Bimolecular reaction rate

In terms of the collision theory a bimolecular reaction rate is written as... 

The observant reader will notice that the rate of conversion of A is initially the same as the first-order process shown. However, as A and B decrease, the bimolecular reaction rate slows appreciably, and the decrease in A concentration tails off more slowly than the first-order process. As illustrated in the inset, however, when [Bo] = 10 X [Aq], the second-order and first-order reactions are virtually indistinguishable. In fact, plots of ln[Ao] versus time look virtually linear (/.c., pseudo-first-order) when [Bo] 3[Aq]. 

Bimolecular Reaction Rate Constant for the End Functional Groups During Network Formation... 

Before discussing these points in detail, it is worthwhile to consider how the diffusion equation for relative motion of two species is developed from a reduction of the diffusion equation describing the motion of both species separately. It introduces some of the complexities to the many-body problem and, at the same time, shows an interesting parallel to the theory of bimolecular reaction rates in the gas phase [475]. 

In summary, the QRRK theory result for the observed bimolecular reaction rate constant fcbimol was given by Eq. 10.198 as... 

From the nature of the relation between k and E, it is now permissible to take an average of k for the whole reaction and to investigate the relation between this and the constant value of E. Substitution of such an average value of k in the equation for bimolecular reaction rates leads to a value of E equal to 22,000 calories. This agrees well with the value 21,000 found from the temperature coefficient. In this calculation the molecular diameter of the chlorine monoxide molecule is taken 4-8 x 10 8 cm. Rankine does not give values for this gas, but the value is inferred from the size of the molecules of gases of similar structure. 

Shoup, G. Lipari, and A. Szabo, Diffusion-controlled bimolecular reaction rates- the effect of rotational diffusion and orientation, Biophys. J. 36, 697-714 (1981). 

A bimolecular reaction rate is proportional to the frequency of collisions between the two molecules of the reacting species. It is known from kinetic theory that the frequency of collisions between two like molecules, A, is proportional to [A]2, and the frequency of collisions between an A and a B molecule is proportional to the product of the concentrations, [A] [B]. If the species whose molecules collide are starting materials in limited concentrations, the reaction is second-order. This reaction follows the rate equation of either type (5) or (2), Table 20-1. 

This is known as the Stern-Volmer equation, where ksv is the bimolecular reaction rate constant (in units of M-1 s-1). From an experimental viewpoint, the plot of k0bS as a function of [Q] is a straight line, of slope ksv the lifetime decreases as a function of [Q], The global luminescence intensity of M as a function of [Q] is equal to ... 

Oxidation rate constant k, for gas-phase second order rate constants, k0H for reaction with OH radical, kN03 with N03 radical and k03 with 03 or as indicated, data at other temperatures see reference k0H < 1.2 5 x 10 16 cm3 molecule-1 s-1 at 298 K (relative rate method, Cox et al. 1976) k0H = 1 x 10 16 cm3 s-1 bimolecular reaction rate, with a reported tropospheric lifetime of greater than 330 yr (Cox et al. 1976 quoted, Callahan et al. 1979)... 

A full six-dimensional PES for the HOCO system has been developed by Schatz and coworkers, particularly L. B. Harding (Kudla et al. 1992 Schatz et al. 1987). This proved to be challenging because of numerous local minima and transition states, and consequently the development of the PES has taken several years. Many points were calculated by using large scale ab initio techniques and the surface was adjusted to reconcile a broad array of experimental data such as nascent product excitations, HOCO decomposition rates, overall bimolecular reaction rates, barrier heights, enthalpy changes, HOCO structural properties, and inelastic scattering data. This PES has been used in several computational studies of the reaction dynamics that employ classical (Kudla and Schatz 1991 Kudla... 

In contrast, when An 0 one finds considerable disagreement in the literature concerning relative gas and liquid equilibrium (and rate) constants, even in the absence of solvation effects. Depending on the particular theoretical treatment, bimolecular reaction rate constants have been estimated to be somewhere between 2 and 100 times faster in solution than in the gas phase (6). However, the very limited experimental data available indicate that there is little, if any, systematic difference between bimolecular rate constants in the two phases (6d). 

The rate constants presented in Table 2 suggest that the N03 radical reactions with the gas-phase PCBs and PCDFs have upper limits to the rate constants for any reaction of fcabs < 10-15 cm3 molecule-1 s-1 (N02 concentrations in the troposphere are sufficiently low that the contribution of any N02-dependent reaction is encompassed within the upper limit to the bimolecular reaction rate constant fcabs). While dibenzo-p-dioxin and 1-chlorodibenzo-p-dioxin react with the N03 radical by the N02-dependent mechanism shown in Scheme 1 and equation (13), the N02 concentrations in the troposphere are sufficiently low that the effective bimolecular rate constants, fca(fcc[N02] + fcd[02])/fcb, are below the upper limits to the rate constants fcabs. 

In equation (1) K y is referred to as the Stern-Volmer constant Equation (1) applies when a quencher inhibits either a photochemical reaction or a photophysical process by a single reaction. <1>° and M° are the quantum yield and emission intensity (radiant exitance), respectively, in the absence of the quencher Q, while <1> and M are the same quantities in the presence of the different concentrations of Q. In the case of dynamic quenching the constant K y is the product of the true quenching constant kq and the excited state lifetime, t°, in the absence of quencher, kq is the bimolecular reaction rate constant for the elementary reaction of the excited state with the particular quencher Q. Equation (1) can therefore be replaced by the expression (2)... 

Solvent Viscosity Dependence of Bimolecular Reaction Rate Constant of the Excited 9-Cyanoanthracene Quenched by 1,3-Cyclohexadiene... 

For reactions of specified activation enthalpies, for example, 40 kJ moP, the bimolecular reaction rate depends on the activation entropy. Thus, a negative entropy of 60 J K moP slows the reaction 1000-fold, that is, lowers the preexponential Arrhenius factor y4 from 10 to lO M s". ... 

The bimolecular reaction rate for particles constrained on a planar surface has been studied using continuum diffusion theory " and lattice models. In this section it will be shown how two features which are not taken account of in those studies are incorporated in the encounter theory of this chapter. These are the influence of the potential K(R) and the inclusion of the dependence on mean free path. In most instances it is expected that surface corrugation and strong coupling of the reactants to the surface will give the diffusive limit for the steady-state rate. Nevertheless, as stressed above, the initial rate is the kinetic theory, or low-friction limit, and transient exp)eriments may probe this rate. It is noted that an adaptation of low-density gas-phase chemical kinetic theory for reactions on surfaces has been made. The theory of this section shows how this rate is related to the rate of diffusion theory. 

The effect of pressure and system non—idealities on bimolecular reaction rates has been incorporated by using fugacity coefficients and compressibility... 

Finally, it is shown that a process involving mobile ions can also contribute to hysteresis. After fast formation of complexes between a polaron and a counter ion, subsequent capture of a further polaron leads to complexes formed by a bipolaron and a counter ion. They can attract slowly moving ions. The Langevin bimolecular reaction rate for this process again yields time constants on the time scale of the measurements. 

A computer simulation approach has been derived that allows detailed bimolecular reaction rate constant calculations in the presence of these and other complicating factors. In this approach, diffusional trajectories of reactants are computed by a Brownian dynamics procedure the rate constant is then obtained by a formal branching anaylsis that corrects for the truncation of certain long trajectories. The calculations also provide mechanistic information, e.g., on the steering of reactants into favorable configurations by electrostatic fields. The application of this approach to simple models of enzyme-substrate systems is described. 

The frequency with which two reactive species encounter one another in solution represents an upper bound on the bimolecular reaction rate. When this encounter frequency is rate limiting, the reaction is said to be diffusion controlled. Diffusion controlled reactions play an important role in a number of areas of chemistry, including nucleation, polymer and colloid growth, ionic and free radical reactions, DNA recognition and binding, and enzyme catalysis. 

Note that r and the diffusion coefficient D have cancelled from Equation 2.29, because D is inversely proportional to the molecular radii r /2. Hence the rate constant kd depends only on temperature and solvent viscosity in this approximation. A selection of viscosities of common solvents and rate constants of diffusion as calculated by Equation 2.29 is given in Table 8.3. The effect of diffusion on bimolecular reaction rates is often studied by changing either the temperature or the solvent composition at a given temperature. For many solvents,54-56 although not for alcohols,57 the dependence of viscosity on temperature obeys an Arrhenius equation, that is, plots of log rj versus 1 IT are linear over a considerable range of temperatures and so are plots of log(kdr]/T) versus 1/T.56... 

R Zellner. Bimolecular Reaction Rate Coefficients. W C Gardiner Jr, Springer, New York, 1984. 

Fig. 33. Comparison of the predicted bimolecular reaction rate constant for CIO + CIO -> CI2 + O2 with the experimental values. Solid line is the predicted total value coupling Schemes 1 and 2 dotted line is the contribution from Scheme 2. Symbols are the experimental values as described in Ref 119.

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