Observed First-Order Rate Constants

Pirrang, Liu, and Morehead [22] have elegandy demonstrated the application of saturation kinetics (Michaehs-Menten) to the rhodium(II)-mediated insertion reactions of a-diazo /9-keto esters and a-diazo /9-diketones. Their method used the Eadie-Hofstee plot of reaction velocity (v) versus v/[S] to give and K, the equilibrium constants for the catalytic process. However, they were unable to measure the Michaelis constant (fC ) for the insertion reactions of a-diazo esters because they proved to be too rapid. 

We determined the first-order rate constants for the disappearance of diazo ester 34 with the several rhodium] 11) catalysts. The rate of disappearance of the diazo ester 34 upon exposure to each of the catalysts was monitored at 27 1°C, by following the UV absorbance at 2=265 run. The decrease in absorbance of the starting material was plotted versus time. The approximately linear portion of this direct plot, from 80% to 20% of the absorbance, was used to calculate the first-order rate constant for the disappearance of the diazo ester (Tab. 16.4). 

The observed first-order rate constants, Kobs (s ) for the reaction of the rhodium] 11) complexes with diazo ester 34 varied over a range of 10, in which the pivalate catalyst (entry 3) was almost two orders of magnitude faster than any of the other catalysts studied. The carboxamidate catalysts (entries 8-10) were slower than all the carboxy-lates, while the bridged phosphine catalyst (entry 7) behaved like most of the other car-boxy late s. 

The ratio of the insertion (Tab. 16.5) to the /9-hydride eHmination product (If A) was determined for each of the catalysts. Two factors are beheved to govern the ratio of insertion to elimination (1) the earliness versus lateness of the transition state, and (2) the steric bulk of the ligand on the rhodium carbenoid. 

A reactive carbenoid would have an early transition state, favoring /9-H elimination over 1,5-insertion. We expect that the increased proportion of eHmination observed with rhodium trifluoroacetate (entry 5) is due to the electron-withdrawing nature of the ligand, which makes the carbene carbon more electron-deficient and thus more reactive. 

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The reaction displays simple first-order kinetics, with the observed first-order rate constant being equal to kik2l(k i + k. ... 

Throughout this section attention is restricted to rate equations that include concentrations of only the substrate, H, and OH. The observed first-order rate constant, therefore, contains concentrations of only H and OH (the quantity [OH ] is often replaced by A",v/[H j). The substrate (reactant) may be ionizable. Rate equations containing the concentration of additional solutes (especially catalytic additives) can be developed as shown in Section 6.3. 

The pH-independent plateau from about pH 5 to 9 represents reaction of the acylsalicylate anion. It is obvious from the pH-rate profile that k" is much larger than k. The theoretical equation for k, the observed first-order rate constant, is derived in the usual way from Eq. (6-71). 

Some authors use O] instead of cr as the substituent constant in such correlations.) An example is provided by the aminolysis of phenyl esters in dioxane the substrates RCOOPh were reacted with -butylamine, and the observed first-order rate constants were related to amine concentration by = k2 [amine] kj [amine]. The rate constants kz and k could be correlated by means of Eq. (7-54), the reaction constants being p = +2.14, b = + 1.03 (for A 2) and p = -1-3.03,8 = -1-1.08 (for ks). Thus, the two reactions are about equally sensitive to steric effects, whereas the amine-catalyzed reaction is more susceptible to electronic effects than is the uncatalyzed reaction. 

Consider the Al scheme.The observed first-order rate constant is defined by the experimental rate equation... 

Observed first-order rate constants (kobSd) under the conditions of excess ligand over the substrate are shown in Table 1. The uncatalyzed buffer rate constant is k = 1.67... 

Influence of OH concentration on the reaction rate constant. From the dependence of the observed first order rate constant on the sodium hydroxide concentration, shown in Table 3, it can be established that equation (2) holds, where ko represents the contribution due to the unimolecular decomposition process and koH is the contribution due to the base-catalysed process in alkaline medium. 

Several authors have suggested that the pathway may prove to be the most common mechanism in substitution reactions of octahedral complexes generally. However, the D path can be clearly demonstrated in some cases including at least two examples from Co(III) chemistry. The path (I - III - IV, Fig. 7) through the fivecoordinate intermediate would lead, in the case of rate studies in the presence of excess anionic ligand, to observed first-order rate constants governed by equation (13)... 

For the gas-phase unimolecular isomerization of cyclopropane (A) to propylene (P), values of the observed first-order rate constant, kuni, at various initial pressures, P0, at 470 C in a batch reactor are as follows ... 

Table 8. Observed first order rate constants (ko, s.) for the displacement of dioxygen in the hemes Fe(TPP)02L ([79] M = Fe, X = O2) by carbon monoxide (Eq. 4) at - 79 °C in CH2CI2 (30)...

Fig. 11. Dependences of observed first-order rate constants on nucleophile concentration for thiocyanate susbtitution at dichlorobis(ethylmaltolato)meta-1(IV) complexes M(etmalt)2Cl2, and at a series of tin(IV) complexes SnL2Cl2 with L = the ligands whose formulae are shown against the thin line plots. The data refer to reactions in acetonitrile solution at 298.2 K (data from Refs. (264) and (265)).

This may appear to be an unlikely situation to encounter until one recalls that there are a number of reactions involving two isolated and independently reacting centers. One might then anticipate that statistically A , = Ik and that Sg = l/2( + < ). These are precisely the conditions demanded by (1.91). It is worth noting that in this case the observed first-order rate constant is k2-... 

Tab. 16.4 Observed first-order rate constants for the insertion to elimination ratios for the reaction of the rhodium(II) catalysts with diazo ester 32.

If data are available we can calculate whether film resistance to heat transfer is important by the estimate of Eq. 36., and whether film resistance to mass transport is important by comparing the observed first-order rate constant based on the volume of particle with the mass transfer coefficient for that type of flow. 

FIGURE 5.6 Typical plot of observed first-order rate constants for the decay of OH as a function of the initial CINO concentration at 373 K (adapted from Finlayson-Pitts et al., 1986). 

In this system k2 values were easily measured in solutions with large excesses (about 150 1) of catalytic sites over substrate concentrations. Under these conditions the observed first-order rate constant was unaffected by increasing polymer concentration. The variation of k2 over the pH range 3 to 7 is displayed in Fig. 20. It is bell shaped and exhibits a maximum at pH 4.5. This bell-shaped pH-rate profile is similar to that of other model primary amines57,58 63 as well as the enzyme acetoacetate decarboxylase.52... 

Collins and Jameson11 found that for small air bubbles (20 to 100 jzm), varying the particle zeta potential from +30 mV to +60 mV resulted in an order of magnitude change in the observed rate constants for each drop size. Table 9 shows the values of the calculated and observed first-order rate constants for the data of Collins and Jameson obtained when their particles (polystyrene) had the minimum stability (zeta potential + 30 mV). The observed rate constants are much smaller than those calculated from collision theory. Their data indicate that between 1 in 40 to I in 100 collisions results in the particles sticking to bubbles. This is consistent with the particle-collision removal mechanism. 

Many inhibitors with very low dissociation constants appear to have a slow onset of inhibition when they are added to a reaction mixture of enzyme and substrate. This was once interpreted as the inhibitors having to induce a slow conformational change in the enzyme from a weak binding to a tight binding state. But in most cases, the slow binding is an inevitable consequence of the low concentrations of inhibitor used to determine its Ki. For example, consider the inhibition of trypsin by the basic pancreatic trypsin inhibitor. Kx is 6 X 10-14 M and the association rate constant is 1.1 X 106 s-1 M-1 (Table 4.1). To determine the value of Ki, inhibitor concentrations should be in the range of K1, where the observed first-order rate constant for association is (6 X Q U M) X (1.1 X 106 s-1 M-1) that is, 6.6 X 10-8 s 1. The half-life is (0.6931/6.6) X 108 s, which is more than 17 weeks. 

Two kinds of unimolecular decay lifetimes can be described. The first is the true radiative lifetime, i.e., the reciprocal of the rate constant for the disappearance of a species which decays only by fluorescence or phosphorescence. Since values of true fluorescence lifetimes may be calculated from the relationship between these quantities and the / numbers (vide supra) of the corresponding absorption bands, these values are (or at least approximations of them) are, in a sense, available. The second kind of lifetime is the reciprocal of an observed first order rate constant for decay of an excited state which may be destroyed by several competing first-order processes (some of which may be apparent first order) operating in parallel. We suggest that the two kinds of lifetime be distinguished by the systematic use of different symbols, as utilized by Pringsheim (4). 

For the reaction of MOH(n 1)+ with propionic anhydride,200 the Bronsted plot of log kMOH versus the pKa of MOH2n+ follows a smooth curve if the values for HzO and OH- are included (Figure 4). However, if the line is drawn to exclude the fcHj0 value, a Bronsted /3 of ca, 0.25 is obtained. Although kMOH for [Co(NH3)5OH]2+ (3 M s 1) is some 103-fold less than k0H, this reaction will compete favourably at neutral pH with base hydrolysis. At pH 7 where the cobalt(III) complex exists almost completely as the MOH2+ species the observed first order rate constant for nucleophilic attack by OH would be ca. 10-4 s 1. AIM solution of [Co(NH3)5OH]2+ would give a value of kobs 2.5 s 1, a rate acceleration of > 104-fold. Since the effective concentration of a nucleophile in the intramolecular reaction could be ca. 102 M, rate accelerations of 10° are possible. The role of the metal ion in such reactions is to provide an effective concentration of an efficient nucleophile at low pH. 

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