The Einstein-Smoluchowski Equation

The Einstein-Smoluchowski Equation Relates Diffusion and Friction 

Integrate both sides and use Equation (3.3) for the reversible work w = - j f dx. The minus sign here is because the force acts downwards while the positive x direction is upwards. This gives 

Because the system is in equilibrium, the Boltzmann distribution law also applies, and the quantity on the right side of Equation (18.48) must also equal cxp -iv/kT). Equating the exponents of these two expressions gives the Einstein-Smoluchowski equation. 

This relationship shows how to determine the friction coefficient if you know D. Because depends on the sizes and shapes of particles, measuring the diffusion constant D giv es some information about the structures of particles. 

For example, solving fluid flow equations (which w e don t do here see [4 ) gives Stokes s law for spherical particles, 

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The transport of dissolved species in a solvent occurs randomly through movement of the Brownian type. The particles of the dissolved substance and of the solvent continuously collide and thus move stochastically with various velocities in various directions. The relationship between the mobility of a particle, the observation time r and the mean shift (jc2) is given by the Einstein-Smoluchowski equation (in three-dimensional case)... 

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 simplified form of the Einstein-Smoluchowski equation gives for radicals 1 and 2... 

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. 

Fraction of Ions Traveling the Mean Square Distance in the Einstein-Smoluchowski Equation... 

It is experimentally observed that the counter begins to register within a few seconds of the termination of the instantaneous current pulse. Suppose, however, that one attempted a theoretical calculation based on the Einstein-Smoluchowski equation... 

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. 

It has further been shown (Section 4.2.6) that in the case of a one-dimensional random walk, depends on time according to the Einstein-Smoluchowski equation... 

The numerical coefficient 5 has entered here only because the Einstein-Smoluchowski equation = 2Dt for a one-dimensional random walk was considered. In general, it is related to the probability of the ion s jumping in various directions, notjust forward and backward. For convenience, therefore, the coefficient will be taken to be unity, in which case... 

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.)... 

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] ... 

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

The mobility of species 7, defined in Section 2.3.3, is linked to the diffusion coefficient by the Einstein-Smoluchowski equation ... 

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),... 

The Fickian diffusion constant is, on the other hand, related to the hopping of individual particles as sketched by the potential in Fig. 1.9 through the Einstein-Smoluchowski 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. 

It is interesting to inquire why there should be an effect of (n) on the rate of reaction of a small molecule with a polymer coil. We are guided by the discussion of Behzadi and Schnabel (41) concerning the reaction of 0H (produced by pulse radiolysis) with poly(ethylene oxide), poly(vinylpyrrolidone) and dextrane. First, we assume that the coil-small molecule reaction rate can be described by the Einstein-Smoluchowski equation (where Ap 0 and PqE = 1 is assumed for simplicity) ... 

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