Collision distance

The loss term contains contributions from all possible collisions that can deflect molecule i during the time dt. With reference to Fig. 12.13, any molecule j that arrives within the effective collision distance A is assumed to contribute to the loss term. Molecules j approaching i with relative velocity... 

Estimates of X (i.e. when A, is assumed to be 0) using eqn (52) or more complicated versions thereof (78) have turned out to be somewhat less than successful. It is usually difficult to use the spherical approximation (Fig. 4) for the shape of organic molecules, and other, more complex treatments produce problems of their own. Thus the intuitively satisfying model for electron transfer between two aromatic species, parallel orientation of the molecular planes at collision distance, cannot be fitted to the triaxial ellipsoidal model discussed in Section 4. Instead, one has to assume that electron transfer takes place over a considerably larger distance. This expansion of the transition state seems to be fairly constant for different compounds and can be included as an ad hoc (at least at present) parameter in the calculation of X. 

By means of Eq. (34) it is possible to calculate the center-to-center collision distance dxi from a measured diffusion constant D i- If three diffusion constants Dx2, Ms- and Ms can be measured, individual molecular diameters f/j, (I2, and can be obtained if it can be assumed that the collision distances are the arithmetic averages of the molecular diameters involved, as would be the case with hard spheres ... 

For typical values such as we have been using, tab = 4 A, ASc = — K In 3.3 = —2.5 cal/mole-°K, a result which probably underestimates somewhat the entropy decrease in the formation of chemically reacting pairs A-B, since we have not considered the possible loss of internal degrees of freedom of A and B in forming a more closely interacting pair A-B. In addition, the A-B distance in such a pair is probably much closer to bond distances than to the larger Van der Waals or collision distances. 

This expression relates the second-order rate constant, k, for an outer-sphere electron transfer reaction to the free energy of reaction, AG°, with one adjustable parameter, X, known as the reorganization energy. Wis the electrostatic work term for the coulombic interaction of the two reactants, which can be calculated from the collision distance, the dielectric constant, and a factor describing the influence of ionic strength. If one of the reactants is uncharged, Wis zero. In exact calculations, AG should be corrected for electrostatic work. The other terms in equation 46 can be treated as constants (Eberson, 1987) the diffusion-limited reaction rate constant, k, can be taken to be 10 M" is the equilibrium constant for precursor complex formation and Z is the universal collision frequency factor (see Eberson, 1987). 

Increasing Q, would decrease the average minimum distance between a self localized PFq.fast excitation and a possible CTS defect, which suggests that Eq. 4, simply describes the qualitative dependence with distance for the Dexter electron exchange mechanism k j = ko e R, where ko is the maximum rate constant for energy transfer, occurring when donor and acceptor are at the collision distance Ro and R is the separation between donor and acceptor when they are further apart than Ro. 

The advantage of this form is that ggas(r) already contains the short-range structural features of g(r), therefore y(r) depends relatively weakly on r. We may define a mean collision distance R by... 

Since the reactants are neutral, electrostatic effects are negligible and 0 is equal to one. The collision distance a is equal to 400 pm. Thus,... 

For self-exchange reactions such as that in Figure 1.8, Equation 1.18 applies. This is obtained by setting the free energy terms in Equation 1.12 equal to zero. In most cases, the work term W(r) can be calculated. This is the energy required to bring the reactants from effectively infinite separation to within collision distance r, approximated as the... 

The work term W(r), Equation 1.39, is as shown in Equation 1.4, but with the subscript on the collision distance r modified to indicate a self-exchange reaction. 

Note that the diffusion timescale xd contains a characteristic size of an object L. This ties in the length scale of the problem. Similarly, the mean free path can be associated with mean free time between collisions x by the relation l = vx. Note that this is a statistical quantity since collision distances are not fixed. However, the probability p that a particle emerging from a collision travels a distance x without a collision is related to the mean free path as p = exp(-x/ ). Associated with the relaxation time x, is a length scale which is the characteristic size of a volume over which local thermodynamic equilibrium can be defined. Generally, the hierarchy of the length scales is X < < ,. 

Assuming a Poisson distribution of the electroactive speeies, the enhancement fae-tor can be expressed as a power series of a probabihty funetion which is related to the concentration. At low concentrations the probability of finding more than one molecule in a hemisphere with a radius of the molecular collision distance is nearly zero and / = 1. The factor /, and therefore D, increases noticeably at higher concentrations. 

Heterogeneous electrode reactions can be compared with homogeneous kinetics in solution, with regard to mass transport. The second-order rate coefficient for a fast homogeneous reactions in solution, k(hom), which would be observed if diffusion were infinitely fast, can be related to the measured rate coefficient, kob3(hom) by application of Eick s first law in a spherical continuum diffusion field around the reacting molecule. At a collision distance Tab. this corresponds to the average... 

While important, transient effects often pass unnoticed due to limited time resolution of the experimental apparatus or due to the small magnitude of the effect in fact, the phenomenon is more easily detected on slow diffusion processes in viscous media and long fluorescence lifetimes. The non-exponential intensity decay, resulting from a transient effect is described by Eq. (15.62), where a and b depend on diffusional parameters (diffusion coefficient and collision distance) and quencher concentration. 

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