Fractional exponent

Usually, diffusivity and kinematic viscosity are given properties of the feed. Geometiy in an experiment is fixed, thus d and averaged I are constant. Even if values vary somewhat, their presence in the equations as factors with fractional exponents dampens their numerical change. For a continuous steady-state experiment, and even for a batch experiment over a short time, a very useful equation comes from taking the logarithm of either Eq. (22-86) or (22-89) then the partial derivative ... 

Except in simple cases (square and cube roots) radical signs are replaced by fractional exponents. If n is odd,... 

Fractional exponent, dimensionless parameter. Fractal dimension, dimensionless parameter. Scrape method. 

D and fractional exponent a (Table 15) show that the surface of electrochemically polished Cd electrodes is flat and free from components of pseudo-capacitance. The somewhat higher values of D for electrochemically polished high-index planes and for chemically treated electrodes indicate that the surface of these electrodes is to some extent geometrically and energetically inhomogeneous. However, the surface of chemically treated Cd electrodes, in comparison with the surface of mechanically polished or mechanically cut electrodes, is relatively... 

The optimum diameter obtained from equations 5.14 and 5.15 should remain valid with time. The cost of piping depends on the cost power and the two costs appear in the equation as a ratio raised to a small fractional exponent. 

Either coverage dependent terms in the rate coefficients or fractional exponents in the coverage terms must be incorporated. Table I summarizes our analysis of several cases using the equation... 

Be sensitive to possible domain violations, that is, the potential for the optimizer to move variables to values for which the functions are not defined (negative log arguments, negative square roots, negative bases for fractional exponents) or for which the functions that make up the model are not valid expressions of the systems being modeled. 

When an exponent is a fraction, the denominator of this fractional exponent means the root of the base number, and he numerator means a raise of the base to that power ... 

If your calculator has the fraction key and the parentheses keys 0E, then you can calculate fractional exponents on your calculator. To find 32 , enter 0E3E00E000 to get a result of 2. 

According to Eq. (28.2), the compression ratio is raised to a small fractional exponent (viz., 0.23). So even if the compression ratio goes up a lot, the amp load on the motor driver will increase by very little. 

For a number with a fractional exponent, the numerator of the exponent tells you the power to raise the number to, and the denominator of the exponent tells you the root you take. 

This equation reveals that when measurements for fractal objects or processes are carried out at various resolutions, the log-log plot of the measured characteristic 9 (oj) against the scale oj is linear. Such simple power laws, which abound in nature, are in fact self-similar if oj is rescaled (multiplied by a constant), then 9 (oj) is still proportional to oja, albeit with a different constant of proportionality. As we will see in the rest of this book, power laws, with integer or fractional exponents, are one of the most abundant sources of self-similarity characterizing heterogeneous media or behaviors. 

Because the order is the sum of the exponents of the reactants, a first-order reaction must depend only on the concentration of a single reactant (we re going to ignore fractional exponents). An example of such a reaction might be a decomposition reaction with only one reactant. The rate law for such a reaction would be as follows ... 

A recent kinetic study (Wender, Greenfield, Metlin, Markby and Orchin, 21) with benzhydrol as substrate indicates that the rate of the hydrogenation to diphenylmethane increases with increasing concentration of dicobalt octacarbonyl catalyst and is first order with respect to the concentration of substrate. The rate also increases slightly with hydrogen pressure and the correct rate equation probably includes the concentration of hydrogen to a fractional exponent (probably %). 

As seen in Table 2.1, the overall order of an elementary step and the order or orders with respect to its reactant or reactants are given by the molecularity and stoichiometry and are always integers and constant. For a multistep reaction, in contrast, the reaction order as the exponent of a concentration, or the sum of the exponents of all concentrations, in an empirical power-law rate equation may well be fractional and vary with composition. Such apparent reaction orders are useful for characterization of reactions and as a first step in the search for a mechanism (see Chapter 7). However, no mechanism produces as its rate equation a power law with fractional exponents (except orders of one half or integer multiples of one half in some specific instances, see Sections 5.6, 9.3, 10.3, and 10.4). Within a limited range of conditions in which it was fitted to available experimental results, an empirical rate equation with fractional exponents may provide a good approximation to actual kinetics, but it cannot be relied upon for any extrapolation or in scale-up. In essence, fractional reaction orders are an admission of ignorance. 

Characteristic of the rate equations 5.76 and 5.79 is their one-half order with respect to the dissociating reactant, in the case of eqn 5.79 with respect to the coreactants B and C as well. This is an exception to the rule that a reasonably simple mechanism does not give a rate equation with fractional exponents. Conversely, an observed, conversion-independent order of one half is an indication that the reaction might involve fast pre-dissociation. 

Positive or negative fractional exponents of one half or integer multiples of one half are also common in rate equations of chain reactions, where, however, they are caused by binary termination steps rather than fast pre-dissociation (see Chapter 9). 

Apart from chain reactions, the most common occurrence of fractional exponents of one half or integer multiples of one half is in heterogeneous catalysis. For example, in hydrogenation on Group VIII transition metals, hydrogen is adsorbed as atoms, H2 — 2H(ads), producing such behavior. 

One-plus rate equations play a key role in network elucidation. Perhaps the most difficult step in that endeavor is the translation of a mathematical description of experimental results into a correct network of elementary reaction steps. The observed behavior can usually be fitted quite well by a traditional power law with empirical, fractional exponents, at least within a limited range of conditions. This has indeed been standard procedure in times past. However, such equations are highly unlikely to result from a combinations of elementary steps. Their acceptance may be expedient, but as far as network elucidation is concerned they are a dead... 

If fitting a power law requires fractional exponents, a one-plus rate equation with integer exponents should be tried instead. 

The conventional procedure of fitting a rate equation to experimental data is to use a power law reflecting the observed reaction orders. However, while fractional reaction orders may provide an acceptable fit, they cannot be produced by reasonable mechanisms. A better way is to fit the data to "one-plus" rate equations, that is, equations containing concentrations with integer exponents only, but with denominators composed of two or more additive terms of which the first is a "one." Such equations behave much like power laws with fractional exponents but, in contrast to these, can arise from reasonable mechanisms and therefore are more likely to hold over wide ranges of conditions. As an exception, rate equations with constant exponents of one half or integer (positive or negative) multiples of one half can result from chain reactions and reactions initiated by dissociation, and are acceptable if such a mechanism is probable or conceivable. 

The rates of product formation (and reactant consumption) are seen to be of order one half in the initiator or, if the reaction is initiated by a reactant converted in the propagation cycle, the rate equation involves exponents of one half or integer multiples of one half. For an example, see the hydrogen-bromide reaction below. This is one of the exceptions to the rule that reasonably simple mechanisms do not yield rate equations with fractional exponents. [The other exceptions are reactions with fast pre-dissociation (see Section 5.6) and of heterogeneous catalysis with a reactant that dissociates upon adsorption.]... 

If a power-law rate equation requires fractional exponents, one-plus equations with integer exponents should be tried instead. If chain mechanisms or pre-dissociation may be involved, one-plus equations with exponents that are integer multiples of one half should also be tried. 

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