The Limiting First-Order Rate Constant

Only in a relatively few systems are deviations from constancy for the function observed and in even fewer, is a point reached in which is a constant, independent of [L,]. 

All the kinetic features expected for a D mechanism and rate law (4.9) i. e. marked effects of L and Lj on the rate constants, are shown in the comprehensive studies in nonaqueous solution of substitution in low-spin Fe(II) complexes of the type FeN4XY where N4 are planar porphyrins, phthalocyanins and macrocycles and X and Y are neutral ligands, CO, R3P, pyridines etc. Small discrimination factors (Ar, /kj) suggest that the five-coordinated intermediate in these systems is very reactive.There have been problems in the confirmation of curvature in the plots of A o,j/[L ] for classical reactions of a number of aquapent-ammine complexes.  

Examining the relationship between the hydrolysis rate constants (A ,) and the equilibrium constant (Aj) for a series of reactions of the type (4.28) and (4.29) involving charged ligands X has been very helpful in delineating the type of / mechanism. 

For these reactions the general LFER holds log/ , = fllogX, -t b 

Since we shall not obtain the comparable amount of detailed information on the mechanisms of substitution in octahedral complexes from the studies of more complicated substitutions involving chelation and macrocycle complex formation (Secs. 4.4 and 4.5) it is worthwhile summarizing the salient features of substitution in Werner-type complexes. 

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The electron transfer reaction from copper to heme within the ternary protein complex was also studied in solution by stopped-flow spectroscopy. Analysis by Marcus theory of the temperature dependence of the limiting first-order rate constant for the redox reaction (Davidson and Jones, 1996) yielded values for the of 1.1 eV and H b of 0.3 cm , and predicted an electron transfer distance between redox centers which was consistent with the distance seen in the crystal structure. Thus, the electron transfer event is rate-limiting for this redox reaction. Experiments are in progress to determine the validity of the predicted pathways for electron transfer shown in Figure 7. 

Data obtained under these conditions can be fitted using a least-squares procedure based upon the exact solution to the differential equations describing this mechanism [37, 44]. This yields values for the complex dissociation constant Ky, and the limiting first-order rate constant (a minimum value for the second-order rate constant for complex formation can also be obtained from this analysis). Note that AId refers to the interaction between reduced P and oxidized P, a situation that is observable only by kinetic methodology. 

A more detailed kinetic investigation of the Au/Bipy/cytochrome c system was carried out using the rotating ring-disk technique (12). It was found that rate constants for adsorption and desorption of the protein were 3 x cm sec" and 50 sec", respectively. The limiting first-order rate constant within the protein-electrode complex was determined as 50 sec", a reasonable value as compared to that of long-range electron transfer between or within proteins. 

In the absence of acid the Via Vila reaetions are very slow, with observed rate constants two or three orders of magnitude smaller than those obtained in the presence of pro tic acid at the same temperature. All the compounds studied to date have metalated triphenylphosphines, and the equatorial ligands were P(XC6H4)3, (X = H, /7-CH3, p-C, W-CH3, m-Cl). The reaction is first order in the eoncentration of the monometalated species Via. For the acid-catalyzed experiments, the experimental data were fitted to a limiting rate equation of type obs = [H ]/(Xe -I- [H+]) where k is the limiting first-order rate constant and is the equilibrium constant for the aeid-base equilibria before the rate-limiting step. 

Competition experiments again feature prominently in another discussion of the possible role of transient five-coordinate [Co(NH3)5] in induced and in spontaneous aquation of pentaaminecobalt(III) derivatives. " The operation or nonoperation of the D mechanism at various cobalt(III) centers and at penta-cyanoferrate(II) still requires a few experiments providing unambiguous results. Its operation at molybdenum(O)- and tungsten(0)-penta or tetracarbonyl complexes seems more firmly based. The question of its operation at pentacyanoferrates(III) does not seem to have caused much concern. The only recent paper which mentions kinetics of such a reaction, replacement of 2-methyl imidazolate in [Fe "(CN)5(2-Meimid)] ", reports that the limiting first-order rate constant is 2.3 x 10 s at 298 K, but is more preoccupied with redox catalysis by traces of iron(II) than with simple substitution. 

Six-co-ordinate low-spin iron(u) phthalocyanine complexes of the type L2pePc (L = imidazole, pyridine, piperidine, or 2-methylimidazole) reversibly bind carbon monoxide in toluene solution. The reaction has a dissociative mechanism and it is found that the limiting first-order rate constants for the dissociation of L parallel the equilibrium constants for reaction (8), viz. 2-MeIm > pip > py>Im. A substantial... 

The overall study showed that the rate of reaction has a first order dependence on both hydrogen concentration and total [Ir] (up to certain limiting hydrogen and catalyst concentrations) and an inverse dependence on the nitrile concentration. The observed kinetic dependence of the pseudo first order rate constant (k ) for the hydrogenation of C=C in NBR may be summarized by the expression show in Equation (1). 

Experimental. In order to study the nucleophilic properties of 13 it was necessary to add excess I " to the solutions to prevent precipitation of I2. The rate of formation of CoCCN I-3 was followed spectrophotometrically after the I3 " in aliquots of the solution taken at suitable time intervals was reduced to I by arsenite ion. A typical set of experiments was carried out at 40°C. and unit ionic strength, with all solutions containing 0.5/1/ 1 and variable I3 " at a maximum concentration of 0.28M, the approximate upper limit imposed by solubility restrictions. The results are presented in Figure 3 as a plot of k the symbol used for the pseudo first-order rate constant for this system, vs. l/(lf). It is apparent that 13 is a remarkably efficient nucleophile, with a reaction rate considerably greater than that found for I at comparable concentrations. The points in Figure 3 also show detectable deviation from linearity, despite the limited range of 13 " concentration which was available. 

As long as the SSA is valid for the mechanism in Equation 4.7, but regardless of whether it either involves a pre-equilibrium or proceeds via an initial rate-limiting step (or neither), the same prediction is obtained - a first-order rate law will be observed. However, the correspondence between the measured first-order rate constant, k0 iil and mechanistic rate constants is different, and additional evidence is required to distinguish between the alternatives. 

At high pressures the observed first order rate constant is strictly independent of pressure, but if experiments are carried out at low or intermediate pressures then the first order rate constant depends on pressure, and the reaction moves from strict first order kinetics at high pressures to second order at low pressures. At pressures intermediate between these two limits, the reaction shows complex kinetics with no simple order. This requires explanation, see below and Problem 4.17. 

Several oxidants of widely different reduction potentials and kinetic properties react with a superoxorhodium complex L2(H20)Rh002 + in acidic aqueous solutions with essentially the same first-order rate constant provided the concentration of the oxidant is higher than a certain limit.41 Below this minimum concentration, which is different for different oxidants, the rate constant decreases with a decrease in [oxidant]. 

This is precisely the behaviour predicted by the Kira mechanism, provided that the formation of the silene-ROH complex is reversible and the proton transfer steps are rate-limiting. The complete mechanism is shown in Scheme 4, while equation 27 gives the predicted expression for the pseudo-first-order rate constant for decay of the silene, derived assuming the steady-state approximation for the silene-alcohol complex. Equation 27 reduces to the quadratic expression in [ROH] of equation 28 when k c (A h + A h [ROII ), i.e. under the conditions of the equilibrium assumption for the complex. In practice, it is difficult to distinguish between the two situations given by equations 27 and 28. The experimentally determined second- and third-order rate constants roh and k2ROH are defined in equations 29 and 30, respectively, in terms of the mechanism of Scheme 4 and using the... 

Figure 3. Dependence of the observed first-order rate constant of Cun(H 2GGhis) on the trien concentration. Cu11 = 2 X 10 4M, pH 6.9, 1M NaClOJt, 25.0°. At high [trien]T the rate constant is proton transfer rate limited.

In Fig. 11, at high concentrations of ethylene carbonate, the rate constants ks[EC] and kR[EC] for insertion into the EBTHI zirconaaziridine 17q are much greater than kSSR and ksss and insertion occurs more rapidly than the equilibrium can be maintained. The product ratio reflects the equilibrium of 17q, where Keq is 17.2 (Eq. 32) [21]. Beak has called this limit a dynamic thermodynamic resolution pathway [66]. In contrast, at the lowest concentration of ethylene carbonate in Fig. 10, the first-order rate constants kSSR and ksss for diastereomer interconversion are comparable to the effective first-order rate constants for insertion. As Keq is known to be 17.2, ks/kR can be calculated the 53% ee of (S)-amino acid ester 19q (Scheme 9) implies that kslkR<0.19 (Eq. 33) and that the rate constant for insertion kR[EC] into the minor diastereomer is at least five times faster than ks[EC] into the major diastereomer. 

In pure ethane the initial first-order rate constants fall off with decreasing pressure. Sacchse, loc, city found that at about 38 mm Hg the rates were about half their high-pressure limiting values. This is of course quite reasonable in terms of the proposed mechanism. 

Thus the ratio of the high-pressure-limiting first-order rate constants for the systems NO + N2O6 [Eq. (XIII.19.16)] to pure N2O6 [Eq. (XIII.19.14)] should be 1 + d/2fc3. From the data of Mills and Johnston at high pressures... 

Here the reaction shows mass-law inhibition by one of its products X, while the concentration of carbonium ion R+ is at its equilibrium value. One would expect such limiting behavior only in very exceptional cases, since it would imply reaction in a reasonably strongly ionizing solvent at low water content. Very few cases have been observed to fit this limiting behavior, but one example is provided by the hydrolysis of triphenyl methyl chloride in 85 per cent aqueous acetone. The apparent first-order rate constant for hydrolysis is decreased fourfold by 0.01 M NaCl and remains unchanged on the addition of other salts, such as NaC104, not having the common Cl ion. "... 

For redox reactions involving proteins, the actual meaning of the kinetically-determined limiting first-order rate constant for the electron transfer reaction (k3 in Eq. 6) must be... 

Rate constants for interfacial reactions have mainly been determined from experiments using particle suspensions where the concentration of reactive solute is monitored as a function of time. In these experiments, the solid surface is in large excess and the consumption of reactive solute follows first order kinetics. By plotting the pseudo-first-order rate constant against the solid surface area to solution volume ratio, the second-order rate constant can be obtained (from the slope). The main limitation here is that only relatively stable solutes can be studied experimentally. It is not possible to study the reactivity of short-lived species such as radicals using this approach. 

Given that if is often not large, a limiting rate (as expected for Eq. (12)) is often not seen, and Eq. (14) operates. Separation of ku and K is then not accessible from the kinetics. Rarely, the pseudo first-order rate constants have been reported to be a relatively complicated function of [M ]. This is the case for [Co(NH3)5l] + and [RWNHslsI] with Ag+ (32, 174), in which the complications are reported to result from the existence of species such as [((NH3)5MI)2Ag] + in addition to [(NHslsMIAg] . In the presence of excess chloride ion, Hg -promoted aquation can be complicated by separate pathways involving Hg , ... 

The specific rate constants of interest to the ECD and NIMS are dissociative and nondissociative electron attachment, electron detachment, unimolecular anion dissociation, and electron and ion recombination. The reactions that have been studied most frequently are electron attachment and electron and ion recombination. To measure recombination coefficients, the electron concentration is measured as a function of time. The values are dependent on the nature of the positive and negative ions and most important on the total pressure in the system. Thus far few experiments have been carried out under the conditions of the NIMS and ECD. However, the values obtained under other conditions suggest that there is a limit to the bimolecular rate constant, just as there is a limit to the value of the rate constant for electron attachment. The bimolecular rate constants for recombination are generally large, on the order of 10 7 to 10-6 cc/molecule-s or 1014 to 1015 1/mole-s at about 1 atm pressure. Since the pseudo-first-order rate constants are approximately 100 to 1,000 s 1, the positive-ion concentrations in the ECD and NIMS are about 109 ions/cc. 

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