Diffusion-controlled second-order

Radicals are normally very reactive, often reacting under diffusion control (second-order rate constant ca. 109 dm3 mol-1 s 1) with other species (radicals, neutral molecules or ions), Scheme 10.1. However, although radical-radical reactions normally have very high second-order rate constants, radical concentrations are generally very low. Therefore, radical-neutral molecule or radical-ion reactions with smaller rate constants take place preferentially because the concentrations of neutral molecules or ions can be very high. Radical... 

In 1989, Amyes and Jencks reported a study of azide common ion inhibition for solvolysis of a series of ot-azido ethers (Fig. I).23 The rate constants A Hoh for addition of water to the oxocarbenium ions were determined with an assumed diffusion controlled second-order rate constant /caz of 5.0 x 109 M-1 s 1 for attack on the oxocarbenium ion by azide and the observed rate suppression by varying concentrations of added azide, as determined from Equation (1). With Hoh in hand, the lifetime of the oxocarbenium ion was taken as hoh... 

On the other hand, NO reacts with O2 at nearly diffusion-controlled second-order rate to produce peroxynitrite (Eq. (13)) (37). 

If two reactants A and B with radii and r, respectively, diffuse together and react at an interaction distance (r + r ), then theories developed from Brownian motion predict that the diffusion-controlled second-order rate constant is given by... 

The rate of a gas phase reaction, 2A =s> B, is believed controlled by external diffusion and second order surface reaction with only substance A adsorbed to a substantial extent. The rate of diffusion is rd - 0.9(pg-ps), mol/(h)(kg catalyst)... 

One of the most important characteristics of micelles is their ability to take up all kinds of substances. Binding of these compounds to micelles is generally driven by hydrophobic and electrostatic interactions. The dynamics of solubilisation into micelles are similar to those observed for entrance and exit of individual surfactant molecules. Their uptake into micelles is close to diffusion controlled, whereas the residence time depends on the sttucture of the molecule and the solubilisate, and is usually in the order of 10 to 10" seconds . Hence, these processes are fast on the NMR time scale. 

Recovery of dilute acetic acid is achieved by esterification with methanol using a sulfonated resin (Dowex 50w) in a packed distillation column (54). Pure methyl acetate is obtained. This reaction is second order in acetic acid, 2ero order in methanol, and partially diffusion controlled. 

The reaction kinetics approximation is mechanistically correct for systems where the reaction step at pore surfaces or other fluid-solid interfaces is controlling. This may occur in the case of chemisorption on porous catalysts and in affinity adsorbents that involve veiy slow binding steps. In these cases, the mass-transfer parameter k is replaced by a second-order reaction rate constant k. The driving force is written for a constant separation fac tor isotherm (column 4 in Table 16-12). When diffusion steps control the process, it is still possible to describe the system hy its apparent second-order kinetic behavior, since it usually provides a good approximation to a more complex exact form for single transition systems (see Fixed Bed Transitions ). 

In these circumstances a decision must be made which of two (or more) kinet-ically equivalent rate terms should be included in the rate equation and the kinetic scheme (It will seldom be justified to include both terms, certainly not on kinetic grounds.) A useful procedure is to evaluate the rate constant using both of the kinetically equivalent forms. Now if one of these constants (for a second-order reaction) is greater than about 10 ° M s-, the corresponding rate term can be rejected. This criterion is based on the theoretical estimate of a diffusion-controlled reaction rate (this is described in Chapter 4). It is not physically reasonable that a chemical rate constant can be larger than the diffusion rate limit. 

Schmolukowski in 1917, a diffusion-controlled bimolecular reaction in solution at 25 °C can reach a value for th second-order rate constant k as high as 7 x 109 m 1s-1. Nitrosations of secondary aliphatic amines also have rates which are relatively close to diffusion control (see Zollinger, 1995, Sec. 4.1). 

Fig. 3. Reduced time plots, tr = (t/t0.9), for the contracting area and contracting volume equations [eqn. (7), n = 2 and 3], diffusion-controlled reactions proceedings in one [eqn. (10)], two [eqn. (13)] and three [eqn. (14)] dimensions, the Ginstling— Brounshtein equation [eqn. (11)] and first-, second- and third-order reactions [eqns. (15)—(17)]. Diffusion control is shown as a full line, interface advance as a broken line and reaction orders are dotted. Rate processes become more strongly deceleratory as the number of dimensions in which interface advance occurs is increased. The numbers on the curves indicate the equation numbers.

Influence" of ionic charges on second-order diffusion-controlled rate constants in aqueous solution at 25 °C... 

Time-resolved optical absorption spectroscopy experiments have shown that arenesul-fonyl radicals decay with clean second-order kinetics14 the values of 2 k,/a h where s2 is the extinction coefficient at the monitoring wavelength, increased linearly with decreasing viscosity of the solvent, further indicating that reaction 16 is clearly a diffusion-controlled process. 

Several assumptions were made in order to analyze kinetic data in terms of this expression (2). First it was assumed that k 2 m kj, k2 k 3, and kj/k j k /k ( - If). Second it was assumed that the rate constants were independent of the extent of reaction i.e., that all six functional groups were equally reactive and that the reaction was not diffusion controlled. The concentration of polymer hydroxyl functionality was determined experimentally using infrared spectroscopy as described elsewhere (7). A major unknown is the instantaneous concentration of methanol. Fits to the kinetic data were made with a variety of assumptions concerning the methanol concentration. The best fit was achieved by assuming that the concentration of methanol was initally constant but decreased at a rate proportional to the concentration of residual polymer hydroxy groups towards the end of the reaction. As... 

Matheson and Rabani (1965) measured the rate of the reaction eh + eh— H2 + 20H at pH 13 under 100 atmospheres H2 pressure, where all radicals are converted to eh. From a pure second-order decay, the rate constant was determined as 6 x 109 M 1s 1. There are contradictory views on this reaction. According to some, this rate is too low for a diffusion-controlled reaction between like charges, by a factor of -4 (see Farhataziz, 1976). This factor of 4 can be accounted for by spin considerations, since each electron is a doublet but the end product H2is a singlet. To be consistent, then, one has to consider the rate of reaction eh + O—-O2- as normal for diffusion control. 

The cyclohexadienyl radicals decay by second-order kinetics, as proven by the absorption decay, with almost diffusion-controlled rate (2k = 2.8 x 109 M 1 s 1). The cyclohexyl radicals 3 and 4 decay both in pseudo-first-order bimolecular reaction with the 1,4-cyclohexadiene to give the cyclohexadienyl radical 5 and cyclohexene (or its hydroxy derivative) (equation 15) and in a second order bimolecular reaction of two radicals. The cyclohexene (or its hydroxy derivative) can be formed also in a reaction of radical 3 or... 

Evaluation of kinetic data. Rate constants were determined for 2-H exchange from 3-R-4-methylthiazolium ions, catalyzed by D2O (pseudo first order) and DO- (second order).154 The observed rate constants for the pD-independent exchange reaction were corrected for the solvent isotope effect ( h2o/ d2o = 2.6), and the reverse protonation of the carbene by H30+ was assumed to be diffusion-controlled (k = 2 x 1010 M-1 s-1). A similar analysis was performed for the exchange catalysed by DO-. The results agreed nicely, giving pAfa = 18.9 for 213 and p/sfa = 18.0 for thiamine.154 The thiazolium ion 213 seems to be less acidic in water154 than in DMSO152 (Ap/fa = 2.4). Aside from the... 

Many transition metal complexes have been considered as synzymes for superoxide anion dismutation and activity as SOD mimics. The stability and toxicity of any metal complex intended for pharmaceutical application is of paramount concern, and the complex must also be determined to be truly catalytic for superoxide ion dismutation. Because the catalytic activity of SOD1, for instance, is essentially diffusion-controlled with rates of 2 x 1 () M 1 s 1, fast analytic techniques must be used to directly measure the decay of superoxide anion in testing complexes as SOD mimics. One needs to distinguish between the uncatalyzed stoichiometric decay of the superoxide anion (second-order kinetic behavior) and true catalytic SOD dismutation (first-order behavior with [O ] [synzyme] and many turnovers of SOD mimic catalytic behavior). Indirect detection methods such as those in which a steady-state concentration of superoxide anion is generated from a xanthine/xanthine oxidase system will not measure catalytic synzyme behavior but instead will evaluate the potential SOD mimic as a stoichiometric superoxide scavenger. Two methodologies, stopped-flow kinetic analysis and pulse radiolysis, are fast methods that will measure SOD mimic catalytic behavior. These methods are briefly described in reference 11 and in Section 3.7.2 of Chapter 3. 

An alternative electrochemical method has recently been used to obtain the standard potentials of a series of 31 PhO /PhO- redox couples (13). This method uses conventional cyclic voltammetry, and it is based on the CV s obtained on alkaline solutions of the phenols. The observed CV s are completely irreversible and simply show a wave corresponding to the one-electron oxidation of PhO-. The irreversibility is due to the rapid homogeneous decay of the PhO radicals produced, such that no reverse wave can be detected. It is well known that PhO radicals decay with second-order kinetics and rate constants close to the diffusion-controlled limit. If the mechanism of the electrochemical oxidation of PhO- consists of diffusion-limited transfer of the electron from PhO- to the electrode and the second-order decay of the PhO radicals, the following equation describes the scan-rate dependence of the peak potential ... 

The rate of a batch slurry reaction is controlled by diffusion from the bulk liquid to the surface of the catalyst and by a second order reaction on the surface. Equations for the two processes are rd = 0.25(C-Cs)... 

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