Re-evaluation of the century-old Langmuir isotherm for modeling adsorption phenomena in solution
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Estimation of methane adsorption capacity is crucial for the characterization of shale gas reservoirs. The methane adsorption capacity in shales is measured using high-pressure methane adsorption to obtain the adsorption isotherms, which can be fitted by Langmuir model. The determined Langmuir parameters can provide the methane adsorption capacity under actual reservoir conditions. In this study, a prediction model for the methane adsorption in shales was constructed based on 66 samples from 6 basins in China and Western Australia. The model was established in four steps: a model of Langmuir volume at experimental temperature, the temperature dependence of Langmuir volume, a model of Langmuir pressure, the temperature dependence of Langmuir pressure. In the model of Langmuir volume at experimental temperature, total organic carbon (TOC) and clay content (Vsh) were considered. A positive relationship was observed between the TOC and the temperature effect on the Langmuir volume. As the Langmuir pressure is sensitive to various factors, the Langmuir pressure at experimental temperature shows no trend with the TOC, clay content and thermal maturity, but a positive trend with the Langmuir volume. The results of this study can help log analysts to quantify adsorbed gas from well-log data since TOC and Vsh, which are the measure inputs of the introduced models, can be obtained from well-log data as well.
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Irving Langmuir published a seminal paper, on the adsorption of gases at a constant temperature in 1918, which presented one of the most important isotherm equations available in post Langmuir era. Many other useful isotherm equations were suggested both before and after 1918. The 1918 equation was the first where a model for adsorption phenomenon was suggested. In this article, we present a reasonably coherent discussion on Langmuir and some other adsorption isotherms reported in literature focusing on their uses and applications as well as analysing the merits and demerits to commemorate the hundred years of Langmuir's pioneering contribution in surface chemistry.
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Standard isotherm equations do not estimate capacity (Qmax) and distribution coefficient (Kd) for complex or non-Langmuir-shaped isotherm plots. In this study, two mycotoxins, that is, aflatoxin B1 (AfB1) and cyclopiazonic acid (CPA), were mixed with kaolinite and a naturally acidic montmorillonite clay (LPHM) at 25 °C, respectively. Isotherm data gave S-type plots. The data were fitted to the models of Langmuir (LM) and multi-Langmuir (MLM); however, these models did not provide a good fit for data that displayed multisite adsorption or multiple plateaus. While a published modification of the Langmuir equation (QKLM), which defines an effective partition coefficient as a function of the surface coverage, was able to fit simple isotherm plots from all categories (H, L, S, C), it did not fit complex or multisite isotherm plots. Importantly, an equation that enables the estimation of Qmax and Kd for both S-shaped and multisite isotherm plots has not yet been reported. Since the LM, MLM, and QKLM did not provide adequate fitting of the data, several modifications of the LM were developed: shifted Langmuir, shifted squared Langmuir, shifted cubed Langmuir, shifted exponential Langmuir, exponential Langmuir, and shifted modified Langmuir. These equations were used to derive information about the adsorption of mycotoxins to clay and to gain insight into the molecular mechanism(s) and site(s) of adsorption. The objectives of this study were to present a series of modified Langmuir equations that can be used to estimate the Qmax and Kd of a specific adsorption site and to relate Qmax to available adsorption area.
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The Langmuir isotherm is a widely used model for analyzing adsorption equilibrium data. This study evaluated the efficiency and accuracy of all four linear forms of the Langmuir isotherm and its non-linear form using 67 experimental data sets selected from the literature. The results showed that only if all four linear forms simultaneously show high accuracy, then the non-linear form also shows high accuracy, and therefore it can be said that the process probably follows the Langmuir isotherm. On the contrary, when at least one of the four linear forms of the Langmuir isotherm has low accuracy, it means that the non-linear form also has low accuracy, and it can be concluded that this process does not follow the Langmuir isotherm. This research suggests that all four linear forms of the Langmuir isotherm should be evaluated simultaneously to conclude whether the studied system follows the Langmuir isotherm or not. In other words, relying on only one of the four linear forms of the Langmuir isotherm to model adsorption and calculate the Langmuir constant and maximum adsorption capacity is an incomplete approach, contrary to the conventional approach.
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The sorption of Cu(Ⅱ)by loess was investigated experimentally.In general,the measured results were fitted with the conventional Langmuir isotherm very well.However,the further applicability analysis showed that the parameters in Langmuir equation no longer keep their original physical meanings.The parameters in Langmuir equation are characterized more by the precipitation than the adsorption of Cu.
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