Fast exchange approximation

Agarose. In previous work (1 ) it has been shown that the observed relaxation in homogeneous gels is described by the fast exchange approximation. 

Because the phenomenon of lanthanide induced shift results from a fast exchange phenomenon, the linewidths of the peaks are governed by the fast exchange approximation, Equation 2.12 [52,53] ... 

The high-field approximation does not exhibit the contribution due to the production of compounds by the exchange reactions to the electrophoretic pattern. This effect can be illustrated by the fast-exchange approximation which yields the roots... 

Figure 7. The Fermi resonance mechanism within the adiabatic and exchange approximations. F, fast mode S, slow mode B, bending mode.

Now, comparing 7Sf and /Sf ex, we observe also some splitting modifications. We may attribute them to some additional changes in the frequency gap. Indeed, as shown in Section V.B, the full treatment of the Fermi coupling mechanism leads to a displacement of the potential of the fast mode (both in energy and position) that does not appear within the exchange approximation. The fast mode then involves an effective frequency that differs from oo0, which leads to an effective gap which differs from (oo0 — 2oog). 

Applying the exchange approximation and neglecting the zero-point energy terms, we may safely limit the representation of the Hamiltonian Hsf ex within the following reduced base which accounts for the ground states of each mode and for the first (second) overtone of the fast (bending) mode ... 

We have seen that the exchange approximation consists mainly in ignoring the driven part of the fast mode motion, given by... 

As a consequence of the two above points, there is presently no satisfactory theory able to incorporate the specific relaxation of the fast mode and of the bending modes in a model working beyond both the adiabatic and exchange approximations. 

Prom these approximate equations, it follows that exchange rate constants kex = 1/xm are easily obtained in region II and, if the chemical shift difference is large, in region III also. In the very fast exchange, region IV information on water exchange can also be obtained if the interaction between the... 

Thermal dissociation of 02 has a high activation energy and is usually quite slow. However, at the high temperatures in combustion systems where thermal NO is important, both O and 02 are involved in a number of reactions that are fast compared to the thermal NO formation. Due to the fast exchange between O and 02 in these reactions, the partial equilibrium assumption for (R12) is a reasonable approximation. Assuming partial equilibrium, we can relate the oxygen atom concentration to the 02 concentration and the equilibrium constant for the reaction,... 

In protein solutions the water protons may be considered to reside in two different environments, i.e. the bulk water, and the hydration spheres of the protein molecules. If there is fast exchange of protons between these environments a single proton nuclear magnetic resonance will be observed, which corresponds to the average of the resonances in the different environments. Following McConnell (74) the observed longitudinal relaxation time is to a good approximation... 

Under fast exchange (Wi/2/Av 1), a Lorentzian line is observed at the population-weighted mean of va and vb, and an approximate expression for the rate constant based on the observed line width can be used ... 

Halle et al. (1981) measured NMR relaxation for solutions of several proteins as a function of frequency and protein concentration. They estimated hydration by use of a two-state fast-exchange model with local anisotropy and with assumed values of the order parameter and several other variables. The hydration values ranged from 0.43 to 0.98 h for five proteins, corresponding approximately to a double layer of water about a protein. The correlation time for water reorientation was, averaged over the set of proteins, 20 psec, about eight times slower than that for bulk water. A slow correlation time of about 10 nsec was attributed to an ordering of water by protein at very high concentration. 

In contrast with the short-lived MLCT-excited states [Ru(bpy)3] [74], the electrogenerated [Ru(bpy)3]"+ can be conveniently studied by temperature-dependent ESR [42, 50a, b. Analysis of the broad unresolved lines of this and related species reveals electron hopping between the three equivalent ligand sites with activation energies of approximately 10 kJ mol . Similar slow vs. fast exchange phenomena were reported for the extended atom [75] species 17 which contains an encapsulated Na+ ion and mobile spins in the bpy-containing periphery. 

This approximation is valid when A. is much less than the linear dimension of the liquid volume, that is for pores with low curvature. Therefore, experimental relaxation rates from a system in the fast exchange limit can provide information on the ratios of the surface-to-bulk volumes of liquid, given the relaxation rates of surface and bulk liquid are known as well as the value for A. If relaxation values are acquired for different stages of filling, it is possible to determine details of the filling process, like the formation of puddles within the pore structure [Alll, Bool]. 

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