Second-Order Rate Processes

In some texts, the [A] = [B] case is the only one shown for second-order rate processes because it is easy to derive, but actually it is not very general and applies only to cases of reagents which dimerize or to cases where the concentrations have been carefully prepared so that in fact [A] = [B] by initial preparation. We wUl see below that the general case of [A] [B] is not very difficult and much more general. Nevertheless, let us derive this case and see how we could treat the data graphically. 

This is the more general case of a second-order rate process where the concentrations of the two species are not the same. In some texts, this is avoided because of the problem with integrating the rate expression but we show here a special theorem [4] that permits solution of this problem 

We hope this littie proof has not made the process more mysterious than it is in an actual 

using the theorem above we have shown that - 

This is not the most general form because the coefficients of A and B are both 1 but we are ready for an example. Students could look up such integrals in tables or use a computer program to solve such problems but here we offer a way to use a simple calculus trick which is general to many other cases 

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There have been comparatively few kinetic studies of the decompositions of solid malonates [1103]. The sodium and potassium salts apparently melt and non-isothermal measurements indicate second-order rate processes with high values of E (962 125 and 385 84 kJ mole-1, respectively). The reaction of barium malonate apparently did not involve melting and, from the third-order behaviour, E = 481 125 kJ mole-1. 

Solid KC104 oxidizes lauryl aldehyde to lauric acid by a second-order rate process [eqn. (16)] under the influence of ultrasonic irradiation... 

The chemical stability of Guthion phosphorodithioate was investigated in two solvent systems N-methyl-2-pyrrolidone and butyrolactone. Epiclilorohydrin was found to alter the rate of decomposition of the two systems. Kinetic treatment of the data revealed a second-order rate process. Rate constants and activation energies were calculated for each system. 

In the case of the Consolidation coal, a few data were obtained at different temperatures. Figure 5 shows the result of plotting the data as a second order rate process at temperatures... 

The value of AH, 15.2 Kcal/mole, is about half that which Wiser (14) obtained (28.8 Kcal/mole) for dissolution of coal by hydrogen donor solvents in the 350-450°C temperature regime and by the second order rate process. On the other hand, it is about twice the value obtained by Hill, et (15) for low temperature dissolution of coal in a hydrogen donor solvent under the driving force of ultrasonic energy, i.e., 8.7 Kcal/mole. 

The values of pseudo-second-order equation parameters together with correlation coefficients are listed in Table 1. The correlation coefficients for the pseudo-second-order equation were 0.999. The calculated q values also agree very well with the experimental data. This strongly suggests that the biosorption of Cu + onto aminated ephedra waste is most appropriately represented by a pseudo-second-order rate process and the biosorption rate is controlled by chemical biosorption. 

Determination of the cross section for a nuclear reaction product requires measurement of the reaction rate, R = number of events/unit time. As in any two-body collision (second-order rate process), the rate is the product of a target-projectile collision factor cn n and the probability (7 that if a collision occurs, a specific product will be formed, i.e.,... 

Since isoprene was the principal desired product from methyl pentene pyrolysis it was essential to determine its stability under reaction conditions. Isoprene degradation proceeds via second-order rate processes. Isoprene dimer and related compounds formed during the pyrolysis are dependent on the isoprene partial pressure in the reaction system. 

A computer-controlled modulated molecular beam source is used to investigate the kinedcs of the surface reactions which occur when bromine is reactively scattered by Pd(lll). The reaction products are atomic bromine and molecular bromine the latter species arises from an adatom recombination process and gives rise to a product vector modulated at twice the frequency of the incident beam (2u.) By making suitable measurements of the temperature dependence of the product vector phase shifts at w and 2u, the four kinetic paranwteis which characterise the first-order and second-order rate processes are obtained. These are A, =2.5X 10 s , = 177 kJ moP, = 3.6X 10 m s , 13 3 mol. The significance of these values is discussed... 

The average of these two values is 0.587 L/mol min and once again there perhaps should be more confidence in the value from the longer time period of 52 min but the two values are close enough to support the assignment of a second-order rate process. 

For a second-order rate process (such as coagulation), the half-life ti/2 (i.e. the time taken for the initial number of particles. No, to be reduced by 2) is given by, ti/2 = 1/icNo. 

For most color photographic systems, development is the rate determining step, and within that step the formation of semiquinone is the slow process (37). The fate of the highly reactive QDI is deterrnined by the relative rates of a number of competing processes (38). The desired outcome is reaction with ionized coupler to produce dye (eq. 3). Typically, the second-order rate constant for this process with ionized coupler is about 10 to 10 ... 

For two-equivalent couplers where the conversion of the leuco dye to image dye is rapid, the experimentally observed second-order rate constant, k, can be equated with kj, the rate of nucleophilic attack of coupler anion on oxidized developer. Thus when the pH of the process is specified, two parameters, piC and k, can be convenientiy used to characterize the molecular reactivity of a large variety of photographically weU-behaved couplers (40,54). 

We can reach two useful conclusions from the forms of these equations First, the plots of these integrated equations can be made with data on concentration ratios rather than absolute concentrations second, a first-order (or pseudo-first-order) rate constant can be evaluated without knowing any absolute concentration, whereas zero-order and second-order rate constants require for their evaluation knowledge of an absolute concentration at some point in the data treatment process. This second conclusion is obviously related to the units of the rate constants of the several orders. 

What are the second-order rate constants for these processes ... 

The inactivation is normally a first-order process, provided that the inhibitor is in large excess over the enzyme and is not depleted by spontaneous or enzyme-catalyzed side-reactions. The observed rate-constant for loss of activity in the presence of inhibitor at concentration [I] follows Michaelis-Menten kinetics and is given by kj(obs) = ki(max) [I]/(Ki + [1]), where Kj is the dissociation constant of an initially formed, non-covalent, enzyme-inhibitor complex which is converted into the covalent reaction product with the rate constant kj(max). For rapidly reacting inhibitors, it may not be possible to work at inhibitor concentrations near Kj. In this case, only the second-order rate-constant kj(max)/Kj can be obtained from the experiment. Evidence for a reaction of the inhibitor at the active site can be obtained from protection experiments with substrate [S] or a reversible, competitive inhibitor [I(rev)]. In the presence of these compounds, the inactivation rate Kj(obs) should be diminished by an increase of Kj by the factor (1 + [S]/K, ) or (1 + [I(rev)]/I (rev)). From the dependence of kj(obs) on the inhibitor concentration [I] in the presence of a protecting agent, it may sometimes be possible to determine Kj for inhibitors that react too rapidly in the accessible range of concentration. ... 

Greater success in extending kinetic measurements to higher degrees of polymerization has been achieved with polyesterifications catalyzed by a small amount of a strong acid catalyst. The catalyst concentration being constant throughout the process, the second-order rate expression... 

Since an elementary reaction occurs on a molecular level exactly as it is written, its rate expression can be determined by inspection. A unimolecular reaction is first-order process, bimolecular reactions are second-order, and termolecular processes are third-order. However, the converse statement is not true. Second-order rate expressions are not necessarily the result of an elementary bimolecular reaction. While a... 

If the EDA and CT pre-equilibria are fast relative to such a (follow-up) process, the overall second-order rate constant is k2 = eda c e In this kinetic situation, the ion-radical pair might not be experimentally observed in a thermally activated adiabatic process. However, photochemical (laser) activation via the deliberate irradiation of the charge-transfer absorption (hvct) will lead to the spontaneous generation of the ion-radical pair (equations 4, 5) that is experimentally observable if the time-resolution of the laser pulse exceeds that of the follow-up processes (kf and /tBet)- Indeed, charge-transfer activation provides the basis for the experimental demonstration of the viability of the electron-transfer paradigm in Scheme l.21... 

For the series of -branched alkyl radicals, the second-order rate constant in eq 3 is relatively unaffected by steric effects [compare Figure 2 (right)] as expected for an outer-sphere process. In strong contrast, the rate constant kL for ligand substitution in eq 21 is adversely affected by increasing steric effects, as shown in Figure 17. 

Ifcobs is directly proportional to pyridine concentration. Therefore a plot of kobs vs. [pyridine] is linear, with a slope (k ) equal to the second order rate constant for ylide formation, and an intercept (k0) equal to the sum of all processes that destroy the carbene in the absence of pyridine (e.g.) intramolecular reactions, carbene dimerization, reactions with solvent, and, in the case of diazirine or diazo carbene precursors, azine formation. 

We conclude that the reaction is second-order, by the process of elimination. We confirm this conclusion by computing values of the second-order rate constant with the equation... 

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