Power-law rate

Although very often we will not know a priori how a complex reaction proceeds in detail, for the purpose of parameterization it may be advantageous to write the rate as a function of concentrations or partial pressures, in the form of a power rate law ... 

Elementary reactions have integral orders. However, for overall reactions the rate often cannot be written as a simple power law. In this case orders will generally assume non-integral values that are only valid within a narrow range of conditions. This is often satisfactory for the description of an industrial process in terms of a power-rate law. The chemical engineer in industry uses it to predict how the reactor behaves within a limited range of temperatures and pressures. 

Here we illustrate how to use kinetic data to establish a power rate law, and how to derive rate constants, equilibrium constants of adsorption and even heats of adsorption when a kinetic model is available. We use the catalytic hydrodesulfurization of thiophene over a sulfidic nickel-promoted M0S2 catalyst as an example ... 

Overall, catalytic processes in industry are more commonly described by simple power rate law kinetics, as discussed in Chapter 2. However, power rate laws are simply a parameterization of experimental data and provide little insight into the underlying processes. A micro-kinetic model may be less accurate as a description, but it enables the researcher to focus on those steps in the reaction that are critical for process optimization. 

For practical purposes the reaction kinetics are described by a power rate law with reaction orders between 0.5 and 1.0 for ammonia and -0.1 to 1.0 for NOx. Although... 

The SCR catalyst is considerably more complex than, for example, the metal catalysts we discussed earlier. Also, it is very difficult to perform surface science studies on these oxide surfaces. The nature of the active sites in the SCR catalyst has been probed by temperature-programmed desorption of NO and NH3 and by in situ infrared studies. This has led to a set of kinetic parameters (Tab. 10.7) that can describe NO conversion and NH3 slip (Fig. 10.16). The model gives a good fit to the experimental data over a wide range, is based on the physical reality of the SCR catalyst and its interactions with the reacting gases and is, therefore, preferable to a simple power rate law in which catalysis happens in a black box . Nevertheless, several questions remain unanswered, such as what are the elementary steps and what do the active site looks like on the atomic scale ... 

What is a power rate law What do you think is the historical origin of the power rate law ... 

How appropriate is the power rate law for describing the kinetics of a catalytic reaction ... 

If, further, a power rate law of the form of equation 4.1-3 is applicable, then... 

Kinetics There have been few comprehensive studies of the kinetics of selective oxidation reactions (31,32). Kinetic expressions are usually of the power-rate law type and are applicable within limited experimental ranges. Often at high temperature the rate expression is nearly first order in the hydrocarbon reactant, close to zero order in oxygen, and of low positive order in water vapor. Many times a Mars-van Krevelen redox type of mechanism is assumed to operate. 

In a detailed kinetic study, Sridhar and Ruthven [256], using nickel supported on Kieselghur (58% Ni), alumina (14% and 40% Ni) and silica-alumina (5% Ni), showed that over all four catalysts the rates of both hydrogenation and hydrocracking could be correlated according to the power rate law equation... 

Each step in the above reaction may contribute a certain resistance. The over-all rate is usually determined by the so-called rate-controlling step. The sign of electrical charge on each adsorbed particle is determined by measuring work function changes. Reactions I, II, and III can also be expressed in terms of the power rate law as follows ... 

These considerations can be extended to reversible processes. They also apply to single phase, liquid systems. For the case, rather common in heterogeneous catalysts, in which one reactant is in a gas phase and the others and the products are in a liquid phase, application of the principles given above is straightforward provided that there is mass transfer equilibrium between gas phase and liquid phase, i.e., the fugacity of the reactant in the gas phase is identical with its fugacity in the liquid phase. In such case, a power rate law for an irreversible reaction of the form... 

When equation (16.1) is approximated by a power rate law equation the observed orders of the reaction with respect to R and the oxidant add up to nearly unity, consistent with the experimental data. Table 16.3 presents the values of the rate constants for equation (16.1), found from the kinetic data, plus the formal reaction orders. 

It is our belief that the subject can be approached more effectively from a different point of view, also enabling a derivation of a power rate law to be obtained and, moreover, allowing of a rapid estimate of the temperature-independent rate constants Xo or ko. 

Under the conditions of the experiments, the nickel surfaces may be considered as fully covered by hydrogen 0H = 1. Now the temperature-independent rate constants for the three mechanisms can be calculated from the power rate law as given in IV, 2. According to (29), the exponent of the hydrogen pressure in the rate equation will be... 

Such a relation was also found to hold for both cylindrical pellets of calcite and powdered CaC03.21 Cremer and Nitsch,22 in studying the decomposition of CaC03, found that for samples which followed a 2/3 power rate law the pressure dependence of the rate (in m1Bt1) was given by... 

Figure 3 shows the influence of the D2 partial pressure on activity and selectivity. The activity decreases when Pd2 is increased (Figure 3A). Using the power rate law, r=kFfXcpP D2, and keeping kPacp constant, the order of the activity in D2 was determined at (3= -0.85.

Figure 4 shows the influence of the CP partial pressure on the activity and product distribution. The reaction rate is increased by increasing PCp, whereas the selectivity is almost unaffected. Keeping P d2 constant, and using the power rate law, the order of the activity in Pep was determined at a = +0.87. 

As a typical example of this type of reaction, the transformation A) — products may be considered, where the kinetics are described by a simple power rate law of the order n. Since this reaction is completely characterized by specifying the conversion of reactant A. the above system of diflcntial equations (eqs 11-18) may be readily expressed in a convenient, nondimcnsional form. For this purpose, the reactant concentration and the temperature arc related to their corresponding values in the bulk fluid phase (eqs 24 and 25), and the radius coordinate r is divided by the pellet radius R to introduce a dimensionless coordinate (eq 26). 

For single, irreversible reactions obeying simple, integer order power rate laws, this problem can generally be solved analytically. In the case of a first-order reaction in a spherical pellet, the following mass balance is found ... 

In the previous sections, only simple, irreversible reactions have been considered whose kinetics were assumed to obey a simple power rate law of the type r = kc". The reason for this assumption was to have analytical solutions for most of the important problems in order to demonstrate the key effects in a clear manner. Moreover, many heterogeneously catalyzed reactions, although not strictly obeying a power rate law, can nevertheless be described by this kind of rate expression for practical purposes, at least when the concentration range to be covered is not too wide. 

In Fig. 19, calculated curves of the effectiveness factor versus the Weisz modulus are shown for different values of Kpis [91]. For comparison, this diagram also contains the curves corresponding to the results which apply to simple, irreversible power rate laws of zeroth, first and second order. From this figure it is obvious that a strong adsorption of at least one of the products leads to a similar decrease of the effectiveness factor as it is observed in the case of a reversible reaction. 

Beside the convenient representation of the effectiveness factor as a function of the observable Weisz modulus, this diagram has the additional advantage that the error arising from an approximation of the truly hyperbolic form of the rate expression by a simple power rate law with integer order can be estimated. 

However, when the view is restricted to simple, irreversible reactions obeying an nth order power rate law and, if additionally, isothermal conditions arc supposed, then—together with the results of Section 6.2.3—it can be easily understood how the effective activation energy and the effective reaction order will change during the transition from the kinetic regime to the diffusion controlled regime of the reaction. 

Table 2 lists most of the available experimental criteria for intraparticle heat and mass transfer. These criteria apply to single reactions only, where it is additionally supposed that the kinetics may be described by a simple nth order power rate law. The most general of the criteria, 5 and 8 in Table 2, ensure the absence of any net effects (combined) of intraparticle temperature and concentration gradients on the observable reaction rate. However, these criteria do not guarantee that this may not be due to a compensation of heat and mass transfer effects (this point has been discussed in the previous section). In fact, this happens when y/J n [12]. 

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