Rate equations-solids

Figure 27. Interpretation of the saturation parameter. Shown is a Michaelis Menten rate equation (solid line) and the corresponding saturation parameter d (dashed line). For small substrate concentration S Km the reaction acts in the linear regime. For increasing concentrations the saturation parameter d ... 

Figure 2 Kinetics of gas-phase propylene homometathesis at 0°C, catalyzed by (a) perrhenate/silica-alumina activated by SnMe4 (10 mg, 0.83 wt % Re) and (b) MeReOs on HMDS-capped silica-alumina (10 mg, 1.4 wt % Re). Solid lines are curve-fits to the first-order integrated rate equation. Solid squares first addition solid circles second addition open circles third addition of propylene (30 Torr) to the catalyst.

When reactants are distributed between several phases, migration between phases ordinarily will occur with gas/liquid, from the gas to the liquid] with fluid/sohd, from the fluid to the solid between hquids, possibly both ways because reactions can occur in either or both phases. The case of interest is at steady state, where the rate of mass transfer equals the rate of reaction in the destined phase. Take a hyperbohc rate equation for the reaction on a surface. Then,... 

For the radiative mechanism of heat transfer to solids, the rate equation for parallel-surface operations is... 

Rate of Reaction Rate equations of fluid reactions catalyzed by solids are of two main types ... 

This means that the eonversion is proportional to time. Eigure 3-4 shows plots of the zero order rate equations. Examples of zero order reaetions are the intensity of radiation within the vat for photoehemieal reaetions or the surfaee available in eertain solid eatalyzed gas reaetions. 

The experimental study of solid eatalyzed gaseous reaetions ean be performed in bateh, eontinuous flow stirred tank, or tubular flow reaetors. This involves a stirred tank reaetor with a reeyele system flowing through a eatalyzed bed (Figure 5-31). For integral analysis, a rate equation is seleeted for testing and the bateh reaetor performanee equation is integrated. An example is the rate on a eatalyst mass basis in Equation 5-322. 

Chemical reactions obey the rules of chemical kinetics (see Chapter 2) and chemical thermodynamics, if they occur slowly and do not exhibit a significant heat of reaction in the homogeneous system (microkinetics). Thermodynamics, as reviewed in Chapter 3, has an essential role in the scale-up of reactors. It shows the form that rate equations must take in the limiting case where a reaction has attained equilibrium. Consistency is required thermodynamically before a rate equation achieves success over tlie entire range of conversion. Generally, chemical reactions do not depend on the theory of similarity rules. However, most industrial reactions occur under heterogeneous systems (e.g., liquid/solid, gas/solid, liquid/gas, and liquid/liquid), thereby generating enormous heat of reaction. Therefore, mass and heat transfer processes (macrokinetics) that are scale-dependent often accompany the chemical reaction. The path of such chemical reactions will be... 

Note that wq cannot be fixed by detailed balance, the reason being that it contains the information about the energy exchange with the solid which is not contained in the static lattice gas Hamiltonian. However, by comparison with the phenomenological rate equation (1) we can identify it as... 

If the three-parameter Michaelis-Menten equation is divided by C i, it becomes the same as the three-parameter Langmuir-I linshelwood equation where 1/Cm = Ka. Both these rate equations can become quite complex when more than one species is competing with the reactant(s) for the enzyme or active sites on the solid catalyst. 

Those special cases of integration of eq. (1), which have found the most widespread application in the development of rate equations for reactions of solids, are discussed below. 

The above rate equations were originally largely developed from studies of gas—solid reactions and assume that particles of the solid reactant are completely covered by a coherent layer of product. Various applications of these models to kinetic studies of solid—solid interactions have been given. 

RATE EQUATIONS COMMONLY USED IN KINETIC ANALYSES OF ISOTHERMAL REACTIONS OF SOLIDS... 

Rate equations which have found application in kinetic studies of solid phase reactions a... 

There have been few discussions of the specific problems inherent in the application of methods of curve matching to solid state reactions. It is probable that a degree of subjectivity frequently enters many decisions concerning identification of a best fit . It is not known, for example, (i) the accuracy with which data must be measured to enable a clear distinction to be made between obedience to alternative rate equations, (ii) the range of a within which results provide the most sensitive tests of possible equations, (iii) the form of test, i.e. f(a)—time, reduced time, etc. plots, which is most appropriate for confirmation of probable kinetic obediences and (iv) the minimum time intervals at which measurements must be made for use in kinetic analyses, the number of (a, t) values required. It is also important to know the influence of experimental errors in oto, t0, particle size distributions, temperature variations, etc., on kinetic analyses and distinguishability. A critical survey of quantitative aspects of curve fitting, concerned particularly with the reactions of solids, has not yet been provided [490]. 

Note also that we have just introduced the concepts of nuclei and nucleation in our study of solid state reaction processes. Our next step will be to examine some of the mathematics used to define rate processes in solid state reactions. We will not delve into the precise equations here but present them in Appendices at the end of this chapter. But first, we need to examine reaction rate equations as adapted for the solid state. 

Three types of rate equations are shown here. These rate equations ean be used for quite complieated reactions, but a specific method or measurement approach is needed. How we do this is critical to determining accurate estimation of the progress of a solid state reaction. We will discuss suitable methods in another chapter. We now return to the subject of nucleation so that we can apply the rate equations given above to specific cases. First, we examine heterogeneous processes. 

These rate equations C2in be used for quite complicated reactions, but a specific method or approach is needed. Many authors have tried to devise methods for obtaining rate constants and orders of reaction for given solid state reactions. None have been wholly successful, except for Freeman and Carroll (1948). 

Asymptotic Solution Rate equations for the various mass-transfer mechanisms are written in dimensionless form in Table 16-13 in terms of a number of transfer units, N = L/HTU, for particle-scale mass-transfer resistances, a number of reaction units for the reaction kinetics mechanism, and a number of dispersion units, Npe, for axial dispersion. For pore and solid diffusion, = r/rp is a dimensionless radial coordinate, where rp is the radius of the particle. If a particle is bidis-perse, then rp can be replaced by rs, the radius of a subparticle. For preliminary calculations, Fig. 16-13 can be used to estimate N for use with the LDF approximation when more than one resistance is important. 

The mechanism of solid catalysis involves processes of diffusion, formation of loose combinations with the solid and reactions of those combinations. Reactions with enzymes also involve intermediate, temporary combinations with the enzymes. The rate equations that may proposed in particular cases depends on what are believed to be controlling mechanisms. Many such eqautions are considered in Chapter 6. Here only one of the simpler forms will be examined for evaluation of the parameters, namely,... 

Enzymes also are homogeneous catalysts, although they are sometimes attached to solid surfaces without degradation. They possess a different form of rate equation, for which the development may be found in problem P2.03.02. Their behavior is especially sensitive to temperature and to substrate concentration. 

Such processes assume that molecules from a fluid phase in contact with a solid catalytic surface combine chemically with catalyst surface molecules and reaction subsequently proceeds between chemisorbed molecules followed by desorption of the products. A large number of different rate equations with varying numbers of constants can be derived by making various auxiliary assumptions and tested against experimental rate data. Since a more or less plausible mechanism is postulated, the feeling is that a chosen rate equation is somewhat extrapolatable outside an experimental range with greater... 

The mechanism of the solid catalyzed gas phase reaction, A+B o C, at constant pressure and temperature is to be investigated. Write the rate equations for these possible controlling mechanisms. 

Tests were made on the rate of the reaction,C2H4+ HC1 o C2H5C1, in the presence of inert methane and a solid catalyst.The tabulated data are of 104r in lbmol/(h)(lb catalyst) and partial pressures of methane (I), C2H4 (A), HC1 (B) and C2H5C1 (C) in atm. Inlet partial pressures were 10 for methane and one each for ethylene and HC1. The rate equation is believed to be... 

A solid catalyzed gas phase reaction, A2 2B, normally has a rate equation dependent on the partial pressure of the reactant as follows,... 

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