# The Kinetics of Sorption–Desorption Phenomena: Local and Non-Local Kinetic Equations

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## Abstract

**:**

## 1. Introduction

## 2. Local Kinetic Equations

#### 2.1. No Saturation: $k\left(\sigma \right)=k$

#### 2.2. Linear Saturation: $p=1$

#### 2.3. Quadratic Saturation: $p=2$

## 3. Non-Local Kinetic Equations

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

d | Thickness of the cell [m]. |

D | Diffusion coefficient [m${}^{2}$/s]. |

$\delta $ | Delay time [s]. |

$\epsilon $ | Dimensionless delay time. |

$k\left(\sigma \right)$ | Adsorption coefficient [m/s]. |

$n\left(t\right)$ | Bulk density of adsorbable particles [1/m${}^{3}$]. |

${n}_{0}$ | Bulk density of adsorbable particles in thermodynamic |

equilibrium in the absence of adsorption [1/m${}^{3}$]. | |

${n}^{*}$ | Equilibrium bulk density of particles |

in the presence of adsorption [1/m${}^{3}$]. | |

$N=n\left(t\right)/{n}_{0}$ | Reduced bulk density of particles. |

${N}^{*}={n}^{*}/{n}_{0}$ | Reduced equilibrium bulk density of particles |

in the presence of adsorption | |

$r={\sigma}_{M}/{\sigma}_{0}$ | Maximum density of adsorbed particles, ${\sigma}_{M}$, |

in units of surface density of adsorption sites, ${\sigma}_{0}$. | |

$S=\sigma /{\sigma}_{M}$ | Reduced surface density of particles. |

${S}^{*}={\sigma}^{*}/{\sigma}_{M}$ | Reduced equilibrium surface density of particles |

in the presence of adsorption. | |

$\sigma \left(t\right)$ | Surface density of adsorbed particles [1/m${}^{2}$]. |

${\sigma}^{*}$ | Equilibrium value of the surface density of adsorbed |

particles [1/m${}^{2}$]. | |

${\sigma}_{M}={n}_{0}d/2$ | Maximum density of adsorbable particles [1/m${}^{2}$]. |

${\sigma}_{0}$ | Surface density of adsorbing sites [1/m${}^{2}$]. |

$\tau $ | Desorption time [s]. |

${\tau}_{D}={d}^{2}/D$ | Diffusion time [s]. |

${\tau}_{k}=d/2k$ | Intrinsic adsorption time [s]. |

${\tau}_{R}$ | Effective relaxation time [s]. |

$T=t/\tau $ | Dimensionless time. |

$u=\tau /{\tau}_{k}$ | Desorption time, $\tau $, in units of intrinsic adsorption time, ${\tau}_{k}$. |

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**Figure 1.**Dependence of the dimensionless surface density of the steady state versus the dimensionless parameter r. Dashed and dash-dotted lines represent the limiting cases corresponding to small and large r, respectively, as given by Equations (21) and (23). The curves are drawn for $u=0.3$ or $\tau =0.3\phantom{\rule{0.166667em}{0ex}}{\tau}_{\kappa}$, i.e., the adsorption time is approximately three-times faster than the desorption time.

**Figure 2.**Dependence on the dimensionless surface density of adsorbed particles, $S\left(T\right)$, versus the dimensionless time, T, for $u=0.3$, $r=0.5$, (dash-dotted line) and $r=2$, (full line).

**Figure 3.**Relaxation time, in $\tau $ units, versus the dimensionless parameter r. The dashed and dash-dotted lines are the limiting cases of large and small r, respectively. The figure is drawn for $u=0.3$.

**Figure 4.**Plot of dS/dT, given by Equation (31), versus S for $u=0.1$ and $r=0.5$ (dash-dotted), $r=1$ (full), and $r=1.5$ (dashed). The insert shows a closeup of the three curves around $S=0$. Clearly, for all physically meaningful values of r and u, there are three distinct real solutions.

**Figure 5.**Dependence on the dimensionless surface density of adsorbed particles, $S\left(T\right)$, versus the dimensionless time, T, in the case of a quadratic reduction of the adsorption coefficient. The curves were numerically obtained for $u=0.3$ and three values of r: $r=0.1$ (dash-dotted), $r=5$ (full), and $r=10$ (dashed).

**Figure 6.**Comparison (in dimensionless units) of the time dependencies of S relevant to cases where saturation is absent (dash-dotted), with saturation linear in S (full) and saturation quadratic in S (dashed). The curves were drawn for $u=0.3$ and $r=10$.

**Figure 7.**Time evolution (in dimensionless units) of S for $\epsilon =5\phantom{\rule{0.166667em}{0ex}}{\epsilon}_{c}$ (dash-dotted), $\epsilon =0.9\phantom{\rule{0.166667em}{0ex}}{\epsilon}_{c}$ (full), and $\delta =0$ (dashed), with $u=0.1$. From the inset, where it is reported the evolution for small T, it is evident that in the case $\epsilon =0$ the initial boundary condition on $dS/dT$ is not satisfied, as a consequence from the fact that in this case the differential equation passes from the second to the first order.

**Figure 8.**Time evolution (in dimensionless units) of the surface density of adsorbed particles in the presence of linear (dashed), and quadratic (dash-dotted), saturation effect. The curve are drawn for $u=0.1$, $r=100$ and $\epsilon =5\phantom{\rule{0.166667em}{0ex}}{\epsilon}_{c}$. Observed in the inset, comparison between the time evolution of S in the absence of saturation effect (full), with linear (dashed), and quadratic (dash-dotted) saturation effect.

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**MDPI and ACS Style**

Barbero, G.; Scarfone, A.M.; Evangelista, L.R.
The Kinetics of Sorption–Desorption Phenomena: Local and Non-Local Kinetic Equations. *Molecules* **2022**, *27*, 7601.
https://doi.org/10.3390/molecules27217601

**AMA Style**

Barbero G, Scarfone AM, Evangelista LR.
The Kinetics of Sorption–Desorption Phenomena: Local and Non-Local Kinetic Equations. *Molecules*. 2022; 27(21):7601.
https://doi.org/10.3390/molecules27217601

**Chicago/Turabian Style**

Barbero, Giovanni, Antonio M. Scarfone, and Luiz R. Evangelista.
2022. "The Kinetics of Sorption–Desorption Phenomena: Local and Non-Local Kinetic Equations" *Molecules* 27, no. 21: 7601.
https://doi.org/10.3390/molecules27217601