# Applying Linear Forms of Pseudo-Second-Order Kinetic Model for Feasibly Identifying Errors in the Initial Periods of Time-Dependent Adsorption Datasets

^{1}

^{2}

## Abstract

**:**

^{2}(adj-R

^{2}), high reduced χ

^{2}(red-χ

^{2}), and high Bayesian information criterion (BIC) values. After removing these points, the experimental data were adequately fitted with the PSO model. Statistical analyses demonstrated that the nonlinear method must be used for modelling the PSO model because its red-χ

^{2}and BIC were lower than the linear method. Type 1 has been extensively applied in the literature because of its very high adj-R

^{2}value (0.9999) and its excellent fitting to experimental points. However, its application should be limited because the potential errors from experimental points are not identified by this type. For comparison, the other kinetic models (i.e., pseudo-first-order, pseudo-nth-order, Avrami, and Elovich) are applied. The modelling result using the nonlinear forms of these models indicated that the fault experimental points from the initial periods were not detected in this study.

## 1. Introduction

^{2+}ions onto biosorbent (derived from maize straw modified with succinic anhydride) rapidly occurred under the first period of adsorption kinetics, with approximately 70.0–96.6% of total Cd

^{2+}ion solutions (C

_{o}= 100, 300, and 500 mg/L at pH = 5.8) being removed within 1 min. Zbair et al. [4] reported that the adsorption process of bisphenol A or diuron by Argan-nutshell-based hydrochar reached a fast equilibrium within 4 min of contact. An analogous result has been reported by many scholars who invested in the adsorption of Cd

^{2+}ions onto orange-peel-derived biochar [5] and rhodamine B dye onto white-sugar-based activated carbon [6]. Therefore, some authors [3,5,6] took a very short time (1 min, 2 min, etc.) for adsorption kinetics instead of a longer one (i.e., starting from 10 min) [7].

_{t}) decrease, and the amount of adsorbate adsorbed by an adsorbent at times (q

_{t}) increases. This means that the values q

_{t}at the initial periods might be higher than those expected if researchers do not conduct experiments carefully. They can be defined as the error points in datasets and affect the results of modelling adsorption on kinetics. It might be hard to verify those doubtful results by existing techniques, and this is a current limitation in this field.

_{t}= 0 at t = 0, and q

_{t}= q

_{t}at t = t), the nonlinear form of the PSO model is obtained as Equation (2) [16].

_{2}(kg/(mol × min)) is the rate constant of this model; q

_{e(2)}and q

_{t}(mol/kg) are the amounts of pollutant in the solution adsorbed by the material at equilibrium and any time t (min) is obtained from Equation (3); C

_{o}and C

_{t}(mol/L) are the concentrations of the pollutant at the beginning and any time t (min); m (kg) is the dry mass of the material; and V (L) is the volume of the pollutant used in the study of the adsorption kinetics.

^{2}or r

^{2}value obtained from this type is very close to 1 [2,12].

^{2}; Equation (10)) and adjusted determination coefficient (adj-R

^{2}; Equation (11)). They include chi-squared (χ

^{2}; Equation (12)), reduced chi-square (red-χ

^{2}; Equation (13)), and Bayesian information criterion (BIC; Equation (14)) [2,12,16] A model with higher adj-R

^{2}value and lower red-χ

^{2}indicates a better fitting than the others. Because of the difference among the parameters of the models used, the magnitude (or the absolute value) of ΔBIC (BIC of model 1–BIC of model 2) is used for selecting the best fitting model [2,16].

_{exp}is q

_{t}(mol/kg) obtained from experiments; y

_{exp–mean}is an average of y

_{exp}values used for modelling; y

_{model}is q

_{t}calculated based on the PSO model; N is the number of experimental points; P is the number of parameters of the model (I.e., P = 2 for the PSO model); and DOF is the degrees of freedom.

^{th}-order, Avrami, and Elovich ones. Commercial activated carbon and paracetamol (PRC) were used as the target adsorbent and adsorbate.

## 2. Experimental Conditions and Procedures

^{2}/g and 0.670 cm

^{3}/g, respectively. The experiment of kinetics adsorption was conducted in triplicate, and the result of q

_{t}was averaged. The adsorption condition was maintained at 25 °C, pH = 7.0, and initial PRC (3.45 mmol/L). The solutions (1.0 M NaOH and 1.0 M HCl) were used to adjust the pH of PRC solution before and during the adsorption process.

_{t}value was calculated based on the mass balance equation (Equation (3)). The blank samples (without the presence of CAC in the PRC solution) were conducted simultaneously.

## 3. Pseudo-Second-Order Model

_{t}values are nearly identical at this region) presents in the kinetic curve (the plot of q

_{t}vs. t) [2].

^{2}value was only 0.8645, suggesting that the adsorption process was not well-described by the PSO model. Clearly, the application of the nonlinear method did not provide information as to where the experimental points of time-dependent adsorption are errors.

^{2}value of 0.9999 (Table 1). A similar conclusion was obtained for Type 6. In contrast, the fault experimental points (denoted as Point 1 and Point 2 in the corresponding figures) were well identified by the others (Types 2, 3, 4, and 5). This is because their R

^{2}values only ranged from 0.4571 to 0.5627 (Table 1).

**Table 1.**Parameters of the pseudo-second-order model obtained from the nonlinear form and its six linear forms.

Result of Modelling | Result of Re-Modelling | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Full Raw Data (n = 16) * | Manipulated Data (n = 14) ** | Revisited Data (n = 16) *** | |||||||||||||

k_{2} | q_{e(2)} | adj-R^{2} | χ^{2} | BIC | k_{2} | q_{e(2)} | adj-R^{2} | χ^{2} | BIC | k_{2} | q_{e(2)} | adj-R^{2} | χ^{2} | BIC | |

1. Nonlinear form | |||||||||||||||

0.2034 | 1.419 | 0.8645 | 2.2 × 10^{−2} | −61.6 | 0.0635 | 1.501 | 0.9972 | 4.2 × 10^{−4} | −113.3 | 0.0635 | 1.501 | 0.9985 | 3.65 × 10^{−4} | −131.0 | |

2. Linear forms | |||||||||||||||

Type 1 | 0.0711 | 1.504 | 0.9999 | 3.1 × 10^{−2} | −52.1 | 0.0624 | 1.504 | 0.9999 | 4.6 × 10^{−4} | −104.4 | 0.0625 | 1.504 | 0.9999 | 4.0 × 10^{−4} | −121.9 |

Type 2 | 0.856 | 1.309 | 0.5627 | 3.4 × 10^{−2} | −50.8 | 0.0701 | 1.489 | 0.9793 | 6.0 × 10^{−4} | −100.8 | 0.0633 | 1.504 | 0.9999 | 4.0 × 10^{−4} | −121.9 |

Type 3 | 0.746 | 1.348 | 0.4571 | 3.1 × 10^{−2} | −52.3 | 0.0673 | 1.495 | 0.9731 | 6.0 × 10^{−4} | −100.8 | 0.0641 | 1.500 | 0.9953 | 3.9 × 10^{−4} | −122.1 |

Type 4 | 0.3498 | 1.418 | 0.4571 | 2.6 × 10^{−2} | −55.2 | 0.0656 | 1.498 | 0.9731 | 6.0 × 10^{−4} | −100.8 | 0.0638 | 1.501 | 0.9953 | 3.9 × 10^{−4} | −122.1 |

Type 5 | 0.4543 | 1.382 | 0.5627 | 2.6 × 10^{−2} | −54.8 | 0.0684 | 1.492 | 0.9793 | 6.0 × 10^{−4} | −100.8 | 0.0632 | 1.505 | 0.9999 | 4.0 × 10^{−4} | −121.7 |

Type 6 | 0.0804 | 0.703 | 0.9109 | 6.3 × 10^{−1} | −3.60 | 0.0804 | 0.672 | 0.9053 | 6.3 × 10^{−4} | −92.9 | 0.0805 | 0.750 | 0.9173 | 5.1 × 10^{−1} | −6.938 |

_{t}= 0.741 and 0.789 mol/kg); and *** Case 3: results obtained after using the new Points 1 (q

_{t}= 0.131 mol/kg; Table 3) and 2 (0.484 mol/kg; Table 3) estimated from the PSO model (k

_{2}= 0.0635 and q

_{e(2)}= 1.501); the unit of q

_{e}(mol/kg) and k

_{2}(kg/(mol × min)); q

_{e,exp}(1.495 ± 0.0061 mol/kg) obtained from the experiment.

**Table 2.**Parameters of some models obtained based on the three cases: full raw data, manipulated data, and revisited data.

Unit | Result of Modeling (Using the Nonlinear Method) | |||
---|---|---|---|---|

Full Raw Data (n = 16) * | Manipulated Data (n = 14) ** | Revisited Data (n = 16) *** | ||

q_{e,exp} | mol/kg | ~1.495 | ~1.495 | ~1.495 |

1. Pseudo-second-order (PSO) model | ||||

q_{e(2)} | mol/kg | 1.419 | 1.501 | 1.501 |

k_{2} | kg/(mol × min) | 0.2034 | 0.0635 | 0.0635 |

adj-R^{2} | - | 0.8645 | 0.9972 | 0.9985 |

red-χ^{2} | - | 2.16 × 10^{−2} | 4.20 × 10^{−4} | 3.65 × 10^{−4} |

BIC | - | −61.6 | −113.3 | −131.1 |

2. Pseudo-first-order (PFO) model | ||||

q_{e(1)} | mol/kg | 1.368 | 1.436 | 1.428 |

k_{1} | 1/min | 0.1792 | 0.0512 | 0.0576 |

adj-R^{2} | - | 0.7506 | 0.9597 | 0.9710 |

red-χ^{2} | - | 3.98 × 10^{−2} | 6.09 × 10^{−3} | 6.86 × 10^{−3} |

BIC | - | −51.3 | −85.6 | −81.16 |

3. Elovich model | ||||

α | mol/(kg × min) | 126.1 | 1364 | 2.913 |

α | mg/(g × min) | 19,073 | 206,235 | 440.4 |

β | kg/mol | 9.299 | 11.11 | 6.335 |

adj-R^{2} | - | 0.9435 | 0.9480 | 0.8700 |

red-χ^{2} | - | 9.03 × 10^{−3} | 7.75 × 10^{−3} | 3.08 × 10^{−2} |

BIC | - | −76.5 | −81.5 | −55.6 |

4. Avrami model | ||||

q_{e(AV)} | mol/kg | 1.536 | 1.496 | 1.472 |

k_{AV} | 1/min | 0.0973 | 0.0618 | 0.0510 |

n_{AV} | - | 0.2913 | 0.4571 | 0.6312 |

adj-R^{2} | - | 0.9801 | 0.9995 | 0.9935 |

red-χ^{2} | - | 3.17 × 10^{−3} | 7.35 × 10^{−5} | 1.54 × 10^{−3} |

BIC | - | −95.4 | −162.1 | −107.7 |

5. Pseudo-n^{th}-order (PNO) model | ||||

q_{e(PNO)} | mol/kg | 1.933 | 1.522 | 1.515 |

k_{PNO} | kg^{m}^{−1}/(min × mol^{m}^{−1}) | 0.0268 | 0.0659 | 0.0631 |

m | - | 7.299 | 2.247 | 2.134 |

adj-R^{2} | - | 0.9605 | 0.9979 | 0.9986 |

red-χ^{2} | - | 6.31 × 10^{−3} | 3.21 × 10^{−4} | 3.25 × 10^{−4} |

BIC | - | −83.8 | −137.0 | −134.1 |

_{t}= 0.741 and 0.789 mol/kg) from the raw datasets; and *** Case 3: results obtained after removing Points 1−2 and submitting the new Point 1 (q

_{t}= 0.131 mol/kg; Table 3) and Point 2 (0.484 mol/kg; Table 3) estimated from the PSO model (k

_{2}= 0.0635 and q

_{e(2)}= 1.501) into the raw datasets.

**Table 3.**Comparison of the result estimation of some q

_{t}values at the initial periods obtained by different kinetic models and q

_{t(exp)}from the experiment.

Time (min) | q_{t(exp)} | q_{t} Values Estimated Based on the Models | Difference (%) between q_{t(exp)} and q_{t(model)} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

PFO | PSO | PNO | Elovich | Avrami | PFO | PSO | PNO | Elovich | Avrami | ||

1 min (Point 1) | 0.741 | 0.072 | 0.131 | 0.151 | 0.866 | 0.042 | 164.7 | 140.0 | 132.4 | −15.7 | 178.7 |

5 min (Point 2) | 0.789 | 0.324 | 0.484 | 0.525 | 1.011 | 0.197 | 83.4 | 47.8 | 40.2 | −24.7 | 120.0 |

15 | 0.933 | 0.770 | 0.883 | 0.905 | 1.110 | 0.517 | 19.1 | 5.43 | 3.03 | −17.40 | 57.4 |

30 | 1.090 | 1.127 | 1.112 | 1.114 | 1.173 | 0.855 | −3.290 | −1.965 | −2.129 | −7.258 | 24.2 |

45 | 1.188 | 1.293 | 1.217 | 1.211 | 1.209 | 1.076 | −8.446 | −2.441 | −1.928 | −1.770 | 9.846 |

60 | 1.247 | 1.369 | 1.278 | 1.268 | 1.235 | 1.221 | −9.327 | −2.389 | −1.635 | 1.005 | 2.116 |

^{2}values being 0.9972 (for the nonlinear method), and 0.9731−0.9793 (for the linear form of Type 2–5; Table 1). Figure 1b shows that the experimental data (after removing Points 1 and 2 in Figure 1a) were well described by the nonlinear form of the PSO model (adj-R

^{2}= 0.9972). Although the evaluation of regular residuals can be used to identify the potential errors within a dataset [11], it is not helpful for many cases. For example, Figure 1c shows that the outlier identification is only visible for Point 1.

^{2}values did not change before (adj-R

^{2}= 0.9999) and after manipulation (0.9999; Table 1). The difference between the k

_{2}(0.0711 kg/(mol × min)) value of full data and the k

_{2}value (0.0624 kg/(mol × min)) of manipulated data obtained from Type 1 was only 13% (Table 1). However, the difference was remarkable for Type 2 (170%), Type 3 (167%), Type 4 (137%), and Type 5 (148%). A similar percentage of the high difference was reported for the case of using the nonlinear method (105%). The result suggested that the use of the linear form (Type 1) of the PSO model can achieve a very high adj-R

^{2}value and well describe the experimental data (Figure S1). However, it is very hard to verify whether the parameters of the PSO model are highly believable and accurate. In the literature, some authors have analysed the problems of the linear form of Type 1 [2,21]. They found that although the R

^{2}value of Type 1 is very high (even reaching 1), it is a spurious correlation [2,21].

^{2}and BIC) were applied. The results (Table 1) demonstrated that the values of red-χ

^{2}and BIC obtained from the nonlinear method were lower than those from the linear method. Therefore, the utilization of the nonlinear method can minimize some error functions during the modelling process. A similar conclusion has been reported by some scholars [2,18,19].

_{2}value).

## 4. Other Common Adsorption Kinetic Models

^{th}-order model (herein called the PNO model) (Equation (16)) [22,23] (also known as the general-order kinetic model (Equation (17)) [16] and the Avrami-fractional-order model (Equation (18)) [26]. More detailed information on these models has been reported in some documents [16].

_{t}(mol/kg) is defined in Equation (2); k

_{1}(1/min), k

_{PNO}(kg

^{m}

^{−1}/(min × mol

^{m}

^{−1})), and k

_{AV}(1/min) are the adsorption rate constants of the PFO, PNO, and Avrami models, respectively; q

_{e(1)}, q

_{e(PNO)}, and q

_{e(AV)}are the adsorption capacities of CAC towards paracetamol at equilibrium (mol/kg) estimated by the PFO, PNO, and Avrami models, respectively; m and n

_{AV}are the exponents of the PNO and Avrami models; α (mol/(kg × min)) and β (kg/mol) are the initial rate and the desorption constant, respectively, of the adsorption kinetics of the Elovich model.

_{e(1)}and k

_{1}. The parameter q

_{e(1)}must be selected before applying the linear method. However, it is not easy to select q

_{e(1)}appropriately. If q

_{e(1)}is lower than (or equal to) q

_{t}, the result of calculating ln(q

_{e(1)}− q

_{t}) is an error (#NUM!). By selecting q

_{e(1)}slightly higher than the highest value of q

_{t}, the result of fitting the linear method is provided in Figure S2. Clearly, the errors in the experimental datasets of time-dependent adsorption (i.e., Points 1 and 2) were not detected through the linear fitting of the PFO model (Figure S2).

## 5. Re-Modelling Using Points 1 and 2 Estimated from the PSO Model

_{t}at 1 min (Point 1) and 5 min (Point 2) are estimated. The results (Table 3) indicated a remarkable difference between the q

_{t}value obtained from the experiment (q

_{t(exp)}) and the q

_{t}value estimated from the model (q

_{t(model)}). Taking the case of the PSO model as a typical example, the values q

_{t(exp)}at 1 min and 5 min (0.741 and 0.789 mol/kg) are overwhelming higher than those obtained from the PSO model (0.131 and 0.484 mol/kg, respectively). The differences between q

_{t(exp)}and q

_{t(model)}are 140% for Point 1 and 47.8% for Point 2 (Table 3). The result confirms again that they (Points 1 and 2) were errors and outliers.

_{t}values for Point 1 (0.131 mol/kg) and Point 2 (0.484 mol/kg) are estimated by the PSO model. After submitting these estimated points into the datasets, the results of re-modelling are provided in Figure S4 and Table 2 (revisited data). The fitting model followed the decreasing order: the PNO model (BIC = −134.1) > PSO model (−131.1) > Avrami model (−107.7) > PFO model (−81.16) > Elovich model (−55.6). The results indicate that the Elovich and Avrami models can be considered empirical equations used for physically fitting the experiential data of time-dependent adsorption. In particular, the initial rate α of the Elovich model (Table 2) indicated a remarkable change of 19,073 mg/(g × min) for the full raw data (n = 16), 206,235 mg/(g × min) for the manipulated data (n = 14), and 440.4 mg/(g × min) for the manipulated data (n = 16). The exponent n

_{AV}of the Avrami model that does not provide information on the overall order of the adsorption process only serves an empirically adjusted parameter. Unlike those models, the PNO model (especially its exponent m) is helpful for initially checking whether the datasets are feasible or not. The overall order of the adsorption process (m) was 2.134, which is close to the second order.

_{t}= 0.131 mol/kg) and Point 2 (0.484 mol/kg)) the datasets, the results of re-modelling using six linear forms of the PSO model are provided in Figure S5 and Table 1. As expected, the errors (Points 1 and 2) in the experimental datasets are not observed in Types 1–5, suggesting that the application of the linear form (i.e., Types 1–5) of the PSO model can help to identify these potential errors in the initial adsorption period.

^{2}= 0.9999) when considering three cases: the full raw data (n = 16), manipulated data (n = 14), and manipulated data (n = 16) in Table 1; therefore, it is impossible to verify whether there are errors in the datasets or not. Type 6 should not be used because it must first estimate the unknown q

_{e}value as in the case of the linear form of the PFO model. Figure 5 provides a brief summary of the validation and re-verification processes of the time-dependent adsorption data. Adsorption kinetic curves (q

_{t}vs. t) must indicate an equilibrium region (as showed in Figure 5).

_{3}-enriched materials occurred very fast at the first period of adsorption. The primary adsorption was surface precipitation such as Cd(CO

_{3}) and/or (Cd,Ca)CO

_{3}[5,29]. This mechanism was known as nonactivated chemisorption that often occurs very rapidly and has low activation energy [29,30]. It might not be necessary to recheck the initial periods of the time-dependent adsorption datasets. This is because it is the nature of nonactivated chemisorption [3,30].

## 6. Conclusions

_{t}vs. t indicates two regions: kinetics and equilibrium (equilibrium is a plateau region).

^{2}= 0.9999; n = 14), and revisited data (adj-R

^{2}= 0.9999; n = 16). However, it (adj-R

^{2}or r

^{2}) has been denoted as a spurious correlation in some cases. The application of Type 1 for calculating the parameters (q

_{e(2)}and k

_{2}) should be avoided because it is hard to validate the accuracy of these parameters. For Type 6, the q

_{e}value must be firstly estimated (a similar case of the linear form of the PFO model); therefore, its parameters obtained did not bring a high accuracy.

## Supplementary Materials

_{t}= 0.131 mol/kg) and Point 2 (0.484 mol/kg) estimated from the PSO model to the datasets (Note: the modelling results for the case of the revisited data are provided in Table 2); Figure S5. Re-modelling results using the six linear forms of the PSO model after submitting Point 1 (0.131 mol/kg) and Point 2 (0.484 mol/kg) estimated from the PSO model to the six datasets (Note: the modelling results for the case of the revisited data are provided in Table 1).

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Adsorption kinetics of paracetamol by commercial activated carbon fitted by the PSO model.

**Figure 2.**The linear forms (Type 1 and Type 2) of the pseudo-second-order model (

**a**,

**c**) before (n = 16) and (

**b**,

**d**) after (n = 14; Points 1 and 2 removed) data manipulated.

**Figure 3.**The linear forms (Type 3 and Type 4) of the pseudo-second-order model (

**a**,

**c**) before (n = 16) and (

**b**,

**d**) after (n = 14; Points 1 and 2 removed) data manipulated.

**Figure 4.**The linear forms (Type 5 and Type 6) of the pseudo-second-order model (

**a**,

**c**) before (n = 16) and (

**b**,

**d**) after (n = 14; Points 1 and 2 removed) data manipulated.

**Figure 5.**Processes for validation and re-verification of time-dependent adsorption data (Note: adsorption kinetic curves (q

_{t}vs. t) must indicate two regions: kinetic and equilibrium; equilibrium is a plateau region).

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**MDPI and ACS Style**

Tran, H.N.
Applying Linear Forms of Pseudo-Second-Order Kinetic Model for Feasibly Identifying Errors in the Initial Periods of Time-Dependent Adsorption Datasets. *Water* **2023**, *15*, 1231.
https://doi.org/10.3390/w15061231

**AMA Style**

Tran HN.
Applying Linear Forms of Pseudo-Second-Order Kinetic Model for Feasibly Identifying Errors in the Initial Periods of Time-Dependent Adsorption Datasets. *Water*. 2023; 15(6):1231.
https://doi.org/10.3390/w15061231

**Chicago/Turabian Style**

Tran, Hai Nguyen.
2023. "Applying Linear Forms of Pseudo-Second-Order Kinetic Model for Feasibly Identifying Errors in the Initial Periods of Time-Dependent Adsorption Datasets" *Water* 15, no. 6: 1231.
https://doi.org/10.3390/w15061231