Equilibrium constant desorption

Note that in the final desorption step the equilibrium constant for adsorption of AB equals 1 /K4 whereas for the other adsorption steps it is defined as... 

Having estimated the sticking coefficient of nitrogen on the Fe(lll) surface above, we now consider the desorption of nitrogen, for which the kinetic parameters are readily derived from a TPD experiment. Combining adsorption and desorption enables us to calculate the equilibrium constant of dissociative nitrogen adsorption from... 

Suppose we successfully measured the sticking coefficient and the activation energy for adsorption of a certain molecule, as well as the rate of desorption. Is it then possible to estimate the equilibrium constant for adsorp-tion/desorption ... 

There are three approaches that may be used in deriving mathematical expressions for an adsorption isotherm. The first utilizes kinetic expressions for the rates of adsorption and desorption. At equilibrium these two rates must be equal. A second approach involves the use of statistical thermodynamics to obtain a pseudo equilibrium constant for the process in terms of the partition functions of vacant sites, adsorbed molecules, and gas phase molecules. A third approach using classical thermodynamics is also possible. Because it provides a useful physical picture of the molecular processes involved, we will adopt the kinetic approach in our derivations. 

If one takes the ratio of the pseudo rate constant for adsorption to that for desorption as an equilibrium constant for adsorption (K), equation 6.2.4 can be written as... 

Kinetic Term The designation kinetic term is something of a misnomer in that it contains both rate constants and adsorption equilibrium constants. For thfe cases where surface reaction controls the overall conversion rate it is the product of the surface reaction rate constant for the forward reaction and the adsorption equilibrium constants for the reactant surface species participating in the reaction. When adsorption or desorption of a reactant or product species is the rate limiting step, it will involve other factors. 

The vacancy coverage, 9V, which is initially equal to 0.075, rapidly decreases during the initial period of NO exposure but then very slowly increases. This behavior can be attributed to the following factors. The first is that 0V in equilibrium with 0.10 atm of H2 is larger than 0V in equilibrium with 0.0028 atm of NO. Calculating the equilibrium constants for H2 and NO adsorption and desorption of these gases, given in Table I, one... 

In a typical SPR experiment real-time kinetic study, solution flows over the surface, so desorption of the guest immobilized on the surface due to this flow must be avoided.72 In the first stage of a typical experiment the mobile reactant is introduced at a constant concentration ([H]0) into the buffer flowing above the surface-bound reactant. This favors complex association, and the progress of complex formation at the surface is monitored. The initial phase is then followed by a dissociation phase where the reactant is removed from the solution flowing above the surface, and only buffer is passed over the surface to favor dissociation of the complex.72 74 The obtained binding curves (sensograms) contain information on the equilibrium constant of the interaction and the association and dissociation rate constants for complex formation (Fig. 9). 

Here FMON and - mon represent the available vacant sites and surface sites occupied by B, respectively, of the first monolayer on a solid absorbate. The equilibrium constant KB for the reaction is given by the ratio of the rate constant for k.d for adsorption and k for desorption... 

The principle we have applied here is called microscopic reversibility or principle of detailed balancing. It shows that there is a link between kinetic rate constants and thermodynamic equilibrium constants. Obviously, equilibrium is not characterized by the cessation of processes at equilibrium the rates of forward and reverse microscopic processes are equal for every elementary reaction step. The microscopic reversibility (which is routinely used in homogeneous solution kinetics) applies also to heterogeneous reactions (adsorption, desorption dissolution, precipitation). 

The dependences of pH and C-potential on the adsorbed amount of M(H20)2+ at the total metal ion concentrations of 3 x10-3 mol dm-3 are shown in Figures 7 and 8, respectively. The amount adsorbed for each M2+ increases with the pH, and the inflection points are shifted toward the lower pH region in the order of Co2+, Zn2+, Pb2+, Cu2+, which corresponds to the order of the hydrolysis constant of metal ions. To explain the M2+-adsorption/desorption, Hachiya et al. (16,17) modified the treatment of the computer simulation developed by Davis et al. (4). In this model, M2+ binds coordina-tively to amphoteric surface hydroxyl groups. The equilibrium constants are expressed as... 

This is called a Langmuir adsorption isotherm for a species A, and the function 9a( Pa) isi shown in Figure 7-23. The KjS are the adsorption-desorption equilibrium constants for species A and B. By historical convention we call these the adsorption isotherms. Before i we proceed let us note that this is a true thermodynamic equilibrium relation so that... 

The relation (Pa) is called the adsorption isotherm. It is used to detennine surface areas of solids and catalysts as well as to determine the adsorption-desorption equilibrium constant Ka- This is measured by determining the amount of a gas that can be adsorbed by a known weight of solid, as shown in Figure 7-24. 

The products of the surface reaction adsorbed are subsequently desorbed into the gas phase. The rate of desorption of C is exactly the opposite in sign to the rate of adsorption of C and the desorption equilibrium constant Ktx. is the reciprocal of the adsorption equilibrium constant Kc. For the desorption of C, according to... 

In these relations, Ki denotes the equilibrium constant of reaction step i. For the numerical evaluation of the model, it is assumed that the backward reaction of step lb has the same transition state as the transition state for the re-desorption of A2 in Model 1, and that the entropy of the molecular precursor on the surface is negligible. The results are shown in Figure 4.37. It is observed that the model predicts that catalysts of much larger reactivity (more negative AEt) will be optimal for reactions where the diatomic molecule is strongly bound to the surface before the dissociation. 

For enzyme reactions K is the traditional Michaelis constant. For a heterogeneous surface on which adsorption and desorption but no reaction occurs (k2 = 0) K is simply an equilibrium constant for adsorption. (Actually we are not being as economical as we could be in this non-dimensionalization of the equations. We could have divided throughout by K instead of introducing the pressure scale p°, and eqn (12.9) would then have read... 

The constant K characterizes the equilibrium adsorption/desorption between a bare surface site and an adsorbate-covered one (i.e., reaction 11.39). In the higher-adsorption / desorption processes (e.g., reaction 11.41) the adsorption (left-hand side) and desorption (right-hand side) sites are already adsorbate covered such reactions are physically very similar no matter what the particular number of adsorbed layers i is involved. Therefore the approximation is made that... 

Physically, this is essentially the same process as the adsorption/desorption in reaction 11.41 when i becomes large. Assuming that the governing equilibrium constant is thus the same as in Eq. 11.47, the steady-state expression result is... 

In the following discussion we will concentrate on the surface reaction, adsorption, and desorption. The complications induced by the transport phenomena will be ignored. In order to develop an expression for the overall rate, the surface concentrations, AL, BL, etc., are related to the concentrations of the reactants in the bulk phase by an "Equilibrium constant". For example ... 

If the adsorption of A is the rate determining step in the sequence of adsorption, surface reaction and desorption processes, then equation 3.71 will be the appropriate equation to use for expressing the overall chemical rate. To be of use, however, it is first necessary to express CA, Cv and Cs in terms of the partial pressures of reactants and products. To do this an approximation is made it is assumed that all processes except the adsorption of A are at equilibrium. Thus the processes involving B and P are in a state of pseudo-equilibrium. The surface concentration of B can therefore be expressed in terms of an equilibrium constant KB for the adsorption-desorption equilibrium of B ... 

Later, it became clear that the concentrations of surface substances must be treated not as an equilibrium but as a pseudo-steady state with respect to the substance concentrations in the gas phase. According to Bodenstein, the pseudo-steady state of intermediates is the equality of their formation and consumption rates (a strict analysis of the conception of "pseudo-steady states , in particular for catalytic reactions, will be given later). The assumption of the pseudo-steady state which serves as a basis for the derivation of kinetic equations for most commercial catalysts led to kinetic equations that are practically identical to eqn. (4). The difference is that the denominator is no longer an equilibrium constant for adsorption-desorption steps but, in general, they are the sums of the products of rate constants for elementary reactions in the detailed mechanism. The parameters of these equations for some typical mechanisms will be analysed below. 

This leads to some complicated differential equations which are usually solved numerically. To simplify things, let us assume that the surface reaction (Eq. (2.24)) is the rate-determining step, while the adsorption and the desorption steps are at equilibrium (i.e., the net change in Eqs. (2.4) and (2.6) is zero). In this case, Eq. (2.26) apply, where KA and /adsorption equilibrium constants for A and B, respectively. 

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