## What is/are First Order Model?

First Order Model - A high R2 value, low root mean square error (RMSE), and mean absolute percentage error (MAPE) suggest that the Cr (VI) adsorption follows the pseudo-first-order model.^{[1]}The Langmuir and pseudo-first-order model were fitting these adsorption characteristics well.

^{[2]}Meanwhile, comparison results show that the global LPV modeling approach can describe the nonlinear battery dynamics adequately with first-order model, which makes it a promising method for control-oriented applications.

^{[3]}In vitro release of the oxaprozin-loaded microemulsion best fit the first-order model, while the microemulsion preparation had a certain sustained release effect.

^{[4]}32% and outperformed a traditional first-order model in every case, as well as more complex models in certain cases.

^{[5]}952) compared with the pseudo-first-order model (R2 ≥ 0.

^{[6]}The kinetic curves of the betalain degradation of water samples depicted a first-order model, indicating the alteration of a violet colouration of betalains from beetroot waste for 5–7 days.

^{[7]}The kinetic of the removal process was examined and it was found that the reaction obeys a pseudo-first-order model and the intraparticle diffusion is not the sole mechanism dominating the reaction.

^{[8]}To date, the replica framework has been discussed for first-order models, whereby elementary replica constituents are single neurons with independent Poisson inputs.

^{[9]}The adsorption equilibrium data and contact time data were best fitted with Langmiur and Pseudo-first-order model, respectively.

^{[10]}The kinetic data was well described by the Pseudo first-order model indicating that physicosorption is the predominant mechanism.

^{[11]}Kinetic data fit well with the pseudo-first-order model which shows that physisorption is favoured than chemisorption.

^{[12]}The kinetic data were consistent with the pseudo-first-order model.

^{[13]}In the kinetic study, the results for catalysts showed that the experimental data were fitted the pseudo-first-order model.

^{[14]}The adsorption kinetic data is fitted by the pseudo-first-order model with R2 > 0.

^{[15]}The kinetic studies revealed that the adsorption of Sr2+ onto MnO2-110 followed the pseudo-first-order model whereas the adsorption equilibrium data obeyed the Freundlich and Sips model.

^{[16]}Adsorption kinetics showed reasonable conformity with the pseudo-first-order model, where isothermal adsorption followed a Sips model.

^{[17]}We recently introduced the notion of twin-width, a novel graph invariant, and showed that first-order model checking can be solved in time f(d, k)n for n-vertex graphs given with a witness that the twin-width is at most d, called d-contraction sequence or d-sequence, and formulas of size k [Bonnet et al.

^{[18]}The adsorption process depends largely on the BPA concentration and the results fitted well with the pseudo-first-order model.

^{[19]}About kinetic adsorption, the pseudo-first-order model showed satisfactorily represented.

^{[20]}The kinetic study shows that our results fit well with the modified pseudo-first-order model.

^{[21]}The kinetics of Rh-B dye photodegradation process correlates well with the Pseudo-first-order model.

^{[22]}This study comprehensively investigated plastic, aqueous, and operating variables in the flotation removal of polyethylene terephthalate (PET) and polystyrene (PS) MPs, assisted by numerous bench-scale experiments and a first-order model with rectangular distribution of floatability.

^{[23]}Kinetic analyses indicated that photocatalytic degradation rate followed the first-order model.

^{[24]}To estimate and compare the resistance of the target aggregates in solutions with different salinities, a pseudo-first-order model that describes the rupture degree as a function of shear rate increments obtains the characteristic shear rate.

^{[25]}The equilibrium data and kinetics were well described by Langmuir and pseudo-first-order models, respectively.

^{[26]}The optimal conditions for the rice bran/chitosan hydrogel beads to adsorb reactive blue 4 were pH 3, a dosage of 40 mg, at 50 °C, for 7 h of adsorption, and the kinetic and isothermal adsorption data were consistent with the pseudo-first-order model and the Langmuir isotherm model, respectively.

^{[27]}Additionally, a kinetic study was carried out using a first-order model, and the results showed that the kinetic properties are compatible with this model.

^{[28]}The first-order model proposed by Seol & Jirka (J.

^{[29]}The kinetics of the photocatalytic degradation process was well described by Langmuir-Hinshelwood’s pseudo-first-order model (kapp = 0.

^{[30]}The SF, MG, CR, and MO dyes’ uptake reactions are in agreement with the kinetic behavior of the pseudo-first-order model and the equilibrium properties of the Langmuir model.

^{[31]}999) as compared to pseudo-first-order model.

^{[32]}Nonlinear first-order models represent the magnitude and rate saturation of the actuators.

^{[33]}The pseudo-first-order model demonstrated a greater prediction capacity for the system.

^{[34]}The adsorption of metal ions and pigments onto activated bentonite clay under UBM was quite well by the pseudo-first-order model.

^{[35]}The kinetic mechanism for ILs/W systems had followed from first-order model and for O/W systems followed Higuchi zero-order model.

^{[36]}In this test, first-order model was modified to solve this problem.

^{[37]}According to the kinetics study results, the adsorption process obeys the pseudo-first-order model.

^{[38]}They have different thermal stability, so that heat inactivation kinetics of crude peroxidase extracts from radicles does not fit the first-order model.

^{[39]}Coefficients estimated by conventional, modified, and residual first-order models ranged from 0.

^{[40]}The first-order model was selected due to high linear fitting regression.

^{[41]}Also, the degradation processes are fitted well with the first order model.

^{[42]}The transesterification data agreed well with pseudo-first order model with highest rate constant value of 2.

^{[43]}Furthermore, it is shown that the degradation kinetics can be described by pseudo-first order model.

^{[44]}The results show that the pseudo-first order model is the one that best describes the biosorption of the AR73.

^{[45]}With respect to the first order modeling of chlorine decay and pathogen inactivation, chlorine concentrations are found to be significantly distinct for seasonal variations in water supply to maintain 3-log inactivation of Giardia cysts.

^{[46]}Also, the adsorption behaviour of crystal violet in electrocoagulation was also studied and the isothermal and kinetic models were proposed to be the Dubinin-Radushkevich model and pseudo-first order model.

^{[47]}The experimental data fitted the first order model and the inactivation kinetic study of PME was completed with the calculations of the activation energy and volume of activation.

^{[48]}The proposed model can preserve sharp features and simultaneously suppress the staircase effects in smooth regions which overcomes the drawback of the first order models.

^{[49]}For the bisector of the diagram at A and B>8, the critical value is stabilized and set at the level characteristic of the first order model.

^{[50]}

## pseudo second order

The pseudo-second-order version yields a higher fit to the experimental facts than the pseudo-first-order model.^{[1]}Kinetic date followed pseudo-second-order rate kinetic better than pseudo-first-order model.

^{[2]}Also, its adsorption mechanism was estimated by kinetic models (pseudo-first-order model, pseudo-second-order model, and intraparticle diffusion).

^{[3]}The pseudo-first-order model and pseudo-second-order model were used to analyze the adsorption data, and the results showed that the adsorption process conforms to the kinetic model.

^{[4]}Phosphate adsorption showed a close agreement between the experimental and theoretical capacities predicted by the pseudo-second-order model, whereas ibuprofen fitted to a first-order model.

^{[5]}The adsorption equilibrium of adsorbent was reached in 420 min with initial solution concentrations at 1 mg/L, 10 mg/L, and 50 mg/L, and the process better fitted to the nonlinear pseudo-second-order model than to the nonlinear pseudo-first-order model.

^{[6]}Kinetic data best fitted pseudo-second-order model and not the pseudo-first-order model.

^{[7]}In addition, the pseudo-first-order model, pseudo-second-order model, and Elovich model all fitted the adsorption process well, which indicated that both physical adsorption and chemical adsorption existed simultaneously, in which physical adsorption played a dominant role and chemical adsorption played a minor role.

^{[8]}Adsorption kinetics was studied by using pseudo-first-order model, pseudo-second-order model, Elovich model, liquid film diffusion model, modified Freundlich equation, and Bangham equation.

^{[9]}The adsorption kinetics of MB and Pb2+ on pristine spiky balls followed the pseudo-second-order and the pseudo-first-order models (R2 = 0.

^{[10]}Compared with the Lagergren pseudo-first-order model, the pseudo-second-order model was found to explain the adsorption kinetics better.

^{[11]}The experimental data for CBR and CBS were in good agreement with the pseudo second-order model, whereas the pseudo first-order model provided a better fit for CBL.

^{[12]}The results of LDH-CO3 fitted well with the pseudo -first-order model while the pseudo -second-order model fitted with SDS-LDH results.

^{[13]}The adsorption isotherm and kinetic model results showed that the Langmuir and the pseudo-second-order were well-fitted to the adsorption experimental data better than the Freundlich and pseudo-first-order models for both Cu2+ and Ni2+ with their maximum adsorption capacity of 15.

^{[14]}To predict the behaviors of the competitive and non-competitive adsorption process of ions onto hydrogels, the experimental adsorption data were analyzed by the pseudo-first-order model and the pseudo-second-order model.

^{[15]}By conducting kinetic studies, it was found that the PNP adsorption on the PAN–ACF surface followed a pseudo-first-order model, while the PAN–ACF/TiO2 and PAN–ACF/ZnO obeyed a pseudo-second-order model, and all samples followed the Langmuir adsorption isotherm.

^{[16]}Adsorption kinetic, pseudo-second-order model is more preferable than pseudo-first-order model because R2 value is 0.

^{[17]}Compared with the pseudo-first-order model, the pseudo-second-order model could better describe the adsorption kinetics.

^{[18]}Kinetic models highlighted include the Lagregren’s pseudo-first-order model (PFO), pseudo-second-order model (PSO), intra-particle diffusion model, Elovich model, and Bangham’s pore diffusion model.

^{[19]}The pseudo-first order, pseudo-second order, and intra-particle diffusion equation were used to evaluate the kinetic data and the drug adsorption process followed the pseudo-first-order model well.

^{[20]}Furthermore, the adsorption kinetics indicated that the PDA-SF data were better fitted by the pseudo-second-order model than the pseudo-first-order model.

^{[21]}The dynamic adsorption behavior of CV and RB4 can be represented well by the pseudo-second-order model and pseudo-first-order model, respectively.

^{[22]}Four kinetic models were applied to better comprehend the evolution of the colour: Lagergren's first-order model, Peleg's pseudo-second-order model, an intra-particle diffusion model and a parabolic diffusion model.

^{[23]}Adsorption kinetics showed that the lead and copper adsorption followed the pseudo-second-order model while cadmium suited with the pseudo-first-order model.

^{[24]}It is intriguing that the adsorption kinetics fits well with both pseudo-second-order kinetic model and pseudo-first-order model before 60 s, while only fits well with pseudo-second-order adsorption model in the whole adsorption process.

^{[25]}The removal of Cr(VI) was better fitted by pseudo-second-order model than pseudo-first-order model.

^{[26]}Pseudo first-order model, pseudo second-order model, and intraparticle diffusion model were applied to describe the adsorption processes.

^{[27]}It was observed that the obtained data were fitted more accurately with the pseudo-second-order model than the pseudo-first-order model.

^{[28]}Furthermore, the pseudo-second-order model was found to be more accurate than the pseudo-first-order model for the study of oil absorption kinetics.

^{[29]}The obtained experimental data were used for kinetics determination using three model equations: a pseudo first-order model, a pseudo second-order model, and the intraparticle diffusion model.

^{[30]}The Pseudo-Second-Order model was fitted better onto kinetic data compared to the Pseudo-First-Order model.

^{[31]}The pseudo-second-order model described the adsorption kinetic data better than the pseudo-first-order model, while kinetic studies showed that the adsorption process included external surface adsorption and intraparticle diffusion, which was likely the rate-limiting step.

^{[32]}The first stage with faster adsorption rate followed the pseudo-first-order model, while the second stage fitted the pseudo-second-order model better.

^{[33]}Adsorption isotherms of AMX onto CeO2@SiO2 were fitted by Langmuir, Freundlich, and two-step adsorption models, while the adsorption kinetics of AMX achieved a better fit by the pseudo-second-order model than the pseudo-first-order model.

^{[34]}Kinetic investigations displayed that the adsorption of metallic ions was more likely correlated with the pseudo-first-order model than the pseudo-second-order model and diffusion model.

^{[35]}The amount of phosphate adsorbed by these two muds correspond better with the Freundlich isotherm and pseudo-second-order models than with the Langmuir isotherm and pseudo-first-order models.

^{[36]}The pseudo-second-order model justified the experimental data of the adsorbents, while the pseudo-first-order model only fitted the MgO-NPs data.

^{[37]}Kinetics adsorption of methylene blue and phenol was studied using the pseudo-first-order model and the pseudo-second-order model.

^{[38]}The banana peel data and its activated form were best described by the pseudo-second-order model indicating chemisorption process while the remaining adsorbents followed the pseudo-first-order model indicating physisorption process.

^{[39]}Kinetic studies showed that a pseudo second order model was more suitable than the pseudo first order model.

^{[40]}Kinetics study revealed that the pseudo-second order model fitted better than the first order model.

^{[41]}Pseudo second-order kinetic model explained the experimental data better than Pseudo-first order model and intra particle diffusion model.

^{[42]}The pseudo-first order model was found to be applicable compared to the pseudo-second order kinetic analysis, with intra-particle diffusion mechanism involved in the removal process.

^{[43]}The pseudo-second order kinetic model was better fitted to the kinetic experimental data than the Elovich and pseudo-first order models.

^{[44]}The pseudo-second order model can describe the removal kinetics of Hg(II) more accurately than the pseudo-first order model.

^{[45]}The pseudo-first order model was found to be the most suitable for the kinetic data of NaMMT loaded hydrogels while that of SpMMT containing hydrogels followed the pseudo-second order kinetics.

^{[46]}The data for adsorption of As(III) showed a better fit too pseudo first order model of reaction kinetics while the data for As(V) fitted better to pseudo second order model.

^{[47]}The experimental data better fitted to pseudo-second order than first order and pseudo-first order model.

^{[48]}The adsorption kinetics of the dye on the surface of the composite GO/ (PVP-AAc) were studied using of pseudo-first order model and pseudo-second order model.

^{[49]}Remazol Yellow kinetic assays and modelling of the experimental data using the pseudo-first and pseudo-second order kinetic models demonstrated a better adjustment to the pseudo-first order model with a calculated uptake capacity of 14.

^{[50]}

## second order model

Based on a literature review of studies of confirmatory factor analysis (CFA), three types of models were tested: first-order models with method factors and covariances, a second-order model with method factors and covariances, and nested models with method factors and covariances.^{[1]}This work aimed to investigate whether the second-order models with separate Phase I and Phase II components of HR response can achieve better fitting performance compared to the first-order models that do not delineate the two phases.

^{[2]}In our technique, RSM was implemented on an all-maximization problem as a case-study process; in which case, first-order models (FOMs) for the responses were fitted using 2k factorial designs until the FOMs proved to be inadequate, while uniform precision rotatable central composite design was used to obtain second-order models (SOMs) for the respective responses in the event of model inadequacy of the FOMs.

^{[3]}The aim is to guide whether it’s worth investing towards developing a second-order model instead of a first-order model with respect to prediction accuracy considering modelling complexity, experiments required and the computational cost.

^{[4]}In comparison, the kinetic model showed that the first-order model was better than the second-order model.

^{[5]}4 1/d respectively for BOD5 and COD were obtained by the first order model, while the corresponding values for the second order model were K=0.

^{[6]}It is shown that the second order model in general yields significant improvements over the first order model.

^{[7]}We consider macroscopic partial differential equation models of two types: (i) First Order Models: equilibrium models, scalar models expressing car mass conservation; and (ii) Second Order Models: dynamic models, $$2 \times 2$$ hyperbolic systems expressing mass conservation as well as vehicle acceleration rules.

^{[8]}

## adsorption kinetics study

The adsorption kinetics study revealed that the adsorption data matched with a nonlinear pseudo-first-order model.^{[1]}The better adjustment of the experimental data in the adsorption kinetics study was adjusted using the pseudo-first-order model, being k1 = 0.

^{[2]}The adsorption kinetics study referred to satisfactory adsorption of Pb(Ⅱ) via pseudo-first-order model and pseudo-second-order model, while pseudo-second-order model was the best model to describe adsorption of Zn(Ⅱ) onto NCRM.

^{[3]}

## kinetic data respectively

Various non-linear isotherm and kinetic models were used to find plausible mechanisms involved in the adsorption, and it was found that the Langmuir and pseudo-first-order models show the best agreement with isotherm and kinetic data, respectively.^{[1]}Langmuir and pseudo-first-order models present a better fit to the isotherm and kinetic data, respectively, and the maximum Langmuir adsorption capacity was found to be 601.

^{[2]}The Langmuir model and pseudo-first-order model were the best to fit the equilibrium and kinetic data, respectively.

^{[3]}

## intraparticle diffusion model

The experimental data adjusted to the pseudo-first-order model, and the intraparticle diffusion model suggested a multi-stage process.^{[1]}Meanwhile, the intraparticle diffusion model was best at predicting the kinetic data at adsorbate concentration of 300 and 450 mg/L, while the pseudo-first-order model emerged as the best fit at 600 mg/L concentration.

^{[2]}

## intra particle diffusion

The kinetic data of As(III) followed the intra-particle diffusion model while As(V) fitted the pseudo-first-order model.^{[1]}

## Pseudo First Order Model

Kinetic studies showed that a pseudo second order model was more suitable than the pseudo first order model.^{[1]}According to the findings, the pseudo first order model was introduced to study photodegradation kinetics.

^{[2]}Pseudo first order model describes the reaction well since R2 (0.

^{[3]}The data for adsorption of As(III) showed a better fit too pseudo first order model of reaction kinetics while the data for As(V) fitted better to pseudo second order model.

^{[4]}The reactions all fitted pseudo first order model.

^{[5]}The dynamic process of CIP removal was monitored by UV-vis spectrophotometry, and can be well predicted by a pseudo first order model.

^{[6]}Reaction kinetics were studied and fitted well with pseudo first order model because of the mesoporous structure of polyaniline (PANI).

^{[7]}993) were fitted to pseudo first order models.

^{[8]}Kinetic studies showed that a pseudo second order model was more suitable than the pseudo first order model.

^{[9]}Kinetics followed pseudo first order model with rate constant (k) of 0.

^{[10]}The adsorption equilibrium reached within 25 and 180 min as the initial solution concentration increased from 10 to 300 mg/L, and the data fitted well using nonlinear pseudo first order model with determination coefficient (R2) in between 0.

^{[11]}Also, the kinetic models were followed pseudo second order model and pseudo first order model for GFP and GFM, respectively.

^{[12]}Whereas the reduction kinetics obeyed a pseudo first order model.

^{[13]}

## Simple First Order Model

This paper gives an introduction to the basic controllers with an integral action for regulation and tracking of simple first order models with compensation of acting disturbances.^{[1]}The simple first order model must be replaced with representing the LC as an orientation changing, anisotropic uniaxial layer.

^{[2]}