DOI 10.1140/epje/i2004-10030-4 Eur. Phys. J. E 15, 3–8 (2004) T HE EUROPEAN P HYSICAL JOURNAL E Statistical interpretation of the kinetic equation in the adsorption problem L.R. Evangelista 1 and G. Barbero 2, a 1 Departamento de F´ ısica, Universidade Estadual de Maring´ a, Avenida Colombo, 5790 - 87020-900, Maring´ a, Paran´ a, Brazil 2 Dipartimento di Fisica del Politecnico and INFM, Corso Duca degli Abruzzi, 24 - 10129 Torino, Italy Received 9 February 2004 and Received in final form 14 June 2004 / Published online: 14 September 2004 – c  EDP Sciences / Societ` a Italiana di Fisica / Springer-Verlag 2004 Abstract. A statistical interpretation for the adsorption phenomenon is presented in the framework of the Maxwell-Boltzmann statistics. A model for an adsorbing surface is proposed and the connection between the phenomenological parameters, entering in the kinetic equation at the boundary surfaces, with the microscopic model is derived. PACS. 66.10.-x Diffusion and ionic conduction in liquids – 68.43.-h Chemisorption/physisorption: adsor- bates on surfaces – 68.43.Mn Adsorption/desorption kinetics 1 Introduction Recently, the adsorption-desorption phenomenon has at- tracted the interest of several groups in different do- mains [1–9]. All the calculations are usually performed by solving the drift-diffusion equation with the kinetic equation at the limiting surface describing the adsorption- desorption effect. The adsorption phenomenon requires that a molecule loses sufficient energy during its collision with the surface that it can trap into the physiosorption well. When the atom or molecule strikes the surface, it excites vibrational modes in the surface and if the energy exchange is greater than the initial collision energy, it will be trapped in the well. We can calculate the adsorption rate for a bulk molecule just in front of the surface by means of the kinetic theory. The number of molecules in- ciding on the surface, per unit area, per unit time, is the normal component of the density of current defined as the product of the bulk density of particles, just in front of the surfaces, times the velocity of the particles. The adsorp- tion rate is now given by the product of the incoming flux with the suitable defined sticking coefficient. This kind of analysis holds when the molecules from the liquid are adsorbed on the surface when they hit an empty place, and during the adsorption phenomenon the molecule does not change its structure. In this framework the adsorption phenomenon can be treated as a first-order chemical re- action and from the discussion reported above, the rate of adsorption is proportional to the density just in front of the surface. This is the so-called Langmuir’s approx- imation [10]. The desorption phenomenon is the reverse a e-mail: giovanni.barbero@polito.it of the adsorption phenomenon and requires that an ad- sorbed molecule gains sufficient energy from the surface to break its bounding with the surface. If in the desorp- tion phenomenon the adsorbate remains intact on the sur- face, and desorbs reversibly then the desorption rate is just proportional to the surface coverage, i.e. to the sur- face density of already adsorbed particles. In this case, the desorption is of first order. Of course, more complicated situations, in which the structure of the molecule changes during the adsorption phenomenon, can also occur. For instance, if two incident molecules X are adsorbed as a molecule XX, the adsorption rate is proportional to the square of the density. In the same manner, the desorption rate strongly depends on the order of the chemical reac- tion describing the phenomenon [10,11]. In the present investigation, we limit our analysis to the case in which the adsorption-desorption phenomenon can be described by first-order chemical reaction, because we are mainly in- terested in the description of the adsorption-desorption of dyes from the surfaces. Furthermore, we assume that the system is far from reaching the saturation in the covering ratio. Obviously, our analysis can be extended to consider more complicated cases. With our simplifying hypothe- ses, the kinetic equation at the surfaces is written in the form of a balance of the adsorption effect proportional to the density of the particles just in front of the surface, and the desorption effect proportional to the already ad- sorbed density of particles. The aim of our paper is to show that this equation is in agreement with Maxwell- Boltzmann statistical mechanics. We will present also a simple model to deduce separately the phenomenological coefficients entering in the kinetic equation.