Bullen et al. A revised pseudo-second order kinetic model 1
A revised pseudo-second order kinetic model for adsorption,
sensitive to changes in sorbate and sorbent concentrations
1*
Jay Bullen;
1
Sarawud Saleesongsom; and
1,2*
Dominik J. Weiss
1
Department of Earth Science and Engineering, Imperial College London, London SW7 2AZ, United Kingdom
2
Civil and Environmental Engineering, Princeton University, United States of America
*Corresponding authors:
Email: j.bullen16@imperial.ac.uk; d.weiss@imperial.ac.uk
Bullen et al. A revised pseudo-second order kinetic model 2
Graphical Abstract
Keywords
Adsorption kinetics; kinetic model; Lagergren; Ho and McKay; intraparticle diffusion; particle size
Abstract
Much contemporary research considers the development of novel sorbents for the removal of toxic
contaminants. Whilst these studies often include experimental adsorption kinetics, modelling is
normally limited to application of the pseudo-second order (PSO) rate equation, which provides no
sensitivity towards changes in experimental conditions and thus no predictive capability. We
demonstrate a relatively simple modification of the PSO model, with the final form dqt/dt = k’C
t
(1-
(q
t
/q
e
))^2 where k’=k
2
*(q
e
*^2)/C
0
*. We demonstrate that unlike the PSO model, this new rate
equation provides first-order dependence upon initial sorbate concentration (observed
experimentally as x
̄
=0.829±0.417), whilst rate constant k’ is significantly less sensitive to changes in
C
0
and C
s
than PSO rate constant k
2
. We demonstrate that this model improves predictive capacity
towards changes in C
0
and C
s
, particularly when q
e
is calculated using the Langmuir or Freundlich
adsorption isotherm. Finally, we explore how the new rate constant, k’, responds to changes in
sorbent morphology, identifying that particle radius is a better constraining parameter than surface
area. In this new equation, the conditionality of the rate constant upon experimental conditions is
significantly decreased, facilitating better comparison of new results with the literature.
Bullen et al. A revised pseudo-second order kinetic model 3
1. Introduction
There is a great wealth of recent literature concerning the development of novel sorbent materials for
the remediation of contaminated water. This includes composite materials offering superior stability
1
, ease of separation from effluent
2
3
, or multifunctional capabilities such as photocatalytic activity
4
5
. Understanding adsorption kinetics is important for identifying the minimum duration needed for
batch treatments and estimating maximum flow rates for column or continuous-flow treatments
6
.
Environments will be encountered with different concentrations of contaminants (with the initial
concentration denoted as C
0
), where different amounts of sorbent (with sorbent concentration
denoted as C
s
) are required to achieve desired filter device life-times. Predicting the range of
experimental conditions under which novel sorbents will be deployed is challenging, and laboratory
experiments can only hope to capture the approximate environments in which new sorbents will
operate. It is thus desirable to have an element of predictive capability towards understanding
adsorption kinetics.
There are two main classes of kinetic adsorption models, differing in their mechanistic description:
diffusion-controlled and adsorption-controlled. A summary of the major kinetic models in each
category is given in Table 1. For diffusion-controlled models, external diffusion is rarely considered
rate limiting
7
. Instead, adsorption rates are controlled by intraparticle diffusion, with a diffusivity
parameter reflecting the rate at which sorbate diffuses through the porous spaces within sorbent
particles
7
. This category includes the Crank model (1956)
8
which approximates the sorbent as
spherical particles and the concentration of sorbate at the surface is constant
7
. Adsorption kinetics
can often be linearised using the Weber-Morris model (1963)
9
, where q
t
is proportional to t
1/2
10
.
Adsorption-controlled kinetic models consider formation of the surface complex to be the rate
determining step. The original adsorption-controlled model was the Lagergren, or pseudo-first order
model (1898)
11
, with a reaction rate that is first order with respect to (q
e
-q
t
), where q
e
is the
concentration of sorbate adsorbed at equilibrium, and q
t
is the concentration of sorbate adsorbed at
time t
10
. The Elovich equation (1934)
12
is another adsorption-control kinetic model, where the
reaction rate decreases exponentially with increasing adsorption progress
10
.
At present, the pseudo-second order (PSO) rate equation, popularised by Ho and Mckay (1999)
13
, is
probably the most popular model used to describe adsorption kinetics
14
, especially for new and novel
sorbent materials. The PSO rate equation is the same as the earlier Lagergren model, save that the
dependence of reaction rate upon (q
e
-q
t
) is second-order rather than first-order. The PSO rate
equation takes the form:
Equation 1
where t is time (minutes), q
t
is the amount of sorbate adsorbed at time t (mg g
-1
), k
2
is the pseudo-
second order rate constant (g mg
-1
min
-1
), and q
e
is the amount of sorbate adsorbed at equilibrium
(mg g
-1
)
13
. The vital limitation of the PSO is that it does not explicitly include the variables C
0
and C
s
(the initial concentration of sorbate and sorbent, respectively) and therefore lacks sensitivity to
changes in C
0
and C
s
. The PSO therefore does not allow us to identify appropriate sorbent
Bullen et al. A revised pseudo-second order kinetic model 4
concentrations for water treatment design under different contaminant concentrations. Table 1
demonstrates that the lack of sensitivity towards C
t
and C
s
is common to most adsorption kinetic
models – they are without predictive capacity, and their parameters are valid only for the particular
experimental conditions under which they were calculated.
Furthermore, k
2
is only valid for the experimental conditions under which it was determined, and so
comparison of literature studies with different experimental values of C
0
and C
s
can lead to false
conclusions
15
. The ability to normalise rate constants to C
0
and C
s
would thus be another major
advantage, allowing improvements in sorbent engineering to be truly identified and validated.
The aim of this work was to develop a kinetic adsorption model that could be used to predict changes
in adsorption kinetics as a function of sorbate and sorbent concentration. Furthermore, we explored
how this model could provide normalised rate constants for better comparison between different
sorbents. Our motivation towards both these aims was to help researchers consider the operating
conditions under which their sorbents may be best used, e.g. comparative modelling of batch and
continuous-flow systems
6
.
We used the PSO model as a starting point, due to it being well-known and ubiquitous, and capable
of describing most kinetic data, despite doubts over whether adsorption or diffusion-controlled
models are more mechanistically appropriate
16
. To achieve these aims, we (1) collected a wide range
of literature where the influence of C
0
and C
s
on adsorption kinetics were investigated; (2) determined
the relationship between C
0
and C
s
on adsorption kinetics within this data set; (3) built the observed
C
0
and C
s
dependence into a modified form of the PSO model; (4) verified that the rate constants of
this new model are less conditional than PSO rate constant k
2
; (5) used the modified model to predict
experimental data; (6) explored the influence of surface morphology on the remaining variance
between rate constants in the literature.
Bullen et al. A revised pseudo-second order kinetic model 5
Table 1: A selection of kinetic models widely used to describe adsorption. The table illustrates that in most of these kinetic
models
is sensitive to neither changes in C
0
nor changes in C
s
, and are thus limited in predictive capability. C
0
is the initial
concentration of aqueous sorbate, C
t
is the concentration of aqueous sorbate at time t, C
e
is the concentration of aqueous
sorbate at equilibrium, q
t
is the quantity of sorbate adsorbed at time t (e.g. mg g
-1
), q
e
is the quantity of sorbate adsorbed at
equilibrium (mg g
-1
), k is the rate constant, a is the desorption constant, α is the initial adsorption rate, D is molecular diffusion
coefficient of the sorbate in solution, r is the radial coordinate, ρ is particle density, ε
r
is particle porosity, N is the local
concentration of sorbate in the adsorbed phase (presumably equivalent to q
t
), D
s
is surface diffusivity (assumed constant), θ
t
is surface coverage at time t, θ
e
is surface coverage at equilibrium, u
e
is the relative sorbate uptake at equilibrium, with
, k
a
is a rate constant for adsorption. q
e
*
2
and C
0
*
2
denote the values of q
e
and C
0
used to determine k
2
from a given
experimental data set.
Model
Equation
Is the rate equation
sensitive to changes in C
t
?
Is the rate equation
sensitive to changes in
C
s
?
Reference
Intraparticle diffusion-controlled models
Crank model
Yes
No
17
Simplified
Crank model
No
(model assumes constant
surface concentration of
sorbate)
No
7
Web and
Morris
No
No
7
Adsorption-controlled models
Pseudo-first
order
No
No
(unless q
e
is replaced
with a variable term)
10
Pseudo-
second order
No
No
(unless q
e
is replaced
with a variable term)
10
Elovich
equation
No
No
10
Second-order
rate equation
(aqueous
phase)
Yes
(second order)
No
10
Integrated
kinetic
Langmuir
model
Yes
(but C
0
only and not C
t
)
Yes
(θ
e
and u
e
will change
with C
s
)
18
This work
where
Yes
(1
st
order)
Yes
(q
e
is calculated at each
time interval using
adsorption isotherms)
This work
Bullen et al. A revised pseudo-second order kinetic model 6
2. Experimental
2.1. Data sets
Literature sources that experimentally investigated the influence of C
0
, C
s
or particle size upon
adsorption kinetics were compiled (references are given in the Supplementary Information). Studies
using oxyanions, metal cations and organic dyes as the sorbate were included, to ensure our findings
would be general and non-sorbate-specific. Both mineral sorbents and organic sorbents (activated
carbon and chitosan) were included, however zeolites and metal-organic frameworks (MOFs) were
not since the sorption mechanism of sorbate trapping within cages might present inconsistent results.
None of the literature sources found gave any mechanistic account or mathematical explanation for
observed differences in adsorption kinetics due to varying C
0
, C
s
or particle size. The majority of
literature sources used the PSO equation to model adsorption kinetics.
In total 79 literature sources with around 200 kinetic experiments were collected. To investigate the
influence of initial sorbate concentration, 11 literature sources with a combined 43 kinetic
experiments were collected. For initial sorbent concentration, 11 literature sources with 43
experiments were collected. For particle size, 15 data sets from 6 literature sources, with a combined
47 experiments were collected. A data set was considered to be all kinetic experiments using the same
sorbate-sorbent system within a single literature source.
2.2. Calculation of initial rates
To investigate the dependency of adsorption kinetics on C
0
, C
s
and particle radius, the method of initial
rates was used
19
. Experimental kinetic data from the literature was tabulated and initial reaction rates
were determined. Many data sets lacked good resolution for the initial stages of adsorption, e.g. in
~30% of all experiments reaction progress had already exceeded 50% when the first data point after
mixing sorbent with sorbate was collected. Calculating the initial rate (the rate of reaction at t=0) using
the slope of q
t
versus time at the earliest data points would lead to a systematic under-calculation of
the initial reaction rate, as early curvature in adsorption kinetics would be missed (Figure 1). This error
would become more significant the later that initial adsorption data is collected. Determination of
initial adsorption kinetics using the slope of q
t
versus t would also give a random error associated with
calculating rates from two data points only.
To provide a better estimation of the initial rate, the PSO model was fit to each data set and the rate
calculated at t=0. PSO parameters (k
2
and q
e
) were determined using the linearised form of the PSO
rate equation, which is as follows:
Equation 2
Adsorption kinetic profiles were plotted as
against t and the linear regression obtained using the
LINEST function in Excel. q
e
was obtained via the relationship
where m is the gradient of the
Bullen et al. A revised pseudo-second order kinetic model 7
linear regression, and k
2
via
where c is the y-intercept. The initial rate of adsorption
was calculated through simplifying Equation 1 as:
Initial rate =
Equation 3
Parameters were normalised to the same units for ease of comparison: k
2
– g mg
-1
min
-1
; q
e
– mg g
-1
;
and initial rate – mg g
-1
min
-1
. The advantages in determining the initial rate of adsorption using the
PSO model compared with the initial slope of q
t
versus time is demonstrated in Figure 1. In the
example, the initial rate was three times greater when calculated using the PSO model compared with
the initial slope (4.4 versus 1.3 mg g
-1
min
-1
). In this experiment, the first data point collected after
mixing gave a value of q
t
= 4 mg g
-1
. Given that q
e
= 5.4 mg g
-1
, adsorption progress had already reached
75% completion when the first data was collected, and it is therefore not surprising that the slope of
q
t
versus time gives a significantly lower initial rate compared with the PSO model: the PSO includes
curvature within its interpolation of early adsorption kinetics (interpolating between t=0 and 3
minutes). Not only are systematic errors reduced, but the PSO method also reduces the random error
due to uncertainties in determination of sorbate concentrations, since most kinetic experiments gave
at least 5 data points with which to fit the PSO parameters, rather than just two data points used in
the initial slope method.
(a)
(b)
Figure 1: Illustration of how calculation of initial rates using the slope between the first two data points of q
t
as a function of
t gives a systematically lower result than when fitting the PSO model using the linearised plot of t/q
t
as a function of t. The
data presented is from the Cr(VII)/Mg-Al-CO
3
data set with 10 mg L
-1
sorbate and 1.5 g L
-1
sorbent
20
. The initial rate was
calculated to be 8.5 mg g
-1
min
-1
when using the PSO method with only the first two data points.
2.1. Determining the order of reaction
The order of reaction with respect to C
0
, C
s
or particle radius was determined by calculating the
gradient of log(initial rate) versus log(independent variable)
19
, where each data point represents a
single kinetic experiment within a given literature study where the influence of either C
0
or C
s
on
adsorption kinetics was investigated.
Bullen et al. A revised pseudo-second order kinetic model 8
Equation 4
For each data set, the order of reaction with respect to the independent variable was calculated using
the LINEST formula in Microsoft Excel. The error of each data set was calculated as the standard error
of the slope. A generalised order of reaction was calculated as the average (mean, x
̄
) and median (x
͂
)
order of reaction between all data sets collected, with errors reported as the standard deviation
between all data sets. The dependencies of k
2
and k’ upon C
0
, C
s
and r were determined using the
same method, substituting initial rates for k
2
or k’.
2.2. Modelling
As the modified rate equation was not easily linearised, this new model was simulated using Microsoft
Excel. The quantity of sorbate adsorbed at the n
th
data point was calculated using the following
formula:
Equation 5
With the concentration of aqueous sorbate remaining, C
t
, decreasing according to the following
equation:
Equation 6
The time interval between data points,
or Δt, was minimised until the magnitude of Δt
had no significance on the observed kinetics.
Bullen et al. A revised pseudo-second order kinetic model 9
3. Results and Discussion
3.1. Influence of C
0
and C
s
upon adsorption rates
We first aimed to establish the influence that initial sorbent concentration, C
0
, and sorbent
concentration, C
s
, have on adsorption kinetics. We collected a range of experimental data, with
oxyanions, metal cations and organic dyes represented as sorbates, as well as single component
minerals, composite systems and activated carbon represented as sorbents. We calculated the initial
rate of reaction (interpolated from the linearised PSO plot) and calculated the order of reaction with
respect to C
0
and Cs as the gradient in the slope of log(initial rate) as a function of log(C
0
or C
s
). In total
11 data sets with a combined 43 kinetic experiments were collected to investigate the influence of C
0
.
For C
s
, 11 data sets with a combined 43 kinetic experiments were collected.
The data compiled from the literature generally shows a linear relationship between initial sorbate
concentration and initial rate (normalised to mass, mg g
-1
min
-1
). This is shown in Figure 2a where the
average result for the gradient between log(initial rate) and log(C
0
) was x
̄
=0.829±0.417, x
͂
=0.801.
Despite large uncertainties, the average and median results were significantly closer to first-order than
zero-order or second-order. This is intuitive for both diffusion and adsorption-controlled mechanisms,
as twice as much sorbate should lead to twice as much sorbate flux from sorbent surface into pores,
and collisions with the sorbent surface should be twice as frequent.
Similarly, a first order dependence of reaction rate on C
s
was observed in the compiled data sets. This
is shown in Figure 2b where the average gradient was x
̄
=1.18±0.61, x
͂
=1.15. This is also intuitive for
both diffusion and adsorption-controlled mechanisms, as when C
s
is doubled, the total surface area is
doubled, and the overall flux of sorbate entering sorbent pores is doubled, whilst the rate of collisions
between sorbate and sorbent surface is also doubled. When normalised to mass (mg g
-1
min
-1
) the
initial rate was zero-order with respect to C
s
, as expected. Again, despite large statistical errors, the
relationship between initial rate and C
s
was significantly closer to first-order (when rate is normalised
to mg L
-1
min
-1
) than both zero-order and second-order. For both C
0
and C
s
, the median result was very
close to the average, indicating that the analysis was not significantly skewed by any outlying data
sets.
Bullen et al. A revised pseudo-second order kinetic model 10
(a)
(b)
Figure 2: The initial rate of adsorption as a function of (a) initial sorbate concentration (C
0
) and (b) initial sorbent
concentration (C
s
). Each data point represents a single kinetic experiment – each experiment was fitted with the pseudo-
second order model and the rate at time t=0 calculated as discussed in the experimental section. Kinetic experiments are
grouped into data sets, where all conditions except for (a) C
0
or (b) C
s
were kept constant. The gradients of most data sets
align with the 1:1 gradient (grey guidelines given for visual reference) indicating that the adsorption reaction is first order
with respect to the independent variables C
0
and C
s
.
3.2.
Conditionality of k
2
and limitations to predictive capabilities of the PSO
To predict the sensitivity of adsorption kinetics as a function of changes to C
0
and C
s
, and to facilitate
comparison of adsorption kinetics within the literature, it is necessary that the rate constant
determined under one set of experimental conditions can used to model further experiments differing
in C
0
and C
s
. Having identified a first-order dependence of reaction rate on C
0
and C
s
, we aimed to
identify the influence that independent variables C
0
and C
s
have on k
2
, using the same data sets.
Bullen et al. A revised pseudo-second order kinetic model 11
Whilst section 3.1 demonstrated that adsorption kinetics are approximately first order with respect
to C
0
(Figure 2a), the PSO rate constant, k
2
, is inversely proportional to C
0
, i.e. doubling initial sorbate
concentration results in the rate constant k
2
decreasing by a factor of 2 (Figure 3a). The inverse
relationship between k
2
and C
0
is explained by the second order dependence of PSO on the absolute
concentration of available surface sites, (q
e
-q
t
), and is demonstrated mathematically in the following
discussion.
(a)
(b)
Figure 3: Pseudo-second order (PSO) rate constant k
2
as a function of (a) initial sorbate concentration (C
0
) and (b) initial
sorbent concentration (C
s
). Grey lines indicate gradients and thus reaction orders of (a) -1 and (b) +1.
Many studies of adsorption kinetics operate under conditions where the sorbent remains unsaturated
at equilibrium. This is natural as the authors wish to investigate the conditions under which their
sorbent successfully removes contaminants. At low values of C
e
, a linear relationship is observed
between C
e
and q
e
, known as Henry’s adsorption isotherm
21
. If we approximate the adsorption
isotherm as a linear relationship between C
e
and q
e
, and let coefficient a represent the factor increase
in C
0
between experiments (1) and (2), we get the following:
Bullen et al. A revised pseudo-second order kinetic model 12
Equation 7
where coefficient a is equal not just to the quotient of C
0(2)
and C
0(1)
, but also the quotient of reaction
rates, given the first order dependence of reaction rate upon C
0
observed in section 3.1. q
e(2)
can thus
be substituted for the product of coefficient a and the old q
e(1)
:
Equation 8
Rearrangement of Equation 1 gives the following:
Equation 9
which at time t=0 reduces to:
Equation 10
Substitution of q
e(2)
with aq
e(1)
as per Equation 8 gives:
Equation 11
Substituting initial rate (2) for the product of a and initial rate (2) as per Equation 7 gives:
Equation 12
This mathematically demonstrates that k
2
is inversely proportional to C
0
, to a first approximation,
when the sorbent is unsaturated and adsorption can be represented by the linear adsorption
isotherm. This was observed experimentally, as whilst the reaction rate only doubles with a doubling
of C
0
(Figure 2a), the PSO rate constant, k
2
, conditional to the value of C
0
used, decreased by a factor
of two. The final results for the order of reaction with respect to C
0
were x
̄
=-0.761 ±0.663, x
͂
=-0.765,
N=11 data sets and 43 kinetic experiments. The standard deviation is large, nearly including x=0 for a
zero-order relationship between k
2
and C
0
.
Most data sets showed first-order dependence of k
2
upon C
s
(Figure 3b). This can also be explained by
considering adsorption isotherms and the term (q
e
-q
t
)
2
. If coefficient b indicates the factor increase in
C
s
between two experiments, then:
Equation 13
Bullen et al. A revised pseudo-second order kinetic model 13
When the sorbent is saturated and C
e
>>(C
s
*q
e
), the adsorption isotherm approaches a plateau and
increasing C
s
has limited effect on q
e
.
Equation 14
Since a zero-order dependence of rate (when normalised to mass, mg g
-1
min
-1
) upon C
s
was identified
in section 3.1, then:
Equation 15
From Equation 10, we obtain:
Equation 16
C
s
does not enter the equations, and thus k
2
is predicted to have zero order dependence upon C
s
at
high values of C
e
where the adsorption isotherm reaches a plateau.
We can also consider the opposite case, being the low C
e
region of the adsorption isotherm. When
C
e
<<(C
s
*q
e
) and the majority of sorbate is removed by adsorption at equilibrium regardless of sorbent
concentration, and increasing C
s
causes a decrease in q
e
as limited sorbate is divided across a larger
surface area. In this case:
Equation 17
If Equation 15 based upon our observations from section 3.1 is still valid, then:
Equation 18
Which gives a second-order dependence of k
2
upon C
s
.
It therefore follows that the low and upper bounds of k
2
dependence upon C
s
are zero and second-
order respectively. These bounds fit the majority of data sets, as shown in Figure 3b. The final results
for the order of reaction with respect to C
0
were x
̄
=1.16 ±1.28, x
͂
=1.39, N=11 data sets and 43 kinetic
experiments. Zero, first and second-order dependencies between k
2
and C
s
are all included within the
standard deviation. Out of 11 data sets, three showed a relationship between k
2
and C
s
outside of the
predicted constraints: methylene blue/raffia fibres demonstrated a dependency of 3.1 ±0.2, and the
two Hg(II)/biosorbent data sets gave a dependency of -1.1 ±0.2.
Having demonstrated the conditionality of PSO rate constant k
2
with respect to C
0
and C
s
, we
investigated the implications of this conditionality with regards to predictive adsorption kinetic
Bullen et al. A revised pseudo-second order kinetic model 14
modelling. For predictive modelling, a single rate constant should be able to model experiments
differing in C
0
and C
s
. We used a single value of k
2
, being the average value of k
2
for all kinetic
experiments within a given data set, to model experiments with different initial conditions (C
0
and C
s
).
The unique value of q
e
determined for each kinetic experiment was kept for modelling. All parameters
are tabulated in the supplementary information. Figure 4 shows that the resulting PSO model cannot
accurately describe data sets differing in C
0
or C
s
when a fixed value for rate constant k
2
is used. The
implication is that (a) the PSO cannot be used to predict changes in adsorption kinetics as a function
of C
0
and C
s
, and (b) comparison of the PSO rate constant, k
2
, between different literature studies, is
only valid if PSO parameters were determined under the same experimental conditions (C
0
and C
s
).
(a)
(b)
(c)
(d)
Figure 4: Demonstrating the lack of predictive capability in the PSO model due to the C
0
and C
s
conditionality of k
2
. A selection
of literature data sets were modelled using PSO kinetics with a single value of k
2
, calculated as the average value of PSO rate
constant, k
2
, across all kinetic experiments within the given data set. (a) and (b) are data sets wherein C
0
is varied whilst (c)
and (d) are data sets where C
s
is varied. Experimental data is from
22
23
24
20
.
Bullen et al. A revised pseudo-second order kinetic model 15
3.3. Development of a kinetic model sensitive to changes in C
0
and C
s
To provide sensitivity when predicting changes in adsorption kinetics due to changes in C
0
and C
s
, we
modified the PSO rate equation, giving a new pseudo-second order kinetic model, as follows. Firstly,
to remove the inverse relationship between the rate constant upon C
0
(Figure 3a), the right-hand side
of the equation must be made proportional to C
0
(Figure 2a) and not proportional to C
0
2
, as the
unmodified PSO equation approximates to through the term (q
e
-q
t
)
2
(Equation 1). This problem is
solved by replacing (q
e
-q
t
)
2
, which provides second order dependence on the absolute amount of
adsorption capacity remaining, with a new term providing second order dependence on the relative
amount of adsorption capacity remaining. The term used herein is of the form
. This
modification of Equation 1 gives the following:
Equation 19
where
2
. Here, q
e
*
denotes the equilibrium concentration of adsorbed sorbate in the
particular kinetic experiment used to calculate k
2
. At time t=0, this new term,
, always
returns a value of 1, independent of the sorbate concentration.
The first order dependence of rate upon the sorbate concentration can then be explicitly defined,
giving the final equation:
Equation 20
where
with C
0
*
denoting the initial sorbate concentration in the kinetic experiment used
to calculate k
2
. Modification of the equation to this form, with second order dependence on the
relative concentration of unused adsorption capacity rather than the absolute concentration, results
in no change to adsorption kinetics when simulated under the same experimental conditions.
Figure 2b demonstrated a first-order dependence of initial rate upon C
s
when normalised to volume
(mg L
-1
min
-1
) and zero order when normalised to sorbent mass (mg g
-1
min
-1
). The current modified
equation is dependent on the relative availability of adsorption capacity rather than the absolute
availability. Any changes in q
e
due to varying C
s
do not affect the term
)
2
at t=0, as it continues
to reduce to unity. Therefore, when the rate equation is normalised to mg g
-1
min
-1
(i.e.
rather
than
) no modification of the rate equation for C
s
is required. Our modified rate equation, Equation
20, is similar to the adsorption-only form of the kinetic Langmuir model (kLm), which at high surface
coverage is first order with respect to C
t
and second order to
)
25
18
.
Bullen et al. A revised pseudo-second order kinetic model 16
3.4. Confirming that k’ is less conditional than k
2
A predictive adsorption kinetic model, sensitive to changes in C
0
and C
s
, requires a rate constant that
is unaffected by either independent variable. Since the ideal rate constant is therefore entirely
unaffected by changes to C
0
and C
s
, the slope of log(initial rate) as a function of log(C
0
) or log(C
s
) should
return a gradient of zero, indicating zero-order dependence. We calculated the dependency of k’ upon
C
0
and C
s
identically as for k
2
in section 3.2. We then validated whether k’ is less conditional than k
2
by
identifying whether Δlog(k’)/Δlog(C
0
or C
s
) or Δlog(k
2
)/Δlog(C
0
or C
s
) returns the gradient closes to zero.
The visual comparison of k
2
and k’ dependencies upon C
0
and C
s
is given in Figure 5. k
2
displayed an
inverse dependence on C
0
with x
̄
=-0.761±0.663, x
͂
=-0.765, and an approximately first-order
dependence on C
s
with x
̄
=1.16 ±1.28, x
͂
=1.39. The new rate constant k’ was significantly closer to zero-
order dependency than any other relationship with regards to both C
0
and C
s
. For C
0
the results were
x
̄
=-0.28±0.53, x
͂
=-0.38 for C
0
dependency, and x
̄
=0.04±0.61, x
͂
=-0.02 for C
s
dependency. The standard
deviation values of k’ were also smaller than for k
2
, indicating that the relationships between
independent variables and rate constants were most consistent across data sets for k’ than k
2
. These
results suggest that the new rate constant k’ is less conditional than k
2
, and that the dependency of
adsorption kinetics upon C
0
and C
s
has been captured by the new rate equation, at least in part, if not
fully.
Figure 5: Mathematical demonstration of reduced sensitivity to C
0
and C
s
in the new rate constant k’ compared with k
2
. Where
the rate constant is not influenced by changes in C
0
and C
s
, a gradient of the log(rate constant) versus either log(C
0
) or log(C
s
)
will return a value of zero (indicating that the rate constant does not change). A value of +1 indicates a first order relationship,
-1 indicates an inverse relationship and so on. The white shaded area of the plot indicates experimental data sets where k’
was less sensitive to changes in C
0
and C
s
than k
2
. The grey shaded area indicates data sets where k’ varied more with changes
in C
0
and C
s
than k
2
.
3.5. Use of adsorption isotherms for modelling adsorption kinetics
The PSO model traditionally uses a unique, fixed value of q
e
to model each adsorption kinetic
experiment. However, q
e
is sensitive to both C
0
and C
s
, and so the single value of q
e
obtained under
Bullen et al. A revised pseudo-second order kinetic model 17
one set of experimental conditions cannot be used to predict adsorption kinetics once C
0
and C
s
are
changed. To provide appropriate sensitivity, the modified kinetic model must therefore have a q
e
term
that is sensitive to sorbate concentration. This can be achieved by replacing q
e
with adsorption
isotherms such as the Langmuir or Freundlich models
21
. Using adsorption isotherms, q
e
can either be
set as the concentration of adsorbed sorbate at equilibrium (as per convention), or q
e
can be
recalculated at each point in time, giving an out-of-equilibrium value of q
e
which decreases throughout
the kinetic experiment as adsorption progress increases. In this second case, the term C
e
within the
adsorption isotherm would be replaced with C
t
. Huang et al. previously demonstrated that this second
case actually gives better account of the true driving force of the reaction during the initial stages of
adsorption
26
. Recalculating ‘q
e
’ using C
t
in this way is also more appropriate when modelling column
systems, where C
t
<<C
e
for the majority of the experiment. Langmuir and Freundlich adsorption
isotherm data from the literature was incorporated into the modified rate equation (Equation 19), and
a selection of data sets were then modelled using this approach, presented in section 3.6.
3.6. Validating improvements in the modified equation
In order to validate our modified model, we used Equation 19 to model the same selection of data
sets previously modelled using the PSO model with an average value of k
2
(Figure 4). A single value of
k’ was used to model all kinetic experiments within each data set, taken as the average value of k’
across the data set. All parameters are tabulated in the supplementary information. The results are
given in Figure 6 and in all cases the goodness of fit (represented by R
2
) was improved compared with
when predicting adsorption kinetics using the PSO model with a value of k
2
taken as the average
between all kinetic experiments within the data set. The results were R
2
=0.9735 vs. 0.9619 for
As(III)/HFO (with C
0
varied), R
2
=0.8225 vs. 0.5248 for As(V)/Fe
2
O
3
(C
0
varied), R
2
=0.8833 vs. 0.7575 for
Cd/Fe
2
O
3
(C
s
varied) and R
2
=0.9751 vs. 0.9321 for Cr(VI)/Mg-Al-CO
3
(C
s
varied). This is especially
significant given that in section 3.2 a unique value of q
e
determined by experimental fitting was used
for the PSO model, whilst for the modified kinetic model, q
e
was calculated using a single adsorption
isotherm (Langmuir or Freundlich) for all kinetic experiments within the data set. The modified model
thus not only provided a better fit than the unmodified pseudo-second order model, but provided a
better fit whilst using fewer fitting parameters.
Bullen et al. A revised pseudo-second order kinetic model 18
(a)
(b)
(c)
(d)
Figure 6: Modelling adsorption kinetics using the modified model. For each data set a single value of rate constant k’ was
used (chosen as the average value of k’ between all kinetic experiments within the given data set) and q
e
was determined for
all points in time, within all kinetic experiments using the same adsorption isotherm parameters. The single kinetic model was
used to model experiments differing in C
0
(a and b) and C
s
(c and d). The Freundlich adsorption isotherm was used for all
experiments in (a) and (d), and the Langmuir adsorption isotherm for all experiments in (c) and (d). Experimental data is from
22
23
24
20
.
The q
t
values predicted by the PSO and modified models were cross-calibrated against the values of
q
t
observed experimentally (Figure 7). Using the unmodified PSO with a single value of k
2
but unique
experimentally-fitted values of q
e
gave a cross-calibration slope of 1.0770±0.0186 (R
2
=0.9638), i.e. an
error of 7.7% and uncertainty of 1.7% (Figure 7a). The cross-calibration between experiment and
model was improved when using the modified kinetic model, with a slope of 0.9598±0.0061 (R
2
=
0.9949) – an error of 4.0% and an uncertainty of 0.6%, both lower than in the unmodified PSO model.
In the logarithmic form (Figure 7b) the cross-calibration slope of the modified model versus
experiment was closer to unity with a gradient of 1.0117±0.0063 and R
2
= 0.9957, indicating that the
linear regression in the previous panel (Figure 7a) is skewed by the few data points at high q
t
values.
Again, there is a better goodness of fit in the modified model compared with the original PSO
(slope=1.0104±0.0106 and R
2
=0.9880). The modified rate equation presented in this work therefore
goes some way towards providing sensitivity towards changes in C
0
and C
s
.
Bullen et al. A revised pseudo-second order kinetic model 19
(a)
(b)
Figure 7: Calibration curves between the experimentally observed q
t
and q
t
predicted by the original PSO and modified kinetic
models in (a) absolute form and (b) logarithmic form. Again, a single averaged value of k’ was used to model all kinetic
experiments in a single data set. N = 132 data points were included in the calibration plots. Shown are data points for the
original PSO model (blue open squares) and the modified model (black filled circles) along with their linear regressions (blue
dotted and black solid lines respectively). The red dashed line indicates the one-to-one slope for a perfect model.
Using the original PSO model, two unique parameters (k
2
and q
e
) were required for each kinetic
experiment within each data set, and no predictive capability was offered. In the modified model,
one kinetic parameter, rate constant k’, along with two adsorption isotherm parameters (K
L
and q
max
for Langmuir or K
F
and n for Freundlich), can be used to model multiple kinetic experiments, offering
some (though clearly imperfect) predictive capability.
3.7. Influence of sorbent morphology on rate constant k’
Whilst rate constants were successfully normalised to differences in experimental conditions C
0
and
C
s
by converting the PSO model with rate constant k
2
to the modified rate equation with rate constant
k’, values of k’ calculated for given sorbate-sorbent combinations still showed significant variation
across the literature (Supplementary Information). One important factor not yet discussed is sorbent
morphology, i.e. particle size, porosity and surface area, which all vary depending on the preparation
of the given sorbent sample. We thus investigated whether sorbent morphology accounts for the
remaining variance in rate constants.
3.7.1. Surface area
The influence of surface area on the initial rate of reaction could not be probed as fully as the influence
of C
0
and C
s
was in section 3.1, due to a lack of literature investigating the effects of surface area on
adsorption kinetics. However, an example of the influence of surface area on initial kinetics is given in
Figure 8. Instead, different literature sources with the same sorbate-sorbent combinations were
compared on the basis of surface area.
Bullen et al. A revised pseudo-second order kinetic model 20
Figure 8: Example of the influence of surface area on initial kinetics. Experimental data from Di et al.
27
.
A correlation between increasing surface area and k’ was identified from a comparison of the compiled
literature data (Figure 9). For arsenic oxyanions with mineral sorbents this correlation was weak: the
average gradient of log(k’) as a function of log(surface area) was just 0.404±0.472 (R
2
=0.287 only, N=5
sorbate-sorbent combinations). The correlation was more significant when considering all arsenic
oxyanion-sorbent combinations as a single data set, giving a gradient of 0.564±0.192 (R
2
=0.291, N=23
data sets). Whilst the upper bound of log(k’) values increased with increasing surface area, the lower
bound changed little, and this generated the large uncertainty in the gradient and the source for low
R
2
values.
Similar results were found for activated carbon/chitosan data sets: whilst the upper bound of log(k’)
increased with surface area, the lower bound changed little. The average gradient was 0.150±0.167
(R
2
=0.043, N=4 sorbate-sorbent combinations), whilst the combination of all data into a single data
set gave a gradient of 0.197±0.535 (R
2
=0.292, N=20). The fluoride/Al
2
O
3
and methylene blue/TiO
2
systems showed the strongest correlations. For fluoride/Al
2
O
3
the gradient was 4.36±2.49 (R
2
=0.435,
N=6), and for methylene blue/TiO
2
the gradient was 0.97±0.81 (R
2
=0.419, N=3).
High surface areas can be achieved both by reducing particle radius and by introducing greater
porosity. The poor goodness of fit in the linear regression might be due to a convolution of particle
radius and porosity effects within the measured surface area, meaning that surface area is an
inappropriate choice for constraining variables. It is also worth noting that the PSO rate constant k
2
tended to decrease with increasing surface area (Supplementary Information). This is explained by the
term (q
e
-q
t
)
2
within the PSO rate equation and the fact that q
e
(mg g
-1
) is roughly proportional to
surface area.
Bullen et al. A revised pseudo-second order kinetic model 21
(a)
(b)
(c)
Figure 9: Greater values of rate constant k’ were obtained with increasing surface area, demonstrated for (a) arsenic oxyanion
and mineral sorbent systems, (b) activated carbon and chitosan systems, and (c) fluoride/Al
2
O
3
and methylene blue/TiO
2
.
Each data point within a given data set refers to a separate study, so despite sharing a chemical formula, the sorbents within
a data set may differ in sorbent morphology. Grey lines indicate the 1:1 gradient between log(k’) and log(surface area).
3.7.2. Particle size
In the intraparticle diffusion model, adsorption kinetics are controlled by the rate at which sorbate
diffuses within sorbent pores
28
. One form of the intraparticle diffusion kinetic model is as follows:
Equation 21
where D is the effective diffusivity of the sorbate, and r is the radius of spherical sorbent particles
29
.
In the intraparticle diffusion model the rate of adsorption is thus proportional to the inverse of r
2
.
Bullen et al. A revised pseudo-second order kinetic model 22
The influence of particle size on adsorption kinetics has been investigated more thoroughly in the
literature than the influence of surface area, so an analysis of initial rates was performed, with the
dependence of initial rate, k’ and k
2
upon particle radius illustrated in Figure 10 (N=14 data sets with
37 kinetic experiments). Due to the wide variety of techniques used to report particle size, values of r
have a large degree of uncertainty: the literature reported particle size as the size fractionation
achieved through mesh sieving (either upper bounds, lower bounds, or both); the particle size
distribution of sorbent suspensions determined by dynamic light scattering (itself strongly dependent
on pH); and from SEM and TEM images.
(a)
(b)
Figure 10: The relationship between (a) the initial rate and (b) k’ and particle radius. Each data set is sourced from a separate
study, i.e. all data points within a given data set uses the same sorbent, with the same morphology. Grey lines are a visual
reference for the 1:1 gradient. The relationship between particle radius and k
2
is presented in the Supplementary Information.
Log(initial rate) as a function of log(
) gave an average gradient of 1.25±0.57 (x
͂
=1.31). For log(k
2
) the
gradient was only 0.982±0.891 (x
͂
=1.08). The new rate constant, as log(k’), gave a gradient of 1.12±0.63
(x
͂
=1.15), lying between those of log(initial rate) and log(k
2
). This again suggests that the modified
model gives a better description of kinetic sensitivities to experimental conditions, in this case, particle
size. With a gradient of 1.25±0.57, the dependence of initial rate upon particle radius was thus
determined to be closer to first-order than second-order, in contradiction to the (
) dependency of
rate given by the intraparticle diffusion model (Equation 21).
Having determined the empirical relationship between initial rate, k’ and particle radius using the
method of initial rates, we then investigated whether adsorption kinetic data reported in the literature
using the same sorbate-sorbent concentrations could be rationalised on the basis of particle radius.
The relationship between particle radius and log(k’) was weaker when incorporating multiple
literature sources into a single sorbate-sorbent data set, compared with when using a single literature
source where the influence of particle radius was explicitly investigated (Figure 11).
As discussed, using a single literature source for each sorbate-sorbent combination, an average
gradient of 1.25±0.57 was obtained (median=1.31, N=14 data sets ad 37 kinetic experiments),
indicating a first-order, or greater than first-order, dependence of k’ upon 1/r. However, when
comparing the same sorbate-sorbent combinations but between different literature sources, the
dependency was only 0.405±0.341 (x
͂
=0.353, N=12 data sets and 64 sources) indicating a relationship
that is weaker than first-order (Figure 11a-d). In the first case, literature studies investigating the
influence of particle size upon adsorption kinetics primarily achieved a range of sorbent particle sizes
Bullen et al. A revised pseudo-second order kinetic model 23
by size fractionation (sieving) which often maintained surface area and porosity
30
. In the second case
where sorbate-sorbent combinations are compared between different literature sources, despite
having the same chemical formula, sorbents were synthesised under different experimental
conditions and the surface morphology and porosity cannot be assumed constant. Combining all data
points, when not categorised into sets of the same sorbate-sorbent combination, a gradient of
0.329±0.095 (N=12 data sets and 64 sources) was obtained, not dissimilar to the gradient obtained
when data was catalogued into same sorbate/sorbent combination sets.
With each sorbate-sorbent combination considered as a separate data set, the average order of
reaction was 0.404±0.472 with respect to surface area, and 0.405±0.341 with respect to 1/r.
Combining all data points (with different sorbate-sorbent combinations) into a single data set gave an
average order of reaction of 0.564±0.192 with respect to surface area, and 0.329±0.095 with respect
to 1/r. In both cases the uncertainties in the linear regression between log(k’) and log(r
-1
) were smaller
than the regression between log(k’) and log(surface area) both on an absolute and relative basis.
Particle radius is thus a better constraining parameter for adsorption kinetics than surface area, and
whilst rate constant k’ could not be normalised across literature data into a single unifying value on
the basis of particle radius alone (Figure 11a-d), particle size is clearly an important factor in any
discussion or cross-comparison of adsorption kinetics and normalised rate constants such as k’.
Bullen et al. A revised pseudo-second order kinetic model 24
(a)
(b)
(c)
(d)
Figure 11: The relationship between k’ and particle radius for (a) arsenic oxyanions/mineral sorbents, (b) activated carbon
and chitosan sorbents, and (c) F
-
/Al
2
O
3
and methylene blue/TiO
2
systems. Each data point within a given data set is a separate
study within the literature, i.e. despite sharing a chemical formula, data points within the set may differ in morphology. The
solid and dashed black lines in figure (d) presents the average gradient of all data points combined into a single data set
(0.329±0.051).
Bullen et al. A revised pseudo-second order kinetic model 25
4. Conclusions
This work demonstrates that the PSO kinetic model can be made sensitive towards changes in initial
sorbate concentration and sorbent concentration through modification into the form
)
2
where
. This model provides a better fit to experimental data varying in C
0
and C
s
than
the unmodified PSO with fewer fitting parameters, and therefore provides a limited amount of
predictive capability. The new rate equation is similar to an adsorption-only form of the kinetic
Langmuir model (kLm), which at high surface coverage is first order with respect to C
t
and second
order to
)
25
18
, however with fewer parameters it is simpler, and may be more useful for the
non-expert. Furthermore, normalising conditional rate constants k
2
to our new k’ is recommended to
remove some of the error that arises when comparing literature adsorption kinetics data where
experimental conditions are different
15
. Finally, we explored the influence of sorbent morphology
upon rate constant k’ and demonstrated that particle size is a better constraining parameter than
surface area. Particle radius was, however, insufficient to fully account for the variance remaining
between experimentally determined values of rate constant, k’. This study indicates that the modified
kinetic model and its rate constant are better equipped for comparing new sorbents with literature
data than the original pseudo-second order model and k
2
, with k’ being easily calculated from PSO
parameters through the expression
.
5. Acknowledgements
The authors acknowledge support from the Engineering Physical Sciences Research Council (EPSRC)
[grant number EP/N509486/1].
Bullen et al. A revised pseudo-second order kinetic model 26
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