This study included an evaluation of the phosphate retention efficiency on the material as a function of the phosphate concentration in the initial solution (0.5 mg/L, 1.0 mg/L, and 2.0 mg/L), sorbent grain size (1.0-1.6 mm, 1.6-2.5 mm, and 2.0-5.0 mm), and process temperature (5 \({}^{\circ}\)C, 10 \({}^{\circ}\)C, 15 \({}^{\circ}\)C, 20 \({}^{\circ}\)C, and 25 \({}^{\circ}\)C). It was found that an increase in the process temperature and the phosphate concentration in the solution favored sorption, while the effect of the sorbent grain size was ambiguous. It was determined that sorption can be described well using the Langmuir linearization of the Langmuir model. Thermodynamic analysis and the separation coefficient suggest that phosphorus sorption on Rockfos\({}^{\textcircledR}\) material is primarily based on chemisorption, and the process is endothermic and spontaneous over the entire temperature range. The determined parameters of the tested material, especially the qmax (maximum sorption capacity), provide a basis for the design of a filter for removing phosphate from wastewater, assuming that the load is equal to the inflow to the filter and adheres to the specified requirements for treated wastewater. w +
Footnote †: journal: applied sciences  
0000-0002-2300-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-00000-0000-0000-00000-00000-00000-00000-00000-00000-00000-000000-00000-000000-000000-000000-000000-000000-000000-00000  
the use of circulating cycles [10]. One method of extracting phosphorus from a stream of flowing wastewater is to filter it through a specific material. In this process, apart from various types of flow physical phenomena [11], chemical reactions of phosphate ions with cations constituting the adsorbent, one of the most important roles is played by the adsorption process [12]. Additionally, sorption materials saturated with previously retained phosphorus can be used as an agricultural fertilizer [13]. Sorption is a process that has been known for many years and is used in a wide range of chemical-engineering process lines [14], water treatments [15; 16], and wastewater treatments [17; 18; 19; 20; 21]. Published studies on the parameters and phenomena occurring during sorption concern isotherms, kinetics, and the removal efficiencies of various contaminants. The most common studies concern the sorption of lead [22; 23; 24; 25], nickel [26], cadmium [27], surfactants [28], dyes [18], etc. The independent variables that affect sorption efficiency the most are the grain sizes of the deposits [29], process temperature [22], water pH [23; 25; 26; 27], and, most importantly, the type of adsorbent [30; 31; 32]. In batch studies, the dose of the adsorbent is a critical parameter [25; 27]. Sufficient results have been obtained using amorphous calcium silicate hydrates (A-CSHs) with a molar ratio of calcium to silicon greater than 2 [33] or a single-phase calcium silicate hydrate (C-S-H) [34]. Extensive research has been conducted concerning opoka [35] and a derived material called Polonite\({}^{\text{\textregistered\char 13}}\)[36; 37]. A similar material is a product called Rockfos\({}^{\text{\textregistered\char 13}}\), which is formed after firing carbonate-silica rock (opoka) at a high temperature (980 \({}^{\circ}\)C) [38]. To date, Rockfos\({}^{\text{\textregistered\char 13}}\) material has been used to improve the efficiency of removing phosphorus from effluents flowing out from hydrolytic treatment plants [39; 40; 41].