Translational diffusion coefficient equation

Sun Y-P and Saltiel J 1989 Application of the Kramers equation to stiibene photoisomerization in / -alkanes using translational diffusion coefficients to define microviscosity J. Phys. Chem. 93 8310-16... 

It is possible to get molecular weight from the sedimentation coefficient if we assiune a conformation or if we combine with other measurements, namely the translational diffusion coefficient via the Svedberg equation [50]... 

For example, in the case of PS and applying the Smoluchowski equation [333], it is possible to estimate the precipitation time, fpr, of globules of radius R and translation diffusion coefficient D in solutions of polymer concentration cp (the number of chains per unit volume) [334]. Assuming a standard diffusion-limited aggregation process, two globules merge every time they collide in the course of Brownian motion. Thus, one can write Eq. 2 ... 

Photon correlation spectroscopy (PCS) has been used extensively for the sizing of submicrometer particles and is now the accepted technique in most sizing determinations. PCS is based on the Brownian motion that colloidal particles undergo, where they are in constant, random motion due to the bombardment of solvent (or gas) molecules surrounding them. The time dependence of the fluctuations in intensity of scattered light from particles undergoing Brownian motion is a function of the size of the particles. Smaller particles move more rapidly than larger ones and the amount of movement is defined by the diffusion coefficient or translational diffusion coefficient, which can be related to size by the Stokes-Einstein equation, as described by... 

Hydrodynamic properties, such as the translational diffusion coefficient, or the shear viscosity, are very useful in the conformational study of chain molecules, and are routinely employed to characterize different types of polymers [15,20, 21]. One can consider the translational friction coefficient, fi, related to a transport property, the translational diffusion coefficient, D, through the Einstein equation, applicable for infinitely dilute solutions ... 

In dynamic light scattering (DLS), or photon correlation spectroscopy, temporal fluctuations of the intensity of scattered light are measured and this is related to the dynamics of the solution. In dilute micellar solutions, DLS provides the z-average of the translational diffusion coefficient. The hydrodynamic radius, Rh, of the scattering particles can then be obtained from the Stokes-Einstein equation (eqn 1.2).The intensity fraction as a function of apparent hydrodynamic radius is shown for a triblock solution in Fig. 3.4. The peak with the smaller value of apparent hydrodynamic radius, RH.aPP corresponds to molecules and that at large / Hs,Pp to micelles. 

The Svedberg equation relates 5 to the molar mass of the particles, M2, their translational diffusion coefficient, D (related to the frictional force exerted on the particle), and to (dp/dc2)kl ... 

At infinite particle dilution, the translational diffusion coefficient is given by Einstein s equation... 

The physical properties of these polymeric dendrimers have been studied to some extent. Intrinsic viscosity measurements combined with MW afford values of according to Eq. (5). Alternatively, the translational diffusion coefficient leads to Rh according to Eq. (6). These equations may well be applicable, since it is observed that Rn and Rh scale with the 1/3 power of MW in support of the equal density hard-sphere assumption [88]. 

A measurement of physical parameters in solution for isolated macromolecules provides a manner by which the shape of a macromolecule can be determined. The approximate dimensions and axial ratio or radius can be calculated by applying Equations (4.3) through (4.17). As shown in Figure 4.10, the particle scattering factor for collagen molecules depicted in Figure 4.9 is more sensitive to bends than is the translational diffusion coefficient. 

The r-average translational diffusion coefficient l> is calculated from the equation Dj = V/q2. For a collection of identical spheres undergoing ordinary Brownian motion in solution. 

Because the magnitude of the correlation function in the short X limit is inversely proportional to the concentration of fluorophores, FCS can be used to follow changes in concentration. The diffusion time Xd yields the translational diffusion coefficient. If the fluorophore undergoes intramolecular dynamics or photophysical processes, then Equation 15 must be modified accordingly (27). Analysis of the autocorrelation with a modified fitting function can then provide dynamic information about these processes. 

Calculation with the variation of translational diffusion coefficients showed that the nonlinear effects lessened effects of translational diffusion (Fig. 2). In the linearized equation, solvent relaxed quickly for a large translational diffusion coefficient. That is, a value of S(t) at t=(2DR) decreased with the increase in the translational diffusion coefficient for the linearized equation. For calculation with the nonlinear Smoluchowski-Vlasov equation (1), however, a value of S(t) at t=(2DR) increased slightly with the translational diffusion coefficient. Thus, the ratio of a nonlinear value to a linear value increased with the translational diffusion coefficient. 

Figure 2. Values of S(t) at t=(2DR)" for some translational diffusion coefficients. Circles show values by the nonlinear equation for z=0- l, diamonds show values by the linearized equation, triangles show ratios between them.

Saltiel, J., Sun, Y.-P. (1989), Application of the Kramers Equation to Stilbene Photoisomerization in n-Alkanes Using Translational Diffusion Coefficients to Define Microviscosity," J. Phys. Chem. 93, 8310. 

Under these circumstances, the apparent rate at which Q appears to move through the film from electrode to the outer boundary of the film depends upon the rate of the electron-transfer reaction between P and Q. Considerations of analogous reactions in homogeneous solution showed that such a process is equivalent to diffusion (76, 77). The apparent diffusion coefficient observed for a species. Dp, is composed of contributions from the physical movement of the species (governed by its translational diffusion coefficient, D) and the electron-transfer process. When bimolecular kinetics apply and the species can be considered as points, then Dp can be estimated from the Dahms-Ruff equation. 

These equations may also be applicable to the depolarized Rayleigh scattering of cylindrically symmetric small molecules in solution. In such cases one must usually use interferometric detection since the rotation is generally fast enough to give frequency shifts of the order of wavenumbers (see Section 7.8). In such cases the contribution of the translational diffusion coefficient to the I H spectrum is usually negligible. 

Equation (8.8.22) should be compared with that for a rod [Eq. (8.7.13)]. Note that the Sq(x) contribution to the spectrum depends only on the translational diffusion coefficient of the coil while the term labeled S2 is the first appreciable term which depends on the intramolecular motion. Note that it depends only on the intramolecular relaxation time of the k = 1 normal mode. 

Caroline and co-workers have recently reported measurements of translational diffusion coefficients in solutions of PS in two mixed-solvent systems at or near theta conditions. In the solvent CCb-methanol (85), they observed the diffusion theta state, defined when the coefficient y of Equation 41 equals 0.5, to occur at 25°C and a volume fraction of CCI4, (fyCCU = 0.8025. In this system there is strong preferential adsorption of the polymer for CCI4, and it is not possible to define a true theta state such that y = a = V2 and A2 = 0 simultaneously. Under diffusion theta conditions, the concentration dependence of Dt apparently is closely described by the Pyun-Fixman hard-sphere model. In the mixed solvent benzene—2 propanol, polystyrene exhibits a true theta condition at T = 25.5°C and (benzene) = 0.04. Frost and Caroline confirmed that y = 0.5 within experimental error in this system (86) and report that values of the parameter fcf are scattered between the extreme values corresponding to the predictions of Yamakawa (and Imai) and the soft-sphere model of Pyun-Fixman (or the Freed theory). 

The translational diffusion coefficient of a chain molecule is measurable using quasi-elastic light scattering (23, 24). It can be calculated from an appropriate chain model using the Kirkwood (25) equation... 

Determination of D is the first step in studying macromolecular coils by fractal analysis. D is usually estimated by finding the exponents in the Mark-Kuhn-Houwink type equation, which relate the characteristic viscosity [r ], the translational diffusion coefficient Dq, or the rate sedimentation coefficient Sq) with the molecular weight (M) of polymers [3] ... 

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