Surface Area Estimation: Replacing the Brunauer–Emmett–Teller Model with the Statistical Thermodynamic Fluctuation Theory

Abstract

Surface area estimation using the Brunauer–Emmett–Teller (BET) analysis has been beset by difficulties. The BET model has been applied routinely to systems that break its basic assumptions. Even though unphysical results arising from force-fitting can be eliminated by the consistency criteria, such a practice, in turn, complicates the simplicity of the linearized BET plot. We have derived a general isotherm from the statistical thermodynamic fluctuation theory, leading to facile isotherm fitting because our isotherm is free of the BET assumptions. The reinterpretation of the monolayer capacity and the BET constant has led to a statistical thermodynamic generalization of the BET analysis. The key is Point M, which is defined as the activity at which the sorbate–sorbate excess number at the interface is at its minimum (i.e., the point of strongest sorbate–sorbate exclusion). The straightforwardness of identifying Point M and the ease of fitting by the statistical thermodynamic isotherm have been demonstrated using zeolite 13X and a Portland cement paste. The adsorption at Point M is an alternative for the BET monolayer capacity, making the BET model and its consistency criteria unnecessary. The excess number (i) replaces the BET constant as the measure of knee sharpness and monolayer coverage, (ii) links macroscopic (isotherms) to microscopic (simulation), and (iii) serves as a measure of sorbate–sorbate interaction as a signature of sorption cooperativity in porous materials. Thus, interpretive clarity and ease of analysis have been achieved by a statistical thermodynamic generalization of the BET analysis.

Cooperative Sorption on Heterogeneous Surfaces

“…In IUPAC nomenclature, a 2 is p / p 0 and ⟨ n 2 ⟩ is n . Quantifying statistical interactions via fluctuation is common to a well-established approach to solutions, macromolecules, and colloids. − Note that our goal is to fit the overall shape of an isotherm based on statistical thermodynamics; hence, our focus is mainly on the functional shape of the isotherm in the finite a 2 region, where ⟨ n 2 ⟩ is the dominant contribution to the surface excess …”

“…38−41 Note that our goal is to fit the overall shape of an isotherm based on statistical thermodynamics; hence, our focus is mainly on the functional shape of the isotherm in the finite a 2 region, where ⟨n 2 ⟩ is the dominant contribution to the surface excess. 37 Sorption Polynomial. The key to the statistical thermodynamic approach to modeling isotherms is the sorption polynomial, which is the generalization of the binding polynomial used widely in protein− ligand binding.…”

“… 38 − 41 Note that our goal is to fit the overall shape of an isotherm based on statistical thermodynamics; hence, our focus is mainly on the functional shape of the isotherm in the finite a 2 region, where ⟨ n 2 ⟩ is the dominant contribution to the surface excess. 37 …”

“…In contrast to the previous model-based approaches, our goal is to establish a universal approach, founded on two postulates of (i) the finite ranged nature of the interface and (ii) the existence of several different types of microscopic interfacial subsystems that act independently at sorption, to model all IUPAC isotherm types. In our recent papers, we have proposed a universal approach to interpret sorption isotherms based on quantifying the underlying sorbate–sorbent and sorbate–sorbate interactions via the fluctuation theory. ,, This has led to two general approaches to model isotherms: cluster expansion of the sorbate–sorbate interaction and the cooperative sorption theory, thereby covering IUPAC Types I, II, and V. In this paper, we will generalize our cooperative sorption theory to model heterogeneous and stepwise isotherms. On the basis of statistical thermodynamics, we propose the two universal measures of cooperative sorption: (i) the sorbate cluster number and (ii) the free energy of transferring the cluster from the saturated sorbate vapor to the interface.…”

Preparation and redistribution mechanism of dimethyldichlorosilane catalyzed by the AlCl3/ZSM-5(5T)@MIL-53(Al) core–shell catalyst

“…In the past decade, many researchers have developed Materials of Institute Lavoisier (MIL) frameworks (MILs) 17–19 with large specific surfaces and high thermal stability for various catalytic purposes. Particularly, the chemical stability of MIL-53(Al) 20,21 is high, and even its Brunauer–Emmett–Teller 22 (BET)-specific surface reaches up to 1140 m 2 g −1 . 23 This three-dimensional network structure is connected by a one-dimensional parallel chain of [AlO 4 (OH) 2 ] octahedrons and terephthalic acid.…”