Double-exponential form

Ligand release in the reaction of bis(A/-alkylsalicylaldiminato)zinc(II) complexes with ammonium ions in acetonitrile follows first-order kinetics, albeit in double exponential form for the t-butyl complex. The first bond to break is Zn-0 (317). 

Following the same procedure, a similar double exponential form for the cumulative failure distribution F a) was obtained by Ray and Chakrabarti (1985a) and was finally reduced (fitted) to a Weibull form (Weibull 1951). Later, Duxbury et al. (1986) obtained this generic form (1.25) using Lifshitz argument for the estimate of the size of the most dangerous defect for small disorder (p 1). 

Beale and Duxbury also determined the cumulative failure distribution probability F E ) and found very good agreement with the Gumbel double exponential form (2.69). 

While the double exponential form characterizes (approximately) the decays of the 205 cm-1 bands in the high-energy spectra, decays in the vibrationally relaxed region of the spectra are different These other decays, an example of which is shown in Fig. 29, are devoid of any early time spike, but have finite rise times and long-component lifetimes matching those of the long components of Fig. 28. This behavior is consistent with the decay of vibrationally relaxed fluorescence in the dissipative regime of IVR. 

The previous decomposition of the data relaxation suggests a double exponential form of the OKE response function and hence of the related correlator, = Al e -f A2 e 2, The two relaxation times, t and T2, are compatible with the DSE picture. As reported before, the diffusive relaxation is in general described by a multiexponential decay [42], which for an asymmetric rotator reduces to a biexponential if the principal axes of the rotational diffusion... 

To illustrate the use of time-resolved data for sensing, assume that each form (free and bound) has a unique decay time, rF and rb- The intensity decay of a mixture is a double-exponential... 

Note also that Eq. (5.2) is equivalent to the common Laplace transform. A comparison of double-exponential and distributional analyses is represented in Figure 5.1. The distribution function shows width about central values which the double-exponential fit cannot express because of its mathematical form. Here the appearance of central values may partially be a consequence of the model functions assumed in the solution. Nevertheless, width directly... 

It can be seen that the coupling of the formation and decay processes increases with the width of the flash. In an intermediate case, the time dependence of the absorbance change will have the functional form of a double exponential, A exp(—f/i ) + B exp (—tlx"). One lifetime will be close to the lifetime of the transient species and the other to the lifetime of the pump. In the most unfavorable conditions, the functional form will be a single exponential with nearly the lifetime of the pump. The determination of the lifetime of a transient species formed by the decay of transformation of an excited state offers a similar difficulty. The reduction of methylviologen, MV2 +, by the metal to ligand charge transfer (MLCT) state19 of the Re(I) complex and the reoxidation of the produced radical, MV +, are illustrated in Equations 6.57-6.59. 

The hydraulic model of Fig. 4.5.3 can be described by a pair of simultaneous differential equations with a solution in the form of a double exponential expression (Schaefer et al. 1981) ... 

B—>A photo-orientation is observed only for individualizable isomers, and its evolution is described by a double-exponential behavior in the form of... 

Equation (11) is the transition-state equation for electrochemical rates (i is, of course, proportional to d and concentration of reactants in the double layer at the electrode interface in the usual way ) and is obviously equivalent to the Tafel equation in exponential form [Eq. (4)]. From Eq. (11) it is seen that the Tafel slope for a simple electron transfer process is RT/pF, i.e., b is linear in temperature. We shall return later to a more critical examination of Eq. (11) insofar as energy and entropy components of the free energy of activation are concerned. 

Figure 2.6 Measured intermediate scattering functions of a w/o-droplet microemulsion for the system D2 0/n-octane-d- 8/CioE4. The four curves were obtained at four different q values close to the minimum of the droplet form factor. The solid lines are double exponential fits with only two adjustable parameters. The translational diffusion coefficient was determined using PCS (see Fig. 2.5) and used as input for the analysis of the NSE data. (Figure redrawn with data taken from Ref. [67].)...

Computer) Starting the double reptation relationship equation (3-116) for a bimodal mixture, use an iterative process to estimate the molecular-weight ratio of the two components of the blend with the relaxation data listed in file MW-Blend. TXT in the CD. This data is for a 50 50 blend. Assume exponential forms for the F(t, M,). 

At pH 3.2, on the other hand, the decay becomes markedly nonexponential (see Fig. 2.3a). Modeling by a function of the form of Equation 2.6 was not statistically adequate [90] the value is 5.75 and a marked deviation in the residuals is apparent. This infers that additional fluorescent excited states are present under these conditions. Analysis of the data in terms of a double-exponential function as described by Equation 2.7... 

---