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Surface barriers and symmetry of adsorption and desorption processes

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Abstract

Adsorption and desorption of hydrocarbons in a realistic model (2496 atoms) of ZSM-5 zeolite (MFI), including an external surface and a reservoir for molecules, have been studied using classical molecular dynamics, with special focus on surface barriers. Different degrees of surface blocking have been modeled according to experimental observations. Using previous molecular dynamics results, from the analysis of adsorption and desorption path lengths, we demonstrate that surface barriers are symmetric, i.e. with equal paths for desorption and adsorption, in agreement with the principle of microscopic reversibility and in contradiction with a model proposed recently. A new thermodynamic analysis confirms the symmetry of adsorption/desorption paths.

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Abbreviations

D:

(Transport or Fickian) diffusivity (cm2s−1)

K:

Surface permeance (cm·s−1), defined by Flux = k·Δc (‘c’ is concentracion)

ka and kd :

Adsorption and desorption rate constants

K:

Equilibrium constant

µz,g,ads,des µ+ :

Chemical potential (z = surface, g = gas, +  = transition state, ads/des = adsorption, desorption)

F:

Helmholtz free energy

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Acknowledgements

J. K. and D.M.R. acknowledge financial support by the Alexander von Humboldt Foundation and the Fonds der Chemischen Industrie. G.S. thanks MICINN of Spain for funding through projects RTI2018-101784-B-I00, RTI2018-101033-B-I00, SEV-2016–0683. G.S. thanks ASIC-UPV for computational facilities.

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Authors and Affiliations

  1. Instituto de Tecnologia Quimica U.P.V.-C.S.I.C. Universidad Politecnica de Valencia. Avenida Los Naranjos S/N, Valencia, 46022, Spain

    German Sastre

  2. Faculty of Physics and Earth Sciences, University of Leipzig, Linnéstraße 5, Leipzig, 04103, Germany

    Jörg Kärger

  3. Department of Chemical and Biological Engineering, University of Maine, Orono, ME, USA

    Douglas M. Ruthven

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  1. German Sastre
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Appendix: The Teixeira et al. model formulated in terms of transition state theory

Appendix: The Teixeira et al. model formulated in terms of transition state theory

See (Fig. 6).

Fig. 6
figure 6

Schematic representation of Teixeira et al. model in terms of transition state theory

Full size image

Gas Phase: \({\mu }_{g}= {\mu }_{g }^{o}\)+ \(RTln{c}_{g}\)

External Surface: \({\mu }_{1}\,=\, {\mu }_{1 }^{o}\)+\(RTln{c}_{1}\)  = \({\mu }_{g }^{o}\)+ \(RTln{c}_{g}\) (Equilibrium between gas and surface).

c1/cg = \(exp\left[\frac{{\mu }_{g }^{o}-{\mu }_{1 }^{o} }{RT}\right]\)  = K1.

Intra-crystalline: \({\mu }_{z}= {\mu }_{z }^{o}\)+ \(RTln{c}_{z}\)

Ads. Rate =  kac1 = \({k}_{a}{c}_{g }exp\left[\frac{{\mu }_{g }^{o}-{\mu }_{1 }^{o} }{RT}\right]\) Des. Rate =  kdcz.

Trans. State Theory: \({k}_{a}\)= \(\left(\frac{kT}{h}\right)exp\left(\frac{{\mu }_{1}^{0}- {\mu }^{*}}{RT}\right)\) \({k}_{d}\)=\(\left(\frac{kT}{h}\right)exp\left(\frac{{\mu }_{z}^{0}- {\mu }^{*}}{RT}\right)\)

Net Rate = kac1−kdcz = \(\left(\frac{kT}{h}\right)exp\left(\frac{- {\mu }^{*}}{RT}\right)\left[{c}_{g}{e}^{\frac{{\mu }_{g}^{o}}{RT}}- {c}_{z}{e}^{\frac{{\mu }_{z}^{o}}{RT}}\right]\) as c1/cg = \(exp\left[\frac{{\mu }_{g }^{o}-{\mu }_{1 }^{o} }{RT}\right]\)

At equilibrium: \(K= \frac{{c}_{z}^{eq}}{{c}_{g}}\) = \(exp\left(\frac{{\mu }_{g}^{o}- {\mu }_{z}^{o}}{RT}\right)\)  = \(exp\left(\frac{-\Delta {F}^{o}}{RT}\right)\)

The final equation is exactly the same as for direct mass transfer between the gas and intra-crystalline phases. The intermediate state (c1, μ1) cancels out because, at equilibrium, c1/cg = exp \(\left[\frac{{\mu }_{g }^{o}-{\mu }_{1 }^{o} }{RT}\right]\).

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Sastre, G., Kärger, J. & Ruthven, D.M. Surface barriers and symmetry of adsorption and desorption processes. Adsorption 27, 777–785 (2021). https://doi.org/10.1007/s10450-020-00260-1

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