Gas-phase species equation

In these equations the independent variable x is the distance normal to the disk surface. The dependent variables are the velocities, the temperature T, and the species mass fractions Tit. The axial velocity is u, and the radial and circumferential velocities are scaled by the radius as F = vjr and W = wjr. The viscosity and thermal conductivity are given by /x and A. The chemical production rate cOjt is presumed to result from a system of elementary chemical reactions that proceed according to the law of mass action, and Kg is the number of gas-phase species. Equation (10) is not solved for the carrier gas mass fraction, which is determined by ensuring that the mass fractions sum to one. An Arrhenius rate expression is presumed for each of the elementary reaction steps. 

The molar production rate of gas-phase species by heterogeneous reaction is given by sk Equation 7.82 serves as an implicit boundary condition for the gas-phase species equations. 

Pollard and Newman" have also studied CVD near an infinite rotating disk, and the equations we solve are essentially the ones stated in their paper. Since predicting details of the chemical kinetic behavior is a main objective here, the system now includes a species conservation equation for each species that occurs in the gas phase. These equations account for convective and diffusive transport of species as well as their production and consumption by chemical reaction. The equations stated below are given in dimensional form since there is little generalization that can be achieved once large chemical reaction mechanisms are incorporated. 

Eley-Rideal mechanisms If the mechanism involves a direct reaction between a gas-phase species and an adsorbed intermediate (Eley-Rideal step, reaction 8.4-5), the competition between the reactants for surface sites does not occur. From equations 8.4-6 and -21, since one reactant does not have to adsorb on a site in order to react,... 

In Equation 3, e and m are the impinging electron energy and mass, (e) is the reaction cross section, and / (e) is the electron energy distribution function. Of course, if an accurate expression for fie) and if electron collision cross sections for the various gas phase species present are known, k can be calculated. Unfortunately, such information is generally unavailable for the types of molecules used in plasma etching. 

Considering a general differential control volume, use a conservation law and the Reynolds transport theorem to write a species conservation equation for gas-phase species A in general vector form. Considering that the system consists of the gas phase alone, the droplet evaporation represents a source of A into the system. 

Equation 9.72 introduces a great deal of nomenclature at once. Chemical species are indexed by k, with K being the total number of species (later, when we generalize the kinetics to multiple phases, the variable Kg is used for the number of gas-phase species) reactions are indexed by the variable i, with / being the total number of reactions in the mechanism the name of species k is represented by X v ki is the stoichiometric coefficient of species k in the forward direction of reaction i is the stoichiometric coefficient of species k in the reverse direction of reaction i. 

By far the largest contribution to a gas-phase species entropy comes from translational motion. Equation 8.98 provided a means to calculate this contribution ... 

Often the thermochemical properties, for example, p,°, are known for the gas-phase species. An alternate derivation is to equate Eqs. 9.58 and 11.76 at equilibrium, leading to the following expression for Kp ... 

The time-rate-of-change of surface species k due to heterogeneous reaction is given by Eq. 11.102. As discussed above, the effects of surface chemistry must be accounted for as boundary conditions on gas-phase species through flux-matching conditions such as Eq. 11.123. For a transient simulation, a differential equation for the site fraction Zk of surface species k can be written... 

At steady state the surface species concentations have to adjust themselves consistent with the adjacent gas-phase species concentrations such that the condition sk = 0 is satisfied. In a steady-state reacting flow simulation, such as discussed in Sections 6.2 and 7.7, the surface-species governing equations are taken to be... 

In this equation m is the mass of the system and Sk is the net molar production rate of species by heterogeneous reaction. Note that the summation runs only over the gas-phase species, since it is only these species that can supply or remove mass from the channel. Note further that there may be circumstances (for example purely catalytic systems) where the surface chemistry does not supply or remove mass from the flow (i.e., = 0). 

The plug-flow equations involve the chemical production rates of gas-phase species by surface reaction sk. In general, since the surface reactions involve both gas-phase and surface species, the evaluation of sk depends on the gas-phase composition and the surface composition. Although neither the surface composition nor the production rates of surface species appear directly in the plug-flow equations, the needed gas-phase production rates cannot be evaluated until the surface composition is known. Therefore the surface composition along the channel walls must be determined simultaneously with the solution of the gas-phase plug-flow problem. 

At the stagnation surface, the axial velocity serves as the boundary condition for the continuity equation. For a nonreactive surface, u = 0 for a reactive surface, u = Mst. At the stagnation surface, the pressure-curvature equation itself is solved, as is the axial-momentum equation at the face just above the surface. Therefore neither requires a boundary condition at the surface. When the surface is reactive, the flux of gas-phase species to and from the surface is determined from the heterogeneous surface chemistry. The boundary condition is implemented as mass fluxes into (or out of) the half control volume that spans from the stagnation surface to half the distance to the first interior grid point (Fig. 17.14). 

Here, s represents a free surface site and subscripts a and g represent adsorbed and gas-phase species, respectively. It is assumed that CO adsorbs on two sites prior to dissociation on the same sites and that hydrogen competes for the same sites. It is assumed that all steps prior to the rate-determining steps are in equilibrium. Reactions (21) and (25) were assumed to occur only in one direction and to have the same rates. The assumption that equation (21) is not in equilibrium would appear to be the main difference between the approach of Van Meerten et al. and that put forward by Ross.181 Van Meerten et al proceeded to consider a series of different equations which they then attempt to fit to their data. They showed that the best fit to their data was obtained if the rate determining step was the combination of CHa and Ha... 

With the equation for R-j we obtain the source terms representing heterogeneous chemistry equations for the solid and gas phase species. Those source terms are discussed briefly in (13). 

Examples of thermochemical considerations of cupric enolates include the study of the binding of Cu + with kojic acid (16), a cyclic a-ketoenol. Comparison was made between the divalent cations of U02 +, Cu +, Zn +, Ni +, Co +, Cd +, Ca + where these metals are listed in decreasing order of binding constants over 6 powers of 10. In this case carbon-bonded metal seems most unreasonable because it would ruin the chelation as well as any aromaticity in the pyrone ring. It is admittedly an assumption that pyrones are aromatic. There are no one-ring pyrones for which there are enthalpy of formation data for gas phase species, as opposed to the benzoannelated compounds coumarin (I7)i07a, I07b chromone (is) " " "and xanthone (19) . Plausible, but unstable, Cu(II) enolates eliminate copper and form the 1,4-dicarbonyl compounds as shown in equation 8. 

The radical cations of cyclopropanes are of interest for numerous reasons. The first reason is that because Koopmans theorem equates (to within a sign) the various ionization potentials of a molecule to its various orbital energies, the ionization process thus gives direct information about the electronic (i.e. orbital) interactions in the molecule (see, for example. Ref. 62). Photoelectron spectroscopy on gas phase species is the method of choice for these investigations. We refer the reader to Chapter 5 by Ballard for a more complete discussion of the method and will quote only a few results. 

The chief assumption made in the development presented previously was that the gas enviromnent is known and essentially constant throughout the reactor. This assumption simplified the treatment considerably, since an analysis based on the solid phase alone could be used to describe the bed behavior. Many practical systems using large-particle beds fluidized with a large excess of gas conform to this situation. If vigorously fluidized beds of fine particles are involved, however, the composition of the gas seen by the solid would vary. The previous simple analysis would then be inapplicable. In these instances, the conservation equations for the gas-phase species... 

Equations relating the total concentrations of these aqueous-phase species with the corresponding equilibrium concentrations of the gas-phase species can be derived similarly to those for S(IV). 

The proposed reaction mechanism includes adsorption, dissociation, surface reaction and desorption steps. The adsorbed species are denoted by H, D and HD, where is an active empty site on the surface. The rates of the elementary steps are given by the following equations for the gas phase species and surface species respectively... 

To make a connection between AH p as defined in the gas phase, to the chemistry in solution, correction increments for solvation enthalpies have to be included. For the example of water, the above gas phase equilibrium (equation 2) is formulated for the hydrated species, eq. 4 ... 

This second-order ODE for 4 a(9) with split boundary conditions, given by equations (19-11) and (19-12), cannot be solved numerically until one invokes stoichiometry and the mass balance with diffusion and chemical reaction to relate the molar densities of aU gas-phase species within the pores of the catalytic pellet... 

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