Order, of a rate equation

The order of a rate equation equals the sum of the exponents of the concentration terms in... 

Concentration-Dependent Term of a Rate Equation 17 which for a first-order reaction becomes simply... 

With a view to the study of electrode kinetics, we mention here briefly the fact that the quantity [SF ] contains both first-order parameters, F and X, and second-order parameters, FF, Xj, and Xj.. The latter provide essential information about (a) the potential dependence of the rate constant and (b) the stoichiometric order of the rate equation, that cannot be obtained from the first-order parameters. This will be discussed further in Sect. 4. 

If a large excess of one or more of the reactants is used, such that the concentration of that reactant changes hardly at all during the course of the reaction, the effective order of the reaction is reduced. Thus, if in carrying out a reaction which is normally second-order with a rate equation <3lA - k2CACB an excess of B is used, then CB remains constant and equal to the initial value C80. The rate equation may then be written 9lA = A, CA where = kiCBo and the reaction is now said to be pseudo-first-order. 

We have told you what sorts of starting materials and conditions favour El or E2 reactions, but we haven t told you how we know this. El and E2 differ in the order of their rate equations with respect to th.e base, so one way of finding out if a reaction is El or E 2 is to plot a graph of the variation of rate with base concentration. But this can be difficult with El reactions because the base (which need be only very weak) is usually the solvent. More detailed evidence for the differences between reaction mechanisms comes from studying the rates of elimination in substrates that differ only in that one or more of the protons have been replaced by deuterium atoms. These differences are known as kinetic isotope effects. 

The constancy of k when experimental values are put into an integrated rate equation can be the criterion of the order of a rate function. 

If we return now to the question of the uniqueness of the rate parameters determined from thermal desorption measurements, we see that all of the analytical methods depend on the assumption of a rate equation whose validity, in general, is not tested. In particular, when there are adsorbate—adsorbate (lateral) interactions, or where desorption occurs via a precursor state, the coverage dependence in the pre-exponential term is not a simple function and the concept of reaction order is not meaningful. 

In the next two sections we shall look in some detail at how experiments can be designed, and how the resulting data can be analysed, to obtain the form of a rate equation for a chemical reaction under a given set of experimental conditions. In subsequent sections we shall see that it is the values of partial orders of reaction, together with the value of the rate constant and the way in which it varies with temperature, that enable us to propose detailed mechanisms for reactions such as those in Table 4.2, among many others. 

Much of the language used for empirical rate laws can also be appHed to the differential equations associated with each step of a mechanism. Equation 23b is first order in each of I and C and second order overall. Equation 23a implies that one must consider both the forward reaction and the reverse reaction. The forward reaction is second order overall the reverse reaction is first order in [I. Additional language is used for mechanisms that should never be apphed to empirical rate laws. The second equation is said to describe a bimolecular mechanism. A bimolecular mechanism implies a second-order differential equation however, a second-order empirical rate law does not guarantee a bimolecular mechanism. A mechanism may be bimolecular in one component, for example 2A I. 

The units of the reaetion rate eonstant k vary with the overall order of the reaetion. These units are those of a rate divided by the nth power of eoneentration as evident from Equations 3-13 and 3-14. 

Figure 3-7 gives plots of Equations 3-54 and 3-58, respeetively. Consider the seeond order reaetion 2A-I-B—produets, whieh is first order with respeet to both A and B, and therefore seeond order overall. The rate equation is ... 

Integrating the rate equation is often diffieult for orders greater than 1 or 2. Therefore, the differential method of analysis is used to seareh the form of the rate equation. If a eomplex equation of the type below fits the data, the rate equation is ... 

Improbable as a zero-order reaction may seem on the basis of what has been said thus far, let us consider the possibility of this rate equation ... 

The isolation technique showed that the reaction is first-order with respect to cin-namoylimidazole, but treatment of the pseudo-first-order rate constants revealed that the reaction is not first-order in amine, because the ratio k Jc is not constant, as shown in Table 2-2. The last column in Table 2-2 indicates that a reasonable constant is obtained by dividing by the square of the amine concentration hence the reaction is second-order in amine. For the system described in Table 2-2, we therefore find that the reaction is overall third-order, with the rate equation... 

This procedure constitutes an application of the steady-state approximation [also called the quasi-steady-state approximation, the Bodenstein approximation, or the stationary-state hypothesis]. It is a powerful method for the simplification of complicated rate equations, but because it is an approximation, it is not always valid. Sometimes the inapplicability of the steady-state approximation is easily detected for example, Eq. (3-143) predicts simple first-order behavior, and significant deviation from this behavior is evidence that the approximation cannot be applied. In more complex systems the validity of the steady-state approximation may be difficult to assess. Because it is an approximation in wide use, much critical attention has been directed to the steady-state hypothesis. 

The global rate of the process is r = rj + r2. Of all the authors who studied the whole reaction only Fang et al.15 took into account the changes in dielectric constant and in viscosity and the contribution of hydrolysis. Flory s results fit very well with the relation obtained by integration of the rate equation. However, this relation contains parameters of which apparently only 3 are determined experimentally independent of the kinetic study. The other parameters are adjusted in order to obtain a straight line. Such a method obviously makes the linearization easier. 

According to the definition given, this is a second-order reaction. Clearly, however, it is not bimolecular, illustrating that there is distinction between the order of a reaction and its molecularity. The former refers to exponents in the rate equation the latter, to the number of solute species in an elementary reaction. The order of a reaction is determined by kinetic experiments, which will be detailed in the chapters that follow. The term molecularity refers to a chemical reaction step, and it does not follow simply and unambiguously from the reaction order. In fact, the methods by which the mechanism (one feature of which is the molecularity of the participating reaction steps) is determined will be presented in Chapter 6 these steps are not always either simple or unambiguous. It is not very useful to try to define a molecularity for reaction (1-13), although the molecularity of the several individual steps of which it is comprised can be defined. 

The time required to convert a given fraction of the limiting reagent is a characteristic of the rate equation. A comparison of successive half-times, or any other convenient fractional time, reveals whether a reaction follows any simple-order rate law. Thus, the ratio of the time to reach 75 percent completion to that for 50 percent is characteristic of the reaction order. Values of this ratio for different orders are as follows ... 

The order of a reaction cannot in general be predicted from the chemical equation a rate law is an empirical law ... 

Equation 8b is plotted in Fig. 13.14. We see that the concentration of the reactant decreases rapidly at first but then changes more slowly than a first-order reaction with the same initial rate. This slowing down of second-order reactions has important environmental consequences because many pollutants degrade by second-order reactions, they remain at low concentration in the environment for long periods. Equation 8a can be written in the form of a linear equation ... 

The two forms of inhibition can occur together. Their combined eflects are modeled by changing the denominator of the rate equation. For an irreversible, first-order reaction, a suitable rate equation is... 

Second, the symmetry properties of one of the processes (the Berry step) are analysed. The operators associated with it are shown to commute with the elements of a cyclic group of order ten. Because of the structure of the multiplication table, the same is true for the operators associated with the other stereoisomerization processes. The solution of the rate equations for any process are derived from these properties (Sections IV and V). 

For example, experimental studies show that the rate law for the reaction of O3 with NO2 to give N2 O5 and O2 is first order in each reactant 2 NO2 + O3 N2 O5 + O2 Experimental rate = [N02 ][03 ] Notice that for this reaction, the order of reaction with respect to NO2 is 1, whereas the stoichiometric coefficient is 2. This shows that the order of a reaction for a particular species cannot be predicted by looking at the overall balanced equation. We describe additional examples in Section 15-1. 

Although the two cases represent entirely different reaction mechanisms, the overall rate of reaction maintains the same form with respect to its dependence on reactant concentration. Measurements of the kinetics would in both cases reveal the reaction to be first order in [R]. In general, it is not possible to prove that a mechanism is correct on the basis of kinetic measurements, as one can almost always find a modified mechanism leading to the same behavior of the rate equation. It is often possible, however, to exclude certain mechanisms on the basis of kinetic measurements. 

Quantitative information can be drawn from such plots. For the a-th order kinetics the slope is the reaction order a and the intercept is In k. For the catalytic reaction considered above with the surface reaction as the rate-limiting process, linearization of the rate equation (5.4-112) leads to ... 

Reaction order. Partial reaction order can be estimated by studying the reaction rate at surplus of all reactants but the one for which the order is to be evaluated. The concentrations of the reactants present in excess will not change significantly during the course of reaction and may be assumed to be constant. A rate equation of the form (5.4-117) then changes into ... 

The order of the above reaction is, therefore, 1.5 + 0.5 = 2. This is typical of situations where the order of reaction and the molecularity of the reaction are the same. It may, however, be noted that the form of rate law, which determines the order of a reaction, can only be derived by actual experiment, and that may or may not be equal to the molecularity of the reaction as provided by the equation representing that reaction. Thus, a general reaction... 

The order of the reaction is 2 + 1 = 3, whereas the molecularity of the reaction, as given by the equation is 4. This reaction can be treated as a representative example which shows that the order of a reaction is strictly an experimental quantity, being concerned solely with the manner in which the rate depends on concentration. In other words, the order of a reaction should be regarded as a mathematical convenience and not as a fundamental property of the reaction. It must be mentioned here that the order of a reaction corresponds to the... 

Equation (3) is in the form of a differential equation describing a first-order kinetic process, and, as a result, drug absorption generally adheres to first-order kinetics. The rate of absorption should increase directly with an increase in drug concentration in the GI fluids. 

A certain element of confusion is to be met with both in textbooks, and in the literature, over the use and meaning of the terms order (cf. p. 39) and molecularity as applied to reactions. The order is an experimentally determined quantity, the overall order of a reaction being the sum of the powers of the concentration terms that appear in the rate equation ... 

The variation of reaction rate with temperature follows the Arrhenius equation, which we have used to study the rate of chemical reactions in the interstellar medium ISM (Section 5.4, Equation 5.9), and can be applied to the liquid phase or reactions occurring on surfaces. Even the smallest increases in temperature can have a marked effect on the rate constants, as can be seen in the increased rate of chemical reactions at body temperature over room temperature. Considering a reaction activation energy that is of the order of a bond energy, namely 100 kJ mol-1, the ratio of the rate constants at 310 K and 298 K is given by ... 

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