Abstract

The functional shape of a sorption isotherm is determined by underlying molecular interactions. However, doubts have been raised on whether the sorption mechanism can be understood in principle from analyzing sorption curves via a range of competing models. We have shown recently that it is possible to translate a sorption isotherm to the underlying molecular interactions via rigorous statistical thermodynamics. The aim of this paper is to fill the gap between the statistical thermodynamic theory and analyzing experimental sorption isotherms, especially of microporous and mesoporous materials. Based on a statistical thermodynamic approach to interfaces, we have derived a cooperative isotherm, as a generalization of the Hill isotherm and our cooperative solubilization model, without the need for assumptions on adsorption sites, layers, and pore geometry. Instead, the statistical characterization of sorbates, such as the sorbate-interface distribution function and the sorbate number distribution, as well as the existence of statistically independent units of the interface, underlies the cooperative sorption isotherm. Our isotherm can be applied directly to literature data to reveal a few key system attributes that control the isotherm: the cooperative number of sorbates and the free energy of transferring sorbates from the saturated vapor to the interface. The sorbate–sorbate interaction is quantified also via the Kirkwood–Buff integral and the excess numbers.

Highlights

  • Microporous and mesoporous materials,[1,2] such as activated carbons,[3−5] porous silica,[6] and metal−organic frameworks,[7−9] are powerful adsorbents

  • One approach is to understand the functional shape of an isotherm based on the underlying molecular interactions influenced by interfacial geometry and pore sizes.[1−9] Here, we first clarify why understanding cooperative sorption isotherms has been challenging despite, or because of, the many isotherm models that have been proposed.[4,10,11]

  • We have shown that sorbate−sorbate interaction, which has been considered to play an important role in the functional shape of an isotherm,[4,5,71−73] can be quantified directly from an isotherm’s derivative.[65] (We emphasize that the sorbate−sorbate interaction, which takes place at the interface, is mediated by the interface.) From the experimental data on water vapor adsorption on microporous and mesoporous carbons,[3,4,66,74] the underlying sorbate−sorbate interaction has been quantified. (This is analogous to the cosolvent−cosolvent interaction, when enhanced by the solute, which leads to the cooperative onset of solubilization.61,75−77)

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Summary

Introduction

Microporous and mesoporous materials,[1,2] such as activated carbons,[3−5] porous silica,[6] and metal−organic frameworks,[7−9] are powerful adsorbents. They are important because of their many industrial applications and because of the challenges they pose to understanding their sorption capacities from a molecular basis.[1−9]. There are, broadly speaking, three classes of isotherm models: empirical, semi-empirical, and physical.[12,13] The empirical models can fit experimental data; yet, since they are not based on a physical basis, they cannot be used to understand sorption mechanisms. The goodness of the fit alone cannot be used to conclude the superior realism of one model over the rest.[33]

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