Rate constant hard-sphere

This result is identical to the hard-sphere rate constant Eq. 10.82 derived from the simple collision theory introduced in Section 10.2. 

The collisional stabilization (de-activation) efficiency is assumed to be unity, which is consistent with the strong collision assumption. Thus the stabilization rate constant ks is equal to the hard-sphere rate constant kHS, and Eq. 10.120 becomes... 

The strong collision assumption is often invoked to equate ks with the hard-sphere rate constant kns This approximation assumes that every collision of C (n) with another molecule M will completely stabilize (deactivate) the excited molecule. It fact different collision partners are more or less effective in such deactivation. A collision efficiency fi is introduced to account for this effect ... 

This reaction rate constant expression bears some similarity to a hard-sphere rate constant however, an additional term q /AnsoRkBT) results from the long range attractive interaction, and is in general the dominant contribution to the reaction rate constant for reactions of this type. 

The simplest approach to computing the pre-exponential factor is to assume that molecules are hard spheres. It is also necessary to assume that a reaction will occur when two such spheres collide in order to obtain a rate constant k for the reactants B and C as follows ... 

To a first approximation, the activation energy can be obtained by subtracting the energies of the reactants and transition structure. The hard-sphere theory gives an intuitive description of reaction mechanisms however, the predicted rate constants are quite poor for many reactions. 

An intrinsic surface is built up between both phases in coexistence at a first-order phase transition. For the hard sphere crystal-melt interface [51] density, pressure and stress profiles were calculated, showing that the transition from crystal to fluid occurs over a narrow range of only two to three crystal layers. Crystal growth rate constants of a Lennard-Jones (100) surface [52] were calculated from the fluctuations of interfaces. There is evidence for bcc ordering at the surface of a critical fee nucleus [53]. 

Clusters larger than 25 all appear to react without further dramatic variations. The magnitude of the large cluster rate constants are a few percent of the hard sphere collision cross... 

The foundations of the theory of flocculation kinetics were laid down early in this century by von Smoluchowski (33). He considered the rate of (irreversible) flocculation of a system of hard-sphere particles, i.e. in the absence of other interactions. With dispersions containing polymers, as we have seen, one is frequently dealing with reversible flocculation this is a much more difficult situation to analyse theoretically. Cowell and Vincent (34) have recently proposed the following semi-empirical equation for the effective flocculation rate constant, kg, ... 

FIGURE 1.22. Solvent reorganization energies derived from the standard rate constants of the electrochemical reduction of aromatic hydrocarbons in DMF (with n-Bu4N+ as the cation of the supporting electrolyte) uncorrected from double-layer effects. Variation with the equivalent hard-sphere radii. Dotted line, Hush s prediction. Adapted from Figure 4 in reference 13, with permission from the American Chemical Society. 

Now, for hard sphere model the rate constant is given by... 

The above discussion shows that very good agreement of observed and calculated exchange rate constants can be obtained using the semiclassical formalism. In the bimolecular reactions discussed in this paper the reactants were treated as hard spheres and an outer-sphere mechanism was assumed. If the... 

The same relationships apply, in principle, to the concerted pathway (60). However, the intrinsic barrier is now so high, because of the contribution of bond breaking, that the possibility of observing a region that is counterdiffusion controlled is quite unlikely since the rate constant of the forward reaction would then be immeasurably small in most cases. At any rate, even if one conceives that such a situation might occur, and AGq then determined would be so different from that of the outer sphere electron transfer (59) in the stepwise pathway that the confusion would hardly be possible. 

Collision theory is based on the concept that molecules behave like hard spheres during a collision of two species, a reaction may occur. To estimate a rate constant for a bimolecular reaction between reactants A and B based on this theory, one needs first to calculate the number of collisions occurring in a unit volume per second (ZA1 ) when the two species, A and B, having radii rA and ru, are present in concentrations jVa and Aru, respectively. From gas kinetic theory, this can be shown to be given by Eq. (I) ... 

An alternative to the hard-sphere collision rate constant in Eq. 10.155 is used for the case of a Lennard-Jones interaction potential between the excited molecule (1) and the collision partner (2) characterized by a cross section a 2 and well depth en... 

Evaluate the Lennard-Jones collisional rate constant, ku at this temperature. How large a correction to the hard-sphere value does this make ... 

Since any quenching action is a bimolecular process, it is essential that the molecules M and Q should be in relatively close contact, but not necessarily in hard sphere (van der Waals) contact. Theoretical models lead to the distance dependence of the quenching rate constants as exponentials or sixth powers of r, the centre-to-centre distance of M and Q. Since these distance dependences are very steep, it is convenient to define a critical interaction distance r at which the quenching efficiency is, this being the distance at which 50% of the molecules M decay with emission of light (or undergo a chemical reaction) and 50% are quenched by some nearby molecule Q. 

The diffusion equation for the rate constant, eqn. (4.13), assumes that the molecules must come within van der Waals (hard sphere) contact for reaction to take place. This is of course a reasonable assumption when chemical bonds are made or broken in the reactants, but does not apply to long range processes such as energy transfer and possibly electron transfer. 

The diffusional rate constant kD is calculated on the basis of the Debye-Hiickel theory (Equation 6.107), where the distance tr is the sum of A and B radii in the hard-sphere approximation. 

The pre-exponential factor of a bimolecular reaction is related to the reaction cross-section (see Problem 2.3). A relation that is fairly easy to interpret can be obtained within the framework of transition-state theory. Combining Eqs (6.9) and (6.54), we can write the expression for the rate constant in a form that gives the relation to the (hard-sphere) collision frequency ... 

Notes Calculated from modified direct count procedures. Estimated from experimental values of P1/2, the pressure where the pseudounimolecular rate constant is 1/2 its high pressure limiting value, assuming hard-sphere or modified hard-sphere collision diameters. ... 

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