Experimental rate equations

In Section 1.2 we distinguished between elementary and complex reactions. We now make a distinction between simple and complicated rate equations. A simple rate equation has the form of Eq. (1-11). A complicated rate equation has a form different from Eq. (1-11) it may be a sum of terms like that in (1-11), or it may have quantities in the denominator. We have seen that there is no necessary relationship between the complexity of the reaction and the form of the experimental rate equation. Simple rate equations are treated in Chapter 2 and complicated rate equations in Chapter 3. 

The experimental rate equation was found to be —d ]ldt = A obs[I"], the plots being linear over the course of the reactions. Therefore,... 

Suppose a reaction is found to have the experimental rate equation... 

However, for this elementary reaction the experimental rate equation is... 

Because the experimental rate equation is v = kSi, the first-order rate constant k becomes... 

This reaction scheme agrees with the experimental rate equation, with k = k 2ftk 2 i/k-126 and k = k 21/k-126- The rate constant at a high [SO42-] would plateau at the value of Jk 26. On the other hand, one can consider ion-pair formation ... 

Table III. Experimental rate equation for methanol carbonylation catalyzed by Ni-activated carbon...

A mechanism represented by Equations 5, 6, 7, and 8 could be applied to cobalt (III), but the rate-limiting step would have to be the first substitution reaction to account for the experimental rate equation (Equation 2). It is known that cobalt (III) complexes are substitution inert (6, 23) unless significant amounts of cobalt(II) are present (I, 8, 23), and hence one could visualize the first and slow step as follows ... 

In recent years the Coal Research Laboratory has been investigating the kinetics and isotherm behavior of methanol sorption on coal (6, 7, 10) along with the sorption of other vapors on coal (6) and of polar vapors on swelling gels (9, 10). Methanol sorption was shown to be reversible on coal, and its sorption behavior supports the model of coal as a gel or mixture of gels in its physical structure. All indications (I, 6, 7) are that its interaction is with specific and a fixed number of sites for a particular coal sample. Although the sorption of methanol is reversible, coal exhibits sorption behavior which is interpreted in terms of an irreversible swelling of the coal gel upon initial exposure to methanol vapor. As a result of these studies, an isotherm and experimental rate equation for the sorption and desorption were derived that fit the observed data. The isotherm derived for methanol sorption on coal was ... 

The reactions of butane-2,3-diol by HCF in alkaline medium using Ru(III) and Ru(VI) compounds as catalysts leads to similar experimental rate equations for both the reactions. The mechanism involves the formation of a catalyst-substrate complex that yields a carbocation for Ru( VI) or a radical for Ru(III) oxidation. The role of HCF is in catalyst regeneration. The rate constants of complex decomposition and catalyst regeneration have been determined.89 A probable mechanism invoving formation of an intermediate complex has been proposed for the iridium(III)-catalysed oxidation of propane- 1,2-diol and of pentane-1,5-diol, butane-2,3-diol, and 2-methylpentane-2,4-diol with HCF.90-92 The Ru(VIII)-catalyzed oxidation some a-hydroxy acids with HCF proceeds with the formation of an intermediate complex between the hydroxy acid and Ru(VIII), which then decomposes in the rate-determining step. HCF regenerates the spent catalyst.93... 

Chemical reactions that do not follow simple reaction order kinetics are called complex reactions. The experimental rate equations provide crucial evidence for deducing their mechanism. A convenient... 

In eqn. (8), the acid catalyzed reactions of HS and S2 are formulated as first-order decompositions of H2S and HS- (uncatalyzed). Consequently, a distinction between bimolecular proton transfer to a substrate and unimolecular decomposition of the conjugate acid of the substrate is not possible solely on the basis of the experimental rate equation. For both mechanisms, that represented by eqn. (7) as well as that represented by eqn. (8), the same equation is obeyed for the dependence of k on the hydrogen ion concentration, viz. 

The difference between this order in H2 and the experimentally observed value of 1.5 is attributed to the role of H2 as a third body. Moreover, if the reasonable assumption that k 2 7 is made then the two experimental rate equations are in fair agreement with each other and the theoretically derived expression. 

The ratio k i/k2 describes the partitioning of the tetrahedral intermediate between the reactant and the product states. The experimental rate equation is shown in Eq. (12)... 

The way in which the chain-reaction mechanism explains the experimental rate equation (9.103) can be shown as follows. The rate of disappearance of hydrogen is equal to -... 

The apparent increase in rate may arise from the increase in surface as carbon was removed from the original particle. Although the conventional experimental rate equation for the carbon-steam reaction (6) states that steam is a part of the rate equation, it further states that steam also inhibits the rate. The test procedure used here maintains inlet steam at a fixed velocity and in large excess thus any effect of the level of steam conversion should be negligible on the integral rates calculated here. In the runs shown in Figure 2, the maximum use of steam by carbon in a 5-min. period varied from 3.6 to 8.5% of the total steam available. Of... 

The above discussion has been based on simple qualitative ideas about how an elementary reaction may occur. The way to test this picture, of course, is to see if rates of reaction measured experimentally, using different concentrations of each reactant and at different temperatures, show the same predicted behaviour. For this purpose the experimental rate equations for a few elementary reactions involving two reactant species are given in Table 4.1. In each case the experimental rate constant is denoted by the symbol k. Comparison of the form of the experimental rate equations in Table 4.1 with Equation 4.6 makes it clear that there is a good agreement between theory and experiment. Other details also help to confirm this conclusion. For example, the experimental rate constant for the reaction between potassium atoms and Br2 molecules is found to be independent of temperature, suggesting that the energy barrier to reaction is effectively zero. By contrast, the rate constants for the other two reactions (in Table 4.1) are markedly temperature dependent. 

It is important to emphasize that the relationship is empirical in that it represents a generalization of the simplest mathematical way of representing the experimental rate equations for each of the reactions studied. The relationship is based simply on the results of observation and experiment. 

It is often, but not always, the case that the partial orders of reaction turn out to be small integers. If the partial order for a reactant is either 1 or 2, then the reaction is referred to as being first-order or second-order in that particular reactant. The most frequently observed values of overall order n are also 1 and 2 and the corresponding reactions are then referred to as being, respectively, first- and second-order processes. An overall order of reaction can only be defined for a reaction that has an experimental rate equation corresponding to the general form given in Equation 4.8. 

A few selected examples of reactions with experimental rate equations of the form in Equation 4.8 are given in Table 4.2. 

Table 4.2 Experimental rate equations determined under isothermal conditions...

---