Development of a comprehensive analytical solution for modeling adsorption kinetics and equilibrium

ABSTRACT

Adsorption has been traditionally modeled using equilibrium models like the Langmuir and the Freundlich isotherm models, and kinetic models such as pseudo-first and pseudo-second order equations. However, most of these equations are not in an Indepedant, and single-equation form. A novel mathematical framework that combines adsorption kinetics and isotherm equations to an independant, single-equation form is presented herin. The adsorption isotherm equations were first solved using the mass balance equations to derive an independant equation, eliminating the intermediate variables. This derivation was done for the commonly used adsorption isotherms such as Langmuir, Freundlich, Langmuir-Freundlich and Linear isotherms. The simplified equations can be used for predicting solid phase adsorption (q_e) and aqueous phase concentration (C_e) in terms of equilibrium constants, and was called as Equilibrium Adsorption Solution Equation (EASE). This EASE equation was further integrated into adsorption kinetic equations to give a solution which can also predict kinetic adsorption. This framework was named as Combined Adsorption Kinetic and Equilibrium (CAKE) equation and was further validated with experimental data. The CAKE equation offers the following advantages: By using this combined equation, both the kinetics and equilibrium concentrations of the batch adsorption systems can be predicted. Further, the intermediate variables are eliminated, and the final equation involving only model constants such as maximum adsorption capacity (q_max), affinity constant (k_a), initial concentration (C_o), and kinetic rate constants (k_t) are required to to calculate pollutant concentration at any given time. The model was validated using an experimental dataset of 2,4-Dichlorophenol adsorption studies on activated carbon. The CAKE and EASE models fitted the experimental datasets very well with an R² of 0.85, and a normalized absolute percentage error (NAPE) of 7.5%., thus validating the developed equations.

Graphical abstract

KEYWORDS:

• Adsorption isotherms
• adsorption kinetics
• Langmuir
• Freundlich

1. Introduction

Adsorption is a convenientl approach which is commonly employed to treat various pollutants, such as organic and inorganic pollutants. The fundamental process in adsorption involves the use of high surface area adsorbents for the removal of contaminants. Some of the common adsorbents used for their good adsorptive properties are activated carbon^[1], montmorillonite,^[2] biochar,^[3] nanoparticles,^[4] and biomass^[5] etc. The literature on adsorption suggests the use of adsorption equilibrium Isotherms and Kinetic models for simulating the equilibrium of adsorption and the kinetics of the adsorption, respectively^[1,6–8]

1.1. Equilibrium models

Adsorption equilibrium models such as Langmuir isotherm represent mathematical equations that describe the behavior of adsorbate molecules on the adsorbent surface.^[9–11] These isotherm models contribute to simulating the interaction occurring between the solute and adsorbant when the adsorption system is at equilibrium. Some of the popularly used isotherms are viz Langmuir, Freundlich, Temkin, Redlich Peterson model, etc^[12–15] , and are listed below in Table 1. Each model provides insights into the correlation among the adsorbate concentration (C_o), surface coverage, and other appropriate parameters while basing its conclusions on specific adsorption process.^[16] The desired model to choose is dependant on the system, adsorption behavior and is at the discretion of the individual researcher. However, the choice of isotherm model is also based on experimental results and the individual adsorption system under consideration. In numerous scientific, industrial, and environmental applications, these isotherm models are necessary for the evaluation and optimization of adsorption processes. The following are some of the conventional adsorption equilibrium isotherm models, as listed in Table 1.

Table 1. Single and multiparameter adsorption isotherm models.

Sl No.	Isotherm models	Non-linear equations	Parameters & constants	Importance	Ref.
One Parameter model
1.	Henry Isotherm	${\mathrm{q}}_{\mathrm{e}}={\mathrm{K}}_{{\mathrm{H}}_{\mathrm{e}}}.{\mathrm{C}}_{\mathrm{e}}$	${\mathrm{K}}_{{\mathrm{H}}_{\mathrm{e}}}$	It is a linear form of adsorption equation and the simplest form. It is widely used to investigate adsorption at relatively low concentrations.	^[Citation17]
Two Parameter model
1.	Langmuir Isotherm	${\mathrm{q}}_{\mathrm{e}}={\mathrm{q}}_{\mathrm{m}}\frac{{\mathrm{k}}_{\mathrm{L}}{\mathrm{C}}_{\mathrm{e}}}{1+{\mathrm{k}}_{\mathrm{L}}{\mathrm{C}}_{\mathrm{e}}}$	kL, qm	It is used to investigate mono-layer adsorption capacity. The adsorption is a function of the surface sites available for adsorbate. The surface further has an equal distribution of energy to attract adsorbate.	^[Citation18]
2.	Freundlich Isotherm	$\begin{array}{c}{\mathrm{q}}_{\mathrm{e}}={\mathrm{k}}_{\mathrm{f}}.{({\mathrm{C}}_{\mathrm{e}})}^{1/\mathrm{n}}\end{array}$	kf, n	It is applicable to adsorption over heterogeneous absorbent surfaces. The exponential spread of the active site and the respective energies is also described.	^[Citation19]
3.	Temkin Isotherm	${\mathrm{q}}_{\mathrm{e}}=\frac{\mathrm{RT}}{{\mathrm{b}}_{\mathrm{T}}}.\ln \left({\mathrm{A}}_{\mathrm{T}}.{\mathrm{C}}_{\mathrm{e}}\right)$	AT, bT	It considersthe impact of secondary interactions between adsorbate/adsorbate by adhering to the simple assumption that adsorption heat (ΔHads) of adsorbed molecules in the layer declines linearly as a result of increased surface coverage.	^[Citation20]
4.	Sips Isotherm	${\mathrm{q}}_{\mathrm{e}}={\mathrm{q}}_{\mathrm{ms}}.\left(\frac{{\mathrm{K}}_{\mathrm{s}}{\mathrm{C}}_{\mathrm{e}}^{{\mathit{\beta }}_{\mathrm{s}}}}{1+{\mathrm{K}}_{\mathrm{s}}{\mathrm{C}}_{\mathrm{e}}^{{\mathit{\beta }}_{\mathrm{s}}}}\right)$	Ks, βs	It is a combination of Langmuir and Freundlich isotherms, which can identify the heterogeneity of the surface at high concentrations. Some models can be used in lower-concentration, and the Sips model is one of them.	^[Citation21]
Three Parameter model
1.	Langmuir-Freundlich	${\mathrm{q}}_{\mathrm{e}}=\mathit{\theta }=\frac{{\mathrm{q}}_{\mathrm{s}}}{{\mathrm{q}}_{\mathit{\alpha }}}=\frac{{\left({\mathrm{k}}_{\mathrm{a}}\left(\frac{{\mathrm{C}}_{\mathrm{S}}}{{\mathrm{C}}_0}\right)\right)}^{\mathrm{z}}}{1+{\left({\mathrm{k}}_{\mathrm{a}}\left(\frac{{\mathrm{C}}_{\mathrm{S}}}{{\mathrm{C}}_0}\right)\right)}^{\mathrm{z}}}$	ka z Cs	It is a form of Langmuir isotherm that is applicable on a heterogeneous surface. In this case, mostly Quasi-Gaussian distribution is inferred.	^[Citation22]
2.	Brunauer, Emmett, Teller Isotherm	${\mathrm{q}}_{\mathrm{e}}=\frac{\mathrm{BC}{\mathrm{Q}}_{\mathrm{m}}}{\left({\mathrm{C}}_{\mathrm{s}}-{\mathrm{C}}_{\mathrm{e}}\right)\left[1+\left(\mathrm{B}-1\right)\left(\text{{{{rm{C}}\_{rm{e}}}}}/\text{{{{rm{C}}\_{rm{S}}}}}\right)\right]}$	B, Cs, C	It almost follows the Langmuir isotherm assumptions but is a multilayered adsorption system. It implements a Langmuir isotherm in each adsorption layer.	^[Citation23]
3.	Toth Isotherm Model	${\mathrm{q}}_{\mathrm{e}}=\frac{{\mathrm{K}}_{\mathrm{T}}{\mathrm{C}}_{\mathrm{e}}}{{\left({\mathrm{a}}_{\mathrm{T}}+{\mathrm{C}}_{\mathrm{e}}\right)}^{\frac{1}{\mathrm{t}}}}$	KT, aT, t	It is a model that can function for both low and high concentrations and also a function of time (t). This model is another modified form of Langmuir isotherm which reduces discrepancies involved in experimental and anticipated data.	^[Citation16]
4.	Redlich Peterson Model	${\mathrm{q}}_{\mathrm{e}}=\frac{{\mathrm{k}}_{\mathrm{RP}}{\mathrm{C}}_{\mathrm{e}}}{1+{\mathit{\alpha }}_{\mathrm{RP}}{\mathrm{C}}_{\mathrm{e}}^{\beta }}$	kRP, ∝RP, β	The isotherm brings together Langmuir and Freundlich models. However, does not depict mono-layer adsorption. The model also has numerator expression which can approximate to Henry’s law for low-concentration adsorbate solution.	^[Citation24]
5.	Khan Isotherm Model	${\mathrm{q}}_{\mathrm{e}}=\frac{{\mathrm{q}}_{\mathrm{s}}{\mathrm{b}}_{\mathrm{k}}{\mathrm{C}}_{\mathrm{e}}}{{\left(1+{\mathrm{b}}_{\mathrm{k}}{\mathrm{C}}_{\mathrm{e}}\right)}^{{\mathrm{a}}_{\mathrm{k}}}}$	bK, ak ,qs	It is a general form of adsorption model that can be used for adsorption from pure solutions representing both Langmuir and Freundlich isotherms. In addition, the model can also be used for binary and ternary adsorption systems.	^[Citation25]
6.	Modified Langmuir Freundlich Isotherm Model	${\mathrm{q}}_{\mathrm{e}}={\mathrm{q}}_{\mathrm{m}}\frac{{\left[{\mathrm{c}}_{\mathrm{e}}{\mathrm{k}}_{\mathrm{a}}\left(\mathrm{pH}\right)\right]}^{\mathrm{n}1}}{1+{\left[{\mathrm{c}}_{\mathrm{e}}{\mathrm{k}}_{\mathrm{a}}\left(\mathrm{pH}\right)\right]}^{\mathrm{n}1}}$	ka, pH, n1, ${\mathrm{q}}_{\mathrm{m}}$	The MLF model expands on the Langmuir-Freundlich model , and can be used even if the pH is changing.	^[Citation26]

1.2. Kinetic models

Unlike adsorption isotherms as discussed in subsection 1.1, kinetic models are employed in unsteady state adsorption studies, when adsorption has not reached equilibirum, These models include the pseudo-first-order, pseudo-second-order, Elovich model, Intraparticle diffusion and Avrami model^[27–29] as listed in Table 2. The first two models, the Pseudo-first-order model and the Pseudo-second order model, listed in Table 2 interpret the rate at which pollutants get transferred onto a solid phase and aid in recognizing physisorption/chemisorption. The third model (Avrami model) emphasizes multilayer adsorption kinetics with both physical and chemical processes and is specifically applicable to systems involving chemisorption and surface heterogeneity.^[34] The fourth model called the Elovich model, represents a two-step process involving chemisorption followed by a diffusion-controlled stage. The next model (Ritchies second-order model) is usually used to quantify initial particle loading.^[35] The subsequent models (Intraparticle diffusion and Boyd’s model) focus on the transport of adsorbate molecules within the pores of the adsorbent, offering insights into diffusion as a rate-controlling step. In addition, the Weber-Morris model sheds light on the role and significance of intraparticle diffusion in the overall adsorption process.^[33] Kinetic models play a crucial role in understanding and optimizing adsorption dynamics, aiding in effectively removing organic pollutants from diverse environmental and industrial contexts. Researchers often utilize a combination of these models to capture the complex kinetics and design efficient adsorption systems. In the following subsection 1.3 specifics of kinetic-equilibrium models are discussed.

Table 2. Different adsorption kinetic models and their significance.

Sl. No	Models	Non-linear form	Significance	Ref.
Empirical models
1.	Pseudo-first-order	${\mathrm{q}}_{\mathrm{t}}={\mathrm{q}}_{\mathrm{e}}\left(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}\right)$	This model is employed when the early stages of the adsorption process are of primary interest or when the adsorption follows a linear process.	^[Citation5]
2.	Pseudo-second-order	$q_{\mathrm{t}}=\left(\frac{{\mathrm{k}}_2.{\mathrm{q}}_{\mathrm{e}}^2.\mathrm{t}}{1+{\mathrm{k}}_2.{\mathrm{q}}_{\mathrm{e}}.\mathrm{t}}\right)$	When a precise description of adsorption kinetics is required, adsorption deviates from linearity with time, and this model provides insights into the actual mechanism and interaction forces involved in the adsorption process.	^[Citation30]
3.	Avrami model	${\mathrm{q}}_{\mathrm{t}}={\mathrm{q}}_{\mathrm{e}}.{\left(1-{\mathrm{e}}^{-{\mathrm{k}}_{\mathit{AV}}\mathrm{t}}\right)}^{{\mathrm{n}}_{\mathit{AV}}}$	This model is frequently employed in studies where phase change is observed, especially in nucleation and growth kinetics. The value of nAV =1,2,3 determines the transformation that fits one-, two- and three-dimensional interface movement.	^[Citation31]
4.	Elovich model	${\mathrm{q}}_{\mathrm{t}}=\frac{1}{\mathit{\beta }}ln\left(1+\mathit{a\beta }\mathrm{t}\right)$	This model can be used when adsorption over heterogeneous adsorbent surfaces has to be determined.	^[Citation28]
5.	Ritchie’s second order	${\mathrm{q}}_{\mathrm{t}}=\left(\frac{\alpha q_{\mathrm{\infty }}\mathrm{t}}{1+\mathit{\alpha t}}\right)$	This model is used when adsorption is predominantly controlled by the occupation of active sites, where a single adsorbate ion or molecule has the capacity to occupy multiple active sites (n). The RSO equation stands apart from the non-linear structure of the PSO model. The model finds frequent application in studying the adsorption of gases or liquids onto solid surfaces.	^[Citation32]
6.	Pseudo-n^th -order-(PNO) model	${\mathrm{q}}_{\mathrm{t}}={\mathrm{q}}_{\mathrm{e}}\left(1-\frac{1}{\left(1+\left({\mathrm{n}}_{\mathrm{s}}-1\right){\mathrm{q}}_{\mathrm{e}}^{{\mathrm{n}}_{\mathrm{s}}-1}{\mathrm{k}}_{{\mathrm{n}}_{\mathrm{s}}}{\mathrm{t}}^{\left(\frac{1}{{\mathrm{n}}_{\mathrm{s}}-1}\right)}\right)}\right)$ (n≠1)	The PNO model mainly represents the adsorption processes where the order factor is between 1 and 2 or larger than 2. Unfortunately, the PNO model is an empirical equation without specific physical meanings.
Diffusion models
1.	Intraparticle diffusion	${\mathrm{q}}_{\mathrm{t}}={\mathrm{k}}_{\mathit{pi}}.{\mathrm{t}}^{0.5}+{\mathrm{C}}_{\mathrm{i}}$	This model investigates different stages of adsorption, such as external, internal diffusion and equilibrium.	^[Citation33]
2.	Boyd’s external diffusion equation	${\mathrm{q}}_{\mathrm{t}}={\mathrm{q}}_{\mathrm{\infty }}.\left(1-{\mathrm{e}}^{-\mathrm{r}.\mathrm{t}}\right)$	This model has frequent applications on kinetic studies to quantify rate constant. The assumption that, film diffusion is rate-limiting step in the initial stage of adsorption	^[Citation32]

1.3. Kinetic-equilibrium models

After a thorough examination of literature, it was possible to identify a few of the investigations wherein kinetic adsorption equation forms were investigated from isotherms such as Langmuir and Freundlich. For example, Liu and their team examined empirical kinetics equation forms, that is, first and second-order kinetic Langmuir kinetics (LK) by providing adequate conditions that could simplify LK to PFO and PSO. The analysis also emphasized that the simplification was a function of initial concentration (C_o).^[36] Further, an integrated-order rate equation was proposed to describe the kinetics of the adsorption of methylene blue on carbon behavior of the adsorption activity. The study reported that mixed form/order of kinetic equation could illustrate the kinetics data better as compared to the existing rate equations.^[37] In continuation of the previous study, Marczewski derived and analyzed integrated kinetic Langmuir equation (IKL). They provided a complete analytical solution of the Langmuir kinetic equation and compared it with mixed order rate equations. The study also examined the impact of new terms equilibrium coverage (θ_qc) and relative equilibrium uptake (u_eq) and could determine the adsorption behavior from the product of equilibrium coverage and relative equilibrium uptake, that is, Langmuir batch equilibrium factor (f_eq). In addition, the analysis and comparison of the new model (IKL) with the n^th order, mixed 1,2-order, and multi-exponential allows one to understand fully the Langmuir kinetics. Further, the analysis yielded a similar expression as that of pseudo-first-order (PFO) and pseudo-second-order (PSO) kinetic equations. The expression of IKL was also extended to include some governing parameters, such as lateral interactions and heterogeneity.^[38] The reported equation exhibited good agreement with experimental data as well.^[39] On the other hand, 8 new models (Langmuir Kinetic) especially for homogenous adsorption were reported. These models were capable of illustrating the kinetic and adsorption behavior of varied adsorbents and were found to be more generalized as compared to existing ones. Further, the models could also quantify the maximum adsorption capacity (mg/g).^[40] Further, four unique models that can describe two-rate adsorption which are solely based on realistic processes such as adsorbing of two-species, two-sites with the difference in binding energy, re-arrangement, and bilayer adsorption modes was investigated. The said models could realistically account for systems influenced by either mass transport or activation limitations or the transition between the two. These models had practical applications in elucidating the adsorption kinetics of surfactants, proteins, and biomolecular systems, especially in cases involving surfaces with both hydrophilic and hydrophobic sites.^[41] An equation was derived on modification of Langmuir isotherm for adsorption processes. Here, the rate is controlled by diffusive transport phenomena referred to as Diffusion-Controlled Langmuir Kinetics (DCLK), assuming that the macroscopic forward rate of adsorption is inversely related to the square root of time. A new study by Salvestrini reported a “factorized differential Langmuir rate” form. The study examined different conditions under which the differential form would reduce to either the PFO or PFO kinetic model. They also proposed a graphical method and evaluated it with experimental data. From their investigation, the authors were also able to appropriately express the equation derived by^[38] as a function of time (t), initial pollutant concentration (C_o), adsorbent dosage (m), adsorption rate constants (k₁ or k₂), adsorption capacity (q_max).^[42] In another research, the authors^[43] improvised the conventional rate (PFO/PSO) and Vermeulen equation by introducing a new parameter, that is, fractional surface coverage (θ) (fractal-like approach) for heterogeneous surfaces. The study also proposed a separation factor (R_H) primarily to describe fractal-like Langmuir form. To validate the proposed expression/model, data on nitrate adsorption on PAN/AC was used. The stated outcomes showed an accurate estimation of equilibrium time by modified PSO (R²=.998). The improvisation of the previously developed equation^[42] was further reported and was called as “diffusion-controlled Langmuir kinetics” (DCLK) model. The model worked on the assumption that forward adsorption rate as an inverse function of time^1/2 (t^1/2) was assumed. The model could better describe the kinetics of adsorption as compared to the existing models. Further, the kinetics (PFO/PSO) using Statistical Rate Theory (SRT) at close to equilibrium condition (C_o>> C_e and θ_e≈1) was explored. The model could also prove that fractional coverage is a linear function of t^1/2 for initial adsorption.^[44] Unlike the investigation by^[40] on homogenous adsorption, modified kinetic equation for a solid-liquid interface assumes a heterogeneous chemical reaction. Considering this equation as a base, a working equation was also developed which was able to provide an independent variable, that is, adsorption rate constants. The investigation could also distinguish between PFO and PSO based on three dimensionless numbers.^[45] Further, a general kinetic model called as mixed order model (MO) was developed by^[46] which was able to describe the whole adsorption process with accuracy. In addition, the MO model was able to describe both the PFO and PSO kinetic of any given adsorption system. The above previous studies have limitation due to the complexity of the methods used (by using integral or differential methods mainly with Langmuir isotherm as base form).

1.4. Research gap

The Scopus survey tool was used to determine the scope of adsorption and kinetic equations.^[47] The Scopus survey for the keywords “Isotherm” AND “Kinetic” AND “Adsorption” displayed a total of 36,026 published articles in 50 years (from 1976–2024). Firstly, from Fig. 1a, it can be observed that there was a continuous exponential rise in publications between the years 2004 to 2024 (from <100 articles per year to nearly 3600 articles per year). Figure 1b displays the list of domains in which adsorption, isotherm and kinetics are extensively used which was ≈ 75% (except red colored portion). Thus, it means, most of the adsorption studies involve the implication of either isotherm or kinetic model separately.

Figure 1: Scopus survey of publications in (a) adsorption domain (b) subject wise publication in adsorption using isotherm and kinetics models.

Second, from Subsection 1.3 (literature), it is evident that there are minimal studies where combined kinetics and equilibrium approaches were undertaken. There is insufficient readily available compilation of combined kinetic and equilibrium models which could aid in quantifying q_max, t_e, C_t. Consequently, it is clear that there is considerable need for such research work.

Thirdly, keeping in mind the practical applicability in treatment systems of large-scale, it is not necessary to carry out the adsorption process till equilibrium duration. Monitoring the remaining concentration and determining equilibrium time is a prolonged process. Therefore, when an adsorbent is prepared/commercially available, we need to know the efficiency of that adsorbent and the adsorption capacity for a particular pollutant (say 2,4-DCP) for a limited amount of time, especially in continuous processes. For this purpose readily available, simplified and solved equations are limited.

Thus, there is a need for combined adsorption kinetic equation using isotherm (Langmuir, Freundlich, Langmuir-Freundlich), which can estimate,

1. Equilibrium adsorption capacity (q_max) and the respective concentration (C_t) are not available.

2. Saturation/equilibrium time (t_e)

With this fundamental research gap in mind, we proposed the CAKE approach herein, which can forecast the adsorption (q_t) at any given point of time (t), for any initial concentration (C_o) of the pollutant for a known volume (V) of the system and known mass of adsorbent (m).

The objectives of the present investigation were to (i) Develop a simplified solution for adsorption equilibrium equation using isotherm equations and mass balance. The equation was called Equilibrium Adsorption Solution Equation (EASE) and (ii) Combine the equilibrium EASE form with kinetic equations (PFO/PSO) to develop the combined kinetic-equilibrium equations labeled as CAKE equation (iii) Validate the developed CAKE and EASE equations with experimental data.

2. Development of models

2.1. Derivation of combined adsorption kinetic equilibrium models

Before proceeding to the derivation there are some of the assumptions which are used in the equations. Assumptions: (a) The rate of adsorption reactions (k₁, k₂) would remain constant for all the concentrations. (b) Temperature of the closed system is constant. (c) Equilibrium is reached for the isotherm experiment within the specified equilibrium time. (d) No chemical reaction is occurring in the system other than the adsorption.

For Langmuir isotherm, the derivation follows these steps:

1. Initially, isotherm equations were used to arrive at the equilibrium concentration C_e.

2. Further, the obtained C_e was substituted in the equation representing experimental adsorption and simplified.

3. The simplified equation is solved for q_e.

Similar steps were also taken by considering Freundlich isotherm and Langmuir-Freundlich isotherm and were substituted in the experimental adsorption capacity equation and solved for q_e for all the cases. The derivation of all the CAKE models is presented in the following sub-sections.

2.2. Derivation of combined isotherm-kinetic models for Langmuirs isotherm

2.2.1. Combining Langmuir and pseudo first/second order models

For the Langmuir adsorption isotherm, the expression for adsorption capacity is given by Eq. 1

(1) $\begin{array}{c}{\mathrm{q}}_{\mathrm{e}}=\left(\frac{{\mathrm{q}}_{\mathrm{m}}{\mathrm{k}}_{\mathrm{L}}{\mathrm{C}}_{\mathrm{e}}}{1+{\mathrm{k}}_{\mathrm{L}}{\mathrm{C}}_{\mathrm{e}}}\right)\end{array}$(1)

Solving for C_e gives,

${\mathrm{C}}_{\mathrm{e}}=\frac{{\mathrm{q}}_{\mathrm{e}}}{{\mathrm{q}}_{\mathrm{m}}.{\mathrm{k}}_{\mathrm{L}}-{\mathrm{k}}_{\mathrm{L}}.{\mathrm{q}}_{\mathrm{e}}}$

(2) $\begin{array}{c}{\mathrm{C}}_{\mathrm{e}}=\frac{{\mathrm{q}}_{\mathrm{e}}}{{\mathrm{k}}_{\mathrm{L}}\left({\mathrm{q}}_{\mathrm{m}}-{\mathrm{q}}_{\mathrm{e}}\right)}\end{array}$(2)

Mass balance suggests that ${\mathrm{C}}_{\mathrm{o}}={\mathrm{C}}_{\mathrm{e}}+{\mathrm{q}}_{\mathrm{e}}\frac{\mathrm{m}}{\mathrm{V}}$

We have an expression for adsorption capacity from mass balance, as shown below in Eq. 3

(3) ${\mathrm{q}}_{\mathrm{e}}=\left({\mathrm{C}}_{\mathrm{o}}-{\mathrm{C}}_{\mathrm{e}}\right)\frac{\mathrm{V}}{\mathrm{m}}$(3)

Solving for C_o gives,

${\mathrm{q}}_{\mathrm{e}}.\frac{\mathrm{V}}{\mathrm{m}}={\mathrm{C}}_{\mathrm{o}}-{\mathrm{C}}_{\mathrm{e}}$

(4) $\begin{array}{c}{\mathrm{C}}_{\mathrm{o}}={\mathrm{C}}_{\mathrm{e}}+{\mathrm{q}}_{\mathrm{e}}.\frac{\mathrm{m}}{\mathrm{V}}\end{array}$(4)

Substituting value C_e from Eq. 2 in Eq. 4

(5) $\begin{array}{c}{\mathrm{C}}_{\mathrm{o}}=\frac{{\mathrm{q}}_{\mathrm{e}}}{{\mathrm{k}}_{\mathrm{L}}\left({\mathrm{q}}_{\mathrm{m}}-{\mathrm{q}}_{\mathrm{e}}\right)}+\frac{\mathrm{m}}{\mathrm{V}}.{\mathrm{q}}_{\mathrm{e}}\end{array}$(5)

Simplifying further,

(6) $\begin{array}{c}{\mathrm{C}}_{\mathrm{o}}.{\mathrm{k}}_{\mathrm{L}}.{\mathrm{q}}_{\mathrm{m}}-{\mathrm{C}}_{\mathrm{o}}.{\mathrm{k}}_{\mathrm{L}}.{\mathrm{q}}_{\mathrm{e}}={\mathrm{q}}_{\mathrm{e}}+{\mathrm{q}}_{\mathrm{e}}.\frac{\mathrm{m}}{\mathrm{V}}.{\mathrm{k}}_{\mathrm{L}}.{\mathrm{q}}_{\mathrm{m}}-{\mathrm{q}}_{\mathrm{e}}^2.\frac{\mathrm{m}}{\mathrm{V}}.{\mathrm{k}}_{\mathrm{L}}\end{array}$(6)

Rewriting Eq. 6 in the form.

ax² + bx + c = 0

We get,

(7) $\begin{array}{c}{\mathrm{q}}_{\mathrm{e}}^2.\frac{\mathrm{m}}{\mathrm{V}}.{\mathrm{k}}_{\mathrm{L}}-{\mathrm{q}}_{\mathrm{e}}\left[1+\frac{\mathrm{m}}{\mathrm{V}}.{\mathrm{k}}_{\mathrm{L}}.{\mathrm{q}}_{\mathrm{m}}+{\mathrm{C}}_{\mathrm{o}}.{\mathrm{k}}_{\mathrm{L}}\right]+{\mathrm{C}}_{\mathrm{o}}.{\mathrm{k}}_{\mathrm{L}}.{\mathrm{q}}_{\mathrm{m}}=0\end{array}$(7)

Solving for ${\mathrm{q}}_{\mathrm{e}}$ from Eq. 7 by using quadratic equation solution as below,

$\mathrm{x}=\left(\frac{-\mathrm{b}\pm \sqrt{{\mathrm{b}}^2-4\mathrm{ac}}}{2\mathrm{a}}\right)$

The solution of Eq. 7 is as given below.

(8) $\begin{array}{c}{\mathrm{q}}_{\mathrm{e}}=\frac{\left(\mathit{\hspace{0.17em}}1+\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}{\mathrm{k}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}+}{\mathrm{C}}_{\mathrm{o}}\mathit{\hspace{0.17em}}{\mathrm{k}}_{\mathrm{L}}\right)\pm \sqrt{{\left(1+\frac{\mathrm{m}}{\mathrm{V}}{\mathrm{k}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}+}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\right)}^{2\mathit{\hspace{0.17em}}}-\mathit{\hspace{0.17em}}4\mathit{\hspace{0.17em}}\frac{\mathrm{m}}{\mathrm{V}}{{\mathrm{k}}_{\mathrm{L}}}^2{\mathrm{C}}_{\mathrm{o}}{\mathrm{q}}_{\mathrm{m}}}}{2\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}{\mathrm{k}}_{\mathrm{L}}}\end{array}$(8)

Further simplifying Eq. 8, we get

(9) ${\mathrm{q}}_{\mathrm{e}}=\frac{\left({\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V}\right)\pm \sqrt{-4\mathrm{m}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}^2\mathrm{m}{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{(-{\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}-\mathrm{V}-{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V})}^2}}{2{\mathrm{k}}_{\mathrm{L}}\mathrm{m}}$(9)

Eq. 9 is called Equilibrium Adsorption Solution Equation-Langmuir (EASE-L).

We can also determine equation for equilibrium concentration by substituting “q_e” in Eq. 4 with Eq. 9

(10) ${\mathrm{C}}_{\mathrm{e}}={\mathrm{C}}_{\mathrm{o}}-\left(\frac{\left({\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V}\right)\pm \sqrt{-4\mathrm{m}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}^2\mathrm{m}{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{(-{\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}-\mathrm{V}-{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V})}^2}}{2{\mathrm{k}}_{\mathrm{L}}\mathrm{m}}\right)\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}$(10)

Therefore, Eq. 10 can be labeled as Equilibrium Concentration derived from Langmuir model (EC-L)

The kinetic pseudo first order (PFO) equation is expressed as in Eq. 11

(11) $\begin{array}{c}{\mathrm{q}}_{\mathrm{t}}={\mathrm{q}}_{\mathrm{e}}\left(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}\right)\end{array}$(11)

Substituting EASE-L from Eq. 9 in Eq. 11, we get the combined adsorption kinetic-equilibrium (CAKE) equation for pseudo-first -order kinetics as

(12) $\begin{array}{c}{\mathrm{q}}_{\mathrm{t}}=\left(\frac{\left({\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V}\right)\pm \sqrt{-4\mathrm{m}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}^2\mathrm{m}{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{(-{\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}-\mathrm{V}-{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V})}^2}}{2{\mathrm{k}}_{\mathrm{L}}\mathrm{m}}\right)\left(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}\right)\end{array}$(12)

From Eq. 12, the concentration at time “t” can be given as below.

(13) ${\mathrm{C}}_{\mathrm{t}}={\mathrm{C}}_{\mathrm{o}}-\left(\frac{\left({\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V}\right)\pm \sqrt{-4\mathrm{m}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}^2\mathrm{m}{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{(-{\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}-\mathrm{V}-{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V})}^2}}{2{\mathrm{k}}_{\mathrm{L}}\mathrm{m}}\right)\left(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}\right)\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}$(13)

The equation for PSO is expressed as in Eq. 14,

(14) ${\mathrm{q}}_{\mathrm{t}}=\left(\frac{{\mathrm{k}}_2{\mathrm{q}}_{\mathrm{e}}^2\mathrm{t}}{1+{\mathrm{k}}_2{\mathrm{q}}_{\mathrm{e}}\mathrm{t}}\right)$(14)

Substituting q_e from Eq. 9 in Eq. 15, we get the CAKE equation for Langmuir-PSO

(15) ${\mathrm{q}}_{\mathrm{t}}=\left(\frac{{\mathrm{k}}_2{\left(\frac{\left({\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V}\right)\pm \sqrt{-4\mathrm{m}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}^2\mathrm{m}{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{(-{\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}-\mathrm{V}-{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V})}^2}}{2{\mathrm{k}}_{\mathrm{L}}\mathrm{m}}\right)}^2\mathrm{t}}{1+{\mathrm{k}}_2{\left(\frac{\left({\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V}\right)\pm \sqrt{-4\mathrm{m}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}^2\mathrm{m}{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{(-{\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}-\mathrm{V}-{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V})}^2}}{2{\mathrm{k}}_{\mathrm{L}}\mathrm{m}}\right)}^{\mathit{\hspace{0.17em}}}\mathrm{t}}\right)$(15)

Similarly, concentration at time “t” can be obtained for PSO,

(16) ${\mathrm{C}}_{\mathrm{t}}={\mathrm{C}}_{\mathrm{o}}-\left(\frac{{\mathrm{k}}_2{\left(\frac{\left({\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V}\right)\pm \sqrt{-4\mathrm{m}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}^2\mathrm{m}{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{(-{\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}-\mathrm{V}-{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V})}^2}}{2{\mathrm{k}}_{\mathrm{L}}\mathrm{m}}\right)}^2\mathrm{t}}{1+{\mathrm{k}}_2\left(\frac{\left({\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V}\right)\pm \sqrt{-4\mathrm{m}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}^2\mathrm{m}{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{(-{\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}-\mathrm{V}-{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V})}^2}}{2{\mathrm{k}}_{\mathrm{L}}\mathrm{m}}\right)\mathrm{t}}\right)\frac{\mathrm{m}}{\mathrm{V}}$(16)

Eq. 12 and Eq. 15 are CAKE equations for PFO and PSO as obtained from Langmuir Isotherm, respectively.

2.3. Freundlich isotherm: derivation of CAKE and EASE equations

2.3.1. Combining Freundlich and pseudo First/Second order models

Similar derivation can be done for Freundlich isotherm equation, which is given by the expression Eq. 17

(17) $\begin{array}{c}{\mathrm{q}}_{\mathrm{e}}={\mathrm{k}}_{\mathrm{f}}[\left({\mathrm{C}}_{\mathrm{e}}{)}^{1/\mathrm{n}}\right]\end{array}$(17)

Taking the logarithm function on both sides of Eq. 17

$\log {\mathrm{q}}_{\mathrm{e}}=\log \left[{\mathrm{k}}_{\mathrm{f}}.{\left({\mathrm{C}}_{\mathrm{e}}\right)}^{1/\mathrm{n}}\right]$

Solving for C_e, we get

$\log {\mathrm{q}}_{\mathrm{e}}\mathit{\hspace{0.17em}}=\log {\mathrm{k}}_{\mathrm{f}}\mathit{\hspace{0.17em}}+\mathit{\hspace{0.17em}}\frac{1}{\mathrm{n}}\log ({\mathrm{C}}_{\mathrm{e}})$

$\mathrm{n}\left[\log {\mathrm{q}}_{\mathrm{e}}\mathit{\hspace{0.17em}}-\log {\mathrm{k}}_{\mathrm{f}}\mathit{\hspace{0.17em}}\right]\mathit{\hspace{0.17em}}=\log ({\mathrm{C}}_{\mathrm{e}})$

$\log {\left(\frac{{\mathrm{q}}_{\mathrm{e}}}{{\mathrm{k}}_{\mathrm{f}}}\right)}^{\mathrm{n}}=\log ({\mathrm{C}}_{\mathrm{e}})$

(18) $\begin{array}{c}{\mathrm{C}}_{\mathrm{e}}={\left(\frac{{\mathrm{q}}_{\mathrm{e}}}{{\mathrm{k}}_{\mathrm{f}}}\right)}^{\mathrm{n}}\end{array}$(18)

Substituting value C_e from Eq. 18 in Eq. 17

(19) $\begin{array}{c}{\mathrm{C}}_{\mathrm{o}}={\left(\frac{{\mathrm{q}}_{\mathrm{e}}}{{\mathrm{k}}_{\mathrm{f}}}\right)}^{\mathrm{n}}+{\mathrm{q}}_{\mathrm{e}}.\frac{\mathrm{m}}{\mathrm{V}}\end{array}$(19)

For fractional n, we can make “n” close to a nearest whole number and calculate K_f,

For “n” =1 in the Eq. 19 we get,

$\begin{array}{c}{\mathrm{C}}_{\mathrm{o}}=\left(\frac{{\mathrm{q}}_{\mathrm{e}}}{{\mathrm{k}}_{\mathrm{f}}}\right)+{\mathrm{q}}_{\mathrm{e}}.\frac{\mathrm{m}}{\mathrm{V}}\end{array}$

Therefore, Eq. 19 changes and rearranged as below,

$\begin{array}{c}\left(\frac{{\mathrm{q}}_{\mathrm{e}}}{{\mathrm{k}}_{\mathrm{f}}}\right)+{\mathrm{q}}_{\mathrm{e}}.\frac{\mathrm{m}}{\mathrm{V}}-{\mathrm{C}}_{\mathrm{o}}=0\end{array}$

For “n” = 1, q_e is given by

$\left(\frac{{\mathrm{q}}_{\mathrm{e}}}{{\mathrm{k}}_{\mathrm{f}}}\right)+{\mathrm{q}}_{\mathrm{e}}.\frac{\mathrm{m}}{\mathrm{V}}-{\mathrm{C}}_{\mathrm{o}}=0$

Or ${\mathrm{q}}_{\mathrm{e}}=\left(\frac{{\mathrm{k}}_{\mathrm{f}}{\mathrm{C}}_{\mathrm{o}}}{1+\frac{\mathrm{m}}{\mathrm{V}}\ast {\mathrm{k}}_{\mathrm{f}}}\right)$

Similarly, substituting “n”=2 in the above Eq. 19 we get

(20) $\begin{array}{c}{\left(\frac{{\mathrm{q}}_{\mathrm{e}}}{{\mathrm{k}}_{\mathrm{f}}}\right)}^2+\left(\frac{{\mathrm{q}}_{\mathrm{e}}}{{\mathrm{k}}_{\mathrm{f}}}\right).\frac{{\mathrm{k}}_{\mathrm{f}}\mathrm{m}}{\mathrm{V}}-{\mathrm{C}}_{\mathrm{o}}=0\end{array}$(20)

Eq. 20 is in the quadratic form and the solution for q_e is shown below,

(21) ${\mathrm{q}}_e\mathit{\hspace{0.17em}}=\frac{\mathit{\hspace{0.17em}}1\mathit{\hspace{0.17em}}}{2}{\mathrm{k}}_{\mathrm{f}}^2\left(\sqrt{\frac{4{\mathrm{C}}_{\mathrm{o}}}{{\mathrm{k}}_{\mathrm{f}}^2}+\frac{{\mathrm{m}}^2}{{\mathrm{V}}^2}}-\frac{\mathrm{m}}{\mathrm{V}}\right)\mathit{\hspace{0.17em}}$(21)

Now, substituting EASE-F from Eq. 21 in Eq. 11 and Eq. 14

The Eq. 22 is CAKE form for PFO.

(22) $\begin{array}{c}{\mathbf{q}}_{\mathbf{t}}=\left(\frac{1}{2}{\mathbf{k}}_{\mathbf{f}}^2\left(\sqrt{\frac{4{\mathbf{C}}_{\mathbf{o}}}{{\mathbf{k}}_{\mathbf{f}}^2}+\frac{{\mathbf{m}}^2}{{\mathbf{V}}^2}}-\frac{\mathbf{m}}{\mathbf{V}}\right)\right)(1-{\mathbf{e}}^{-{\mathbf{k}}_1\mathbf{t}})\end{array}$(22)

Similarly, for PSO another equation (Eq. 23) can be obtained as below,

(23) $\begin{array}{c}{\mathbf{q}}_{\mathbf{t}}=\left(\frac{{\mathbf{k}}_2{\left(\frac{1}{2}{\mathbf{k}}_{\mathbf{f}}^2\left(\sqrt{\frac{4{\mathbf{C}}_{\mathbf{o}}}{{\mathbf{k}}_{\mathbf{f}}^2}+\frac{{\mathbf{m}}^2}{{\mathbf{V}}^2}}-\frac{\mathbf{m}}{\mathbf{V}}\right)\right)}^2\mathbf{t}}{1+{\mathbf{k}}_2\left(\frac{1}{2}{\mathbf{k}}_{\mathbf{f}}^2\left(\sqrt{\frac{4{\mathbf{C}}_{\mathbf{o}}}{{\mathbf{k}}_{\mathbf{f}}^2}+\frac{{\mathbf{m}}^2}{{\mathbf{V}}^2}}-\frac{\mathbf{m}}{\mathbf{V}}\right)\right)\mathbf{t}}\right)\end{array}$(23)

Eq. 22 and Eq. 23 are CAKE equations (PFO and PSO) as obtained from Freundlich Isotherm for the quadratic form of Eq. 20.

Now, substituting n=3 in the above Eq. 19 we get

(24) ${\left(\frac{{\mathrm{q}}_{\mathrm{e}}}{{\mathrm{k}}_{\mathrm{f}}}\right)}^3+\left(\frac{{\mathrm{q}}_{\mathrm{e}}}{{\mathrm{k}}_{\mathrm{f}}}\right).\frac{{\mathrm{k}}_{\mathrm{f}}\mathrm{m}}{\mathrm{V}}-{\mathrm{C}}_{\mathrm{o}}=0$(24)

Eq. 24 is cubic nature and the solution for q_e can be given as below,

(25) ${\mathrm{q}}_{\mathit{e}}=\frac{1}{3}{\mathrm{k}}_{\mathrm{f}}\left(\sqrt[3]{\frac{27{\mathrm{C}}_{\mathrm{o}}\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^2}}}{2}}-\frac{3\sqrt[3]{2}{\mathrm{k}}_{\mathrm{f}}\mathrm{m}}{\sqrt[3]{\left(27{\mathrm{C}}_{\mathrm{o}}+\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^3}}\right)}\mathrm{V}}\right)$(25)

Eq. 21 and Eq. 25 are called as Equilibrium Adsorption Solution Equation-Freundlich (EASE-F).

(26) ${\mathrm{C}}_{\mathrm{e}}={\mathrm{C}}_{\mathrm{o}}-\frac{1}{3}{\mathrm{k}}_{\mathrm{f}}\frac{\mathrm{m}}{\mathrm{V}}\left(\sqrt[3]{\frac{27{\mathrm{C}}_{\mathrm{o}}\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^2}}}{2}}-\frac{3\sqrt[3]{2}{\mathrm{k}}_{\mathrm{f}}\mathrm{m}}{\sqrt[3]{\left(27{\mathrm{C}}_{\mathrm{o}}+\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^3}}\right)}\mathrm{V}}\right)\mathit{\hspace{0.17em}}$(26)

Eq. 26 is Equilibrium Concentration derived from Freundlich model (EC-F). Similarly, we have CAKE equations for the cubic form of Eq. 24 as given below,

(27) ${\mathrm{q}}_{\mathrm{t}}=\frac{1}{3}{\mathrm{k}}_{\mathrm{f}}\left(\sqrt[3]{\frac{27{\mathrm{C}}_{\mathrm{o}}\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^2}}}{2}}-\frac{3\sqrt[3]{2}{\mathrm{k}}_{\mathrm{f}}\mathrm{m}}{\sqrt[3]{\left(27{\mathrm{C}}_{\mathrm{o}}+\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^3}}\right)}\mathrm{V}}\right)(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}})$(27)

(28) ${\mathrm{C}}_{\mathrm{t}}={\mathrm{C}}_{\mathrm{o}}-\frac{1}{3}{\mathrm{k}}_{\mathrm{f}}\frac{\mathrm{m}}{\mathrm{V}}\left(\sqrt[3]{\frac{27{\mathrm{C}}_{\mathrm{o}}\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^2}}}{2}}-\frac{3\sqrt[3]{2}{\mathrm{k}}_{\mathrm{f}}\mathrm{m}}{\sqrt[3]{\left(27{\mathrm{C}}_{\mathrm{o}}+\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^3}}\right)}\mathrm{V}}\right)\left(1-{\mathrm{e}}^{-\mathrm{kt}}\right)$(28)

(29) ${\mathbf{q}}_{\mathbf{t}}=\left(\frac{{\mathbf{k}}_2{\left\{\frac{1}{3}{\mathbf{k}}_{\mathbf{f}}\left(\sqrt[3]{\frac{27{\mathbf{C}}_{\mathbf{o}}\sqrt{729{\mathbf{C}}_{\mathbf{o}}^2+\frac{108{\mathbf{k}}_{\mathbf{f}}^3{\mathbf{m}}^3}{{\mathbf{V}}^2}}}{2}}-\frac{3\sqrt[3]{2}{\mathbf{k}}_{\mathbf{f}}\mathbf{m}}{\sqrt[3]{\left(27{\mathbf{C}}_{\mathbf{o}}+\sqrt{729{\mathbf{C}}_{\mathbf{o}}^2+\frac{108{\mathbf{k}}_{\mathbf{f}}^3{\mathbf{m}}^3}{{\mathbf{V}}^3}}\right)}\mathbf{V}}\right)\right\}}^2\mathbf{t}}{1+{\mathbf{k}}_2\frac{1}{3}{\mathbf{k}}_{\mathbf{f}}\left(\sqrt[3]{\frac{27{\mathbf{C}}_{\mathbf{o}}\sqrt{729{\mathbf{C}}_{\mathbf{o}}^2+\frac{108{\mathbf{k}}_{\mathbf{f}}^3{\mathbf{m}}^3}{{\mathbf{V}}^2}}}{2}}-\frac{3\sqrt[3]{2}{\mathbf{k}}_{\mathbf{f}}\mathbf{m}}{\sqrt[3]{\left(27{\mathbf{C}}_{\mathbf{o}}+\sqrt{729{\mathbf{C}}_{\mathbf{o}}^2+\frac{108{\mathbf{k}}_{\mathbf{f}}^3{\mathbf{m}}^3}{{\mathbf{V}}^3}}\right)}\mathbf{V}}\right)\mathbf{t}}\right)$(29)

Similarly, we have CAKE equations for the cubic form of eq. 24 as given in eq. 27 and eq. 29

(30) $\begin{array}{rl}& {\mathbf{C}}_{\mathbf{t}}={\mathbf{C}}_{\mathbf{o}}-\frac{\mathbf{m}}{\mathbf{V}}\left(\frac{{\mathbf{k}}_2{\left\{\frac{1}{3}{\mathbf{k}}_{\mathbf{f}}\left(\sqrt[3]{\frac{27{\mathbf{C}}_{\mathbf{o}}\sqrt{729{\mathbf{C}}_{\mathbf{o}}^2+\frac{108{\mathbf{k}}_{\mathbf{f}}^3{\mathbf{m}}^3}{{\mathbf{V}}^2}}}{2}}-\frac{3\sqrt[3]{2}{\mathbf{k}}_{\mathbf{f}}\mathbf{m}}{\sqrt[3]{\left(27{\mathbf{C}}_{\mathbf{o}}+\sqrt{729{\mathbf{C}}_{\mathbf{o}}^2+\frac{108{\mathbf{k}}_{\mathbf{f}}^3{\mathbf{m}}^3}{{\mathbf{V}}^3}}\right)}\mathbf{V}}\right)\right\}}^2\mathbf{t}}{1+{\mathbf{k}}_2\frac{1}{3}{\mathbf{k}}_{\mathbf{f}}\left(\sqrt[3]{\frac{27{\mathbf{C}}_{\mathbf{o}}\sqrt{729{\mathbf{C}}_{\mathbf{o}}^2+\frac{108{\mathbf{k}}_{\mathbf{f}}^3{\mathbf{m}}^3}{{\mathbf{V}}^2}}}{2}}-\frac{3\sqrt[3]{2}{\mathbf{k}}_{\mathbf{f}}\mathbf{m}}{\sqrt[3]{\left(27{\mathbf{C}}_{\mathbf{o}}+\sqrt{729{\mathbf{C}}_{\mathbf{o}}^2+\frac{108{\mathbf{k}}_{\mathbf{f}}^3{\mathbf{m}}^3}{{\mathbf{V}}^3}}\right)}\mathbf{V}}\right)\mathbf{t}}\right)\end{array}$(30)

2.4. Langmuir-Freundlich (Sips isotherm)

2.4.1. Combining Langmuir-freundlich and pseudo First/Second order models

Similar derivation can be done for Langmuir-Freundlich isotherm, which is given by Eq. 31

(31) $\begin{array}{c}{\mathrm{q}}_{\mathrm{e}}=\left(\frac{{\mathrm{q}}_{\mathrm{m}}{\left({\mathrm{k}}_{\mathrm{g}}{\mathrm{C}}_{\mathrm{e}}\right)}^{{\mathrm{n}}_{\mathrm{L}}}}{1+{\left({\mathrm{k}}_{\mathrm{g}}{\mathrm{C}}_{\mathrm{e}}\right)}^{{\mathrm{n}}_{\mathrm{L}}}}\right)\end{array}$(31)

The EASE equation can be written as

(32) ${\mathrm{q}}_{\mathrm{e}}=\left(\frac{{\mathrm{k}}_{\mathrm{g}}\mathit{m}{n}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}-\sqrt{-4{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}^2{\mathrm{m}\mathrm{n}}_{\mathrm{L}}^2{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{\left({\mathrm{k}}_{\mathrm{g}}\mathit{m}{n}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}\right)}^2}}{2{\mathrm{k}}_{\mathrm{g}}\mathit{m}{n}_{\mathrm{L}}}\right)$(32)

Eq. 32 can be labeled as Equilibrium Adsorption Solution Equation-Langmuir-Freundlich (EASE-LF). In addition to this, Equilibrium concentration derived from Langmuir-Freundlich model (EC-LF) can be represented as given below in Eq. 33.

(33) ${\mathrm{C}}_{\mathrm{e}}={\mathrm{C}}_{\mathrm{o}}-\frac{\mathrm{m}}{\mathrm{V}}\left(\frac{{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}-\sqrt{-4{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}^2\mathrm{m}{{\mathrm{n}}_{\mathrm{L}}}^2{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{\left({\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}\right)}^2}}{2{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}}\right)\mathit{\hspace{0.17em}}$(33)

Further, the above forms can be used in PFO and PSO equations.

(34) $\begin{array}{c}{\mathrm{q}}_{\mathrm{t}}=\left(\frac{{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}-\sqrt{-4{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}^2{\mathrm{m}\mathrm{n}}_{\mathrm{L}}^2{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{\left({\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}\right)}^2}}{2{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}}\right)\left(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}\right)\end{array}$(34)

(35) ${\mathrm{C}}_{\mathrm{t}}={\mathrm{C}}_{\mathrm{o}}-\frac{\mathrm{m}}{\mathrm{V}}\left(\frac{{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}-\sqrt{-4{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}^2\mathrm{m}{{\mathrm{n}}_{\mathrm{L}}}^2{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{\left({\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}\right)}^2}}{2{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}}\right)\left(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}\right)\mathit{\hspace{0.17em}}$(35)

(36) ${\mathrm{q}}_{\mathrm{t}}=\left(\frac{{\mathrm{k}}_2{\left(\frac{{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}-\sqrt{-4{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}^2{\mathrm{m}\mathrm{n}}_{\mathrm{L}}^2{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{\left({\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}\right)}^2}}{2{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}}\right)}^2\mathrm{t}}{1+{\mathrm{k}}_2\left(\frac{{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}-\sqrt{-4{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}^2{\mathrm{m}\mathrm{n}}_{\mathrm{L}}^2{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{\left({\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}\right)}^2}}{2{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}}\right)\mathrm{t}}\right)$(36)

(37) ${\mathrm{C}}_{\mathrm{t}}={\mathrm{C}}_{\mathrm{o}}-\frac{\mathrm{m}}{\mathrm{V}}\left(\frac{{\mathrm{k}}_2{\left(\frac{{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}-\sqrt{-4{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}^2{\mathrm{m}\mathrm{n}}_{\mathrm{L}}^2{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{\left({\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}\right)}^2}}{2{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}}\right)}^2\mathrm{t}}{1+{\mathrm{k}}_2.\left(\frac{{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}-\sqrt{-4{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}^2{\mathrm{m}\mathrm{n}}_{\mathrm{L}}^2{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{\left({\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}\right)}^2}}{2{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}}\right)\mathrm{t}}\right)$(37)

Eq. 34 and Eq. 36 are the CAKE (PFO and PSO) equations obtained from LF Isotherm.

2.5. Linear isotherm/Henry’s equation

2.5.1. Combining linear isotherm with pseudo First-/Second-order models

At low concentrations, the Linear isotherm (Henry’s equation), can be used and is given by,

${\mathrm{q}}_{\mathrm{e}}={\mathrm{K}}_{\mathrm{a}}.{\mathrm{C}}_{\mathrm{e}}$

Or

${\mathrm{C}}_{\mathrm{o}}={\mathrm{C}}_{\mathrm{e}}+{\mathrm{K}}_{\mathrm{a}}.{\mathrm{C}}_{\mathrm{e}}.\frac{\mathrm{m}}{\mathrm{V}}$

${\mathrm{C}}_{\mathrm{e}}=\frac{{\mathrm{C}}_{\mathrm{o}}}{1+{\mathrm{K}}_{\mathrm{a}}.\frac{\mathrm{m}}{\mathrm{V}}}$

(38) ${\mathrm{q}}_{\mathrm{e}}=\frac{{\mathrm{K}}_{\mathrm{a}}.{\mathrm{C}}_{\mathrm{o}}}{1+{\mathrm{K}}_{\mathrm{a}}.\frac{\mathrm{m}}{\mathrm{V}}}$(38)

Eq 38 is the EASE form for the Henry’s isotherm, and corresponding EC eqn is given above, Substituting the q_e term in Eq. 11 and Eq. 14 we get,

${\mathrm{q}}_{\mathrm{t}}=\frac{{\mathrm{K}}_{\mathrm{a}}{\mathrm{C}}_{\mathrm{o}}}{1+\mathit{\hspace{0.17em}}{\mathrm{K}}_{\mathrm{a}}\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}}\left(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}\right)$

Or

${\mathrm{q}}_{\mathrm{t}}={{\mathrm{K}}_{\mathrm{d}}}^{\prime }{\mathrm{C}}_{\mathrm{o}}$

Where, ${{\mathrm{K}}_{\mathrm{d}}}^{\prime }$ is apparent distribution coefficient for kinetic problems in linear isotherm.

Also solving for C_t gives

(39) $\left({\mathrm{C}}_{\mathrm{o}}-{\mathrm{C}}_{\mathrm{t}}\right).\frac{\mathrm{V}}{\mathrm{m}}=\frac{{\mathrm{K}}_{\mathrm{a}}{\mathrm{C}}_{\mathrm{o}}}{1+{\mathrm{K}}_{\mathrm{a}}\frac{\mathrm{m}}{\mathrm{V}}}\left(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}\right)$(39)

$\mathrm{or}$

(39b) ${\mathrm{C}}_{\mathbf{t}}\mathit{\hspace{0.17em}}={\mathrm{C}}_{\mathrm{o}}-\mathit{\hspace{0.17em}}\frac{{\mathrm{K}}_{\mathrm{a}}{\mathrm{C}}_{\mathrm{o}}\frac{\mathrm{m}}{\mathrm{V}}}{1+\mathit{\hspace{0.17em}}{\mathrm{K}}_{\mathrm{a}}\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}}\left(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}\right)$(39b)

Similarly, solving for q_t gives

(40) $\begin{array}{c}{\mathrm{q}}_{\mathrm{t}}=\left(\frac{{\mathrm{k}}_2{\left(\frac{{\mathrm{K}}_{\mathrm{a}}{\mathrm{C}}_{\mathrm{o}}}{1+\mathit{\hspace{0.17em}}{\mathrm{K}}_{\mathrm{a}}\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}}\right)}^2\mathrm{t}}{1+\mathit{\hspace{0.17em}}{\mathrm{k}}_2\left(\frac{{\mathrm{K}}_{\mathrm{a}}{\mathrm{C}}_{\mathrm{o}}}{1+\mathit{\hspace{0.17em}}{\mathrm{K}}_{\mathrm{a}}\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}}\right)\mathrm{t}}\right)\end{array}$(40)

And C_t is given for pseudo-second-order kinetics by following equation,

(41) ${\mathrm{C}}_{\mathrm{t}}={\mathrm{C}}_{\mathrm{o}}-\frac{\mathrm{m}}{\mathrm{V}}\left(\frac{{\mathrm{k}}_2{\left(\frac{{\mathrm{K}}_{\mathrm{a}}{\mathrm{C}}_{\mathrm{o}}}{1+\mathit{\hspace{0.17em}}{\mathrm{K}}_{\mathrm{a}}\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}}\right)}^2\mathrm{t}}{1+\mathit{\hspace{0.17em}}{\mathrm{k}}_2\left(\frac{{\mathrm{K}}_{\mathrm{a}}{\mathrm{C}}_{\mathrm{o}}}{1+\mathit{\hspace{0.17em}}{\mathrm{K}}_{\mathrm{a}}\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}}\right)\mathrm{t}}\right)$(41)

Eq. 40 is CAKE form for pseudo-second-order kinetics.

Eq. 39 and Eq. 40 are modified CAKE equations (PFO and PSO) as obtained from the Linear/Henry form of the isotherm.

Similarly, C_t is given by

(42) ${\mathrm{C}}_{\mathrm{t}}={\mathrm{C}}_{\mathrm{o}}-\frac{\mathrm{m}}{\mathrm{V}}\left(\frac{{\mathrm{k}}_2{\left(\frac{{\mathrm{K}}_{\mathrm{a}}{\mathrm{C}}_{\mathrm{o}}}{1+\mathit{\hspace{0.17em}}{\mathrm{K}}_{\mathrm{a}}\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}}\right)}^2\mathrm{t}}{1+\mathit{\hspace{0.17em}}{\mathrm{k}}_2\left(\frac{{\mathrm{K}}_{\mathrm{a}}{\mathrm{C}}_{\mathrm{o}}}{1+\mathit{\hspace{0.17em}}{\mathrm{K}}_{\mathrm{a}}\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}}\right)\mathrm{t}}\right)$(42)

At high concentrations (saturation range), ${\mathrm{C}}_{\mathrm{e}}$ is given by the derivation below, we can approximate as, q_e can be approximated as q_m

${\mathrm{q}}_{\mathrm{e}}\mathit{\hspace{0.17em}}={\mathrm{q}}_{\mathrm{m}}$

(43) ${\mathrm{C}}_{\mathrm{e}}=\mathit{\hspace{0.17em}}{\mathrm{C}}_{\mathrm{o}}-{\mathrm{q}}_{\mathrm{m}}\ast \frac{\mathrm{m}}{\mathrm{V}}$(43)

(44) ${\mathrm{C}}_{\mathrm{t}}=\mathit{\hspace{0.17em}}{\mathrm{C}}_{\mathrm{o}}-{\mathrm{q}}_{\mathrm{m}}\ast \frac{\mathrm{m}}{\mathrm{V}}$(44)

For pseudo-first-order kinetics

(45) ${\mathrm{q}}_{\mathrm{t}}={\mathrm{q}}_{\mathrm{m}}\left(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}\right)$(45)

For pseudo-second-order kinetics,

(46) ${\mathrm{q}}_{\mathrm{t}}=\left(\frac{{\mathrm{k}}_2{\mathrm{q}}_{\mathrm{m}}^2\mathrm{t}}{1+{\mathrm{k}}_2{\mathrm{q}}_{\mathrm{m}}\mathrm{t}}\right)$(46)

Table 3 provides a summary and comprehensive list of all CAKE models for q_e or solid phase adsorption at equilibrium , while Table 4 provides equations governing equilibrium aqueous concentration (C_e). Furthermore Table 5 presents, equations to determine the concentration (C_t) at a specific time (t) in batch adsorption studies.

Table 3. List of all the kinetic and equilibrium models.

Sl.No	EASE EQUATIONS FOR qe FOR DIFFERENT ISOTHERMS	CAKE MODEL
1.	(Equilibrium Adsorption Solution Equation-Langmuir) ${\mathbf{q}}_{\mathbf{e}}=\frac{\left({\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V}\right)\pm \sqrt{-\mathit{\hspace{0.17em}}4\mathrm{m}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}^2\mathrm{m}{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{(-{\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}-\mathrm{V}-{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V})}^2}}{2{\mathrm{k}}_{\mathrm{L}}\mathrm{m}}$	${\mathrm{q}}_{\mathrm{t}}=\left(\frac{\left({\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V}\right)-\sqrt{-\mathit{\hspace{0.17em}}4\mathrm{m}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}^2\mathrm{m}{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{(-{\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}-\mathrm{V}-{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V})}^2}}{2{\mathrm{k}}_{\mathrm{L}}\mathrm{m}}\right)\left(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}\right)$
2.		${\mathrm{q}}_{\mathrm{t}}=\left(\frac{{\mathrm{k}}_2\ast {\left(\frac{\left({\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V}\right)-\sqrt{-\mathit{\hspace{0.17em}}4\mathrm{m}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}^2\mathrm{m}{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{(-{\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}-\mathrm{V}-{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V})}^2}}{2{\mathrm{k}}_{\mathrm{L}}\mathrm{m}}\right)}^2\ast \mathrm{t}}{1+\mathit{\hspace{0.17em}}{\mathrm{k}}_2\ast {\left(\frac{\left({\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V}\right)-\sqrt{-\mathit{\hspace{0.17em}}4\mathrm{m}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}^2\mathrm{m}{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{(-{\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}-\mathrm{V}-{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V})}^2}}{2{\mathrm{k}}_{\mathrm{L}}\mathrm{m}}\right)}^{\mathit{\hspace{0.17em}}}\ast \mathrm{t}}\right)$
3.	(Equilibrium Adsorption Solution Equation-Freundlich) ${\mathbf{q}}_{\mathit{e}}=\frac{\mathit{\hspace{0.17em}}1\mathit{\hspace{0.17em}}}{3}{\mathrm{k}}_{\mathrm{f}}\left(\sqrt[3]{\frac{27{\mathrm{C}}_{\mathrm{o}}\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^2}}}{2}}-\frac{3\sqrt[3]{2}{\mathrm{k}}_{\mathrm{f}}\mathrm{m}}{\sqrt[3]{\left(27{\mathrm{C}}_{\mathrm{o}}+\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^3}}\right)}\mathrm{V}}\right)$	${\mathrm{q}}_{\mathrm{t}}=\left(\frac{\mathit{\hspace{0.17em}}1\mathit{\hspace{0.17em}}}{3}{\mathrm{k}}_{\mathrm{f}}\left(\sqrt[3]{\frac{27{\mathrm{C}}_{\mathrm{o}}\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^2}}}{2}}-\frac{3\sqrt[3]{2}{\mathrm{k}}_{\mathrm{f}}\mathrm{m}}{\sqrt[3]{\left(27{\mathrm{C}}_{\mathrm{o}}+\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^3}}\right)}\mathrm{V}}\right)\right)(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}})$
4.		${\mathrm{q}}_{\mathrm{t}}=\left(\frac{{\mathrm{k}}_2\ast {\left(\frac{\mathit{\hspace{0.17em}}1\mathit{\hspace{0.17em}}}{3}{\mathrm{k}}_{\mathrm{f}}\left(\sqrt[3]{\frac{27{\mathrm{C}}_{\mathrm{o}}\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^2}}}{2}}-\frac{3\sqrt[3]{2}{\mathrm{k}}_{\mathrm{f}}\mathrm{m}}{\sqrt[3]{\left(27{\mathrm{C}}_{\mathrm{o}}+\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^3}}\right)}\mathrm{V}}\right)\mathit{\hspace{0.17em}}\right)}^2\mathrm{t}}{1+\mathit{\hspace{0.17em}}{\mathrm{k}}_2\ast \left(\frac{\mathit{\hspace{0.17em}}1\mathit{\hspace{0.17em}}}{3}{\mathrm{k}}_{\mathrm{f}}\left(\sqrt[3]{\frac{27{\mathrm{C}}_{\mathrm{o}}\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^2}}}{2}}-\frac{3\sqrt[3]{2}{\mathrm{k}}_{\mathrm{f}}\mathrm{m}}{\sqrt[3]{\left(27{\mathrm{C}}_{\mathrm{o}}+\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^3}}\right)}\mathrm{V}}\right)\right)\mathrm{t}}\right)$
5.	(Equilibrium Adsorption Solution Equation-Langmuir-Freundlich) ${\mathbf{q}}_{\mathbf{e}}=\left(\frac{{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}-\sqrt{-4{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}^2\mathrm{m}{{\mathrm{n}}_{\mathrm{L}}}^2{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{\left({\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}\right)}^2}}{2{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}}\right)$	${\mathrm{q}}_{\mathrm{t}}=\left(\frac{{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}-\sqrt{-4{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}^2\mathrm{m}{{\mathrm{n}}_{\mathrm{L}}}^2{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{\left({\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}\right)}^2}}{2{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}}\right)\left(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}\right)$
		$\begin{array}{rl}& {\mathrm{q}}_{\mathrm{t}}=\left(\frac{{\mathrm{k}}_2{\left(\frac{{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}-\sqrt{-4{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}^2\mathrm{m}{{\mathrm{n}}_{\mathrm{L}}}^2{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{\left({\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}\right)}^2}}{2{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}}\right)}^2\mathrm{t}}{1+\mathit{\hspace{0.17em}}{\mathrm{k}}_2\left(\frac{{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}-\sqrt{-4{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}^2\mathrm{m}{{\mathrm{n}}_{\mathrm{L}}}^2{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{\left({\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}\right)}^2}}{2{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}}\right)\mathrm{t}}\right)\end{array}$
6.	(Equilibrium Adsorption Solution Equation-Henry’s form) $q_e=\frac{K_a\ast C_o}{1+K_a\ast \frac{m}{V}}$	${\mathrm{q}}_{\mathrm{t}}=\frac{{\mathrm{K}}_{\mathrm{a}}{\mathrm{C}}_{\mathrm{o}}}{1+\mathit{\hspace{0.17em}}{\mathrm{K}}_{\mathrm{a}}\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}}\left(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}\right)$
		${\mathrm{q}}_{\mathrm{t}}=\left(\frac{{\mathrm{k}}_2{\left(\frac{{\mathrm{K}}_{\mathrm{a}}{\mathrm{C}}_{\mathrm{o}}}{1+\mathit{\hspace{0.17em}}{\mathrm{K}}_{\mathrm{a}}\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}}\right)}^2\mathrm{t}}{1+\mathit{\hspace{0.17em}}{\mathrm{k}}_2\left(\frac{{\mathrm{K}}_{\mathrm{a}}{\mathrm{C}}_{\mathrm{o}}}{1+\mathit{\hspace{0.17em}}{\mathrm{K}}_{\mathrm{a}}\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}}\right)\mathrm{t}}\right)$

Table 4. List of equations for C_e the equilibrium concentration (EC equations derived from EASE).

Sl.No	Isotherm used	Equilibrium Concentration Equation
1.	Langmuir Isotherm	${\mathrm{C}}_{\mathrm{e}}={\mathrm{C}}_{\mathrm{o}}-\left(\frac{\left({\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V}\right)-\sqrt{-4\mathrm{m}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}^2\mathrm{m}{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{(-{\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}-\mathrm{V}-{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V})}^2}}{2{\mathrm{k}}_{\mathrm{L}}\mathrm{m}}\right)\frac{\mathrm{m}}{\mathrm{V}}$
2.	Freundlich Isotherm	${\mathrm{C}}_{\mathrm{e}}={\mathrm{C}}_{\mathrm{o}}-\frac{1}{3}{\mathrm{k}}_{\mathrm{f}}\left(\sqrt[3]{\frac{27{\mathrm{C}}_{\mathrm{o}}\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^2}}}{2}}-\frac{3\sqrt[3]{2}{\mathrm{k}}_{\mathrm{f}}\mathrm{m}}{\sqrt[3]{\left(27{\mathrm{C}}_{\mathrm{o}}+\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^3}}\right)}\mathrm{V}}\right)\frac{\mathrm{m}}{\mathrm{V}}$
3.	Langmuir -Freundlich Isotherm	${\mathrm{C}}_{\mathrm{e}}={\mathrm{C}}_{\mathrm{o}}-\left(\frac{{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}-\sqrt{-4{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}^2\mathrm{m}{{\mathrm{n}}_{\mathrm{L}}}^2{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{\left({\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}\right)}^2}}{2{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}}\right)\frac{\mathrm{m}}{\mathrm{V}}$
4.	Henry’s isotherm (Linear model)	${\mathrm{C}}_{\mathrm{e}}={\mathrm{C}}_{\mathrm{o}}-\left(\frac{{\mathrm{K}}_{\mathrm{a}}.{\mathrm{C}}_{\mathrm{o}}}{1+{\mathrm{K}}_{\mathrm{a}}.\frac{\mathrm{m}}{\mathrm{V}}}\right)\frac{\mathrm{m}}{\mathrm{V}}$

Table 5. List of equations for C_t (concentration at desired time) derived from CAKE models.

Sl.No	Isotherm used	Concentration Equation at desired time
1.	Langmuir Isotherm	${\mathrm{C}}_{\mathrm{t}}={\mathrm{C}}_{\mathrm{o}}-\frac{\mathrm{m}}{\mathrm{V}}\left(\frac{\left({\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V}\right)\pm \sqrt{-4\mathrm{m}{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}^2\mathrm{m}{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{(-{\mathrm{k}}_{\mathrm{L}}\mathrm{m}{\mathrm{q}}_{\mathrm{m}+}-\mathrm{V}-{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{L}}\mathrm{V})}^2}}{2{\mathrm{k}}_{\mathrm{L}}\mathrm{m}}\right)\left(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}\right)$
		${\mathrm{C}}_{\mathrm{t}}={\mathrm{C}}_{\mathrm{o}}-\frac{1}{3}{\mathrm{k}}_{\mathrm{f}}\frac{\mathrm{m}}{\mathrm{V}}\left(\sqrt[3]{\frac{27{\mathrm{C}}_{\mathrm{o}}\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^2}}}{2}}-\frac{3^3\sqrt[]{2}{\mathrm{k}}_{\mathrm{f}}\mathrm{m}}{\sqrt[3]{\left(27{\mathrm{C}}_{\mathrm{o}}+\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^3}}\right)}\mathrm{V}}\right)\left(1-{\mathrm{e}}^{-\mathrm{kt}}\right)$
2.	Freundlich Isotherm	${\mathrm{C}}_{\mathrm{t}}={\mathrm{C}}_{\mathrm{o}}-\frac{1}{3}{\mathrm{k}}_{\mathrm{f}}\frac{\mathrm{m}}{\mathrm{V}}\left(\sqrt[3]{\frac{27{\mathrm{C}}_{\mathrm{o}}\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^2}}}{2}}-\frac{3^3\sqrt[]{2}{\mathrm{k}}_{\mathrm{f}}\mathrm{m}}{\sqrt[3]{\left(27{\mathrm{C}}_{\mathrm{o}}+\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^3}}\right)}\mathrm{V}}\right)\left(1-{\mathrm{e}}^{-\mathrm{kt}}\right)$
		${\mathrm{C}}_{\mathrm{t}}={\mathrm{C}}_{\mathrm{o}}-\frac{\mathrm{m}}{\mathrm{V}}\left(\frac{{\mathrm{k}}_2{\left\{\frac{1}{3}{\mathrm{k}}_{\mathrm{f}}\left(\sqrt[3]{\frac{27{\mathrm{C}}_{\mathrm{o}}\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^2}}}{2}}-\frac{3^3\sqrt[]{2}{\mathrm{k}}_{\mathrm{f}}\mathrm{m}}{\sqrt[3]{\left(27{\mathrm{C}}_{\mathrm{o}}+\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^3}}\right)}\mathrm{V}}\right)\right\}}^2\mathrm{t}}{1+{\mathrm{k}}_2\frac{1}{3}{\mathrm{k}}_{\mathrm{f}}\left(\sqrt[3]{\frac{27{\mathrm{C}}_{\mathrm{o}}\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^2}}}{2}}-\frac{3^3\sqrt[]{2}{\mathrm{k}}_{\mathrm{f}}\mathrm{m}}{\sqrt[3]{\left(27{\mathrm{C}}_{\mathrm{o}}+\sqrt{729{\mathrm{C}}_{\mathrm{o}}^2+\frac{108{\mathrm{k}}_{\mathrm{f}}^3{\mathrm{m}}^3}{{\mathrm{V}}^3}}\right)}\mathrm{V}}\right)\mathrm{t}}\right)$
3.	Langmuir -Freundlich Isotherm	${\mathrm{C}}_{\mathrm{t}}={\mathrm{C}}_{\mathrm{o}}-\frac{\mathrm{m}}{\mathrm{V}}\left(\frac{{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}-\sqrt{-4{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}^2\mathrm{m}{{\mathrm{n}}_{\mathrm{L}}}^2{\mathrm{q}}_{\mathrm{m}}\mathrm{V}+{\left({\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}+\mathrm{V}+{\mathrm{C}}_{\mathrm{o}}{\mathrm{k}}_{\mathrm{g}}{\mathrm{n}}_{\mathrm{L}}\mathrm{V}\right)}^2}}{2{\mathrm{k}}_{\mathrm{g}}\mathrm{m}{\mathrm{n}}_{\mathrm{L}}}\right)\left(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}\right)$
		$\frac{{\mathrm{K}}_{\mathrm{a}}{\mathrm{C}}_{\mathrm{o}}\frac{\mathrm{m}}{\mathrm{V}}}{1+\mathit{\hspace{0.17em}}{\mathrm{K}}_{\mathrm{a}}\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}}\left(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}\right)$
4.	Henry’s Isotherm	${\mathrm{C}}_{\mathrm{t}}\mathit{\hspace{0.17em}}={\mathrm{C}}_{\mathrm{o}}-\mathit{\hspace{0.17em}}\frac{{\mathrm{K}}_{\mathrm{a}}{\mathrm{C}}_{\mathrm{o}}\frac{\mathrm{m}}{\mathrm{V}}}{1+\mathit{\hspace{0.17em}}{\mathrm{K}}_{\mathrm{a}}\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}}\left(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}\right)$
		${\mathrm{C}}_{\mathrm{t}}={\mathrm{C}}_{\mathrm{o}}-\frac{\mathrm{m}}{\mathrm{V}}\left(\frac{{\mathrm{k}}_2{\left(\frac{{\mathrm{K}}_{\mathrm{a}}{\mathrm{C}}_{\mathrm{o}}}{1+\mathit{\hspace{0.17em}}{\mathrm{K}}_{\mathrm{a}}\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}}\right)}^2\mathrm{t}}{1+\mathit{\hspace{0.17em}}{\mathrm{k}}_2\left(\frac{{\mathrm{K}}_{\mathrm{a}}{\mathrm{C}}_{\mathrm{o}}}{1+\mathit{\hspace{0.17em}}{\mathrm{K}}_{\mathrm{a}}\frac{\mathrm{m}}{\mathrm{V}}\mathit{\hspace{0.17em}}}\right)\mathrm{t}}\right)$

2.6. Error functions

Different error functions, that is, the sum of squares error (SSE),^[48] mean absolute percentage error (MAPE) and normalized average percentage error (NAPE) were used in the present investigation for evaluating the models. The purpose of using these error functions was to determine the best fit between experimental (q_exp) and model (q_mod) data (in the Freundlich-PFO/PSO model). The lower the value of SSE, MAPE and NAPE,^[49] the higher the accuracy of the model. The expressions for the error functions are given below,

(47) $\mathrm{SSE}=\sum _{\mathrm{i}=1}^{\mathrm{N}}{\left({\mathrm{q}}_{\exp }-{\mathrm{q}}_{\mathrm{mod}}\right)}^2$(47)

(48) $\mathrm{MAPE}=\frac{1}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}\left(\frac{{\mathrm{q}}_{\exp }-{\mathrm{q}}_{\mathrm{mod}}}{{\mathrm{q}}_{\exp }}\right)$(48)

(49) $\mathrm{NAPE}=\frac{1}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}\left(\frac{{\mathrm{q}}_{\exp }-{\mathrm{q}}_{\mathrm{mod}}}{{\mathrm{q}}_{\mathrm{m}}}\right)$(49)

3. Materials and methods

3.1. Preparation of adsorbate

The experimental study used analytical grade 2,4-dichlorphenol (C₆H₄OCl₂), orthophosphoric acid (H₃PO₄), hydrochloric acid (HCl) and sodium bicarbonate (NaOH). To prepare 1000 mg/L of solution, 1g of 2,4-dichlorophenol (2,4-DCP) was added to 1000 ml of distilled water and stirred well. The desired concentration was further prepared by diluting the stock solution. The concentration of the adsorbate solution was measured in a UV-Vis spectrophotometer at a wavelength (λ) of 284 nm.^[4]

3.2. Preparation of adsorbent

An agricultural waste, Cassia fistula pod (CFP) shell was used as a precursor of activated carbon. The shell was separated, washed, and dried under the sun. Further, the dried shells were pulverized and sieved through 425 microns. The fine particles were then collected and mixed with H₃PO₄ in equal ratios (w/v) and allowed overnight. The impregnated material was carbonized for 60 minutes in muffle furnace at 500°C in the presence of nitrogen supply (inert atmosphere). The carbonized material/activated carbon was cooled, washed using distilled water, dried and further used for adsorption studies labeled as Cassia fistula pod activated carbon (CFPAC).

3.3. Optimization of operational parameters

Batch experiments were performed to find the best conditions. The parameters such as initial concentration (25-500mg/L), dosage (0.2g/L-4g/L), temperature (20–50°C) and agitation speed (50-200rpm) were considered for the optimization. The pH (2–12) of the solution was adjusted using standard acid and base solution of HCl and NaOH (0.1N-1N).

Initially, the experiments were conducted in a 250 mL standard conical flask with a stopper. A working volume (V) of 50 ml for varied initial concentrations (C_o) as mentioned above, an adsorbent dosage(m) of 50 mg, working speed of 150rpm, and temperature at 30°C in a thermostat till equilibrium was considered. Once equilibrium was attained the samples were collected in clean, dry vials. Further, samples are centrifuged to reduce interference of particles in the sample and analyzed using a UV-visible spectrophotometer (Shimadzu, Japan).

3.4. Isotherm study and model fit

The isotherm studies were performed in batch mode at optimized conditions for different initial concentrations ranging from 25-500mg/L till equilibrium time (2 hours). The experimental data were further fitted to isotherm models such as Langmuir isotherm,^[10] Freundlich isotherm,^[50] and LF-model^[51] as listed in Table 1. Furthermore, the error functions are also calculated to identify the best isotherm model.

3.5. Kinetic study and model fit

In kinetic studies, samples were collected and analyzed at different intervals of time until equilibrium is achieved.^[18] The data obtained from kinetic studies were modeled using linear and non-linear forms of equations.^[9] The details of the kinetic models used for the study are listed in Table 2.

3.6. Non-linear optimization

Microsoft Excel solver was used for non-linear optimization by minimizing the errors. GRG non-linear solution method was used to minimize the sum of squares of errors.

4. Experimental results

4.1. Equilibrium studies of 2,4-DCP adsorption

The experimental data of adsorption of 2,4-DCP was fitted to different isotherm models to obtain different isotherm parameters (q_m, k_L, k_f, 1/n, k_g, n_L). The Fig. 2 below shows plots of different isotherms considered for the study.

Figure 2: Plot of various isotherms for the adsorption of 2,4-DCP.

[Co= 25-500mg/L, V=50ml, m= 30mg, pH=2, T=20°C, Speed = 120rpm]

The graphs depict that, Langmuir-Freundlich model is better fitted with experimental data. The value of n_L approaching unity which means monolayer adsorption is dominating the adsorption of 2,4-DCP.^[52] Fig. 3 shows plots of comparison of model values with experimental data. The details of error values of different EASE models are listed Table 6.

Figure 3: Comparison of experimental results with new EASE models.

[Co= 25-500mg/L, V=50ml, m= 30mg, pH=2, T=20°C, Speed = 120rpm]

Table 6. Comparison of error values of all EASE models.

Sl.No	Model fit values	EASE Model
		EASE-Langmuir	EASE-Freundlich	EASE-Langmuir Freundlich
1.	R^2	0.9954	0.9702	0.9954
2.	MAPE	3.4001	4.1971	3.2803
3.	NAPE	1.9031	3.2843	1.8859

The EASE equation could provide a good fit as observed in Table 6. The order of fit was found to be EASE-LF>EASE-L>EASE-F which was exclusively based on all the error functions listed in subsection 2.3.

4.2. Application of CAKE model for 2,4-DCP adsorption

As mentioned in the methodology, data pertaining to isotherms and kinetics experiments were obtained for adsorption of 2,4-DCP on Cassia fistula (CF) pods derived activated carbon (CFPAC). The model parameters for the present investigation are listed in Table 7. The obtained experimental plots are compared with the modeled values. The comparison was based on values of R² and mean absolute percentage error (MAPE).

Table 7. Summary of model parameters (Isotherm & kinetic) used in the study.

Sl. No	Model	Parameter	Values/constants	Unit
1.	Langmuir Isotherm	qm	374.84	mg/g
		kL	0.0454	L/mg
2.	Freundlich isotherm	kf	68.014	L/mg
		1/n	0.304	constant
3.	Langmuir-Freundlich Isotherm	qm	441.52	mg/g
		kg	0.028	L/mg
		nL	0.699	constant (heterogeneity index)
4..	Pseudo-First-Order Kinetics	k1	0.2739	L/min
5.	Pseudo-Second-Order Kinetics	k2	.0058	g/mg min

4.2.1. Langmuir CAKE model (PFO and PSO)

Figure 4 shows the adsorption kinetics of 2,4-DCP onto CFPAC at different initial solute concentrations. The graph also incorporates a comparison of experimental data with CAKE Langmuir models (PFO and PSO) values. From Fig. 4a, it can be visualized that, the equilibrium time as obtained from the new kinetic Langmuir model i.e., Eq. 12 for lower concentrations say 25ppm and 50ppm was approximately 20mins, which was similar to the experimental values. The experimental values were forecasted well by the CAKE equations, irrespective of the concentration chosen. Further, mean absolute percentage error (MAPE) was found to vary from 5–7%. Similarly, from Fig. 4b, the predicted equilibrium time was also 20mins as found after applying Eq. 15. In this case, the value of MAPE varied from 6–10%. Based on the error %, it could be concluded that the model fit was better for Langmuir-PFO compared to Langmuir-PSO. The dotted portion of the graph indicates experimental data and continuous lines indicate modeled data.

Figure 4: Illustration of experimental and predicted values a) Langmuir-PFO b) Langmuir-PSO model.

4.2.2. Freundlich CAKE model (PFO and PSO)

Figure 5a,(b) demonstrates the plot of adsorption kinetics. In addition, the graph incorporates a comparison of experimental (PFO and PSO) with the new model i.e., the Freundlich CAKE model. It can be visualized that the equilibrium time as obtained from the new kinetic model i.e., Eq. 27 for all the concentrations was approximately 20mins which was in congruence with the experimental values. The modified Freundlich-PFO is given below. However, it was possible to fit experimental data with the least MAPE ranging 4–7% from the new model using different n and k_f values. Further, in the case of Freundlich-PSO (Eq. 29), the equilibrium time was approximately 30mins. The average % error varied from 5–6.5% indicating a good fit for lower concentrations. The R²_PSO was found to be greater as compared to R²_PFO. Thus, the model fit was better for Freundlich-PSO. The dotted portion of the graph indicates experimental data and continuous lines indicate predicted data.

Figure 5: Illustration of experimental and predicted values a) LF-PFO b) LF-PSO

4.2.3. Langmuir-freundlich CAKE model (PFO and PSO)

Figure 6 denotes the chart of experimental and predicted values using the new model i.e., LF-kinetic model. In the case of LF-PFO Fig. 6a, it can be observed that the equilibrium time achieved by using i.e., Eq. 34 was nearly 20mins for all the concentrations. An early equilibrium in 15min was observed in the computed time similar to the experimental values. The interesting part here was it was possible to get the best fit with the least % average error in the range of 6–9% for the new model for the same constant (n_L and k_g) and the lowest NAPE value ranging from 2.99% to 7.4%. Further, in Fig. 6b in the case of LF-PSO using Eq. 36, an equilibrium time of 25mins was observed. The deviation of 5 mins w.r.t experimental data was observed.

Figure 6: Illustration of experimental and predicted values a) LF-PFO b) LF-PSO.

The NAPE of 4.102 and 7.39% was obtained with PFO and PSO, respectively. Thus, indicating that the LF-PSO model fit was better for FL-PSO. The error values for different CAKE models are listed in Table 8. The dotted portion of the graph indicates experimental data and continuous lines indicate predicted data.

Table 8. Error values for different models.

Sl.No	Model fit values	CAKE Model
		CAKE-Langmuir		CAKE-Freundlich		CAKE-Langmuir Freundlich
		PFO	PSO	PFO	PSO	PFO	PSO
1.	R^2	0.8859	0.8316	0.7496	0.8268	0.8748	0.9001
2.	MAPE	6.395	10.812	9.276	8.895	6.3766	9.2564
3.	NAPE	4.452	15.253	7.252	6.660	4.1021	7.393

5. Discussion

Our literature survey showed that both the Langmuir model and the Freundlich model are the extensively employed isotherm models in more than 95% cases.^[18,39] Moreover, for simulating adsorption kinetics, the pseudo-first order and pseudo-second order models are frequently employed. However, there are limited equations combining the isotherms and kinetics.

A new kinetic-equilibrium modeling approach has been presented in this research article, wherein, the adsorption isotherm model and kinetic model are linked by following simple mathematical steps (see Subsection 2.2). This linkage was done for Langmuir model,^[36,53] Freundlich and LF-models. The advantage of the developed CAKE and EASE models is that a single set of parameters can be used for both types of models, isotherms and kinetics unlike earlier investigations.^[42,45,46]

Earlier investigations have proposed an integrated kinetic Langmuir (IKL) equation to describe the typical kinetic behavior of the adsorption activity.^[38] Further, in another research, the conventional rate equations were improvised by introducing a new parameter (θ) called fractional surface coverage,^[43] which is difficult to measure directly. Also, the preceding investigations lack ease of usability (e.g. Langmuir rate equation’s solution^[36]). In some cases, it is difficult to determine fraction surface coverage at a given time. The absence of a clear single equation has made it challenging to use the adsorption models in chemical industries. Most chemical industries mainly operate in the kinetic regime. In such cases intermediate parameters such as q_e are difficult to determine.

For example, the Pseudo-second order equation is given by,

${\mathrm{q}}_{\mathrm{t}}={\mathrm{q}}_{\mathrm{e}}\left(1-{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}\right)$

But q_e is not known either; hence the determination of q_t becomes hard.

${\mathrm{q}}_{\mathrm{e}}=\frac{{\mathrm{q}}_{\mathrm{m}}{\mathrm{k}}_{\mathrm{L}}{\mathrm{C}}_{\mathrm{e}}}{1+{\mathrm{k}}_{\mathrm{L}}{\mathrm{C}}_{\mathrm{e}}}$

“q_e” can be determined using the isotherm equation as above. However, again the problem is that C_e is not known. In such cases, the value of q_e cannot be found since C_e is not available readily and has to be measured. This is similar to chicken and egg? which came first puzzle. To disentangle this puzzle, we have distinctly solved for q_e in terms of constants such as q_m, V, k_t, and k_L, which are constants which need to be determined only once. Thus, the measurement of C_e for every C_o is eliminated. Thus, the listed equations (as mentioned in Table 5) have been formulated to enhance their practical applicability. By presenting these equations in an independant, clear and usable formr, we aim to facilitate their seamless utilization in various chemical processes and applications aspects.^[36,41,44]

Also in this article, an experimental study on the adsorption of 2,4-DCP on CFPAC was conducted and utilized for validation of the derived equations. In the kinetic investigation, the PSO model gave a better fit than the PFO model. When considering CAKE models, the Langmuir-PFO and Langmuir-PSO models gave the best overall fits to experimental data (Table 8). This suggests that the behavior was single-layer homogenous adsorption. The model’s superiority was in the order, of LF-PSO>Langmuir-PFO> LF-PFO > Langmuir-PSO> Freundlich-PSO > Freundlich-PFO. The better fit of LF-PSO was credited to the fact that the LF equation represents an improvement over the conventional Langmuir and Freundlich models. This was due to the presence of additional heterogeneity term in LF model. Freundlich-PFO isotherm gave highest errors, which suggests that isotherm did not follow the Freundlich model, and a mono-layer adsorption was likely.

A linear isotherm CAKE equation was also developed. This is particularly useful at low concentrations. Concentrations of contaminants in groundwater usually are low levels, where use of linear isotherm is warranted. This form of CAKE and EASE can be used for problems in several other domains where pollutants are at ppb levels.

The combined adsorption Kinetics and Equilibrium (CAKE) approach showed the clear failure of Freundlich models, which is not visible in Equilibrium Freundlich isotherms alone. An early equilibrium was observed for this experiemntal system in the models, and the equilibirum time was about 20 minutes. Thus, the CAKE models can give a better clarity on the proper choice of adsorption isotherms, which are suited for kinetic and equilibrium scenarios. The MAPE and NAPE error analysis was performed to determine the goodness of fit of the CAKE modeling framework (Table 8). The CAKE approach would be especially useful for industries and field-scale applications where kinetics processes predominate. This is because CAKE gives a single equation which can be incorporated into software like MS Excel to easily give predictions at desired times. The use of a single equation also makes it usable in various other contaminant transport softwares as well.

6. Conclusions

To summarize, this study has introduced a novel analytical modeling framework designated as Combined Adsorption Kinetic and Equilibrium (CAKE) and Equilibrium Adsorption Solution Equation (EASE). The equations were formulated through the utilization of existing adsorption isotherm, mass balance and kinetic rate equations, specifically PFO/PSO. The incorporation of various isotherms, including Langmuir, Freundlich, and LF into the kinetic models, enhances the usefulness of the modeling framework. The obtained models were used to estimate equilibrium concentration (C_t), equilibirum time (t_e) and adsorption capacity (q_t) at any time. The CAKE models estimated t_e= 20mins for the concentration 25ppm and 50ppm, which was in congruence with our experimental data. Notably, the LF-PFO equation emerges as the most effective model in portraying the adsorption kinetics of 2,4-DCP onto CFPAC, based on higher values of R². The CAKE models also gave a better clarity on the proper choice of adsorption isotherms and suggested that the Freundlich model did not suit the data relatively well. The exact selection of a suitable model proves pivotal for accurately estimating both the equilibrium time (t_e) and adsorption capacity (q_max). Ranking the order of fit, LF-PSO surpasses other models, followed by Langmuir-PFO, LF-PFO, Langmuir-PSO, Freundlich-PSO, and Freundlich-PFO. The superiority of LF-PSO was attributed to the fact that the LF equation represents an improvement over the conventional Langmuir and Freundlich models. Also, error analysis showed that the CAKE and EASE models demonstrated lower errors, as quantified by mean absolute percentage error (MAPE) of 8.5%, Normalized absolute percentage error (NAPE) of 7.5% and R² of 0.85.

The development of these computational models carries substantial practical implications for the chemical industry. For instance, these models can help optimize operations by reducing the time, chemicals and energy , as the predictions can be made for kinetic adsorption processes for q_t and C_t (Table 3, Table 5). Also, the mass of adsorbent required can be minimized by predicting adsorption capacity q_t (Table 3) ,thus saving the excess usage of adsorbent. This not only improves efficiency but also has the potential to save costs and enhance the overall usability of adsorption processes. Ultimately, this work serves as a valuable contribution to the field of adsorption, offering more accurate and efficient methods for modeling and predicting adsorption behavior.

For the future prospects, researchers can adopt the developed CAKE and EASE models to simulate adsorption of pollutants in field scale wastewater treatment systems. Furthermore, one can formulate combined equations by considering other isotherm models and kinetic equations. Also, artificial intelligence and machine learning systems can be developed with the experimental data and predictions from CAKE equations.

Nomenclature

${\mathrm{q}}_{\mathrm{e}}$	=	Equilibrium adsorption capacity of adsorbents, (mg/g)
${\mathrm{C}}_{\mathrm{e}}$	=	Equilibrium concentration of adsorbates, (mg/L)
${\mathrm{C}}_0$	=	Initial concentration of adsorbates, (mg/L)
${\mathrm{K}}_{{\mathrm{H}}_{\mathrm{e}}}$	=	Henry constant (L/g)
qm	=	Maximum adsorption capacity of adsorbent, (mg/L)
kL	=	Langmuir constant, (L/mg)
${\mathrm{k}}_{\mathrm{F}}$	=	Freundlich constant, (L/mg)
n	=	Magnitude of adsorption driving force
AT	=	Heat of adsorption (kJ/mol)
bT	=	Equilibrium binding energy (J/mol)
Ks,	=	Sips equilibrium constant (L/g)
βs	=	Sips model exponent (unitless)
qmS	=	Sips maximum adsorption capacity (mg/g)
Cs	=	Saturated solute concentration (mg/L)
z	=	Factor of heterogeneity
B, C	=	Interaction with the surface pertaining to BET constant
KT	=	Toth Equilibrium constant, (L/mg)
aT	=	Affinity co-efficient dependent on temperature (L/mg)
t	=	Toth constant (Heterogeneity of adsorption)
kRP	=	Redlich peterson constant (L/g)
∝RP	=	Redlich Peterson constants (L/g)β
$\mathit{\beta }$	=	Redlich Peterson exponential constant
bk	=	Khan isotherm model constant (unitless)
ak	=	Khan isotherm exponent (unitless)
qs	=	Khan isotherm maximum adsorption capacity (mg/g)
ka	=	Affinity constant (unitless)
pH	=	pH of solution
n1	=	Index of heterogeneity (unitless)
kg	=	Affinity co-efficient for adsorption, (L/mg)
qt	=	Adsorption capacity at time “t”
qe	=	Adsorption capacity at equilibrium (mg/g)
k1	=	Pseudo first-order rate constant (min−1)
k2	=	Pseudo-second-order rate constant (g/mg/min)
kAV	=	Avrami Kinetic constant (min−1)
nAV	=	Avrami exponent (unitless)
kpi	=	Diffusion constant (mg/g min)
Ci	=	Thickness of boundary layer (mg/g)
a	=	Initial adsorption rate (mg/g/min−1)
βt	=	Desorption constant (g/mg)
qexp	=	Experimental adsorption capacity (mg/g)
qmod	=	Model adsorption capacity (mg/g)
V	=	Volume of solution (mg/g)
m	=	Dosage of adsorbent (mg)
r	=	Rate coefficient (min−1)
∝	=	nth order rate constant (min−1)
q∞	=	Equilibrium adsorption capacity at infinite time (mg/g)
ns	=	Active sites occupied by an adsorbate

Authors contributions

Praveengouda Patil: Conceptualization, methodology, Investigation, Data curation, Validation, Formal analysis, modeling.

Bhojaraja Mohan: Data collection, survey, methodology, and investigation.

Gautham Jeppu: Project administration, Conceptualization, Validation, Resources, Supervision, review & editing.

Chikmagalur Raju Girish: Project administration, Validation, Resources, Supervision, review & editing.

Availability of data and materials

All the data are available and can be provided to the editorial team on request.

Consent to Publish

We declare that there is no conflict of interest associated with this manuscript. As the corresponding author, we confirm that the manuscript has been proofread and approved for submission by all named authors.

Mr. Praveengouda Patil

Mr. Bhojaraja Mohan

Dr. Gautham Jeppu

Dr. Chikmagalur Raju Girish (corresponding author)

Novelty Statement

1. A new modeling framework connecting kinetic and equilibrium equations called the Combined Adsorption Kinetic and Equilibrium (CAKE) approach for adsorption kinetics and adsorption equilibrium is developed.

2. Various forms of CAKE equations are developed by considering different combinations of kinetic and equilibrium models, such as Langmuir, Freundlich, Langmuir-Freundlich and Linear isotherms.

3. Isotherm and Kinetic constants are incorporated into the above-developed forms and the adsorption capacity (q_t), concentration (C_t) at a given time were predicted for an experimental dataset of the 2,4-Dichlorophenol adsorption system, thus validating the models.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The current work did not receive any form of funding from the agencies.