Temkin adsorption conditions

Mechanism Rate deteimining step Langmuir adsorption condition 2h 2h 0 1 Temkin adsorption conditions... 

A similar analysis can be performed for other mechanisms of hydrogen entry, and the various relationships for both Langmuir and nonactivated Temkin adsorption conditions are given in Table 1. 

Temkin isotherm. Condition number 2 is removed and it is assumed that the energetically most favorable sites are occupied first. The Temkin isotherm corresponds to supposing that the adsorption enthalpy changes linearly with pressure. 

Very often the fall in heat of adsorption is more nearly linear than logarithmic, and it is this type of behavior that led to the derivation of the Temkin adsorption isotherm. The isotherm is, in fact, derived from the Langmuir adsorption isotherm by inserting the condition that the heat of adsorption decreases linearly with surface coverage. Such an effect can arise from repulsive forces on a uniform surface or from surface heterogeneity of the surface. 

The analysis is straightforward in cases where adsorbate interaction effects do not play a role. This is the case either under Langmuirian adsorption conditions or in the saturation limit, 6>o 1. The former is an unrealistic assumption, since it is known that Temkin conditions prevail, illustrated in the seminal work of Sepa et al. (1981). Under the latter assumption, one can proceed to determine values of the effective transfer coefficient in the Butler-Volmer equation. Each quasi-equilibrium step, represented by a factor K in vqrr, contributes an amount of ( — to... 

Using the values of kinetic parameters proposed in the seminal paper by Sepa et al. (1981), gives ac - yh+ = -1 /2, under both Temkin- and Langmuir-type adsorption conditions. This means that the net effect of increasing solution phase potential and decreasing proton concentration on the rate of the ORR and on Teiec is negative. In other words, an increase in proton concentration C + at the pore wall along with a decrease in has a net positive impact on electrochemical performance. It is, thus, of primordial importance to understand the factors that influence the proton affinity of the channel. 

On the left-hand side, the lower of the two curves represents the condition where 0H is low (and negligible). This condition pertains to an overpotential just above RT/F in value. The H is relatively tightly adsorbed and the two adsorbed H atoms react relatively slowly. As the electrode potential becomes more negative, 0 increases. Because of the Temkin equation AGe = AGe + r0, the increase of r0 for the upper curve makes AGg, the free energy of adsorption of the //atoms, less negative. The binding of H atoms to the surface is less tight and the Hads + Hads — H2 reaction increases in rate. 

Other modeling efforts include soil acidification models of the macroscopic type that account for the process of S04 sorption in different ways. These approaches, which assume equilibrium conditions to prevail, include the adsorption isotherm, solubility product, and anion exchange. Prenzel (1994) discussed the various limitations of the above approaches in their capability to account for changes in pH. Recently, Fumoto and Sverdrup (2000) used a constant capacitance approach to describe the pH dependency of S04 sorption isotherms in an andisol. Other modeling efforts of S04 isotherms were reported by Gustafsson (1995) in a spodosol. Such isotherm models are of the equilibrium type and include linear and Temkin types of models. 

There is an apparent discrepancy between the treatment of electrode kinetics under Temkin conditions, at intermediate values of the coverage, and the results shown in Fig. 141(b) for the adsorption pseudocapacitance in the same region. For the purpose of calculating the kinetic parameters, we have assumed that 0 is a linear function of potential. This is a valid assumption, as we can see in Fig. 21. Yet such a linear dependence of 6 on should give rise to a constant value... 

We saw earlier (cf. Section 19) that the potential dependence of adsorption of intermediates formed by charge transfer affects the kinetics of electrode reactions. We have worked out the kinetic parameters for a few mechanisms under so-called Langmuir and Temkin conditions (i.e., when the Langmuir and the Temkin isotherms are applicable, respectively). Here we shall derive the appropriate kinetic equations for the combined adsorption isotherm. 

The relationship of 0 or In 0 to V follows from the adsorption isotherm that applies to the chemisorption of the intermediate a Langmuir relation does not always apply, as we indicate in Section X on reaction order of elec-trocatalytic processes. For example, the Temkin isotherm for the condition 0.1 < 0 < 0.9, in the form... 

Fig. 16. Estimated surface coverage of hydrogen and ethylene coadsorption under Temkin conditions. Ethylene adsorption is assumed to occur on 4 ( , a ) or 2 (b, b ) sites. Hydrogen surface coverage, curves (o, b). Ethylene surface coverage, curves (a, b ). 2 A HCIO4 electrolyte P of ethylene, 1 atm.

The condition for equilibrium is that the rates of adsorption and desorption are equal. Isotherms may be obtained by equating these rates. Three theoretical isotherms, those of Langmuir (1918), Freun-dlich (1926) and Temkin (Brunauer et al. 1942) are important. Only the Langmuir isotherm is presented here because it is the one most widely used in work related to gas-solid catalytic systems. 

The non-linearity is due to the apparent standard free-energy of hydrogen adsorption. Under these conditions, the Frumkin-Temkin (F-T) corrections should be applied to both, the discharge and the recombination currents [18,19]. The modified set of charging and recombination currents are ... 

Thus. Eq. (94) looks something like the Elovich equation and was developed here as the kinetic equation corresponding to a Temkin isotherm for adsorption equilibria. This agrees with common experimental observations. We see that the Elovich equation should apply at conditions where the desorption rate can be neglected, since we developed it by generalizing the adsorption rate term in Eq. (77) and neglecting the desorption rate term in Eq. (78). 

Behaviors of the 17 vs. log i relation expected under various experimental conditions are reproduced in Figure 9 for the case of the positive Temkin isotherm. The shape is naturally dependent upon the adsorption isotherm, but several conclusions below can be drawn without having precise information concerning the isotherm. Generally speaking, the use of the Temkin isotherm yields smoother curves, and possibly better fit with experiment, than the use of the Langmuir isotherm. ... 

Consider the scheme given above [equations (64)-(66)] for formic acid oxidation. The rate-determining step (65) may be regarded as an adsorption step, since it results in an increase in the number of sites occupied on the surface. The corresponding rate equation, under Temkin conditions will be... 

From equilibrium in the first adsorption step (64) one obtains under Temkin conditions, at intermediate values of the coverage,... 

This isotherm may be derived from kinetic considerations for intermediate surface coverages (0.2 < 0 < 0.8), but it does not lend itself to multicomponent adsorption and also fails to predict the limiting conditions of — 0 when Pa 0 and 9 — when 00. Even though it was used for correlating the kinetics of ammonia synthesis, the Temkin isotherm has not found much use in the kinetic analysis of solid-catalyzed gas-phase reactions. 

A simplified model of equilibrium surface suggests that the DR behaviour is observed in low-pressure adsorption on patchwise, weakly heterogeneous surfaces which were grown in equilibrium conditions and hence were quenched at the adsorption temperature. At higher pressures, these surfaces should exhibit the Freundlich behaviour, while in the case of strong heterogeneity adsorption should be described by the Temkin isotherm. The three classic empirical isotherms, Freundlich, Dubinin-Radushkevich, Temkin, seem therefore to be related to adsorption on equilibrium surfaces, and the explanation of these experimental behaviours can be seen as a new chapter of the theory of adsorption the theory of physical adsorption on equilibrium surfaces. 

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