Recently, Drs. Mirosław Grzesik and Krzysztof A. Szymonski commented on our paper published in the journal of Industrial & Engineering Chemistry Research entitled “Erroneous Application of Pseudo-Second Order Adsorption Kinetics Model: Ignored Assumptions and Spurious Correlations”.(1) We appreciate this discussion as the inappropriate use of various equations was what prompted us to analyze this problem and write the article. All references to Figures and Equations below refer to the original paper.(1) Grzesik and Szymonski comment that the pseudo-second-order adsorption kinetics model (eq 1),(1,2) known as Blanchard’s equation, is an empirical equation. In their manuscript, Blanchard et al. developed the equation by assuming the adsorption process (or the ion-exchange process) was second order with respect to the adsorption sites and the parameters in the model were related to this process.(3) Thus, there is a physical basis for this equation, and under appropriate assumptions, this equation can be used, and the trends be modeled accordingly. We used a similar set of assumptions as those of Blanchard, namely, (i) adsorption is reaction controlled, (ii) constant liquid concentration, and (iii) no desorption. In numerous publications, these assumptions are overlooked and/or not satisfied when this model is applied, as was discussed in our article. To satisfy the first assumption of reaction control, the initial liquid concentration of the adsorbate (C_A0) must be sufficient such that the diffusion rate exceeds the adsorption reaction rate. A large value of C_A0 relative to the initial adsorption amount also is beneficial to satisfy the second assumption; that is, the change in the liquid concentration (i.e., ΔC) will be close to zero. To meet the third assumption, only the initial kinetics data should be considered because only at this period is the concentration of adsorbed species essentially zero, so that the desorption rate (k_–1C_AS2) is also essentially zero. The essence of the comments by Grzesik and Szymonski is the same as the one described here and in the paper. Under such circumstances in which the three assumptions are satisfied, the equilibrium adsorption capacity (q_e) will be essentially the max adsorption capacity (q_m) because the plateau of the isotherm has been reached but the liquid concentration of the adsorbate (C_A) is still large. Unlike a kinetics model which describes the process at various distances from equilibrium, the derivation of the Langmuir isotherm is based on reaching adsorption equilibrium, so that the desorption process cannot be eliminated and a full version of eq 3 must be considered in developing the isotherm model.(4) The second reason we wrote this paper was because of the potential spurious correlations obtained when a linearized form of the pseudo-second-order model (eq 7) was used for the fitting process. For clarification, linearization has different meanings in calculus and statistics. In the former, a function is simplified into the form of y = ax + b with approximation by taking the first two terms of the Taylor series expansion. In our paper, we used the statistical definition in which linearization means the conversion of a nonlinear model into the form of y = ax + b without any approximation.(5) We agree with the comment that adsorption kinetics data at shorter times with corresponding larger changes in the adsorbed amount (q_t) should be used (which is also in agreement with satisfying the assumptions of the model, as stated above). This comment was summarized in Figure 2a and the corresponding discussion in the text. Briefly, as indicated in the figure, if the variation in adsorption time was much higher than the variation in adsorbed amount (e.g., variation of t was 10 times the variation of q_t), the pseudo-second order model seemingly fits noncorrelated data (r_qt,t = 0) with high values of R² when a linearized form of this model was used (r_t/qt,t ≈ 1). Using a nonlinear version (eq 6), however, would not create such inconsistency. Further, the application of random data, and simulated data generated from Weber-Morris, pseudo-first-order and pseudo-third-order kinetics data demonstrated that R² values close to 1 could be obtained by using the linear form of the pseudo-second-order model (t/q_t vs t, eq 7) for those kinetics data. Overall, the (statistically) linearized form of this model should be used with caution and, as stated by Grzesik and Szymonski, the data should be carefully obtained, and various methods should be applied to verify the quality/merit/validity of fit. The authors declare no competing financial interest. The authors declare no competing financial interest.
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中文翻译：

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回复“关于'错误应用伪二阶吸附动力学模型：忽略假设和虚假相关性'的评论”

最近，博士。Mirosław Grzesik 和 Krzysztof A. Szymonski 对我们发表在《工业与工程化学研究》杂志上的论文发表了评论题为“错误应用伪二阶吸附动力学模型：忽略假设和虚假相关”。（1）我们感谢这次讨论，因为各种方程的不当使用促使我们分析这个问题并写了这篇文章。以下所有对图表和方程的引用均指原始论文。(1) Grzesik 和 Szymonski 评论称，称为 Blanchard 方程的伪二级吸附动力学模型 (eq 1),(1,2) 是一个经验方程。在他们的手稿中，布兰查德等人。通过假设吸附过程（或离子交换过程）对于吸附位点是二阶的，并且模型中的参数与该过程相关，从而开发了该方程。（3）因此，该方程有物理基础，并在适当的假设下，可以使用这个方程，并相应地对趋势进行建模。我们使用了一组与 Blanchard 相似的假设，即（i）吸附受反应控制，（ii）恒定液体浓度，以及（iii）无解吸。在许多出版物中，这些假设在应用此模型时被忽略和/或不满足，正如我们在文章中所讨论的。为了满足反应控制的第一个假设，吸附质的初始液体浓度（C _A0 ) 必须足以使扩散速率超过吸附反应速率。相对于初始吸附量较大的C _A0值也有利于满足第二个假设；即，在液体浓度的变化（即，Δ Ç）将接近零。为了满足第三个假设，只应考虑初始动力学数据，因为只有在此期间吸附物质的浓度基本为零，因此解吸速率 ( k _–1 C _AS2) 也基本上为零。Grzesik 和 Szymonski 的评论的本质与这里和论文中描述的相同。在满足这三个假设的情况下，平衡吸附容量 ( q _e ) 将基本上是最大吸附容量 ( q _m )，因为已经达到等温线的平台但被吸附物的液体浓度 ( C _A) 还是很大的。与描述离平衡不同距离的过程的动力学模型不同，Langmuir 等温线的推导基于达到吸附平衡，因此解吸过程无法消除，在开发等温线模型时必须考虑方程 3 的完整版本.(4) 我们写这篇论文的第二个原因是因为在拟合过程中使用伪二阶模型（方程 7）的线性化形式时获得了潜在的虚假相关性。为了澄清起见，线性化在微积分和统计学中具有不同的含义。在前者中，一个函数被简化为y = ax + b的形式通过采用泰勒级数展开式的前两项来近似。在我们的论文中，我们使用统计定义，其中线性化意味着将非线性模型转换为y = ax + b的形式，没有任何近似。 (5) 我们同意较短时间的吸附动力学数据对应较大的评论。改变吸附量（q_吨）应使用（这也是在协议与满足模型的假设，如上面所述）。该评论在图 2a 和文本中的相应讨论中进行了总结。简而言之，如图所示，如果吸附时间的变化远高于吸附量的变化（例如，吨是的变化的10倍q_吨），伪二阶模型似乎适合非相关数据（[R _QT，吨= 0）具有高的值- [R ²当使用该模型的线性化形式（ř_{吨/ QT，吨}≈1）。然而，使用非线性版本 (eq 6) 不会产生这种不一致。此外，随机数据的应用，以及从 Weber-Morris、伪一级和伪三阶动力学数据生成的模拟数据的应用表明，通过使用伪方程的线性形式可以获得接近 1 的R ²值。二阶模型 ( t / q _tvs t , eq 7) 对于那些动力学数据。总体而言，应谨慎使用该模型的（统计）线性化形式，并且如 Grzesik 和 Szymonski 所述，应仔细获取数据，并应应用各种方法来验证拟合的质量/优点/有效性。作者声明没有竞争性经济利益。作者声明没有竞争性经济利益。
本文引用了 5 篇其他出版物。