Langmuir 等温线、Freundlich 等温线和其他线性等温线等分析等温线模型通常用于为广泛的吸附研究建模吸附数据集。大多数这些研究都认为 pH 值是固定的。然而，pH 是一个变化很大的重要参数。因此，为一组实验开发的模型参数不能用于 pH 值不同的另一种情况。可以模拟 pH 变化的表面络合模型是复杂的、难以使用的多参数模型。我们早期开发的修改后的 Langmuir-Freundlich (MLF) 等温线可以模拟 pH 依赖性吸附在针铁矿涂层砂上。然而，它只在针铁矿覆盖的沙子上进行了砷吸附测试。所以，铬吸附数据集被认为可以扩展其他金属离子的 MLF 等温线。两种不同的吸附剂，即。选择椰子根活性炭 (CoAC) 和棕榈雄花活性炭 (PaAC) 使用 MLF 等温线模型对 Cr(VI) 进行吸附建模。开发了一种改进的建模策略来拟合 MLF 等温线，它只需要一个 pH 与吸附数据集，而不是不同 pH 值下的多个等温线。新方法可以令人满意地模拟各种实验数据集的 pH 依赖性吸附。PaAC 和 CoAC 的最大吸附容量分别为 88.64 (mg/g) 和 100.1 (mg/g)。该模型的亲和常数（选择使用 MLF 等温线模型对 Cr(VI) 进行吸附建模。开发了一种改进的建模策略来拟合 MLF 等温线，它只需要一个 pH 与吸附数据集，而不是不同 pH 值下的多个等温线。新方法可以令人满意地模拟各种实验数据集的 pH 依赖性吸附。PaAC 和 CoAC 的最大吸附容量分别为 88.64 (mg/g) 和 100.1 (mg/g)。该模型的亲和常数（选择使用 MLF 等温线模型对 Cr(VI) 进行吸附建模。开发了一种改进的建模策略来拟合 MLF 等温线，它只需要一个 pH 与吸附数据集，而不是不同 pH 值下的多个等温线。新方法可以令人满意地模拟各种实验数据集的 pH 依赖性吸附。PaAC 和 CoAC 的最大吸附容量分别为 88.64 (mg/g) 和 100.1 (mg/g)。该模型的亲和常数（新方法可以令人满意地模拟各种实验数据集的 pH 依赖性吸附。PaAC 和 CoAC 的最大吸附容量分别为 88.64 (mg/g) 和 100.1 (mg/g)。该模型的亲和常数（新方法可以令人满意地模拟各种实验数据集的 pH 依赖性吸附。PaAC 和 CoAC 的最大吸附容量分别为 88.64 (mg/g) 和 100.1 (mg/g)。该模型的亲和常数（K a ) 对于 PaAC 数据集被发现为 0.007 (L/mg)，对于 CoAC 数据集则为 0.0106(L/mg) 和 0.004 (L/mg)。计算出拟合的平均R 2值，PaAC 为 0.98，CoAC 为 0.85。模型拟合的平均均方根误差 (RSME) 为 0.07（小于 10%）。这种建模策略需要较少的实验数据，并且不需要高级表征研究。因此，这项研究表明，MLF 等温线可以扩展到其他污染物和不同的吸附剂，以模拟 pH 依赖性吸附。
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A simplified modeling procedure for adsorption at varying pH conditions using the modified Langmuir–Freundlich isotherm
Analytical isotherm models such as Langmuir isotherm, Freundlich isotherm, and other linear isotherms are commonly used for modeling adsorption datasets for a wide range of adsorption studies. Most of these studies consider pH to be fixed. However, pH is an important parameter that varies widely. Hence, the model parameters developed for one set of experiments cannot be used in another scenario where the pH is different. Surface complexation models that can simulate pH changes are complex, multi-parameter models that are difficult to use. The modified Langmuir–Freundlich (MLF) isotherm developed earlier by us could simulate pH-dependent adsorption on goethite-coated sands. However, it has only been tested for arsenic adsorption on goethite-coated sands. Therefore, chromium adsorption datasets were considered to extend this MLF isotherm for other metal ions. Two different adsorbents, viz. coconut root activated carbon (CoAC) and palm male flower activated carbon (PaAC), were selected for the adsorption modeling of Cr(VI) using the MLF isotherm model. An improved modeling strategy was developed for fitting the MLF isotherm, which required only a single pH versus adsorption dataset, instead of several isotherms at different pH values. The new methodology could simulate the pH-dependent adsorption satisfactorily for various experimental datasets. The maximum adsorption capacity was 88.64 (mg/g) and 100.1 (mg/g) for PaAC and CoAC, respectively. The affinity constant for this model (Ka) was found to be 0.007 (L/mg) for PaAC dataset and 0.0106(L/mg) and 0.004 (L/mg) for the CoAC dataset. The average R2 values of fitting were calculated and found to be 0.98 for PaAC and 0.85 for CoAC. The average root mean square error (RSME) of the fitting of the model was 0.07 (less than 10%). This modeling strategy required less experimental data and did not require advanced characterization studies. Therefore, this study indicates that the MLF isotherm can be extended to other contaminants and for different adsorbents to model the pH-dependent adsorption.
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