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

Abstract

Magnetic nanosorbents efficiently capture substances, particularly basic dyes, and can be easily recovered using a magnetic field in water treatment. Adsorption is a cost-effective and highly efficient method for basic dye removal. This study compared eight nonlinear kinetic adsorption models using Microsoft Excel 2023, which provided a detailed analysis and statistical results comparable to advanced programs like MATLAB and OriginPro. The Fractal Like-Pseudo First Order (FL-PFO) model showed the best fit for the kinetic adsorption model, closely predicting experimental data at 33.09 mg g⁻¹. This suggests that the diffusion rate of basic dye within the magnetic nanosorbent pores is a crucial factor. The statistical parameters confirmed the suitability of these kinetic adsorption models for describing the observed behavior. Overall, Microsoft Excel emerged as a reliable tool for predicting adsorption behavior using various kinetic models for basic dye removal, offering a wide range of functions for diverse applications, including environmental monitoring and modeling. Corrected Akaike’s information criterion was used to determine the optimal model. It found the lowest AIC_corrected value of about −3.8479 for the FL-PFO kinetic model, while the Elovich kinetic adsorption model had the highest AIC_corrected value of 29.6605. This indicates that the FL-PFO kinetic model effectively correlated the kinetic data. It can be concluded that Microsoft Excel’s accessibility, familiarity, and broad range of capabilities make it a valuable resource for many aspects of environmental engineering.

1. Introduction

Magnetic nanosorbents are a type of nanomaterial that possess magnetic properties and are used for various applications, including water and wastewater treatment [1]. These magnetic nanosorbents consist of magnetic nanoparticles, which are typically composed of iron oxide or other magnetic materials and functionalized with specific surface groups or coatings with activated carbon [2] to enhance their adsorption capabilities. Magnetic nanosorbents have several advantages such as high adsorption capacity [3], magnetic separation, reusability/regeneration, targeted adsorption, and scalability (can be synthesized in various sizes and shapes). However, it should be realized that there exist potential challenges and risks associated with their use. These might include the need for careful disposal of the nanosorbents to prevent environmental contamination, the potential toxicity [4] of the iron oxide magnetic nanoparticles used, and the development of efficient and cost-effective synthesis methods [5]. Generally, magnetic nanosorbents have shown great promise in water and wastewater treatment applications, providing the efficient and targeted removal of contaminants. Ongoing research and the development of magnetic nanosorbents aim to optimize their performance, increase their stability and minimize potential environmental impacts. High adsorption capacity and easy and efficient separation from water using external magnetic fields are the key focus areas of much research. As magnetic nanosorbents have high surface areas and can be functionalized with specific adsorbent materials or coatings, they can effectively remove a wide range of contaminants from water, including heavy metals [6], organic pollutants [7], dyes [8], and other harmful substances. They have a high adsorption capacity, which allows for the efficient removal of contaminants even at low concentrations [9].

It is well known that basic dyes are a type of synthetic dye that are commonly used in various industries, including textiles, paper, and some types of inks, which are typically alkaline or basic in nature with a pH greater than 7 [10]. In addition, basic dyes also have vibrant and bright coloration properties that make them visually appealing for staining purposes [11]. Moreover, basic dyes are less expensive compared to other types of dyes used in biological staining [12], such as fluorescent dyes. On the other hand, while basic dyes have their advantages, they also have limitations such as non-specific staining, lack of specificity, poor light stability, potential toxicity, limited color options, and interference with subsequent staining. However, it is significant to note that the use and disposal of these dyes can have environmental impacts such as water pollution, the production of chemical waste, energy consumption, and resource depletion. The pollution concerns associated with basic dyes are especially relevant in the textile industry [13]. In the textile industry, the dyeing process often involves large amounts of water and chemicals, including basic dyes. The improper disposal of dye wastewater can lead to the contamination of local water bodies. To address these pollution concerns, various regulations, sustainable practices, and technologies have been developed to minimize the environmental impact of the dyeing processes. This includes the use of eco-friendly dyes, more efficient dyeing methods, and wastewater treatment processes to reduce pollution. The common methods of removal or treatment of basic dyes in wastewater include physical methods, chemical methods [14], biological methods [15], and advanced treatment [16]. However, the choice of treatment depends on the specific context, the type and concentration of the dye pollutants, and the environmental regulations that are in place. Physical methods (filtration, adsorption, sedimentation, and coagulation–flocculation) are good options for treatment due to their cost-effectiveness and easy operation. Adsorption has been a widely used technique for removing basic dyes due to its high efficiency, versatility, selectivity, reusable adsorbents, low energy requirements, adsorbent variety, ease of operation, reduced chemical usage [17,18], and removal of non-biodegradable compounds [18], which prevent contaminant spread and environmental compatibility.

To understand and describe the rate adsorption process (physisorption and chemisorption or both sorption processes), kinetics is an essential concept that should be considered. The common kinetic adsorption models used for water and wastewater treatment include Pseudo-First-Order (PFO), Pseudo-Second-Order (PSO), Fractal Like-Pseudo First Order (FL-PFO), Fractal Like-Second Order (FL-PSO), General (rational) Order, Elovich, Diffusion-Chemisorption, and Avrami Fractionary-Order (AV) [19,20]. These kinetic adsorption models are valuable tools in the field of adsorption science and engineering. They help researchers and engineers understand and predict how rate adsorption processes will behave under different conditions, which is essential for designing efficient adsorption process systems for various applications, including wastewater treatment, air purification, and chemical separation. Interpreting the kinetic adsorption models is crucial for a comprehensive understanding of adsorption mechanisms and for the design and optimization of effective processes for removing basic dyes onto magnetic nanosorbents from wastewater sources. A certain publication critiques the Lagergren equation for its empirical nature, lack of predictive ability, and inconsistency [21]. When examining adsorption regulated by the Henry regime, the Pseudo-First-Order (PFO) model shows inconsistency, as it aligns with the Linear Driving Force (LDF) model solely under the condition of a constant adsorbate concentration in the bulk (C₀ ≈ C_e); i.e., in linear systems, the kinetic rate constants of both equations can only be correlated [22]. Linear regression analysis is a valuable tool that has been widely used to evaluate kinetic adsorption model parameters and assess the goodness of fit of kinetic adsorption models [23,24]. However, linearization is a mathematical technique used to simplify parameter estimation and model fitting but should not alter the inherent characteristics of the experimental data and predicted data or the distribution of errors of kinetic adsorption. In other words, regrettably, the transformation of kinetic data to fit into the linear forms of the model frequently introduces uncertainty and bias, often done carelessly [25]. Therefore, nonlinear kinetic adsorption is essential for considering alternative modeling approaches such as nonlinear regression [26], which can provide accurate parameter estimates and better capture the nonlinear behavior of the system without altering the distribution of errors in the data [27,28]. Generally, error functions have been widely applied for the goodness of fit of kinetic adsorption models such as the Residual Sum of Squares Error (ERRSQ/SSE), the Chi-square (χ2), the coefficient of determination (R-sq), the adjusted nonlinear coefficient of determination (Adj R-sq), the Average Relative Error (ARE), the Hybrid Fractional Error Function (HYBRID), Marquardt’s Percent Standard Deviation (MPSD), and Root Mean Square Error (RMSE) Sum of Absolute Errors (EABS) and Normalized Standard Deviation (NSD) [29,30]. Additionally, Akaike’s information criteria are widely recognized as reliable benchmarks for model selection, renowned for their predictive accuracy and straightforward decision-making basis [31]. These error functions can be applied through advanced software tools such as Microsoft Excel (Windows 10 and Microsoft 365 2020), MATLAB (version R2019), and OriginPro (version 2019) and even through Minitab (version 16) and SigmaPro (version 8). However, the MATLAB, OriginPro, Minitab, and SigmaPro programs are not available with built-in functions of statistics. On the other hand, Microsoft Excel is a readily available software tool that offers built-in statistical functions and kinetic adsorption models that users prefer.

This research highlights the significance of magnetic nanosorbents in a particular research context of the sorption properties of basic dyes that can be manipulated or controlled using a magnetic field. The purpose of this study would likely require rigorous experimental validation, thorough analysis, and potentially pilot-scale testing to demonstrate the effectiveness of the developed magnetic nanosorbent composite in practical water treatment scenarios. The treatment performance and efficiency evaluation were assessed through kinetic adsorption models and statistical algorithms using Microsoft Excel Solver’s built-in functions. How well the magnetic nanosorbent composite performed in removing basic dyes from wastewater was assessed by parameters such as adsorption capacity, removal efficiency, or other relevant metrics. After obtaining parameter values for the kinetic adsorption models through nonlinear fitting, the models were compared using the software MATLAB and OriginPro (free trial), which allow for more sophisticated data analysis and visualization compared to Microsoft Excel’s spreadsheet-based program tools.

2. Materials and Methods

2.1. Experimental Dataset

Experimental data for kinetic adsorption studies are typically obtained through laboratory experiments (Figure 1) designed to measure the adsorption behavior of contaminants over time, as shown in

Table 1. Experimental data at time t for basic dyes obtained from a batch system that adsorbed onto the magnetic nanosorbent.

t, t at the Equilibrium Adsorption (min)	qt,exp at Time t (mg g−1)
0	0.00
5	11.20
10	15.31
15	18.16
20	20.60
30	25.32
40	27.88
50	30.55
60	31.84
90	32.50
120	32.63
150	32.76
180	32.76
210	32.76
240	32.76

[2]. The laboratory data of adsorption kinetics were collected from a previous study performed by using a magnetic nanosorbent (derived from waste macadamia nut shells and magnetite nanoparticles) as an adsorbent with basic dye (methylene blue) used as an adsorbate. This magnetic nanosorbent was successful in the removal of basic dyes in wastewater and was simple to handle and separate during the treatment process, saving the cost of operation. The data were performed via batch sorption, which was achieved subsequent to the equations as presented below.

$${\mathrm{q}}_{\mathrm{t}}=\frac{{\mathrm{C}}_0{-\mathrm{C}}_{\mathrm{e}}}{\mathrm{m}}\mathrm{V}$$

(1)

The quantity ‘q_t’ (mg g⁻¹) represents the amount of basic dye adsorbed at a given time ‘t’. ‘C₀’ stands for the initial concentration, while ‘C_e’ denotes the equilibrium concentration of methylene blue in milligrams per liter (mg L⁻¹). ‘V’ represents the volume of the raw water used in a prior study (the synthetic solution was derived from methylene blue), measured in liters (L). Finally, ‘m’ stands for the mass of the prepared magnetic nanosorbent, measured in grams (g).

2.2. Kinetic Adsorption Models

It is important to note that the choice of kinetic adsorption model depends on the specific adsorbent contaminant system being studied. Different models may be more appropriate for different scenarios, and it is essential to validate the chosen model against experimental data to ensure its accuracy and reliability. Additionally, it is worth considering that adsorption kinetics can be influenced by various factors such as temperature, pH, concentration, and the properties of the adsorbent and contaminant. Kinetic adsorption models are mathematical models used to describe and predict the adsorption behavior of contaminants (such as basic dyes) in water and wastewater treatment processes over time. These models are based on the assumption that adsorption follows a certain rate equation or kinetic mechanism. The kinetic adsorption models that this research focused on can be listed as [32,33,34,35,36,37,38]:

$$\mathrm{Pseudo}-\mathrm{first}-\mathrm{order} \left(\mathrm{PFO}\right){=\mathrm{q}}_{\mathrm{t}}{=\mathrm{q}}_{\mathrm{e}}\left[{1-\exp (-\mathrm{k}}_1\times \mathrm{t})\right]$$

(2)

$$\mathrm{Pseudo}-\mathrm{second}-\mathrm{order} \left(\mathrm{PSO}\right){=\mathrm{q}}_{\mathrm{t}}=\frac{{\mathrm{k}}_2{\times \mathrm{q}}_{\mathrm{e}}{ }^2\times \mathrm{t}}{{1+(\mathrm{k}}_2{\times \mathrm{q}}_{\mathrm{e}}\times \mathrm{t})}$$

(3)

$$\mathrm{Fractal} \mathrm{like}-\mathrm{pseudo} \mathrm{first} \mathrm{order} \left(\mathrm{FL}-\mathrm{PFO}\right){=\mathrm{q}}_{\mathrm{t}}{=\mathrm{q}}_{\mathrm{e}}.\left(1-\exp \left({-\mathrm{k}}_1{.\mathrm{t}}^{\mathsf{\alpha }}\right)\right)$$

(4)

$$\mathrm{Fractal} \mathrm{like}-\mathrm{pseudo} \sec \mathrm{ond} \mathrm{order} \left(\mathrm{FL}-\mathrm{PSO}\right){=\mathrm{q}}_{\mathrm{t}}=\frac{{\mathrm{k}}_2{.\mathrm{q}}_{\mathrm{e}}{ }^2{.\mathrm{t}}^{\mathsf{\alpha }}}{{1+\mathrm{k}}_2{.\mathrm{q}}_{\mathrm{e}}{.\mathrm{t}}^{\mathsf{\alpha }}}$$

(5)

$$\mathrm{General} \left(\mathrm{rational}\right){ \mathrm{order}=\mathrm{q}}_{\mathrm{t}}{=\mathrm{q}}_{\mathrm{e}}-\frac{{\mathrm{q}}_{\mathrm{e}}}{{\left[{\mathrm{t}\times \mathrm{k}}_{\mathrm{r}}{\times \mathrm{q}}_{\mathrm{e}}^{\mathrm{n}-1}\times \left(\mathrm{n}-1\right)+1\right]}^{1/(\mathrm{n}-1)}}$$

(6)

$${\mathrm{Elovich}=\mathrm{q}}_{\mathrm{t}}=\frac{1}{\mathsf{\beta }}\times \ln (1+\mathsf{\alpha }\times \mathsf{\beta }\times \mathrm{t})$$

(7)

$${\mathrm{Diffusion}-\mathrm{Chemisorption}=\mathrm{q}}_{\mathrm{t}}=\frac{{\mathrm{q}}_{\mathrm{e}}{\times \mathrm{k}}_{\mathrm{DC}}{\times \mathrm{t}}^{1/2}}{{\mathrm{k}}_{\mathrm{DC}}{\times \mathrm{t}}^{1/2}{+\mathrm{q}}_{\mathrm{e}}}$$

(8)

$${\mathrm{Avrami} \mathrm{fractionary}-\mathrm{order}=\mathrm{q}}_{\mathrm{t}}=\mathrm{qe}\times \left[{1-\exp (-(\mathrm{k}}_{\mathrm{av}}{\times \mathrm{t}\left)\right)}^{{\mathrm{n}}_{\mathrm{av}}}\right]$$

(9)

2.3. Error Function Statistic Tools for Kinetic Adsorption Model

The error function statistic tool is a mathematical method commonly used in water and wastewater treatment to evaluate the performance of treatment processes. It is used to assess the accuracy of predictions or measurements by comparing the expected or desired values with the actual values obtained during the treatment process. The error function statistic provides a quantitative measure of the deviation between the predicted or target values and the observed values. It is important to note that the choice of statistical tools may vary depending on the specific circumstances and assumptions of the adsorption process. The selection of the best-fitting kinetic model depends on the specific system and experimental conditions. Different kinetic adsorption models may be more appropriate for different adsorption processes and statistical results. Statistical results can be used to validate the best-fitting kinetic adsorption model and evaluate its predictive performance of the adsorption process. The most commonly used error function statistics in water and wastewater treatment are shown in Equations (10)–(18) that were used and evaluated to define the kinetic adsorption characteristics of basic dyes adsorbed onto the magnetic nanosorbent. They provide a measure of the average magnitude of errors, disregarding their direction. The statistical formulas for calculating them are as follows [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48].

$$\mathrm{Sum} \mathrm{of} \mathrm{Squares} \mathrm{Error} (\mathrm{ERRSQ}/\mathrm{SSE})={\sum }_{\mathrm{i}=1}^{\mathrm{n}}{{(\mathrm{q}}_{\mathrm{e}, \exp }-{ \mathrm{q}}_{\mathrm{e}, \mathrm{cal}})}^2$$

(10)

$${\mathrm{Chi}-\mathrm{square} (\mathsf{\chi }}^2)={\sum }_{\mathrm{i}=1}^{\mathrm{n}}\frac{{({\mathrm{q}}_{\mathrm{e}, \mathrm{cal}}-{\mathrm{q}}_{\mathrm{e}, \exp })}^2}{{\mathrm{q}}_{\mathrm{e}, \exp }}$$

(11)

$${\mathrm{Coefficient} \mathrm{of} \mathrm{determination} (\mathrm{R}}^2)=1-\frac{\sum _{\mathrm{n}=1}^{\mathrm{n}}{({\mathrm{q}}_{\mathrm{e}, \exp }-{ \mathrm{q}}_{\mathrm{e}, \mathrm{cal}})}^2}{\sum _{\mathrm{n}=1}^{\mathrm{n}}{({\mathrm{q}}_{\mathrm{e}, \exp }-\overline{{ \mathrm{q}}_{\mathrm{e}, \exp }})}^2}$$

(12)

$$\mathrm{Average} \mathrm{Relative} \mathrm{Error} (\mathrm{ARE})=\frac{100}{\mathrm{n}}.{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left(\frac{{(\mathrm{q}}_{\mathrm{e}, \exp }-{\mathrm{q}}_{\mathrm{e}, \mathrm{cal}})}{{\mathrm{q}}_{\mathrm{e}, \exp }}\right)$$

(13)

$$\mathrm{Root} \mathrm{Mean} \mathrm{Square} \mathrm{Error} (\mathrm{RMSE})=\sqrt{\frac{{\sum }_{\mathrm{n}=1}^{\mathrm{n}}{({\mathrm{q}}_{\mathrm{e}, \exp }-{\mathrm{q}}_{\mathrm{e}, \mathrm{cal}})}^2}{\mathrm{n}-\mathrm{p}}}$$

(14)

$$\mathrm{Hybrid} \mathrm{Fractional} \mathrm{Error} \mathrm{Function} (\mathrm{HYBRID})=\frac{100}{\mathrm{n}-\mathrm{p}}.{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left(\frac{{(\mathrm{q}}_{\mathrm{e}, \exp }-{\mathrm{q}}_{\mathrm{e}, \mathrm{cal}})}{{\mathrm{q}}_{\mathrm{e}, \exp }}\right)$$

(15)

$$\mathrm{Marquardt}’\mathrm{s} \mathrm{Percent} \mathrm{Standard} \mathrm{Deviation} (\mathrm{MPSD})=100.\sqrt{\frac{1}{\mathrm{n}-\mathrm{p}}.{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left(\frac{{(\mathrm{q}}_{\mathrm{e}, \exp }-{\mathrm{q}}_{\mathrm{e}, \mathrm{cal}})}{{\mathrm{q}}_{\mathrm{e}, \exp }}\right)}^2}$$

(16)

$$\mathrm{Normalized} \mathrm{Standard} \mathrm{Deviation} (\mathrm{NSD})=100.\sqrt{\frac{1}{\mathrm{n}-1}.{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left(\frac{{(\mathrm{q}}_{\mathrm{e}, \exp }-{\mathrm{q}}_{\mathrm{e}, \mathrm{cal}})}{{\mathrm{q}}_{\mathrm{e}, \exp }}\right)}^2}$$

(17)

$$\mathrm{Sum} \mathrm{of} \mathrm{Absolute} \mathrm{Errors} (\mathrm{EABS})={\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left|{(\mathrm{q}}_{\mathrm{e}, \mathrm{cal}}-{\mathrm{q}}_{\mathrm{e}, \exp })\right|}_{\mathrm{i}}$$

(18)

2.4. Microsoft Excel Minimized Solver Error Functions and Kinetic Adsorption Models

Microsoft Excel was used as a tool for the determination of the nonlinear kinetic adsorption models for this study. While Excel is primarily known as spreadsheet software, it offers various mathematical and statistical functions that can be utilized for data analysis and curve fitting to evaluate the best-fitting of the nonlinear kinetic adsorption models. Microsoft365, 2021, was used to determine nonlinear kinetic adsorption models and error functions. However, it is worth mentioning that while Microsoft Excel can be used for nonlinear kinetic adsorption modeling, it may have limitations compared to specialized software designed specifically for advanced curve fitting and analysis. If the analysis requires more complex modeling techniques or advanced statistical methods, consideration should be given to using dedicated scientific software such as MATLAB and OriginPro for justifying the analysis results. To evaluate the statistical results from the Microsoft Excel Solver spreadsheet, the data fitting was estimated with MATLAB and OriginPro via the formulated algorithm as presented in Figure 2.

To solve the experimental data (Table 1) of kinetic adsorption obtained from laboratory experiments, eight kinetic adsorption models (Section 2.2) and nine statistical error functions (Section 2.3) were used for analysis via the Microsoft Excel spreadsheet function, as shown in

Table 2. Input of the experimental data and run of Microsoft Excel Solver’s spreadsheet-based program for the PFO kinetic adsorption model as an example model for data fitting.

Row/ Column	Time, t	qt,exp	qt,cal	Residual	Residual2	Upper, CI	Lower, CI	Abbreviation	Factor Result	Abbreviation	Statistical Result
	A	B	C	D	E	F	G	H	I	J	K
1	0	0.0000	0.0000	0.0000	0.0000	0.7006	−0.70	SSR	17.465	SSE	17.465
2	5	11.1995	7.9189	−3.2805	10.7619	8.6196	7.22	k1	0.056	Chi-sq	1.284
3	10	15.3123	13.9111	−1.4013	1.9636	14.6117	13.21	qe,PFO	32.546	ARE	1.928
4	15	18.1638	18.4452	0.2814	0.0792	19.1459	17.74	Mean of qt,exp	25.135	RMSE	1.159
5	20	20.6035	21.8762	1.2727	1.6198	22.5768	21.18	df	13.000	HYBRID	2.249
6	30	25.3165	26.4368	1.1204	1.2552	27.1375	25.74	SE of qt,exp	0.324	MPSD	19.087
7	40	27.8788	29.0481	1.1693	1.3674	29.7488	28.35	R-square	0.9877	NSD	111.692
8	50	30.5507	30.5433	−0.0073	0.0001	31.2440	29.84	Critical t	2.160	EABS	10.114
9	60	31.8384	31.3994	−0.4390	0.1927	32.1001	30.70	CI	0.701
10	90	32.4954	32.3310	−0.1644	0.0270	33.0316	31.63	Adjust R-square	0.9867
11	120	32.6268	32.5058	−0.1210	0.0146	33.2065	31.81
12	150	32.7582	32.5387	−0.2196	0.0482	33.2393	31.84
13	180	32.7582	32.5448	−0.2134	0.0455	33.2455	31.84
14	210	32.7582	32.5460	−0.2122	0.0450	33.2466	31.85
15	240	32.7582	32.5462	−0.2120	0.0450	33.2468	31.85

q_t,exp = $I$3*(1 − EXP(−$I$2*A1)) Array; SSR = SUM(E1:E15) Array; Mean of exp, q_t = AVERAGE(B1:B15) Array; Df = COUNT(B1:B15) − COUNT(I2:I3) Array; SE of exp, q_t = SQRT(SUM((B1:B15 − C1:C15)^2)/I5) Array; R-square = 1 − (SUM((B1:B15 − C1:C15)^2)/(SUM((B1:B15-I4)^2))) Array; Critical t = TINV(0.05,I5) Array; CI = I6*I8 Array; Adjust R-square = (1 − ((COUNT(B1:B15) − 1)/(COUNT(B1:B15) − 2))*(1 − I7)) Array; SSE = SUM((C1:C15 − B1:B15)^2) Array; Chi-sq = SUM(((C2:C15 − B2:B15)^2)/(B2:B15)) Array; ARE = (100/(COUNT(B2:B15)))*(SUM((B2:B15 − C2:C15)/(B2:B15))) Array; RMSE = SQRT(SUM((B1:B1 − C1:C15)^2)/(COUNT(C1:C15) − 1)) Array; HYBRID = (100/((COUNT(B2:B15)) − (COUNT(I2:I3))))*(SUM((B2:B15 − C2:C15)/(B2:B15))) Array; MPSD = 100*SQRT((1/COUNT(B2:B11 − I2:I3)*(SUM((B2:B15 − C2:C15)/(B2:B15))^2))) Array; NSD = 100*(SQRT(SUM(((B1:B15-C1:C15))^2)/(COUNT(B1:B15) − 1))) Array; EABS = SUM(ABS((C1:C15 − B1:B15))) Array.

and

Table 3. Experimental dataset for time t, q_t,exp, q_t,cal, Residual, Residual², SSR, k₁, and q_e.

Row/ Column	Time, t	qt,exp	qt,cal	PFO Model (Equation (2))	Residual	Residual2	Abbreviation	Factor Results	Abbreviation	Statistic Results
	A	B	C		D	E	H	I	J	K
1	0	0.0000	0.0000	$I$3*(1 − EXP(−$I$2*A1))	0.0000	0.0000	SSR	10,376.070	SSE	10,376.070
2	5	11.1995	0.2754	$I$3*(1 − EXP(−$I$2*A2))	−10.6556	113.5427	k1	0.300	Chi-sq	358.083
3	10	15.3123	0.4425	$I$3*(1 − EXP(−$I$2*A3))	−14.6472	214.5405	qe,PFO	0.700	ARE	97.213
4	15	18.1638	0.5438	$I$3*(1 − EXP(−$I$2*A4))	−17.4716	305.2554	Mean of qt,exp	25.135	RMSE	5.044
5	20	20.6035	0.6053	$I$3*(1 − EXP(−$I$2*A5))	−19.9052	396.2180	df	13.000	HYBRID	113.416
6	30	25.3165	0.6651	$I$3*(1 − EXP(−$I$2*A6))	−24.6165	605.9744	SE of qt,exp	7.538	MPSD	962.364
7	40	27.8788	0.6872	$I$3*(1 − EXP(−$I$2*A7))	−27.1788	738.6879	R-square	−6.3269	NSD	2722.403
8	50	30.5507	0.6953	$I$3*(1 − EXP(−$I$2*A8))	−29.8507	891.0619	Critical t	2.160	EABS	367.419
9	60	31.8384	0.6983	$I$3*(1 − EXP(−$I$2*A9))	−31.1384	969.6002	CI	16.285
10	90	32.4954	0.6999	$I$3*(1 − EXP(−$I$2*A10))	−31.7954	1010.9485	Adjust R-square	−6.8905
11	120	32.6268	0.7000	$I$3*(1 − EXP(−$I$2*A11))	−31.9268	1019.3218
12	150	32.7582	0.7000	$I$3*(1 − EXP(−$I$2*A12))	−32.0582	1027.7296
13	180	32.7582	0.7000	$I$3*(1 − EXP(−$I$2*A13))	−32.0582	1027.7296
14	210	32.7582	0.7000	$I$3*(1 − EXP(−$I$2*A14))	−32.0582	1027.7296
15	240	32.7582	0.7000	$I$3*(1 − EXP(−$I$2*A15))	−32.0582	1027.7296

. Table 2 demonstrates the input of experimental data and the run of Microsoft Excel Solver’s spreadsheet-based program using the PFO kinetic adsorption model as an example model for data fitting. Table 3 provides the Excel codes for the PFO kinetic adsorption model using a Microsoft Excel-based program. In this study, the PFO kinetic adsorption model was used as a case study in solving via Microsoft Excel Solver’s spreadsheet-based program. The related kinetic adsorption and statistical formula can be followed in the steps detailed below.

The applications of the Microsoft Excel Solver Function spreadsheet provide a flexible and versatile tool for solving a wide range of optimization problems encountered in engineering and data analysis, as follows.

2.4.1. Experimental Dataset

As shown in Table 2, the experimental data of t at the equilibrium adsorption (min) and q_t,exp at time t (mg g⁻¹) were put in column A, rows A1–A15, and column B, rows B1–B15, respectively. In this case, the study had 15 datasets for kinetic adsorption at time t. The calculation via the Microsoft Excel Solver Function spreadsheet depends on the dataset of each experiment. In terms of column C, rows C1–C15 are the q_t,cal at time t (mg g⁻¹) of basic dye adsorbed onto nanosorbent materials. Column C, rows C1–C15 were calculated using the Solver Function of Equation (2) (PFO kinetic adsorption model), as shown in Figure 3.

Notice that the PFO equation has k1 (row H2) and q_e (row H3) selected as critical factors for the nonlinear regression analysis of the kinetic adsorption models. Therefore, k1 and q_e had inserted the estimated values of 0.3 (row I2) and 0.7 (row I3) for running the algorithm, respectively. After that, the q_t,cal was computed using the Solver Function of PFO (see in Figure 3, column C). It can be seen that the values of the q_t,cal calculated data did not come close to the q_t,exp experimental data in the comparison of columns B and C. Column D (residual) was calculated as column B minus column C (row B1- row C1), while column E (Residual²) is the square of each row of column D. Residual² is one of the critical information components for evaluating the nonlinearity of the kinetic adsorption models, which was used to compute the Sum of Square Residual (SSR, row I1, SUM (E1:E15), 10,376.070). Henceforward, the solver parameters of the function can be accessed by clicking the tools bar’s “Data” button in the Solver toolbar. This opens a dialog box that can set various options for nonlinear kinetic adsorption-fitting regression analysis via the solving method of the GRG Nonlinear model, as shown in Figure 3.

2.4.2. Microsoft Excel Solver’s Parameter Function

To evaluate the nonlinear of kinetic adsorption models through the Microsoft Excel Solver Function, the data of Table 3 in column A (t), column B (q_t,exp) and column C (q_t,cal) were plotted as shown in Figure 3a. Next, the red line (time t: predicted data, q_t,cal) was selected and then clicked the tools bar “Data” with the chosen Solver Function, as presented in Figure 3b. For the Solver parameter, the objective was set as “$S$1, 10,376.070” with the changing variable cells inserted as “$S$2:$S$3, 0.3000:0.7000” and a Solving method selected as GRG Nonlinear. After clicking the Solve button, the Solver results appeared, and then “Keep Solver Solution” and “OK” were clicked, as shown in Figure 3d. The Solver had converged to the current solution, and all constraints were satisfied. It can be seen in Figure 3c that the predicted data followed and closed to the experimental data. All data results were changed as appropriate in Table 2.

2.4.3. Evaluating the Statistical Results

The numerical outcomes obtained from the statistical results were derived from data analysis using statistical methods (Equations (10)–(18)) via the Microsoft Excel Solver function spreadsheet-based program. This provided information about the patterns, relationships, or trends present in the data and helped in drawing meaningful conclusions or making informed decisions about the kinetic adsorption model for the controlling mechanism. To confirm the statistical results using the Microsoft Excel Solver function spreadsheet-based program, they were compared with MATLAB and OriginPro (free trial), which are both powerful software tools that are commonly used for data analysis, visualization, and scientific plotting [49]. While MATLAB and OriginPro share some similarities, they have distinct features and are often preferred in different contexts. However, this research only focused on the approach of how to apply nonlinear models through the Microsoft Excel Solver function spreadsheet-based program as a sample tool for minimizing the statical error functions and kinetic adsorption model. In terms of methodology for analysis, MATLAB and OriginPro are not detailed in this report, and we merely present the statistical results in comparison with the Microsoft Excel Solver Function.

3. Results and Discussion

3.1. Substantiation of Kinetic Adsorption Model Data via Microsoft Excel Solver Function

Experimental data were collected through actual observations and measurements of the removal of basic dye using the magnetic nanosorbent in a batch system. These data points were obtained by conducting controlled experiments or observations under specific conditions [2]. Experimental data were considered the most reliable form of data as they directly reflect what happens in the adsorption mechanism process. However, predicted data can still be valuable for making informed decisions and predictions in situations where conducting experiments may be impractical or time-consuming. The key difference between experimental and predicted data is the source of information. Experimental data are obtained through direct observations and measurements, while predicted data are generated using models and algorithms to estimate future or unobserved value. Predicted data with the consideration of the “kinetic adsorption model” refers to using mathematical models or simulations that take into account the kinetics (rate) of the adsorption process to estimate or forecast the behavior of the system. These models can be based on empirical relationships, theoretical equations, or a combination of both. Therefore, the Microsoft Excel Solver Function was used as a based program to predict the adsorption mechanism of eight kinetic adsorption models (PFO, FSO, Elovich, Diffusion-Chemisorption, FL-PFO, FL-PSO, General (rational) Order, and Avrami Fractionary-Order). In addition, the error function (the goodness of fit) has commonly been used in the context of analyzing and quantifying errors or uncertainties in different kinetic adsorption models, and it can be computed numerically using various algorithms, which are shown and described in Section 2.2. In this study, the goodness of fit was solved to obtain the statistical results at the same time as analyzing the kinetic adsorption models. Table 2, Table 3,

Table 4. Experimental data and PSO nonlinear kinetic adsorption statical results from the Microsoft Excel Solver’s parameter function.

Row/ Column	Time, t	qt,exp	qt,cal	Residual	Residual2	Upper, CI	Lower, CI	Abbreviation	Factor Result	Abbreviation	Statistical Result
	A	B	C	D	E	F	G	H	I	J	K
1	0	0	0.0000	0.0000	0.0000	2.5791	−2.5791	SSR	18.528	SSE	18.528
2	5	11.1995	10.2650	−0.9344	0.8731	12.8441	7.6859	k2,PSO	0.002	Chi-sq	0.733
3	10	15.3123	15.9556	0.6432	0.4137	18.5347	13.3765	qe,pso	35.804	ARE	−0.154
4	15	18.1638	19.5723	1.4085	1.9838	22.1514	16.9932	Mean of qt,exp	25.135	RMSE	1.194
5	20	20.6035	22.0741	1.4706	2.1626	24.6532	19.4950	df	13.000	HYBRID	−0.180
6	30	25.3165	25.3092	−0.0073	0.0001	27.8883	22.7301	SE of qt,exp	1.194	MPSD	1.527
7	40	27.8788	27.3105	−0.5683	0.3230	29.8896	24.7314	R-square	0.9869	NSD	115.040
8	50	30.5507	28.6707	−1.8799	3.5341	31.2498	26.0916	Critical t	2.160	EABS	13.680
9	60	31.8384	29.6554	−2.1830	4.7654	32.2345	27.0763	CI	2.579
10	90	32.4954	31.4560	−1.0394	1.0803	34.0351	28.8769	Adjust R-square	0.9859
11	120	32.6268	32.4409	−0.1859	0.0346	35.0200	29.8618
12	150	32.7582	33.0620	0.3038	0.0923	35.6411	30.4829
13	180	32.7582	33.4894	0.7312	0.5346	36.0685	30.9103
14	210	32.7582	33.8016	1.0433	1.0885	36.3807	31.2224
15	240	32.7582	34.0395	1.2813	1.6417	36.6186	31.4604

q_t,exp = ($I$2*($I$3^2)*A1)/(1 + $I$3*$I$2*A1) Array; SSR = SUM(E1:E15) Array; Mean of exp, q_t = AVERAGE(B1:B15) Array; df = COUNT(B1:B15) − COUNT(I2:I3) Array; SE of exp, q_t = SQRT(SUM((B1:B15 − C1:C15)^2)/I5) Array; R-square = 1 − (SUM((B1:B15 − C1:C15)^2)/(SUM((B1:B15-I4)^2))) Array; Critical t = TINV(0.05,I5) Array; CI = I6*I8 Array; Adjust R-square = (1 − ((COUNT(B1:B15) − 1)/(COUNT(B1:B15) − 2)) * (1 − I7)) Array; SSE = SUM((C1:C15 − B1:B15)^2) Array; Chi-sq = SUM(((C2:C15 − B2:B15)^2)/(B2:B15)) Array; ARE = (100/(COUNT(B2:B15)))*(SUM((B2:B15 − C2:C15)/(B2:B15))) Array; RMSE = SQRT(SUM((B1:B1 − C1:C15)^2)/(COUNT(C1:C15) − 1)) Array; HYBRID = (100/((COUNT(B2:B15)) − (COUNT(I2:I3))))*(SUM((B2:B15 − C2:C15)/(B2:B15))) Array; MPSD = 100*SQRT((1/COUNT(B2:B11 − I2:I3)*(SUM((B2:B15 − C2:C15)/(B2:B15))^2))) Array; NSD = 100*(SQRT(SUM(((B1:B15-C1:C15))^2)/(COUNT(B1:B15) − 1))) Array; EABS = SUM(ABS((C1:C15 − B1:B15))) Array.

,

Table 5. Experimental data and Elovich nonlinear kinetic adsorption statical results from the Microsoft Excel Solver’s parameter function.

Row/ Column	Time, t	qt,exp	qt,cal	Residual	Residual2	Upper, CI	Lower, CI	Abbreviation	Factor Result	Abbreviation	Statistical Result
	A	B	C	D	E	F	G	H	I	J	K
1	0	0.0000	0.0000	0.0000	0.0000	5.2795	−5.2795	SSR	77.637	SSE	77.637
2	5	11.1995	13.6427	2.4432	5.9693	18.9221	8.3632	αe	10.592	Chi-sq	3.055
3	10	15.3123	17.4592	2.1469	4.6091	22.7387	12.1798	βe	0.168	ARE	−1.732
4	15	18.1638	19.7657	1.6020	2.5663	25.0452	14.4863	Mean of qt,exp	25.135	RMSE	2.444
5	20	20.6035	21.4238	0.8203	0.6730	26.7033	16.1444	df	13.000	HYBRID	−2.021
6	30	25.3165	23.7824	−1.5341	2.3535	29.0618	18.5029	SE of qt,exp	2.444	MPSD	17.146
7	40	27.8788	25.4670	−2.4118	5.8169	30.7464	20.1875	R-square	0.9452	NSD	235.488
8	50	30.5507	26.7784	−3.7722	14.2297	32.0579	21.4990	Critical t	2.160	EABS	29.564
9	60	31.8384	27.8524	−3.9860	15.8880	33.1319	22.5730	CI	5.279
10	90	32.4954	30.2470	−2.2484	5.0555	35.5264	24.9675	Adjust R-square	0.9410
11	120	32.6268	31.9498	−0.6770	0.4584	37.2292	26.6703
12	150	32.7582	33.2722	0.5140	0.2642	38.5517	27.9927
13	180	32.7582	34.3535	1.5953	2.5449	39.6330	29.0740
14	210	32.7582	35.2682	2.5100	6.3002	40.5477	29.9888
15	240	32.7582	36.0609	3.3027	10.9078	41.3404	30.7815

q_t,exp = ($I$2*($I$3^2)*A1)/(1 + $I$3*$I$2*A1) Array; SSR= SUM(E1:E15) Array; Mean of exp, q_t = AVERAGE(B1:B15) Array; df = COUNT(B1:B15) − COUNT(I2:I3) Array; SE of exp, q_t = SQRT(SUM((B1:B15 − C1:C15)^2)/I5) Array; R-square = 1 − (SUM((B1:B15 − C1:C15)^2)/(SUM((B1:B15-I5)^2))) Array; Critical t = TINV(0.05,I5) Array; CI = I6*I8 Array; Adjust R-square = (1 − ((COUNT(B1:B15) − 1)/(COUNT(B1:B15) − 2)) * (1 − I7)) Array; SSE = SUM((C1:C15 − B1:B15)^2) Array; Chi-sq = SUM(((C2:C15 − B2:B15)^2)/(B2:B15)) Array; ARE = (100/(COUNT(B2:B15)))*(SUM((B2:B15 − C2:C15)/(B2:B15))) Array; RMSE = SQRT(SUM((B1:B1 − C1:C15)^2)/(COUNT(C1:C15) − 1)) Array; HYBRID = (100/((COUNT(B2:B15)) − (COUNT(I2:I3))))*(SUM((B2:B15 − C2:C15)/(B2:B15))) Array; MPSD = 100*SQRT((1/COUNT(B2:B11 − I2:I3)*(SUM((B2:B15 − C2:C15)/(B2:B15))^2))) Array; NSD = 100*(SQRT(SUM(((B1:B15-C1:C15))^2)/(COUNT(B1:B15) − 1))) Array; EABS = SUM(ABS((C1:C15 − B1:B15))) Array.

,

Table 6. Experimental data and Diffusion-Chemisorption nonlinear kinetic adsorption statical results from the Microsoft Excel Solver’s parameter function.

Row/ Column	Time, t	qt,exp	qt,cal	Residual	Residual2	Upper, CI	Lower, CI	Abbreviation	Factor Result	Abbreviation	Statistical Result
	A	B	C	D	E	F	G	H	I	J	K
1	0	0.0000	0.0000	0.0000	0.0000	4.8965	−4.8965	SSR	66.781	SSE	66.781
2	5	11.1995	14.1201	−2.9206	8.5299	19.0165	9.2236	kDC	9.001	Chi-sq	2.936
3	10	15.3123	17.7717	−2.4593	6.0483	22.6681	12.8752	qe,DC	47.309	ARE	−2.421
4	15	18.1638	20.0712	−1.9075	3.6384	24.9677	15.1748	Mean of qt,exp	25.135	RMSE	2.266
5	20	20.6035	21.7488	−1.1453	1.3118	26.6453	16.8524	df	13.000	HYBRID	−2.825
6	30	25.3165	24.1424	1.1741	1.3785	29.0388	19.2459	SE of qt,exp	2.266	MPSD	23.969
7	40	27.8788	25.8374	2.0414	4.1672	30.7339	20.9410	R-square	0.9528	NSD	218.404
8	50	30.5507	27.1377	3.4129	11.6482	32.0342	22.2413	Critical t	2.160	EABS	27.528
9	60	31.8384	28.1848	3.6536	13.3491	33.0812	23.2883	CI	4.896
10	90	32.4954	30.4430	2.0524	4.2125	35.3394	25.5465	Adjust R-square	0.9450
11	120	32.6268	31.9699	0.6569	0.4315	36.8664	27.0735
12	150	32.7582	33.1030	−0.3448	0.1189	37.9995	28.2066
13	180	32.7582	33.9924	−1.2341	1.5231	38.8888	29.0959
14	210	32.7582	34.7173	−1.9590	3.8378	39.6137	29.8208
15	240	32.7582	35.3245	−2.5663	6.5856	40.2209	30.4280

q_t,exp = ($I$3*$I$2*(A1^(1/2)))/($I$2*(A1^(1/2))+$I$3) Array; SSR= SUM(E1:E15) Array; Mean of exp, q_t = AVERAGE(B1:B15) Array; df = COUNT(B1:B15) − COUNT(I2:I3) Array; SE of exp, q_t = SQRT(SUM((B1:B15 − C1:C15)^2)/I5) Array; R-square = 1 − (SUM((B1:B15 − C1:C15)^2)/(SUM((B1:B15-I4)^2))) Array; Critical t = TINV(0.05,I5) Array; CI = I6*I8 Array; Adjust R-square = (1 − ((COUNT(B1:B15) − 1)/(COUNT(B1:B15) − 2)) * (1 − I7)) Array; SSE = SUM((C1:C15 − B1:B15)^2) Array; Chi-sq = SUM(((C2:C15 − B2:B15)^2)/(B2:B15)) Array; ARE = (100/(COUNT(B2:B15)))*(SUM((B2:B15 − C2:C15)/(B2:B15))) Array; RMSE = SQRT(SUM((B1:B1 − C1:C15)^2)/(COUNT(C1:C15) − 1)) Array; HYBRID = (100/((COUNT(B2:B15)) − (COUNT(I2:I3))))*(SUM((B2:B15 − C2:C15)/(B2:B15))) Array; MPSD = 100*SQRT((1/COUNT(B2:B11 − I2:I3)*(SUM((B2:B15 − C2:C15)/(B2:B15))^2))) Array; NSD = 100*(SQRT(SUM(((B1:B15-C1:C15))^2)/(COUNT(B1:B15)−1))) Array EABS = SUM(ABS((C1:C15 − B1:B15))) Array.

,

Table 7. Experimental data and FL-PFO nonlinear kinetic adsorption statical results from the Microsoft Excel Solver’s parameter function.

Row/ Column	Time, t	qt,exp	qt,cal	Residual	Residual2	Upper, CI	Lower, CI	Abbreviation	Factor Result	Abbreviation	Statistical Result
	A	B	C	D	E	F	G	H	I	J	K
1	0	0.0000	0.0000	0.0000	0.0000	1.6507	−1.6507	SSR	6.887	SSE	6.887
2	5	11.1995	9.8172	−1.3822	1.9105	11.4679	8.1666	αFL-PFO	0.804	Chi-sq	0.368
3	10	15.3123	15.1939	−0.1185	0.0140	16.8446	13.5432	k1,FL-PFO	0.096	ARE	0.404
4	15	18.1638	18.9705	0.8067	0.6508	20.6211	17.3198	qe,FL-PFO	33.089	RMSE	0.758
5	20	20.6035	21.7786	1.1751	1.3810	23.4293	20.1280	Mean of qt,exp	25.135	HYBRID	0.514
6	30	25.3165	25.6126	0.2961	0.0877	27.2632	23.9619	df	12.000	MPSD	3.266
7	40	27.8788	28.0133	0.1345	0.0181	29.6639	26.3626	SE of qt,exp	0.758	NSD	70.139
8	50	30.5507	29.5780	−0.9726	0.9460	31.2287	27.9274	R-square	0.9951	EABS	7.592
9	60	31.8384	30.6255	−1.2129	1.4710	32.2762	28.9749	Critical t	2.179
10	90	32.4954	32.1841	−0.3113	0.0969	33.8347	30.5334	CI	1.651
11	120	32.6268	32.7343	0.1075	0.0115	34.3849	31.0836	Adjust R-square	0.9943
12	150	32.7582	32.9433	0.1851	0.0343	34.5940	31.2927
13	180	32.7582	33.0269	0.2687	0.0722	34.6776	31.3763
14	210	32.7582	33.0617	0.3035	0.0921	34.7123	31.4110
15	240	32.7582	33.0765	0.3183	0.1013	34.7272	31.4259

q_t,exp = $I$4*(1 − (EXP(−$I$3*(A1^$I$2)))) Array; SSR= SUM(E1:E15) Array; Mean of exp, q_t = AVERAGE(B1:B15) Array; df = COUNT(B1:B15) − COUNT(I2:I4) Array; SE of exp, q_t = SQRT(SUM((B1:B15 − C1:C15)^2)/I6) Array; R-square = 1 − (SUM((B1:B15 − C1:C15)^2)/(SUM((B1:B15-I5)^2))) Array; Critical t = TINV(0.05,I6) Array; CI = I7*I9 Array; Adjust R-square = (1 − ((COUNT(B1:B15) − 1)/(COUNT(B1:B15) − 3)) * (1 − I8)) Array; SSE = SUM((C1:C15 − B1:B15)^2) Array; Chi-sq = SUM(((C2:C15 − B2:B15)^2)/(B2:B15)) Array; ARE = (100/(COUNT(B2:B15)))*(SUM((B2:B15 − C2:C15)/(B2:B15))) Array; RMSE = SQRT(SUM((B1:B1 − C1:C15)^2)/(COUNT(C1:C15) − 3)) Array; HYBRID = (100/((COUNT(B2:B15)) − (COUNT(I2:I4))))*(SUM((B2:B15 − C2:C15)/(B2:B15))) Array; MPSD = 100*SQRT((1/COUNT(B2:B11 − I2:I4)*(SUM((B2:B15 − C2:C15)/(B2:B15))^2))) Array; NSD = 100*(SQRT(SUM(((B1:B15-C1:C15))^2)/(COUNT(B1:B15) − 1))) Array; EABS = SUM(ABS((C1:C15 − B1:B15))) Array.

,

Table 8. Experimental data and FL-PSO nonlinear kinetic adsorption statical results from the Microsoft Excel Solver’s parameter function.

Row/ Column	Time, t	qt,exp	qt,cal	Residual	Residual2	Upper, CI	Lower, CI	Abbreviation	Factor Result	Abbreviation	Statistical Result
	A	B	C	D	E	F	G	H	I	J	K
1	0	0.0000	0.0000	0.0000	0.0000	2.5671	−2.5671	SSR	16.658	SSE	16.658
2	5	11.1995	9.3616	−1.8379	3.3777	11.9287	6.7945	αFL-PSO	1.130	Chi-sq	0.810
3	10	15.3123	15.5317	0.2193	0.0481	18.0987	12.9646	k2,FL-PSO	0.002	ARE	0.409
4	15	18.1638	19.5089	1.3451	1.8092	22.0759	16.9418	qe,FL-PSO	34.887	RMSE	1.178
5	20	20.6035	22.2272	1.6237	2.6364	24.7942	19.6601	Mean of qt,exp	25.135	HYBRID	0.520
6	30	25.3165	25.6471	0.3306	0.1093	28.2141	23.0800	df	12.000	MPSD	3.305
7	40	27.8788	27.6811	−0.1977	0.0391	30.2482	25.1141	SE of qt,exp	1.178	NSD	109.079
8	50	30.5507	29.0165	−1.5342	2.3538	31.5835	26.4494	R-square	0.9882	EABS	12.553
9	60	31.8384	29.9546	−1.8838	3.5486	32.5217	27.3876	Critical t	2.179
10	90	32.4954	31.5961	−0.8993	0.8087	34.1632	29.0291	CI	2.567
11	120	32.6268	32.4453	−0.1816	0.0330	35.0123	29.8782	Adjust R-square	0.9863
12	150	32.7582	32.9592	0.2010	0.0404	35.5263	30.3922
13	180	32.7582	33.3017	0.5435	0.2954	35.8688	30.7347
14	210	32.7582	33.5453	0.7871	0.6196	36.1124	30.9783
15	240	32.7582	33.7269	0.9687	0.9384	36.2940	31.1599

q_t,exp = ($I$3*($I$4^2)*(A1^$I$2))/(1 + ($I$3*$I$4*(A1^$I$2))) Array; SSR= SUM(E1:E15) Array; Mean of exp, q_t = AVERAGE(B1:B15) Array; df = COUNT(B1:B15) − COUNT(I2:I4) Array; SE of exp, q_t = SQRT(SUM((B1:B15 − C1:C15)^2)/I6) Array; R-square = 1 − (SUM((B1:B15 − C1:C15)^2) /(SUM((B1:B15-I5)^2))) Array; Critical t = TINV(0.05,I6) Array; CI = I7*I9 Array; Adjust R-square = (1 − ((COUNT(B1:B15) − 1)/(COUNT(B1:B15) − 3)) * (1 − I8)) Array; SSE = SUM((C1:C15 − B1:B15)^2) Array; Chi-sq = SUM(((C2:C15 − B2:B15)^2)/(B2:B15)) Array; ARE = (100/(COUNT(B2:B15)))*(SUM((B2:B15 − C2:C15)/(B2:B15))) Array; RMSE = SQRT(SUM((B1:B1 − C1:C15)^2)/(COUNT(C1:C15) − 3)) Array; HYBRID = (100/((COUNT(B2:B15)) − (COUNT(I2:I4))))*(SUM((B2:B15 − C2:C15)/(B2:B15))) Array; MPSD = 100*SQRT((1/COUNT(B2:B11 − I2:I4)*(SUM((B2:B15 − C2:C15)/(B2:B15))^2))) Array; NSD = 100*(SQRT(SUM(((B1:B15 − C1:C15))^2)/(COUNT(B1:B15)−1))) Array; EABS = SUM(ABS((C1:C15 − B1:B15))) Array.

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Table 9. Experimental data and Avrami Fractionary-Order nonlinear kinetic adsorption statical results from Microsoft Excel Solver’s parameter function.

Row/ Column	Time, t	qt,exp	qt,cal	Residual	Residual2	Upper, CI	Lower, CI	Abbreviation	Factor Result	Abbreviation	Statistical Result
	A	B	C	D	E	F	G	H	I	J	K
1	0	0.0000	0.0000	0.0000	0.0000	2.6286	−2.6286	SSR	17.466	SSE	17.466
2	5	11.1995	7.9144	3.2851	10.7917	10.5430	5.2858	kav	0.378	Chi-sq	1.286
3	10	15.3123	13.9044	1.4079	1.9823	16.5330	11.2758	qe,av	32.550	ARE	1.937
4	15	18.1638	18.4380	−0.2742	0.0752	21.0666	15.8094	nav	0.147	RMSE	1.206
5	20	20.6035	21.8692	−1.2657	1.6021	24.4978	19.2407	Mean of qt,exp	25.135	HYBRID	2.466
6	30	25.3165	26.4317	−1.1152	1.2438	29.0603	23.8031	df	12.000	MPSD	15.659
7	40	27.8788	29.0452	−1.1664	1.3605	31.6738	26.4166	SE of qt,exp	1.206	NSD	111.693
8	50	30.5507	30.5423	0.0084	0.0001	33.1709	27.9137	R-square	0.9877	EABS	10.084
9	60	31.8384	31.3999	0.4386	0.1923	34.0284	28.7713	Critical t	2.179
10	90	32.4954	32.3337	0.1617	0.0262	34.9622	29.7051	CI	2.629
11	120	32.6268	32.5092	0.1176	0.0138	35.1378	29.8806	Adjust R-square	0.9856
12	150	32.7582	32.5422	0.2160	0.0467	35.1708	29.9136
13	180	32.7582	32.5484	0.2098	0.0440	35.1770	29.9198
14	210	32.7582	32.5496	0.2087	0.0435	35.1781	29.9210
15	240	32.7582	32.5498	0.2084	0.0434	35.1783	29.9212

q_t,exp = $I$3*(1 − (EXP(-($I$2*A1))^$I$4)) Array; SSR= SUM(E1:E15) Array; Mean of exp, q_t = AVERAGE(B1:B15) Array; df = COUNT(B1:B15) − COUNT(I2:I4) Array; SE of exp, q_t = SQRT(SUM((B1:B15 − C1:C15)^2)/I6) Array; R-square = 1 − (SUM((B1:B15 − C1:C15)^2)/(SUM((B1:B15-I5)^2))) Array; Critical t = TINV(0.05,I6) Array; CI = I7*I9 Array; Adjust R-square = (1 − ((COUNT(B1:B15) − 1)/(COUNT(B1:B15) − 3)) * (1 − I8)) Array; SSE = SUM((C1:C15 − B1:B15)^2) Array; Chi-sq = SUM(((C2:C15 − B2:B15)^2)/(B2:B15)) Array; ARE = (100/(COUNT(B2:B15)))*(SUM((B2:B15 − C2:C15)/(B2:B15))) Array; RMSE = SQRT(SUM((B1:B1 − C1:C15)^2)/(COUNT(C1:C15) − 3)) Array; HYBRID = (100/((COUNT(B2:B15)) − (COUNT(I2:I4))))*(SUM((B2:B15 − C2:C15)/(B2:B15))) Array; MPSD = 100*SQRT((1/COUNT(B2:B11 − I2:I4)*(SUM((B2:B15 − C2:C15)/(B2:B15))^2))) Array; NSD = 100*(SQRT(SUM(((B1:B15-C1:C15))^2)/(COUNT(B1:B15)−1))) Array; EABS = SUM(ABS((C1:C15-B1:B15))) Array.

and

Table 10. Experimental data and General (rational) Order nonlinear kinetic-adsorption statical results from Microsoft Excel Solver’s parameter function.

Row/ Column	Time, t	qt,exp	qt,cal	Residual	Residual2	Upper, CI	Lower, CI	Abbreviation	Factor Result	Abbreviation	Statistical Result
	A	B	C	D	E	F	G	H	I	J	K
1	0	0.00000	0.00000	0.00000	0.0000	2.1692	−2.1692	SSR	11.894	SSE	11.894
2	5	11.19945	8.93986	2.25959	5.1058	11.1091	6.7706	kGen	0.019	Chi-sq	0.736
3	10	15.31235	14.92560	0.38675	0.1496	17.0948	12.7564	qe,Gen	33.312	ARE	0.888
4	15	18.16378	19.08772	−0.92394	0.8537	21.2569	16.9185	nGen	1.348	RMSE	0.996
5	20	20.60349	22.07370	−1.47021	2.1615	24.2429	19.9045	Mean of qt,exp	25.135	HYBRID	1.130
6	30	25.31646	25.92949	−0.61303	0.3758	28.0987	23.7603	df	12.000	MPSD	7.175
7	40	27.87881	28.19517	−0.31637	0.1001	30.3644	26.0260	SE of qt,exp	0.995	NSD	92.174
8	50	30.55066	29.61534	0.93532	0.8748	31.7846	27.4461	R-square	0.9916	EABS	9.855
9	60	31.83840	30.55165	1.28676	1.6557	32.7209	28.3824	Critical t	2.179
10	90	32.49542	31.98413	0.51129	0.2614	34.1533	29.8149	CI	2.169
11	120	32.62682	32.57076	0.05606	0.0031	34.7400	30.4015	Adjust R-square	0.9902
12	150	32.75822	32.85548	−0.09726	0.0095	35.0247	30.6863
13	180	32.75822	33.01052	−0.25230	0.0637	35.1797	30.8413
14	210	32.75822	33.10226	−0.34403	0.1184	35.2715	30.9330
15	240	32.75822	33.16007	−0.40184	0.1615	35.3293	30.9909

q_t,exp = ($I$3 − ($I$3/((A1*$I$2*($I$3^($I$4 − 1))*($I$4 − 1)+1)^(1/($I$4 − 1))))) Array; SSR= SUM(E1:E15) Array; Mean of exp, q_t = AVERAGE(B1:B15) Array; df = COUNT(B1:B15) − COUNT(I2:I4) Array; SE of exp, q_t = SQRT(SUM((B1:B15 − C1:C15)^2)/I6) Array; R-square = 1 − (SUM((B1:B15 − C1:C15)^2)/(SUM((B1:B15-I5)^2))) Array; Critical t = TINV(0.05,I6) Array; CI = I7*I9 Array; Adjust R-square = (1 − ((COUNT(B1:B15) − 1)/(COUNT(B1:B15) − 3)) * (1 − I8)) Array; SSE = SUM((C1:C15 − B1:B15)^2) Array; Chi-sq = SUM(((C2:C15 − B2:B15)^2)/(B2:B15)) Array; ARE = (100/(COUNT(B2:B15)))*(SUM((B2:B15 − C2:C15)/(B2:B15))) Array; RMSE = SQRT(SUM((B1:B1 − C1:C15)^2)/(COUNT(C1:C15) − 3)) Array; HYBRID = (100/((COUNT(B2:B15)) − (COUNT(I2:I4))))*(SUM((B2:B15 − C2:C15)/(B2:B15))) Array; MPSD = 100*SQRT((1/COUNT(B2:B11 − I2:I4)*(SUM((B2:B15 − C2:C15)/(B2:B15))^2))) Array; NSD = 100*(SQRT(SUM(((B1:B15-C1:C15))^2)/(COUNT(B1:B15) − 1))) Array; EABS = SUM(ABS((C1:C15 − B1:B15))) Array.

present the values of the experimental data and nonlinear kinetic adsorption statical results achieved from the Microsoft Excel Solver’s parameter function along with the Excel code of regression statistic.

To summarize, the nonlinear regression of kinetic adsorption models involves fitting mathematical equations to experimental data to describe the adsorption process. The goodness-of-fit measure is then used to assess how well the model fits the data and how effectively it explains the adsorption behavior. The best-selected model often chooses the kinetic adsorption model with the best goodness-of-fit metrics to gain insights into the specific adsorption system under study via Microsoft Excel Solver’s parameter-built function that is available in Microsoft Office. Using MATLAB and OriginPro for evaluating parameter values in the context of the nonlinear regression of kinetic adsorption models in comparison with Microsoft Excel Solver’s parameter-built function showed elevated levels; that is a common practice in scientific research and data analysis. The comparison of the parameter results and statistical results is explained in Section 3.2.

3.1.1. Two-Parameter Kinetic Adsorption Models

By following this framework and using the built functionalities offered by Microsoft Office, the prediction can effectively analyze experimental adsorption data, estimate model parameters, and evaluate the goodness of fit for various kinetic adsorption models. Table 2, Table 3, Table 4, Table 5 and Table 6 are the predictions of the two-parameter kinetic adsorption models: the PFO, PSO, Elovich, and Diffusion-Chemisorption models. These kinetic adsorption models indicate the parameter results such as k₁ (0.056 min⁻¹), q_e,cal,PFO (32.546 mg g⁻¹), k₂ (0.002 g mg⁻¹ min⁻¹), q_e,cal,PSO (35.804 mg g⁻¹), α_e (10.592 mg g⁻¹ min⁻¹), β_e (0.168 g mg⁻¹), k_DC (9.001 mg g⁻¹ min⁻ⁿ), and q_e,cal,DC (47.309 mg g⁻¹). k₁ is the Pseudo-First-Order rate constant (min⁻¹). q_{e,cal, PFO} is the predicted adsorbate adsorbed onto the adsorbent at equilibrium (mg g⁻¹) for the PFO kinetic adsorption model. k₂ is the Pseudo-Second-Order rate constant (g mg⁻¹ min⁻¹). q_{e,cal, PSO} is the predicted adsorbate adsorbed onto the adsorbent at equilibrium (mg g⁻¹) for the PSO kinetic adsorption model. α_e is initial adsorption rate of the Elovich model (mg g⁻¹ min⁻¹). β_e is the Elovich desorption constant (g mg⁻¹). K_DC is the Diffusion-Chemisorption constant (mg g⁻¹ min⁻ⁿ). q_e,cal,DC is the predicted adsorbate adsorbed onto the adsorbent at equilibrium (mg g⁻¹) for the Diffusion-Chemisorption kinetic adsorption model. In addition, column C (q_t,cal) presents the values of q_t at time t, which were computed from the built function of the two-parameter kinetic adsorption modes. It can be seen in Equations (2), (3), (7), and (8) or in the formula array as presented in Table 2, Table 3, Table 4, Table 5 and Table 6: (PFO = $I$3*(1 − EXP(−$I$2*A1)), PSO = ($I$2*($I$3^2)*A1)/(1 + $I$3*$I$2*A1), Elovich = (1/($I$3))*(LN(1 + ($I$2*$I$3*A1))), and Diffusion-Chemisorption = ($I$2*($I$3^2)*A1)/(1 + $I$3*$I$2*A1), respectively. In Table 2, Table 3, Table 4, Table 5 and Table 6, it can be observed that the q_t,cal calculations were different, which affected the statistical and model parameter results. It is suggested that the statistical and model parameter results depend on the built-in function and the kinetic adsorption model. To determine the best-fitting nonlinear kinetic model for adsorption data, their goodness of fit was evaluated using a commonly used error function, and lower values of these error functions indicate better fits, as shown in Table 2, Table 3, Table 4, Table 5 and Table 6 in column H–K. The results of the two-parameter kinetic adsorption models found that the PFO model (Table 2) indicates SSR (17.465), k₁ (0.056), q_e,PFO (32.546), R-square (0.9877), Adjust R-square (0.9867), SSE (17.465), Chi-sq (1.284), ARE (1.982), RMSE (0.078), HYBRID (2.249), MPSD (19.807), NSD (111.692), and EABS (10.114). It can be seen that the coefficient of determination R-square and the adjusted R-square was close to 1 for a fitted kinetic model. This indicates that the model provides an excellent fit to the experimental data. In other words, the values close to 1 (approaching 1) mean that the model’s predictions are very close to the actual experimental data points, suggesting that the chosen kinetic model is an excellent representation of the adsorption process in comparison to all two parameter kinetic models. The PFO (Lagergren) kinetic adsorption model suggests that the adsorption process occurs uniformly or in a well-mixed manner on surfaces like the homogeneous site. It means that the concentration of the basic dye (molecules or particles being adsorbed) was relatively constant throughout the fluid in contact with the surface of the magnetic nanosorbent. However, the PFO model is not able to explain all adsorption mechanisms or basic dye removal due to the controlling mechanism being affected by the experimental condition [50]. In addition, the PFO model is not applicable to the entirety of the adsorption reaction, as the adsorption rate declines until it reaches maximum capacity, resulting in zero rate at equilibrium [21,51]. Therefore, various kinetic models and different water pollutants were developed, and three-parameter kinetic adsorption models were required in the interpretation of the adsorption mechanism, as the data shows in Table 7, Table 8, Table 9 and Table 10.

3.1.2. Three-Parameter Kinetic Adsorption Models

Table 7, Table 8, Table 9 and Table 10 presents three-parameter kinetic adsorption models for experimental data and nonlinear kinetic adsorption statical results from Microsoft Excel Solver’s parameter function. The three-parameter kinetic adsorption models are mathematical expressions used to describe the process of adsorption, which is the adherence of molecules or ions from a fluid (basic dyes) onto a surface. These models incorporate three parameters to represent the adsorption process more accurately than simpler two-parameter models. The model parameter kinetic adsorption models such as q_t, q_e, k₁, k₂, α, k_av, n_av, k_Gen, n_Gen (Column I), and statistical results (column K) were computed via Microsoft Excel Solver’s parameter function, as available in Microsoft Excel. The parameter kinetic adsorption models were evaluated and analyzed for goodness of fit. It can be seen that the nonlinear regression results of FL-PFO stipulate the best fitting due to a mathematical model fitting of a set of experimental or observed data. FL-PFO indicated small values of R-square (0.9951, close to 1), Adjust R-square (0.9943, close to 1), SSE (6.887), Chi-sq (0.368), ARE (0.404), RMSE (0.758), HYBRID (0.514), MPSD (3.266), NSD (70.139), and EABS (7.592). General (rational) Order nonlinear kinetic-adsorption statical results were also provided as the best fitting, as they were better than FL-PSO and Avrami Fractionary-Order, respectively. The FL-PFO indicated that diffusion through micropores is the rate-controlling mechanism [52,53]; it suggests that the rate at which a substance (such as an adsorbate, basic dyes) moves or diffuses through small micropores or mesopores within a solid material/magnetic nanosorbent is the limiting factor in the overall process [54]. This finding can have implications in various fields, including adsorption processes, chemical reactions, and transport phenomena [55]. This could lead to a slower overall adsorption process, especially if the micropores are highly tortuous or if the substance has a relatively low diffusion coefficient. This might involve using materials with larger pores, modifying the surface chemistry to reduce interactions, or adjusting the process conditions to promote more efficient transport. Therefore, it is always a valuable insight when the rate-controlling mechanism is identified, as it provides actionable information for improving process efficiency or designing materials. Figure 4 illustrates the potential mechanisms that could occur during the adsorption process of Methylene Blue (MB)/basic dye onto the magnetic nanosorbent material.

However, it is important to note that while these three-parameter models can provide a better fit to experimental data as compared to simpler models, adsorption processes can be quite complex and may involve multiple mechanisms. The selection of an appropriate model should be based on the experimental system and the underlying assumptions that best match the observed behaviour.

3.2. Comparison of Kinetic Adsorption Model Parameters and Error Functions with Advanced Program Tools

Nonlinear two- or three-parameter kinetic adsorption models are mathematical expressions used to describe adsorption processes where two or three parameters are involved and the relationship between these parameters is nonlinear. These models are used to fit experimental adsorption data and understand the dynamics of adsorption systems where the rate of adsorption or the amount of adsorbate adsorbed changes nonlinearly with time. Fitting these nonlinear models to experimental data requires specialized nonlinear regression techniques. Software tools and optimization algorithms can be used to estimate the parameters of the models that best describe the observed data. It is important to note that the selection of the appropriate model and the interpretation of the parameter values should be done carefully, taking into consideration the physical and chemical characteristics of the system being studied. The various software tools for advanced calculations and statistical analysis, such as Microsoft Excel, MATLAB, and OriginPro, have strengths in different areas of data analysis and modeling. All the tools offer a variety of mathematical and statistical functions for data manipulation, analysis, and modeling; this depends on the specific needs of analyses-performing operations such as curve fitting, regression analysis, hypothesis testing, data transformation, and more. Microsoft Excel is a widely used spreadsheet software that is versatile for data analysis, calculations, and simple statistical analysis. It can perform calculations, create charts and graphs, and use functions for basic statistics like mean, median, and standard deviation. However, Excel may have limitations when it comes to more complex mathematical operations or specialized statistical analyses. Therefore, the built-in function of Microsoft Excel can be useful for various calculations and statistical analyses that are evaluated in comparison with MATLAB and OriginPro such as basic mathematical functions, statistical functions regression and trend analysis, and array formulas.

Table 11. Comparison of using various software tools for the kinetic adsorption models.

Model Parameter	Parameter Values for the Kinetic Adsorption Models
	Microsoft Excel	MATLAB	OriginPro
PFO
qe,PFO	32.55	32.55	32.55
k1,PFO	0.056	0.056	0.056
R-sq.	0.9877	0.9877	0.9877
Adj R-sq.	0.9867	0.9867	0.9867
SSE	17.47	17.47	17.47
Chi-sq.	1.284	-	1.284
ARE	1.928	-	-
RMSE	1.159	1.159	-
HYBRID	2.249	-	-
MPSD	19.087	-	-
NSD	111.692	-	-
EABS	10.114	-	-
PSO
qe,PSO	35.80	35.80	35.80
k2,PSO	0.002	0.002	0.002
R-sq.	0.9869	0.9869	0.9869
Adj R-sq.	0.9859	0.9859	0.9859
SSE	18.53	18.53	18.53
Chi-sq.	0.733	-	0.733
ARE	−0.154	-	-
RMSE	1.194	1.194	-
HYBRID	−0.180	-	-
MPSD	1.527	-	-
NSD	115.040	-	-
EABS	13.680	-	-
Elovich
αe	10.59	10.59	10.59
βe	0.168	0.168	0.168
R-sq.	0.9452	0.9452	0.9452
Adj R-sq.	0.9410	0.9410	0.9410
SSE	77.64	77.64	77.64
Chi-sq.	3.055	-	3.055
ARE	−1.732	-	-
RMSE	2.444	2.444	-
HYBRID	−2.021	-	-
MPSD	17.146	-	-
NSD	235.488	-	-
EABS	29.564	-	-
Diffusion-Chemisorption
qe,DC	47.31	47.31	47.31
kDC	9.001	9.002	9.001
R-sq.	0.9528	0.9528	0.9528
Adj R-sq.	0.9450	0.9450	0.9450
SSE	66.78	66.78	66.78
Chi-sq.	2.936	-	2.936
ARE	−2.421	-	-
RMSE	2.267	2.267	-
HYBRID	−2.825	-	-
MPSD	23.969	-	-
NSD	218.404	-	-
EABS	27.528	-	-
FL-PFO
αFL-PFO	0.804	0.804	0.803
k1,FL-PFO	0.096	0.096	0.097
qe,FL-PFO	33.09	33.09	33.09
R-sq.	0.9951	0.9951	0.9951
Adj R-sq.	0.9943	0.9943	0.9943
SSE	6.89	6.89	6.89
Chi-sq.	0.368	-	0.368
ARE	0.404	-	-
RMSE	0.758	0.758	-
HYBRID	0.514	-	-
MPSD	3.266	-	-
NSD	70.139	-	-
EABS	7.592	-	-
FL-PSO
αFL-PSO	1.13	1.13	1.13
k2,FL-PSO	0.002	0.002	0.002
qe,FL-PSO	34.88	34.88	34.88
R-sq.	0.9883	0.9883	0.9882
Adj R-sq.	0.9863	0.9863	0.9863
SSE	16.66	16.66	16.66
Chi-sq.	0.810	-	0.810
ARE	0.409	-	-
RMSE	1.178	1.178	-
HYBRID	0.520	-	-
MPSD	3.305	-	-
NSD	109.079	-	-
EABS	12.553	-	-
Avrami Fractionary-Order
qe,av	32.55	32.55	32.55
kav	0.378	0.378	0.236
nav	0.147	0.147	0.236
R-sq.	0.9877	0.9877	0.9877
Adj R-sq.	0.9856	0.9856	0.9856
SSE	17.47	17.47	17.47
Chi-sq.	1.286	-	1.286
ARE	1.937	-	-
RMSE	1.206	1.206	-
HYBRID	2.466	-	-
MPSD	15.585	-	-
NSD	111.693	-	-
EABS	10.084	-	-
General (rational) Order
qe,Gen	33.34	33.31	33.34
kGen	0.019	0.019	0.019
nGen	1.348	1.348	1.348
R-sq.	0.9916	0.9916	0.9916
Adj R-sq.	0.9902	0.9902	0.9902
SSE	11.89	11.89	11.89
Chi-sq.	0.736	-	0.736
ARE	0.888	-	-
RMSE	0.9956	0.9956	-
HYBRID	1.430	-	-
MPSD	7.175	-	-
NSD	92.174	-	-
EABS	9.855	-	-

Note: The sign of “-” means not available in the software tool.

presents a comparison of using three software tools (Microsoft Excel, MATLAB and OriginPro) for the kinetic adsorption models. Virtually identical results in statistical analysis, particularly when fitting models, ensure that the fit is good. Achieving identical results across different statistical software or tools can be challenging due to variations in algorithms, numerical precision, and implementation details. In practice, what often matters more than getting identical results is getting results that are consistent and meet the objectives of the experimental analysis. If the fits are reasonably close and provide meaningful insights, the model may be considered acceptable even if the results are not exactly identical. Therefore, two- or three-parameter kinetic adsorption models were brought to be discussed and explained using the statistical results and parameters of each kinetic adsorption model, as shown in Table 11.

Microsoft Excel, MATLAB and OriginPro provide all the model parameters with the R-sq, Adj R-sq, and SSE statistical results. The results found that the R-sq and Adj R-sq were different for all kinetic models, while the SSE and all the model parameter results were very much the same. In terms of Chi-sq statistical results, it was found that MATLAB did not provide a good software tool. On the other hand, Microsoft Excel and MATLAB provided the RMSE, while OriginPro is not available in the software tool.

In a comparison of the model parameters, FL-PFO gave a q_e (33.09 mg g⁻¹), which was close to the experimental data (32.76 mg g⁻¹) and closer to the experimental data than in the other kinetic adsorption models. The R-sq (0.9951) and Adj R-sq (0.9943) were close to 1 with fewer error functions such as SSE (6.89), Chi-sq (0.368), ARE (0.404), RMSE (0.758), HYBRID (0.514), MPSD (3.266), NSD (70.139), and EABS (7.592), which indicated the goodness of fit. It can be concluded that Microsoft Excel fulfilled all statical results and model parameters; however, the built-in function of the error functions is the main point. These functions are often used in optimization algorithms to fine-tune model parameters to minimize errors. Microsoft Excel can be a valuable tool for statistical analysis and modeling, but it is important to ensure that it is used correctly and to have an understanding of the role of error functions in assessing and improving kinetic adsorption models. Familiarity with error functions is essential for effective model evaluation and parameter optimization.

3.3. Application of Akaike’s Information Criterion for the Selected Kinetic Adsorption Model

In this study, it has been widely observed in the literature that increasing the number of parameters in a model tends to lead to better correlations. However, we have employed Akaike’s information criterion (AIC) [56,57], as suggested by Akaike in 1998, to assess the effectiveness of adsorption kinetic models while disregarding the influence of the number of parameters. In this context, the model with the lowest AIC value is considered to be the most suitable. To evaluate AIC, we require knowledge of the error Sum of Squares (SSE, as shown Table 11), which is computed by summing the squares of the differences between the experimental and predicted values. This can be mathematically expressed as follows

$$\mathrm{AIC}=\mathrm{N}.\ln \left(\frac{\mathrm{SSE}}{\mathrm{N}}\right)+2\mathrm{K}$$

(19)

Here, ‘K’ denotes the number of parameters, and ‘N’ represents the number of data points. Upon examining Table 11, it is evident that the FL-PFO method yields lower AIC values. However, if the dataset contains fewer than 40 data points, it is recommended to employ AIC_corrected. This correction is defined as follows:

$$\mathrm{A}\mathrm{I}{\mathrm{C}}_{\mathrm{corrected}}=\mathrm{AIC}+\frac{2\mathrm{K}(\mathrm{K}+1)}{\mathrm{N}-\mathrm{K}\mathrm{-}1}$$

(20)

Using corrected Akaike’s information criterion (AIC_corrected), it is determined that the FL-PSO kinetic adsorption model is the most effective, with a value of −3.4879 (closer to zero as compared to other AIC_corrected values, Figure 5, and

Table 12. Akaike’s information criterion for eight kinetic adsorption models.

Kinetic Adsorption Models	N	K	SSE	AIC	AICcorrected
PFO	15	2	17.47	6.2865	7.2865
PSO	15	2	18.53	7.1701	8.1701
Elovich	15	2	77.64	28.6605	29.6605
Diffusion-Chemisorption	15	2	66.78	26.4003	27.4003
FL-PFO	15	3	6.89	−5.6697	−3.4879
FL-PSO	15	3	16.66	7.5744	9.7562
Avrami Fractionary-Order	15	3	17.47	8.2865	10.4683
General (rational) Order	15	3	11.89	2.5147	4.6965

), while the Elovich kinetic adsorption model displayed the highest value of 29.6605. This underscores the efficiency of the FL-PSO kinetic adsorption model in correlating the kinetic data. In this study, it examined the adsorption of basic dye compounds. It found that basic dyes primarily undergo both physical and chemisorptions [36]. The FL-PFO inherently combines these features, allowing it to accurately correlate both energetically homogeneous and heterogeneous solid surface systems, providing another reason for its exceptional performance. The model proves highly effective in characterizing kinetics in this specific context.

Figure 6 was plotted to analyze the data visually and to compare the shape, trends, and patterns in the experimental data with those in the predicted data. In addition to visual inspection, it can perform quantitative assessments such as calculating correlation coefficients, the Root Mean Square Error (RMSE), or other statistical measures to quantify the agreement between the two datasets, as indicated in Table 11. These graphs show the plot of the FL-PFO kinetic adsorption model that confirms adsorption mechanism. It can be concluded that Microsoft Excel provides an easy software tool that offers a wide range of applications in various fields, especially environmental engineering, to assess environmental impacts, evaluate scenarios, and make predictions based on different input conditions.

4. Conclusions

Conducting kinetic adsorption studies on nanosorbents is a valuable area of research that helps researchers understand how quickly substances are adsorbed onto nanomaterials along with the adsorption mechanism. Basic dye (methylene blue) was chosen as a model solute to predict the adsorption capacities of an adsorbent material, which is a common and valuable approach in adsorption studies. Kinetic adsorption studies provide insights into the material’s adsorption capabilities and can inform its potential use in practical applications. Applying nonlinear regression for standardizing experimental data is a useful approach when dealing with complex relationships that cannot be adequately captured by linear models through Microsoft Excel’s built-in function. It allows users to fit a nonlinear mathematical function to experimental data, estimate parameters, and potentially standardize the data within the modeling process to gain insights into the underlying processes. Among the kinetic adsorption models, FL-PFO (33.09 mg g⁻¹ of predicted data, close to the experimental data of about 32.76 mg g⁻¹) was demonstrated to be the best-fitting, which was closer to the experimental data as compared to other kinetic adsorption models and had fewer error function results. This suggests that the rate at which a substance diffuses through small micropores or mesopores within a nanosorbent is the limiting factor in the overall process. An AIC_corrected value of −3.487 for the FL-PFO kinetic model indicates a favorable model fit. This insightful analysis is highly recommended for evaluating any adsorbate–adsorbent system. It aids in the selection of an appropriate kinetic model and significantly improves the correlation of adsorption data, thereby enhancing the understanding of the adsorption process. In conclusion, these findings suggest that Microsoft Excel, when equipped with the Solver Function and suitable spreadsheet methods, can be a robust tool for predicting adsorption behavior using various kinetic adsorption models for basic dye removal. This information can be valuable for researchers and practitioners in fields such as environmental science, chemistry, and materials science where adsorption studies are common.