Half-integer rate laws

This problem illustrates how half-integer rate laws arise. Also, it shows how the rate coefficient k for a reaction can be related to other parameters k2 and in this case). 

There is one important exception Certain types of chain reactions and reactions involving dissociation produce exponents of one half or integer multiples of one half in power-law or one-plus rate equations (see Sections 5.6, 9.2, and 10.3.1). Such exponents should be accepted if found not to vary with conversion and if there is good reason to believe that a mechanism of this kind may be operative. 

Equation (3.3) is called the rate law or rate equation for Reaction (3.1). The exponents m and n are generally integers or half-integers m is called the order of the reaction with respect to A, and n the order with respect to B, etc. The overall order of the reaction is m + w +.... The orders m, n,. .. must be determined experimentally, because, in general, they cannot be predicted theoretically or deduced from Reaction (3.1). 

Table 14 shows the calculation of the reaction rate, the time law, and the half-life depending on the reaction s order. The order results from the sum of the exponents of the concentrations. The number does not necessarily have to be an integer. The half-life states in which time half of the reactants is converted into the products. Reaction rate constants k are 1012 to 10"11 L/s for first order reactions and 1010 to 10"11 L/(mol s) for second order reactions. 

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. 

Rate equations of multistep reactions often are not power laws. Reaction orders therefore may vary with concentrations, and attempts at accurate determination would be futile. Unless reaction orders are integers, or integer multiples of one half in special cases, only their ranges (such as between zero and plus one) are of interest. 

Reactions with fast pre-dissociation may, but need not, lead to fractional reaction orders of one half or integer multiples of one half or non-power law rate equations involving such exponents. 

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. 

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. 

---