Quasi-equilibrium conditions

Eqs. (1,4,5) show that to determine the equilibrium properties of an adsorbate and also the adsorption-desorption and dissociation kinetics under quasi-equilibrium conditions we need to calculate the chemical potential as a function of coverage and temperature. We illustrate this by considering a single-component adsorbate. The case of dissociative equilibrium with both atoms and molecules present on the surface has recently been given elsewhere [11]. 

For the equihbrium properties and for the kinetics under quasi-equilibrium conditions for the adsorbate, the transfer matrix technique is a convenient and accurate method to obtain not only the chemical potentials, as a function of coverage and temperature, but all other thermodynamic information, e.g., multiparticle correlators. We emphasize the economy of the computational effort required for the application of the technique. In particular, because it is based on an analytic method it does not suffer from the limitations of time and accuracy inherent in statistical methods such as Monte Carlo simulations. The task of variation of Hamiltonian parameters in the process of fitting a set of experimental data (thermodynamic and... 

Figure 28. (a) Mass spectrum of protonated water clusters H+(H20) (n = 4-45) at 119 K and 0.3 torr He in a flow tube reactor. Note the prominence of H3O+(H2O>20 even under quasi-equilibrium conditions, (b) Mass-spectrometric abundance of OH-(H20)n produced under thermal conditions. Note a magic number at n = 20, though not as prominent as for the case of H30+ hydrates. Taken with permission from ref. 92. 

Mechanisms 1 and 2 are included in the model that is used here for comparison with experimental data. Interface recombination and dark current effects are not included however, the experimental data have been adjusted to exclude the effects of dark current. To include the additional bulk and depletion layer recombination losses, the diffusion equation for minority carriers is solved using boundary conditions relevant to the S-E junction (i.e., the photocurrent is linearly related to the concentration of minority carriers at the interface). Using this boundary condition and assuming quasi-equilibrium conditions (flat quasi-Fermi levels) ( 4 ) in the depletion region, the following current-voltage relationship is obtained. 

In principle, a continuous procedure can be used to construct the isotherm under quasi-equilibrium conditions the pure adsorptive is admitted (or removed) at a slow and constant rate and a volumetric or gravimetric technique used to follow the variation of the amount adsorbed with increase (or decrease) in pressure. A carrier gas technique, making use of conventional gas chromatrographic equipment, may be employed to measure the amount adsorbed provided that the adsorption of the carrier gas is negligible. In all types of measurement involving gas flow it is essential to confirm that the results are not affected by change in flow rate and to check the agreement with representative isotherms determined by a static method. 

In principle, the continuous procedure, where the adsorption takes place continuously and slowly, under quasi-equilibrium conditions, meets the above requirement of reversibility (Rouquerol et al. 1972). In this experiment, the basic experimental quantities from which one wishes to derive the differential enthalpy of adsorption are the rate of adsorption, f°, and the corresponding heat flow, [Pg.46]

More flexibility (including the possibility of determining the desorption branch) is obtained, at the expense of stability, by the continuous gas-flow controlled procedure (Venero and Chiou, 1988), presented in Figure 3.9. Here, the flow of adsorptive is set at a pre-determined value and then controlled by a loop including the flowmeter and the leak-valve. With a thermal mass flowmeter of good quality, flow rates can be correctly controlled down to c. 5 cm3 h-1 With microporous adsorbents, and also when a low specific surface area necessitates the use of large amounts of sample, the flow rate may prove to be a limitation (i.e. not low enough to ensure the required quasi-equilibrium conditions). 

Starting from the Poisson equation and assuming Gaussian spatial distributions for ions and intrablob electrons with the dispersions and ae, prove that ae — cn <3C a and estimate A a = ae —on numerically. Assume the initial number of ion-electron pairs in the blob are no 30 and a, 40 A and e - is the dielectric permitivity of the medium. Note that at r a, out-diffusion flux of electrons is compensated for by their drift in the electric field of the ions (quasi-equilibrium condition). 

This equation can be used to determine V if the differential capacity C is extrapolated to zero frequency, namely under the quasi-equilibrium conditions required to apply the electrocapillary equation for a thermodynamic estimate of dE/dE (see Eq. 12) ... 

In his treatment of the photoresponse in the semiconductor, Wilson assumed, as had Memming, that quasi-equilibrium conditions obtained across the depletion layer, i.e. the product np is a constant. 

The quasi-equilibrium condition (mass-action law) for the reaction with this stoichiometry is... 

The dissociation of I2 in the first step is very fast, the trimolecular second step is slow. The quasi-equilibrium condition for the first step is... 

Using this relationship to replace pY in the quasi-equilibrium condition... 

As first pointed out by Briggs and Haldane [30], the assumption of quasiequilibrium in the first step is inconveniently restrictive. They relaxed that postulate by replacing the quasi-equilibrium condition 8.15 with the Bodenstein approximation for the trace intermediate X ... 

With eqns 8.42 and 8.43 for coefficients involving lumped reactants and with C0 Ccat and the quasi-equilibrium condition C0CPh /CMt s KaU one finds Xxi to be... 

The quasi-equilibrium condition for the inhibition reaction is ATinh s Cs /CACinh and the material balance for A is CEA = CA + Cs. Combination of these conditions shows that the fraction of total A that is still free to react with the catalyst is... 

Rouquerol et al. (11, 12) have recently described the experimental determination of entropies of adsorption by applying thermodynamic principles to reversible gas-solid interactions. Theoretically, the entropy change associated with the adsorption process can only be measured in the case of reversible heat exchange. The authors showed how isothermal adsorption microcalorimetry can be used to obtain directly and continuously the integral entropy of adsorption by a slow and constant introduction of adsorbate under quasi-equilibrium conditions (11) or by discontinuous introduction of the adsorbate in an open system (12). 

Clearly, under quasi-equilibrium conditions T = Tq, P = Pq, m = fiQi, and d9 = 0, so that the standard form... 

For localized 1/1 adsorption, AE is related to the inflection point of a F(E) isotherm at fc- constant, corresponding to the potential of the most positive peak, AFp, in cyclic or linear sweep voltammograms under quasi-equilibrium conditions (slow sweep rates). This peak potential corresponds to the formation of a first Meads overlayer on S. The formation of a second Meads overlayer on top of the first one can be considered as Me adsorption on an UPD modified substrate. 

Organic Surface ratio of different types of molecules in mixed self-assembled monolayers formed in quasi-equilibrium conditions 118... 

The kinetics of all the other steps do not deviate significantly from either steady-state or quasi-equilibrium conditions (see later discussion). 

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