Power-function models

In the model-specification stage, we concentrate upon two types of reaction-rate models, the power-function model... 

A similar procedure may also be used with the power-function models. For example, if we are considering the reaction (B14)... 

From a consideration of either Eqs. (113) or (114) (K3), it is evident that a saddle point is predicted from the fitted rate equation. This could eliminate from consideration any kinetic models not capable of exhibiting such a saddle point, such as the generalized power function model of Eq. (1) and the several Hougen-Watson models so denoted in Table XVI. 

The Fractional Power or Power Function model can be expressed as ... 

When the real process generates power-law data, alternatively a sum of exponentials and power function models may be used. But ... 

Table I summarizes the results of previous investigations on catalytic dehydrogenation of n-butane. In this table 2 models were used to correlate dehydrogenation rate data by various investigators one is a power function model, and the other is a Langmuir-Hinshelwood model. The power function model can be obtained by applying the mass action law to describe rate data. Thus, the model presents the dependence of partial...

In their pioneering work Dodd and Watson (3) correlated the dehydrogenation data by Langmuir-Hinshelwood rate models and found that a dual-site surface rate-controlling model is most plausible. Noda and co-workers (21) and more recently Carra and his colleagues (2) obtained essentially the same results as Dodd and Watson (3) for the chromia-alumina catalyzed dehydrogenation of n-butane. As Table I shows, somewhat different values for the orders of power function models are quoted in the references using this method of correlation. 

LH Langmuir-Hinshelwood model, PF Power function model. b In the present investigation the values of n and p were computed by utilizing the rate data reported. 

Table II shows the results of butenes hydrogenation reactions. An interesting feature of the power function model for the hydrogenation reaction is the pressure dependencies obtained. As shown in Table II some experimental results indicated consistently that the hydrogenation rate is proportional to the 0.5 power of hydrogen and butenes partial pressure. This may imply, as discussed by Bond (I), that hydrogen is dissociated upon adsorption and that butenes are adsorbed weakly on a portion of active sites owing to the steric hindrance of adsorbing butenes molecules.

Combining both the dehydrogenation and hydrogenation rate expressions Lyubarskii (IS) obtained the following power function model to describe the net rate of dehydrogenation of n-butane. 

An attempt was made to explore the possibilities for forming empirical rate models which have the following properties (a) thermodynamic consistency, (b) agreement with the power function model, (c) agreement with the rate equation obtained by measuring the apparent stoichiometric number, (d) simultaneous and adequate representation of the dehydrogenation, hydrogenation, and intermediate data. 

Power function models, on the other hand, directly utilize the concept of reaction order. Unlike homogeneous reactions, the reaction orders encountered in solid-catalyzed reactions can be negative or positive, integer, fractional, or zero moreover, product concentrations may also appear in the rate equation. Due to the simplicity of their form, power function models are considerably easier to handle and integrate than the full LHHW expressions and are preferred especially if the reaction is affected by diffusional limitations. These models cannot, however, be used to discriminate between other than grossly different mechanisms, and they are reliable only within the limits of the reaction conditions used to obtain the kinetic data. 

The definite advantages to the use of both formulations under appropriate conditions are evident however, care must be taken not to apply either of the models arbitrarily. Estimation of reaction orders is desirable in a number of applications, and LHHW equations can be reduced to power law form by making the most abimdant surface intermediate (MASI) approximation, if the fractional surface coverage of one adsorbed intermediate is much greater than all others under reaction conditions [21]. Alternatively, catalytic rate data analysis can be started by expressing the rate in terms of a power function model and then... 

Utilization of LHHW and/or power function models together with linear and nonlinear regression techniques to fit rate data for purposes of model discrimination and parameter estimation... 

Several reports provide information on the relationship of PbB to EP in children. Soldin et al. (2003) compared EP (as ZPP) and PbB using a database of 4,908 admissions, 0—17 years of age in 2001—2002, and compared the measurements to those of 10 years earlier. PbB had declined to a quite low range of mean, 2.2—3.3 pg/dl, with a median of 3 pg/dl. ZPP means ranged between 21.1 and 21.6 pg/dl, with median 21—27 pg/dl. These low PbBs were linked to EP through a complex power function model. Counter et al. (2007) evaluated the relationship of PbB to the ZPP/heme ratio in 78 Ecuadorian children having a PbB mean of 26.4 pg/dl and a mean ratio of 152 pmol/mol Hb. WMle there was a statistically significant association of PbB to the ratio, an even stronger association was calculated for log ZPP. 

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