Decay rate exponential coefficient

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... 

There should exist a correlation between the two time-resolved functions the decay of the fluorescence intensity and the decay of the emission anisotropy. If the fluorophore undergoes intramolecular rotation with some potential energy and the quenching of its emission has an angular dependence, then the intensity decay function is predicted to be strongly dependent on the rotational diffusion coefficient of the fluorophore.(112) It is expected to be single-exponential only in the case when the internal rotation is fast as compared with an averaged decay rate. As the internal rotation becomes slower, the intensity decay function should exhibit nonexponential behavior. 

Figure 6. Deviation of the decay rate coefficient caused by flow and diffusion in the reactor for an exponentially decaying signal.

The decay rate of the exponential is DK x, where D is the translational diffusion coefficient and... 

The dec8y rate of the order-parameter fluctuations is proportional to the thermal diffusivity in case of pure gases near the vapor-liquid critical point and is proportional to the binary diffusion coefficient in case of liquid mixtures near the critical mixing point (6). Recently, we reported (7) single-exponential decay rate of the order-parameter fluctuations in dilute sugercritical solutions of liquid hydrocarbons in CO for T - T 10 C. This implied that the time scales associated with thermal diffusion and mass diffusion are similar in these systems. 

Single exponential kinetics occurs in the basic situation when the decay rate at time t is proportional to the population N(t) via a constant coefficient k,... 

If, in a freely vibrating system that contains a loss mechanism, the system energy is seen to decay at a rate that is proportional to the instantaneous energy W, i.e., -dW/dt=a W, it follows that the decay is exponential, W = Wo6". (This is the case for a linear system in which the loss coefficient itself is independent of vibration amplitude.) The decay coefficient is a = C0T, where CO =... 

The higher the barrier, the larger the exponential coefficient p and the more dramatically the electron transfer rate decays with distance. By a fortunate coincidence of units, the P in " is approximated by the square root of the barrier height in eV. Thus for typical biological redox centers that must overcome a barrier of about 8eV to be ionized in a vacuum, we can estimate the P for exponential decay of electron transfer in vacuum to be about 2.8 Much less of a barrier is presented by a surrounding organic... 

Calculation of the decay rate X and half-life period t at a fixed temperature step. At each temperature step, we have built the exponential reliability distribution, i.e., the probability of the non nucleation event within timelength t (P(E>t). The Ln[P(E>t)] were plotted versus timelength t and the data were fitted by a straight line passing by the origin (Fig. 6 correlation coefficients of the fits ranging between 0.84 and 0.99, Table 3). 

From here the decay rates are obtained by minimizing with respect to A the coefficient of —t in the exponential, that gives 7n = nA + G(A ), where A is defined by the condition G(A ) = —n. For small deviations from the asymptotic Lyapunov exponent, that is appropriate for n not too large, we can use the approximation G(A) = (A — A°°)2/(2A) to see that the dominant contributions come from values of A smaller than the asymptotic one, and obtain... 

Forces between DNA Double Helices. The repulsion and attraction of DNA is the molecular interaction most studied to date. In simple salts, repulsion is again exponential with decay rates of 2.8 3.3 A (II). Unlike forces between polysaccharides, the coefficient of the force depends on the type of cationic counterion, even though electrostatic double layer repulsion is low enough to suggest that the helix is largely neutralized by ion association. 

This function is the sum of two exponentials, one whose decay rate depends solely on the translational diffusion coefficient and another whose decay rate depends on both the translational diffusion coefficient and the kinetic-relaxation rate 1/rr. The strength of the term containing the kinetic relaxation time depends on the difference between the polarizabilities of the molecule in states 1 and 2, as we surmised in Section (6.1). In most cases the purely diffusive contribution which contains the square of theaverage polarizability increment will contribute much more strongly than the second term. This result should be contrasted with the electrophoretic case in the slow exchange limit [Eq. (6.3.4)]. In the electrophoretic case ka and kb separately appear in the expressions, whereas in the zero field case only the combination tr appears. 

For multimodal particle size distributions, the correlation function is the sum of exponentials, each with a decay rate proportional to the average diffusion coefficient of a size mode. To analyse C(r) in this case, a non-linear regres-siaverage particle size. This qiproach is often limited to bimodal distributions due to limitations in signal-to-noise ratio. 

Concentration fluctuations in polymer solutions that are in thermal equilibrium are well understood. The intensity of the polymer concentration fluctuations is proportional to the osmotic compressibility and the fluctuations decay exponentially with a decay rate determined by the mass-diffusion coefficient D. Probing these fluctuations with dynamic light scattering provides a convenient way for measuring this diffusion coefficient Z) [ 1],... 

N denotes the number of active (growing) nuclei. The time y represents the time the nucleus got activated. The exponent m gives the dimension of nuclei growth. The law of nucleation can be postulated in various ways, such as unimolecular decay law. The left-hand side of the equation origins from Avrami s treatment for the nuclei overly. It gives the relation between the extended rate of conversion and the true rate of conversion. The pre-exponential coefficient includes several constants grouped together. 

Distribution of Particle Size It is aU but impossible that every solute molecule or particle has exactly the same hydrodynamic radius in a given solution. There is always a dishibution in as illustrated in Figure 3.16. The peak position and width of the distribution vary from sample to sample. The distribution in leads to a distribution in the diffusion coefficient and therefore a distribution in the decay rate T of gi(T). Then, gi(T) is not a simple exponential decay. 

The problem of relating the pre-exponential coefficients ay to the experimental pre-exponential coefficients Ay is solved here by using the ratios of the coefficients (because Ay = 5, Oy, being 5, a constant for a given measurement, ai i/aj2 — Aj j/ A, 2) However, this solution leaves us with only three experimental values, the two decay times and the A //A 2 ratio (the A2j/A2 2 ratio equals —1, i.e., Eqs. 15.31 and 15.32 are not independent), for the four unknowns (rate constants). There are several methods to obtain the fourth piece of information, the most common being the measurement of the lifetime of A in the absence of reaction (1/k ), when possible. From the A/y/A/,2 ratio one obtains,... 

Despite the simplification to a number of six unknowns (smaller than the seven equations obtained from the fluorescence decays), there are still problems, because the fluorescence decays of the two excimers cannot be measured independently from each other (due to strong overlap of the emission spectra of Ei and E2). Thus, the pre-exponential coefficients of the excimer decays are linear combinations of A2,j and A3 j, and their splitting implies knowledge of the emission spectra and the radiative rate constants of the two excimers (see below). The splitting is not simple because the emission spectra of Ei and E2 nearly overlap, and thus the fluorescence decays of [lPy(3)lPy] do not substantially change along the excimer band (see pre-exponential coefficients at 480 and 520 nm in Fig. 15.15). 

In light-scattering experiments the decay rate of the order parameter is measured. Experiments performed by Chang et al. (1986) on mixtures of carbon dioxide and ethane show that the diffusion coefficient associated with the exponential decay (6.5) of the order-parameter fluctuations again satisfies the Stokes-Einstein law (6.21). However, the diffusion coefficient cannot simply be identified with either the thermal diffusivity a or the mass diffusivity D12, since the order-parameter fluctuations now incorporate fluctuations in both the density and the concentration. 

Both for very small coupling strengths r,a in the model and for very large r,a, the effective transmission coefficient decreases to zero as r,a decreases to zero or increases to infinity. In the perturbation theory limit P,a p (rp 1), one obtains isolated resonances with Lorentzian line shape and exponential decay rate coefficients given by equation (58) ... 

Coupling to these low-frequency modes (at n < 1) results in localization of the particle in one of the wells (symmetry breaking) at T = 0. This case, requiring special care, is of little importance for chemical systems. In the superohmic case at T = 0 the system reveals weakly damped coherent oscillations characterised by the damping coefficient tls (2-42) but with Aq replaced by A ft-If 1 < n < 2, then there is a cross-over from oscillations to exponential decay, in accordance with our weak-coupling predictions. In the subohmic case the system is completely localized in one of the wells at T = 0 and it exhibits exponential relaxation with the rate In k oc - hcoJksTY ". 

Selected entries from Methods in Enzymology [vol, page(s)] Analysis of GTP-binding/GTPase cycle of G protein, 237, 411-412 applications, 240, 216-217, 247 246, 301-302 [diffusion rates, 246, 303 distance of closest approach, 246, 303 DNA (Holliday junctions, 246, 325-326 hybridization, 246, 324 structure, 246, 322-324) dye development, 246, 303, 328 reaction kinetics, 246, 18, 302-303, 322] computer programs for testing, 240, 243-247 conformational distribution determination, 240, 247-253 decay evaluation [donor fluorescence decay, 240, 230-234, 249-250, 252 exponential approximation of exact theoretical decay, 240, 222-229 linked systems, 240, 234-237, 249-253 randomly distributed fluorophores, 240, 237-243] diffusion coefficient determination, 240, 248, 250-251 diffusion-enhanced FRET, 246, 326-328 distance measurement [accuracy, 246, 330 effect of dye orientation, 246, 305, 312-313 limitations, 246,... 

Equation (5.47) shows that the velocity autocorrelation function , v(t )-v(t), decays exponentially with time. The rate of decay is determined by the friction coefficient / (= 1 /b-m), that is, by particle mass and mobility. 

The ratio -ln[yp(r)]/T = 1 describes first-order decay that is unaffected by mass transport. When yp is calculated by Eq. 6 the ratio will not equal 1, and will express the deviation between the case of the measured first-order rate constant with flow and diffusion and the ideal case of no flow and diffusion. Figure 6 shows a plot of -ln[yp(r)]/T vs. z for the case when reaction zone at t = 0. The parameters are those from an investigation of the reaction flash photolysis of CF2ClBr in the presence of 02 and NO, where the reaction of CF2C102 radicals with N02 was studied [41]. For reference, rd = 0.1024 corresponds to a total pressure of 1 torr. Figure 6 clearly shows that at low pressures the deviation from exponential decay occurs at shorter times, z = kt, than at higher pressures. This is due to the pressure dependence of the diffusion coefficient. 

When 0 k, Eq. 16 reduces to ca(t) =ct0e kt, and the kinetics can be observed unperturbed by the source residence time. However, when this inequality is not fulfilled, (16) predicts an approximately exponential rise of c, which passes through a maximum and then decays away. If 0 is known or can be determined, then (16) can be used to fit data and extract the first-order rate coefficient. The value of 0 could be experimentally determined by introducing an unreactive species into the ion source as a step function. The transient response is given by... 

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