Diffusion-controlled bimolecular rate constant

By comparing time-resolved and steady-state fluorescence parameters, Ross et alm> have shown that in oxytocin, a lactation and uterine contraction hormone in mammals, the internal disulfide bridge quenches the fluorescence of the single tyrosine by a static mechanism. The quenching complex was attributed to an interaction between one C — tyrosine rotamer and the disulfide bond. Swadesh et al.(()<>> have studied the dithiothreitol quenching of the six tyrosine residues in ribonuclease A. They carefully examined the steady-state criteria that are useful for distinguishing pure static from pure dynamic quenching by consideration of the Smoluchowski equation(70) for the diffusion-controlled bimolecular rate constant k0,... 

An important use of Brownian dynamics is for computation of the diffusionTControlled bimolecular rate constant. The diffusion equations can be related to the reaction rate via the flux at the r ctive surface. Smoluchowski showed that, for two spherical reactants with no interparticle forces, the analytical result for the diffusion-controlled bimolecular rate constant k is... 

Bimolecular association rate constant Rate constant for dissociation Rate constant for Dexter energy transfer Rate constant for diffusion-controlled reactions Rate constant for fluorescence... 

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). 

A minor component, if truly minute, can be discounted as the reactive form. To continue with this example, were KCrQ very, very small, then the bimolecular rate constant would need to be impossibly large to compensate. The maximum rate constant of a bimolecular reaction is limited by the encounter frequency of the solutes. In water at 298 K, the limit is 1010 L mol-1 s"1, the diffusion-controlled limit. This value is derived in Section 9.2. For our immediate purposes, we note that one can discount any proposed bimolecular step with a rate constant that would exceed the diffusion-controlled limit. 

The rate of MV formation was also dependent on pH. The bimolecular rate constant, as calculated from the first order rate constant of the MV build-up and the concentration of colloidal particles, was substantially smaller than expected for a diffusion controlled reaction Eq. (10). The electrochemical rate constant k Eq. (9) which largely determines the rate of reaction was calculated using a diffusion coefficient of of 10 cm s A plot of log k vs. pH is shown in Fig. 24. 

Mn2(CO)9 reacted with CO at a rate well below the diffusion-controlled limit (77), and the bimolecular rate constant was solvent dependent [It =... 

Photosensitization of diaryliodonium salts by anthracene occurs by a photoredox reaction in which an electron is transferred from an excited singlet or triplet state of the anthracene to the diaryliodonium initiator.13"15,17 The lifetimes of the anthracene singlet and triplet states are on the order of nanoseconds and microseconds respectively, and the bimolecular electron transfer reactions between the anthracene and the initiator are limited by the rate of diffusion of reactants, which in turn depends upon the system viscosity. In this contribution, we have studied the effects of viscosity on the rate of the photosensitization reaction of diaryliodonium salts by anthracene. Using steady-state fluorescence spectroscopy, we have characterized the photosensitization rate in propanol/glycerol solutions of varying viscosities. The results were analyzed using numerical solutions of the photophysical kinetic equations in conjunction with the mathematical relationships provided by the Smoluchowski16 theory for the rate constants of the diffusion-controlled bimolecular reactions. 

The experimental and simulation results presented here indicate that the system viscosity has an important effect on the overall rate of the photosensitization of diary liodonium salts by anthracene. These studies reveal that as the viscosity of the solvent is increased from 1 to 1000 cP, the overall rate of the photosensitization reaction decreases by an order of magnitude. This decrease in reaction rate is qualitatively explained using the Smoluchowski-Stokes-Einstein model for the rate constants of the bimolecular, diffusion-controlled elementary reactions in the numerical solution of the kinetic photophysical equations. A more quantitative fit between the experimental data and the simulation results was obtained by scaling the bimolecular rate constants by rj"07 rather than the rf1 as suggested by the Smoluchowski-Stokes-Einstein analysis. These simulation results provide a semi-empirical correlation which may be used to estimate the effective photosensitization rate constant for viscosities ranging from 1 to 1000 cP. 

Pick s laws describe the interactions or encounters between noninteracting particles experiencing random, Brownian motion. Collisions in solution are diffusion-controlled. As is discussed in most physical chemistry texts , by applying Pick s Pirst Law and the Einstein diffusion relation, the upper limit of the bimolecular rate constant k would be equal to... 

M -sec k The corresponding upper limit value for a bimolecular rate constant in the gas phase is about 10 M -sec k Thus in solutions, bimolecular rate constants cannot exceed 10 -10 M -sec since diffusion control takes over from collision control. 

A reaction whose rate is limited (or controlled) only by the speed with which reactants diffuse to each other. For a ligand binding to a protein, the bimolecular rate constant for diffusion-limited association is around 10 M s. The enzyme acetylcholinesterase has an apparent on-rate constant of 1.6 x 10 M s with its natural cationic substrate acetylcholine, and the on-rate constant of about 6 X 10 with acetylselenoylcholine and about... 

Thus, bimolecular rate constant depends only on the viscosity and the temperature of the solvent. The calculated rate constants for diffusion-controlled bimolecular reactions in solution set the upper limit for such reactions. 

The maximum value for the bimolecular rate constant occurs when the activation energy act is zero and the steric factor is 1. The rate is then said to be diffusion-controlled, and it is equal to the encounter frequency of the molecules. Assuming that the reacting molecules are uncharged spheres of radius rA and rB, the encounter frequency may be calculated as... 

Favorable proton transfers between electronegative atoms such as O, N, and S are extremely fast. The bimolecular rate constants are generally diffusion-controlled, being 1010 to 10" s-1 A/-1 (Table 4.2). For example, the rate constant for the transfer of a proton from H30+ to imidazole, a favorable transfer since imidazole is a stronger base than H20, is 1.5 X 1010 s 1 M l (Table 4.3). The rate constant for the reverse reaction, the transfer of a proton from the imidazolium ion to water, may be calculated from the difference in their p a s by using the following equations ... 

According to the Smoluchowski theory of diffusion-controlled bimolecular reactions in solutions, this constant decreases with time from its kinetic value, k0 to a stationary (Markovian) value, which is kD under diffusional control. In the contact approximation it is given by Eq. (3.21), but for remote annihilation with the rate Wrr(r) its behavior is qualitatively the same as far as k(t) is defined by Eq. (3.34)... 

Broadly speaking, dynamic electron transfer involves two steps. The first step is the diffusion controlled formation of an encounter complex between the electron/hole acceptor molecule and the particle. The second step is the electrochemical interfacial charge-transfer, which may be characterized by a rate constant kct. Albery et al. [143] and Gratzel et al. [129] have independently arrived at the same result, relating the observed effective bimolecular rate constant for hole/electron acceptor oxidation/re-duction, /cobs (m3mor s I) to reactant diffusion and A ct. 

A.2.2. Diffusion-Controlled Rate Constant Recently, we have calculated the diffusion-controlled (i.e., attainable maximum) rate constant of ET at an OAV interface [49]. Figure 8.8 shows models for diffusion-controlled bimolecular reactions (a) in homogeneous solution and (b) at an O/W interface. 

The different pathways are reflected by the magnitudes of the bimolecular rate constants at 25 °C L = MeCN, 2 7.6 X 10 dm mofr s L = P(OMe)s, k2 8.9 X 10 dm mol s. The laPer value is only about one order of magnitude slower than the diffusion controlled limit. These studies show the value of time-resolved IR spectroscopy for obtaining kinetic data. 

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