Modelos de isotermas de adsorção classificação, significado físico, aplicação e método de solução

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Chemosphere 258 (2020) 127279
Contents lists avai
Chemosphere
journal homepage: www.elsevier .com/locate/chemosphere
Review
Adsorption isotherm models: Classification, physical meaning,
application and solving method
Jianlong Wang a, b, *, Xuan Guo a
a Laboratory of Environmental Technology, INET, Tsinghua University, Beijing, 100084, PR China
b Beijing Key Laboratory of Radioactive Waste Treatment, Tsinghua University, Beijing, 100084, PR China
h i g h l i g h t s
* Corresponding author. Full post address: Energy S
hua University, Beijing, 100084, PR China.
E-mail address: wangjl@tsinghua.edu.cn (J. Wang)
https://doi.org/10.1016/j.chemosphere.2020.127279
0045-6535/© 2020 Elsevier Ltd. All rights reserved.
g r a p h i c a l a b s t r a c t
� The derivation and physical meaning
of 13 adsorption isotherms were
analyzed.
� The application of adsorption
isotherm models were evaluated and
summarized.
� The model validity evaluation equa-
tions were discussed based on
literature.
� A user interface for solving isotherms
was developed based on Excel
software.
a r t i c l e i n f o
Article history:
Received 29 April 2020
Received in revised form
28 May 2020
Accepted 30 May 2020
Available online 10 June 2020
Handling Editor: Y Yeomin Yoon
Keywords:
Adsorption
Isotherm models
Physical meaning
Solving method
User interface
a b s t r a c t
Adsorption is widely applied separation process, especially in environmental remediation, due to its low
cost and high efficiency. Adsorption isotherm models can provide mechanism information of the
adsorption process, which is important for the design of adsorption system. However, the classification,
physical meaning, application and solving method of the isotherms have not been systematical analyzed
and summarized. In this paper, the adsorption isotherms were classified into adsorption empirical iso-
therms, isotherms based on Polanyi’s theory, chemical adsorption isotherms, physical adsorption iso-
therms, and the ion exchange model. The derivation and physical meaning of the isotherm models were
discussed in detail. In addition, the application of the isotherm models were analyzed and summarized
based on over 200 adsorption equilibrium data in literature. The statistical parameters for evaluating the
fitness of the models were also discussed. Finally, a user interface (UI) was developed based on Excel
software for solving the isotherm models, which was provided in supplemental material and can be
easily used to model the adsorption equilibrium data. This paper will provide theoretical basis and
guiding methodology for the selection and use of the adsorption isotherms.
© 2020 Elsevier Ltd. All rights reserved.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2. Classification and physical meanings of the isotherm models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
cience Building, INET, Tsing-
.
mailto:wangjl@tsinghua.edu.cn
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J. Wang, X. Guo / Chemosphere 258 (2020) 1272792
2.1. Adsorption empirical isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.1. Linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.2. Freundlich isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.3. Redlichepeterson (ReP) isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.4. Sips isotherm model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.5. Toth isotherm model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.6. Temkin isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2. Adsorption models based on Polanyi’s potential theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1. Dubinin-Radushkevich (D-R) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2. Dubinin-Astakhov (D-A) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3. Chemical adsorption models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.1. Langmuir model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.2. Volmer isotherm model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4. Physical adsorption models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4.1. BET model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4.2. Aranovich model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5. Ion exchange isotherm model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3. Applications of the isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4. Statistical parameters for the evaluation of the isotherm models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5. Solving methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6. Concluding remarks and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Declaration of competing interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Supplementary data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1. Introduction
Adsorption process is a surface phenomenon in which adsor-
bates transfer onto adsorbents. Over the past decades, adsorption
technology has been widely applied for the water and wastewater
treatment because it is low-cost, efficient, simple, and environ-
mentally friendly. The adsorption mechanisms mainly include
chemical adsorption corresponding to the formation of chemical
bonds, physical adsorption related to the van der Waals force, and
the ion exchange (Fig.1). The knowledge of adsorptionmechanisms
is of great importance to design the adsorbents and the adsorption
systems. The adsorptionmechanisms have been investigated by the
modeling of the adsorption equilibrium data, the characterization
of adsorbent before and after adsorption, the molecular dynamics
study, and the density functional theory (DFT) calculation (Zhuang
et al., 2020a; Ahmad et al., 2019; Ghazi et al., 2018). Among these
methods, modeling of adsorption data by isotherm models is the
most convenient and widely used one. In addition, the adsorption
isotherm models can provide information of the maximum
Fig. 1. The possible adso
adsorption capacity, which is significant in the evaluation of the
performance of the adsorbents.
A variety of isotherms have been applied in adsorption systems,
such as the Langmuir model (Langmuir, 1916, 1918), linear model,
the Freundlich model (Freundlich, 1906), the Sips model (Sips,
1948), the Temkin model (Temkin and Pyzhev, 1940), and the
Brunauer, Emmett, and Teller (BET) model (Brunauer et al., 1938).
Among thesemodels, the linear, Freundlich, Sips, Temkin, and some
other models are empirical models, which lack of effective theo-
retical support. The adsorption mechanisms cannot be obtained
through these models. Thus, the derivations and physical meanings
of these models should be investigated. In addition, the isotherms
have been classified according to the number of model parameters
(Foo and Hameed, 2010; Song and Shen, 2005; Ayawei et al., 2017;
Al-Ghouti and Da’ana, 2020) or the shapes of the isotherms (S-, L-,
H-, and C-shaped) (Limousin et al., 2007). However, the classifica-
tion based on the number of parameters lacks of theoretical basis.
For example, the Langmuir and BET isotherms are two parameters
models, they represent the chemical adsorption and physical
rption mechanisms.
Fig. 2. The adsorption mechanisms revealed by the linear model.
J. Wang, X. Guo / Chemosphere 258 (2020) 127279 3
adsorption, respectively. The mechanism information cannot be
provided by the number of model parameters. The classification
based on the shapes of the models has also the limitation, because
the majority of adsorption equilibrium data of liquid-solid systems
are L-shaped. Therefore, the isotherms should be classified more
reasonable based on their physical meanings.
To the best of our knowledge, the physical meaning and the
classification of these isotherms are not thoroughly studied.
Moreover, several isotherms are used in incorrect or unsuitable
forms sometimes. For example, the adsorption potential ε of the
Dubinin-Radushkevich (D-R) model has been estimated in wrong
form of ε ¼ RT ln (1 þ1/Ce), Where: Ce (mg∙L�1) is the equilibrium
concentration; R (8.314 Jmol�1∙K�1) is the universal gas constant; T
(K) is the temperature) (Tang et al., 2018; Hu and Zhang, 2019;
Bezzina et al., 2020). The most frequently used form of the BET
model, which was reported by Foo and Hameed (2010), Staudt et al.
(2013), Petkovska (2014), Saadi et al., and so on, was proved to have
poor estimations in the model parameters in liquid-solid system
(Ebadi et al., 2009).
The majority of published papers used linear regression method
to estimate the model parameters (Foo and Hameed, 2010). The
linear regression method is simple and convenient. However, the
linearization of the adsorption models can change the independent
and dependent variables, and introduce the propagated errors. In
this sense, the estimation of the model parameters is inaccurate
and biased (Ho, 2006; Kumar and Sivanesan, 2006; Bolster and
Hornberger, 2007). The accurate calculations of model parameters
can be given by the nonlinear regression, but it is more complicated
than the linear regression method. Therefore, convenient methods
for solving the nonlinear isotherms should be developed.
The objectives of this review paper were to classify the
adsorption isotherms according to their physical meaning and to
thoroughly analyze the isotherm models. Firstly, the classification,
derivation, and physical meaning of the isotherms were analyzed;
secondly, the application of the isotherm models was summarized
based on literature; thirdly, the statistical parameters which can
evaluate the fitness were analyzed; and finally, a software was
compiled in Excel to solve the nonlinear isotherms.
2. Classification and physical meanings of the isotherm
models
Isotherm refers to the relationship between the equilibrium
adsorbate concentrations in the liquid-phase and the equilibrium
adsorption amount on the solid-phase at a certain temperature.We
can model the equilibrium adsorption data by the isotherms, and
investigate the adsorption information, such as the adsorption
mechanisms, the maximum adsorption capacity, as well as the
properties of adsorbents by the isotherms.
In this section, thirteen isotherms were classified based on their
theoretical derivation and physical meaning as empirical models,
isotherm models based on the Polanyi’s potential theory, chemical
adsorption models, physical adsorption models, and ion exchange
models, according to their physical meanings. The adsorption
empirical isotherms (such as the linear, Freundlich, Sips and Toth
models) are characterized with lacking specific physical meaning.
The isotherm models based on the Polanyi’s potential theory (the
D-R and Dubinin-Astakhov (D-A) models) are semi-empirical
models, which can be used in the modeling of the adsorption of
porous materials. The chemical, physical, and ion exchange models
are theoretical models with rigorous deduction and specific phys-
ical meanings. The chemical isotherms describe the monolayer
adsorption process, the physical isotherms represent the multi-
layer adsorption, while the ion exchange isotherms can model
the ion exchange adsorption process. The derivations andmeanings
of the isotherm models will be analyzed in this section.
2.1. Adsorption empirical isotherms
2.1.1. Linear model
The linear model (Henry’s law) has the following form:
qe ¼KCe (1)
where qe (mg$g�1) and Ce (mg$L�1) are adsorbed amount and
adsorbate concentrations at equilibrium, K (L$g�1) is the partition
coefficient.
The linear isothermmodel has been used to represent the
partition of adsorbates between solid and liquid phases. The
mechanisms of partition processes are the electrostatic in-
teractions, the van der Waals interactions, as well as the hydro-
phobic interactions (Guo et al., 2019a; 2019b).
The deduction of the linear model is explained as follows.
Based on Langmuir theory (Langmuir, 1916, 1918), the adsorp-
tion and desorption rate are described by Eqs. (2) and (3):
ra ¼ kað1� qÞCt (2)
rd ¼ kdq (3)
where ra (mg$g�1$h�1) and rd (mg$g�1$h�1) are adsorption and
desorption rate, respectively, ka (L$g�1$h�1) and kd (mg$g�1$h�1)
are adsorption and desorption rate constants, respectively, q is the
coverage rate of the adsorption sites (at adsorption equilibrium
q ¼ qe), Ct (mg$L�1) is the adsorbate concentration at time t.
If the coverage rate q ≪ 1, Eq. (2) is simplified to Eq. (4):
ra ¼ kaCt (4)
At adsorption Equilibrium:
qe ¼ kakd
Ce (5)
qe equals to qe/qm. Let K equals to qmka/kd, Eq. (5) is transformed
to Eq. (1).
The adsorption mechanisms revealed by the linear model is
shown in Fig. 2. Based on the deduction of the linear model, the
linear model represents the condition that the coverage ratio of the
adsorption sites is low. Therefore, the linear model can represent
the monolayer adsorption at low initial adsorbate concentrations
C0. Khan et al. (2019) suggested that the Langmuir model approx-
imated to Henry’s lawwhen the pressure is low in the adsorption of
gas on solid, which was similar with our results.
J. Wang, X. Guo / Chemosphere 258 (2020) 1272794
2.1.2. Freundlich isotherm
The Freundlich model is used to represent nonlinear adsorption
phenomenon (Freundlich, 1906). It is one of the most widely used
isotherm in adsorption. The linear and nonlinear forms of the
Freundlich model is given by the following equations:
qe ¼KFC1=ne (6)
logqe ¼ logKF þ
1
n
logCe (7)
where KF (L1/n$mg1�1/n$g�1) and n are constants, the Freundlich
model will reduce to the linear model when n ¼ 1.
The nonlinear Freundlich model (Eq. (6)) can be solved by
nonlinear regression analysis. Eq. (7) is easily to be solved by
plotting log qe versus log Ce. However, the propagated errors are
generated in the linearization process, which lead to the inaccurate
estimations of parameters (Guo and Wang, 2019a). In this paper,
the nonlinear method is recommended in the calculation of the
parameters, which is given in following section.
The Freundlich model has been regarded as an empirical
equation without physical meaning. In many published papers, the
Freundlich isotherm was applied to represent the multi-layer
adsorption on heterogamous surfaces (Zaheer et al., 2019; Wang
et al., 2017). In 1947, Halsey and Taylor derived the Freundlich
model from the Langmuir isotherm:
The adsorption and desorption rate are described by Eqs. (2) and
(3). At adsorption equilibrium, ra ¼ rd:
qe
1� qe ¼
ka
kd
Ce ¼ bðqÞCe (8)
In which q (mg$g�1) is the adsorbed amount given by Eq. (10),
b(q) is described by the following Equation (Halsey and Taylor,
1947):
bðqÞ¼A0e
q
RT (9)
q¼ � qmLlnq (10)
where, qmL (mg$g�1) is the maximum adsorption capacity, A0 is the
constant.
Substitution of Eqs. (9) and (10) into Eq. (8) yields:
qe
1� qe ¼A0e
q
RTCe (11a)
ln
qe
1� qe ¼ lnA0Ce �
qmL
RT
lnqe (11b)
When qe z 0.5, Eq. (11b) is simplified to Eq. (12):
qe ¼A
RT
qml
0 C
RT
qmL
e (12)
By definition of KF ¼ qmA0RT/qmL and n¼ qmL/RT, Eq. (12) becomes
Eq. (6).
Thus, the Freundlich model describes the adsorption condition
at which the equilibrium coverage fraction is about 50%.
Ezzati (2019) derived the pseudo-first-order (PFO) model from
the Freundlich isotherm. The PFO model can describe the diffu-
sional adsorption (Guo andWang, 2019c). Therefore, the Freundlich
model can also describe the physical adsorption process.
Based on the above, both the chemical adsorption with about
50% coverage fraction and the physical adsorption can be repre-
sented by the Freundlich model.
2.1.3. Redlichepeterson (ReP) isotherm
The ReP model is an empirical hybrid model of the Langmuir
and Freundlich models, which has been frequently applied in the
homogeneous or heterogeneous adsorption processes. The ReP
isotherm model can be described by Eq. (13) (Redlich and
Peterson, 1959):
qe ¼ KRPCe
1þ aRPCge
(13)
where KRP (L$g�1) and aRP (Lg$mg-g) are constants, g is the exponent
(0 ¼ g � 1). We can see from Eq. (13) that when g equals to 1, the
ReP model reduces to the Langmuir model (Eq. (26)), and when g
equals to 0 or Ce approaches to 0, it will reduce to the linear model
(Eq. (1)). In addition, if Ce approaches to infinite, qe z (KRP/aRP)
Ce
(1�g), which reduces to the Freundlich model (Eq. (6)).
2.1.4. Sips isotherm model
The Sips model is another hybrid model combining the Lang-
muir and Freundlich models (Sips, 1948). According to Ebadi et al.
(2015), Sips model is the most applicable 3-parameter isotherm
model for monolayer adsorption. Sips model can describe the ho-
mogeneous or heterogeneous systems. The non-linear Sips
isotherm model is presented by Eq. (14).
qe ¼ qmsKSC
ns
e
1þ KSCnse
(14)
where qms (mg$g�1) is the maximum adsorbed amount, Ks (Lns$mg-
ns) and ns are the Sips constants.
The Sips model becomes the Langmuir model when ns ¼ 1, and
becomes the Freundlich model at low C0. However, the Sips model
doesn’t satisfy the Henry’s law at low C0. The Sips model can be
derived as following:
If one adsorbate molecule can be adsorbed on 1/ns adsorption
sites, the adsorption and desorption rate can be described by Eqs.
(15) and (16) (Ho et al., 2002):
ra ¼ kað1� qÞ1=nsCt (15)
rd¼ kdq1=ns (16)
At adsorption equilibrium, ra ¼ rd:
kað1� qeÞ1=nsCe ¼ kdq1=nse (17)
Rearrangement of Eq. (17) yields:
qe ¼
�
ka
kd
�
Cnse
1þ
�
ka
kd
�
Cnse
(18)
By definition of KS ¼ ka/kd, and qe ¼ qe/qms, Eq. (18) becomes Eq.
(14), the general form of the Sips model.
Therefore, the Sips model represents the monolayer adsorption
of one adsorbate molecule onto 1/ns adsorption sites (Fig. 3).
2.1.5. Toth isotherm model
This model is developed to widen the application of the Lang-
muir model in heterogeneous systems (Eq. (19)) (Toth, 1971). It
assumes that the adsorption energies of most adsorption sites are
smaller than mean energy (Ho et al., 2002).
Fig. 3. The adsorption mechanisms revealed by the Sips isotherm model.
J. Wang, X. Guo / Chemosphere 258 (2020) 127279 5
qe ¼ KTCe�
aT þ Cze
�1=z (19)
where KT (mg$g�1) is the constant, aT (mgz$L-z) is the Toth constant,
z is a component that describes the degree of heterogeneity of the
adsorption systems. z is temperature independent, while the value
of aT increases with the increase of temperature (Rudsinski and
Everetta, 1992). When z ¼ 1, Toth model becomes Langmuir
isotherm. Larger deviation of z from 1 indicates that the adsorption
system is more heterogeneous.
2.1.6. Temkin isotherm
The Temkin model presumes that adsorption is a multi-layer
process (Temkin and Pyzhev, 1940). Extremely high and low con-
centrations values of the adsorbate in liquid phase are ignored.
Yang (1993) derived the statistical mechanical expression for the
Temkin isotherm and substituted the derived equation into the
Clapeyron-Clausius equation, confirmed that the differential heat of
adsorption was linear decreased with increasing coverage.
The Temkin model is presented by Eq. (20) (Temkin and Pyzhev,
1940):
qe ¼RTb lnðACeÞ (20)
where A (L∙g�1) and b (J∙mol�1) are the constants.
2.2. Adsorption models based on Polanyi’s potential theory
Polanyi’s potential theory assumes that the adsorption system
contains a “adsorption space”, where the molecules lose potential
energies. The potential energies are temperature independent, and
increase in the spaces closing to the adsorbent. The highest po-
tential energy is reached in the pores or cracks inside the adsorbent
(Schenz and Manes, 1975; Polanyi, 1932).
2.2.1. Dubinin-Radushkevich (D-R) model
The D-R model was proposed as an empirical isotherm to
represent the adsorption of vaporson solids (Dubinin and
Radushkevich, 1947). The D-R model is developed according to
Polanyi’s theory and the assumption that the distribution of pores
in adsorbent follows the Gaussian energy distribution (Polanyi,
1932; Gil and Grange, 1996; Dąbrowski, 2001). The nonlinear D-R
model is presented as following:
qe ¼ qmD�Re�KDRε
2
(21)
ε¼RTln Cs
Ce
(22)
where qmD-R (mg$g�1) is the maximum adsorbed amount, KDR
(mol2$kJ�2) is the model constant, ε (kJ$mol�1) is the adsorption
potential based on the Polanyi’s potential theory, Cs (mg$L�1) is the
solubility of adsorbates.
It should be noticed that ε has been incorrectly calculated in
many published papers as shown in Eq. (23):
ε¼RTln
�
1þ 1
Ce
�
(23)
Hu and Zhang (2019) demonstrated that Eq. (23) is a distinct
misconception. The terms ε and RT have the same dimension, and
the term (1 þ1/Ce) is meaningless because the dimensions of 1 and
1/Ce are inconsistent.
Themean free energy (E, kJ$mol�1) can be calculated by Eq. (24):
E¼ 1ffiffiffiffiffiffiffiffiffiffiffi
2KDR
p (24)
E is frequently applied to determine whether the adsorption is
dominated by physical process (E < 8 kJ mol�1) or chemical process
(8 < E < 16 kJ mol�1) (Chabani et al., 2006).
2.2.2. Dubinin-Astakhov (D-A) model
The D-A model (Eq. (25)) was developed as a more generalized
version of the D-R model (Dubinin and Astakhov, 1971).
qe ¼ e
�
�
�
ε
EDA
�nDA�
(25a)
qe ¼ qmD�Ae
�
�
�
ε
EDA
�nDA�
(25b)
where qmD-A (mg$g�1) is the maximum adsorbed amount, ε
(kJ$mol�1) is the adsorption potential, which can be computed by
Eq. (22), EDA (kJ$mol�1) is the characteristics energy, nDA is a con-
stant related to the percent of pore filling (Chen and Yang, 1994).
The theoretical basis of the D-A model is the Polanyi’s potential
theory (Polanyi, 1932). The D-A model is a semi-empirical model,
which can be used to investigate the micropore structures of the
adsorbent. Chen and Yang (1994) have derived the D-A model
based on the statistical mechanical principles. According to Chen
and Yang (1994), for adsorption of adsorbate onto adsorbent with
micropores and mesopores, the general isotherm model was
deduced to the D-A model when the equilibrium coverage fraction
was much greater than Ce/Cs. Cheng and Hu (2016) demonstrated
that the D-A model could be applied as a general model for the
adsorption of acetylene on MOFs. The D-A model can represent the
adsorption in homogeneous microporous systems.
2.3. Chemical adsorption models
The chemical adsorption isotherm models consider the mono-
layer adsorption process that the adsorbate molecules are adsorbed
in the adsorption sites of the adsorbents. The following studied
chemical adsorption models (the Langmuir and the Volmer
isotherm models) are theoretical models with specific physical
meanings and reasonable derivations.
J. Wang, X. Guo / Chemosphere 258 (2020) 1272796
2.3.1. Langmuir model
The most commonly applied Langmuir isotherm was raised to
represent the gas-solid adsorption (Langmuir, 1916, 1918). The
nonlinear and linear Langmuir models are presented as following:
qe ¼ qmKLCe1þ KLCe
(26)
Ce
qe
¼ Ce
qm
þ 1
KLqm
(27)
where KL (L$mg�1) is the ratio of the adsorption rate and desorption
rate, qm (mg$g�1) is the maximum adsorption capacity estimated
by the Langmuir model.
Eq. (26) is solved by the nonlinear regression method. Plotting
Ce/qe versus Ce can solve the linearized Langmuir model (Eq. (27)).
The Langmuir model can be linearized by 4 forms. Other forms of
the linearized Langmuir model (1/qe ¼ (1/(qmKL)*1/Ce þ 1/qm,
qe ¼ qm - (1/KL)*(qe/Ce), and qe/Ce ¼ qmKL - qeKL)) as well as the
comparison of linearization methods were discussed by Guo and
Wang (2019a). Langmuir-1 (Eq. (27)) could provide the most ac-
curate estimations of the parameters among the linearization
methods, which were similar with the nonlinear method. However,
the estimations of the Langmuir constants by the linearization
methods were inaccurate and biased, the error (%) of the estimated
parameters could reach up to 40% (Guo and Wang, 2019a). Even
though Langmuir-1 (Eq. (27)) can give relatively accurate estima-
tions of the model parameters, the performance of Langmuir-1 is
still poorer than the nonlinear method. Foo and Hameed (2010)
also concluded that the nonlinear method represented a powerful
tool, which avoided the drawbacks of linearization process. The
nonlinear method for the Langmuir isotherm is provided in the
following section.
Webber and Chakkravorti (1974) suggested to calculate sepa-
ration factor (RL):
RL ¼
1
1þ KLC0
(28)
The values of RL > 1, RL ¼ 1, and RL < 1 reflect that the adsorption
is unfavorable, linear, and favorable, respectively.
To better understand the mechanisms revealed by the Langmuir
model, the assumptions and deductions of the Langmuir model are
reviewed as follows.
The basic assumptions of the Langmuir isotherm are: (1)
monolayer adsorption; (2) the distribution of adsorption sites is
homogeneous; (3) the adsorption energy is constant; and (4) the
interaction between adsorbate molecules is negligible.
The adsorption and desorption rate are described by Eqs. (2) and
(3). At adsorption equilibrium, Ct and q is replaced by the
Fig. 4. The adsorption mechanisms reveal
equilibrium adsorbate concentration Ce and the equilibrium
coverage rate qe, and the adsorption rate equals to the desorption
rate:
ra ¼ rd (29)
Simultaneous Eqs. (2), (3) and (29) yield:
qe ¼ kaCekaCe þ kd
(30)
qe is the ratio of qe and qm. By definition of KL ¼ ka/kd, Eq. (30)
transforms to Eq. (26), which is the standard form of Langmuir
model.
Thus, Langmuir model describes equilibrium condition of
monolayer homogeneous adsorption (Fig. 4). ra is directly propor-
tional to (1 - q) and Ct. rb is only directly proportional to q. qe rep-
resents the coverage ratio of the whole adsorption system,
therefore, the term “homogeneous” means the macroscopic ho-
mogeneous adsorption. For most adsorption processes, the adsor-
bent materials are homogeneous in macroscopic view, and the
solution is homogeneous with agitation. Therefore, even though
the adsorbent materials (such as microplastics, activated carbons
from natural sources, modified mineral, and shale) have irregular
shapes and non-uniform surfaces inmicroscope, the adsorption can
also be represented by the Langmuir isotherm (Guo and Wang,
2019b; Mondal and Majumder, 2019; Liu et al., 2018). The mono-
layer adsorption on the surfaces and in the pores inside the
adsorbent can also be modeled by Langmuir model. This may help
to explain the results that the equilibrium data can be adequately
represented by Langmuir isotherm, while diffusion is the rate-
controlling-step. For example, Suzaki et al. (2017) concluded that
Langmuir model described the adsorption isotherm of metals in
fixed-bed columns, and the adsorption rate limiting step was in-
ternal diffusion. Marin et al. (2014) demonstrated that the rate
limiting step of dye onto SD-2 was internal diffusion, while the
Langmuir model was adopted to model the process.
2.3.2. Volmer isotherm model
The Volmer model is a distributed monolayer adsorption model,
which assumes that the adsorbate molecules can move on the
surfaces of adsorbents, and the interactions between adsorbates are
negligible. It is presented as following (Volmer, 1925):
bVCe ¼
qe
1� qee
qe
1�qe (31)
where bv (L$mg�1) is the affinity constant, qe ¼ qe/qmV (qmV
(mg$L�1) is the maximum adsorbed amount estimated by the
Volmer model).Rearrangement of Eq. (31) yields:
ed by the Langmuir isotherm model.
J. Wang, X. Guo / Chemosphere 258 (2020) 127279 7
bVCe ¼
qe
qmV � qe
e
qe
qmV�qe (32)
Afonso et al. (2016) derived the Volmer model according to
adsorption kinetics:The adsorption rate raV:
raV ¼
Ctffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pMRT
p að1� qÞSVp (33)
where raV (mol$kg�1$s�1) is the adsorption rate, M is the molar
mass, Sv is the specific surface areas of adsorbents, a is the pro-
portion of the successful impacts and total impacts on the surfaceof
the adsorbent, p is the probability that the intermolecular area is
sufficient formolecules to arrive the surfaces, p¼ exp(-q/(1-q))).The
desorption rate rdV:
rdV ¼ k0nVe�
EA
RT (34)
where rdV (mol$kg�1$s�1) is the desorption rate, EA (J$mol�1) is the
desorption energy, k0 (s�1) is the maximum desorption frequency
reached at infinite T, nV (mol$kg�1) is the specific adsorbed
amount.At adsorption equilibrium, raV ¼ rdV:
Ceffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pMRT
p að1� qeÞSVe�
qe
1�qe ¼ k0nVe�
EA
RT (35)
Considering that qe ¼ nV/nm (nm (mol$kg�1) is the maximum
specific adsorbed amount) and bV ¼ aSV=
�
nmk0exp
� � EART�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pMRT
p �
,Eq. (35) becomes Eq. (31).
2.4. Physical adsorption models
The theoretical physical adsorption isotherm models simulate
the multi-layer adsorption process. The Van Der Waals force is the
main driving forces of the physical adsorption. In this section, the
Brunauer, Emmett, and Teller (BET) and the Jovanovich isotherm
Fig. 5. The adsorption mechanisms revealed by the BET model.
qe¼
qmBETKBET1Ce
h
1�ðnBETþ1ÞðKBET2CeÞnBETþnBET ðKBET2CeÞnBETþ1
i
ð1�KBET2CeÞ
�
1þ
�
KBET1
KBET2
�1
�
KBET2Ce�
�
KBET1
KBET2
�
ðKBET2CeÞnBETþ1
�
(36)
models were reviewed.
2.4.1. BET model
BET model was proposed to represent adsorption of gas to
multimolecular layers (Brunauer et al., 1938). It is a theoretical
multi-layer physical adsorption model (Fig. 5). It has been applied
for calculation the specific areas and the pore size distribution of
the porous materials (Duong, 1998). The basic presumptions of BET
isotherm are that the adsorption is multi-layer homogeneous
adsorption, the adsorption energy in the first layer is different with
other layers, and for each layer, the adsorption rate equals to the
desorption rate. Ebadi et al. (2009) presented the application of the
BET model in liquid-solid systems (Eq. (36)). Detailed derivation of
this model was provided in Ebadi et al. (2009).where KBET1
(L$mg�1) and KBET2 (L$mg�1) are adsorption equilibrium parame-
ters in first and upper layers, nBET is the number of adsorption
layers, qmBET (mg$g�1) is the maximum monolayer adsorbed
amount.
For n ¼ 1, BET isotherm reduces to Langmuir isotherm. For
n ¼ ∞, Eq. (36) is simplified to Eq. (37) (Ebadi et al., 2009):
qe ¼ qmBETKBET1Ceð1� KBET2CeÞ½1� KBET2Ce þ KBET1Ce�
(37)
The BET model has another non-linear form (Eq. (38)), which is
the most familiar form adopted by literatures, such as Foo and
Hameed (2010), Staudt et al. (2013), and Petkovska (2014), and
Saadi et al..
qe ¼ qmBETCBETCe
ðCs � CeÞ
�
1þ ðCBET � 1Þ CeCs
� (38)
where CBET is the constant, Cs (mg$L�1) is the solubility of the
adsorbate. Cs can be calculated as an adjustable parameter or be
treated as a constant taken from solubility data.
Ebadi et al. (2009) compared the three methods in the appli-
cations of the BET model: (a) using Eq. (38) where Cs is a constant
taken from solubility data, (b) using Eq. (38) where Cs is an
adjustable parameter, and (c) using Eq. (37). The results indicated
that the application of Eq. (38) to liquid-solid adsorption system led
to poor estimations of the model parameters. The correct form of
the BET model should be Eqs. (36) and (37).
2.4.2. Aranovich model
The Aranovich isotherm is a theoretically corrected poly-
molecular adsorption isotherm, which contains two parameters
and can be used to model the adsorption with broader range of
adsorbate concentrations (Aranovich, 1992). This model is correctly
used in the determination the surface areas of porous adsorbents
(Aranovich, 1992). The basic assumptions of the Aranovich
isotherm are that the surfaces of the adsorbent are flat and ho-
mogeneous, only the “nearest neighbors” interact, and the
desorption energy depends on number of the layers. This model
can solve the problem caused by not taking lateral interactions into
account and the prohibition of vacancies in the adsorbate. The
Aranovich isotherm can be described by the following equation
(Aranovich, 1992; Duong, 1998):
qe ¼
qmACA
Ce
CsAffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�
1� CeCsA
�s �
1þ CA CeCsA
� (39)
where qmA (mg$g�1) is the maximum adsorbed amount, CA is the
J. Wang, X. Guo / Chemosphere 258 (2020) 1272798
Aranovich constant, CsA (mg$L�1) is the adsorbates monolayer
saturation concentration. This model will change to the Henry’s law
at low C0. Detailed derivation of the Aranovich isotherm could be
seen in Aranovich (1992).
2.5. Ion exchange isotherm model
The exchange of adsorbate ion MeA with ZAþ or ZAþ charge onto
adsorbent with MeB ion with ZBþ or ZBþ charge is described by the
following Equations (K�onya and Nagy, 2009; Nagy et al., 2016):
zBMe
ZAþ
A þ zAMeZBþB � S4zAMeZBþB þ zBMeZAþA � S (40)
zBMe
ZA�
A þ zAMeZB�B � S4zAMeZB�B þ zBMeZA�A � S (41)
where S is the surface adsorption sites.
The equilibrium constant KB,A of Eqs. (40) and (41) is calculated
by Eq. (42):
KB; A ¼
qZBA c
ZA
B
qZAB c
ZB
A
(42)
Where qA (mg$g�1) and qB (mg$g�1) are the adsorbate con-
centrations of MeA and MeB in the solid adsorbent, respectively, cA
(mg$L�1) and cB (mg$L�1) are the adsorbate concentrations of MeA
and MeB in solution.
For homovalent exchange process (ZA ¼ ZB) (Nagy et al., 2016):
KB; A ¼
qAcB
qBcA
(43)
cA
qA
¼1
x
�
cA þ
cB
KB; A
�
(44)
cB
qB
¼1
x
�
cB þKB; AcA
�
(45)
where x (mg$L�1) is the amount of concentrations of MeA and MeB
ions on solid phase.
For monovalent and bivalent ions exchange process (ZA ¼ 1,
ZB ¼ 2) (Nagy et al., 2016):
KB; A ¼
q2AcB
qBc2A
(46)
KA;B ¼
qBc2A
q2AcB
(47)
cA
qA
¼ 1
xmono
�
cA þ
2
KB; A
qAcB
cA
�
(48)
cB
qB
¼ 1
xmono
 
2cB þ
1
KA; B
c2A
ðxmono � 2qBÞ
!
(49)
where xmono (mg$L�1) is the exchange sites, xmono ¼ qA þ 2qB.
3. Applications of the isotherms
The isotherms are shown in Table 1. The linearized isotherms are
not suitable to be used to estimate the parameters because the
estimations are biased and inaccurate, as reviewed in previous
sections. Therefore, we only present the nonlinear forms of the
adsorption isotherms (except for the linear isotherm model).
The best-fit isotherms for adsorption of metals, dyes, pharma-
ceuticals, and other types of organic pollutants onto biosorbents
and abiotic adsorbents are presented in Tables 2e5. The best-fit or
optimum isotherm refers to the isothermwhich can best model the
experimental data, with high value of the coefficient of determi-
nation (R2) or low values of other statistical parameters, such as
nonlinear chi-square (c2) and residual sum of squares error (SSE).
The numbers of the optimum isotherm models are summarized
in Fig. 6 based onmore than 200 adsorption data sets (Abazari et al.,
2019; Ahmed et al., 2017; Ahsan et al., 2018; Akinpelu et al., 2019;
Arami et al., 2008; Asgaria et al., 2019; Barman et al., 2018; Basu
et al., 2019; Beekaroo and Mudhoo, 2011; Bhatti et al., 2016;
Bhosle et al., 2016; Bhowmik et al., 2018; Bouras et al., 2017; Cao
et al., 2019; Chaabane et al., 2020; Chaari et al., 2019;
Chakraborty et al., 2018; Chen and Wang, 2007a, 2007b; 2008a,
2008b; 2009, 2011; 2012a, 2012b; 2016a, 2016b; Chen et al., 2014,
2016; 2019, 2020a; 2020b; Cheng and Hu, 2016; Cheng et al., 2019;
Costa et al., 2015; da Silva and Pietrobelli, 2019; Dayanidhi et al.,
2020; Deniz and Kepekci, 2016; de Sousa et al., 2018; Dursun,
2006; El-Zahhar et al., 2014; Guo and Wang, 2019b; Guo et al.,
2019a; Hamza et al., 2018; Haque et al., 2010; Hodson et al., 2017;
Hossain et al., 2012; Hu et al., 2019; Igwe and Abia, 2007; Gamoudi
and Srasra, 2019; Karmakara et al., 2019; Lebron et al., 2019; Li et al.,
2019; Lim et al., 2020; Liu et al., 2020a, 2020b; Luo andWang, 2018;
Maged et al., 2020; Mahmoodi et al., 2010; Mahmoud et al., 2017;
Mallek et al., 2018; Mirsoleimani-azizi et al., 2018; Mnasri-Ghnimi
and Frini-Srasra, 2019; Nagy et al., 2016; Ngabura et al., 2018;
Nguyen et al., 2016; Oveisi et al., 2018; Pan et al., 2005, 2009a;
2009b, 2009c; Phasuphan et al.,2019; Pi et al., 2017; Ping et al.,
2006; Reddad et al., 2002; Ringot et al., 2007; Saeidi et al., 2020;
Sarma et al., 2018; Sebastian et al., 2019; Selvakumar and
Rangabhashiyam, 2019; Shahawy and Heikal, 2018; Shikuku et al.,
2018; Silva et al., 2020; Singh et al., 2018; Souza et al., 2018;
Subramanyam and Das, 2014; Sun et al., 2019; Tang et al., 2018;
Thang et al., 2019; Tian et al., 2018, 2020; Torabian et al., 2014;
Tosun, 2012; Wan et al., 2019; Wang, 1999a, 1999b; 2002; Wang
and Chen, 2006, 2009; 2014; Wang and Guo, 2020; Wang and
Shih, 2011; Wang and Wang, 2016, 2019; Wang and Zhuang, 2017,
2019a; 2019b, 2019c; 2020; Wang et al., 2000a, 2000b; 2001, 2016;
2018; Wu et al., 2013, 2018; 2019; Xing et al., 2016, 2019; 2020;
Xing andWang, 2016; Xu et al., 2020; Xu andWang, 2017; Xue et al.,
2019; Yin et al., 2017; Yu and Wang, 2016; Yu et al., 2016, 2017;
Zango et al., 2020; Zazycki et al., 2017; Zhang et al., 2016a, 2016b;
2020a, 2020b; Zhao et al., 2012; Zhou et al., 2010; Zhu and Wang,
2017; Zhu et al., 2012, 2014; Zhuang and Wang, 2019a, 2019b;
2019c; Zhuang et al., 2018a, 2018b; 2018c, 2018d; 2019a, 2019b;
2020a, 2020b; _Z�ołtowska-Aksamitowska et al., 2018). We mainly
focus on the literatures published in recent 5 years, but some
models (such as the D-R, D-A, BET, ReP models) are not frequently
used in recent years, therefore, some literatures published in earlier
years are also included.
As shown in Tables 2e5 and Fig. 6, the Langmuir model is the
most commonly applied optimum isotherm to represent the data of
metals ions, dyes, pharmaceuticals, as well as other types of organic
pollutants onto adsorbents, followed by the Freundlich model. One
reason is that these two models are most frequently adopted in the
adsorption studies, owing to the simplicity of the method (linear
regression method). In addition, the Langmuir model represents
homogeneous monolayer adsorption, and the adsorption systems
are homogeneous in macroscopic view for most adsorption pro-
cesses, as concluded in previous sections. The third reason is that a
large number of the adsorption processes is monolayer chemi-
sorption which relate to the surface bonding (Zhuang et al., 2020b;
Table 1
Summary of the adsorption isotherm models.
Classification Model’s name Model equation References
Adsorption empirical isotherm models Linear isotherm model (Henry’s law) qe ¼ KCe e
Freundlich isotherm model qe ¼ KFC1=ne Freundlich (1906)
RedlichePeterson (ReP) isotherm model
qe ¼ KRPCe
1þ aRPCge
Redlich and Peterson (1959)
Sips isotherm model
qe ¼ qmsKSC
ns
e
1þ KSCnse
Sips (1948)
Toth isotherm model
qe ¼ KTCeðaT þ CZe Þ1=z
Toth (1971)
Temkin isotherm model
qe ¼ RTb lnðACeÞ
Temkin and Pyzhev (1940)
Adsorption models based on the
Polanyi’s
potential theory
Dubinin-Radushkevich (D-R) isotherm
model
qe ¼ qmD�Re�KDRε2 ; ε ¼ RTln
Cs
Ce
Dubinin and Radushkevich (1947)
Dubinin-Astakhov (D-A) isotherm model
qe ¼ qmD�Ae
�
�
�
ε
EDA
�nDA�
; ε ¼ RTln Cs
Ce
Dubinin and Astakhov (1971)
Chemical adsorption models Langmuir isotherm model
qe ¼ qmKLCe1þ KLCe
Langmuir, (1916), 1918
Volmer isotherm model
bVCe ¼
qe
qmV � qe
e
qe
qmV � qe
Volmer (1925)
Physical adsorption models BET isotherm model (n ¼ ∞) qe ¼
qmBETKBET1Ce
ð1� KBET2CeÞ½1� KBET2Ce þ KBET1Ce�
Brunauer et al. (1938)
Aranovich isotherm model
qe ¼
qmACA
Ce
CsAffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�
1� Ce
CsA
�s �
1þ CA
Ce
CsA
�
Aranovich (1992)
Ion exchange isotherm model Homovalent ion exchange model cA
qA
¼ 1
x
�
cA þ
cB
KB; A
�
cB
qB
¼ 1
x
ðcB þ KB; AcAÞ
K�onya and Nagy, (2009); Nagy et al.,
(2016)
Monovalent and bivalent ions exchange
model
cA
qA
¼ 1
xmono
�
cA þ
2
KB; A
qAcB
cA
�
cB
qB
¼ 1
xmono
 
2cB þ
1
KA; B
c2A
ðxmono � 2qBÞ
!
J. Wang, X. Guo / Chemosphere 258 (2020) 127279 9
Manaa et al., 2020). The Temkin, Sips, and ReP empirical models
are also applied in some adsorption processes (Tables 2e5 and
Fig. 6). Theoretically, the Sips and ReP models can provide better
predictions for the equilibrium data than the Langmuir model,
because they are 3-parameter models, which combine the Lang-
muir and Freundlich models and are more flexible in parameter
estimations. However, the complications of the nonlinear regres-
sion method make it difficult to solve the Sips and ReP models.
Therefore, the applications of the Sips and RePmodels are less than
the Langmuir model. The linear model can successfully describe the
adsorption systems with low C0 (Tables 4 and 5). Guo et al. (2019a)
reported that the adsorption of SMT on microplastics with
C0 ¼ 0e12 mg L�1 could be modeled by the linear isotherm. Zhou
et al. (2010) found that the isotherm of PFOA and PFOS on acti-
vated sludge was linear at low values of C0 (0.08e0.63 mm mol$L�1
and 0.046e0.93 mm mol$L�1, respectively). Zhang et al. also sug-
gested that the adsorption isotherm of 9-nitroanthracene onto
microplastics was linear when C0 ¼ 10e500 mg L�1. These results
are coincidence with the theoretical analysis of the linear isotherm
studied in previous section. The D-R model was used to model the
adsorption of metals, dyes, and pharmaceuticals onto biosorbents
and abiotic adsorbents (Tables 2e4 and Fig. 6). However, it has been
used in incorrect form, as reviewed in the previous section. The
correct form of the D-R model is provided in Table 1. The ion ex-
change isotherm was applied in the adsorption of metals ions.
(Nagy et al., 2016). As shown in Tables 2e5 and Fig. 6, the appli-
cations of other types of isotherm models, such as the D-A, BET,
Toth, and Volmer models, are limited. The possible reason is that
the nonlinear solving methods of these models are complicated.
Therefore, we develop a convenient UI to solve these nonlinear
models based on Excel, which is introduced in the following
section.
Based on the theoretical analysis in above sections, it can be
seen that different isotherm models represent different adsorption
mechanisms. However, in most cases, we do not know the possible
adsorption mechanisms. Therefore, we should use isotherms as a
useful tool to investigate the adsorption mechanisms, instead of
understanding themechanisms and thenmodeling the equilibrium
data with a certain isotherm. We recommend to fit the equilibrium
data by the chemical, physical and other theoretical isotherm
models, and obtain the optimum isotherm by judging the statistical
parameters.
4. Statistical parameters for the evaluation of the isotherm
models
The statistical parameters (error functions) used to evaluate the
fitness of the isotherms are depicted in Table 6 and Fig. 7, based on
(Sebastian et al., 2019; Basu et al., 2019; Ngabura et al., 2018; Souza
et al., 2018; Wang et al., 2016; Bhatti et al., 2016; Dursun, 2006;
Cheng et al., 2019; Hu et al., 2019; Zazycki et al., 2017; Dayanidhi
et al., 2020; Cao et al., 2019; Zhuang et al., 2018a; b; Mahmoud
et al., 2017; Reddad et al., 2002; Igwe and Abia, 2007; Hossain
et al., 2012; Guo and Wang, 2019b; X. Zheng et al., 2020; Tang
et al., 2018; Hodson et al., 2017; Xing et al., 2016; Chaabane et al.,
2020; Asgaria et al., 2019; Mnasri-Ghnimi and Frini-Srasra, 2019;
Lim et al., 2020; Liu et al., 2020b; Bhosle et al., 2016; Nagy et al.,
2016; Tosun, 2012; Singh et al., 2018; Selvakumar and
Table 2
Applications of the adsorption isotherms in the adsorption of metals.
Adsorbate Adsorbent type Adsorbent Adsorption conditions Optimum
isotherm
Model parameters Statistical parameters References
Cd(II) Biosorbents Magnetite nanoparticles
synthesi � zed from Hevea bark
200 rpm, C0 ¼ 10e20 ppm Langmuir
Freundlich
qmax ¼ 37.03 (mg$g�1); KL ¼ 0.23
(L$mg�1)
KF ¼ 6.9 (L1/n$mg1�1/n$g�1); 1/n ¼ 0.90
R2 ¼ 0.98
R2 ¼ 0.99
Sebastian et al. (2019)
Na(I) 200 rpm, C0 ¼ 10e20 ppm Langmuir qmax ¼ 3.95 (mg$g�1); KL ¼ 0.44
(L$mg�1)
R2 ¼ 0.97
Cr(III) Cymbopogon flexuosus
immobilized
in sodium alginate beads
298 K, C0 ¼ 10e200 mg L�1 Freundlich KF ¼ 13.4 (L1/n$mg1�1/n$g�1); n ¼ 1 R2 ¼ 0.99 Basu et al. (2019)
Zn(II)Durian peel 120 rpm, 303 K, C0 ¼ 10
e200 mg L�1
Temkin b ¼ 0.5675 kJ mol�1; A¼ 0.0653 L mg�1 R2 ¼ 1;
SSE ¼ 0.7816
Ngabura et al. (2018)
Zn(II) 120 rpm, 313 K, C0 ¼ 10
e200 mg L�1
Temkin b ¼ 0.4223 kJ mol�1; A¼ 0.0662 L mg�1 R2 ¼ 1; SSE ¼ 2.5226
Zn(II) 120 rpm, 323 K, C0 ¼ 10
e200 mg L�1
Temkin b ¼ 0.7369 kJ mol�1; A¼ 0.0715 L mg�1 R2 ¼ 1; SSE ¼ 3.7837
Cd(II) Malpighia emarginata D.C. seed
fiber
300 rpm Langmuir qmax ¼ 0.124 (mg$g�1); KL ¼ 0.095
(L$mg�1)
R2 ¼ 0.9998 Souza et al. (2018)
Cr 300 rpm Langmuir qmax ¼ 0.118 (mg$g�1); KL ¼ 0.066
(L$mg�1)
R2 ¼ 0.9993
Cu(II) 300 rpm Langmuir qmax ¼ 0.095 (mg$g�1); KL ¼ 0.078
(L$mg�1)
R2 ¼ 0.9996
Ni(II) 300 rpm Langmuir qmax ¼ 0.081 (mg$g�1); KL ¼ 0.063
(L$mg�1)
R2 ¼ 0.9995
Pb(II) Gelation with alginate C0 ¼ 100e1000 mg L�1 Langmuir qmax ¼ 435.3 (mg$g�1); KL ¼ 0.046
(L$mg�1)
R2 ¼ 0.977 Wang et al. (2016)
Cu(II) C0 ¼ 100e1000 mg L�1 Langmuir qmax ¼ 167.1 (mg$g�1); KL ¼ 0.011
(L$mg�1)
R2 ¼ 0.882
Zr(IV) Citrus peel biomass C0 ¼ 10e100 mg L�1 Langmuir qmax ¼ 68.4774 (mg$g�1); RL ¼ 0.00544 R2 ¼ 0.997 Bhatti et al. (2016)
Pb(II) Pretreated Aspergillus niger 150 rpm Langmuir qmax ¼ 34.69 (mg$g�1); KL ¼ 0.021
(L$mg�1)
ε% ¼ 1.3 Dursun (2006)
Pb(II) 150 rpm ReP aRP ¼ 0.066 (dm3 mg�1)g, KRP ¼ 1.65
(dm3 mg�1); g ¼ 0.93
ε% ¼ 3.3
Sr(II) Modified mercerized bacterial
cellulose membrane
150 rpm, 303.15 K, C0 ¼ 5
e900 mg L�1
Langmuir qmax ¼ 44.86 (mg$g�1); KL ¼ 0.085
(L$mg�1)
R2 ¼ 0.996 Cheng et al. (2019)
Sr(II) Algal sorbent derived from
Sargassum horneri
100 rpm, pH ¼ 5 Sips qms ¼ 1.72 (mmol$g�1); KS ¼ 1.56
(Lns$mmol-ns); ns ¼ 1.70
R2 ¼ 0.990; c2 ¼ 0.089;
RMSE ¼ 0.051; AICc ¼ �81.6
Hu et al. (2019)
Au Activated carbon (AC) 250 rpm Langmuir qmax ¼ 7.86 (mg$g�1); KL ¼ 0.8987
(L$mg�1); RL ¼ 0.059
R2 ¼ 0.9895 Zazycki et al. (2017)
V(Ⅳ) Eggshell powder (ES) C0 ¼ 100e500 mmol L�1 Freundlich KF ¼ 1.09 (L1/n$mmol1�1/n$g�1); 1/
n ¼ 0.6707
R2 ¼ 0.9988 Dayanidhi et al. (2020)
Fe(II) C0 ¼ 100e500 mmol L�1 Freundlich KF ¼ 1.27 (L1/n$mmol1�1/n$g�1); 1/
n ¼ 0.8551
R2 ¼ 0.9947
Fe(III) C0 ¼ 100e500 mmol L�1 Freundlich KF ¼ 1.59 (L1/n$mmol1�1/n$g�1); 1/
n ¼ 0.8546
R2 ¼ 0.9974
Pb(II) Multi-pore activated carbons
(MPAC)
175 rpm Langmuir qmax ¼ 1.34 (mmol$g�1); KL ¼ 37.05
(L$mmol�1)
R2 ¼ 0.993 Cao et al. (2019)
Ni(II) 175 rpm Langmuir qmax ¼ 0.97 (mmol$g�1); KL ¼ 22.53
(L$mmol�1)
R2 ¼ 0.983
U(Ⅵ) Modified chitosan beads 150 rpm, pH ¼ 6, T, 298 K,
C0 ¼ 10e600 mg L�1
Langmuir qmax ¼ 117.65 (mg$g�1); KL ¼ 0.14
(L$mg�1)
R2 ¼ 0.9995 Zhuang et al. (2018a)
Cd(II) Heat-inactivated marine
Aspergillus flavus
C0 ¼ 0.02e0.2 mol L�1, pH 7.0 Langmuir qmax ¼ 3333.333 (mg$g�1); RL ¼ 0.874 R2 ¼ 0.9992 Mahmoud et al. (2017)
Pb(II) Sugar beet pulp C0 ¼ 0.2 � 10�3e2.5 � 10-3 M Langmuir qmax ¼ 0.356 (mmol$g�1); KL ¼ 16.9
(L$mmol�1)
R2 ¼ 0.987 Reddad et al. (2002)
Cu(II) C0 ¼ 0.2 � 10�3e2.5 � 10-3 M Langmuir qmax ¼ 0.333 (mmol$g�1); KL ¼ 3.73
(L$mmol�1)
R2 ¼ 0.984
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Cd(II) C0 ¼ 0.2 � 10�3e2.5 � 10-3 M Freundlich KF ¼ 0.185 (L1/n$mmol1�1/n$g�1); 1/
n ¼ 0.26
R2 ¼ 0.975
Cd(II) Rice husk 850 mm pH ¼ 7.5 D-R e R2 ¼ 0.9988 Igwe and Abia (2007)
Pb(II) Rice husk 450 mm pH ¼ 7.5 D-R e R2 ¼ 0.9765
Zn(II) Rice husk 450 mm pH ¼ 7.5 D-R - R2 ¼ 0.9956
Cu(II) Palm oil fruit shells 120 rpm, C0 ¼ 1e35 mg L�1 Sips
ReP
Toth
qs ¼ 45.275; KS ¼ 1.328; ns ¼ 0.898
aRP ¼ 2.054; KRP ¼ 59.502; g ¼ 0.938
KT ¼ 34.902; aT ¼ 0.608; z ¼ 0.813
R2 ¼ 0.994; c2 ¼ 3.797;
R2 ¼ 0.994; c2 ¼ 4.4244;
R2 ¼ 0.994; c2 ¼ 4.4043;
Hossain et al. (2012)
Sr(II) Abiotic
adsorbents
Polyethylene (PE) 160 rpm, C0 ¼ 0e10 mg L�1 Freundlich KF ¼ 0.225 (L1/n$mg1�1/n$g�1); n ¼ 1.35 R2 ¼ 0.989; c2 ¼ 10.8; SSE ¼ 390;
AICc ¼ 40.1
Guo and Wang (2019b)
Cu(II) L-cysteine (Cys) intercalated
MgAl-layered double hydroxide
(MgAl-Cys-LDH)
pH ¼ 5, C0 ¼ 5e500 mg L�1 Langmuir qmax ¼ 58.07 (mg$g�1); KL ¼ 0.1328
(L$mg�1)
R2 ¼ 0.9735 X. Zheng et al., 2020
Pb(II) pH¼ 5.73, C0¼ 20e1400mg L�1 Langmuir qmax ¼ 186.2 (mg$g�1); KL ¼ 0.0033
(L$mg�1)
R2 ¼ 0.9557
Cd(II) pH ¼ 5.85, C0 ¼ 5e500 mg L�1 Langmuir qmax ¼ 93.11 (mg$g�1); KL ¼ 0.2190
(L$mg�1)
R2 ¼ 0.9511
Pb(II) Porous inorganic polymer
microspheres
250 shakes/min, C0 ¼ 50
e150 mg L�1
Langmuir
D-R
qmax ¼ 621.12 (mg$g�1); KL ¼ 7.69
(L$mg�1)
qmD-R ¼ 620.76 (mg$g�1);
KDR ¼ 2.36 � 10�8 (mol2$kJ�2)
R2 ¼ 0.998
R2 ¼ 0.995
Tang et al. (2018)
Zn(II) High density polyethylene
(HDPE)
220 rpm, C0 ¼ 0.1e10 mg L�1 Freundlich ln KF ¼ 5.49; 1/n ¼ 0.43 R2 ¼ 0.72 Hodson et al. (2017)
Zn(II) Woodland soil 220 rpm, C0 ¼ 0.1e10 mg L�1 Freundlich ln KF ¼ 5.76; 1/n ¼ 0.65 R2 ¼ 0.94
Co(II) Nanoscale zero valent iron
(ZVI)/graphene (GF) composite
pH ¼ 5.7 Langmuir qmax ¼ 131.58 (mg$g�1) e Xing et al. (2016)
Cu(II) Functionalized graphene oxide
sheets GO-EDA-CAC-BPED
293 K, pH ¼ 7 Langmuir qmax ¼ 3.891 (mmol$g�1); KL ¼ 0.074
(L$mmol�1)
R2 ¼ 0.988 Chaabane et al. (2020)
Ni(II) 293 K, pH ¼ 7 Langmuir qmax ¼ 3.508 (mmol$g�1); KL ¼ 3.392
(L$mmol�1)
R2 ¼ 0.990
Co(II) 293 K, pH ¼ 7 Langmuir qmax ¼ 3.401 (mmol$g�1); KL ¼ 0.086
(L$mmol�1)
R2 ¼ 0.992
Cs(I) Metal organic framework
(MOF)
200 rpm Langmuir qmax ¼ 86.2 (mg$g�1); KL ¼ 0.01
(L$mg�1)
R2 ¼ 0.98 Asgaria et al., 2019
Sr(II) 200 rpm Langmuir qmax ¼ 58.47 (mg$g�1); KL ¼ 0.01
(L$mg�1)
R2 ¼ 0.99
Cd(II) Pillared clays C0 ¼ 10e100 mg L�1 Langmuir qmax ¼ 7e24 (mg$g�1); KL ¼ 0.749
e3.346 (L$mg�1)
R2 ¼ 0.997e0.998 Mnasri-Ghnimi and Frini-
Srasra (2019)
Pt(IV) Metal-organic frameworks of
MIL-101(Cr)eNH2
120 rpm, C0 ¼ 0e1600 mg L�1 Langmuir qmax ¼ 277.6 (mg$g�1); KL ¼ 0.018
(L$mg�1)
R2 ¼ 0.9696 Lim et al. (2020)
Pd(II) 120 rpm, C0 ¼ 0e1600 mg L�1 Langmuir qmax ¼ 140.7 (mg$g�1); KL ¼ 0.003
(L$mg�1)
R2 ¼ 0.9703
As(III) Fe0/COFs 300 rpm, C0 ¼ 1.08e4.59 mg L�1 Freundlich - R2 ¼ 0.990 Liu et al. (2020b)
Sr(II) Phosphonate-functionalized
polymer
pH ¼ 3 Freundlich - R2 ¼ 0.969 Bhosle et al. (2016)
Co Bentonite clay 1440 rpm, pH ¼ 6.5 Ion
exchange
KAB ¼ 0.64; xmono ¼ 9 � 10�4(mg$L�1) e Nagy et al. (2016)
Mn 1440 rpm, pH ¼ 6.5 Ion
exchange
KAB ¼ 0.85; xmono ¼ 6.2 � 10�4(mg$L�1) e
Hg 1440 rpm, pH ¼ 2.76 Ion
exchange
KAB ¼ 1.02; xmono ¼ 4.2 � 10�4(mg$L�1) e
Ammonium Clinoptilolite 200 rpm, C0 ¼ 30e250 mg L�1,
pH ¼ 4.5
ReP
Temkin
aRP ¼ 0.084; KRP ¼ 0.863; g ¼ 0.905
b ¼ 748.566; A ¼ 0.469
R2 ¼ 0.999
R2 ¼ 0.999
Tosun (2012)
Ammonium 200 rpm, C0 ¼ 30e250 mg L�1,
pH ¼ 4.5
ReP aRP ¼ 0.585; KRP ¼ 2.217; g ¼ 0.701 R2 ¼ 0.994
Ammonium 200 rpm, C0 ¼ 30e250 mg L�1,
pH ¼ 4.5
ReP aRP ¼ 0.610; KRP ¼ 2.387; g ¼ 0.709 R2 ¼ 0.993
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Table 3
Applications of the adsorption isotherms in the adsorption of dyes.
Adsorbate Adsorbent
type
Adsorbent Adsorption conditions Optimum
isotherm
Model parameters Statistical
parameters
References
Rhodamine-B (RB) Biosorbents Banana peel powder e Langmuir qmax ¼ 1.6572e3.8804 (mg$g�1); KL ¼ 0.0658e0.0 3 (L$mg�1) R2 ¼ 0.9459
e0.9740
Singh et al. (2018)
Rhodamine-B (RB) Modified biosorbent from Kappaphycu
alvarezii (EKA)
C0 ¼ 10e50 mg L�1 Langmuir qmax ¼ 112.35 (mg$g�1); KL ¼ 0.001 (L$mg�1) R2 ¼ 0.9459 Selvakumar and
Rangabhashiyam (2019)
Modified biosorbent from Gracilaria
Salicornia (EGS)
C0 ¼ 10e50 mg L�1 Langmuir qmax ¼ 105.26 (mg$g�1); KL ¼ 0.1 (L$mg�1) R2 ¼ 0.9459
Modified biosorbent from Gracilaria
edulis (ECG)
C0 ¼ 10e50 mg L�1 Langmuir qmax ¼ 97.08 (mg$g�1); KL ¼ 0.01 (L$mg�1) R2 ¼ 0.9459
Congo red (CR) Magnetic mycelial pellets 120 rpm, C0 ¼ 25
e5000 mg L�1
Sips qmax ¼ 101.8 (mg$g�1) e Q. Zhang et al., 2016a,
2016b
Sunset yellow (SY) Modified skin of Iron stick yam
(ISY@PEI)
pH ¼ 2, C0 ¼ 100
e500 mg L�1
Langmuir qmax ¼ 476.20 (mg$g�1); KL ¼ 10.21 (L$mg�1) R2 ¼ 0.9840 Y. Zhang et al., 2016a,
2016b
Lemon yellow (LY) pH ¼ 2, C0 ¼ 100
e500 mg L�1
Langmuir qmax ¼ 138.92 (mg$g�1); KL ¼ 6.24 (L$mg�1) R2 ¼ 0.9821
Acid Red 14
(AR14)
Mesoporous egg shell membrane (ESM) 200 rpm ReP KRP ¼ 293714.7 (L$g�1); aRP ¼ 349186.2 (Lg$mg-g); ¼ 0.897703 R2 ¼ 0.992 Arami et al. (2008)
AcidBlue 92
(AB92)
200 rpm ReP KRP ¼ 758653.5 (L$g�1); aRP ¼ 481025.2 (Lg$mg-g); ¼ 0.633962 R2 ¼ 0.998
Congo red (CR) Modified Glossogyne tenuifolia leaves C0 ¼ 100e500 mg L�1 Freundlich KF ¼ 2.268 (L1/n$mg1�1/n$g�1); n ¼ 3.457 R2 ¼ 0.994 Yang and Hong (2018)
malachite green
(MG)
C0 ¼ 100e500 mg L�1 Freundlich KF ¼ 1.548 (L1/n$mg1�1/n$g�1); n ¼ 22.23 R2 ¼ 0.997
Congo red (CR) Penicillium janthinellum sp. strain (P1) 120 rpm, C0 ¼ 25
e500 mg L�1
Langmuir qmax ¼ 344.83 (mg$g�1); KL ¼ 0.1261 (L$mg�1) R2 ¼ 0.9991 Wang et al. (2015a)
Basic Blue 41
(BB41)
Canola hull 200 rpm, C0 ¼ 25
e100 mg L�1
Langmuir qmax ¼ 67.5676 (mg$g�1); KL ¼ 1.4095 (L$mg�1) R2 ¼ 0.9963 Mahmoodi et al., 2010
Basic Red 46
(BR46)
200 rpm, C0 ¼ 25
e100 mg L�1
Langmuir qmax ¼ 49.0196 (mg$g�1); KL ¼ 0.0165 (L$mg�1) R2 ¼ 0.9964
Basic Violet 16
(BV16)
200 rpm, C0 ¼ 25
e100 mg L�1
Langmuir qmax ¼ 25 (mg$g�1); KL ¼ 0.1973 (L$mg�1) R2 ¼ 0.9892
Sulfur blue 15
(SB15)
Acidithiobacillus thiooxidans 200 rpm, pH ¼ 7.7,
C0 ¼ 200e2000 mg L�1
Langmuir qmax ¼ 1428.6 (mg$g�1); KL ¼ 0.00464 (L$mg�1) R2 ¼ 0.9982 Nguyen et al. (2016)
Reactive-azo (RA) Pistachio by-product C0 ¼ 50e2000 mg L�1 Sips qms ¼ 109.535 (mg$g�1); KS ¼ 0.0111.409 (Lns$mg-n 1/ns ¼ 1.409 R2 ¼ 1 Deniz and Kepekci
(2016)
Congo red (CR) Aspergillus carbonarius M333 250 rpm, C0 ¼ 20
e125 mg L�1
Langmuir qmax ¼ 99.01 (mg$g�1); KL ¼ 0.036 (L$mg�1) R2 ¼ 0.988 Bouras et al. (2017)
Penicillium glabrum Pg1 250 rpm, C0 ¼ 20
e125 mg L�1
Langmuir qmax ¼ 101.01 (mg$g�1); KL ¼ 0.079 (L$mg�1) R2 ¼ 0.998
Methylene Blue Fucus vesiculosus ZnCl2 250 rpm, C0 ¼ 50
e700 mg L�1
Sips qms ¼ 1380.465 (mg$g�1); KS ¼ 0.0130.513 (Lns$mg- ; ns ¼ 0.513 R2 ¼ 0.979 Lebron et al. (2019)
Spirulina maxima ZnCl2 250 rpm, C0 ¼ 50
e700 mg L�1
Sips qms ¼ 316.393 (mg$g�1); KS ¼ 0.0261.248 (Lns$mg-n ns ¼ 1.248 R2 ¼ 0.998
Chlorella pyrenoidosa ZnCl2 250 rpm, C0 ¼ 50
e700 mg L�1
Sips qms ¼ 210.844 (mg$g�1); KS ¼ 0.0411.019 (Lns$mg-n ns ¼ 1.019 R2 ¼ 0.997
Congo red (CR) Cashew nut shell (CNS) 120 rpm, C0 ¼ 20
e100 mg L�1
ReP
Sips
Toth
KRP ¼ 5.548 (L$g�1); aRP ¼ 3.186 (Lg$mg-g); g ¼ 0.6 qms ¼ 14.29
(mg$g�1); KS ¼ 0.102 (Lns$mg-ns); ns ¼ 1.841
KT ¼ 67.39 (L$g�1), aT ¼ 1.013 (mgz$L-z), z ¼ 0.178
R2 ¼ 0.999
R2 ¼ 0.999
R2 ¼ 0.999
Reactive Red 158
(RR158)
Cocos nucifera L. Shell Powder 200 rpm, adsorbent
dosage: 12.5 g/L
D-R qmD-R ¼ 0.0035 (mg$g�1); KDR ¼ �0.01 (mol2$kJ�2) R2 ¼ 0.999 Beekaroo and Mudhoo
(2011)
200 rpm, adsorbent
dosage: 7.5 g L�1
Temkin b ¼ �194.931; A ¼ 0.010 R2 ¼ 1
200 rpm, adsorbent
dosage: 10 g L�1
Temkin b ¼ �1394.244; A ¼ 0.050 R2 ¼ 0.999
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91
g
g
s);
ns)
s);
s);
38
Methyl orange
(MO)
Abiotic
adsorbents
Magnetic zeolite imidazolate
framework-67 composites (MZIF-67)
150 rpm Freundlich KF ¼ 0.124 (L1/n$mg1�1/n$g�1); 1/n ¼ 2.113 R2 ¼ 0.995 Xue et al. (2019)
Direct blue 86
(DB-86)
150 rpm Langmuir qmax ¼ 0.085 (mg$g�1); KL ¼ 303.03 (L$mg�1) R2 ¼ 0.996
Methyl orange
(MO)
Porous metal-organic framework (MOF)
MIL-53
C0 ¼ 5e200 ppm Langmuir qmax ¼ 57.9 (mg$g�1) e Haque et al. (2010)
Porous metal-organic framework (MOF)
MIL-101
C0 ¼ 5e200 ppm Langmuir qmax ¼ 114 (mg$g�1) e
Basic Yellow 28
(BY28)
Smectite rich natural clay e Sips
Langmuir
qms ¼ 76.12 (mg$g�1); 1/ns ¼ 0.451
qmax ¼ 76.92 (mg$g�1); KL ¼ 0.119 (L$mg�1)
R2 ¼ 0.99
R2 ¼ 0.996
Chaari et al. (2019)
Cristal Violet (CV) Tunisian Smectite Clay 300 rpm, C0 ¼ 12.5
e100 mg L�1
Toth e R2 ¼ 0.999 Hamza et al. (2018)
Yellow B2R Chia seeds (Salvia hispanica) oil
extraction
150 rpm, 303 K, C0 ¼ 50
e675 mg L�1
Toth e R2 ¼ 0.9927 da Silva and Pietrobelli
(2019)
Bromophenol blue
(BPB)
Polymer-clay composite P(AAm-AA)-
Kao
Room temperature Freundlich 1/n ¼ 0.347 R2 ¼ 0.989 El-Zahhar et al. (2014)
Methyl orange
(MO)
Fe2O3/Mn3O4 nanocomposite 200 rpm, Adsorbent dose
0.25 g L�1
Langmuir qmax ¼ 322.58 (mg$g�1); KL ¼ 0.256 (L$mg�1) R2 ¼ 0.9972;
c2 ¼ 1.634
Bhowmik et al. (2018)
200 rpm, Adsorbent dose
1 g L�1
Langmuir qmax ¼ 121.95 (mg$g�1); KL ¼ 0.956 (L$mg�1) R2 ¼ 0.999;
c2 ¼ 0.331
Acid Orange 7
(AO7)
Modified porous cellulose-based
microsphere
C0 ¼ 20e90 mg mL�1 Langmuir qmax ¼ 257.67 (mg$g�1) R2 ¼ 0.9909 Wan et al. (2019)
C0 ¼ 20e90 mg mL�1 Langmuir qmax ¼ 178.99 (mg$g�1) R2 ¼ 0.9605
Reactive dyes
black 5 (RDBK
5)
MIL-101-Cr metal organic framework 293 K, 150 rpm Langmuir qmax ¼ 450 (mg$g�1); KL ¼ 0.05 (L$mg�1) R2 ¼ 0.93 Karmakara et al., 2019
Reactive dyes blue
2 (RDB 2)
293 K, 150 rpm Langmuir qmax ¼ 435 (mg$g�1); KL ¼ 0.08 (L$mg�1) R2 ¼ 0.96
Basic Red 46
(BR46)
Metal-organic framework NH2-MIL-
125(Ti)
In the presence of
ultrasonic irradiation
Langmuir qmax ¼ 1250 (mg$g�1); KL ¼ 1.431 (L$mg�1) R2 ¼ 0.995 Oveisi et al. (2018)
Basic Blue 41
(BB41)
In the presence of
ultrasonic irradiation
Langmuir qmax ¼ 1429 (mg$g�1); KL ¼ 1.290 (L$mg�1 R2 ¼ 0.993
Methylene Blue
(MB)
In the presence of
ultrasonic irradiation
Langmuir qmax ¼ 833 (mg$g�1); KL ¼ 1.09 (L$mg�1) R2 ¼ 0.953
Methyl orange
(MO)
HDPyþ modified clay 2700 rpm, C0 ¼ 10
e1000 mg mL�1
Langmuir qmax ¼ 277.27 (mg$g�1); KL ¼ 0.29 (L$mg�1) R2 ¼ 0.99 Gamoudi and Srasra
(2019)
Indigo carmine
(IC)
2700 rpm, C0 ¼ 10
e1000 mg mL�1
Langmuir qmax ¼ 326.31 (mg$g�1); KL ¼ 0.01 (L$mg�1) R2 ¼ 0.98
Procion Red MX5B
(MX5B)
Montmorillonite Mt 303 K, C0 ¼ 20
e150 mg mL�1
Temkin
Freundlich
b ¼ 2.48 kJ mol�1; A ¼ 0.21 L mg�1
KF ¼ 0.31 (L1/n$mg1�1/n$g�1); 1/n ¼ 0.62
R ¼ 0.99
R ¼ 0.99
Sarma et al. (2018)
Montmorillonite Mt 1 303 K, C0 ¼ 20
e150 mg mL�1
Temkin
Freundlich
b ¼ 2.75 kJ mol�1; A ¼ 0.19 L mg�1
KF ¼ 0.37 (L1/n$mg1�1/n$g�1); 1/n ¼ 0.6
R ¼ 0.99
R ¼ 0.99
Montmorillonite Mt 2 303 K, C0 ¼ 20
e150 mg mL�1
Temkin
Freundlich
b ¼ 2.94 kJ mol�1; A ¼ 0.18 L mg�1
KF ¼ 0.4 (L1/n$mg1�1/n$g�1); 1/n ¼ 0.6
R ¼ 0.99
R ¼ 0.99
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Table 4
Applications of the adsorption isotherms in the adsorption of pharmaceuticals.
Adsorbate Adsorbent
type
Adsorbent Adsorption conditions Optimum
isotherm
Model parameters Statistical parameters References
Ciprofloxacin (CIP) Biosorbents Magnetic biosorbents 298.15 K, 220 rpm Langmuir qmax ¼ 527.93 (mg
(L$mg�1)
R2 ¼ 0.9911 Zheng et al. (2020)
318.15 K, 220 rpm Langmuir qmax ¼ 485.17 (mg
(L$mg�1)
R2 ¼ 0.9894
Tetracyclines (TC) Sulfonated spent coffee waste
(SCWeSO3H)
e Langmuir qmax ¼ 473.93 (mg
(L$mg�1)
R2 ¼ 0.988 Ahsan et al. (2018)
Diclofenac (DCF) Biochar C0 ¼ 100e20,000 mg L�1, pH ¼ 7 Langmuir qmax ¼ 7.25 � 103
KL ¼ 5.37 � 10�3 (
R2 ¼ 0.998 Li et al. (2019)
Trimethoprim (TMP) Biochar C0 ¼ 100e400,000 mg L�1, pH ¼ 7 Langmuir qmax ¼ 2.08 � 103
KL ¼ 1.17 � 10�3 (
R2 ¼ 0.987
Ibuprofen (IBP) Wood apple biochar (WAB) C0 ¼ 1e45 mg mL�1 Freundlich KF ¼ 1.353 (L1/n$m
n ¼ 1.603
R2 ¼ 0.957 Chakraborty et al. (2018)
Steam activated wood apple
biochar (WASAB)
C0 ¼ 1e45 mg mL�1 Langmuir qmax ¼ 12.658 (mg 7
(L$mg�1)
R2 ¼ 0.966
Fluoxetine Spent coffee ground (SCG) 170 rpm, pH ¼ 9 Sips qms ¼ 14.31 (mg$g
(Lns$mg-ns); 1/ns ¼
R2 ¼ 0.996 Silva et al. (2020)
Pine bark 170 rpm, pH ¼ 9 Sips qms ¼ 6.53 (mg$g� g-
ns); 1/ns ¼ 6.74
R2 ¼ 0.991
Cork waste 170 rpm, pH ¼ 9 Sips qms ¼ 4.74 (mg$g� g-
ns); 1/ns ¼ 3.87
R2 ¼ 0.990
Tetracycline (TC) Magnetic nano-scale biosorbent
(Fe3O4/MFX)
C0 ¼ 40e120 mg mL�1, pH ¼ 6 Langmuir qmax ¼ 1.47 (mg$g
(L$mg�1)
R2 ¼ 0.986 Pi et al. (2017)
Ochratoxin A (OA) Beta-glucanes 400 rpm BET qmBET ¼ 0.24 (mg$
(L$mg�1)
HYBRID ¼ 5.2701 � 10�3 Ringot et al. (2007)
Yeast cell wall fraction (LEC) 400 rpm BET qmBET ¼ 0.040 (mg 4
(L$mg�1)
HYBRID ¼ 5.5069 � 10�3
Ibuprofen (IBP) Modified chitin pH ¼ 6, C0 ¼ 250e2000 mg L�1 Langmuir qmax ¼ 400.39 (mg
(L$mg�1)
R2 ¼ 0.962 _Z�ołtowska-
Aksamitowska et al.,
2018Acetaminophen (ACT) pH ¼ 6, C0 ¼ 250e2000 mg L�1 Freundlich KF ¼ 5.173 (L1/n$m 23 R2 ¼ 0.972
Ibuprofen (IBP) Chitosan-modified waste tire
crumb rubber
e Freundlich KF ¼ 5.21 � 10�9 ( 1/
n ¼ 6.349
R2 ¼ 0.9949 Phasuphan et al. (2019)
Diclofenac (DFC) e Freundlich KF ¼ 0.056(L1/n$m
n ¼ 2.78
R2 ¼ 0.9878
Naproxen e Freundlich KF ¼ 0.165 (L1/n$m
n ¼ 3.647
R2 ¼ 0.9863
Tetracycline (TC) Human hair-derived high surface
area material (HHC)
120 rpm, C0 ¼ 25e355 mg mL�1 Langmuir qmax ¼ 128.52 (mg
(L$mg�1)
R2 ¼ 0.9899 Ahmed et al. (2017)
120 rpm, C0 ¼ 25e355 mg mL�1 Langmuir qmax ¼ 210.18 (mg 8
(L$mg�1)
R2 ¼ 0.9948
Chlortetracycline
(CTC)
Calcium-rich biochar C0 ¼ 40e2000 mgmL�1, pH ¼ 6 ± 1 Freundlich KF ¼ 22.98 (L1/n$m 91 R2 ¼ 0.982 Xu et al. (2020)
Dimetridazole (DMZ) Biomass carbon foam pellets
(BCFPs)
C0 ¼ 40e600 mg mL�1 D-R qmD-R ¼ 0.0012 (m 28
(mol2$kJ�2)
R2 ¼ 0.9954 Sun et al. (2019)
Sulfamethazine (SMT) Polyamide (PA) 160 rpm, C0 ¼ 0e12 mg L�1 Linear K ¼ 38.7 (L$kg�1) R2 ¼ 0.982; c2 ¼ 0.0186 Guo et al. (2019a)
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$g�1); KL ¼ 0.053
$g�1); KL ¼ 0.02
$g�1); KL ¼ 0.05
(mg$g�1);
L$mg�1)
(mg$g�1);
L$mg�1)
g1�1/n$g�1); 1/
$g�1); KL ¼ 0.0858
�1); KS ¼ 1.78
2.54
1); KS ¼ 15.8 (Lns$m
1); KS ¼ 1.92 (Lns$m
�1); KL ¼ 4.78
g�1); CBET ¼ 9.549
$g�1); CBET ¼ 19.92
$g�1); KL ¼ 0.0017
g1�1/n$g�1); n ¼ 2.
L1/n$mg1�1/n$g�1);
g1�1/n$g�1); 1/
g1�1/n$g�1); 1/
$g�1); KL ¼ 0.0168
$g�1); KL ¼ 0.0045
g1�1/n$g�1); n ¼ 1.
ol$g�1); KDR ¼ 0.00
Abiotic
adsorbents
Polypropylene (PP) 160 rpm, C0 ¼ 0e12 mg L�1 Linear K ¼ 15.1 (L$kg�1) R2 ¼ 0.918; c2 ¼ 0.132
Polystyrene (PS) 160 rpm, C0 ¼ 0e12 mg L�1 Linear K ¼ 21.0 (L$kg�1) R2 ¼ 0.996; c2 ¼ 0.073
Cephalosporin C (CEP-
C)
Aged Polystyrene (PS) 160 rpm, Freshwater, C0 ¼ 0
e10 mg L�1
Linear K ¼ 21.0 (L$kg�1) Adj R2 ¼ 0.989 Guo and Wang (2019b)
160 rpm, simulated seawater,
C0 ¼ 0e10 mg L�1
Langmuir qmax ¼ 473.93 (mg$g�1); KL ¼ 0.05
(L$mg�1)
Adj R2 ¼ 0.988
Diclofenac (DFC) Covalent organic frameworks (COFs) 160 rpm at room temperature,
C0 ¼ 10e200 mg L�1
Sips qms ¼ 109 (mg$g�1); KS ¼ 0.0140.56
(Lns$mg-ns); ns ¼ 0.56
R2 ¼ 0.997 Zhuang et al., 2020a,
2020b
Sulfamethazine (SMT) 160 rpm at room temperature,
C0 ¼ 10e200 mg L�1
Sips qms ¼ 113.2 (mg$g�1); KS ¼ 0.0270.91
(Lns$mg-ns); ns ¼ 0.91
R2 ¼ 0.995
Amoxicillin (AMX) Metal-organic framework (MOF;
[Zn6(IDC)4(OH)2(Hprz)2]n)
pH ¼ 7, C0 ¼ 10e90 mg L�1 Langmuir qmax ¼ 486.4 (mg$g�1); KL ¼ 0.126
(L$mg�1)
R2 ¼ 0.9845 Abazari et al. (2019)
Tetracycline (TC) Metal-organic framework (MOF-5) e Langmuir qmax ¼ 232.558 (mg$g�1); KL ¼ 0.377
(L$mg�1)
R2 ¼ 0.9979
Mirsoleimani-azizi et al.,
2018
Modification of natural bentonite clay
(BC)
C0 ¼ 5e200 mg L�1 Freundlich KF¼ 10.76 (L1/n$mg1�1/n$g�1); n¼ 2.079 R2 ¼ 0.989; SSE ¼ 164.1 Maged et al. (2020)
Thermally activated bentonite (TB) C0 ¼ 5e200 mg L�1 Freundlich KF¼ 21.13 (L1/n$mg1�1/n$g�1); n¼ 1.944 R2 ¼ 0.997; SSE ¼ 163.5
Nalidixic acid (NA) Montmorillonite pH ¼ 4 Linear K ¼ 1.68 (L$g�1) R2 ¼ 0.996 Wu et al. (2013)
Kaolinite pH ¼ 4 Freundlich KF ¼ 0.45 (L1/n$mg1�1/n$g�1); 1/n ¼ 0.28 R2 ¼ 0.98
Tetracycline
Hydrochloride (TH)
Hierarchical porous ZIF-8 e Freundlich KF ¼ 46.2 (L1/n$mg1�1/n$g�1) R2 ¼ 0.965 Chen et al. (2019)
Chloramphenicol (CP) Hierarchical porous ZIF-8 e Freundlich KF ¼ 1.78 (L1/n$mg1�1/n$g�1) R2 ¼ 0.992
Azithromycin (AZM) Faujasite-type zeolites 1 (FAU1) C0 ¼ 0e400 mg L�1, pH ¼ 6.5 Langmuir qmax ¼ 8.5 (mg$g�1); KL ¼ 29.93
(L$mg�1)
R2 ¼ 0.847 de Sousa et al. (2018)
Faujasite-type zeolites 1 (FAU2) C0 ¼ 0e400 mg L�1, pH ¼ 6.5 Freundlich KF ¼ 9.4 (L1/n$mg1�1/n$g�1); 1/n ¼ 0.106 R2 ¼ 0.831
Sulfamethoxazole
(SMX)
Faujasite-type zeolites 1 (FAU1) C0 ¼ 0e400 mg L�1, pH ¼ 6.5 Freundlich KF ¼ 26.10 (L1/n$mg1�1/n$g�1); 1/
n ¼ 1.571
R2 ¼ 0.998
Faujasite-type zeolites 1 (FAU2) C0 ¼ 0e400 mg L�1, pH ¼ 6.5 Freundlich KF ¼ 105.46 (L1/n$mg1�1/n$g�1); 1/
n ¼ 1.847
R2 ¼ 0.994
Tetracycline (TC) Montmorillonite 180 rpm, C0 ¼ 0.1e8 mmol L�1 Langmuir qmax ¼ 1.06 (mmol$g�1); KL ¼ 5.74
(L$mmol�1)
R2 ¼ 0.99 Wu et al. (2019)
Ciprofloxacin (CIP) 180 rpm, C0 ¼ 0.1e8 mmol L�1 Langmuir qmax ¼ 0.51 (mmol$g�1); KL ¼ 11.1
(L$mmol�1)
R2 ¼ 0.99
Sulfachloropyridazine
(SCP)
Iron-modified clay 303 ± 1 K, C0 ¼ 1e2 mg L�1 Temkin b ¼ 9.029 kJ mol�1; A ¼ 2.924 L mg�1 R2 ¼ 0.998 Shikuku et al. (2018)
Sulfadimethoxine
(SDM)
303 ± 1 K, C0 ¼ 1e2 mg L�1 Temkin b ¼ 9.878 kJ mol�1; A ¼ 3.175 L mg�1 R2 ¼ 0.794
Sulfamethazine (SMT) Graphene 150 rpm, 298 K, C0 ¼ 2e100 mg L�1 Langmuir qmax ¼ 91.08 (mg$g�1); KL ¼ 0.201
(L$mg�1)
R2 ¼ 0.969 Zhuang et al. (2018b)
150 rpm, 308 K, C0 ¼ 2e100 mg L�1 Langmuir qmax ¼ 87.79 (mg$g�1); KL ¼ 0.238
(L$mg�1)
R2 ¼ 0.996
150 rpm, 318 K, C0 ¼ 2e100 mg L�1 Langmuir qmax ¼ 104.93 (mg$g�1); KL ¼ 0.234
(L$mg�1)
R2 ¼ 0.991
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Table 5
Applications of the adsorption isotherms in the adsorption of other types of organic pollutants.
Adsorbate Adsorbent
type
Adsorbent Adsorption conditions Optimum
isotherm
Model parameters Statistical parameters References
Oil & Grease Biosorbents Phragmites australis e Freundlich KF ¼ 10.84 (L1/n$mg1�1/n$g�1); 1/n ¼ 0.62 R2 ¼ 0.990 Shahawy and
Heikal (2018)BOD C0 ¼ 3500e20900 mg L�1 Langmuir qmax ¼ 2036.66 (mg$g�1); KL ¼ 0.00124 (L$mg�1) R2 ¼ 0.99
COD C0 ¼ 7300e43600 mg L�1 Langmuir qmax ¼ 4385.96 (mg$g�1); KL ¼ 0.000455 (L$mg�1) R2 ¼ 0.995
Perfluorooctanoate
(PFOA)
Activated carbon felts (ACFs) VS 120 rpm, pH ¼ 7 Langmuir qmax ¼ 0.0649 (mmol$m2); KL ¼ 0.0198 (L$mmol �1) R2 ¼ 0.998 Saeidi et al. (2020)
Perfluorooctanesulfonate
(PFOS)
120 rpm, pH ¼ 7 Langmuir qmax ¼ 0.154 (mg$g�1); KL ¼ 0.187 (L$mg�1) R2 ¼ 0.999
Perfluorooctanoate
(PFOA).
Activated sludge 150 rpm, C0 ¼ 0.08
e0.63 mmol L�1
Linear K ¼ 150e350 (L$kg�1) - Zhou et al. (2010)
Perfluorooctanesulfonate
(PFOS)
150 rpm, C0 ¼ 0.046
e0.93 mmol L�1
Linear K ¼ 200e4050 (L$kg�1) -
Phenol (Ph) Granulated cork 40 rpm Freundlich
Langmuir
KF ¼ 0.02 (L1/n$mg1�1/n$g�1); 1/n ¼ 0.86 qmax ¼ 0.92
(mg$g�1); KL ¼ 0.016 (L$mg�1)
R2 ¼ 0.98
R2 ¼ 0.98
Mallek et al.
(2018)
2-chlorophenol (2-CP) Freundlich
Langmuir
KF ¼ 0.05 (L1/n$mg1�1/n$g�1); 1/n ¼ 0.81 qmax ¼ 1.54
(mg$g�1); KL ¼ 0.029 (L$mg�1)
R2 ¼ 0.99
R2 ¼ 0.99
2-nitrophenol (2-NP) Langmuir qmax ¼ 5.09 (mg$g�1); KL ¼ 0.011 (L$mg�1) R2 ¼ 0.99
2,4-dichlorophenol (2,4-
DCP)
Freundlich KF ¼ 0.20 (L1/n$mg1�1/n$g�1); 1/n ¼ 0.80 R2 ¼ 0.99
Phenol (Ph) Chicken manure biochar 250 rpm, C0 ¼ 10
e200 mg L�1, pH ¼ 7.0
Langmuir qmax ¼ 106.2 (mg$g�1); KL ¼ 7.53 (L$mg�1) R2 ¼ 0.98 Thang et al. (2019)
2,4-dinitrophenol (DNP) Langmuir qmax ¼ 148.1 (mg$g�1); KL ¼ 6.45 (L$mg�1) R2 ¼ 0.99
Acenapthene Urban wood waste activated carbon 308 K, C0 ¼ 2e10 mg L�1 Temkin b ¼ 6.99 kJ mol�1; A ¼ 14.43 L mg�1 R2 ¼ 0.93 Barman et al.
(2018)Naphthalene Temkin b ¼ 0.25 kJ mol�1; A ¼ 2.64 � 1010 L mg�1 R2 ¼ 1
Methanol Activated carbon 207C e D-A qmD-A ¼ 0.15; nDA ¼ 1.72 R2 ¼ 0.9901 Zhao et al. (2012)
Activated carbon 207 E A e D-A qmD-A ¼ 0.28; nDA ¼ 2.08 R2 ¼ 0.9865
Activated carbon WS-480 e D-A qmD-A ¼ 0.27; nDA ¼ 1.78 R2 ¼ 0.9942
9-Nitroanthracene Abiotic
adsorbents
Polyethylene (PE) 60 rpm, C0 ¼ 10
e500 mg L�1
Linear K ¼ 34.00 (L$g�1) R2 ¼ 0.9818 J. Zheng et al.,
2020
Polypropylene (PP) 60 rpm, C0 ¼ 10
e500 mg L�1
Langmuir qmax ¼ 1.16 � 103 (mg$g�1); KL ¼ 0.04 (L$g�1) R2 ¼ 0.8802
Polystyrene (PS) 60 rpm, C0 ¼ 10
e500 mg L�1
Linear K ¼ 24.81 (L$g�1) R2 ¼ 0.9617
Naphthalene Functionalized multiwall carbon
nanotubes (MWCNT-OH)
C0 ¼ 5e30 mg L�1 Langmuir KL ¼ 277.78 (L$mg�1) R2 ¼ 0.9703 Akinpelu et al.
(2019)Fluorene C0 ¼ 5e30 mg L�1 Langmuir KL ¼ 1428.6 (L$mg�1) R2 ¼ 0.9517
Benzo[b]fluoranthene SieMCMe41 mesoporous
molecular sieve
150 rpm, C0 ¼ 150
e1000 mg L�1
Langmuir qmax ¼ 149.44 (mg$g�1); KL ¼ 0.012 (L$g�1) R2 ¼ 0.918 Costa et al., 2015
Benzo[k]fluoranthene 150 rpm, C0 ¼ 150
e1000 mg L�1
Langmuir qmax ¼ 147.09 (mg$g�1); KL ¼ 0.011 (L$g�1) R2 ¼ 0.942
Benzo[a]pyrene 150 rpm, C0 ¼ 150
e1000 mg L�1
Langmuir qmax ¼ 164.69 (mg$g�1); KL ¼ 0.014 (L$g�1) R2 ¼ 0.946
PCB-28 Coreeshell superparamagnetic
Fe3O4@b-CD composites
Room temperature Langmuir qmax ¼ 40.01 (mmol$kg�1); KL ¼ 12232.31
(L$mmol�1)
e Wang et al.
(2015b)
PCB-52 Room temperature Langmuir qmax ¼ 30.32 (mmol$kg�1);KL ¼ 380.96 (L$mmol�1) e
Perfluorooctanesulfonate
(PFOS)
Alumina 150 rpm, C0 ¼ 40
e400 mg L�1
Langmuir qmax ¼ 0.252 (mg$m2); KL ¼ 0.0587 (L$mg �1) R2 ¼ 0.938 Wang and Shih
(2011)
Perfluorooctanoate
(PFOA)
150 rpm, C0 ¼ 40
e400 mg L�1
Langmuir qmax ¼ 0.157 (mg$m2); KL ¼ 0.00908 (L$mg �1) R2 ¼ 0.977
Perfluorooctanoate
(PFOA)
Polyaniline nanotubes (PASNTs) 308 K Langmuir qmax ¼ 1651 (mg$g�1); KL ¼ 0.0716 (L$mg�1) R2 ¼ 0.9884
Perfluorooctanesulfonate
(PFOS)
308 K Langmuir qmax ¼ 1100 (mg$g�1); KL ¼ 0.06745 (L$mg�1) R2 ¼ 0.9928
Bisphenol A (BPA) Fe3O4@b-CD-CDI 298 K Langmuir qmax ¼ 52.68 (mg$g�1); KL ¼ 0.153 (L$mg�1) R2 ¼ 0.9910 Liu et al. (2020a)
318 K Langmuir qmax ¼ 47.03 (mg$g�1); KL ¼ 0.115 (L$mg�1) R2 ¼ 0.9753
Chrysene MIL-88(Fe) 200 rpm Langmuir qmax ¼ 42.378 (mg$g�1); KL ¼ 3.746 (L$mg�1) R2 ¼ 0.8665; RMSE ¼ 0.006; Zango et al. (2020)
NH2-MIL-88(Fe) 200 rpm Langmuir qmax ¼ 44.050 (mg$g�1); KL ¼ 0.820 (L$mg�1) R2 ¼ 0.988; RMSE ¼ 0.002;
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Ph
en
ol
N
at
u
ra
l
so
il
e
R
e
P
K
R
P
¼
2.
35
1
(L
$g
�
1
);
a R
P
¼
0.
05
36
9
(L
g $
m
g-
g )
;
g
¼
0.
96
34
R
2
¼
0.
99
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;
SS
E
¼
4.
70
02
2;
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R
ID
¼
3.
77
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02
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br
am
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ya
m
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d
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(2
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¼
5.
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¼
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;
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;
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(C
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¼
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m
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m
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¼
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.(
20
06
)
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tr
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(C
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¼
0.
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7
J. Wang, X. Guo / Chemosphere 258 (2020) 127279 17
Rangabhashiyam, 2019; Q. Zhang et al., 2016a, 2016b; Arami et al.,
2008; Yang and Hong, 2018; Wang et al., 2015a; b; Mahmoodi
et al., 2010; Nguyen et al., 2016; Deniz and Kepekci, 2016;
Bouras et al., 2017; Lebron et al., 2019; Beekaroo and Mudhoo,
2011; Xue et al., 2019; Haque et al., 2010; Chaari et al., 2019;
Hamza et al., 2018; da Silva and Pietrobelli, 2019; El-Zahhar et al.,
2014; Bhowmik et al., 2018; Wan et al., 2019; Karmakara et al.,
2019; Oveisi et al., 2018; Gamoudi and Srasra, 2019; Sarma
et al., 2018; Zheng et al., 2020; Ahsan et al., 2018; Li et al., 2019;
Chakraborty et al., 2018; Silva et al., 2020; Pi et al., 2017; Ringot
et al., 2007; _Z�ołtowska-Aksamitowska et al., 2018; Phasuphan
et al., 2019; Ahmed et al., 2017; Xu et al., 2020; Sun et al., 2019;
Guo et al., 2019a; Zhuang et al., 2020a, 2020b; Abazari et al., 2019;
Mirsoleimani-azizi et al., 2018; Maged et al., 2020; Wu et al.,
2013; Chen et al., 2019; de Sousa et al., 2018; Wu et al., 2019;
Shikuku et al., 2018; Shahawy and Heikal, 2018; Saeidi et al.,
2020; Zhou et al., 2010; Mallek et al., 2018; Thang et al., 2019;
Barman et al., 2018; Zhao et al., 2012; J. Zheng et al., 2020;
Akinpelu et al., 2019; Costa et al., 2015; Wang and Shih, 2011; Liu
et al., 2020a; Zango et al., 2020; Subramanyam and Das, 2014;
Torabian et al., 2014; Cheng and Hu, 2016; Ping et al., 2006). We
can see from Fig. 7 that the widely used statistical parameters are
the R2, adjust coefficient of determination (AdjR2), c2, SSE, root
mean square error (RMSE), and hybrid fractional error function
(HYBRID). The R2 values are most frequently calculated statistical
parameter. 81% of the reviewed references have adopted R2 to
evaluate the fitting results, because the R2 values can be simply
calculated by the Origin, SPSS, Excel software and so on. However,
the differences in the R2 values are small in statistics. For example,
the R2 values of the Sips and Langmuir models were 0.99 and
0.996 in the adsorption of BY28 onto Smectite rich natural clay
(Chaari et al., 2019). This result makes it difficult to determine the
optimum isotherms. Therefore, other statistical parameters also
should be calculated to evaluate the fitness. The calculation
equations of the statistical parameter are summarized in Table 6.
In following section, the above statistical parameters can be
calculated by a convenient UI.
5. Solving methods
We developed a convenient UI based on Excel to provide a
useful tool for solving the nonlinear adsorption isotherm models
and calculating the model parameters and statistical parameters.
The flow chart of the UI is provided in Fig. 8 (a). This Excel is
attached in the supplemental material. Prior to use, please
download the Excel and open it in editable view. Solver Add-in
should be activated. The way to activate Solver Add-in is
explained as following: select file, then go to options, select Add
ins, select Solver Add-in, and press OK. In addition, please ensure
that the references in Fig. 8 (b) are all added.
The instructions for this UI are depicted in detail in Fig. 9.
Three points should be noticed: (a) when use the Temkin, D-R and
D-A models, (0, 0) should not be input; (b) the independent and
dependent variables of the Volmer model are qe and Ce, respec-
tively; and (c) the ion exchange model is not provided in the UI,
because the values of ZA and ZB are different in different
adsorption processes, and the model equations are distinct for
different values of ZA and ZB.
To test the accuracy of the estimations of the parameters, two
sets of adsorption equilibrium data in Guo and Wang (2019d) are
tested by UI developed in this paper and by Origin (2018) soft-
ware. The results fitted by this UI are presented in Fig. 10, the
model parameters estimated by the UI and the Origin software
are summarized in Table 7. The UI successfully solved the
Fig. 6. Applications of isotherms (Nmodel is the number of optimum isotherms in literatures, L - Langmuir model, F - Freundlich model, T - Temkin model, S - Sips model, IE - ion
exchange model): (a): all types of adsorbate; (b): metals ions; (c) dyes; (d) pharmaceuticals; and (e) other types of organic pollutants. The left figures are the total amount of
adsorbate on all types of adsorbents. The right figures show the amount of adsorbate on biosorbents and abiotic adsorbents, respectively.
J. Wang, X. Guo / Chemosphere 258 (2020) 12727918
Table 6
Statistical parameters.
Error function Abbreviation Equation Description
Coefficient of determination R2
R2 ¼
P ðqmean � qcalÞ2P ðqcal � qmeanÞ2 þP ðqcal � qexpÞ2
qexp (mg$L�1): experimental adsorption capacity;
qmean (mg$L�1): average value of experimental adsorption capacity,
qcal (mg$L�1): calculated adsorption capacity;
Nexp: number of data points;
Npara: number of parameters.
Adjust coefficient of determination Adj R2
AdjR2 ¼ 1� ð1 � R2Þ ðNexp � 1ÞðNexp � Npara � 1Þ
Nonlinear chi-square c2
c2 ¼P ðqexp � qcalÞ2Þ
q2cal
Residual sum of squares error SSE SSE ¼P ðqexp � qcalÞ2
Root Mean Square Error RMSE
MSE ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
Nexp
X
ðqexp � qcalÞ2
s
Hybrid fractional error function HYBRID
HYBRID ¼ 100
Nexp � Npara
X qexp � qcal
qexp
Fig. 7. Applications of the statistical parameters.
Fig. 8. Flow char (a) and re
J. Wang, X. Guo / Chemosphere 258 (2020) 127279 19
isotherms and provided the estimations of the parameters. The
estimated parameters by the UI and the Origin software are almost
the same, which indicate that the UI can give accurate and reliable
calculations of the models. In addition,