Exponential decay function

Diffusion and Mass Transfer During Leaching. Rates of extraction from individual particles are difficult to assess because it is impossible to define the shapes of the pores or channels through which mass transfer (qv) has to take place. However, the nature of the diffusional process in a porous soHd could be illustrated by considering the diffusion of solute through a pore. This is described mathematically by the diffusion equation, the solutions of which indicate that the concentration in the pore would be expected to decrease according to an exponential decay function. 

X 10 years old, this implies that the content of the reservoir today is about half of what it was when the Earth was formed. The probability density function of residence time of the uranium atoms originally present is an exponential decay function. The average residence time is 6.5 x 10 years. (The average value of... 

Figure 3.18. Time dependence of the peak position of the 1570 cm Raman band of Sj trans-stilbene in chloroform solution (filled triangle). The time dependence of the anti-Stokes/Stokes intensity ratio is also shown with open circles. The best fit of the peak position change with a single-exponential function is shown with a solid curve, while the best fit of the anti-Stokes/Stokes intensity ratio is shown with a dotted curve. The obtained lifetime for both single-exponential decay functions was 12ps. (Reprinted with permission from reference [78]. Copyright (1997) American Chemical Society.)...

Equation (75) shows that (u(t) is an exponentially decaying function for long times with a decay constant /p. For very massive B particles M N mN with M/mN = q = const, the decay rate should vary as 1 /N since p = mNq/ (q + 1). The time-dependent friction coefficient (u(t) for a B particle interacting with the mesoscopic solvent molecules through repulsive LJ potentials... 

It is obvious that (21) is equivalent to a stretched exponential decay function of the general form... 

It is interesting to note that the fast emission dynamics of Ag(0) (shown in Figure 21.7) differs from that of Au(0) j [101]. The decay curve for (Au(0) j could be reasonably fit by a two-exponential decay function with time constants of 74 fs, 5.5 ps and relative amplitudes 0.95, 0.05, respectively (best fit curve shown in Figures 21.5 and 21.7). The Ag nanoparticles initial (71 fs with 0.91 amplitude) and final (5.3 ps with 0.01 amplitude) decay components were similar to those of gold however, an additional component of 650 fs (with 0.08 ampli-... 

A variety of extrapolation algorithms have been applied to the sequences generated by the correlation-consistent cc-pVnZ basis sets [12, 51-55], Dunning and his colleagues had initially suggested fitting their calculations to an exponentially decaying function [12, 51, 52],... 

As already discussed, modelling this multiple exponential decay function with the numerical cell model gives valuable information about cell morphology and membrane permeability. Similar information is available... 

The mixture of these compounds at 0.5 xM per component was equilibrated with a 5 pM concentration of the protein target, then the reaction was quenched with excess staurosporine (100 pM) and analyzed using ALIS every 7 min. The measured protein-ligand complex MS responses were normalized and fit to the exponential decay function above, as shown in Fig. 3.16. The raw data fit the ex-... 

The concentration of solids in the upper lean region can be reasonable represented by an exponential decay function which starts from the value in the lower region and falls to the limiting value in an infinitely high vessel. This is the value for pneumatic conveying. 

Another approach is the characterization of peaks with a well-defined model with limited parameters. Many models are proposed, some representative examples will be deaaib i. Wefl known is the Exponentially Modified Gaussian (EMG) peak, i.e. a Gaussian convoluted with an exponential decay function. Already a few decades ago it was recognized that an instrumental contribution such as an amplifier acting as a first-order low pass system with a time constant, will exponentially modify the... 

A one-dimensional random walk is not necessarily symmetric with respect to jumps toward the right and toward the left. If the chemical potential gradient is sufficiently weak we may still approximate the jump length distribution by an exponentially decaying function, but distinguish that toward the right from that toward the left. 

Figure 3 shows representative single-wavelength absorbance transients for three dyes electrostatically bound to colloidal Sn02. The transients correspond to photoinitiated bleaching and recovery of the respective MLCT absorbances. From Fig. 3, it is clear that (1) injection is rapid in comparison to backET, (2) back ET is complex kinetically, but (3) the complex recovery rates depend strongly on the identity of the dye, at least in the first few hundred nanoseconds of the recovery. In order to isolate the shorter-time recovery kinetics, transients were fit, somewhat pragmatically, to a bi-exponential decay function ... 

We have made a note of the hydrodynamic interactions and other interactions to draw attention to an important fact. That is, the analysis of the DLS data is often quite complex, and a simple use of the single-exponential decay function and the Stokes-Einstein relation is not always sufficient, although many instruments available on the market use such an analysis and report an effective size for the particles in the dispersion. 

Figure 7. In (a) is shown a typical exponential-decay function. The lifetime is 230/ sec. (b) is the semilog plot of the same function.

Once the scaling relation of Eq. (39) is known, the molar mass distribution can, at least in principle, be obtained from a Laplace inversion of the multi-exponential decay function as defined in Eq. (40). At this point, the differences between PCS and TDFRS stem mainly from the different statistical weights and from the uniform noise level in heterodyne TDFRS, which does not suffer from the diverging baseline noise of homodyne PCS caused by the square root in Eq. (38). 

In the PFPE Zdol model, due to the polarity of endgroups induced by the hydroxyl group, the atomistic interaction is different from that in backbone beads. Here, the polarity interaction is assumed to occur within a short range, and is modeled as an exponential decay function. The potential function among endbeads is... 

In order to apply these equations to a femtosecond pump-probe experiment, an additional assumption has to be made regarding the shape of the time resolved signal. We wish to account for the finite relaxation time of the transient polarisation and so the signal must be described by a double convolution of an exponential decay function with the pump and probe intensity envelope functions. We will assume a Gaussian peak shape so that the convolution may be calculated analytically. As we will see, the experimental results require two such contributions, and hence, the following function will be used to fit the experimental data... 

The fractions of protons decaying according to relaxation functions Rx and R2 are given by fj and f2. In molten polymers this relationship has long been exploited to provide a measure of the crosslink density in many polymer systems [64-83]. The form of the decay functions has been the subject of much discussion, however, it is often observed that Rj and R2 can be approximated by simple exponential decay functions. It is generally accepted that the protons with short relaxation times are those directly attached to or adjacent to crosslink points. As an example Figure 13.4 shows the decays of transverse... 

Here, Eq is the permittivity of free space. For a simple Debye-type relaxation process, Eq. (4), and owing to the incorrect representation of the high-frequency limit inherent in any expression for K (to) consistent with an exponential decay function for the electric moment, one obtains for the high-frequency limit of a(to) from Eqs. (4) and (6) (30) ... 

The integral form of Eq. (4.18) (Kaelble, 1971) shows that e is an exponential decay function of t/( /G). The dimensions of r /G reduce to seconds (Appendix 4) and the equation reaches a limiting l/e (0.37emax) in t = ti/G seconds. The retardation time (/rd) is the time required for emax of a Voigt-Kelvin fluid (Fig. 2a) to be reduced to 37% of emax after t has been removed (Barnes et al., 1989 Seymour and Carraher, 1981). A long retardation time is characteristic of a more elastic than viscous fluid. 

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