Surprising Use of the Business Innovation Bass Diffusion Model To Accurately Describe Adsorption Isotherm Types I, III, and V

Abstract

Using adsorption isotherm data to determine heats of adsorption or predict mixture adsorption using the ideal adsorbed solution theory (IAST) relies on accurate fits of the data with continuous, mathematical models. Here, we derive an empirical two-parameter model to fit isotherm data of IUPAC types I, III, and V in a descriptive way based on the Bass model for innovation diffusion. We report 31 isotherm fits to existing literature data covering all six types of isotherms, various adsorbents, such as carbons, zeolites, and metal–organic frameworks (MOFs), as well as different adsorbing gases (water, carbon dioxide, methane, and nitrogen). We find several cases, especially for flexible MOFs, where previously reported isotherm models reached their limits and either failed to fit the data or could not sufficiently be fitted due to stepped type V isotherms. Moreover, in two instances, models specifically developed for distinct systems are fitted with a higher R 2 value compared to the models in the original reports. Using these fits, it is demonstrated how the new Bingel–Walton isotherm can be used to qualitatively assess the hydrophilic or hydrophobic behavior of porous materials from the relative magnitude of the two fitting parameters. The model can also be employed to find matching heats of adsorption values for systems with isotherm steps using one, continuous fit instead of partial, stepwise fits or interpolation. Additionally, using our single, continuous fit to model stepped isotherms in IAST mixture adsorption predictions leads to good agreement with the results from the osmotic framework adsorbed solution theory that was specifically developed for these systems using a stepwise, approximate fitting, which is yet far more complex. Our new isotherm equation accomplishes all of these tasks with only two fitted parameters, providing a simple, accurate method for modeling a variety of adsorption behavior.

Calcium l-Malate and d-Tartarate Frameworks as Adjuvants for the Sustainable Delivery of a Fungicide

“…Finally, N 2 adsorption isotherms at 77 K displayed type-III profiles for both UPMOF-1 and -2, indicating the bulk nonporosity of both materials (Figure S15). This was expected and in line with previous literature reports on MOFs constructed from low-molecular weight (LMW) organic acids and/or alkaline earth metals. ,, Indeed, the flexibility of the l -malate and d -tartarate linkers can cause framework distortion toward dense, nonporous structures. ,, …”

Fitting Adsorption Isotherms with Symbolic Regression

“…The values of these fit parameters are listed in Table for pressure ( P , in bar) and amount adsorbed ( q *, in mol/kg). Figure shows the SR fit curve and a fit of the Bingel–Walton isotherm alongside the experimental data measured by Andres-Garcia et al The Bingel–Walton isotherm is given as follows: q * = q max [ 1 − exp ( − false( a + b false) P ) 1 + b a .25em exp ( − false( a + b false) P ) ] Here, q max is the saturation capacity of the adsorbent, and a and b are fit parameters.…”

“…This sharp step is more difficult to capture, and the original authors used a first-order spline interpolation to fit these data. Later, Bingel and Walton applied the Bingel–Walton isotherm to fit this experimental data, with reasonable results . DataModeler fit this isotherm in the following form: q * = a + b P + c [ d − tan − 1 false( e − P false) ] 2 + f { g − tan − 1 false[ tan − 1 ( h − P ) false] } This equation must be constrained to force zero amount adsorbed at zero pressure.…”

“…However, conventional isotherm models often fail to give good fits to experimental data, especially for stepped isotherms, which are common in vapor adsorption and in flexible metal–organic frameworks (MOFs). This has led to the development of physics-based − and semiempirical models, − which seek to describe stepped isotherms, as well as many other approaches listed by Bingel and Walton . Difficulty in modeling nonconventional adsorption isotherms is a barrier to the implementation of adsorption models.…”

Systematic Investigation of CO2 Adsorption Energetics in Metal–Organic Frameworks Based on Imidazolyl Linkers

“…To date, multiple studies investigate the synthesis and application of MOFs for molecular sieving and storage applications, however, knowledge of framework descriptors influencing thermochemistry of guest–host interactions in MOFs is lacking . With the increasing popularity in the use of MOFs for molecular storage applications, it is essential to identify the effect of the choice of metal and ligand on the energetic drive for the confinement of guest molecules. − Such an understanding could permit the identification of structural descriptors for energetic stabilization of framework interaction with confined molecules.…”