Determination of Isosteric Heat of Adsorption by Quenched Solid Density Functional Theory

Abstract

The heat of adsorption is one of the most important parameters characterizing energetic heterogeneity of the adsorbent surface. Heats of adsorption are either determined directly by calorimetry or calculated from adsorption isotherms measured at different temperatures using the thermodynamic Clausius-Clapeyron equation. Here, we present a method for calculating the isosteric heat of adsorption that requires as input only a single adsorption isotherm measured at one temperature. The proposed method is implemented with either nonlocal (NLDFT) or quenched solid (QSDFT) density functional theory models of adsorption that are currently widely used for calculating pore size distributions in various micro- and mesoporous solids. The pore size distribution determined from the same experimental isotherm is used for predicting the isosteric heat. The QSDFT method has advantages of taking into account two factors contributing to the structural heterogeneity of adsorbents: the molecular level roughness of the surface and the pore size distribution. The method is illustrated with examples of low temperature nitrogen and argon adsorption on selected samples of carbons of different degree of graphitization and MCM-41 mesoporous silicas of different pore size. The isosteric heat predictions from the NLDFT and QSDFT methods are compared against relevant experiments and the results of Monte Carlo (MC) simulations, with good agreement found in the cases where the surface model adequately reflects the pore surface roughness. Analyses with the QSDFT method show that the isosteric heat of adsorption significantly depends of the molecular level roughness of the adsorbent surface, which is ignored in NLDFT and MC models. The proposed QSDFT method with further verification can be used for calculating the isosteric heat as an additional parameter characterizing the adsorbent surface in parallel with routine calculations of the pore size distribution from a single adsorption isotherm.

Isosteric Heat: Comparative Study between Clausius–Clapeyron, CSK and Adsorption Calorimetry Methods

“…The figure presents curves with very similar behavior: initially, they decay from approximately a value of −48 kJ•mol −1 to a value of −27 kJ•mol −1 where later they reach a plateau. This zone is very characteristic for solids of this type and is typical of thermal behavior of the isostere on a heterogeneous surface [22,31,33]. In this area the values evaluated by the three methods are very similar, however, with the low standard deviation of the adsorption calorimetry (A-Cal), it is observed that the indirect methods always underestimated the heat.…”

“…In this area the values evaluated by the three methods are very similar, however, with the low standard deviation of the adsorption calorimetry (A-Cal), it is observed that the indirect methods always underestimated the heat. In the area where the plateau is reached, which ranges from 1 to 3.5 mmol•g −1 , corresponds to isosteric heats that occur on homogeneous surfaces [22,31]. Subsequently, there is an increase in the enthalpic value of isosteric heat of −32 to −48 kJ•mol −1 , which is probably due to interactions of the N2 molecule within the porous structure of MOF-199.…”

“…As mentioned by Tian et al [5,6,[21][22][23][24][25][26], the correct determination of isosteric heat is fundamentally associated with defining the adsorbed phase. It was J. W. Gibbs who introduced this concept for heterogeneous systems in one of his first works in thermodynamics.…”

“…It was J. W. Gibbs who introduced this concept for heterogeneous systems in one of his first works in thermodynamics. From the point of view of the definitions of Gibbs to describe the properties of a surface and the interface, the latter can be represented as a two-dimensional dividing surface without any volume [5,[21][22][23][24][25][26][27][28][29][30]; isosteric heat is often interpreted as the energy to transfer an adsorbate molecule from the bulk of the fluid phase to the adsorbed phase under some fixed thermodynamic conditions of the system such as temperature, total volume and amount of adsorbent [26][27][28][29][30][31]. Some authors have analyzed that the Gibbsian formalism predicts non-physical isosteric heat under maximum thermodynamic conditions of the adsorption isotherm.…”

“…Some authors in recent works [22,[24][25][26] have developed very rigorous thermodynamic studies to deduce isosteric heat without inconsistencies and that can be used for practical applications [21,31,38]. Among other very interesting methods that have also been applied to assess isosteric heat, the outstanding work of Chakraborty et al [23][24][25][26] should be mentioned, who developed a theory to determine the isosteric heat of adsorption between an adsorbate (vapor) and an adsorbent (solid) based on the thermodynamics of chemical equilibrium, Maxwell's relations and the entropy of the adsorbed phase.…”