Chemical diffusion time

Fig. 4. Variation of autocorrelation function with changes in the equilibrium constant in the fast reaction limit. A and B have the same diffusion coefficients but different optical (fluorescence) properties. A difference in the fluorescence of A and B serves to indicate the progress of the isomerization reaction the diffusion coefficients of A and B are the same. The characteristic chemical reaction time is in the range of 10 4-10-5 s, depending on the value of the chemical relaxation rate that for diffusion is 0.025 s. For this calculation parameter values are the same as those for Figure 3 except that DA = Z)B = lO"7 cm2 s-1 and QA = 0.1 and <9B = 1.0. The relation of CB/C0 to the different curves is as in Figure 3. 

The linear and nonlinear optical properties of one-dimensional conjugated polymers contain a wealth of information closely related to the structure and dynamics of the ir-electron distribution and to their interaction with the lattice distorsions. The existing values of the nonlinear susceptibilities indicate that these materials are strong candidates for nonlinear optical devices in different applications. However their time response may be limited by the diffusion time of intrinsic conjugation defects and the electron-phonon coupling. Since these defects arise from competition of resonant chemical structures the possible remedy is to control this competition without affecting the delocalization. The understanding of the polymerisation process is consequently essential. 

Recall that we are assuming faem "C faff (°r fax, if turbulent flow). Anyone who has carefully observed a laminar diffusion flame - preferably one with little soot, e.g. burning a small amount of alcohol, say, in a whiskey glass of Sambucca - can perceive of a thin flame (sheet) of blue incandescence from CH radicals or some yellow from heated soot in the reaction zone. As in the premixed flame (laminar deflagration), this flame is of the order of 1 mm in thickness. A quenched candle flame produced by the insertion of a metal screen would also reveal this thin yellow (soot) luminous cup-shaped sheet of flame. Although wind or turbulence would distort and convolute this flame sheet, locally its structure would be preserved provided that faem fax. As a consequence of the fast chemical kinetics time, we can idealize the flame sheet as an infinitessimal sheet. The reaction then occurs at y = yf in our one dimensional model. 

G(t) decays with correlation time because the fluctuation is more and more uncorrelated as the temporal separation increases. The rate and shape of the temporal decay of G(t) depend on the transport and/or kinetic processes that are responsible for fluctuations in fluorescence intensity. Analysis of G(z) thus yields information on translational diffusion, flow, rotational mobility and chemical kinetics. When translational diffusion is the cause of the fluctuations, the phenomenon depends on the excitation volume, which in turn depends on the objective magnification. The larger the volume, the longer the diffusion time, i.e. the residence time of the fluorophore in the excitation volume. On the contrary, the fluctuations are not volume-dependent in the case of chemical processes or rotational diffusion (Figure 11.10). Chemical reactions can be studied only when the involved fluorescent species have different fluorescence quantum yields. 

Tj-hem TD = to2/4Dt the chemical relaxation time is much larger than the characteristic diffusion time so that there is no chemical exchange during diffusion through the excitation volume. The autocorrelation function is then given by... 

As discussed in detail in Sections C.3.d and C.3.e, the fastest atmospheric reactions of S02 are believed to be with H202 and perhaps with Os at higher pH values. Under extreme conditions of large fog droplets (—10 yu,m) and very high oxidant concentrations, the chemical reaction times may approach those of diffusion, particularly in the aqueous phase. In this case, mass transport may become limiting. However, it is believed that under most conditions typical of the troposphere, this will not be the case and the chemical reaction rate will be rate determining in the S(IV) aqueous-phase oxidation. 

Many experimental studies of the rates of oxidation of S(IV) in solution have used either bulk solutions or droplets that are very large compared to those found in the atmosphere. In addition, reactant concentrations in excess of atmospheric levels have often been used for analytical convenience. The use of large droplets increases the diffusion times, whereas higher reactant concentrations speed up the aqueous-phase chemical reaction rates. The combination of these two factors can lead to a situation where the rates of the diffusion processes, either of the gas to the droplet surface or more likely within the aqueous phase itself, become comparable to, or slower than, the chemical reaction rate. If this is not recognized, the observed rates may be attributed in error to the intrinsic chemical reaction rate. 

If finite chemical reaction times are put into the columnar diffusion flame theory (76), burning rates are predicted to be linearly proportional to pressure at low pressure and independent of pressure (plateau burning) at high pressure. Based on this model, von Elbe et al. (97) proposed the simple equation ... 

The above models describe a simplified situation of stationary fixed chain ends. On the other hand, the characteristic rearrangement times of the chain carrying functional groups are smaller than the duration of the chemical reaction. Actually, in the rubbery state the network sites are characterized by a low but finite molecular mobility, i.e. R in Eq. (20) and, hence, the effective bimolecular rate constant is a function of the relaxation time of the network sites. On the other hand, the movement of the free chain end is limited and depends on the crosslinking density 82 84). An approach to the solution of this problem has been outlined elsewhere by use of computer-assisted modelling 851 Analytical estimation of the diffusion factor contribution to the reaction rate constant of the functional groups indicates that K 1/x, where t is the characteristic diffusion time of the terminal functional groups 86. ... 

Hints and Help Estimate the transport time by assuming that the chemicals diffuse from a reservoir of constant concentration into the liner that at time t = 0 is assumed to be uncontaminated. Equation 25-41 may give you an idea how to calculate the effective diffusion coefficient through the liner. Select the most critical compound among the four. What is the criterion ... 

Over molecular length scales, the diffusion distances become very short (< 1 nm) so that only very rapid events can be influenced by these short diffusion times. Necessarily, this limits the number of systems to only relatively few, where the rate at which the reactants can approach one another is slow or comparable with the rate at which the reactants react chemically with each other. Some typical systems which have been studied are discussed in Sect. 2. The Smoluchowski [3] theory of reactions in solution, which occur at a rate limited solely by how fast the reactants can approach each other (sufficiently closely to react chemically almost instantaneously) is discussed in Sect. 3. If the chemical reaction is not so rapid, the observed rate of reaction may be influenced by both the rate of approach and the rate of subsequent chemical reaction. Collins and Kimball [4], and later Noyes [5], have extended the Smoluchowski theory (1917) to consider this situation (Sect. 4). In light of these quantitative theoretical models of diffusion-limited rate processes, some of the more recent and careful experiments on diffusion-controlled reactions in solution are considered briefly in Sect. 5. As the Smoluchowski theory... 

Since these considerations are independent of the nature of the sample, the results are valid for all crystals which are exposed to a sudden change of intensive state variables. The meaning of the chemical diffusion coefficient D must, however, be carefully investigated in each case (see Section 5.4.4). At 1000°C, Dv for simple transition-metal oxides is on the order of 10 7 cm2/s. This gives for cubic samples of 10-3 cm3 a defect relaxation time of approximately 1 h according to Eqn. (5.86). 

Figure 5-11 illustrates the results of an oxide interdiffusion experiment. Clearly, the transport coefficients are not single valued functions of composition. From the data, one concludes that for a given composition, the chemical diffusion coefficients depend both on time and location in the sample [G. Kutsche, H. Schmalzried (1990)]. Let us analyze this interdiffusion process in the ternary solid solution Co. O-Nq. O, which contains all the elements necessary for a phenomenological treatment of chemical transport in crystals. The large oxygen ions are almost immobile and so interdiffusion occurs only in the cation sublattice of the fee crystal. When we consider the following set ( ) of structure elements... 

These assumptions, however, oversimplify the problem. The parent (A,B)0 phase between the surface and the reaction front coexists with the precipitated (A, B)304 particles. These particles are thus located within the oxygen potential gradient. They vary in composition as a function of ( ) since they coexist with (A,B)0 (AT0<1 see Fig. 9-3). In the Af region, the point defect thermodynamics therefore become very complex [F. Schneider, H. Schmalzried (1990)]. Furthermore, Dv is not constant since it is the chemical diffusion coefficient and as such it contains the thermodynamic factor /v = (0/iV/01ncv). In most cases, one cannot quantify these considerations because the point defect thermodynamics are not available. A parabolic rate law for the internal oxidation processes of oxide solid solutions is expected, however, if the boundary conditions at the surface (reaction front ( F) become time-independent. This expectation is often verified by experimental observations [K. Ostyn, et al. (1984) H. Schmalzried, M. Backhaus-Ricoult (1993)]. 

Figure 15-10. a) Electrochemical device for the determination of chemical diffusion in mixed conductors (Ag2+aS). b) Course in time of 5 (resp. emf of Ag/Ag[ probes), measured at probe n. 6 = starting composition. 

CARS-CS experiments have been reported in the low-concentration limit ((N) <<1) on freely diffusing submicron-sized polymer spheres of different chemical compositions using both the E-CARS [162, 163] and the polarization-resolved CARS [163] detection scheme for efficient nonresonant background suppression. These experiments have unambiguously demonstrated the vibrational selectivity of CARS-CS, the dependence of its ACF amplitude on the particle concentration, (N), the dependence of lateral diffusion time, Tp, on the sphere size, and the influence of the microviscosity on its Brownian motion. 

Other intrinsic characteristic parameters of LAPS have been investigated by different research groups such as the chemical response time, the surface-state densities and zeta potential (for Si3N4), and the minority carrier diffusion length for resolution estimations. For a more detailed description of these experimental set-ups, see, e.g., Refs. [57-61],... 

TDFRS allows for experiments on a micro- to mesoscopic length scale with short subsecond diffusion time constants, which eliminate almost all convection problems. There is no permanent bleaching of the dye as in related forced Rayleigh scattering experiments with photochromic markers [29, 30] and no chemical modification of the polymer. Furthermore, the perturbations are extremely weak, and the solution stays close to thermal equilibrium. 

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