Reaction orders fractional exponents

Equation (1.20) is frequently used to correlate data from complex reactions. Complex reactions can give rise to rate expressions that have the form of Equation (1.20), but with fractional or even negative exponents. Complex reactions with observed orders of 1/2 or 3/2 can be explained theoretically based on mechanisms discussed in Chapter 2. Negative orders arise when a compound retards a reaction—say, by competing for active sites in a heterogeneously catalyzed reaction—or when the reaction is reversible. Observed reaction orders above 3 are occasionally reported. An example is the reaction of styrene with nitric acid, where an overall order of 4 has been observed. The likely explanation is that the acid serves both as a catalyst and as a reactant. The reaction is far from elementary. 

With chemical reactions, the exponents in a rate expression are usually integers. However, the exponents can be fractions or even negative depending on the complexity of the reaction. Reaction order should not be confused with molecularity. Order is an empirical concept whereas molecularity refers to the actual molecular process. However, for elementary reactions, the reaction order equals the molecularity. See Chemical Kinetics Molecularity First-Order Reactions Rate Constants... 

The variable k represents the rate constant. Note the order of each reactant is 1. The reaction order, which describes the order of the entire reaction, can be determined by adding the order of each reactant. For instance, in this example each reactant is first order (meaning each has an understood exponent of 1). The reaction order is the sum of the exponents, or 1 + 1=2. This is a second-order reaction. Most reactions have an order of 0, 1, or 2, but some have fractional orders or larger numbers (though these are quite rare). The order of the reaction must be determined experimentally. Unlike equilibrium expressions, the exponents have nothing to do with the coefficients in the balanced equations. 

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 ... 

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. 

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. 

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.]... 

The reaction orders in a composite event can include negative values, as illustrated by the last exponent in Eq. (2.5-55). Fractional orders are also possible, as long as they correspond to integer orders in the rate-controlling step and thus to whole numbers of each kind of elementary particle in the activated complex. 

For cellulase, typically 0.15 < r < 0.7. This fractional reaction order is characteristic of many effects by cellulase, not just production of glucose. The fractional reaction order indicates a diminishing return on increasing enzyme dosage. The relationship between extent of hydrolysis and reaction time is also expressed by a fractional exponent in time, which indicates a loss of enzyme effectiveness over time. 

This reaction is zero order in A, first order in B, and fu-st order overall. The exponent zero tells us that the rate of this reaction is independent of the concentration of A. Note that reaction order can also be a fraction. 

The reaction order is the exponent in the rate equation or the power to which the concentration or partial pressure must be raised to fit the data. When the exponents are integers or half-integer values, such as A, 1, I A, 2, they may offer clues about the mechanism of the reaction. For example, if the gas-phase reaction of A with B appears to be first order to A and first order to B, this is consistent with the collision theory. The number of collisions per unit volume per unit time depends on the product of the reactant concentrations, and a certain fraction of the collisions will have enough energy to cause reaction. This leads to the following equation ... 

Although the exponents in a rate law are sometimes the same as the coefficients in the balanced equation, this is not necessarily the case, as Equations 14.9 and 14.11 show. For any reaction, the rate law must be determined experimentally. In most rate laws, reaction orders are 0, 1, or 2. However, we also occasionally encounter rate laws in which the reaction order is fractional (as is the case with Equation 14.11) or even negative. 

The exponent is named the reaction order with respect to reactant A, and b is the reaction order with respect to reactant B. For simple reactions a and b integers (1 or 2). In complex reactions the reaction order can be fractional and even negative. The order with respect to each reactant is a particular order. The overall reaction order n is equal to the sum of exponents with respect to all reactants n = S ,-. Usually = 1 or 2, rarely 3. The idea of order for the complex reaction has somewhat different sense. The particular order with respect to a certain reactant characterizes the influence of the concentration of this reactant on the overall reaction rate. This influence can change depending on the concentration of this or other reactants. 

Note 6.2.- Exponent n of equation [6.13] should never be confused with a reaction order, which is defined by speed-concentration relationships and not speed-fractional extent. It is the same for coefficient k, which can be called a kinetic coefficient. It should not be confused with a speed coefficient. Hence there are some cases where Arrhenius law is not followed. 

Related to the preceding is the classification with respect to oidei. In the power law rate equation / = /cC C, the exponent to which any particular reactant concentration is raised is called the order p or q with respect to that substance, and the sum of the exponents p + q is the order of the reaction. At times the order is identical with the molecularity, but there are many reactions with experimental orders of zero or fractions or negative numbers. Complex reactions may not conform to any power law. Thus, there are reactions of ... 

In general, each concentration has some exponent (here, y and z). Each exponent is called the order of the reaction with respect to that particular species. In Equation, y is the order of reaction with respect to species A, and z is the order with respect to species B. When the value of y is 1, the reaction is called first order in A when the value of z is 2, the reaction is called second order in B, and so on. Orders of reaction are small integers or simple fractions. The most common orders are 1 and 2. The sum of the exponents is known as the overall order of the reaction. 

The order of the reaction, n, can be defined as n = a + b. Extended to the general case, the order of a reaction is the numerical sum of the exponents of the concentration terms in the rate expression. Thus if a = b = 1, the reaction just described is said to be second-order overall, first-order relative to A, and first-order relative to B. In principle, the numerical value of a or b can be integral or fractional. 

In Eq. (16-4), k is called the rate constant, the exponents a, b, c, and x are called orders. The orders are usually integers, but may be fractions such as 1/2 or 2/3. A positive order a means that the rate increases with [A], a negative order c means that the rate decreases with increasing [C], If the rate depends on [X] and x > 0, we say that X is a catalyst which increases the rate if x < 0, we say that X is an inhibitor which decreases the rate. The sum of all the individual orders is called the overall order of the reaction. Despite the apparent similarity of an empirical rate law to an equilibrium constant expression, the orders are not necessarily equal to stoichiometric coefficients. 

It occasionally happens that the observed exponents deviate from integers or simple rational fractions by more than experimental error. A possible explanation is that two or more simultaneous mechanisms are in competition, in which case the observed order should lie between the extremes predicted by the individual mechanisms. A possible alternative explanation is that no single reaction step is effectively rate controlling. 

---