Diffusion equation Einstein-Smoluchowski

Perrin model and the Johansson and Elvingston model fall above the experimental data. Also shown in this figure is the prediction from the Stokes-Einstein-Smoluchowski expression, whereby the Stokes-Einstein expression is modified with the inclusion of the Ein-stein-Smoluchowski expression for the effect of solute on viscosity. Penke et al. [290] found that the Mackie-Meares equation fit the water diffusion data however, upon consideration of water interactions with the polymer gel, through measurements of longitudinal relaxation, adsorption interactions incorporated within the volume averaging theory also well described the experimental results. The volume averaging theory had the advantage that it could describe the effect of Bis on the relaxation within the same framework as the description of the diffusion coefficient. 

The fundamental theory of electron escape, owing to Onsager (1938), follows Smoluchowski s (1906) equation of Brownian motion in the presence of a field F. Using the Nemst-Einstein relation p = eD/kRT between the mobility and the diffusion coefficient, Onsager writes the diffusion equation as... 

The Einstein-Smoluchowski equation, derived in Appendix 4.1, relates the mean thermal displacement, X, to the diffusion coefficient and mean lifetime. For a surface ... 

The Peclet number compares the effect of imposed shear (known as the convective effect) with the effect of diffusion of the particles. The imposed shear has the effect of altering the local distribution of the particles, whereas the diffusion (or Brownian motion) of the particles tries to restore the equilibrium structure. In a quiescent colloidal dispersion the particles move continuously in a random manner due to Brownian motion. The thermal motion establishes an equilibrium statistical distribution that depends on the volume fraction and interparticle potentials. Using the Einstein-Smoluchowski relation for the time scale of the motion, with the Stokes-Einstein equation for the diffusion coefficient, one can write the time taken for a particle to diffuse a distance equal to its radius R, as... 

Now, everything falls into place We set out to study the laws of random walk by using the simple model of Fig. 18 and found the Bernoulli coefficients. We then saw that for large n (which is equivalent to large times), the Bernoulli coefficients can be approximated by a normal distribution whose standard deviation, a, grows in proportion to the square root of time, tm (Eq. 18-3). And now it turns out that the solution of the Fick s second law for unbounded diffusion is also a normal distribution. In fact, the analogy between Eqs. 18-3b and 18-17 gave the basis for the law by Einstein and Smoluchowski (Eq. 18-17) that we used earlier (Eq. 18-8). The expression (2Dt)U2 will also show up in other solutions of the diffusion equation. 

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]. 

Einstein-Smoluchowski equation — Relationship between diffusion coefficient D, average width of a jump A of a microscopic species (atom, ion, molecule) involved in diffusion, and average time r between two jumps... 

It is possible to calculate diffusion coefficients by computing the mean square displacement distance and dividing by 6t. [The basic relation here is the Einstein-Smoluchowski equation (Section 4.2.6)]. The values are surprisingly good and are shown in Table 2.26. 

The diffusion coefficient D has appeared in both the macroscopic (Section 4.2.2) and the atomistic (Section 4.2.6) views of diffusion. How does the diffusion coefficient depend on the structure of the medium and the interatomic forces that operate To answer this question, one should have a deeper understanding of this coefficient than that provided hy the empirical first law of Tick, in which D appeared simply as the proportionality constant relating the flux / and the concentration gradient dc/dx. Even the random-walk intapretation of the diffusion coefficient as embodied in the Einstein-Smoluchowski equation (4.27) is not fundamental enough because it is based on the mean square distance traversed by the ion after N stqis taken in a time t and does not probe into the laws governing each stq) taken by the random-walking ion. 

At first the results arising from the Einstein-Smoluchowski equation ( = 2Dt) may seem difficult to understand. Thus, the diffusion considered in the equation is random. Nevertheless, the equation tells us that there is net movement in one direction arising from this random motion. Furthermore, it allows us to calculate how far the diffusion front has traveled. Is there something curious about randomly moving particles covering distances in one direction Comment constructively on this apparently anomalous situation. 

The Einstein-Smoluchowski equation, = 2Dt, gives a measure of the mean-square displacements of a diffusing particle in a time t. There is the mean-square distance traveled by most of the ions. Common observation using dyes or scents shows that diffusion of some particles occurs far ahead of the diffusion front represented by the = 2Dt equation. Determine the distance of this Einstein-Smoluchowski diffusion front for a colored ion diffusing into a solution for 24 hr (D = 3.8 x 10 cm s ). Determine for the same solution how far the farthest 1% of the total diffused material diffused in the same time. Discuss how it is possible that one detects perfume across the space of a room in (say) 30 s. 

The Einstein-Smoluchowski equation, = 2Dt, is a phenomenological equation derived for diffusion along one coordinate. (For example, after the release of a barrier, along a tube containing a liquid.) However, it also applies to any medium. Suppose, now, that metal ions, (e.g., Pt) are deposited on a Pd substrate. Calculate how far the Pt would diffuse into the Pd in 6 weeks. (The diffusion coefficient of Pt into Pd can be estimated from other data as 9 x 10- at295 K.)... 

This form is useful in the generalization of Eq. (485) to fractional diffusion. The investigation of the diffusion equation (485) began when Louis Bachellier (Jules Poincare s student) wrote his thesis in 1900. It was called The Theory of Speculations and was devoted to the evolution of the stock market. Many of the most famous scientists have contributed to our knowledge of diffusion processes, amongst them Einstein, Langevin, Smoluchowski, Fokker, Planck, Levy, and others. 

Self-diffusion is the random translational motion of ensembles of particles (molecules or ions) originating from their thermal energy. It is well known that diffusion, which is closely related to the molecular size of the diffusing species, is given by the Einstein-Smoluchowski equation, Eq. (6.1) [8] ... 

Fig.V-4. A schematic representation of diffusion in the derivation of the Einstein-Smoluchowski equation...

E21.30(b) The Einstein-Smoluchowski equation related the diffusion constant to the unit jump distance and lime... 

The relative importance of migration and diffusion can be gauged by comparing Ud with the steady-state migrational velocity, u, for an ion of mobility Wj in an electric field (Section 2.3.3). By definition, v = where % is the electric field strength felt by the ion. From the Einstein-Smoluchowski equation, (4.2.2),... 

Given the central role of the expulsion rate constant for micellar stability, formation, and dissociation, it is essential to determine the physical governing factors and functional form. Aniansson and Wall based their calculations [54] on a general diffusion in an external potential. In this approach, the diffusion coefficient, D(r) is dependent on the position, r, due to the potential V(r). In a sphero-symmetric system, we can imagine that the diffusion of a unimer only depends on the distance, r, from the origin and this problem can be summarized in a Einstein-Smoluchowski type equation ... 

The photoredox properties of porphyrin molecules at interfaces can be studied by time-resolved spectroscopic and photoelectrochemical techniques. A key difference between both approaches is that the latter only probes molecules located at distances from the interface not larger than the characteristic diffusion length of the excited state. According to the Einstein-Smoluchowski equation, the... 

This equation is called the Einstein-Smoluchowski equation. The function P(x,t) satisfies the diffusion equation with... 

The average move during time At is measured f is the frictional coefficient and No is Avogadro s number. If Ax is identified with X and At with t, just comparing with the Einstein-Smoluchowski equation 5.146 simply provides the diffusion constant D. 

The difficulties mentioned can be resolved to some extent by recognizing that Eq. 1 holds only in the limit of extreme dilution - a situation wherein solute/solute interactions are negligible and the diffusion and mobility relate through solvent properties. In this limit, statistical treatments combine with force balances to yield an Einstein-Smoluchowski relation [9,10] (for electrolytic diffusion this is also sometimes called the Nemst-Einstein equation [11])... 

The Einstein-Smoluchowski Equation Relates Diffusion and Friction... 

The reptation model, like the Rouse model, supposes that the friction involved in dragging the chain through its tube is proportional to the chain length, 5 = N i, Equation (33.33). The diffusion constant Dtubo for the chain moving through the tube is given by the Einstein-Smoluchowski relation, Equa-... 

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