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The adsorption sites (heavy dots) are equivalent and can have unit occupancy. Also, the adsorbates are immobile on the surface. The Langmuir adsorption model explains adsorption by assuming an adsorbate behaves as an ideal gas at isothermal conditions. According to the model, adsorption and desorption are reversible processes.
The Hertz–Knudsen equation describes the non-dissociative adsorption of a gas molecule on a surface by expressing the variation of the number of molecules impacting on the surfaces per unit of time as a function of the pressure of the gas and other parameters which characterise both the gas phase molecule and the surface: [1] [2]
The adsorption rate is dependent on the temperature, the diffusion rate of the solute (related to mean free path for pure gas), and the energy barrier between the molecule and the surface. The diffusion and key elements of the adsorption rate can be calculated using Fick's laws of diffusion and Einstein relation (kinetic theory).
is the rate constant for surface adsorption, m 3 ·mol −1 ·s −1 k − 1 {\displaystyle k_{-1}} is the rate constant for surface desorption, s −1 C S {\displaystyle C_{\mathrm {S} }} is highly related to the total surface area of the adsorbent: the greater the surface area, the more sites and the faster the reaction.
Equation (1) is an adsorption isotherm and can be plotted as a straight line with / [(/)] on the y-axis and = / on the x-axis according to experimental results. This plot is called a BET plot . The linear relationship of this equation is maintained only in the range of 0.05 < p / p 0 < 0.35 {\displaystyle 0.05<{p}/{p_{0}}<0.35} .
The original formulation was for molecules adsorbing from the gas phase and the equation was later extended to adsorption from the liquid phase by comparison with molecular dynamics simulations. [2] For use in adsorption from liquids the equation is expressed based on solute density (molecules per volume) rather than the pressure.
But real adsorption is often done much faster than this infinite time limit i.e. the concentration gradient, decay of concentration at the sub-surface, is only partially formed before the surface has been saturated or flow is on to maintain a certain gradient, thus the adsorption rate measured is almost always faster than the equations have ...
Source: [2] If a solid body is modeled by a constant field and the structure of the field is such that it has a penetrable core, then = ′ [ ()] ′ [ ()]. Here ′ is the position of the dividing surface, = is the external force field, simulating a solid, is the field value deep in the solid, = /, is the Boltzmann constant, and is the temperature.