Reaction coefficients

Other conventions for treating equiUbrium exist and, in fact, a rigorous thermodynamic treatment differs in important ways. Eor reactions in the gas phase, partial pressures of components are related to molar concentrations, and an equilibrium constant i, expressed directiy in terms of pressures, is convenient. If the ideal gas law appHes, the partial pressure is related to the molar concentration by a factor of RT, the gas constant times temperature, raised to the power of the reaction coefficients. 

Vibrational excitation by electron impact of the background neutrals is an important process, because it is a major cause of energy loss for the electrons [reactions SVl (SiH4 stretching mode), SV2 (SIHt bending mode), and HV in Table II]. Moreover, the density of the vibrationally excited molecules has been reported to be important [211]. However, information about reaction coefficients of vibrationally excited molecules is scarce [192]. Here, only the vibrational excitation of SiHa and Ht is included [212, 213]. 

The plasma-wall interaction of the neutral particles is described by a so-called sticking model [136, 137]. In this model only the radicals react with the surface, while nonradical neutrals (H2, SiHa, and Si H2 +2) are reflected into the discharge. The surface reaction and sticking probability of each radical must be specified. The nature (material, roughness) and the temperature of the surface will influence the surface reaction probabilities. Perrin et al. [136] and Matsuda et al. [137] have shown that the surface reaction coefficient of SiH3 is temperature-independent at a value of = 0.26 0.05 at a growing a-Si H surface in a... 

Information about the surface reaction coefficients of radicals Si H2 +i where n > 1 is scarce. Because the structure of these radicals is similar to that of SiH3, the same surface reaction coefficients are used. It is assumed that if Si H2 i+1 radicals recombine at the surface with a hydrogen atom, a Si H2,+2 neutral is formed and is reflected into the discharge. Another possibility is the surface recombination of Si,H2 +i radicals with physisorbed Si ,H2m + i radicals at the surface. Matsuda et al. [137] have shown that the probability of surface recombination of SiHs with physisorbed SiH3 decreases with increasing substrate temperature. Doyle et al. [204] concluded that at a typical substrate temperature of 550 K, SiH3 radicals mainly recombine with physisorbed H atoms. 

For the SiH2 radicals the surface reaction coefficients have been taken as 5 = P = 0.8 [192]. This sticking coefficient is large because there is no barrier for insertion of this species into the a-Si H surface. Kae-Nune et al. [217] specify a surface recombination probability of about 1 for atomic hydrogen on an a-Si H surface during deposition that results mainly in recombination of H with an H-atom bounded to the surface. 

A sensitivity study of the influence of the elementary data (i.e., reaction coefficients, cross sections, and transport coefficients) has been performed to determine the importance of specific elementary data, so to guide further research in this area [189]. 

A possible explanation for the difference in tendencies of the deposition rate between experiment and model is that in the model the surface reaction and sticking coefficients of the radicals are taken to be independent of the discharge characteristics. In fact, these surface reaction coefficients may be influenced by the ions impinging on the surface [251]. An impinging ion may create an active site (or dangling bond) at the surface, which enhances the sticking coefficient. Recent experiments by Hamers et al. [163] corroborate this the ion flux increases with the RF frequency. However, Sansonnens et al. [252] show that the increase of deposition rate cannot be explained by the influence of ions only. 

Sensitivity analysis. A possible cause for the discrepancy between experiment and model is an error in the elementary parameters (reaction coefficients, cross sections, and transport coefficients) which are obtained from the literature. With a sensitivity study it is possible to identify the most important processes [189]. 

Nienhuis [189] has used a fitting procedure for the seven most sensitive elementary parameters (reactions SiH4 -t- SiH2 and Si2H6 -I- SiHi, dissociation branching ratio of SiH4, surface reaction coefficient and sticking probability of SiHa, and diffusion coefficients of SiH and H). In order to reduce the discrep-... 

A sticking model is used for the plasma-wall interaction [137]. In this model each neutral particle has a certain surface reaction coefficient, which specifies the probability that the neutral reacts at the surface when hitting it. In case of a surface reaction two events may occur. The first event is sticking, which in the case of a silicon-containing neutral leads to deposition. The second event is recombination, in which the radical recombines with a hydrogen atom at the wall and is reflected back into the discharge. 

Here, v represents the reaction coefficients vwj is the number of moles of water in the reaction to form Aj, v,-j is the number of moles of the basis species Ai, and so on for the minerals and gases. 

If we had chosen to describe composition in terms of elements, we would need to carry the elemental compositions of all species, minerals, and gases, as well as the coefficients of the independent chemical reactions. Our choice of components, however, allows us to store only one array of reaction coefficients, thereby reducing memory use on the computer and simplifying the forms of the governing equations and their solution. In fact, it is possible to build a complete chemical model (excluding isotope fractionation) without acknowledging the existence of elements in the first place ... 

Like all formulations of the multicomponent equilibrium problem, these equations are nonlinear by nature because the unknown variables appear in product functions raised to the values of the reaction coefficients. (Nonlinearity also enters the problem because of variation in the activity coefficients.) Such nonlinearity, which is an unfortunate fact of life in equilibrium analysis, arises from the differing forms of the mass action equations, which are product functions, and the mass balance equations, which appear as summations. The equations, however, occur in a straightforward form that can be evaluated numerically, as discussed in Chapter 4. 

The other variables in Equations 3.32-3.34 are either known values, such as the equilibrium constants K and reaction coefficients v, or, in the case of the activity coefficients y, yj and activities aw, a, values that can be considered to be known. In practice, the model updates the activity coefficients and activities during the numerical solution so that their values have been accurately determined by the time the iterative procedure is complete. 

We pose the problem for the remaining equations by specifying the total mole numbers Mw, Mi, and of the basis entries. Our task in this case is to solve the equations for the values of nw, mt, and - The solution is more difficult now because the unknown values appear raised to their reaction coefficients and multiplied by each other in the mass action Equation 4.7. In the next two sections we discuss how such nonlinear equations can be solved numerically. 

The new reaction coefficients for the species Aj, then, are simply the matrix products of the old coefficients and the transformation matrix ... 

The equilibrium constants for mineral and gas reactions are calculated from their revised reaction coefficients in similar fashion as,... 

To demonstrate mass action, we show that for any possible reaction the activity product Q matches the equilibrium constant K. This step is most easily accomplished by computing log Q as the sum of the products of the reaction coefficients and log activities of the corresponding species. The reaction for the sodium-sulfate ion pair, for example,... 

Notice that we have added the electron to B and B in order to account for the electrical charge on the aqueous species. This incorporation provides a convenient check the electron s reaction coefficient must work out to zero in order for the reaction to be charge balanced. 

The slopes of the lines in the plot give the reaction coefficients for each species and mineral in the overall reaction. Species with negative slopes appear to the left of the reaction (with their coefficients set positive), and those with positive slopes are placed to the right. The reactant plotted on the horizontal axis appears to the left of the reaction with a coefficient of one. If there are additional reactants, these also appear on the reaction s left with coefficients equal to the ratios of their reaction rates nr to that of the first reactant. 

As before, species with negative reaction coefficients v are reactant species, which... 

As described in Chapter 3, v ,/ and so on are the reaction coefficients by which species are made up from the current basis entries. Mass transfer coefficients are not needed for gases in the basis, because no accounting of mass balance is maintained on the external buffer, and the coefficients for the mole numbers Mp of the surface sites are invariably zero, since sites are neither created nor destroyed by a properly balanced reaction. 

Taking sulfide oxidation (Reaction 22.19) as an example, when the fluid mixture reaches 25 °C, there are about 5 mmol of H2S(aq) and 0.6 mmol of 02(aq) in the unreacted fluid, per kg of vent water. The 02(aq) will be consumed first, after about 0.3 mmol of reaction turnover, since its reaction coefficient is two it is the limiting reactant. The thermodynamic drive for this reaction at this temperature is about 770 kJ mol-1. The energy yield, then, is (0.3 x 10-3 mol kg-1) x (770 x 103 J mol-1), or about 230 J kg-1 vent water (Fig. 22.8). In reality, of course, this entire yield would not necessarily be available at this point in the mixing. If some of the 02(aq) had been consumed earlier, or is taken up by reaction with other reduced species, less of it, and hence less energy would be available for sulfide oxidation. 

Chemical reaction coefficient, 25 281,283, 290 combinations of, 25 293t effect on critical value of mass transfer Peclet number, 25 284t Chemical reaction engineering (CRE), 22 330... 

In this section, we first introduce the standard form of the chemical source term for both elementary and non-elementary reactions. We then show how to transform the composition vector into reacting and conserved vectors based on the form of the reaction coefficient matrix. We conclude by looking at how the chemical source term is affected by Reynolds averaging, and define the chemical time scales based on the Jacobian of the chemical source term. 

The results presented above were discussed in terms of the special case of elementary reactions. However, if we relax the condition that the coefficients vfai and uTai must be integers, (5.1) is applicable to nearly all chemical reactions occurring in practical applications. In this general case, the element conservation constraints are no longer applicable. Nevertheless, all of the results presented thus far can be expressed in terms of the reaction coefficient matrix T, defined as before by... 

Figure 5.2. The composition vector c can be partitioned by a linear transformation into two parts c,., a reacting-scalar vector of length /VT and cc, a conserved-scalar vector of length N. The linear transformation is independent of x and t, and is found from the singular value decomposition of the reaction coefficient matrix Y.

For this case, the temperature can be treated as just another chemical species. The reaction coefficients in the final row of Tc correspond to temperature and will be positive (negative) if the reaction is exothermic (endothermic). 

In Section 5.1, we have seen (Fig. 5.2) that the molar concentration vector c can be transformed using the SVD of the reaction coefficient matrix T into a vector c that has Nr reacting components cr and N conserved components cc.35 In the limit of equilibrium chemistry, the behavior of the Nr reacting scalars will be dominated by the transformed chemical source term S. 36 On the other hand, the behavior of the N conserved scalars will depend on the turbulent flow field and the inlet and initial conditions for the flow domain. However, they will be independent of the chemical reactions, which greatly simplifies the mathematical description. 

M1 1 depends on the reference concentration vector c1 1 and the reaction coefficient matrix Y. 

---