HAL Id: insu-03163152
https://insu.hal.science/insu-03163152
Submitted on 9 Mar 2021
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
A Revised Pseudo-Second-Order Kinetic Model for
Adsorption, Sensitive to Changes in Adsorbate and
Adsorbent Concentrations
Jay Bullen, Sarawud Saleesongsom, Kerry Gallagher, Dominik Weiss
To cite this version:
Jay Bullen, Sarawud Saleesongsom, Kerry Gallagher, Dominik Weiss. A Revised Pseudo-Second-Order
Kinetic Model for Adsorption, Sensitive to Changes in Adsorbate and Adsorbent Concentrations.
Langmuir, 2021, 37 (10), pp.3189-3201. �10.1021/acs.langmuir.1c00142�. �insu-03163152�
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 1
1
A revised pseudo-second order kinetic model for adsorption, 2
sensitive to changes in adsorbate and adsorbent 3
concentrations 4
5
1*
Jay C. Bullen;
1
Sarawud Saleesongsom;
2
Kerry Gallagher; and
1,3*
Dominik J. Weiss 6
7
1
Department of Earth Science and Engineering, Imperial College London, London SW7 2AZ, United Kingdom 8
2
Géosciences/OSUR, University of Rennes, Rennes, 35042, France 9
3
Civil and Environmental Engineering, Princeton University, United States of America 10
11
12
*Corresponding authors: 13
Email: j.bullen16@imperial.ac.uk; d.weiss@imperial.ac.uk 14
15
16
Graphical Abstract 17
18
19
Keywords 20
Adsorption kinetics; kinetic model; pseudo-second order; Lagergren; water treatment; particle size 21
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 2
Abstract 22
The development of new adsorbent materials for the removal of toxic contaminants from drinking 23
water is crucial to achieving the United Nations Sustainable Development Goal 6 (clean water and 24
sanitation). The characterisation of these materials includes fitting models of adsorption kinetics to 25
experimental data, most commonly the pseudo-second order (PSO) model. The PSO model, 26
however, provides no sensitivity to changes in experimental conditions such as adsorbate and 27
adsorbent concentrations (C
0
and C
s
) and consequently is not able to predict changes in performance 28
as a function of operating conditions. Furthermore, the experimental conditionality of the PSO rate 29
constant, k
2
, can lead to erroneous conclusions when comparing literature results. In this study, we 30
analyse 108 kinetic experiments from 47 literature sources to develop a relatively simple 31
modification of the PSO rate equation, yielding: 32
Unlike the original PSO model, this revised rate equation (rPSO) demonstrates the first-order and 33
zero-order dependencies upon C
0
and C
s
that we observe empirically. Our new model reduces the 34
residual sum of squares by 66% when using a single rate constant to model multiple adsorption 35
experiments with varying initial conditions. Furthermore, we highlight 36
is more appropriate for literature comparison, highlighting faster kinetics in the adsorption of 37
arsenic onto alumina versus iron oxides. This revised rate equation should find applications in 38
engineering studies, especially since unlike the PSO rate constant k
2
39
not show a counter-intuitive inverse relationship with the increasing reaction rate when C
0
is 40
increased. 41
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 3
Introduction 42
43
There is a wealth of recent literature concerning the development of novel adsorbent materials for 44
the remediation of contaminated water, such as composite materials offering superior stability
1
, 45
ease of separation from the effluent
2
3
, or multifunctional capabilities such as photocatalytic activity 46
4
5
. Energy input typically forms about one third of water treatment plant operation costs
6
, and if 47
energy efficiencies are to be improved then accurate models of adsorption kinetics (including rate 48
constants) are needed to (a) identify the minimum duration needed for batch treatments and (b) 49
estimate maximum flow rates for column or continuous-flow treatments
7
. Laboratory experiments 50
can only partially capture the environments in which new adsorbents will operate, and in practice 51
different concentrations of adsorbent (C
s
) will be needed to treat different concentrations of 52
contaminant in the influent (C
0
). It is thus important that adsorption models are made sensitive to 53
operating conditions, providing predictive capabilities. 54
The pseudo-second order (PSO) rate equation
8
, popularised by Ho and McKay (1999)
9
, is probably 55
the most popular model currently used to describe adsorption kinetics
10
. The PSO rate equation 56
takes the form: 57
Equation 1 58
where t is time (minutes), q
t
is the amount of adsorbate adsorbed per mass of adsorbent at time t 59
(mg g
-1
), k
2
is the pseudo-second order rate constant (g mg
-1
min
-1
), and q
e
is the amount of 60
adsorbate adsorbed at equilibrium (mg g
-1
)
9
. 61
The decrease in the concentration of aqueous adsorbate with time is given by the equation: 62
Equation 2 63
where C
t
is the concentration of aqueous adsorbate at time t (mg L
-1
), C
0
is the initial adsorbate 64
concentration at t=0 (mg L
-1
) and C
s
is the concentration of adsorbent (g L
-1
). 65
The PSO model is popular for several reasons. Firstly, it has a simple mathematical form. Secondly, 66
despite being applied by Ho and McKay as a mechanistic model for the bidentate adsorption of 67
copper onto peat
11
, the PSO model is able to fit kinetic data for a wide range of systems with 68
different reaction mechanisms
9
12
13
(including where diffusion control is to be expected
14
). Thirdly, 69
Equation 1 can be integrated and rearranged to provide linear equations (of the form y=mx+c) from 70
which the model parameters k
2
and q
e
can be easily obtained by linear regression
15
. 71
However, the PSO model has several important limitations due to the absence of adsorbate and 72
adsorbent parameters within the rate equation, with k
2
and q
e
parameters being valid only under 73
the specific experimental conditions under which the PSO model was fitted. The first limitation is 74
that the model cannot predict how adsorption kinetics will change as a function of C
0
and C
s
, limiting 75
its application in engineering or optimisation studies. Furthermore, rate constants from different 76
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 4
literature sources with different experimental conditions cannot be meaningfully compared: greater 77
values of k
2
do not necessarily indicate adsorbents possessing superior adsorption kinetics. 78
The aim of the present study was to modify the popular PSO equation, introducing sensitivity 79
towards changes in C
0
and C
s
, with the objective of both improving predictive capabilities for 80
engineering studies, and normalising rate constants for easier comparison between literature 81
sources. The PSO model does not necessarily reveal insights into the adsorption mechanisms (i.e. 82
whether intraparticle diffusion or chemisorption is the rate determining step)
13
16
, and our aim was 83
similarly to develop an empirical model, rather than a mechanistic model. We thus conducted an 84
empirical analysis of the adsorption kinetics reported by the literature to assess the influence of C
0
85
and C
s
on adsorption rates, and to modify the PSO rate equation accordingly. 86
We used the method of initial rates to determine the order of reaction with respect to both C
0
and C
s
87
(given the possibility for data at later times to disguise the true reaction order
17
, such as when 88
slower surface precipitation processes coincide with monolayer adsorption
18
). We first performed 89
quality control experiments, investigating various methods for calculating initial rates when the 90
availability of early kinetic data is limited (as per many adsorption experiments). We then compiled a 91
wide range of literature data sets wherein multiple adsorption kinetic experiments with different 92
values of C
0
and C
s
are reported (with each data set being a specific adsorbate-adsorbent system) 93
and determined the order of reaction with respect to each variable. We used mineral and organic 94
adsorbents, and metal, inorganic and organic adsorbates, to achieve a model that is generally 95
applicable to a wide range of systems, as per the original PSO model. We built the observed C
0
and 96
C
s
dependence into a revised form of the PSO rate equation (which we refer to as rPSO) and verified 97
that the rate constants given by this new model are more stable with respect to changes in 98
experimental conditions than the PSO rate constant k
2
. Finally, we used two application studies to 99
assess the potential of this revised PSO model to overcome current limitations: (1) describing 100
multiple experiments with varying values of C
0
and C
s
using a single rate constant, and (2) achieving a 101
more meaningful comparison of the adsorption kinetics reported across the literature. 102
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 5
Experimental 103
104
Data sets 105
Literature sources that experimentally investigated the influence of C
0
, C
s
or particle size upon 106
adsorption kinetics were compiled and the experimental data tabulated (referenced in the 107
Supplementary Information: SI Table S1). Both mineral adsorbents and organic adsorbents (activated 108
carbon and chitosan) were included, however zeolites and metal-organic frameworks (MOFs) were 109
not since the sorption mechanism of adsorbate trapping within cages might produce contrary 110
results. None of the literature sources found gave any mechanistic account or mathematical 111
explanation for observed differences in adsorption kinetics due to varying values of C
0
, C
s
or particle 112
size. The majority of the compiled literature used the PSO model to describe adsorption kinetics. 113
In total 47 literature sources with approximately 100 kinetic experiments were collected. This 114
includes: 14 literature sources with early kinetic data (where
) to investigate the 115
influence of early data availability on calculations of the initial rate; 8 literature sources (9 data sets) 116
with a total 37 experiments where C
0
is varied; 6 literature sources (8 data sets) with a total 27 117
experiments where C
s
is varied; and 21 literature sources (25 experiments) for the adsorption of 118
inorganic arsenic onto iron oxide and alumina adsorbents. A data set is considered to be all kinetic 119
experiments using the same adsorbate-adsorbent system within a single literature source. The 120
compiled data sets are available elsewhere
19
. 121
122
Mathematical approaches for the determination of initial rates 123
Three approaches towards the calculation of initial rates were compared: (1) the initial slope, (2) 124
linearised PSO kinetics, and (3) non-linear PSO kinetics. 125
In the initial slope approach, the initial rate (
at t=0) was calculated as the slope between the 126
origin at (0,0) and the earliest available data point at t>0
20
. 127
Initial rates were also calculated using the following linearised form of the integrated PSO rate 128
equation: 129
Equation 3 130
Kinetic profiles were plotted as
as a function of t and the linear regression was obtained using the 131
LINEST function in Excel (where the residual sum of squares between the data points and the linear 132
regression is minimised). The equilibrium adsorption parameter q
e
was obtained via the relationship 133
where m is the slope of the linear regression, and k
2
via
where c is the y-intercept. 134
The initial rate of adsorption was then calculated through the simplification of Equation 1: 135
Initial rate =
136
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 6
Equation 4 137
Uncertainties in k
2
, q
e
and the initial rate were calculated by linear propagation of the standard 138
errors in m and c given by the LINEST function. 139
Finally, initial rates were calculated using non-linear PSO kinetics. Non-linear fitting of the PSO model 140
to experimental data was achieved by using Microsoft Excel to optimise k
2
and q
e
141
values, minimising the sum of squared residuals between the model and experiment. Uncertainties 142
in k
2
and q
e
were calculated using a Monte-Carlo approach with 200 simulations as described by Hu 143
et al.
21
. 144
For all literature sources, parameters were recalculated to the same units for ease of comparison: k
2
145
(g mg
-1
min
-1
); q
e
(mg g
-1
); and the initial rate (mg g
-1
min
-1
). 146
The influence of the availability of early kinetic data on the accuracy and precision of initial rates 147
calculated using these three approaches was evaluated as follows. Fourteen data sets containing 148
early kinetic data were collected (defined as adsorption experiments containing data within the 149
range
). Initial rates were then re-calculated as data points were consecutively removed 150
from the earliest to the latest. A linear regression between the calculated initial rates and the value 151
of
at the first available kinetic data was determined and extrapolated to
to provide a 152
representing if kinetic data were to be collected within an 153
used as a reference value to 154
determine variation in the calculated initial rate as a function of the availability of early kinetic data. 155
A systematic error was calculated using the average error across all data sets and a random error 156
was calculated as the standard deviation in the error across all data sets. The significance of the 157
differences observed in the initial rate errors given by the three mathematical approaches was 158
determined using the paired samples t-test. 159
After evaluating the three approaches (see Results and Discussion), non-linear PSO kinetics were 160
used to determine initial rates in all subsequent calculations. 161
162
Determining the order of reaction 163
The order of reaction with respect to the independent variables C
0
and C
s
was calculated as the slope 164
of log(initial rate) versus log(independent variable)
22
. The order of reaction was calculated for each 165
data set (kinetic experiments using the same adsorbate-adsorbent system within a single literature 166
source), with each data point used within the linear regression representing a single kinetic 167
experiment at a unique value of C
0
or C
s
. This is represented by the equation: 168
Equation 5 169
For each data set, the order of reaction with respect to the independent variable was calculated 170
using the LINEST function in Microsoft Excel. The error for each data set was specified as the 171
standard error of the slope. An average
) order of reaction representing all data sets was 172
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 7
calculated, with errors reported as the standard deviation. The dependencies of k
2
upon C
0
173
and C
s
were determined using the same method, only substituting the initial rate with k
2
. 174
175
Modelling the revised rate equation 176
Our final rate equation, developed and derived in the Results and Discussion (Equation 11, referred 177
to as the rPSO), is: 178
Equation 6 179
L g
-1
min
-1
. As the rPSO rate equation is not easily 180
integrated, experiments were simulated using Microsoft Excel. The quantity of adsorbate adsorbed 181
at the n
th
data point was calculated using the following formula: 182
Equation 7 183
The time interval between data points,
, was reduced 184
no significant effect on the results. The
e
were obtained by non-linear 185
esidual sum of squared residuals 186
between the model and experiment. 187
For all literature sources, parameters were recalculated to the same units for ease of comparison: k
2
188
(g mg
-1
min
-1
); q
e
(mg g
-1
); and the initial rate (mg g
-1
min
-1
). 189
190
KERRR 191
Equation 8 192
where t is time (minutes), q
t
is the amount of adsorbate adsorbed per mass of adsorbent at time t 193
(mg g
-1
), k
2
is the pseudo-second order rate constant (g mg
-1
min
-1
), and q
e
is the amount of 194
adsorbate adsorbed at equilibrium (mg g
-1
)
9
. 195
The decrease in the concentration of aqueous adsorbate with time is given by the equation: 196
Equation 9 197
where C
t
is the concentration of aqueous adsorbate at time t (mg L
-1
), C
0
is the initial adsorbate 198
concentration at t=0 (mg L
-1
) and C
s
is the concentration of adsorbent (g L
-1
). 199
KERRR 200
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 8
201
Application studies 202
In the first application study (evaluating the potential of the rPSO model for predictive applications), 203
6 data sets were used: 3 where C
0
was varied and 3 where C
s
was varied. In the first step, values of 204
the equilibrium adsorption capacity (q
e
) to constrain the model were obtained by fitting kinetic 205
experiments individually using the PSO and rPSO models (optimising the rate constant and q
e
206
simultaneously to minimise the sum of squared residuals). After constraining q
e
to these values, all 207
experiments within the given data set (i.e. experiments with the same adsorbate-adsorbent system 208
but different C
0
or C
s
values) were simultaneously fit with the PSO and rPSO models, using a single 209
rate constant (k
2
, depending upon the model) to describe all experiments. 210
more 211
meaningful comparisons across the literature), 18 experiments reporting the adsorption kinetics of 212
inorganic arsenic onto iron oxide minerals were collated. Both As(V) and As(III) experiments were 213
included, since no significant difference between the adsorption kinetics was observed. A further 7 214
experiments reporting the adsorption of inorganic arsenic onto alumina (Al
2
O
3
) were collected for 215
comparison. The particle radius (r) was taken as reported by each study. 216
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 9
Results and Discussion 217
218
Determination of the influence of experimental conditions (C
0
and C
s
) on the 219
initial rate of adsorption 220
221
Quality control: The calculation of initial rates and setting criteria for the selection 222
of literature data sets 223
Many adsorption experiments reported by the literature begin their collection of kinetic data at high 224
values of
, where a significant proportion of the reaction has already been completed. This may be 225
due to challenges in collecting samples quickly (especially when filtering is required) given that many 226
adsorption reactions reach equilibrium in the minutes timescale. Other possible reasons include a 227
lack of appreciation over the time-scale at which adsorption kinetics are best measured, with many 228
papers fitting kinetic models on timescales when the reaction has already plateaued and reached 229
equilibrium. 230
Whilst we chose to investigate the influence of C
0
and C
s
using the method of initial rates, the lack of 231
literature reporting early stage kinetic data (i.e. low values of
) is a challenge. A popular approach 232
to calculate initial rates is to determine the slope of a line that is tangent to the experimental data 233
curve and passes through the origin at (0,0)
20
. However, the later the first kinetic data is collected, 234
the shallower the slope will be, creating a systematic underestimation of the initial rate. In the most 235
extreme case, an infinite delay before collection of kinetic data will yield a slope of zero and 236
subsequently an initial rate of zero. Application of a kinetic adsorption model allows for the 237
extrapolation of adsorption rates to t=0, however the accuracy of the calculated initial rates depends 238
upon how closely the experimental data obeys the applied model. 239
We therefore conducted a preliminary experiment to investigate how a limited availability of early 240
kinetic data would influence the accuracy and precision of initial rates calculated using the initial 241
slope or using the original PSO model (given that the PSO model is known to approximately describe 242
a wide range of adsorbate-adsorbent systems
12
). These results were used to set quality control 243
criteria for which data sets would be included within the investigation of the influence of C
0
and C
s
244
on adsorption kinetics. 245
The literature search identified fourteen data sets satisfying the criteria that
, a relatively 246
small number, highlighting how most authors fail to collect early kinetic data (SI Figure S1). Initial 247
rates calculated from the slope between the origin and the earliest available data point show the 248
typical systematic underestimation of initial rates when early kinetic data is missing (Figure 1a, with 249
the solid line representing the average of all data sets). In contrast, when calculated using the PSO 250
model, there is no systematic error in the calculation of initial rates when the first kinetic data is 251
collected in the range
(Figure 1b,c). In all three approaches, significant variation 252
between data sets is observed, i.e. an error in the initial rate unique to each data set. This error (with 253
dashed lines representing one standard deviation) is approximately constant in the initial slope 254
approach, being significant even when early kinetic data is available. In contrast, this error is 255
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 10
insignificant using early kinetic data and either of the PSO approaches, however this error increases 256
in magnitude as early data is sequentially removed. 257
Additionally, the absolute values of the initial rates calculated using the earliest possible kinetic data 258
were compared (i.e. 14 initial rate calculations)-test indicating that both linearised 259
PSO kinetics and non-linear PSO kinetics tend to return an initial rate greater than that calculated 260
using the initial slope (with p=0.87 and 0.97 respectively). The increase in initial rates calculated 261
using non-linear PSO kinetics versus linearised PSO kinetics is not significant (p=0.33 using the 14 262
initial rate calculations). However, when comparing the calculated initial rates with all possible data 263
cut-offs (i.e. 115 initial rate calculations, Figure 1d) the differences are more significant: non-linear 264
PSO kinetics return greater initial rates than linearised PSO kinetics (p=1.00). This is due to the 265
biased weighting of linearised PSO kinetics towards data at later times, with the slope of
versus t 266
increasing as equilibrium adsorption is approached, returning smaller values of q
e
and thus giving a 267
smaller initial rate. The difference between initial rates calculated using linearised and non-linear 268
PSO kinetics was observed despite equilibrium adsorption kinetic data from the original literature 269
being excluded in our calculations. 270
Finally, we considered the uncertainties in the initial rates calculated using the two PSO approaches 271
with varying availability of early kinetic data (Figure 1e). The uncertainties in the initial rates 272
calculated using linearised PSO kinetics were calculated from the standard error in the linear 273
regression fit to
versus t, whilst the uncertainties in non-linear PSO kinetics were calculated using 274
synthetic data and 200 Monte-Carlo simulations
21
. Using linearised PSO kinetics, the propagated 275
uncertainty in the initial rate increases exponentially as early kinetic data is removed. This is 276
explained by how late stage adsorption kinetic data is weighted too heavily when fitting linearised 277
PSO kinetics
10
, where the differences in q
t
are smaller relative to the measurement uncertainty: this 278
increases the uncertainty of the linear regression. At
, the average uncertainty in the initial 279
rates calculated using linearised PSO kinetics is 9.5%. In contrast, the uncertainty in the initial rates 280
calculated using non-linear PSO kinetics is essentially independent of the availability of early kinetic 281
data, and at
the average uncertainty is only 5.7%. 282
As highlighted here, ideally the data sets used to explore the influence of experimental parameters 283
such as the adsorbate concentration (C
0
) and the adsorbent concentration (C
s
) should include early 284
kinetic data to reduce the uncertainty in the calculated initial rates. However, literature data 285
reporting early adsorption kinetics is limited. To provide a balance between the accuracy of our 286
initial rate calculations and the collection of a sufficient quantity of data sets for statistical analysis, 287
we set the criterion that data sets must include kinetic data with
. This boundary condition 288
gives an average error in the calculated initial rate of -44 ±22 % for the initial slope approach, +17 289
±38 % using linearised PSO kinetics, and only +3 ±27% for the non-linear PSO kinetics. The results 290
indicate that non-linear PSO kinetics are most appropriate for calculating initial rates, with an 291
insignificant systematic error. A ~30% uncertainty remains, associated with how closely the 292
adsorption kinetics fit to the PSO form (with deviations being both due to inaccurate measurements 293
and real chemical mechanisms). 294
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 11
By considering a literature source reporting two kinetic experiments only, with a different value of C
0
295
or C
s
in each, a 30% error in the initial rate of the first experiment (as per the boundary condition of 296
) will confer an error of <0.5 in the calculated reaction order. In this work, the average value 297
of
in the first available kinetic data was 0.37 for experiments where C
0
is varied, and 0.27 for 298
experiments were C
s
is varied. Furthermore, the number of kinetic experiments in each literature 299
data set was 3-6, indicating that the uncertainties in the calculated initial rates will be <30%. Under 300
these conditions, the calculated orders of reaction will be accurate to the nearest integer value. This 301
was deemed appropriate for the purposes of developing the revised PSO model, given that it is 302
common practice for kinetic adsorption models to use integer values for reaction orders (e.g. the 303
original PSO equation, and the pseudo-first order kinetic model
9
). In principle, however a similar 304
analysis could be made relaxing this condition. 305
Consequently, non-linear fitting of the PSO model was used to determine the initial rate of 306
adsorption in each kinetic experiment in all subsequent work. 307
308
(a)
(b)
(c)
(d)
(e)
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 12
Figure 1: The influence of the limited availability of early kinetic data on the calculation of initial rates of adsorption, 309
assessed by the analysis of 14 literature sources. The influence of removing early kinetic data on the calculated initial rates 310
was assessed using (a) the initial slope, (b) linearised PSO kinetics, and (c) non-linear PSO kinetics. The theoretical ‘true’ 311
initial rate is given by initial rate
0
and the initial rate calculated with the available data given by initial rate
i
. The solid black 312
line represents the average of the 14 data sets, whilst dotted lines indicate one standard deviation. The three approaches 313
are compared, assessing the influence of data availability on (d) the relative error in the initial rate, and (e) the uncertainty 314
in the calculated initial rate. The horizontal error bars in (d) and (e) represent the size of the bins used for grouping data, 315
whilst the vertical error bars indicate the standard deviation calculated between the 14 unique data sets. The data sets 316
listed are in order: (1) Yang et al., 2019
23
, (2) Yang et al., 2001
24
, (3 and 4) Liu and Shen, 2008
25
, (5) Zhu et al. 2016
26
, (6) 317
Yang et al. 2019
27
, (7) Yang et al. 2019
28
, (8) Mohamed et al. 2007
29
, (9) Drenkova-tuhtan et al. 2015
30
, (10) Liu et al. 318
2016
31
, (11) Ornek et al. 2007
32
, (12) Zhan et al. 2018
33
, (13) Ai et al. 2020
34
, (14) Nadiye-tabbiruka and Sejie 2019
35
. 319
320
Determining the influence of C
0
and C
s
upon the rate of adsorption and the PSO 321
rate constant k
2
322
The influence of C
0
and C
s
upon the initial rate of adsorption (calculated as the rate at t=0 using non-323
linear PSO kinetics) is presented in Figure 2. For each data set, the order of reaction was determined 324
from the slope of log(initial rate) versus log(C
0
) or log(C
s
) (Figure 2a,b). Based upon the evaluation of 325
uncertainties in the initial rates calculated using non-linear PSO kinetics (in the previous section), the 326
reaction orders are accurate to the nearest integer value. The data sets tend to show a first-order 327
dependency of initial rate upon the initial adsorbate concentration (C
0
) (Figure 2c), with an average 328
dependency and standard deviation of 0.80 ±0.38, and a median value of 0.67. Of the 9 data sets, 7 329
were closest to first-order dependency, and 2 were closer to zero-order dependency. The sum of 330
normal distributions representing the reaction orders and uncertainties calculated for each data set 331
was approximated by a single normal distribution. (However, a larger number of data sets are 332
needed to verify this). Based upon the standard deviation and assuming a normal distribution in the 333
results, the relationship between the initial rate and C
0
is first-order with a 90% confidence interval 334
(1.65 standard deviations). This is intuitive for both diffusion and adsorption-controlled mechanisms, 335
as twice as much adsorbate should lead to twice as much adsorbate flux from adsorbent surface into 336
pores, and collisions with the adsorbent surface should be twice as frequent
36
9
. 337
A first-order dependency of initial rates (normalised to mg L
-1
min
-1
) with respect to the adsorbent 338
concentration (C
s
) is also observed (Figure 2d), albeit with a wider distribution of results: an average 339
value of 1.11 ±0.33 and a median of 1.07. Of the 8 data sets, 6 were closest to first-order 340
dependency, with 1 data set closer to zero-order and another closer to second-order. Based upon 341
the standard deviation and assuming a normal distribution in the results, the relationship between 342
the initial rate and C
s
is also first-order with a 90% confidence interval (1.65 standard deviations). 343
This is again intuitive for both diffusion and adsorption-controlled mechanisms, as when C
s
is 344
doubled the total surface area available to solution (m
2
L
-1
) is doubled, the overall flux of adsorbate 345
entering adsorbent pores is doubled, and the rate of collisions between adsorbate and total 346
adsorbent surface is also doubled. When normalised to mass (mg g
-1
min
-1
) the initial rate is zero-347
order with respect to C
s
, as expected. 348
349
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 13
(a)
(b)
(c)
(d)
Figure 2: Determining the influence of initial adsorbate concentration (C
0
) and adsorbent concentration (C
s
) on the initial 350
rate of adsorption using literature data sets. The order of reaction was determined from the slope of log(initial rate) as a 351
function of (a) log(C
0
) and (b) log(C
s
), with each data point representing a single kinetic experiment (with unique values of C
0
352
and C
s
). All experiments in a given data set (one literature paper, where all experimental conditions except for either C
0
or C
s
353
are constant) are grouped by colour and symbol, with oxyanions in red, metal cations in blue, and organic dyes in yellow (a 354
legend referencing the literature sources is presented in SI Figure S2). Uncertainties were calculated from the standard error 355
in the slope. The results are alternatively presented using normal distributions with the mean given by the slope and the 356
standard deviation given by the standard error of the slope (c and d). Here, each data set is represented by a dotted line, 357
and the sum of all data sets given by the dashed line. Solid black lines represent the normal distribution obtained by the 358
average of all data sets. 359
360
For both predictive modelling and the comparison of adsorption kinetics between literature sources, 361
it is necessary that rate constants are not affected by the experimental conditions. Whilst adsorption 362
kinetics are typically first-order with respect to C
0
, the PSO rate constant k
2
is inversely proportional 363
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 14
to C
0
(Figure 3a). Consequently, whilst doubling the initial adsorbate concentration typically 364
increases the initial rate of adsorption by a factor of two, counter-intuitively the PSO rate constant k
2
365
will decrease by a factor of two. The average slope of log(k
2
) versus log(C
0
) is -0.73±0.46. The inverse 366
relationship between k
2
and C
0
is explained by the second-order dependence of the PSO model upon 367
the absolute concentration of available adsorption capacity remaining through the term (q
e
-q
t
)
2
. In 368
cases where the adsorbent is unsaturated, increasing C
0
by a factor of two will approximately double 369
q
e
. The parameter (q
e
-q
t
)
2
at t=0 will increase by a factor of four, and consequently k
2
must decrease 370
by a factor of two to achieve the observed doubling of initial rates. 371
(a)
(b)
Figure 3: Dependence of the pseudo-second order (PSO) rate constant k
2
upon (a) initial adsorbate concentration (C
0
), and 372
(b) adsorbent concentration (C
s
). The data is presented as described in Figure 2, with a legend referencing the literature 373
sources presented in SI Figure S2. 374
375
In contrast, a positive relationship between C
s
and k
2
is observed, with an average dependency of 376
1.57±0.85 (Figure 3b). (First and second-order dependencies between k
2
and C
s
are both included 377
within the standard deviation). This is explained by how when C
s
is doubled, q
e
will decrease by a 378
factor of between 0 and 2: zero when the adsorbate is in excess and q
e
is insignificant compared to 379
C
0
, and two when the adsorbent is in excess and q
e
is large relative to C
0
. Consequently, as C
s
380
increases, (q
e
-q
t
)
2
decreases with a zero-to-second-order dependency, and to achieve the zero-order 381
relationship between C
0
and initial rates (mg g
-1
min
-1
), k
2
must also increase with a dependency that 382
is between zero and second-order. 383
384
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 15
Revision of the pseudo-second order (PSO) rate equation to account for 385
changes in adsorbate (C
0
) and adsorbent (C
s
) concentrations
386
387
Modification of the PSO rate equation 388
In this section, we modify the original PSO rate equation to include the appropriate sensitivity 389
towards C
0
and C
s
, meeting the aims of (a) improving the predictive capacity of this model, and (b) 390
normalising rate constants for better comparison across the literature. 391
Firstly, for a given concentration of adsorbent, the total concentration of adsorption surface sites is 392
constant regardless of the value of C
0
. The term within the rate equation used to represent the 393
contribution of adsorption surface site availability towards the rate of reaction should therefore be 394
independent of C
0
. The original PSO rate equation contains a second-order dependence upon the 395
absolute amount of adsorption capacity remaining, (q
e
-q
t
)
2
, which gives the undesirable inverse 396
relationship between the rate constant and C
0
demonstrated in Figure 3a. This term can be replaced 397
with a second-order dependence upon the relative amount of adsorption capacity remaining, 398
, which will always return a value of 1 at time t=0, independent of C
0
. This term therefore 399
describes the contribution of adsorption surface site availability towards the rate of adsorption more 400
appropriately than the original PSO term (q
e
-q
t
)
2
. Here,
401
Langmuir adsorption isotherm model
20
. This modification of Equation 1 gives the following: 402
403
Equation 10 404
where
. 405
The first-order dependence of the reaction rate upon the adsorbate concentration observed 406
experimentally (Figure 2c) is then defined within the rate equation, giving: 407
Equation 11 408
where
. L g
-1
min
-1
. 409
The initial rate of adsorption tends to be zero-order with respect to C
s
(when normalised to 410
adsorbent mass with the units mg g
-1
min
-1
). The original PSO model gives an initial rate that varies 411
with changes in C
s
, due to its second-order dependence upon the absolute adsorption capacity 412
remaining (q
e
-q
t
)
2
and the decrease in q
e
with increasing C
s
. In contrast, since the rPSO depends on 413
the relative adsorption capacity remaining through the term
, this rate equation displays 414
the zero-order dependency of C
s
identified from analysis of the literature. 415
is therefore theoretically independent of changes in C
s
, unlike the original PSO rate constant k
2
. The 416
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 16
rPSO model is similar to the adsorption-only form of the kinetic Langmuir model (kLm), which at high 417
surface coverage is first order with respect to C
t
and second order to
)
37
38
. 418
419
Validation of the rPSO rate equation 420
The removal of experimental conditionality (i.e. the dependency upon C
0
and C
s
) from the revised 421
model was verified using experimental data from the literature. The ideal rate constant is unaffected 422
by the experimental conditions, and subsequently the dependency of the rate constant with respect 423
to C
0
and C
s
should be zero. These dependencies were calculated from the slope of log(rate 424
constant) versus log(C
0
) or log(C
s
). As highlighted by Figure 4a, the original PSO rate constant k
2
is 425
strongly dependent upon the experimental conditions, being inversely proportional to C
0
and 426
second-order with respect to C
s
. The average C
0
dependency is -0.73±0.46, and the average C
s
427
dependency is 1.57±0.79 -order with respect to both 428
0
and C
s
vary less (there is less 429
scattering) than k
2
. The average C
0
dependency is -0.20±0.38 and the average C
s
dependency is 430
0.10±0.45. These results demonstrate that the new
2
, and 431
that the dependency of adsorption kinetics upon C
0
and C
s
is captured by the new model. 432
When the adsorbate is in excess (q
e
<C
0
) the rPSO model approximates the form of the original PSO 433
model. At higher values of q
e
relative to C
0
, the graphical form of the two models deviates, due to 434
the rPSO kinetics decreasing more rapidly than PSO kinetics due to the consumption of the 435
adsorption, which decreases the parameter C
t
. (However, this effect is logical, given that at low C
t
436
values, the rate of adsorption will be limited by the availability of adsorbate). Therefore, rate 437
calculated from PSO parameters k
2
and q
e
when the 438
adsorbate is in excess, using the formula
. When the adsorbent removes the majority of 439
the adsorbate, however, the rPSO model will require re-fitting due to the increasing difference in the 440
graphical form of the PSO and rPSO models. 441
442
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 17
(a)
(b)
Figure 4: Verifying that experimental conditionality (with respect to C
0
and C
s
) is decreased in the rPSO model versus the 443
original PSO model. (a) The dependencies of the original PSO rate constant k
2
and the rPSO rate constant k’ upon C
0
and C
s
444
were calculated from the slope of log plots, as per Figure 3, with the ideal rate constant giving a reaction order or 445
‘dependency’ of zero. The data is shown as normal distributions with the mean and standard deviation set equal to the 446
slope and standard error of the slope in the log plots (with 9 data sets for C
0
and 8 for C
s
). These values are: -0.73±0.46 for 447
k
2
and C
0
; 1.57±0.79 for k
2
and C
s
; -0.20±0.38 for k’ and C
0
; 0.10±0.45 for k’ and C
s
. (b) A comparison of the form of the 448
original PSO equation with the rPSO equation, with different
ratios (with C
s
=1 g L
-1
). The original PSO model is presented 449
in blue and the rPSO model in red, with
equal to 1 (dotted lines), 2 (dashed lines), and 5 (solid lines). 450
451
Example applications 452
453
Application 1: Evaluating the predictive capability of the revised PSO model 454
The first objective of this work was to provide a simple modification of the popular PSO model to 455
introduce predictive capabilities, for the purpose of engineering studies
39
. If the kinetic model is to 456
be used to predict adsorption performance under different conditions, then it is essential that the 457
model parameters obtained experimentally are valid in a range of scenarios. Consequently, we 458
evaluated whether the rPSO model would provide a better fit to experimental data compared 459
against the PSO model, if a single rate constant is used to model multiple experiments with different 460
values of C
0
and C
s
. 461
The original PSO model tends to systematically overestimate q
t
for experiments with high C
0
or low 462
C
s
values and systematically underestimate q
t
for experiments with low C
0
or high C
s
values (Figure 463
5). This is due to the negative and positive relationships between k
2
and C
0
and C
s
respectively as 464
previously discussed, with this relationship being denied when a single value of k
2
is used to model 465
all experiments. The rPSO model gave a better fit to experimental data (i.e. a smaller sum of squared 466
residuals) in 5 of the 6 data sets tested (all panels in Figure 5 except d) and the median decrease in 467
the sum of squared residuals when changing from the PSO to the rPSO model was 66%. 468
469
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 18
(a)
(b)
(c)
(d)
(e)
(f)
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 19
Figure 5: Application study 1: Application of the revised PSO (rPSO) model (solid lines) to describe multiple experiments with 470
a single rate constant, compared against the original PSO model (dotted lines). (a,b,c) present 3 experiments where C
0
is 471
varied, and (d,e,f) present 3 experiments where C
s
is varied. Experimental data was sourced from Manna et al. (2003)
40
, 472
Singh et al. (1996)
41
, Mezenner and Bensmaili (2009)
42
, Debnath et al. (2017)
43
, Lazaridis et al. (2004)
44
, and Shipley et al. 473
(2013)
45
. 474
475
The average relative error of q
t
calculated by the rPSO is just 1±17%, versus -11±27% for the original 476
PSO model (Figure 6a), indicating that the rPSO model provides greater accuracy when modelling 477
adsorption kinetics with changing values of C
0
and C
s
using a single rate constant. Furthermore, the 478
calibration curve of q
t
(model) against q
t
(experiment) is closer to the ideal one-to-one line in the 479
rPSO model, with an R
2
value of 0.999 versus just 0.9217 for the original PSO model (Figure 6b). 480
Considering each experimental data series in turn, the typical slope of the calibration curve was 481
closer to one, with less scattering, in the rPSO model (a slope of 0.99±0.11) compared against the 482
original PSO model (with a slope of 1.23±0.35) (Figure 6c). 483
(a)
(b)
(c)
Figure 6: Application study 1: Cross-calibration of the rPSO model against experimental data (6 literature sources, 22 484
experiments and 198 data points). (a) Box and whisker plot presenting the relative error of q
t
calculated via the original PSO 485
model and the rPSO model. The boxes highlight the 25%, 50% (median) and 75% percentiles, whilst the whiskers represent 486
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 20
the minima and maxima excluding ‘outlier’ data points, defined as those greater than the top of the box plus 1.5 times the 487
interquartile range, or less than the bottom of the box minus 1.5 times the interquartile range. (b) Cross-calibration plot 488
highlighting the goodness of fit against the one-to-one line. Open shapes indicate q
t
values calculated using the original PSO 489
model, whilst filled shapes indicate the rPSO model. Values of R
2
indicate the goodness of fit against the ideal one-to-one 490
line. (c) Comparison of the cross-calibration slopes with each model and each data set. Literature sources are denoted as 491
As(III)/HFO
40
(dark blue squares), As(V)/Fe
2
O
3
41
(light blue circles), HPO
4
2-
/iron hydroxide
42
(dark green diamonds), 492
Cr(VI)/Fe
2
O
3
43
(light green triangles), Cr(VI)/Mg-Al-CO
3
44
(orange squares), and Cd(II)/Fe
2
O
3
45
(red circles). Further results 493
are presented in SI Figure S3. 494
495
appears to be more stable to changes in experimental conditions 496
than the PSO rate constant k
2
, the parameter q
e
is conditional, depending upon C
0
and C
s
. For 497
predictive modelling, this limitation can be rectified by using an adsorption isotherm to predict q
e
498
(such as the Langmuir or Freundlich model
46
). Though a single value of q
e
can be determined for the 499
entirety of each experiment, in scenarios such as a column reactor the equilibrium adsorbate 500
concentration parameter C
e
has diminished physical significance, and it may be better to replace this 501
term with C
t
, recalculating the hypothetical value of q
e
at each point in time. Huang et al. previously 502
demonstrated that this approach can give a better account of the true driving force of the reaction 503
during the initial stages of adsorption
47
. 504
505
Application 2: Comparison of rate constants between different experimental studies 506
expected to be more meaningful and more appropriate 507
than k
2
when comparing the adsorption kinetics reported in the literature using different 508
experimental conditions, since some of the experimental conditionality (towards changes in C
0
and 509
C
s
) is accounted for. To demonstrate the potential application of normalised rate constants towards 510
achieving a meaningful comparison of the adsorption kinetics reported across the literature, we 511
collected 14 and 7 literature sources reporting the kinetics of inorganic arsenic As(V) and As(III) 512
adsorption onto iron oxide and alumina adsorbents respectively. 513
The average value of log(k
2
) for all iron oxide studies is -0.93±1.50is -514
1.05±1.08. In both cases, the standard deviation is large, with more than an order of magnitude 515
variation in k
2
neither model provides a rate constant that is generally valid for iron 516
oxide adsorbents used by different studies. Similarly, the adsorption of inorganic arsenic onto 517
alumina gives average values of log(k
2
) = 0.98±1.84 0.35±1.75. 518
Whilst the variation in both k
2
519
morphology on adsorption kinetics has not been incorporated into either of the PSO and rPSO 520
models. Adsorption kinetics are often faster for adsorbent materials with smaller particles, due to 521
the improved rate of mass transport of the adsorbate to adsorbent surface sites
48
. For instance, in 522
intraparticle diffusion model gives a rate of adsorption that is proportional to r
-1
(where r is particle 523
size)
49
. 524
Whilst a faster reaction is anticipated for small iron oxide particle sizes, plotting log(k
2
) as a function 525
of log(r
-1
) shows a weak inverse relationship (linear regression gives a slope of -0.49±0.20 and 526
R
2
=0.3234) (Figure 7a). This is due to the significant increase in q
e
as the particle size decreases (SI 527
Figure S4) and the inverse relationship between k
2
and q
e
through the term (q
e
-q
t
)
2
as previously 528
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 21
discussed. In contrast, alumina shows the anticipated positive relationship (with a slope of 0.61±0.28 529
and R
2
=0.4783). Since adsorption onto smaller particles is typically faster than onto larger particles, 530
the comparison of k
2
values for different adsorbent sizes and different adsorbent materials (i.e. iron 531
oxides versus alumina) is not useful and is likely to lead to false conclusions. 532
In contrast, we see the anticipated positive
-1
) for both 533
iron oxide and alumina adsorbents (Figure 7b). For the iron oxides we see a slope of 0.44±0.1 with 534
R
2
=0.5211, and for alumina we see a slope of 0.62±0.25 and R
2
=0.5547. The limited goodness of fit in 535
the linear regression is controlled by factors including poor characterisation of the particle size and 536
differences in experimental conditions beyond C
0
and C
s
, such as the pH. The accurate 537
characterisation of particle size is especially challenging, given the range of techniques used by the 538
literature including transmission electron microscopy (TEM), dynamic light scattering (DLS) and sieve 539
fractionation. 540
The results from this 541
alumina versus iron oxides, suggesting that the adsorption of inorganic arsenic is faster onto alumina 542
than onto iron oxides. With more rigorous analysis, such a study would have important implications 543
in the design of engineered solutions for arsenic remediation, i.e. opting to use alumina in place of 544
iron oxides when high flow rates in a column filter are required, or when larger particle sizes are 545
necessary to achieve the desired flow rate and porosity. This highlights that normalisation of the PSO 546
rate constant k
2
547
meaningful comparison of the adsorption kinetics reported by the literature. 548
549
(a)
(b)
Figure 7: Application study 2: Use of the rPSO rate constant k’ to compare literature sources with adsorption kinetics 550
determined under different experimental conditions. Particle size decreases from left to right. Presented are As(V) (red filled 551
shapes) and As(III) (orange filled shapes) adsorption onto iron oxides, and As(V) (dark blue open squares) and As(III) (light 552
blue open circles) adsorption onto alumina. Each data point indicates a different literature source, with a legend given in SI 553
Figure S5. 554
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 22
Conclusions 555
556
This work aimed to modify the popular pseudo-second order (PSO) model of adsorption kinetics to 557
remove experimental conditionality, focusing upon initial adsorbate concentration (C
0
) and 558
adsorbent concentration (C
s
). A revised PSO rate equation (rPSO) was developed from the empirical 559
analysis of 69 kinetic experiments taken from 15 literature sources. The final equation takes the 560
form
. The first application study demonstrates that the rPSO equation allows 561
for a single rate constant to model multiple experiments, differing in the experimental conditions C
0
562
and C
s
, with greater accuracy and a 66% decrease in the sum of squared residuals versus the original 563
PSO model. The second application study demonstrates that the rPSO equation provides a rate 564
constant which is more useful for comparison across the literature than the PSO rate constant k
2
, 565
obeying the anticipated relationship between adsorption rates and particle size. 566
The new rate equation is similar to an adsorption-only form of the kinetic Langmuir model (kLm), 567
which at high surface coverage is first order with respect to C
t
and second order with respect to 568
)
37
38
. However, the rPSO equation is simpler, with fewer fitting parameters needed, and may 569
thus be more useful for the non-expert. The rPSO equation may prove more useful than the original 570
PSO model in engineering studies where operating conditions are likely to vary. 571
One of the reasons for the popularity of the original PSO model is its linearised forms, from which k
2
572
and q
e
parameters can be readily obtained from experimental data. Whilst, we have unfortunately 573
not yet found a way to linearise the rPSO rate equation, it is worth noting that linearisation of the 574
original PSO model often results in a poorer quality of fit versus when using non-linear fitting
14
. 575
Where necessary, the rate constant for this revised model can be quickly obtained from linearised 576
PSO kinetics using the expression
. 577
In our analytical approach, we demonstrate that non-linear fitting of PSO kinetics is more 578
appropriate than linearised PSO kinetics and the initial slope approach when early kinetic data is 579
limited, as is often the case in adsorption experiments. Our literature survey only yielded 9 data sets 580
where C
0
was varied and 8 where C
s
was varied, satisfying our requirements for early kinetic data 581
with
. This highlights a need for experimental work to investigate the influence of these 582
independent variables more systematically, and to ensure that initial adsorption kinetics are 583
adequately captured, given that the method of initial rates is typically considered the superior 584
approach towards determining reaction orders
17
. 585
586
587
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 23
Acknowledgements 588
The authors acknowledge support from the Engineering Physical Sciences Research Council (EPSRC) 589
[grant number EP/N509486/1]. 590
591
Supporting information 592
The supplementary information provides references to all data sets used in this work; figures 593
showing the influence of adsorbent morphology on k
2
; and tabulates all parameters (q
e
, k
2
594
initial rate) calculated in this work. 595
596
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 24
References 597
(1) Boddu, V. M.; Abburi, K.; Randolph, A. J.; Smith, E. D. Removal of Copper(II) and Nickel(II) Ions from 598
Aqueous Solutions by a Composite Chitosan Biosorbent. Sep. Sci. Technol. 2008, 43 (6), 13651381. 599
https://doi.org/10.1080/01496390801940762. 600
(2) Linghu, W.; Yang, H.; Sun, Y.; Sheng, G.; Huang, Y. One-Pot Synthesis of LDH/GO Composites as Highly 601
Effective Adsorbents for Decontamination of U(VI). ACS Sustain. Chem. Eng. 2017, 5 (6), 56085616. 602
https://doi.org/10.1021/acssuschemeng.7b01303. 603
(3) Deng, L.; Shi, Z.; Peng, X. Adsorption of Cr(VI) onto a Magnetic CoFe2O4/MgAl-LDH Composite and 604
Mechanism Study. RSC Adv. 2015, 5 (61), 4979149801. https://doi.org/10.1039/c5ra06178d. 605
(4) apacity and Mechanism of Arsenate and 606
Phosphate on a Bifunctional TiO2-Fe2O3 Bi-Composite. J. Colloid Interface Sci. 2011, 364 (1), 205212. 607
https://doi.org/10.1016/j.jcis.2011.08.023. 608
(5) Chi, S.; Ji, C.; Sun, S.; Jiang, H.; Qu, R.; Sun, C. Magnetically Separated Meso-g-C3N4/Fe3O4: Bifuctional 609
Composites for Removal of Arsenite by Simultaneous Visible-Light Catalysis and Adsorption. Ind. Eng. 610
Chem. Res. 2016, 55 (46), 1206012067. https://doi.org/10.1021/acs.iecr.6b02178. 611
(6) Smith, K.; Liu, S. Energy for Conventional Water Supply and Wastewater Treatment in Urban China: A 612
Review. Glob. Challenges 2017, 1 (5), 1600016. https://doi.org/10.1002/gch2.201600016. 613
(7) Dichiara, A. B.; Weinstein, S. J.; Rogers, R. E. On the Choice of Batch or Fixed Bed Adsorption Processes 614
for Wastewater Treatment. Ind. Eng. Chem. Res. 2015, 54 (34), 85798586. 615
https://doi.org/10.1021/acs.iecr.5b02350. 616
(8) Blanchard, G.; Maunaye, M.; Martin, G. Removal of Heavy Metals from Waters by Means of Natural 617
Zeolites. Water Res. 1984, 18 (12), 15011507. https://doi.org/10.1016/0043-1354(84)90124-6. 618
(9) Ho, Y. S.; McKay, G. Pseudo-Second Order Model for Sorption Processes. Process Biochem. 1999, 34 619
(5), 451465. https://doi.org/10.1016/S0032-9592(98)00112-5. 620
(10) Simonin, J. P. On the Comparison of Pseudo-First Order and Pseudo-Second Order Rate Laws in the 621
Modeling of Adsorption Kinetics. Chem. Eng. J. 2016, 300, 254263. 622
https://doi.org/10.1016/j.cej.2016.04.079. 623
(11) Ho, Y. S.; Mckay, G. The Kinetics of Sorption of Divalent Metal Ions onto Sphagnum Moss Peat. Water 624
Res. 2000, 34 (3), 735742. 625
(12) Ho, Y. Review of Second-Order Models for Adsorption Systems. J. Hazard. Mater. 2006, 136 (April 626
2005), 681689. https://doi.org/10.1016/j.jhazmat.2005.12.043. 627
(13) Plazinski, W.; Dziuba, J.; Rudzinski, W. Modeling of Sorption Kinetics: The Pseudo-Second Order 628
Equation and the Sorbate Intraparticle Diffusivity. Adsorption 2013, 19 (5), 10551064. 629
https://doi.org/10.1007/s10450-013-9529-0. 630
(14) Xiao, Y.; Azaiez, J.; Hill, J. M. Erroneous Application of Pseudo-Second-Order Adsorption Kinetics 631
Model: Ignored Assumptions and Spurious Correlations. Ind. Eng. Chem. Res. 2018, 57 (7), 27052709. 632
https://doi.org/10.1021/acs.iecr.7b04724. 633
(15) Canzano, S.; Iovino, P.; Leone, V.; Salvestrini, S.; Capasso, S. Use and Misuse of Sorption Kinetic Data: A 634
Common Mistake That Should Be Avoided. Adsorpt. Sci. Technol. 2012, 30 (3), 217225. 635
https://doi.org/10.1260/0263-6174.30.3.217. 636
(16) Qiu, H.; Lv, L.; Pan, B. C.; Zhang, Q. J.; Zhang, W. M.; Zhang, Q. X. Critical Review in Adsorption Kinetic 637
Models. J. Zhejiang Univ. Sci. A 2009, 10 (5), 716724. https://doi.org/10.1631/jzus.A0820524. 638
(17) Ollis, D. F. Kinetics of Photocatalyzed Reactions: Five Lessons Learned. Front. Chem. 2018, 6 (378), 17. 639
https://doi.org/10.3389/fchem.2018.00378. 640
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 25
(18) Lützenkirchen, J.; Behra, P. On the Surface Precipitation Model for Cation Sorption at the (Hydr)Oxide 641
Water Interface. Aquat. Geochemistry 1995, 1 (4), 375397. https://doi.org/10.1007/BF00702740. 642
(19) Bullen, J. C.; Saleesongsom, S. Adsorption Kinetics Data Sets, Compiled from the Literature. As Used in 643
-Second Order Kinetic Model for Adsorption, Sensitive to 644
Changes in Adsorbate and Adsorbent Concentration Zenodo. January 15, 2021. 645
https://doi.org/10.5281/ZENODO.4443102. 646
(20) Atkins, P.; Paula, J. De. Atkins’ Physical Chemistry, 9th ed.; Oxford University Press, 2009. 647
https://doi.org/10.1021/ed056pA260.1. 648
(21) Hu, W.; Xie, J.; Chau, H. W.; Si, B. C. Evaluation of Parameter Uncertainties in Nonlinear Regression 649
Using Microsoft Excel Spreadsheet. Environ. Syst. Res. 2015, 4 (1). https://doi.org/10.1186/s40068-650
015-0031-4. 651
(22) Kirby, M. E.; Bullen, J. C.; Hanif, M. D.; Heiba, H. F.; Liu, F.; Northover, G. H. R.; Resongles, E.; Weiss, D. 652
J. Determining the Effect of PH on Iron Oxidation Kinetics in Aquatic Environments: Exploring a 653
Fundamental Chemical Reaction to Grasp the Significant Ecosystem Implications of Iron Bioavailability. 654
J. Chem. Educ. 2019. https://doi.org/10.1021/acs.jchemed.8b01036. 655
(23) Yang, H.; Yu, H.; Fang, J.; Sun, J.; Xia, J.; Xie, W.; Wei, S.; Cui, Q.; Sun, C.; Wu, T. Mesoporous Layered 656
Graphene Oxide/Fe3O4/C3N3S3 Polymer Hybrids for Rapid Removal of Pb2+ and Cd2+ from Water. 657
ACS Omega 2019, 4 (22), 1968319692. https://doi.org/10.1021/acsomega.9b02347. 658
(24) Yang, X. Y.; Al-Duri, B. Application of Branched Pore Diffusion Model in the Adsorption of Reactive Dyes 659
on Activated Carbon. Chem. Eng. J. 2001, 83 (1), 1523. https://doi.org/10.1016/S1385-660
8947(00)00233-3. 661
(25) Liu, Y.; Shen, L. From Langmuir Kinetics to First- and Second-Order Rate Equations for Adsorption. 662
Langmuir 2008, 24 (20), 1162511630. https://doi.org/10.1021/la801839b. 663
(26) orption of Pyridine from Aqueous Solutions by Polymeric 664
Adsorbents MN 200 and MN 500. Part 2: Kinetics and Diffusion Analysis. Chem. Eng. J. 2016, 306, 665
12231233. https://doi.org/10.1016/j.cej.2016.07.087. 666
(27) Yang, K.; Li, Y.; Zheng, H.; Luan, X.; Li, H.; Wang, Y.; Du, Q.; Sui, K.; Li, H.; Xia, Y. Adsorption of Congo 667
Red with Hydrothermal Treated Shiitake Mushroom. Mater. Res. Express 2019, 7 (1). 668
https://doi.org/10.1088/2053-1591/ab5ff3. 669
(28) Yang, H.; Wang, Y.; Bender, J.; Xu, S. Removal of Arsenate and Chromate by Lanthanum-Modified 670
Granular Ceramic Material: The Critical Role of Coating Temperature. Sci. Rep. 2019, 9 (1), 112. 671
https://doi.org/10.1038/s41598-019-44165-8. 672
(29) Mohamed, S. F.; Al-Bakri, I. M.; El Sayed, O. H. Biosorption of Lead (II) by Pre-Treated Biomass of 673
Marine Brown Algae Sargassum Latifolium and Sargassum Asperifolium. Biosci. Biotechnol. Res. Asia 674
2007, 4 (2), 341350. 675
(30) Drenkova-tuhtan, A.; Mandel, K.; Meyer, C.; Schneider, M. Removal and Recovery of Phosphate from 676
Wastewater with Reusable Magnetically Separable Particles. IWA Spec. Conf. Nutr. Remov. Recover. 677
Mov. Innov. into Pract. 2015, No. May. https://doi.org/10.13140/RG.2.1.1260.8721. 678
(31) Liu, J.; Wu, X.; Hu, Y.; Dai, C.; Peng, Q.; Liang, D. Effects of Cu(II) on the Adsorption Behaviors of Cr(III) 679
and Cr(VI) onto Kaolin. J. Chem. 2016, 2016 (Vi). https://doi.org/10.1155/2016/3069754. 680
(32) ngil, I. A. Adsorption of Lead onto Formaldehyde or Sulphuric Acid Treated 681
Acorn Waste: Equilibrium and Kinetic Studies. Biochem. Eng. J. 2007, 37 (2), 192200. 682
https://doi.org/10.1016/j.bej.2007.04.011. 683
(33) Zhan, W.; Xu, C.; Qian, G.; Huang, G.; Tang, X.; Lin, B. Adsorption of Cu(Ii), Zn(Ii), and Pb(Ii) from 684
Aqueous Single and Binary Metal Solutions by Regenerated Cellulose and Sodium Alginate Chemically 685
Modified with Polyethyleneimine. RSC Adv. 2018, 8 (33), 1872318733. 686
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 26
https://doi.org/10.1039/c8ra02055h. 687
(34) Ai, T.; Jiang, X.; Liu, Q.; Lv, L.; Dai, S. Single-Component and Competitive Adsorption of Tetracycline and 688
Zn(Ii) on an NH4Cl-Induced Magnetic Ultra-Fine Buckwheat Peel Powder Biochar from Water: Studies 689
on the Kinetics, Isotherms, and Mechanism. RSC Adv. 2020, 10 (35), 2042720437. 690
https://doi.org/10.1039/d0ra02346a. 691
(35) Nadiye-tabbiruka, M. S.; Sejie, F. P. Preparation of Coal-Kaolinite Nano Composites and Investigation of 692
Their Use to Remove Methyl Orange from Water. 2019, 9 (2), 3743. 693
https://doi.org/10.5923/j.nn.20190902.01. 694
(36) Largitte, L.; Pasquier, R. A Review of the Kinetics Adsorption Models and Their Application to the 695
Adsorption of Lead by an Activated Carbon. Chem. Eng. Res. Des. 2016, 109, 495504. 696
https://doi.org/10.1016/j.cherd.2016.02.006. 697
(37) Azizian, S. Kinetic Models of Sorption: A Theoretical Analysis. J. Colloid Interface Sci. 2004, 276 (1), 47698
52. https://doi.org/10.1016/j.jcis.2004.03.048. 699
(38) Marczewski, A. W. Analysis of Kinetic Langmuir Model. Part I: Integrated Kinetic Langmuir Equation 700
(IKL): A New Complete Analytical Solution of the Langmuir Rate Equation. Langmuir 2010, 26 (19), 701
1522915238. https://doi.org/10.1021/la1010049. 702
(39) Bullen, J. C.; Lapinee, C.; Salaün, P.; Vilar, R.; Weiss, D. J. On the Application of Photocatalyst-Sorbent 703
Composite Materials for Arsenic(III) Remediation: Insights from Kinetic Adsorption Modelling. J. 704
Environ. Chem. Eng. 2020, 8 (5). https://doi.org/10.1016/j.jece.2020.104033. 705
(40) Manna, B. R.; Dey, S.; Debnath, S.; Ghosh, U. C. Removal of Arsenic from Groundwater Using 706
Crystalline Hydrous Ferric Oxide (CHFO). Water Qual. Res. J. Canada 2003, 38 (1), 193210. 707
https://doi.org/10.2166/wqrj.2003.013. 708
(41) Singh, D. B.; Prasad, G.; Rupainwar, D. C. Adsorption Technique for the Treatment of As(V)-Rich 709
Effluents. Colloids Surfaces A Physicochem. Eng. Asp. 1996, 111 (12), 4956. 710
https://doi.org/10.1016/0927-7757(95)03468-4. 711
(42) Mezenner, N. Y.; Bensmaili, A. Kinetics and Thermodynamic Study of Phosphate Adsorption on Iron 712
Hydroxide-Eggshell Waste. Chem. Eng. J. 2009, 147 (23), 8796. 713
https://doi.org/10.1016/j.cej.2008.06.024. 714
(43) Debnath, A.; Bera, A.; Chattopadhyay, K. K.; Saha, B. Facile Additive-Free Synthesis of Hematite 715
Nanoparticles for Enhanced Adsorption of Hexavalent Chromium from Aqueous Media: Kinetic, 716
Isotherm, and Thermodynamic Study. Inorg. Nano-Metal Chem. 2017, 47 (12), 16051613. 717
https://doi.org/10.1080/24701556.2017.1357581. 718
(44) Lazaridis, N. K.; Pandi, T. A.; Matis, K. A. Chromium(VI) Removal from Aqueous Solutions by Mg-Al-CO3 719
Hydrotalcite: Sorption-Desorption Kinetic and Equilibrium Studies. Ind. Eng. Chem. Res. 2004, 43 (9), 720
22092215. https://doi.org/10.1021/ie030735n. 721
(45) Shipley, H. J.; Engates, K. E.; Grover, V. A. Removal of Pb(II), Cd(II), Cu(II), and Zn(II) by Hematite 722
Nanoparticles: Effect of Sorbent Concentration, PH, Temperature, and Exhaustion. Environ. Sci. Pollut. 723
Res. 2013, 20 (3), 17271736. https://doi.org/10.1007/s11356-012-0984-z. 724
(46) Ayawei, N.; Ebelegi, A. N.; Wankasi, D. Modelling and Interpretation of Adsorption Isotherms. J. Chem. 725
2017, 2017. https://doi.org/10.1155/2017/3039817. 726
(47) Huang, Y.; Farooq, M. U.; Lai, S.; Feng, X.; Sampranpiboon, P.; Wang, X.; Huang, W. Model Fitting of 727
Sorption Kinetics Data: Misapplications Overlooked and Their Rectifications. AIChE J. 2018, 64 (5), 728
17931805. https://doi.org/10.1002/aic.16051. 729
(48) Cooney, D. O.; Adesanya, B. A.; Hines, A. L. Effect of Particle Size Distribution on Adsorption Kinetics in 730
Stirred Batch Systems. Chem. Eng. Sci. 1983, 38 (9), 15351541. https://doi.org/10.1007/978-1-4419-731
8074-8_10. 732
Bullen et al. (2021) A revised pseudo-second order kinetic model for adsorption 27
(49) Simonin, J.; Boute, J. Intraparticle Diffusion-Adsorption Model to Describe Liquid/Solid Adsorption 733
Kinetics. Rev. Mex. Ing. Quim. 2016, 15 (1), 161173. 734
735
