Modeling of Phosphate Sorption Process on the Surface of Rockfos^® Material Using Langmuir Isotherms

Abstract

In this study, we aimed to develop a mathematical description of the process of phosphate sorption on Rockfos^® material using the Langmuir isotherm and determine the basic parameters for modeling this process. The Rockfos^® material was formed through the thermal treatment of opoka at 980 °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 °C, 10 °C, 15 °C, 20 °C, and 25 °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^® 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.

1. Introduction

Phosphorus is an element that is found in wastewater in various forms [1]. It most commonly occurs as organic phosphorus from fecal matter and food residues as well as in the form of phosphates as a residue from detergents and from anaerobic transformations of organic phosphorus [2]. Ortho- and monophosphates are predominant in both the dissolved and molecular forms and cumulatively account for about 40% of total P [3]. Due to its strong biogenicity, phosphorus can negatively affect the environment, especially the water quality in lakes and ponds. A significant source of phosphorus in water bodies is wastewater, so legislation on the discharge of wastewater into the environment stipulates stricter criteria, conditioned primarily by the amount of wastewater according to the population equivalent (p.e.) [4]. These characteristics necessitate the use of various methods of phosphorus removal, among which are biological methods, which involve the use of anaerobic and aerobic activated sludge, and chemical methods, which involve the use of chemical compounds that, with phosphates, form insoluble, easily sedimented sludge [5]. The low abundance of phosphorus as a fossil resource with high industrial and agricultural value [6] requires measures to be taken to recover phosphorus from various sources, including wastewater and sewage sludge [7,8,9] and, wherever possible, 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^® [36,37]. A similar material is a product called Rockfos^®, which is formed after firing carbonate–silica rock (opoka) at a high temperature (980 °C) [38].

To date, Rockfos^® material has been used to improve the efficiency of removing phosphorus from effluents flowing out from hydrophytic treatment plants [39,40,41]. Studies analyzing the phosphorus removal efficiency at different firing temperatures and hydraulic loading rates were performed by Jóźwiakowski et al. [42] and Jucherski et al. [43]. These studies showed that a filter material with finer granulation and a longer retention time of the wastewater in the filter provided better phosphorus removal results. It was also shown that when used at a full technical scale in filters at the outflow area of hydrophytic wastewater treatment plants, the material did not provide a significant phosphorus removal effect, at about 30% [44]. Previous studies on the Langmuir isotherms of phosphate sorption on opoka and opoka-based manufactured materials showed inconsistencies with regards to the relevant parameters due to variations in sorbent sourcing [12], process parameters (the initial concentration of phosphorus in the solution, the mass of sorbents, etc.) [45], and the use of different methods for linearizing the general equation [8,33,46,47,48,49].

In this study, we aimed to develop and analyze a mathematical description of the process of phosphate sorption on Rockfos^® material using Langmuir isotherms and determine the corresponding model parameters. In developing the description, the influence of the initial concentration of phosphate in the solution, the sorbent’s grain size, and the process’s temperature were considered. The parameters determined using Langmuir isotherms, especially the maximum absorbency of the sorbent, can provide a basis for the design of final wastewater treatment filters for use in small wastewater treatment plants, including the determination of the volume of the filter bed and the total operating time.

2. Materials and Methods

2.1. Material

The Rockfos^® material used in this study was produced from an opoka rock quarried near Piaski in Lublin Province, Poland. The material was formed by heat treating the quarried rock at 980 °C. It is characterized by a CaO content of 43.336% by weight, a SiO₂ content of 36.047%, and an Al₂O₃ content of approximately 6%. Detailed information is provided in other publications [11,38,44]. Before the tests, the material was sifted through the appropriate sieves, and fractions with grain sizes of 1.0 ÷ 1.6, 1.6 ÷ 2.5, and 2.0 ÷ 5.0 mm were used for testing.

2.2. Method

A 50.00 mg/L stock solution of synthetic phosphate was prepared by dissolving 0.220 g of Chempur’s analytical-grade anhydrous potassium dihydrogen phosphate (KH₂ PO₄) in distilled water in a 1000 mL volumetric flask. The stock solution was then diluted with distilled water to obtain the desired concentration of the experimental working solution, i.e., 0.5, 1.0, and 2.0 mg/L.

Studies of sorption isotherms were carried out in a set of Erlenmeyer flasks (200 mL) in which increasing masses of sorbent were weighed, viz., 0.05, 0.100, 0.200, 0.300, 0.400, 0.500, 0.600, 0.700, and 0.800 g, with granulations of 1.0 ÷ 1.6, 1.6 ÷ 2.5, and 2.0 ÷ 5.0 mm. To each series of flasks, 100 ml of potassium dihydrogen phosphate (KH₂ PO₄) solution at concentrations of 0.5, 1.0 and 2.0 mg/L were added. The samples were shaken in a laboratory shaker and placed in a thermostatic cabinet at 5, 10, 15, 20, and 25 °C for 24 h. The shaking amplitude was 230 rpm. The solutions were then filtered on a blotting paper filter. Aqueous samples were taken from the filtrate, and their total phosphorus content and pH were determined. The total hardness and calcium concentration of the selected samples were also determined. Each series of tests was performed in triplicate, and the arithmetic mean of the results was used for calculations. As a result, analyses were conducted on nine adsorbent samples, three different granulations, three initial KH₂PO₄ concentrations, and five temperatures. A total of 405 experiments were performed.

Total phosphorus was determined using a spectrophotometric method, with the oxidation of the test sample performed in a thermoreactor at 120 °C for 30 min based on the international standard PN-EN ISO 6878: 2006 + Ap1:2010 + Ap2:2010 [50]. In each sample, we also determined the pH value via a potentiometric method using an ORION Star A329 multi-parameter meter (Thermo Scientific, Waltham, MA, USA). The international standard, PN-EN ISO 10523: 2012 [51], was adhered to when measuring the pH values. Total hardness and calcium content were determined using phthalein red and NANOCOLOR tube tests (Macherey-Nagel, Düren, Germany).

2.3. Evaluation of Phosphate Retention Efficiency

The efficiency of phosphate removal from the solution was determined based on the initial concentration (C₀) and the final concentration (after the sorption process) (C_k) using Formula (1):

$$\mathrm{E}\mathrm{f}={\displaystyle \frac{{\mathrm{C}}_{0-}{\mathrm{C}}_{\mathrm{k}}}{{\mathrm{C}}_0}}·100\mathrm{\%},$$

(1)

in which the numerator is the equilibrium concentration, C_e = C₀ − C_k.

2.4. Models

The Langmuir model in the general form and linear form according to the Langmuir, Lineweaver–Burk, Scatchard, and Eadie–Hoffsie linearization methods were used to model isotherms. These models are shown in

Table 1. Langmuir sorption isotherm models in general and linear form.

Lp.	General Form	Linear Form	References
1	$\mathrm{q}={\displaystyle \frac{{\mathrm{q}}_{\mathrm{m}\mathrm{a}\mathrm{x}}·{\mathrm{K}}_{\mathrm{L}}·{\mathrm{C}}_{\mathrm{e}}}{1+{\mathrm{K}}_{\mathrm{L}}·{\mathrm{C}}_{\mathrm{e}}}}$ (3)		[47,49]
2	Linearization according to Langmuir	${\displaystyle \frac{{\mathrm{C}}_{\mathrm{e}}}{\mathrm{q}}}={\displaystyle \frac{1}{{\mathrm{q}}_{\mathrm{m}\mathrm{a}\mathrm{x}}·{\mathrm{K}}_{\mathrm{L}}}}+{\displaystyle \frac{{\mathrm{C}}_{\mathrm{e}}}{{\mathrm{q}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}}$ (4)	[47,49]
3	Linearization according to Lineweaver–Burke	${\displaystyle \frac{1}{\mathrm{q}}}={\displaystyle \frac{1}{{\mathrm{q}}_{\mathrm{m}\mathrm{a}\mathrm{x}}·{\mathrm{K}}_{\mathrm{L}}}}·{\displaystyle \frac{1}{{\mathrm{C}}_{\mathrm{e}}}}+{\displaystyle \frac{1}{{\mathrm{q}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}}$ (5)	[47,49]
4	Linearization according to Scatchard	${\displaystyle \frac{\mathrm{q}}{{\mathrm{C}}_{\mathrm{e}}}}=-{\mathrm{K}}_{\mathrm{L}}·\mathrm{q}+{\mathrm{q}}_{\mathrm{m}\mathrm{a}\mathrm{x}}·{\mathrm{K}}_{\mathrm{L}}$ (6)	[47,49]
5	Linearization according to Eadie–Hofstee	$\mathrm{q}=\left({\displaystyle \frac{-1}{{\mathrm{K}}_{\mathrm{L}}}}\right)·{\displaystyle \frac{\mathrm{q}}{{\mathrm{C}}_{\mathrm{e}}}}+{\mathrm{q}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ (7)	[47,49]

Notation: q_max—maximum sorption capacity, given in mg/g; K_L—Langmuir isotherm constant, given in L/mg.

. The models in linear form were used to calculate the characteristic coefficients of the process, while the model in general form was used for error analysis.

The sorption capacity q in mg/g was calculated based on Equation (2), in which m denotes the adsorbent mass (in g); C_e denotes the equilibrium phosphate (mg/L); and V denotes the sample volume, given in L

$$\mathrm{q}={\displaystyle \frac{{\mathrm{C}}_{\mathrm{e}}·\mathrm{V}}{\mathrm{m}}}$$

(2)

The modeling of the process using Langmuir isotherms was conducted using linear forms, in which graphs with Ce/q = f(q) in the Langmuir linearization, 1/q = f(1/Ce) in the Lineweaver–Burke linearization, q/Ce = f(q) in the Scatchard linearization, and q = f(q/Ce) in the Eadie–Hofstee linearization [47,49] were made. The points on the graphs were approximated via a linear function accounting for the slope coefficients and the point of intersection with the vertical axis. This allowed for the calculation of the characteristic parameters, which are the Langmuir isotherm constant, K_L, and the maximum sorption capacity, i.e., the mass that the analyzed adsorbent, qmax, can bind.

To evaluate the degree of fit of the linear trend function, the coefficient of determination, R², was determined. The separation coefficient, also called the equilibrium parameter, R_L, which is expressed in Formula (8), was also calculated:

$$R_L={\displaystyle \frac{1}{1+K_L·C_0}}$$

(8)

R_L is interpreted as follows: R_L < 1 indicates that adsorption is favorable; R_L ∼ 0 indicates that adsorption is irreversible; R_L = 1 suggests that the adsorption isotherm is linear; and R_L > 1 indicates that adsorption is unfavorable [28]. A considerable number of publications inaccurately indicate that adsorption is irreversible when R_L = 0 [25,26,27,47,52,53]. Such a situation cannot occur, as the denominator must reach infinity. For the high denominator values in Formula (8), the R_L-value will be very small—close to zero. Also highly doubtful is the situation wherein R_L > 1. This can occur only if the product C₀ · K_L < 0 translates into a negative value of the constant K_L because C₀ cannot be harmful.

2.5. Statistical Analysis

An analysis of the fit of the measurement points and individual models was performed using the coefficient of determination, R²; the sum of squares of errors, ERRSQ (The Sum Square of Errors); the composite fractional error function, HYBRD (Hybrid Fractional Error Function); the average relative error, ARE (Average Relative Error); the Fisher F test; and the χ² test [54]. The relevant formulas are shown below.

$$\mathrm{E}\mathrm{R}\mathrm{R}\mathrm{S}\mathrm{Q}=\sum _{\mathrm{i}=1}^{\mathrm{K}}{\left({\mathrm{q}}_{\mathrm{e}\mathrm{x}\mathrm{p}}-{\mathrm{q}}_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}}\right)}^2$$

(9)

$$\mathrm{H}\mathrm{Y}\mathrm{B}\mathrm{R}\mathrm{D}={\displaystyle \frac{100}{\mathrm{k}-\mathrm{p}}}\sum _{\mathrm{i}=1}^{\mathrm{k}}\left({\displaystyle \frac{{\left({\mathrm{q}}_{\mathrm{e}\mathrm{x}\mathrm{p}}-{\mathrm{q}}_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}}\right)}^2}{{\mathrm{q}}_{\mathrm{e}\mathrm{x}\mathrm{p}}}}\right)$$

(10)

$$\mathrm{A}\mathrm{R}\mathrm{E}={\displaystyle \frac{100}{\mathrm{k}}}\sum _{\mathrm{i}=1}^{\mathrm{k}}\left|{\displaystyle \frac{{\mathrm{q}}_{\mathrm{e}\mathrm{x}\mathrm{p}}-{\mathrm{q}}_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}}}{{\mathrm{q}}_{\mathrm{e}\mathrm{x}\mathrm{p}}}}\right|$$

(11)

$$\mathrm{F}={\displaystyle \frac{\left(\mathrm{k}-\mathrm{p}\right)·{\sum }_{\mathrm{i}=1}^{\mathrm{k}}{\left({\mathrm{q}}_{\mathrm{e}\mathrm{x}\mathrm{p}}-\frac{1}{\mathrm{k}}·{\sum }_{\mathrm{i}=1}^{\mathrm{k}}{\mathrm{q}}_{\mathrm{e}\mathrm{x}\mathrm{p}}\right)}^2}{\left(\mathrm{k}-1\right)·{\sum }_{\mathrm{i}=1}^{\mathrm{k}}{\left({\mathrm{q}}_{\mathrm{e}\mathrm{x}\mathrm{p}}-{\mathrm{q}}_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}}\right)}^2}}$$

(12)

$${\mathsf{\chi }}^2=\sum _{\mathrm{i}=1}^{\mathrm{k}}\left({\displaystyle \frac{{\left({\mathrm{q}}_{\mathrm{e}\mathrm{x}\mathrm{p}}-{\mathrm{q}}_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}}\right)}^2}{{\mathrm{q}}_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}}}}\right)$$

(13)

Here, q_exp denotes the sorption obtained from measurements, q_calc denotes the sorption obtained from calculations using the model in general form, k is the number of measurement points, and p is the number of model parameters.

Statistical studies of the effect of temperature on sorption were performed using the GLM (generalized linear model) [55], with the dependent variable “q”, the quantitative variable “C_e”, and temperature as the qualitative variable.

In addition, in order to show whether there are significant differences between the sorption capacity at different process temperatures and Rockfos^® material granulations, an analysis of variance was performed. The normality of distributions in individual groups was tested using the Shapiro–Wilk test, and the homogeneity of variance was tested using Levene’s test. When the tested variables followed a normal distribution and the homogeneity of variance was maintained, differences between groups were analyzed with the ANOVA test. To identify which means differed significantly, post hoc NIR tests were applied. When variables within groups did not follow a normal distribution (according to the Shapiro–Wilk test), differences between groups were analyzed using the nonparametric Kruskal–Wallis ANOVA test. Significant differences between groups were then identified through multiple comparisons of the mean ranks for all samples. Statistical analyses were performed using STATISTICA 13.3.

2.6. Thermodynamic Analysis

A thermodynamic analysis of the process was performed using relationships (14)–(16) [25,53].

$${\mathrm{K}}_{\mathrm{d}}={\displaystyle \frac{\mathrm{q}}{{\mathrm{C}}_{\mathrm{e}}}}$$

(14)

$$\mathsf{\Delta }\mathrm{G}=-\mathrm{R}·\mathrm{T}·\ln {\mathrm{K}}_{\mathrm{d}}$$

(15)

$$\ln {\mathrm{K}}_{\mathrm{d}}={\displaystyle \frac{\mathsf{\Delta }\mathrm{S}}{\mathrm{R}}}-{\displaystyle \frac{\mathsf{\Delta }\mathrm{H}}{\mathrm{R}\mathrm{T}}}$$

(16)

Here, K_d is the equilibrium constant derived from the adsorption capacity, q, and the equilibrium concentration, Ce; ΔG denotes the standard free energy (J/mol); ΔS denotes the standard entropy (J/mol); ΔH denotes the standard enthalpy (J/(mol · K); R denotes the gas constant, 8.314 (J/(mol · K); and T denotes the absolute temperature (K).

The linear relationship (17) was used to determine the relevant parameters:

$$\ln {\mathrm{K}}_{\mathrm{d}}=\mathrm{f}\left({\displaystyle \frac{1}{\mathrm{T}}}\right)$$

(17)

whose directional coefficient is expressed by H/R, and the ordinate of the point of intersection of the vertical axis is indicated by S/R.

3. Results and Discussion

Figure 1, Figure 2 and Figure 3 show the relationship between the sorption capacity “q” of Rockfos^® material and the loss of phosphate concentration in solution. The analysis considered different initial concentrations of phosphate in solution (0.5 mg/L, 1.0 mg/L, and 2.0 mg/L), granulations of sorption material (1.0–1.6 mm, 1.6–2.5 mm, and 2.0–5.0 mm), and selected process running temperatures (5 °C, 15 °C, and 25 °C). Each data series consists of nine points corresponding to different sorbent weights.

The results show that the amount of phosphate removed from the solution increases curvilinearly with increasing sorbent weights. The range of these values is strongly influenced by the initial concentration of phosphate in the solution, an observation that has been confirmed by previous studies [18,29]. At the lowest concentration, 0.5 mg/L, phosphate loss did not exceed 0.4 mg/L, even when using the most considerable sorbent weight (0.800 g). At higher initial phosphate concentrations, namely, 1.0 mg/L and 2 mg/L, maximum phosphate losses reached 0.90 mg/L and 1.24 mg/L, respectively. Preliminary studies of the Rockfos^® material conducted by Kasprzyk et al. [29] showed a significant variation in the sorption capacity of the material depending on the initial concentration of phosphorus in the solution. The sorption capacity of the fine fraction of the material at a low phosphate concentration was 0.9 mg/g [48]. When using wastewater with higher phosphate concentrations, the sorption capacities were 9.6 mg/g and 45.6 mg/g [8,29]. Karaca et al. [56] conducted studies on a natural material, dolomite, and reported the phosphate sorption value to be 51.02 mg/g at the initial concentration in a solution of 60 mg/L. In turn, experiments using distilled water (with an initial phosphorus concentration of 0.28 mg/L), groundwater (0.34 mg/L), tap water (0.34 mg/L), and sewage (0.56 mg/L) revealed the sorption capacity of dolomite to be 0.06, 0.072, 0.05, and 0.05 mgP/g, respectively. The maximum sorption capacities, calculated using the Freundlich and Langmuir models, were 0.12 and 0.093 gP/kg [57], respectively. Column studies using shale conducted by Kpannieu et al. [58] also confirmed the positive relationship between the sorption capacity of the material and the initial concentration of phosphorus in the solution. The maximum sorption capacities for the material were 0.06 and 0.23 mg P/g, with initial concentrations of phosphorus of 5 and 25 mg/L, respectively [58]. It is worth noting, however, that some studies using natural materials came to different conclusions. Cyrus et al. [59,60] achieved a maximum sorption capacity of 0.17 g P/kg for shale at an initial concentration of 1000 mg P/L, which can be compared to 0.50 g P/kg at an initial concentration of 100 mg P/L [59,60].

The effect of the initial adsorbate concentration in a solution on sorption was also reported by Ho et al. [22] for lead sorption on tree fern sorbent; Reczek et al. [24] for lead on activated carbon, clinoptilolite, and manganese-dioxide-modified chalcedonite; and Tran et al. [49] for cadmium on orange peel sorbent. Wang et al. [23] showed that lead sorption on phosphoric-acid-activated carbon increases with increasing pH and decreases with an increase in ionic strength (which occurs during ion exchange).

The effect of grain size on sorption efficiency is shown in Figure 4, using two equilibrium concentration values (about 0.1 mg/L and 0.3 mg/L) and two selected process temperatures (10 °C and 20 °C) as examples. The magnitudes of phosphate sorption were determined by plotting q = f (Ce) (Figure 1, Figure 2 and Figure 3). The points are arranged somewhat chaotically, and it is impossible to decide which grain size favors sorption.

The scattering of the measurement points may result from variations in the content of active compounds in the individual portions of the sorbent used in this study. It should be emphasized that natural rocks were used in this study after being burned, crushed, and separated into fractions. The compositions of individual particles could therefore be different depending on the natural crystallization processes occurring over long periods of time, temperature fluctuations during crystallization, and other disturbances influencing the variability of the shares of individual oxides of different metals. Statistical analysis also showed that at the initial concentration of phosphates in the solution, 0.5 mg/L, the tested fractions of the Rockfos^® material did not differ significantly in terms of phosphate sorption. Statistical analysis also showed that at the initial concentration of phosphates in the solution (0.5 mg/L), the tested fractions of the Rockfos^® material did not differ significantly in terms of phosphate sorption. This finding was consistent across all the analyzed temperatures. At a concentration of 1.0 mg/L, the sorption capacity for granulations of 1.6–2.5 mm and 2–5 mm across the entire temperature range was significantly higher than that for the 1–1.6 mm granulation. In turn, at a concentration of 2.0 mg/L and low process temperatures (5 °C, 10 °C), sorption for the 2–5 mm fraction was significantly lower than that for the 1.6–2.5 mm granulation. At higher process temperatures, no significant differences in phosphate sorption were observed between the granulations.

A study carried out by Kasprzyk et al. [29] concluded that grain size affects sorption efficiency, but a closer examination of the text in this publication does not support this conclusion. The smaller the grains, the larger the surface area, revealing a relationship that can increase sorption capacity [12,29,61]. In chemisorption processes, however, the chemically active parts of the grains will play a more crucial role (rather than the total surface area). Therefore, it can be stated that in the case of materials with a heterogeneous composition, grain sizes within the ranges applied in this study do not affect the amounts of phosphates retained in the chemisorption process.

Temperature is one of the most critical factors affecting the amount of phosphate sorption on Rockfos^® material. This relationship is illustrated in Figure 5 using the example of a 2.0–5.0 mm granulation with an initial phosphate concentration in solution of 2 mg/L.

The results in Figure 5 show an increase in sorption with increasing temperatures. The points corresponding to 5 °C are the lowest, and those corresponding to 25 °C are the highest. To statistically substantiate the effect of temperature on the amount of sorption (q) and equilibrium concentration (C_e), an analysis was performed using the generalized linear model (GLM); sample results are shown in Figure 6.

The model showed a statistically significant effect of temperature on sorption, with a probability of less than 0.001, which is less than the assumed significance level of 0.05. The data in Figure 6 show an apparent effect of temperature on the amount of phosphate sorption on Rockfos^® material. The 5 and 10 °C results were included in one group, “a”, as they differed slightly from each other. In contrast, the average sorption values for the other temperatures were so different from each other that they were modeled in separate groups, “b”, “c”, and “d”. The ANOVA and Kruskal–Wallis ANOVA tests showed that at an initial phosphate concentration of 0.5 mg/L in the solution, there were no significant differences in the sorption capacity of the material across the various process temperatures. However, at higher initial concentrations (1.0 and 2.0 mg/L), the sorption capacity at 20 °C and 25 °C was significantly higher than that observed at lower temperatures. This pattern applied to each granulation of the Rockfos^® material.

In the literature [25,46], these graphs initially (at low sorbent doses) have a strong upward trend, and from a certain mass of sorbent onward, the graph stacks upward horizontally. A strong upward trend in sorption modeled by the Langmuir isotherm with increasing temperatures was shown by Lomowskiy et al. [62], who studied melanin sorption on buckwheat husks but at higher temperatures of 40 °C, 50 °C, and 60 °C.

Figure 7 shows the effects of phosphate removal from the solution as a sorbent weight function at different sorption process temperatures. The points from 1 to 9 on the horizontal axis refer to the increasing sorbent weight, given in Section 2.2. A close proportional relationship between the effects of phosphate removal from solution and sorbent mass can be observed in all of the data series. However, the amplitude of the values is closely related to the temperature at which the sorption process was carried out. At 5 °C, 10 °C, and 15 °C, the course of the curves is similar, and the determined effects range from 1.5–2.6% to 37.2–41.7% (Figure 7). At 20 °C and 25 °C, the effects were noticeably higher and, depending on the sorbent mass, ranged from 1.1–61.0% to 3.1–66.1%, respectively.

The increase in efficiency as a function of temperature is not always so clear-cut. As shown in the study carried out by Hassani et al. [53], efficiency increased with increasing temperatures for black eriochrome-induced color removal and decreased for blue maxilon removal.

Similarly, the sorption of lead on tree ferns [22], cadmium on eucalyptus bark [27], cadmium on orange peel [49], and phosphate on calcium silicate hydrate (C-S-H) increased with increasing temperatures [34,49], as did the sorption of sodium oleate on a quartz surface [63] and melanin on buckwheat husks [62]. The opposite is true during the sorption of gases, as sorption decreases with increasing temperatures (e.g., carbon dioxide on selected coal samples) [64,65]. Inconclusive results have been obtained regarding the sorption of heavy oil in quartz nanopores at high temperatures [66].

Additional measurements of pH, total hardness, and calcium concentration were made to evaluate the influence of ion exchange processes on the sum effect of sorption. The choice of these indicators was dictated by previous reports of a strong impact of opoka on solution pH [8,13,33,34,44,45] and the chemical composition of the rock [11,12,29,44,45,67]. The hardness of the solution increased during the tests; its average value at the most negligible weight of sorbent was 1.56 ± 0.79 °dh, while it was 7.09 ± 1.98 °dh at the most significant weight. The calcium concentration increased from 12.76 ± 5.47 mg Ca/L to 46.27 ± 11.87 mg Ca/L. This indicates a significant role of ion exchange, during which phosphates react with calcium, leading to the precipitation of calcium phosphate as a residue. The chemisorption process cannot be ignored when using Rockfos^® material. A partial confirmation of this reasoning is the increase in pH in the subsequent samples, that is, with increasing sorbent weights. Due to the heterogeneity of the composition of the rocks themselves, the pH changes of the solution were not uniform, although the vast majority varied from about 7.2 to about 8.6. Example pH results are shown in Figure 8.

Figure 8 shows the boundaries of each of the four quadrilles. The area between each series shows the boundaries (lower and upper) of each quadrille in which 10–11 measurement results were contained. These counts are a result of dividing the 45 measurements for each sorbent weight into four parts. For the most part, the pH values with the sorbent weights used did not exceed 9.0. However, in four cases, the pH exceeded this value and even approached 9.5. At such high pH values, the binding of phosphates to the Rockfos^® material during the sorption process is enhanced by the formation of precipitates due to the reaction of calcium ions with phosphates. In wastewater treatment plants, in sludge treatment lines using anaerobic processes at the outflow area of the treatment plant, high pH stimulates sludge precipitation in the form of struvite or hydroxyapatite [68,69]. The irregular points referred to as the A and E series are there because the points on the graph refer to the single values (minimum and maximum) found in the set of 45 measurements, in each of the nine series assigned to the corresponding weights. These pH measurements should not be equated with studies on the effect of solution pH on the sorption effect, as presented in many publications [14,31,53]. In the present study, the pH during the experiment was controlled based on the observations of other researchers [8,12,70] who have shown that the materials obtained through the thermal processing of opoka in contact with water precipitate a significant increase in pH, sometimes to a range that is unacceptable for water or wastewater discharged into the environment.

The results of modeling studies concerning phosphate sorption on grains of Rockfos^® material are shown in

Table 2. Parameters of adsorption Langmuir isotherms.

T, °C		KL [dm3 /g].	qmax [mg/g].	RL -	R2 -	KL [dm3 /g].	qmax [mg/g].	RL -	R2 -	KL [dm3 /g].	qmax [mg/g].	RL -	R2 -
	C0	0.5 mg/dm2				1.0 mg/dm3				2.0 mg/dm3
	LI—Langmuir isotherm in Langmuir linearization [49]
5 °C	1.0 ÷ 1.60	45.74	0.0221	0.04	0.967	54.92	0.0330	0.02	0.975	9.67	0.1157	0.05	0.959
	1.6 ÷ 2.5	81.50	0.0392	0.02	0.987	20.88	0.0399	0.05	0.931	6.23	0.0664	0.07	0.926
	2.0 ÷ 5.0	35.01	0.0400	0.05	0.987	185.17	0.0530	0.01	0.995	12.99	0.0921	0.04	0.978
10 °C	1.0 ÷ 1.60	70.17	0.0208	0.03	0.987	67.98	0.0313	0.01	0.955	17.91	0.1304	0.03	0.980
	1.6 ÷ 2.5	83.40	0.0340	0.02	0.971	13.64	0.0617	0.07	0.904	23.52	0.1107	0.02	0.981
	2.0 ÷ 5.0	37.85	0.0314	0.05	0.957	60.14	0.0578	0.02	0.995	13.38	0.1016	0.04	0.982
15 °C	1.0 ÷ 1.60	28.95	0.0262	0.06	0.939	19.73	0.0722	0.05	0.964	15.48	0.1312	0.03	0.996
	1.6 ÷ 2.5	30.54	0.0246	0.06	0.953	66.17	0.0551	0.01	0.993	17.33	0.1099	0.03	0.988
	2.0 ÷ 5.0	88.00	0.0258	0.02	0.979	42.42	0.0718	0.02	0.995	28.26	0.1066	0.02	0.991
20 °C	1.0 ÷ 1.60	45.60	0.0436	0.04	0.956	111.32	0.0687	0.01	0.976	16.93	0.1179	0.03	0.995
	1.6 ÷ 2.5	24.51	0.0506	0.08	0.981	37.04	0.1309	0.03	0.952	11.87	0.1535	0.04	0.995
	2.0 ÷ 5.0	27.65	0.0353	0.07	0.959	6.65	0.1327	0.13	0.947	32.30	0.1535	0.02	0.982
25 °C	1.0 ÷ 1.60	24.66	0.0443	0.08	0.984	32.87	0.0961	0.03	0.998	11.39	0.1491	0.04	0.992
	1.6 ÷ 2.5	21.44	0.0520	0.09	0.985	19.79	0.1234	0.05	0.994	10.26	0.1650	0.05	0.993
	2.0 ÷ 5.0	21.23	0.0369	0.09	0.917	20.38	0.1246	0.05	0.970	13.25	0.1335	0.04	0.963
	LII—Langmuir isotherm in Lineweaver–Burke linearization [49]
5 °C	1.0 ÷ 1.60	192.08	0.0150	0.01	0.662	218.69	0.0289	0.00	0.496	48.43	0.0850	0.01	0.360
	1.6 ÷ 2.5	63.98	0.0491	0.03	0.908	−26.63	0.0427	−0.04	0.577	−5.99	0.0646	−0.09	0.697
	2.0 ÷ 5.0	94.53	0.0321	0.02	0.889	130.72	0.0533	0.01	0.779	51.95	0.0726	0.01	0.802
10 °C	1.0 ÷ 1.60	184.56	0.0170	0.01	0.571	230.73	0.0286	0.00	0.346	25.33	0.1237	0.02	0.967
	1.6 ÷ 2.5	108.50	0.0412	0.02	0.726	−32.41	0.0746	−0.03	0.387	70.94	0.0994	0.01	0.096
	2.0 ÷ 5.0	348.48	0.0238	0.01	0.268	134.38	0.0546	0.01	0.795	50.19	0.0832	0.01	0.735
15 °C	1.0 ÷ 1.60	166.47	0.0162	0.01	0.815	92.58	0.0601	0.01	0.739	29.04	0.1193	0.02	0.903
	1.6 ÷ 2.5	143.34	0.0159	0.01	0.688	228.17	0.0518	0.00	0.588	46.92	0.0988	0.01	0.888
	2.0 ÷ 5.0	177.66	0.0229	0.01	0.622	92.05	0.0676	0.01	0.679	69.61	0.1005	0.01	0.854
20 °C	1.0 ÷ 1.60	27.88	0.0475	0.07	0.987	31.93	0.0835	0.03	0.967	29.93	0.1079	0.02	0.961
	1.6 ÷ 2.5	46.03	0.0410	0.04	0.824	15.21	0.1427	0.06	0.998	17.06	0.1459	0.03	0.996
	2.0 ÷ 5.0	141.66	0.0253	0.01	0.632	36.33	0.0869	0.03	0.622	24.65	0.1543	0.02	0.856
25 °C	1.0 ÷ 1.60	63.58	0.0337	0.03	0.984	67.71	0.0905	0.01	0.998	32.38	0.1287	0.02	0.992
	1.6 ÷ 2.5	57.85	0.0381	0.03	0.922	34.64	0.1159	0.03	0.968	26.99	0.1428	0.02	0.957
	2.0 ÷ 5.0	109.60	0.0239	0.02	0.113	50.61	0.1145	0.02	0.476	11.51	0.1321	0.04	0.517
	LIII—Langmuir isotherm in Scatchard linearization [49]
5 °C	1.0 ÷ 1.60	77.54	0.0205	0.03	0.523	64.63	0.0350	0.02	0.243	14.77	0.1139	0.03	0.317
	1.6 ÷ 2.5	30.22	0.0595	0.06	0.904	−19.26	0.0375	−0.05	0.303	−5.47	0.0610	−0.10	0.960
	2.0 ÷ 5.0	56.58	0.0371	0.03	0.730	90.73	0.0552	0.01	0.707	28.93	0.0817	0.02	0.641
10 °C	1.0 ÷ 1.60	94.57	0.0205	0.02	0.527	28.65	0.0449	0.03	0.070	20.28	0.1301	0.02	0.823
	1.6 ÷ 2.5	47.07	0.0481	0.04	0.313	−18.55	0.0653	0.06	0.537	2.24	0.3180	0.18	0.005
	2.0 ÷ 5.0	69.49	0.0304	0.03	0.234	92.05	0.0565	0.01	0.717	22.34	0.1133	0.02	0.607
15 °C	1.0 ÷ 1.60	67.28	0.0218	0.03	0.543	46.94	0.0661	0.02	0.552	22.49	0.1256	0.02	0.841
	1.6 ÷ 2.5	60.89	0.0212	0.03	0.479	115.99	0.0542	0.01	0.517	35.11	0.1029	0.01	0.755
	2.0 ÷ 5.0	91.18	0.0265	0.02	0.562	61.06	0.0707	0.02	0.637	52.63	0.1030	0.01	0.767
20 °C	1.0 ÷ 1.60	38.43	0.0455	0.05	0.879	22.34	0.0919	0.04	0.505	24.35	0.1128	0.02	0.868
	1.6 ÷ 2.5	32.21	0.0525	0.06	0.750	12.91	0.1504	0.07	0.826	16.47	0.1472	0.03	0.957
	2.0 ÷ 5.0	53.62	0.0319	0.04	0.464	11.89	0.1214	0.08	0.434	17.58	0.1642	0.03	0.740
25 °C	1.0 ÷ 1.60	42.46	0.0395	0.04	0.851	54.57	0.0926	0.02	0.846	21.42	0.1383	0.02	0.713
	1.6 ÷ 2.5	33.71	0.0475	0.06	0.769	30.17	0.1184	0.03	0.917	18.40	0.1534	0.03	0.757
	2.0 ÷ 5.0	−10.64	−0.023	−0.22	0.032	17.86	0.1330	0.05	0.352	−6.92	0.1102	−0.08	0.899
	LIV—Langmuir isotherm in Eadie–Hofstee linearization [49]
5 °C	1.0 ÷ 1.60	149.25	0.0172	0.01	0.533	263.16	0.0290	0.00	0.317	46.51	0.899	0.01	0.243
	1.6 ÷ 2.5	100.00	0.0474	0.02	0.303	-21.32	0.0400	-0.05	0.904	-5.70	0.0647	-0.10	0.960
	2.0 ÷ 5.0	77.52	0.0345	0.03	0.730	128.21	0.0536	0.01	0.707	45.05	0.0754	0.01	0.641
10 °C	1.0 ÷ 1.60	178.57	0.0179	0.01	0.527	416.67	0.0284	0.00	0.700	24.63	0.1255	0.02	0.823
	1.6 ÷ 2.5	149.25	0.0406	0.01	0.313	34.60	0.0786	0.03	0.537	476.19	0.969	0.00	0.005
	2.0 ÷ 5.0	294.12	0.0246	0.01	0.234	128.21	0.0549	0.01	0.717	45.05	0.0855	0.01	0.607
15 °C	1.0 ÷ 1.60	123.46	0.0184	0.01	0.543	84.75	0.0612	0.01	0.552	26.74	0.1220	0.02	0.841
	1.6 ÷ 2.5	126.58	0.0169	0.02	0.479	222.22	0.0520	0.00	0.517	46.51	0.0994	0.01	0.755
	2.0 ÷ 5.0	161.29	0.0239	0.01	0.562	96.15	0.677	0.01	0.637	68.49	0.1007	0.01	0.767
20 °C	1.0 ÷ 1.60	43.67	0.0439	0.04	0.879	44.25	0.0803	0.02	0.505	28.09	0.1100	0.02	0.868
	1.6 ÷ 2.5	46.95	0.0443	0.04	0.750	15.63	0.1434	0.06	0.826	17.21	0.1459	0.03	0.957
	2.0 ÷ 5.0	116.28	0.0270	0.02	0.464	27.40	0.0984	0.04	0.434	23.87	0.1567	0.02	0.736
25 °C	1.0 ÷ 1.60	50.00	0.0377	0.04	0.851	64.52	0.0911	0.02	0.846	30.03	0.1314	0.02	0.713
	1.6 ÷ 2.5	43.86	0.0440	0.04	0.769	32.89	0.1170	0.03	0.917	24.33	0.1466	0.02	0.757
	2.0 ÷ 5.0	333.33	0.0218	0.01	0.032	50.76	0.1160	0.02	0.352	−7.7101	0.1176	-0.07	0.899

.

The results presented in Table 2 show that the way the general Langmuir model is linearized has a significant impact on the key parameters. These parameters include K_L, i.e., the Langmuir adsorption constant (L/mg) that is related to the apparent energy of sorption [71], and qmax, which represents the maximum sorption capacity on the surface of the deposit grains.

An analysis of the individual values in Table 2 reveals irregular scatterings, which are challenging to describe methodically. However, q_max increases with increasing temperatures. With increasing phosphate concentrations in the solution, as shown in Figure 9, the most important parameter with a practical dimension is the maximum amount of phosphate that can be absorbed by a gram of analyzed material. The average values as a function of temperature depending on the grain size of the adsorbent for all four linearizations are shown in Figure 9.

Upon analyzing the graphs in Figure 9, it can be assumed that in the winter, at wastewater temperatures below 10 °C, 1 g of the test material can absorb 0.6–0.8 mg of phosphate, while in the summer, at 20–25 °C, it can absorb 0.8–1.0 mg. Scientific reports note opoka’s tremendous variation in phosphate absorption capacity and its modifications. The type of opoka and its source are essential aspects. With high proportions of silica (light opoka) and carbonates (heavy opoka), the sorption volumes of phosphate obtained from the Langmuir isotherm were 80 mg P/g and 137–181 mg P/g, respectively [12]. After the firing of opoka, the content of iron oxides is an essential aspect, and this is especially the case for calcium oxides, which, in amounts between 238.6 and 520.1 g of Ca/kg, stimulate the absorption of 12.3–25.5 mg of P-PO₄⁻³/g [35]. Kasprzyk et al. [8,29] studied two materials, rock with high CaCO content₃ (M1) and Rockfos^® material (M2), which showed considerable differences in sorption. The sorption values obtained from the Langmuir isotherm were q_max = 256.41 mg/g for the M1 material and 0.36 mg/g for the M2 material. Cucarella and Renman [45] and Li et al. [46] showed that the type of rock is not the only parameter that affects phosphate sorption; they noted a dependence on sorbent particle size and pH and a robust correlation with phosphate concentrations in solution.

Laboratory results are not always fully translated into the practical performance of sorbents working in the final stage of wastewater treatment plants. Westholm [70], studying phosphate sorption on blast furnace slag, found that, compared to laboratory results from field studies, phosphorus sorption decreased significantly by about twofold.

The calculations made for LI and LII linearization (Figure 9A) show more excellent stabilization, while those obtained for LIII and LIV linearization (Figure 9B) show overestimated results with such substantial scattering that they can hardly be considered reliable. Since these results were obtained from the exact measurements, it can be seen that the linearization method has a significant impact on the conclusions. The above analyses show that sorption analysis using Langmuir linearization should be used, with Lineweaver–Burke constituting a secondary option.

The linearization of the general formula, approximation with a linear function, and, finally, the determination of K_L and q_max naturally introduce errors, partially expressed by the determination coefficient, R², shown for all cases in Table 2. As can be seen, the highest values of R² were obtained from Langmuir linearization, which means that this mode of operation and this way of determining K_L and q_max generates the least significant errors. The two graphs shown in Figure 10 show a comparison of such cases.

To more precisely determine the goodness of fit of the function with respect to the measurement results, error functions were determined for all the cases, described in Table 2 as optimization criteria. The characteristic quantities were determined using Formulas (8)–(12) and are shown in

Table 3. Analysis of model fit error functions.

T, °C			ESRQ	HYBRYD	ARE	Fisher	χ2
5 °C	1.0 ÷ 1.6	0.5	L.I. 3.57 · 10−5	L.I. 5.96 · 10−2	L.I. 18.3	L.I. 7.04	L.I. 4.17 · 10−3
		1.0	L.II. 1.11 · 10−4	L.II. 5.46 · 10−2	L.II. 9.42	L.IV. 1.34	L.II. 3.82 · 10−3
		2.0	L.I. 2.04 · 10−3	L.I. 4.49 · 10−1	L.I. 17.7	L.I. 1.89	L.I. 3.14 · 10−2
	1.6 ÷ 2.5	0.5	L.IV. 5.06 · 10−4	L.IV. 1.53 · 10−3	L.II. 13.2	L.IV. 1.81	L.IV. 1.07 · 10−2
		1.0	L.IV. 2.81 · 10−3	L.II. 4.65 · 10−1	L.II. 18.4	L.IV. 2.29	L.II. 3.25 · 10−2
		2.0	L.II. 1.93 · 10−2	L.II. 1.23	L.II. 16.5	L.II. 3.79	L.II. 8.61 · 10−2
	2.0–5.0	0.5	L.I. 5.98 · 10−5	L.I. 5.09 · 10−1	L.I. 10.6	L.I. 11.7	L.I. 3.56 · 10−3
		1.0	L.IV. 6.60 · 10−5	L.II. 1.97 · 10−2	L.II. 4.26	L.IV. 3.38	L.II. 1.38 · 10−3
		2.0	L.III. 5.28 · 10−4	L.III. 1.28 · 10−1	L.II. 4.26	L.IV. 3.38	L.II. 1.38 · 10−3
10 °C	1.0 ÷ 1.6	0.5	L.I. 3.20 · 10−5	L.I. 4.62 · 10−2	L.I. 12.2	L.III. 4.55	L.I. 3.23 · 10−3
		1.0	L.IV. 1.94 · 10−4	L.II. 9.35 · 10−2	L.II. 12.8	L.IV. 1.05	L.II. 6.55 · 10−3
		2.0	L.III. 9.15 · 10−4	L.IV. 1.22 · 10−1	L.II. 7.29	L.III. 6.36	L.IV. 8.54 · 10−3
	1.6 ÷ 2.5	0.5	L.IV. 2.29 · 10−4	L.IV. 8.84 · 10−2	L.II. 11.5	L.IV. 1.58	L.IV. 6.19 · 10−3
		1.0	L.II. 3.80 · 10−3	L.II. 5.27 · 10−1	L.II. 20.0	L.II. 1.19	L.II. 3.69 · 10−2
		2.0	L.II. 2.16 · 10−3	L.II. 3.90 · 10−1	L.I. 12.5	L.II. 9.59 · 10−1	L.II. 2.73 · 10−2
	2.0–5.0	0.5	L.IV. 1.16 · 10−4	L.II. 6.94 · 10−2	L.II. 13.7	L.IV. 1.23	L.II. 4.86 · 10−3
		1.0	L.IV. 6.63 · 10−5	L.IV. 1.81 · 10−2	L.II. 4.29	L.IV. 3.47	L.IV. 1.27 · 10−3
		2.0	L.IV. 6.23 · 10−4	L.IV. 1.12 · 10−1	L.II. 10.1	L.IV. 2.75	L.IV. 7.85 · 10−3
15 °C	1.0 ÷ 1.6	0.5	L.I. 4.96 · 10−5	L.I. 7.38 · 10−2	L.II. 19.1	L.I. 6.00	L.I. 5.17 · 10−3
		1.0	L.IV. 3.13 · 10−4	L.IV. 6.87 · 10−2	L.II. 6.97	L.IV. 2.12	L.IV. 4.81 · 10−3
		2.0	L.III. 3.76 · 10−4	L.III. 6.36 · 10−2	L.I. 5.95	L.III. 11.3	L.III. 4.45 · 10−3
	1.6 ÷ 2.5	0.5	L.I. 3.45 · 10−5	L.I. 6.46 · 10−2	L.II. 18.9	L.I. 9.13	L.I. 4.52 · 10−3
		1.0	L.IV. 6.12 · 10−5	L.II. 1.75 · 10−2	L.II. 3.85	L.IV. 1.89	L.II. 1.22 · 10−3
		2.0	L.IV. 3.66 · 10−4	L.II. 5.59 · 10−2	L.II. 5.91	L.IV. 4.10	L.II. 3.91 · 10−3
	2.0–5.0	0.5	L.I. 5.76 · 10−5	L.I. 5.52 · 10−2	L.II. 11.1	L.I. 3.71	L.I. 3.86 · 10−3
		1.0	L.IV. 1.13 · 10−4	L.II. 2.87 · 10−2	L.II. 4.77	L.II. 3.23	L.II. 2.01 · 10−3
		2.0	L.IV. 1.59 · 10−4	L.II. 2.29 · 10−2	L.II. 3.13	L.IV. 3.96	L.IV. 1.60 · 10−3
20 °C	1.0 ÷ 1.6	0.5	L.III. 9.77 · 10−5	L.III. 4.35 · 10−2	L.I. 9.70	L.III. 16.4	L.III. 3.05 · 10−3
		1.0	L.IV. 9.48 · 10−4	L.IV. 1.81 · 10−1	L.II. 11.1	L.IV. 2.84	L.IV. 1.27 · 10−2
		2.0	L.III. 3.92 · 10−4	L.III. 7.16 · 10−2	L.II. 6.20	L.III. 10.3	L.III. 5.01 · 10−3
	1.6 ÷ 2.5	0.5	L.I. 9.93 · 10−5	L.I. 6.02 · 10−2	L.I. 11.3	L.I. 17.1	L.I. 4.21 · 10−3
		1.0	L.IV. 1.93 · 10−3	L.II. 2.16 · 10−1	L.II. 9.62	L.IV. 6.54	L.II. 1.52 · 10−2
		2.0	L.III. 3.02 · 10−4	L.II. 3.50 · 10−2	L.II. 3.71	L.III. 26.6	L.II. 2.45 · 10−3
	2.0–5.0	0.5	L.III. 1.42 · 10−4	L.IV. 1.00 · 10−1	L.II. 15.6	L.III. 2.37	L.IV. 7.02 · 10−3
		1.0	L.I. 1.63 · 10−3	L.I. 4.29 · 10−1	L.I. 20.0	L.I. 4.56	L.I. 3.00 · 10−2
		2.0	L.IV. 1.32 · 10−3	L.IV. 1.52 · 10−1	L.II. 8.16	L.IV. 4.64	L.II. 1.07 · 10−2
25 °C	1.0 ÷ 1.6	0.5	L.I. 4.01 · 10−5	L.I. 3.07 · 10−2	L.I. 11.3	L.I. 2.92	L.I. 2.15 · 10−3
		1.0	L.III. 1.14 · 10−4	L.IV. 2.05 · 10−2	L.II. 3.82	L.III. 7.21	L.IV. 1.44 · 10−3
		2.0	L.III. 8.27 · 10−4	L.III. 1.17 · 10−1	L.II. 8.84	L.III. 4.84	L.III. 8.20 · 10−3
	1.6 ÷ 2.5	0.5	L.I. 6.92 · 10−5	L.I. 4.81 · 10−2	L.I. 11.7	L.I. 24.1	L.I. 3.37 · 10−3
		1.0	L.III. 1.94 · 10−4	L.III. 2.75 · 10−2	L.II. 3.96	L.III. 13.8	L.III. 1.92 · 10−3
		2.0	L.III. 9.98 · 10−4	L.III. 1.29 · 10−1	L.II. 8.95	L.III. 5.77	L.III. 9.02 · 10−3
	2.0–5.0	0.5	L.II. 6.52 · 10−4	L.I. 3.32 · 10−1	L.I. 23.3	L.II. 8.87	L.I. 2.33 · 10−2
		1.0	L.IV. 1.32 · 10−3	L.II. 1.73 · 10−1	L.II. 9.06	L.IV. 1.52	L.II. 1.21 · 10−2
		2.0	L.IV. 3.41 · 10−2	L.IV. 2.67	L.IV. 25.9	L.IV. 1.42	L.IV. 1.87 · 10−1

. The values are given in Table 3. The general formula was defined on the basis that there were no additional errors that are inevitable with linearization and approximation.

In Table 3, individual cells are labeled as L with the corresponding Roman numeral, referring to the specific linearization method. For example, in the first row, the lowest ESRQ, HYBRD, ARE, and χ² values were obtained using Langmuir linearization, labeled LI. The same situation applies for Fisher’s test; however, this test determines the best fit when the value is as high as possible. Analyzing all the fits together with the typical and commonly used R² is an excellent way of selecting the best fit for the results and thus, determining the correct values of K_L, q_max, and R_L. However, listing any fit in the table as the best fit should not be misleading since, for ESRQ, one would have to accept results with a multiplier of no more than 10⁻⁵, with multipliers for HYBRD, ARE, Fisher’s test, and the Chi-square test being less than 5 · 10⁻², less than 10, more significant than 5, and under 5 · 10⁻³, respectively.

In the thermodynamic analysis, we used q_max and C₀ to calculate K_d, followed by functional graphs lnK_d = f(1/T). Example relationships for granulations of 1.0–1.6 mm and 2.0–5.0 mm and initial phosphate concentrations in solution of 0.5 and 1.0 mg/L are shown in Figure 11.

The directional coefficients and ordinates of the vertical axis intersection were used as the basis for calculating standard entropy (ΔS), standard enthalpy (ΔH), and standard free energy (ΔG). A summary of the results is shown in

Table 4. Thermodynamic function values.

Granulation, mm	Concentration C0, mg/L	ΔS, J/mol	ΔH, J/(mol · K)	ΔG, J/(mol · K)
1.0–1.6	0.5	77.321	−29,099.00	−7254.18 ÷ −6073.08
1.0–1.6	1.0	108.980	−38,475.53	−7953.77 ÷ −5947.07
1.0–1.6	2.0	1.238	−7015.85	−6720.82 ÷ −6419.69
1.6–2.5	0.5	52.998	−21,633.86	−6346.95 ÷ −5601.56
1.6–2.5	1.0	132.284	−32,214.26	−7603.71 ÷ −5256.99
1.6–2.5	2.0	87.970	−32,214.26	−8151.18 ÷ −6229.76
2.0–5.0	0.5	0.853	−6992.24	−6146.72 ÷ −6673.73
2.0–5.0	1.0	92.028	−32,647.42	−6835.24 ÷ −5440.45
2.0–5.0	2.0	28.528	−15,138.13	−7258.97 ÷ −6816.61

.

The results in the ΔG column are given for the range of temperatures tested. The first number corresponds to 5 °C and the second to 25 °C, and inside the range are values for intermediate temperatures.

Positive entropy values indicate that some structural changes are taking place on the adsorbent’s surface and reveal random interferences at the solid–liquid interface. They confirm an increase in the degree of disordered particles and energy dissipation throughout the temperature range. Negative enthalpy values indicate that sorption is exothermic. Negative free energy values suggest that the phosphate adsorption process is spontaneous over the studied temperature range. It can also be noted that ΔG increases with increasing solution temperatures, which can be explained by adsorption becoming slightly more challenging with increasing temperatures [53].

4. Conclusions

Our analysis of the results of the phosphate sorption tests conducted on Rockfos^® allowed the following conclusions to be drawn.

• The Langmuir isotherm, especially in the linearized form proposed by Langmuir, 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.
• In assessing the degree of fit between the trend line and the survey results, it is insufficient to rely on the R² coefficient of determination since even a few outliers from the main set cause a sharp drop in the R² value. Additional error analyses, such as ESRQ, HYBRD, ARE, Fisher’s test, and Chi-square test (χ²), are instrumental in better illustrating the degree of fit, since they are performed on the model in its general form. At the same time, R²; refers to the linearized function.
• 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.
• The near-zero values of the separation coefficient R_L prove that the sorption of phosphates on Rockfos^® material is irreversible (chemisorption). 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.
• The tested material can be used in practice in wastewater treatment, and maximum sorption capacities of 0.6–0.8 mgPO₄³⁻/g of sorbent in the winter and 0.8–1.0 mgPO₄³⁻/g in the summer can be assumed for designing the mass and adequate time of filter beds.