The Power Law Kinetics

A third way to describe the kinetics of methane oxidation is the descriptive power law  

This approach, although somehow naive in a first glance, is very useful since it provides information about the apparent methane order (m), which usually approaches unity, as well as about the apparent oxygen reaction order (m), which usually approaches zero. What is perhaps more important is that the reaction order (m) can be naturally combined, as a missing piece of puzzle, to the parameters ( app). ( true) and (Ach4) described above. 

Let us be more precise. Equation (16.4), describing the RE kinetics, can be rewritten in the form of (16.12) considering that 

What is more important, according to (16.14), is that there should be a correlation between the apparent ( app) activation energy, the true activation energy ( true) of the reaction, and the heat of adsorption of oxygen (202) via the following relationships  

These values of n can be used in Eq. (16.16), in combination with the values of true and A02 estimated via the RE kinetics, to estimate a novel pp as seen in column 6 of Table 16.1. These novel values tally impressively with the values of app estimated via the first-order kinetics method and are in column 4 of the same table. 

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From the data shown in table 3, it is evident that the effect of Cs promotion on the power law kinetics is twofold First, the reaction order for NH3 is changed to essentially zero, and secondly, the apparent activation energy is higher by more than 20 kJ/mol in the presence of Cs. Contrary to the results obtained by Aika et al. [5], the reaction order for H2 was negative for all catalysts investigated. The positive reaction order for H2 reported by Aika et al. [5] for... 

Reaction order. One of the most widely used (particularly for homogeneous reactions) kinetic expressions is the power law kinetic equation. ... 

We used the experimental data of Miller et a/. (1981) to evaluate rate constants in the power law kinetic expression ... 

The power-law kinetic expression for a reaction that is first-order in the adsorbed gaseous reactant is [106]... 

The power-law kinetic model for a single irreversible reaction is ... 

A] Concentration of A, the limiting reactant k Rate constant in the power-law kinetic model n Reaction order in the power-law kinetic model P Physicochemical property that varies linearly with concentration... 

Analyzing the mechanism of the catalytic reaction allows the identification of the major factors that affect the reactor design. The reaction kinetics is not sensitive to the concentration of the acetic acid, but the presence of some water is necessary to activate the catalyst. On the contrary, ethylene and oxygen are involved in kinetics through a complex adsorption/surface reaction mechanism. The catalyst manifests high activity and selectivity. The power-law kinetics involves only ethylene and oxygen [8] ... 

A satisfactory degree of correlation between theoretical prediction and experimental results is found when benzene hydrogenation is represented by the power-law kinetic model. 

Particle sintering involves atoms that escape and diffuse away from crystallites, eventually being captured by other particles (267), or crystallite diffusion on the surface and subsequent collision and coalescence with larger particles (262). The latter model is unlikely for particles over 50 A (267). Since practical supported catalysts have a broad particle size distribution, a combination of atom and small crystallite diffusion is probable. That more than one process is responsible for sintering is also apparent from the power law kinetics of the aging process (267-264) ... 

Kinetic models for catalytic reactions vary widely in sophistication, generality and accuracy. The simplest, least general and easiest to obtain are the power law kinetic models which are usually strictly empirical and contain a limited number of parameters. Consequently they are valid only in a narrow region of parameters and cannot be safely extrapolated outside this region. Kinetic models which take into consideration the interaction between the gas phase and the solid surface (whether through a reduction-oxidation mechanism or an adsorption-desorption mechanism) are more difficult to formulate and usually contain a large number of parameters. However, they are certainly more reliable, general and accurate than power law kinetics. 

It is clear that B has a poisoning effect (it acts as an inhibitor). Using power law kinetics instead of equation (3.15) will give negative order for component B, and the power law kinetic expression will be empirically valid only over a narrow range of concentrations of components A, B. 

The power law types of rate equations are sometimes good approximations of more fundamental and complex expressions (Bou-dart, 1956). A series of papers and a book have been published by Bohlboro concerning experimental water-gas shift data on iron-based catalysts, and the treatment of this data using the power law kinetic expressions (Boigars and Campbell, 1974 Bohlboro, 1961, 1962, 1963, 1964, 1969). Bohlboro collected experimental data for the shift reaction in the following regions ... 

The power law kinetic equation could be a simplified form of a mechanistic scheme. A summary of some of the reported reaction orders for the partial pressure of hydrogen and carbon monoxide which have been obtained from power law fits by different groups are listed in Table 9. The partial pressure dependencies vary rather widely. The power law fits were obtained for different cobalt catalysts prepared using different supports and methods. The data in Table 9 show that there is not one best power law equation that would provide a good fit for all cobalt catalysts. Brotz [10], Yang et al. [12] and Pannell et al. [13] defined the Fischer-Tropsch rate as the moles of hydrogen plus carbon monoxide converted per time per mass of catalyst (r g+Hj) Wang... 

It is proposed to consider the use of an existing agitated reactor vessel, which can be operated adia-batically at 3 bar (to suppress vaporization), with a liquid volume of 1.1356 m The reaction occurs in a sequence of elementary steps, with the controlling step involving two molecules of PO. The power-law kinetic equation is ... 

Equilibrium is established rapidly in the first step to provide hydrogen free radicals. The sum of the next two steps is the stoichiometric equation. Step two is the slow or rate-controlling step. Thus, the overall reaction rate is not proportional to the product of the hydrogen and toluene concentrations as given by the law of mass action when apphed to the stoichiometric reaction. Instead, the overall reaction rate is proportional to the product of the hydrogen free-radical and toluene concentrations as given in the second elementary step. For the above hydrodealkylation chain reaction, the power-law kinetic equation is derived as follows. Because the first elementary step approaches equilibrium ... 

The power-law kinetic equation for the second elementary step, which determines the overall reaction... 

The rate expressions of the reforming reactions follow the power law kinetics with an Arrhenius term. They may be replaced by other expressions [24]. 

Table 16.1 Kinetic and thermodynamic parameters estimated by the RE, the first-order, and the power law kinetics and their interrelation.

As shown in Figure 2(a), the system with J i> 0 andzo = [X ]o = 0 obeys the power-law kinetics for sufficiently small bt and the steady-state kinetics for sufficiently large bt. Hie crossover time fcross between the two types of kinetics can be estimated by equating eqn [27] and z = 1 (see also Figure 2(a)) ° ... 

More quantitatively, the power-law kinetics z = 3/2)bt) and the steady-state kinetics (z=l) hold for bt<0.2 and bt>2, respectively, within a few percent of error. 

Despite the presence of thermal initiation, the polymerization of styrene mediated by DEPN also shows power-law behavior. This is because the equilibriirm constant K in this system is so large that it takes a long time to reach the steady state, and therefore, the main body of polymerization in this system takes place in a pre-steady-state regime. The rate and PDI equations for the power-law kinetics were quantitatively examined for this system. 

An absolute comparison of theory and experiment was made for the power-law kinetics in the polymerization of styrene with PS-DEPN (25 mM) at 80 °C. All parameters necessary to compute the theoretical values were taken from independent experiments (Table 1). For example, the specific value of fet to be used in this study was determined by... 

Hydrocracking being an irreversible reaction is frequently modeled by the power-law kinetics ... 

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