Net rate of adsorption

Although we include adsorption here following the chapter on mass transfer, we should be clear that it is a very specific process in its fullest fundamental meaning. Adsorption is the process by which molecules in the fluid phase in contact with a solid move to the solid surface and interact with it. Once at the solid surface these molecules may be reversible or irreversible adsorbed, that is, they may come back off the surface to the fluid phase with their full molecular integrity intact, or they may be so strongly boimd that the rate of removal is for all purposes close enough to zero to be considered zero. 

When the discussion turns to removal of some component from a fluid stream by a high surface area porous solid, such as silica gel, which is found in many consumer products (often in a small packet and sometimes in the product itself), then the term adsorption becomes more global and hence ambiguous. The reason for this ironically is that mass transfer may be convoluted with adsorption. In other words the component to be adsorbed must move from the bulk gas phase to the near vicinity of the adsorbent particle, and this is termed external mass transfer. From the near external surface region, the component must now be transported through the pore space of the particles. This is called internal mass transfer because it is within the particle. Finally, from the fluid phase within the pores, the component must be adsorbed by the surface in order to be removed from the gas. Any of these processes, external, internal, or adsorption, can, in principle, be the slowest step and therefore the process that controls the observed rate. Most often it is not the adsorption that is slow in fact, this step usually comes to equilibrium quickly (after all just think of how fast frost forms on a beer mug taken from the freezer on a humid summer afternoon). More typically it is the internal mass transport process that is rate limiting. This, however, is lumped with the true adsorption process and the overall rate is called adsorption. We will avoid this problem and focus on adsorption alone as if it were the rate-controlling process so that we may understand this fundamentally. 

True adsorption is a mass action process rather than a mass transfer process. What this means is that it will occur even in the absence of a concentration gradient between the bulk gas and the surface. It comes about due to the rapid and chaotic motion of the fluid phase molecules, and their impingement on the surface. From the elementary kinetic theory of an ideal gas we can compute the number of molecules impinging upon a surface per unit time per unit area at a given temperature and pressure. It is  

Hence the number of molecules hitting the surface per unit time per unit area is a flux. Also, it is proportional to the pressure of the gas and the mean speed of the gas molecules and to T 2. At room temperature and pressure the impingement frequency of nitrogen is  

nearly one-third of a mole of nitrogen molecules strikes every square centimeter every second. No wonder the time to equilibration of adsorption is so fast  

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We have covered a body of material in this chapter that deals with movement of mass along gradients and between phases. We have examined the commonalities and differences between linear driving forces, net rates of adsorption, and permeation. Each has the common feature that reaction is not involved but does involve transport between apparently well-defined regions. We move now to chemically reactive systems in anticipation of eventually analyzing problems that involve mass transfer and reaction. 

The net rate of adsorption is the difference between the rate of adsorption and the rate of desorption ... 

The net rate of adsorption is equal to the attachment rate minus the detachment one. Taking into account that the ratio KA = k,Jk A is the adsoiption equilibrium constant, we obtain the following ... 

Since P is the final product of reaction it is also necessary to consider the net rate of desorption. In an analogous way to the net rate of adsorption of A, the net rate of desorption of P may be written ... 

Since we are dealing with gaseous molecules, we usually write the rate of adsorption in terms of the partial pressure of A (PA) rather than molar concentration. The net rate of adsorption and desorption is... 

As an example, suppose that a fluid containing particles is in contact with a rotating disk. What will be the net rate of adsorption onto the collector disk Levich (1962) neglected radial variations and solved the usual convective-diffusion equation, taking the concentration as cf at (he disk surface and as cB far away from the disk. If c< is eliminated from his solution by imposing Equation (11) at this surface, with ] = —D dc/dy =B, the result for the net adsorption rate will be... 

From the kinetic theory of gases, the rate of adsorption is dependent on the pressure and the fraction of bare sites (1-0). The rate of desorption is dependent on 9 and on the energy of activation, E (i.e. equivalent to an energy of adsorption expressed as a positive quantity). Equilibrium is obtained for the values of 9 andp for which the rates of adsorption and desorption are equal. Thus, the net rate of adsorption is zero ... 

Hence, the net rate of adsorption (moles per unit area per second) of species i is... 

Since the net rate of adsorption is zero at equilibrium, the equilibrium relationship is ... 

The net rate of adsorption is equal to the rate of molecular attachment to the surface minus the rate of detachment from the surface. If and fe A are the constants of proportionality for the attachment and detachment processes, then... 

At equilibrium, the net rate of adsorption equals zero. Setting the right-hand side of Equation (10-4) equal to zero and solving for the concentration of CO adsorbed on the surface, we get... 

For desorption to occur, two occupied sites must be adjacent, meaning that the rate of desorption is proportional to the product of the occupied-site concentration, (C S) X (O s).The net rate of adsorption can then be expressed as... 

Fig. 4. Net rate of adsorption as a function of impingement rate. Xe at Tq = 300° K on tungsten 80° K. Slope = sticking coefficient intercept = evaporation rate = F-pfA. From ref. 88.

Assuming equilibrium for all adsorption steps (e.g., the surface reaction is rate-limiting), the net rates of adsorption of reactants and product are all zero. 

The process of adsorption is a dynamic one in which molecules of A are constantly bombarding the surface and a fraction of them are adsorbed at active sites, while those already adsorbed may be spontaneously desorbed. The rate at which adsorption takes place is clearly going to be proportional to the concentration of the species above the surface, namely a, and also to the concentration of vacant sites. We may write it as kauc, where ka is a constant depending on temperature and having the dimensions of volume per mole per unit time. The rate at which molecules of A will desorb may be taken to be proportional to the concentration already adsorbed, k d, where ka has the dimensions of the reciprocal of time. Thus the net rate of adsorption, which is the difference between these two, is... 

In this equation is the concentration of A in the gas phase at the catalyst surface. If the resistance to adsorption is negligible with respect to other steps in the overall conversion process, the concentration of A on the catalyst surface is in equilibrium with the concentration of A in the gas phase. The net rate of adsorption, from Eq. (9-14), approaches zero, and the equihbrium concentration of A is given by the expression... 

Here Pes is the interface Pecletnumber defined in terms ofthe interface diffusivity Ds. The diffusive flux on the right-hand side is equal to the net rate of adsorption (minus desorption). Because we have adopted Langmuir kinetics according to (2-150), the dimensionless form of the balance between diffusion to the interface and adsorption-desorption takes the form... 

Where is the net rate of adsorption in kmol/kg catalyst.hr and ka, kj, are the adsorption and desorption constants, respectively. 

For non-porous catalyst pellets the reactants are chemisorbed on their external surface. However, for porous pellets the main surface area is distributed inside the pores of the catalyst pellets and the reactant molecules diffuse through these pores in order to reach the internal surface of these pellets. This process is usually called intraparticle diffusion of reactant molecules. The molecules are then chemisorbed on the internal surface of the catalyst pellets. The diffusion through the pores is usually described by Fickian diffusion models together with effective diffusivities that include porosity and tortuosity. Tortuosity accounts for the complex porous structure of the pellet. A more rigorous formulation for multicomponent systems is through the use of Stefan-Maxwell equations for multicomponent diffusion. Chemisorption is described through the net rate of adsorption (reaction with active sites) and desorption. Equilibrium adsorption isotherms are usually used to relate the gas phase concentrations to the solid surface concentrations. 

The rate of accumulation of adsorbed reactant (A.S) on the internal surface of the catalyst pellet can be equated to the difference between the net rate of adsorption and the rate of surface reaction giving... 

In a closed system, a steady state is established only when the rate of evaporation of the adsorbed layer becomes comparable with the rate of adsorption on the sample. In a flow system, the steady state pressure achieved with the sample adsorbing can be adjusted by altering the flow of gas into the cell. Once a new steady state is reached, VdNj dt = 0. The net rate of adsorption is just equal to the net flow into the cell, and is therefore immediately known, provided the exit speed SE has been previously determined. 

In a flow system, the rate parameters can therefore be determined in two essentially independent ways. (1) From the pressure drop as a function of time, as indicated on the previous page. (2) By measuring the amount of gas desorbed after different adsorption intervals At. Differentiating the curve of n vs At obtained under reproducible flow conditions then yields the net rate of adsorption. This can again be separated into contributions from adsorption and desorption rates by determining the pressure dependence. 

As we will see below, it may be easier to obtain the desired kinetic results by measuring equilibrium constants first. Then, using the equilibrium constants, one is able to obtain a set of the net rates of adsorption/desorption at various temperatures from the same run, and hence calculate the rate constants with their temperature dependencies. 

At higher partial pressures, the behavior becones nonlinear, and more complex models are required to describe the observed equilibrium data. A frequently used model for monomolecular layer adsorption is the Langmuir isotherm equation. This equation is derived from simple mass-action kinetics. It assumes that the surface of the pores of the adsorbent is homogeneous and that the forces of interaction between the adsorbed molecules are negligible. Let/be the fraction of the surface covered by adsorbed molecules. Therefore, 1 -/ is the fraction of the bare surface. Then, the net rate of adsorption is the difference between the rate of adsorption on the bare surface and desorption from the covered surface ... 

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