Time-distance relationship diffusion

As we indicated earlier, diffusion in a solution is important for the movement of solutes across plant cells and tissues. How rapid are such processes For example, if we release a certain amount of material in one location, how long will it take before we can detect that substance at various distances To discuss such phenomena adequately, we must determine the dependence of the concentration on both time and distance. We can readily derive such a time-distance relationship if we first consider the conservation of mass, which is necessary if we are to transform Equation 1.1 into an expression that is convenient for describing the actual solute distributions caused by diffusion. In particular, we want to eliminate Jj from Equation 1.1 so that we can see how c depends on x and t. 

Because Photosystem II tends to occur in the grana and Photosystem I in the stromal lamellae, the intervening components of the electron transport chain need to diffuse in the lamellar membranes to link the two photosystems. We can examine such diffusion using the time-distance relationship derived in Chapter 1 (Eq. 1.6 x je = 4Djtife). In particular, the diffusion coefficient for plastocyanin in a membrane can be about 3 x 10 12 m2 s-1 and about the same in the lumen of the thylakoids, unless diffusion of plastocyanin is physically restricted in the lumen by the appres-sion of the membranes (Haehnel, 1984). For such a D , in 3 x 10-4 s (the time for electron transfer from the Cyt b(f complex to P ), plastocyanin could diffuse about [(4)(3 x 10-12 m2 s-1) (3 x 10-4 s)]1/2 or 60 nm, indicating that this complex in the lamellae probably occurs in relatively close proximity to its electron acceptor, Photosystem I. Plastoquinone is smaller and hence would diffuse more readily than plastocyanin, and a longer time (2 x 10-3 s) is apparently necessary to move electrons from Photosystem II to the Cyt b(f complex hence, these two components can be separated by greater distances than are the Cyt b f complex and Photosystem I. 

Here rmax is the radial distance to the moving boundary and texp is an experimental time relative to some time t0. The diffusion constant D was determined from knowledge of the highest (dc/dr) coordinate measured at different times and by applying the following relationship ... 

As schematically shown in Figure 12.5 the self-diffusion coefficient D can be determined from the PFGE experiment for different values of A. For normal diffusion, for which Fick s law holds 1, D is independent of A and the average square distance the diffusing particles cover during the diffusion time A is directly proportional to A (Einstein relationship) ... 

Tanner s reviews on the migration of Rn in the ground (Tanner, 1964a, 1978, 1980) contain a detailed description of the processes responsible for Rn emanation and migration and an excellent summary of diffusion studies and coefficients therefore this topic will not be discussed here in detail. The emanation processes of recoil and diffusion release Rn into pore spaces of minerals and rocks, permitting diffusion and mechanical transport to move Rn away from its source. Typical diffusion constants of Rn in various media are listed in Table 1 l-I. From the many diffusion studies carried out it is evident that diffusion alone cannot move Rn beyond about 8 m within the lifetime of a Rn atom. A simple approximation of the distance-time relationship is given by = pDt, where S = distance, D = diffusion coefficient and t = time. Thus, assuming a typical diffusion constant of 0.05 cm /sec., after six half-lives or 23 days, only 1.5% of a Rn... 

If samples of two metals widr polished faces are placed in contact then it is clear that atomic transport must occur in both directions until finally an alloy can be formed which has a composition showing die relative numbers of gram-atoms in each section. It is vety unlikely that the diffusion coefficients, of A in B and of B in A, will be equal. Therefore there will be formation of an increasingly substantial vacancy concentration in the metal in which diffusion occurs more rapidly. In fact, if chemically inert marker wires were placed at the original interface, they would be found to move progressively in the direction of slowest diffusion widr a parabolic relationship between the displacement distance and time. 

The normalized steady-state current vs. tip-interface distance characteristics (Fig. 18) can be explained by a similar rationale. For large K, the steady-state current is controlled by diffusion of the solute in the two phases, and for the specific and y values considered is thus independent of the separation between the tip and the interface. For K = 0, the current-time relationship is identical to that predicted for the approach to an inert substrate. Within these two limits, the steady-state current increases as K increases, and is therefore diagnostic of the interfacial kinetics. 

Translational motion is the change in location of the entire molecule in three-dimensional space. Figure 11 illustrates the translational motion of a few water molecules. Translational motion is also referred to as self-diffusion or Brownian motion. Translational diffusion of a molecule can be described by a random walk, in which x is the net distance traveled by the molecule in time At (Figure 12). The mean-square displacement (x2) covered by a molecule in a given direction follows the Einstein-derived relationship (Eisenberg and Crothers, 1979) ... 

The diffusion coefficient D is one-third of the time integral over the velocity autocorrelation function CvJJ). The second identity is the so-called Einstein relation, which relates the self-diffusion coefficient to the particle mean square displacement (i.e., the ensemble-averaged square of the distance between the particle position at time r and at time r + f). Similar relationships exist between conductivity and the current autocorrelation function, and between viscosity and the autocorrelation function of elements of the pressure tensor. 

The kinetics data of the geminate ion recombination in irradiated liquid hydrocarbons obtained by the subpicosecond pulse radiolysis was analyzed by Monte Carlo simulation based on the diffusion in an electric field [77,81,82], The simulation data were convoluted by the response function and fitted to the experimental data. By transforming the time-dependent behavior of cation radicals to the distribution function of cation radical-electron distance, the time-dependent distribution was obtained. Subsequently, the relationship between the space resolution and the space distribution of ionic species was discussed. The space distribution of reactive intermediates produced by radiation is very important for advanced science and technology using ionizing radiation such as nanolithography and nanotechnology [77,82]. 

In the case of mass transport by pure diffusion, the concentrations of electroactive species at an electrode surface can often be calculated for simple systems by solving Fick s equations with appropriate boundary conditions. A well known example is for the overvoltage at a planar electrode under an imposed constant current and conditions of semi-infinite linear diffusion. The relationships between concentration, distance from the electrode surface, x, and time, f, are determined by solution of Fick s second law, so that expressions can be written for [Ox]Q and [Red]0 as functions of time. Thus, for... 

Equation 1 is given in spherical coordinates, thus assuming a spherical shape for the carbon particle, an assumption which accords reasonably well with microscopic observations of the geometry of particles of the experimental carbon. In Equation 1, C represents the H30+ activity in solution t, time r, the radial distance from the particle center D, the diffusion coefficient and S, the H30+ concentration at the surface of the carbon. For the present experiments, the equilibrium relationship between S and C is described in terms of the Freundlich expression... 

Figure 36.1. Nernst-Einstein relationship representing the time for a protein of 14 kD to diffuse through different distances (diffusion coefficient 10-lom2/s).

Simulations of three-component time-dependent diffusion were made based on two slightly different models using Matlab software [33]. Basically, fluid layer thicknesses are predicted, which determine diffusion distances. In this way, the H+ and OH- concentrations are revealed which can be related back to the pH. By the known fluorescence intensity-pH relationship, the quantum yield is thus given. 

With the Einstein relationship =6DA, where is the average squared distance particles with a self diffusion coefficient D cover in a time A, the maximum average size of the domains can be estimated as =5 pm, when we use A=50 milliseconds as the smallest mixing time for which cross-peaks appear, and 10 10 m2/s for D, the value found for... 

In electrochemistry several equations are used that bear Einsteins name [viii-ix]. The relationship between electric mobility and diffusion coefficient is called Einstein relation. The relation between conductivity and diffusion coefficient is called - Nernst-Einstein equation. The expression concerns the relation between the diffusion coefficient and the viscosity and is known as the - Stokes-Einstein equation. The expression that shows the proportionality of the mean square distance of the random movements of a species to the diffusion coefficient and the duration of time is called - Einstein-Smoluchowski equation. A relationship between the relative viscosity of suspension and the volume fraction occupied by the suspended particles - which was derived by Einstein - is also called Einstein equation [ix]. 

The relaxation process may be accompanied by diffusion. Consequently, the mean relaxation time for such kinds of disordered systems is the time during which the relaxing microscopic structural unit would move a distance R. The Einstein-Smoluchowski theory [226,235] gives the relationship between x and R as... 

The three characteristics, or conditions, as they are called, of a particular diffusion process cannot be rediscovered by mathematical argument applied to the differential equation. To get at the three conditions, one has to resort to a physical understanding of the diffusion process. Only then can one proceed with the solution of the (now) total differential equation (4.42) and get the precise functional relationship between concentration, distance, and time. 

Interactions of water protons account for an alteration of the local magnetic field surrounding a paramagnetic center. The observed relaxivity depends on the distance and time of these interactions and the translational diffusion. The interactions are classified as either inner-sphere, which describes protons of water molecules that are bound to the metal center, or outer-sphere, which describes bulk solvent molecules that experience a paramagnetic effect when they diffuse around the metal center. The diamagnetic contribution has a linear relationship... 

Equation 9.4 provides a relationship between time and the distance at which a particular concentration is achieved. When clearance rates are small relative to diffusion rateS/ it states that the distance from the surface (penetration depth) at which a particular concentration C is achieved advances as the square root of time. In other wordS/ to double the penetration of a compound/ the exposure time must quadruple. Equation 9.5 states that/ given sufficient time and negligible plasma concentration/ most compounds will develop a semilogarithmic concentration profile whose slope is determined by the ratio of the clearance rate to the diffusion constant. Note also that the distance over which the concentration decreases to... 

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