Adsorption Kinetics: Classical, Fractal, or Fractional?

Adsorption-limited kinetics may be described by phenomenological pseudo-order models. Such models leverage on the general principle that the rate of change of the adsorbed material depends on some power of its concentration, and their solutions provide the quantity of adsorbed molecules per unit mass of the sorbent material as a function of time. The assumptions made about how the solute molecules (adsorbents) are distributed around the sorbent material and whether or not diffusion effects are present are crucial for defining the rate of change. In the first case, the homogeneous uniform distribution of solute molecules and the absence of diffusion effects are well-described by classical modeling (integer-order derivatives). In the second case, fractal modeling arises from a departure from homogeneous uniform distribution, time is apparently contracted, and diffusion effects are still absent. In the third case, deviation from both conditions leads to fractional modeling; unlike fractal modeling, there are memory effects that exert an action on a limited number of process steps. We present briefly solutions for various classical and fractal kinetic models that describe adsorption. For the first time, we present adsorption kinetics under the framework of fractional calculus. In particular, we provide detailed expressions for pseudo-first-order fractional kinetics, while for higher orders, recursive relations amenable to numerical treatment are given. Application of each model is discussed.