# Why Is the Linearized Form of Pseudo-Second Order Adsorption Kinetic Model So Successful in Fitting Batch Adsorption Experimental Data?

^{*}

## Abstract

**:**

## 1. Introduction

_{o}. The solute is adsorbed by the adsorbent and the amount of adsorbed material per unit mass of adsorbent is denoted as q. The quantity q evolves over time and finally converges to a specific value q

_{e}. The solute concentration C evolution can be found by a simple mass balance as C = C

_{o}− mq/V.

_{o}, m, and V is not possible. The phenomenological models contain physical parameters allowing the estimation of q(t) for different conditions from those in the experiments in which they are derived (i.e., they allow the scale-up of the process).

_{1}, k

_{2}are kinetic constants).

_{e}. This is not physically justified since this value should be related to C through the isotherm and C varies in time. Thus, the main question is: why does the pseudo-second order model offer the best fit to the data despite having no physical justification? This issue has been addressed in the past [1]. However whereas the motivation in [1] is the relative success between pseudo-first and pseudo-second order models, here our motivation is the observation, through our research, that the pseudo-second order model always appears to fit the data better even with respect to sophisticated mechanistic models developed in our laboratory. The need to resolve this issue led to the present study.

## 2. Methods and Results

#### 2.1. Example 1

_{e}= 1 (arbitrary units; q must have the same units). The most straightforward way to deal with the problem is to assume a specific kinetic constant and to generate the data, considering several sampling sequences in time domain. However, there are experimental limitations to such an approach, since, for practical reasons, measurements at very short time intervals are difficult. Instead we assume a reasonable time sampling sequence with the sampling times exponentially distributed (being a good choice for data following both pseudo-first and pseudo-second order kinetics). The sampling/measurement times considered are 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, and 100 (arbitrary units), and the adsorption data are generated through the following pseudo-first order kinetic equation

- (i)
- The Equation (8) is directly fitted to the data of q vs. t (i.e., nonlinear fitting)
- (ii)
- The Equation (9) is fitted to the transformed data t/q vs. t (i.e., linear fitting)

_{1}

^{2}: corresponds to the relation between q data and their nonlinear fit.

_{2}

^{2}: corresponds to the relation between (t/q) data and their linear fit.

_{3}

^{2}: corresponds to the relation between q data and the equation q = t/(a + bt) with a and b derived from linear fitting procedure (ii). The results of the above calculations are presented in Table 1.

_{2}

^{2}and R

_{3}

^{2}are smaller than the nonlinear fitting one R

_{1}

^{2}. This is normal behavior; however, it is associated only with the sampling times accumulated at the fast increasing part of the q curve. Such a situation is not typically encountered in the literature. As sampling times begin to be distributed to the slowly increasing part of the q data, the situation is radically changed. The R

_{1}

^{2}decreases (taking values indicating successful fitting up to α = 0.5) as α increases. However, the linear fitting correlation R

_{2}

^{2}appears to increase with α, implying a perfect fit for α > 0.2. This is certainly misleading, given that the correlation coefficient R

_{3}

^{2}, representing the real quality of the linear fitting, is very low, denoting poor fitting for α > 0.2.

_{2}

^{2}) is larger than the one (R

_{1}

^{2}) of nonlinear fit of q data. However the q vs. t correlation based on linear fitting appears to be very poor. In practice, the sampling times usually cover an extensive range of q, implying that the linear fit procedure always suggests a perfect success of the pseudo-second order kinetic model (even if the data exactly follow the pseudo-first order kinetics). However the success of the linear fitting is fake; it is simply an outcome of the data transformation. The parameters resulting from the linear fitting procedure lead to a very poor direct correlation between q and t. These arguments are not quite new, since they have been proposed by other researchers in recent years [1,33,34]. However it appears that most researchers are not convinced, so additional work is needed in this direction.

#### 2.2. Example 2

_{1}

^{2}). This behavior is expected, and appears to be correct since no kinetic information actually exists for the particular q data. However when an attempt is made to fit the data by the linear fitting model (ii), the resulting R

_{2}

^{2}indicates complete success. The parameters a and b resulting from the linear fitting procedure appear as a function of the noise amplitude ε in Figure 3. The correlation coefficient is presented as (1 − R

_{2}

^{2}) × 20 in order to enhance the resolution of the presentation. The fitted value of b is relatively stable to noise (decreased by 20%) as ε increases from 0 to 0.5. This is not the case for parameter a which increases from 0 to 2.18 under the same circumstances denoting an unstable to noise behavior. However regardless of the stability of the results, the key point is that a successful linear fit appears in all cases (R

_{2}

^{2}is 0.985 even for ε = 0.5). The above example is a clear indication that employing linearized edition of the pseudo-second order model leads to fake successful fits, even when using data with no information for the kinetic behavior of the process.

#### 2.3. Example 3

^{2}designates the difference between the non-linear fitting arising q values and the nominal ones (before addition of the “experimental” error). The scope of this choice is to present the distance between fitted and “exact” values of q. If R

_{1}

^{2}is used instead of R

^{2}the general picture presented in the following would not be different. The values of R

_{1}

^{2}are slightly larger than those of R

^{2}for Case 1 data, whereas R

_{1}

^{2}and R

^{2}are exactly the same for Case 2 data.

^{2}for non-linear fitting appear, versus the “experimental error” parameter ε in Figure 4a,b for Case 1 and Case 2 data, respectively. The corresponding values R

_{2}

^{2}for linear fitting appear separately in Figure 5 since a different y-axis scale is required for them. Let us start to discuss Figure 4 from the evolution of R

^{2}of the nonlinear fitting. Obviously, the sampling time distribution plays a crucial role in the fitting quality. The R

^{2}for Case 1 data decreases to 0.965 at ε = 0.02 whereas the same value is reached for Case 2 data at ε = 0.08. This means that the proper distribution of sampling times stabilizes the nonlinear fitting procedure with respect to experimental error. Regarding linear fitting, it can be noticed that there is some stability with respect to parameter “b” values for both Case 1 and Case 2 data. The error in its derivation does not exceed 10% even for ε = 0.1. On the other hand, the linear fitting procedure is quite unstable with respect to parameter “a” in Case 1 data (the error is already 15% at ε = 0.01) whereas a poor stability appears for Case 2 data (error 25% at ε = 0.1). However, the data presented in Figure 5 suggest that the linear fitting appears to be always perfect (R

_{2}

^{2}larger than 0.9998) for Case 2 data or very good (R

_{2}

^{2}larger than 0.99) for Case 1 data. Once again, it is shown explicitly that the linear fitting procedure leads to misleading results. The additional information acquired by Example 3 shows that the sampling time distribution is crucial for the quality of the nonlinear fitting. The more equidistant in q the sampling times, the more accurate the fitting.

_{2}and to a fitting which almost entirely depends on q

_{e}. Such a situation cannot be observed in the linearization of the pseudo-first order kinetic model.

## 3. Further Discussion

_{o}− C)/m. Substituting the above relation in the pseudo-second order kinetics equation leads to the following form (where C

_{e}= C

_{o}− mq

_{e}/V):

_{o}and integrating the above equation leads to the form:

_{e}/C

_{o}and A = Vk

_{2}C

_{o}/m. There are several reasons for which Equation (11) must be used for the fitting of the pseudo-second order model instead of Equation (5). At first, any uncertainty for the variables m and V are transferred to the data for q, so it is not easy to estimate its effect on the evaluated parameters q

_{e}and k

_{2}. On the other hand, using Equation (11), the C data depend only on concentration measurements and the uncertainty in m; V appears only in the dimensionless parameter A, rendering trivial the assessment of its impact in k

_{2}evaluation. The second point is that for a specific measurement, uncertainty in concentration of the relative C error is small in regions where the relative q error is large. This inverse behavior of relative error is undesirable, since the q error is a calculated and not a measured one. Finally, according to Example 1, the sampling time must be accumulated at the initial part of the q curve. However, in this region the relative error in q is large (compared to the one in C), which jeopardizes the quality of the fitting procedure. It is noted that the pathology described above for fitting of q with the linear form of the second order model does not exist for fitting of the c curve in a linear form. This is why the above pathology has not been identified in Chemical Reaction Engineering literature [35] for fitting data of second order reactions (i.e., reactant concentration data are always employed). However, in any case, the non-linear fitting is preferable. All analysis of the present work is based on the minimization of the sum of the square deviation as fitting criterion. There are of course several alternative criteria to perform the fitting procedure [36]. However. the use of any criterion would not change the general picture created by the examples presented above.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Example 1: Pseudo-second order nonlinear fits (continuous lines) to pseudo-first order data (symbols) for (

**a**) α = 0.05, (

**b**) α = 0.2, (

**c**) α = 0.5, (

**d**) α = 2.

**Figure 2.**Example 1: Pseudo-second order linear fits (continuous lines) to pseudo-first order data (symbols) for α = 0.05 and α = 0.2.

**Figure 3.**Example 2: Pseudo-second order linear fit extracted parameters and corresponding correlation coefficient R

_{2}

^{2}versus noise amplitude parameter ε.

**Figure 4.**Example 3: Pseudo-second order linear fit extracted parameters a and b and nonlinear fit correlation coefficient versus “experimental error” parameter ε (

**a**) Case 1 sampling times, (

**b**) Case 2 sampling times.

**Figure 5.**Example 3: Pseudo-second order linear fit correlation coefficient R

_{2}

^{2}versus “experimental error” parameter ε, Curve 1: Case 1 sampling times, Curve 2: Case 2 sampling times.

α | R_{1}^{2} | R_{2}^{2} | R_{3}^{2} |
---|---|---|---|

0.05 | 0.9968 | 0.9936 | 0.9943 |

0.2 | 0.9888 | 0.9981 | 0.9832 |

0.5 | 0.9848 | 0.9996 | 0.9785 |

2 | 0.9702 | 1 | 0.9056 |

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**MDPI and ACS Style**

Kostoglou, M.; Karapantsios, T.D. Why Is the Linearized Form of Pseudo-Second Order Adsorption Kinetic Model So Successful in Fitting Batch Adsorption Experimental Data? *Colloids Interfaces* **2022**, *6*, 55.
https://doi.org/10.3390/colloids6040055

**AMA Style**

Kostoglou M, Karapantsios TD. Why Is the Linearized Form of Pseudo-Second Order Adsorption Kinetic Model So Successful in Fitting Batch Adsorption Experimental Data? *Colloids and Interfaces*. 2022; 6(4):55.
https://doi.org/10.3390/colloids6040055

**Chicago/Turabian Style**

Kostoglou, Margaritis, and Thodoris D. Karapantsios. 2022. "Why Is the Linearized Form of Pseudo-Second Order Adsorption Kinetic Model So Successful in Fitting Batch Adsorption Experimental Data?" *Colloids and Interfaces* 6, no. 4: 55.
https://doi.org/10.3390/colloids6040055