Cooperativity in Sorption Isotherms

Abstract

We present a general theory of cooperativity in sorption isotherms that can be applied to sorbent/gas and sorbent/solution isotherms and is valid even when sorbates dissolve into or penetrate the sorbent. Our universal foundation, based on the principles of statistical thermodynamics, is the excess number of sorbates (around a probe sorbate), which can capture the cooperativities of sigmoidal and divergent isotherms alike via the ln–ln gradient of an isotherm (the excess number relationship). The excess number relationship plays a central role in deriving isotherm equations. Its combination with the characteristic relationship (i.e., a succinct summary of the sorption mechanism via the dependence of excess number on interfacial coverage or sorbate activity) yields a differential equation whose solution is an isotherm equation. The cooperative isotherm equations for convergent and divergent cooperativities derived from this novel method can be applied to fit experimental data traditionally fitted via various isotherm models, with a clear statistical thermodynamic interpretation of their parameters..

Replacing the Langmuir Isotherm with the Statistical Thermodynamic Fluctuation Theory

“…with the parameters A F and m F (>1) was proposed originally as an empirical relationship. However, from our theory, the physical meaning of the Freundlich model is clear, as can be demonstrated by its straightforward derivation from the following characteristic relationship for the excess number, 24 Table 1. AB Isotherm Parameters Determined from the Langmuir Fitting Not in Line with the Temperature Independence of the Monolayer Capacity for the CO 2 Adsorption on Activated Carbon a,b 28 b The units for B, mg mol −1 , come from m 3 mol −1 for G 22 and m 3 mg −1 for v.…”

“…The isotherm is a plot of ⟨ n 2 ⟩ against the activity of sorbate false( a 2 = c 2 g / c 2 ⊖ false) , where c 2 ⊖ is the concentration of the saturated vapor. According to the following fundamental relationships of the fluctuation sorption theory, N 22 and G 22 are related to the gradient of an isotherm, via ,, true( ∂ .25em ln .25em ⟨ n 2 ⟩ ∂ .25em ln .25em a 2 true) T = N 22 + 1 and true( ∂ ∂ a 2 a 2 false⟨ n 2 false⟩ true) T = prefix− G 22 v Equations and not only reveal the underlying sorbate–sorbate interaction directly from an experimental isotherm but also serve as the isotherm-generating relationships ; a systematic method of deriving isotherm equations. We emphasize here that eqs and are the simplified versions of our general theory applicable to adsorpt...…”

“…Let us demonstrate how an isotherm equation can be derived directly from the isotherm-generating relationship (eq or ) . To do so, let us express G 22 / v in terms of the disorbate ( B ) and trisorbate ( C ) interactions via G 22 v = B + C a 2 Integrating eqs and yields ⟨ n 2 ⟩ = a 2 A − B a 2 − C 2 a 2 2 where, via a comparison of eq with eq , the integration constant A can be linked to the sorbate-interface KBI via , 1 A = c 2 ⊖ false( G s 2 false) a 2 → 0 where c 2 ⊖ is the saturated vapor concentration.…”

“…Note that the Freundlich model, ⟨ n 2 ⟩ = A F a 2 1 / m F with the parameters A F and m F (>1) was proposed originally as an empirical relationship. However, from our theory, the physical meaning of the Freundlich model is clear, as can be demonstrated by its straightforward derivation from the following characteristic relationship for the excess number, N 22 + 1 = 1 m F Combining eq with the N 22 representation of the isotherm-generating relationship (eq ), one arrives at the Freundlich model (eq ). This derivation identifies the sorbate–sorbate interaction underlying the Freundlich model, via N 22 = 1 m F − 1 < 0 The constant negative excess number in eq signifies sorbate–sorbate exclusion.…”

“…Let us demonstrate how an isotherm equation can be derived directly from the isotherm-generating relationship (eq 6 or 7). 24 To do so, let us express G 22 /v in terms of the disorbate (B) and trisorbate (C) interactions via…”