Single exponential approximation, time

When this holds, the kinetic equations reduce to single exponentials. Chipperfield6 demonstrates that approximate adherence to Eq. (4-25) suffices to fit 20 absorbancetime pairs spaced at equal times over the first 75 percent of the reaction with correlation coefficients better than 0.999. 

Even if we consider a single solvent, e g., water, at a single temperature, say 298K, depends on the solute and in fact on the coordinate of the solute which is under consideration, and we cannot take xF as a constant. Nevertheless, in the absence of a molecular dynamics simulation for the solute motion of interest, XF for polar solvents like water is often approximated by the Debye model. In this model, the dielectric polarization of the solvent relaxes as a single exponential with a relaxation time equal to the rotational (i.e., reorientational) relaxation time of a single molecule, which is called Tp) or the Debye time [32, 347], The Debye time may be associated with the relaxation of the transverse component of the polarization field. However the solvent fluctuations and frictional relaxation occur on a faster scale given by [348,349]... 

The NO dissociation rate constants are summarized in Table HI (50) and are smaller than those seen with NO-metmyoglobin complexes. NP2 and NP3 (A ofr 0.1 s ) release NO approximately 10 times more slowly than NPl and NP4 kofs 2-3 s ) at pH 8.0, and the NO release rate for all nitrophorins decreases as the pH is lowered to 5.0. The NO release curves cannot be fit with a single exponential, indicating two off rates at each pH, as previously noted for both recombinant and insect-derived NPl (46), and which has also been recently reported for recombinant NP2 (145). The biphasic kinetics suggest the presence of slowly interconverting conformations. The values obtained are pH and protein dependent, ranging from 2.6 to 0.05 (Table HI) (50), values that are considerably slower than found for sperm whale metmyoglobin... 

The simplest kinetic treatment would be based on the determination of a half-life for the reaction, where 11/2 is the time taken for the turbidity to reach half the maximum value. Since the curves approximate well to a single exponential, it is then possible to define an apparent first-order rate constant such that ... 

Figure 19 Time-dependent PL intensity of ANTPEP (a) (see Fig. 16) and TAPC (b) (see Fig. 18) solid films, (a) The PL excited at Aex = 350nm and detected at 500 nm, the curves approximated by two exponentials for short- and long-time behavior with the decay times ti and t2, respectively (dashed and dotted lines). Single exponential decay with T = 3.2ns is observed in a dilute (10-5M in DCM) ANTPEP solution (solid line), (b) The PL excited at Aex = 337 nm and detected in various emission spectral regions (cf. absorption and PL spectra in Fig. 18) (1) 370nm (t < Ins) (2) 450 (t = 2.3 ns) (3) 525 nm (4.8 ns) and (4) 580 nm (t = 6.4 ns) (after Ref. 74).

Normally, for semiconductors, Csc < CH so CT Csc. Roat may be varied systematically and the decay of j can often be approximated by a single exponential form, i.e. kr 1/RtCi. or kT potentials well positive of V, the long-time transient time-constant t (Rm + Rout)Csc, and a plot of x vs. R]oad (sflin + Rout) is linear, as shown in Fig. 105. Confirmation of this is obtained from the fact that 1/t2 obeys the Mott-Schottky relationship. At potentials close to V, kec becomes much larger and the decay law more complex. 

Finally, the ensemble-averaged quantity < 1F(Z. .. z r, /)>yy gives the probability of donors surviving to time t. Monte Carlo calculations have shown that in many cases the pair survival probability is approximated well by a single exponential except at very short times [79a, 81]. The rate constant /cd is related to the geometric and dif-fusional characteristics of the system by the equation [79a]... 

Reactions that occur between components in the bulk solution and vesicle-bound components, i.e., reactions occurring across the membrane interface, can be treated mathematically as if they were bimolecular reactions in homogeneous solution. However, kinetic analyses of reactions on the surface of mesoscopic structures are complicated by the finiteness of the reaction space, which may obviate the use of ordinary equations of chemical kinetics that treat the reaction environment as an infinite surface populated with constant average densities of reactant molecules. As was noted above, the kinetics of electron-transfer reactions on the surface of spherical micelles and vesicles is expressed by a sum of exponentials that can be approximated by a single exponential function only at relatively long times [79a, 81], At short times, the kinetics of the oxidative quenching of excited molecules on these surfaces are approximated by the equation [102]... 

The decay of d(t) for 9,10-DMA in PAA and PMA at a = 0 is presented in Figure 8. For PMA a Siam of two exponentials is required to obtain a reasonable fit of the data (as indicated by the random distribution of residuals for the two exponential fit) while for PAA a single exponential function is adequate. The fluorescence decay (s(t)) behaviour of 9,10-DMA attached to these polymers is well described by a single exponential function with a lifetime of approximately 13.5 ns and is relatively insensitive to the degree of neutralization of the polymer. The rotational correlation times obtained from an analysis of r(t) (or from d(t) and s(t) (16)) are given in Table II. 

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