Thermodynamic N₂ Vapor Isotherms of Materials: Zeta Adsorption Isotherm Approach

Abstract

For N2 vapor adsorbing on a nanopowder at 77 K, we use a multicomponent canonical ensemble to determine the conditions for equilibrium between a contacting vapor phase and the adsorbed molecular clusters. Provided the substrate variations (heterogeneities) do not cause a phase transition, such as wetting or capillary condensation, we show that the equilibrium conditions allow an expression for an adsorption isotherm to be formed. The isotherm expression contains four constants that are to be obtained empirically from mass-specific adsorption measurements. The specific surface area of a substrate is expressed in terms of the specific number of adsorption sites and the cross-sectional area of an adsorbed N2 molecule on materials. This procedure is investigated by using the respective specific surface areas of different samples of a given material to transform the data into an area-specific basis. We pooled data sets from multiple samples with possibly different specific surface areas to form a single data set. The thermodynamic (zeta adsorption) isotherm is formulated after a regression analysis is performed on the pooled data set. This isotherm is used with the Gibbs adsorption equation to obtain the solid surface energy of a material as a function of the amount adsorbed. Five materials are examined: silica, α-alumina, γ-alumina, carbon black, and graphitized carbon. Their respective solid surface energies are each shown to be a material property.

Specific Surface Area of Nanopowders from Argon Adsorption at 77 and 87 K: Zeta Adsorption Isotherm Approach

“…A multicomponent canonical ensemble was used , to determine the conditions required for the adsorbate to be in statistical equilibrium. This leads to the expression for the chemical potential of an adsorbed component , with ζ molecules, μ ζ SV where a 0 is the number of empty adsorption sites, q ζ ( T ) is the partition function for a component with ζ molecules, and the cluster-size distribution, a ζ , is the number of clusters (or components) with ζ molecules.…”

“…Because the total number of adsorption sites is constant and every molecular cluster is assumed to occupy a site, the following relation must exist The expressions for a 0 ( x V ) and a ζ ( x V ) may be formulated from eqs – in terms of four isotherm constants, ,, M , c is , α, and ζ th . These constants were previously assumed to act as material properties. ,, The distributions a 0 and a ζ at a given x V have the same units as M , and they are given as and The amount of vapor adsorbed per unit area of the substrate (or the thermodynamic isotherm) can be obtained by summing the product ζa ζ ( x V ) over all of the cluster types or components After using eq in eq , one obtains the expression for the amount of vapor adsorbed n SV ( x V ) under isothermal conditions on a substrate that is heterogeneity-free and as a function of x V ,,, where In eqs and , the isotherm constants, except M , are dimensionless. It should be noted that M is defined as the number of adsorption sites on the substrate per unit surface area of the substrate, but the measured amount adsorbed is often obtained per unit substrate mass .…”

“…When this condition is reached, it is assumed that the wetting transition is impending. The method has been applied before for N 2 , heptane, and H 2 O vapors adsorbing on silica , and on silicon. , …”

“…We examine the issue using the zeta adsorption isotherm (ZAI). − The lateral interactions may play a role in the adsorbate–adsorbate and adsorbate–substrate interactions, as suggested. But if they do, then our hypothesis is that the differences would manifest themselves in the isotherm measurements that are made as a function of x V (≡ to the ratio of the fluid-phase pressure, P V , to the saturation pressure of the fluid at the isotherm temperature, T , P s ( T )).…”

“…Gibbs showed that an isotherm function could be used to formulate a first-order differential equation for the surface energy (or tension), γ SV ( x V ), along an isotherm, but to determine an explicit expression for γ SV ( x V ), the surface energy at some point on the isotherm must be known. At 87 K, Ar forms a condensable fluid isotherm, and we may use the methods that were previously developed, ,, but at 77 K, bulk Ar is a solid, and we propose a new method for determining its surface energy at this temperature.…”