Critical of linear and nonlinear equations of pseudo-first order

Critical of linear and nonlinear equations of pseudo-first order and pseudo-second order kinetic models

Hamou Moussout*, Hammou Ahlafi, Mustapha Aazza, Hamid Maghat

Laboratory of Chemistry and Biology Applied to the Environment, Faculty of Sciences, Moulay Ismaïl University, BP 11201-Zitoune, Meknes, 50060, Morocco

Received 30 November 2017; revised 4 April 2018; accepted 5 April 2018 Available online 4 May 2018

Abstract

The experimental adsorption equilibrium of Cd(II) onto chitosan (Cd(II)/CS) and methyl orange onto bentonite (MO/Bt) were studied in batch adsorption experiments at room temperature for an initial concentration of 236.5 mg/L for Cd (II) (pH

¼

5) and 33 mg/L for MO (pH

¼

3). The adsorption rate increases rapidly for t

<

30 min, and the equilibrium is reached after this contact time for both systems. The values of the experimental maximum amount of Cd(II) and MO adsorbed are q

e¼

56.70 and 56.55 mg/g for Cd/CS and MO/Bt, respectively. The obtained experimental data were analysed using the linear and the nonlinear forms of pseudo-first and pseudo-second order kinetic models (LPFO, NLPFO, LPSO, NLPSO). The appropriate model to describe the adsorption kinetics of each system was determined based on the comparison of R

²

and the standard deviation

Dq (%). It was found

that the adsorption process of Cd(II)/CS followed NLPFO and that of MO/Bt can be described by both of NLPSO and LPSO. The results show that the nonlinear forms (NLPSO and NLPFO) are suitable for describing the kinetics adsorption reactions in the liquid phase and the LPSO (q

_t¼

f(1/t) model can also be suitable for some systems, depending on the experimental conditions. Because of q

t

values, determined from these models correspond well to the experimental data as confirmed by the error analysis values of R

²

and

D

q (%), it is noticed that the determination of R

²

alone is insufficient to decide among the kinetic models.

©

2018 The Authors. Production and hosting by Elsevier B.V. on behalf of University of Kerbala. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords:Adsorption kinetics; Linear and nonlinear models; Pseudo-first order; Pseudo-second order

1. Introduction

Environmental degradation due to the release of different pollutants into receiving environments by industrial and agricultural activities has become of great importance. Thus, to maintain our environment in

a good condition, industrial wastewater must be treated before it is discharged. Among several methods cited in the literature for treatment of wastewaters [1 e 3], the adsorption is the most widely used method, in com- parison to other ones, to eliminate both organic and inorganic pollutants [4,5], because it has several ad- vantages regarding cost, efficiency and ease of use [6,7]. Generally, whatever the nature of the used ad- sorbents and the pollutants to be treated, the adsorption process is always controlled by two important aspects:

kinetics and thermodynamics. In order to better

*Corresponding author.

E-mail addresses:moussouthammou@gmail.com,h.moussout@

edu.umi.ac.ma(H. Moussout).

Peer review under responsibility of University of Kerbala.

https://doi.org/10.1016/j.kijoms.2018.04.001

2405-609X/©2018 The Authors. Production and hosting by Elsevier B.V. on behalf of University of Kerbala. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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Karbala International Journal of Modern Science 4 (2018) 244e254

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understand the interactions between adsorbents and adsorbates at equilibrium, several parameters (q

t

: adsorbed amount, t

_eq

: equilibrium time, q

_e

: adsorbed amount at equilibrium, and E

_a

: activation energy) were, thus, determined, using various kinetic and isothermal models, especially those of Lagergren, and Blanchard et al. (1984) and Ho for pseudo-first order (PFO) and pseudo-second order (PSO), respectively [8 e 10]. Langmuir and Freundlich models [11,12], which are based on the solution concentration, were commonly used to describe adsorption isotherms.

In most cases, the kinetic and the isothermal pa- rameters were deduced by a simple confrontation of the experimental data with the linearized equations of these models. Accordingly, the constants can be ob- tained from the slope and the intercept of a straight t line plot, and the appropriate model which fits the experimental data is the one whose coefficient of determination (R

²

) is close to the unity. However, several articles related to this field [13 e 17], have shown that the incorrect application of the linear equations for the kinetic models of PFO and PSO lead to the erroneous values of the intrinsic kinetic param- eters. This problem was related to the various used mathematical expressions and to the unknown value of q

_e

[14,18]. Plazinski et al. [19] portrayed that PSO ' s

wide applicability over PFO does not necessarily stem from a physical basis, but from a mathematical one. On the other hand, The PFO model has been demonstrated to be valid only at the initial stage of adsorption [20].

Similarly, in the case of adsorption isotherms, crit- icisms have been reported concerning the application of the linear equations of the Langmuir and Freundlich isotherms, which are the most often used to predict the maximum adsorption capacity (q

_m

) at equilibrium and the affinity of the adsorbents for the adsorbates.

Vasanth Kumar and Sivanesan [21] recommended that the use of equilibrium data covering the complete isotherm was the best way to obtain the parameters in isotherm expressions; equilibrium data with a partial isotherm was insufficient. Moreover, for the best fit of experimental kinetic and isotherm data, both in batch and column experiment or in other systems involving the transfer of fluids, several studies [22 e 28] have displayed that the application of the nonlinear method is more suitable than the linear one [22,23,29 e 35], because it allows a better adjustment of the different parameters. Recently, the inconsistencies of linearized forms of different models, and their negative impact on parameter values, involved in the liquid phase adsorption process were reviewed [14,29].

The aim of this work is to compare the two forms of the usually used kinetic models (PFO and PSO) for a better interpretation of batch adsorption experiments of Cadmium (Cd (II)) onto chitosan (CS) and methyl orange (OM) onto bentonite (Bt). The results were discussed to support the different criticisms appeared in literature concerning these two models.

2. Experimental part 2.1. Material and methods

The chitosan used in this study was obtained from the deacetylation of chitin extracted from shrimp shells collected in Morocco. Its degree of deacetylation is 76%

as it was described in the previous work [36]. Bentonite was purchased from Rh ^ one-Poulenc (France) and used without any pre-treatment. The salt Cd(NO

₃

)

₂

was purchased from Across Organics (USA) and methyl orange (99%) from Fisher Scientific International Company. All working concentrations of solutions of each pollutant were prepared with distilled water.

A mass m ¼ 0.1 g of each adsorbent, CS or Bt, was first mixed with V ¼ 20 mL of a synthetic solution of Cd(NO

3

)

2

or MO, with initial concentrations C

0

¼ 236.5 and 33 (mg/L) respectively. Then, the mixture was agitated during a given contact time t and Nomenclature

C

₀

Initial concentrations in liquid solution (mg.L

¹

)

C

_e

Equilibrium concentrations in liquid solution (mg.L

¹

)

q

_e

Amount adsorbed at equilibrium (mg.g

¹

) q

_t

Amount adsorbed at time t (mg.g

¹

) V Volume of the solution (L)

m

_ads

Mass of the adsorbent (g) q

_t,exp

Adsorbed amounts experimental q

_t,cal

Adsorbed amounts calculated

k

n

Rate constant for a kinetic with order n k

₁

Rate constant for a kinetic of the pseudo

first order (min

¹

)

k

₂

Rate constant for a kinetic model of pseudo second order (g.mg

¹

.min

¹

)

t

_eq

Equilibrium time (min)

NLPFO Nonlinear pseudo first order model

LPFO Linear pseudo first order model

NLPSO Nonlinear pseudo second order model

LPSO Linear pseudo second order model

SD Standard deviation

at pH ¼ 5for the adsorption of Cd(II) onto CS and pH ¼ 3 and for the adsorption of MO onto Bt. The temperature was controlled by taking the mixtures in a water circulation bath. After a definite adsorption time, the adsorbent was separated from the liquid phase by filtration over 0.45 m m micro-porous membrane. The remaining concentrations of Cd(II) or OM in the so- lution were analysed with ICP (Induced Chemical Plasma) and UV/Visible Spectrophotometer (Shimadzu UV-1240) at l ¼ 500 nm, respectively. The adsorbed amount of each adsorbent was calculated according to the following equation:

q

_e

¼ C

0

C

e

m

ads

V

sol

mg

g

; ð 1 Þ

where q

_e

(mg/g) is the equilibrium concentration of the adsorbate in a solid phase, C

₀

and C

_e

are the initial and the equilibrium concentrations in a liquid phase (mg/L), respectively. m

_ads

is the mass of the adsorbent (g) and V is the volume of solution in L.

Kinetic adsorption experiments were carried out to establish the effect of time on the adsorption process.

The linear and the nonlinear models were used to describe the kinetics curves. Their validities can be determined by the calculation of the standard deviation (SD) D q (%), and the coefficient of determinationR

²

. The best-fit model is the one with the lowest value of SD and the one in which the value of R

²

is closer to unity. The expression of SD equation is given as follows:

D q ð % Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P q

_t;exp

q

_t;cal

q

_t;exp ²

n 1

s

100 ; ð 2 Þ

Where q

_t,exp

and q

_t,cal

are, respectively, the experi- mental and calculated adsorption capacities, and n is the number of data points.

2.2. Theory

The adsorption kinetics reflect the evolution of the adsorption process versus time. The latter is an important parameter, which must be taken into consideration. In addition, in the treatment of aqueous solutions, the adsorption process is intimately depen- dent on the other experimental parameters, such as pH, ionic strength, temperature, concentration of solute, sorbent dose, the texture of adsorbents … , that affect the kinetics of adsorption of an adsorbate onto any adsorbent. Therefore, the contact time where the adsorption process approaches a true equilibrium must be determined according to these parameters.

In batch adsorption systems, several models describing the diffusion of solutes at the surface and in the pores of adsorbent, have been developed (film diffusion model, intra-particular diffusion model, extra-particular diffusion model, pore diffusion model, etc ...) to explain the adsorption kinetics. However, PFO and PSO models have been widely used to describe the rate of adsorption in liquid e solid in- teractions. The expressions of those two models were obtained by integrating the following general equation:

dq

_t

dt ¼ k

_n

ð q

_e

q

_t

Þ

ⁿ

; ð 3 Þ

Where q

_e

and q

_t

are the adsorbate amounts uptake per mass of adsorbent at equilibrium and at any time t (min), respectively; while, k

_n

(1/min) is the constant rate of the pseudo-n-th order kinetic model.

2.2.1. Pseudo first order expression

Lagergren presented the expression of the pseudo first order reaction model for n ¼ 1 [10] as follows:

dq

t

dt ¼ k

1

ð q

e

q

t

Þ; ð 4 Þ

Where q

_e

and q

_t

are the amounts of adsorbate uptake per mass of adsorbent at equilibrium and at any time t (min), respectively, and k

₁

(min

¹

) is the rate constant of the PFO equation.

Integrating equation (4) for boundary conditions (t ¼ 0, q

_t

¼ 0 and t ¼ t, q

_e

¼ q

_t

) leads to the following linear equation:

ln ð q

_e

q

_t

Þ ¼ lnq

_e

k

₁

t ; ð 5 Þ Which can be rearranged in a nonlinearized form:

q

t

¼ q

e

1 e

^k1:t

: ð 6 Þ

2.2.2. Pseudo second order expression

The expression of the PSO adsorption reaction model proposed by Ho et al. [9] was obtained from equation (3) for n ¼ 2:

dq

t

dt ¼ k

₂

ð q

_e

q

_t

Þ

²

; ð 7 Þ

The integration of this equation for the boundary conditions (t ¼ 0, q

_t

¼ 0 and t ¼ t, q

_e

¼ q

_t

) gives the formula below:

q

_t

¼ q

²_e

k

₂

t

q

_e

k

₂

t þ 1 ; ð 8 Þ

Where q

_e

(mg/g) and q

_t

(mg/g) are the adsorbate amount adsorbed at equilibrium and at any t(min), respectively and k

₂

(g/mg min) is the PSO equation constant rate.

Equation (8) can be rearranged to obtain the linear form:

t q

_t

¼ 1

k

₂

q

²_e

þ 1

q

_e

t ; ð 9 Þ

Beside the linear equations given here, several other linearized forms have been used in the interpretation of the experimental data [14].

2.3. Mathematical study

The nonlinear forms of PFO and PSO models (eqs.

(6) and (8)) can be, mathematically considered, as an equivalent to the following equations (10) and (11):

f ð x Þ ¼ a :

1 e

^b:x

; ð 10 Þ

g ð x Þ ¼ c

²

: d : x

c : d : x þ 1 : ð 11 Þ

The study of the two functions f(x) and g(x) leads to the curves represented in Fig. 1. It is observed that for low values of x (x ¼ xe), the PFO and PSO curves are superimposed; whereas, when x increases, the PFO curve turns out to D asymptote; meanwhile, the PSO curve continues to increase, without ultimately moving towards an asymptote. This behaviour is usually ob- tained in the adsorption kinetics experiments [18,20].

McKay et al. also reported that the PFO equation and experimental data did not correlate well within the whole contact time range and was generally appro- priate for the first 20 e 30 min of adsorption process [37]. Mathematical equation (10) is defined continuous

and differentiable on IR

^þ

. For all x value in IR

^þ

as shown below:

f

⁰

ð x Þ ¼ a

1 e

^bx

₀

¼ abe

^bx

> 0 ;

Hence, the function f(x) is strictly increasing on IR

^þ

and represented as follows:

x

lim

/∞

f ð x Þ ¼ lim

x/∞

a

1 e

^bx

¼ a

So, the straight ( D ) line of equation y ¼ a is a horizontal asymptote off(x) in the neighbourhood of þ∞ . As for any value of x in IR

^þ

f ð x Þ a ¼ ae

^bx

3 0, which means that the function f(x) is always below ( D ) asymptote on IR

^þ

. This indicates that the true value of a ¼ q

_e

, cannot be precisely determined.

For the function g(x) (PSO model), which is a homo- graphic, this difficulty does not occur. This finding explains why, in most cases, experimental data for adsorption kinetics followed the linear PSO model rather than the linear PFO model (see Table 1).

3. Results and discussions

Adsorption kinetics studies of Cd(II) onto CS and OM onto Bt were made in batch adsorption systems, according to the experimental conditions presented in section 2.1. Fig. 2 shows the experimental kinetics adsorption of Cd(II) onto CS (Fig. 2a) and MO onto Bt (Fig. 3a) at room temperature. It can be observed that the experimental points can be divided into two regions (I and II). In the region (I), the uptake (q

_t

) of Cd(II) and MO increased rapidly in a similar way for the contact time t ¼ 30 min; then, after (region II), the experi- mental points evolve differently. The q

_t

values increased gradually and become almost constant for the adsorption of Cd(II) onto CS (q

_e

¼ 57 mg/g at saturation); while, in the case of the adsorption of MO onto Bt, the q

_t

values continue to increase as the contact time increases. This difference can be related to the number and the availability of the adsorbing sites over time of the adsorbents.

From the Figs. 2a and 3a, it was noted that the ki- netics curves ’ profiles are similar to those obtained above (Fig. 1), using the NLPFO and NLPSO models.

Thus, the kinetics data of Figs. 2a and 3a were ana- lysed with these models, using the equations (6) and (8). The adsorption kinetic parameters, determined from these equations, are presented in Table 3 and are compared for each case with those that are obtained from the slopes and the intercepts of linear equations of the previous models (LPFO and LPSO).

0 20 40 60 80

0 20 40 60

80 PSO

f(x),g(x)

x

PFO (Δ)

x_e

Fig. 1. Mathematical presentation of the equations(10) and (11).

Table 1

Bibliographic data of the linear modelling of adsorption kinetics.

Adsorbent Adsorbate teq(min) Model R² qexp(mg.g¹) qcal(mg.g¹) Ref.

Polydopamine microspheres Methylene Blue 120 1^st 0.949 90.70 92.68 [38]

2^nd 0.998 99.60

Chitosan/Al2O3/magnetite nanoparticles Methyl Orange 200 1^st 0.839 102.00 46.63 [39]

2^nd 0.999 99.00

Magnetic graphene-carbon nanotube Methylene Blue 30 1^st 0.690 54.60 12.94 [40]

2^nd 0.990 54.64

Polyaniline/graft chitosan Cu(II) 180 1^st 0.962 83.22 53.30 [41]

2^nd 0.994 100.00

Activated carbons from rambutan acid yellow 17 dye 900 1^st 0.958 220.60 212.16 [42]

2^nd 0.993 225.29

Magnetic iron oxide nanoparticles Phosphate 1440 1^st 0.897 5.03 3.03 [43]

2^nd 0.994 5.48

Poly(cyclotriphosphazene-co-4,4⁰-sulfonyldiphenol) nanotubes

Methylene Blue 15 1^st 0.753 69.16 18.56 [44]

2^nd 0.999 74.85

Iron oxide activated redmud (IOARM) Cd(II) 90 1^st 0.994 0.12 0.078 [45]

2^nd 0.997 0.071

Activated carbon Methylene Blue 300 1^st 0.882 270.27 933.30 [46]

2^nd 0.980 263.20

Activated carbon Co(II) 90 1^st 0.010 16.05 1.01 [47]

2^nd 0.990 15.93

Cellulose-graft-polyacrylamide/hydroxyapatite biocomposite hydrogel (EBH)

anthraquinone dye 2880 1^st 0.896 157.20 103.40 [48]

2^nd 0.998 163.4

Silica-multiwall carbon nanotubes Hg(II) 120 1^st 0.942 77.01 38.10 [49]

2^nd 0.999 79.80

Breadnut peel Malachite green 120 1^st 0.935 193.40 80.28 [50]

2^nd 0.997 167.85

Polyacrylamide/cellulose Methylene Blue 240 1^st 0.970 45.00 16.28 [12]

2^nd 0.999 45.41

Agricultural waste (nFe-A) Pb(II) 60 1^st 0.717 822.50 242.26 [51]

2^nd 0.999 833.33

Copper oxide loaded on activated carbon Methylene Blue 30 1^st 0.963 11.10 2.88 [52]

2^nd 0.998 10.32

ZnO/Zn(OH)2-NP-AC Janus Green B 7 1^st 0.410 107.19 2.54310⁸ [53]

2^nd 0.999 120.48

Bamboo biochar acid black 172 300 – e 215.50 e [54]

2^nd 0.987 238.09

Ferromagnetic ordered mesoporous carbon Orange II 300 1^st 0.980 280.00 48.55 [55]

2^nd 0.999 293.16

Chitosan beads Phosphate 180 1^st 0.998 60.60 61.27 [56]

2^nd 1.000 61.16

Biochars Methylene blue 120 1^st 0.954 1.69 1.027 [57]

2^nd 0.995 1.714

Magnetic ZnFe2O4 Acid Red 88 340 1^st 0.962 96.55 61.1 [58]

2^nd 0.997 99.0

Poly(methacryic acid)/zeolite hydrogel Basic yellow 28 2880 1^st 0.994 62.12 30.28 [59]

2^nd 0.755 52.47

Oxide/chitosan fibers Congo red 600 1^st 0.982 121.48 136.27 [60]

2^nd 0.974 144.93

Magnetic CoFe2O4reduced Pb(II) 80 1^st 0.902 122.00 133.9 [61]

2^nd 0.993 123.3

Activated carbon (cow bone) Pb(II) 360 1^st 0.891 42.33 22.925 [62]

2^nd 0.994 45.455

2^nd e 19.19

Zeolites U(VI) 120 1^st 0.995 12.40 1.078 [63]

2^nd 0.993 10.03

La(OH)3-modified phosphate 1800 1^st 0.987 71.70 60.8 [11]

2^nd 0.997 71.3

3.1. Adsorption of Cd(II) onto CS

The plots of NLPFO, NLPSO, LPFO and LPSO models are demonstrated in Fig. 2a, b. It was clearly appearing that the experimental points conform with NLPFO model rather than the NLPSO. Table 3 dis- plays that the values of R

²

is almost equal to 1 and the value of SD ( D q ¼ 0.98%) is very low. In addition, a NLPFO expression reasonably predicts the q

_e

value (q

_e

(exp) z q

_e

(cal) ¼ 57 mg/g). However, the process- ing of the same experimental data by the LPFO and LPSO models leads to a result that contradicts the previous one. From Fig. 2 and Table 3, it is deduced that when LPSO was applied straight line, the value of R

²

is higher than 0.99 and the lower value of SD ¼ 0.012% were obtained. This contradiction does not allow a priori to decide between the two models (NLPFO and LPSO). Indeed, the application of these models to describe the adsorption kinetics in aqueous solution has been the subject of several criticisms

[14,16,27,66,67]. In their conclusions, the authors recommended the use of nonlinear forms (NLPFO and NLPSO) to describe the kinetic of adsorption curves because due to the transformation of nonlinear forms to linear forms, the units of the Y and X axes changed remarkably. The advantage of nonlinear methods is that the error distribution does not alter as it does in linear methods. Table 2 summarises some examples of literature using NL models. In our case, the NLPFO model seems to be the most appropriate to describe the adsorption of Cd (II) onto CS. The obtained values of R

²

, SD and q

_e

(Table 3) did not indicate that the LPSO model properly fitted. However, from Fig. 2, the PSO model can be applied for t < 30 min (region I); while, the NLPFO model describes the adsorption kinetics of Cd (II) on CS over the whole range of time considered (regions I and II) [19]. On the other hand, according to the theoretical approach of Azizian [13], when the initial concentration of solute (C

₀

)is high, like in this study (C

0

¼ 236.5 mg/L), its sorption kinetics better

0 50 100 150 200 250

0 10 20 30 40 50 60

0 20 40 60 80 100 120 140 160 180 200 -6

-4 -2 0 2 4

0 20 40 60 80 100 120 140 160 180 200 0,0

0,5 1,0 1,5 2,0 2,5 3,0 q(mg/g)t 3,5

t(min)

Experimental points NLPFO NLPSO

a

Exprimental points LPFO

ln(qe-qt)

t(min) b

Experimental points LPSO

t/qt(min.g/mg)

t(min)

Fig. 2. Nonlinear (a) and linear (b) PFO and PSO models for the adsorption kinetics of Cd(II) onto CS.

Table 1 (continued)

Adsorbent Adsorbate teq(min) Model R² qexp(mg.g¹) qcal(mg.g¹) Ref.

Olive pomace Basic green 4 120 1^st 0.357 19.55 6.41 [64]

2^nd 1.000 19.58

Magnetic Cellulose/Graphene Oxide Composite Methyl ne Blue 840 1^st 0.747 50.272 13.719 [65]

2^nd 0.997 49.92

0 50 100 150 200 250 0

10 20 30 40 50 60

0 20 40 60 80 100 120 140 160 180 200 -3

-2 -1 0 1 2 3 4 5

0 20 40 60 80 100 120 140 160 180 200 0,0

0,5 1,0 1,5 2,0 2,5 3,0 q(mg/g)t 3,5

t(min)

Exprimental points NLPFO NLPSO

Experimental points LPFO

ln(qe-qt)

t(min)

a b

Experimental points LPSO

t/qt(min.g/mg)

t(min)

Fig. 3. Linear (b) and nonlinear (a) PFO and PSO models for the adsorption kinetics of MO onto Bt.

Table 2

Bibliographic data for nonlinear modelling of adsorption kinetics.

Adsorbant Adsorbat teq(min) Model R² qexp(mg.g¹) qcal(mg.g¹) Ref.

Cellulose acetate blend

[CA/SPS(90/10)þ[BMIM]Cl-NIM]

salicylic acid 60 1^st 0.862 2.12 2.0513 [68]

2^nd 0.990 2.1484

Alginate/carboxymethyl cellulose sodium composite

Uranium 900 1^st 0.981 101.76 115.09 [69]

2^nd 0.973 167.69

NeS-doped mesoporous carbon (NSMC) Ibuprofen 90 1^st 0.972 54.27 52.55 [70]

2^nd 0.997 55.69

Black rice husk ash Chromium 60 1^st 0.996 2.92 2.19 [71]

2^nd 0.999 3.13

Ulvalactuca algae Cadmium 150 1^st 0.979 127.00 114.29 [72]

2^nd 0.931 122.91

Silica nanohollowsphere methylene blue 1440 1^st 0.972 28.00 19.11 [73]

2^nd 0.950 24.39

Pecan nutshell Lead 300 1^st 0.995 179.00 171.40 [74]

2^nd 0.997 196.00

MesoporousNiO nanoparticles Chromium 50 1^st 0.999 4.50 4.48 [75]

2^nd 0.999 4.74

Hydrochar Methylene blue 400 1^st 0.733 96.56 91.29 [76]

2^nd 0.939 98.35

Banana stalk activated carbon Bentazon 1200 1^st 0.984 50.63 48.98 [7]

2^nd 0.998 51.84

Pongamiapinnata Methylene blue 1440 1^st 0.613 75.30 65 .57 [77]

2^nd 0.994 77 .99

Magnetic biochar composite Lead 180 1^st 0.975 3.29 3.17 [78]

2^nd 0.997 3.38

Mesoporous-activated carbon Methylene blue 360 1^st 0.996 101.15 100.13 [79]

2^nd 0.995 103.31

Bamboo charcoals activated with nitric acid direct yellow 161dye 1800 1^st 0.659 1.92 1.66 [80]

2^nd 0.998 1.78

Candida utilis zinc 360 1^st 0.820 23.12 21.41 [81]

2^nd 0.950 23.66

fits the pseudo-first order model, instead of the pseudo- second order kinetics.

3.2. Adsorption of MO onto Bt

The experimental points of adsorption kinetics of MO onto Bt at room temperature is given in Fig. 3. The adsorption rate was rapid during the first 30 min and then continued at a slower rate from 30 to 200 min.

The kinetic data were analysed by the LPFO, LPSO, NLPFO and NLPSO models (Fig. 3). According to Fig. 3a, for the NLPFO model, the theoretical curve does not follow the experimental points, unlike its linear form. However, both of the LPSO and NLPSO models seems to fit well with the experimental points.

Moreover, the coefficient correlation R

²

and D q (%) as well as the calculated q

_e

values (Table 3) confirm the validity of these models. In addition, the same values of K

₂

were found. This finding confirms that in the case of the linearization of the NLPSO form, the kinetic parameters were not affected. Indeed, the advantage of the pseudo-second order equation as an expression estimating the q

_e

values is its small sensitivity for the influence of the random experimental error, as it is illustrated in the mathematics section. Kumar [16]

found that the most popular form of PSO (t/q vs. t) was not the best form, unlike other authors [82,83].

This Difference shows that the accuracy of the line- arized and the nonlinearized forms are dependent on the adsorbent/adsorbate type and experimental condi- tions, as indicated in the current paper.

4. Conclusion

CS and Bt adsorbents were used on Cd(II) and MO removal, respectively, from aqueous solution. The removal of Cd (II) and MO was fast and the equilib- rium was reached in 30 min for both systems. The processing of experimental data by the linear and nonlinear pseudo-first order and pseudo-second order models depicted that the adsorption kinetics of Cd

follows the NLPFO model and that of MO can be described by both the NLPSO and LPSO models.

However, the study has shown that to decide between the linear and non-linear models, for a given system, is sometimes difficult because the kinetic parameters deduced from these models are identical and the error computations (R

²

and SD) are reasonable in both cases.

Taking into account, the debates undertaken in litera- ture, it briefly appears that the nonlinear forms of these models are more suitable than their linear forms for the modeling of the kinetics of adsorption in liquid phase.

However, the nature of the studied systems and the exact determination of the q

_e

value remain crucial variables to decide.

Acknowledgement

This work was supported by MESRSFC and CNRST Rabat-Morocco, within the framework of the PPR2 project.

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