Do Adsorbent Pore Size and Specic Surface Area Affect The Kinetics of Methyl Orange Aqueous Phase Adsorption?

The kinetics of any adsorption reaction gives more information on the rate at which the adsorbate is taken up by the adsorbent, which is responsible for the residence time of an adsorbate uptake at the adsorbent-aqueous phase interface. This study was aimed at determining the effect of pore size as well as specic surface area (SSA) on the kinetics of the uptake of methyl orange (MO). The basis of the analysis was from data on kinetic models sourced from recent literature. ANOVA of the data revealed that statistical signicance was achieved for SSA but not for pore size (at a signicance level of 0.05). This called for a more theoretical perspective on the research data. The kinetics constant for micropores is far higher than for the two selected regimes of the mesopore. For the SSA, 100–10 m 2 /g adsorbents had a higher mean k 2 value. This suggested that adsorbents in the SSA range had pore sizes that favour rapid uptake. However, further studies will be needed to gain a better understanding of how SSA affects adsorption kinetics. The study also discussed the technical limitations that could arise due to the use of kinetic model linearisation.


Introduction
Adsorption is a very popular technique for the sequestration of pollutants such as dyes from the environment [1]. To obtain the optimum result in any adsorption study, the understanding of the following properties of the adsorbent is required; surface chemistry, hydrophilicity, physiochemical properties, proximate analysis, morphology, crystalline structure, and textural properties (pore size, pore volume, surface area, and pore surface) [2]. These characteristics are important in the selection of adsorbents for speci c pollutants [3,4]. The pore size of an adsorbent can be described as the gap or stretch of its pores. The pore size of adsorbents can be divided into three main groups as stated by the International Union of Pure and Applied Chemistry (IUPAC); micropores have pore sizes that are less than 2 nm, mesopores have pore sizes of 2-50 nm, and macropores have pore sizes more than 50 nm [5][6][7][8]. The speci c surface area (SSA) is also one of the most important properties of adsorbent material as it indicates the necessary active sites for adsorption because adsorption is a surface phenomenon [9][10][11]. The surface area is directly proportional to the adsorptive performance of the material [12,13].
Adsorption modelling is a very important aspect of any adsorption study as it applies adsorption models for the interpretation of experimental data to obtain useful information that would aid in the understanding of process mechanisms, process equilibrium-dynamics, predicting answers to operational condition changes and optimizing the adsorption process [14]. Common models used for understanding adsorption studies better are; thermodynamic models, isotherm models and kinetic models [15][16][17][18].
The time at which equilibrium is obtained is an important feature of any adsorption study. The kinetics of any adsorption reaction gives more information on the rate at which the adsorbate is taken up by the adsorbent, which is responsible for the residence time of an adsorbate uptake at the adsorbent-aqueous phase interface [14]. The adsorption kinetics also gives more information on the mechanism as well as a pathway for the reaction [19,20]. The most common kinetic models used in adsorption studies which are classical models that t adsorption data well even with non-linear methods are the pseudo-rst-order (PFO) and the pseudo-second-order (PSO) models. The PFO model assumes that the adsorption process is dependent only on the concentration of adsorbate in the solution that is present at a speci c time while the PSO model assumes that adsorption is a complex physico-chemical process that is dependent on the concentration of the adsorbate present in the system as well as the number of active sites on the adsorbent [21][22][23].
Methyl Orange (MO) (IUPAC: Sodium 4-([4-(dimethylamino) phenyl]diazenyl) benzene-1-sulfonate) is a sulfonated azo anionic dye popularly known and used as a pH indicator in the titrations for acid and as a dye for textiles [24,25]. It has a molecular formula C 14 H 12 N 3 O 3 NaS and a molecular weight of 327.34 g/mol. It is soluble in water and its density and melting point are 1.28 g/cm 3 and > 300 ℃ [26,27]. The molecular size of MO is 1.19 nm × 0.68 nm × 0.37 nm. It has a dissociation constant (pKa) of 3.47 in water at 25 ℃ [28 -30]. MO is a toxic compound that deteriorates water quality and has been banned for use in food products because it is a carcinogen and teratogen due to the presence of aromatic and -N = N-groups inherent in its structure as shown in Fig. 1 [30,31]. It usually gets into the environment in large quantities through the e uent of textile industries. Different techniques have been utilized in the removal of MO from the environment. They include electrocoagulation [32][33][34][35], ozonation [36-38], biological treatment [39], photo-degradation [40][41][42], membrane processes [43], and adsorption [44][45][46]. Due to the low cost and e cacy of adsorption, it is considered one of the best methods for the removal of MO dye from the environment [47][48][49].
This study is aimed at determining the relationship between pore size as well as speci c surface area (SSA) on the kinetics of the uptake of methyl orange (MO) through the PFO and PSO kinetic models. Empirical ndings obtained were analysed and juxtaposed to obtain observations. The reason for using MO in our study is due to its popularity as a pH indicator as well as a textile dye [50,51]. The results of this study would be able to enable researchers to understand the adsorption process of MO better.

Data Collection and description
For our study, published works of literature on the adsorption of Methyl Orange were obtained from the Google Scholar search engine. The search was restricted to articles published in the past ve years (2016-2021) and which reported the pore size of the various adsorbents used. Based on the nature of the data, adsorbent of macropores size is unavailable for MO adsorption hence the study would focus more on micropore versus mesopore. Data on SSA was also added from articles where it was reported as shown in Table 1. The kinetic models used in this study was restricted to the PFO and the PSO models. The K values of the models were reported based on which of them was best-t from the authors' modelling study. The most widely reported index of kinetic modelling accuracy (being the coe cient of determination, R 2 was also reported. The modelling techniques were reported too, L being for linear modelling and NL being for non-linear modelling.

Research Problem
To verify the effect of pore size on adsorption kinetics, the data were analysed in a variety of ways. The key goal was to evaluate how pore size and SSA relates to the kinetic constants. The kinetic constant was used as the basis of the investigation because it is in the true sense an empirical constant. Hence comparison can be made across studies using it which extricates other adsorption factors. Furthermore, only methyl orange was used because it also helps to narrow down the factors by eliminating the effects of adsorbate properties on the solution chemistry. The study will try to evaluate if there was there a relationship from empirical investigations between pore size and SSA against MO uptake kinetics. The study also discusses the in uence of the modelling technique (whether linear or non-linear) on adsorption kinetics [52].

Statistical analysis of the data
From Table 1, the pseudo-second-order kinetics was best-t in most of the cases. Hence its kinetics constant (k 2) is the selected basis for this investigation. Firstly, before the data can be used for this analysis, we rst need to verify if there is any statistical relationship between the pore size and speci c surface area against the kinetic constant. This was done by the one-way ANOVA and descriptive statistics such as CV, R 2 and RMSE. From Table 2, statistical signi cance (at a threshold of Prob > F being 0.05) was achieved for SSA but not for pore size. What this simply means is that a holistic consideration of the data revealed speci c trends between SSA and the kinetics constant while this was not so for the case of pore size. It is also unsurprising that the descriptive statistics are also correspondingly poorer for pore size. This does not however invalidate the data since the statistical signi cance only veri es if the results in the dataset are due to chance or some speci c factors. A more theoretical perspective will be needed to properly consider the data. 3.2 Theoretical consideration of the effect of pore size on MO adsorption kinetics Empirical evidence cannot be summarily dismissed due to a lack of statistical relationship. Statistics try to evaluate trends based on a holistic computation of the dataset. This does not consider some known theoretical understanding of adsorption. For this case, we are considering the 2 nm pore size threshold. This will be used to differentiate between the micropore and the mesopore. Based on this, we can then summarise the data as shown in Table 3. Considering magnitudes, it is observed that the kinetics for micropores is far higher than for the two selected regimes of the mesopore. We believe this is a bias induced by the uncharacteristic results obtained by Chaukura, Murimba [127]

Theoretical consideration of the effect of SSA on MO adsorption kinetics
In this section, we investigate the possible effects of SSA on MO adsorption kinetics. A simple summary of the results is presented in Table 4.
For the SSA, 100-10 m 2 /g adsorbents had a higher mean k 2 value. This suggests that adsorbents in the SSA range had pore sizes that favour rapid uptake (which we have already earlier observed to be micropores). The SSA of a material is only closely related to its total pore volume. Pore diameters are controlled more by the method and parameter of adsorbent preparation and on the intrinsic nature of the feedstock. Hence, further studies will be needed to gain a better understanding of how SSA affects the adsorption kinetics (if at all it does).

Technical issues with kinetics modelling
In this section, we discuss the technical issues around the modelling of adsorption kinetics, albeit from a mathematical perspective. In Table 1, we reported the technique used by various studies for modelling adsorption. A majority of the studies used the linear modelling technique while others employed non-linear modelling. Tran, You [2] explained that the kinetics of adsorption can be very rapid. In some cases, over 90% of the adsorbate can be removed within the rst 5 minutes of the process. Most experiments usually employed a protracted contact time because the adsorption process becomes signi cantly less rapid as equilibrium is approached. Simonin [136] has explained that serious biases that favour the PSO model are created because of this extended portion of the contact time when kinetics is slow and equilibrium is close. Such vast differences in adsorption kinetics can be di cult to capture by a linearised model. This led to the issues raised by Lima, Sher [137]. Lima, Sher [137] observed that researchers seem too con dent in employing linear techniques because of their perceived similarity with non-linear modelling when comparing their R 2 and adjusted R 2 . This did not take into cognisance the dissimilarities and inconsistencies in the obtained rate constants [138,139]. In some scenarios, studies with PFO as best-ts were erroneously assigned as having PSO as best-t. It is by consequence not surprising that most of the studies in Table 1 have the PSO as best-t. Besides the use of just the R 2 for determining goodness of t, Aniagor, Igwegbe [22] recommended including other accuracy statistics in making this decision (which are still not commonly employed in literature in this respect). Increment in the degree of freedom of a given kinetics data set will unfairly favour the model t. Hence this suggestion.

Conclusion
Some unique observations were derived in this study, albeit MO adsorption. ANOVA of the data revealed that statistical signi cance (at a threshold of Prob > F being 0.05) was achieved for SSA but not for pore size. This does not however invalidate the data since the statistical signi cance only veri es if the results in the dataset are due to chance or some speci c factors thereby requiring a more theoretical perspective. It was observed that the kinetics for micropores is far higher than for the two selected regimes of the mesopore. Based on the molecular size of MO, the compound would quickly ll up a pore of the average diameter of 2 nm as opposed to larger mesopores. This quickness is what is captured by the rate constant. For the SSA, 100-10 m 2 /g adsorbents had a higher mean k 2 value. This suggested that adsorbents in the SSA range had pore sizes that favour rapid uptake (which we have already earlier observed to be micropores). However, further studies will be needed to gain a better understanding of how SSA affects the adsorption kinetics (if at all it does).

Disclosure statements
Con ict of Interest: The authors declare that there are no con icts of interest.
Funding: There was no external funding for the study.
Compliance with Ethical Standards: This article does not contain any studies involving human or animal subjects.

Figure 1
Chemical structure of methyl orange