Einstein’s diffusion laws

Let us first consider isothermal diffusion. Einstein s diffusion law can be written... 

Einstein s diffusion law relates the average distance travelled in Brownian motion, L, to the time by means of the Fickian diffusion coefficient (in this case an average value, since the radical is... 

As expected, the larger the diffusion coefficient, the lower the drag force. Of course, Einstein s diffusion law can be combined with Stokes equation for/ and the resulting equation is called Stokes-Einstein law (Problem 8.1). Together with the equation for the Brownian displacement, it was used by Perrin for early, rather accurate calculations of the Avogadro number. 

In combination with a general force balance, Einstein s diffiision law results in Equation 8.5b, which permits the estimation of the mass of each particle. Thus, upon combining diffiision experiments (for obtaining D) and sedimentation (gravitational) experiments (for obtaining ), we can estimate the mass of colloidal particles without any assumption about their shape. Finally, due to Einstein s diffiision law Df= kgT), the ratio f/fo is equal to DqID, where Do is the diffusion coefficient of a system containing the equivalent unsolvated spheres. 

By equating Fiek s seeond law and the Stokes-Einstein equation for diffusivity, Smoluehowski (1916,1917) showed that the eollision frequeney faetor takes the form... 

Pick s laws describe the interactions or encounters between noninteracting particles experiencing random, Brownian motion. Collisions in solution are diffusion-controlled. As is discussed in most physical chemistry texts , by applying Pick s Pirst Law and the Einstein diffusion relation, the upper limit of the bimolecular rate constant k would be equal to... 

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. 

By comparing this result with Fick s first law (Eq. 18-6), we get the Stokes-Einstein relation between the diffusivity in aqueous solutions and the solution viscosity q ... 

The critical nucleus of a new phase (Gibbs) is an activated complex (a transitory state) of a system. The motion of the system across the transitory state is the result of fluctuations and has the character of Brownian motion, in accordance with Kramers theory, and in contrast to the inertial motion in Eyring s theory of chemical reactions. The relationship between the rate (probability) of the direct and reverse processes—the growth and the decrease of the nucleus—is determined from the condition of steadiness of the equilibrium distribution, which leads to an equation of the Fourier-Fick type (heat conduction or diffusion) in a rod of variable cross-section or in a stream of variable velocity. The magnitude of the diffusion coefficient is established by comparison with the macroscopic kinetics of the change of nuclei, which does not consider fluctuations (cf. Einstein s application of Stokes law to diffusion). The steady rate of nucleus formation is calculated (the number of nuclei per cubic centimeter per second for a given supersaturation). For condensation of a vapor, the results do not differ from those of Volmer. 

The diffusion coefficient of a suspended material is related to the frictional coefficient of the particles by Einstein s law of diffusion ... 

For Brownian motion, the collision frequency function is based on Fick s first law with the particle s diffusion coefficient given by the Stokes-Einstein equation. The Stokes-Einstein relation states that... 

The methods described so far for studying self-diffusion are essentially based on an observation of the diffusion paths, i.e. on the application of Einstein s relation (eq 3). Alternatively, molecular self-diffusion may also be studied on the basis of the Fick s laws by using iso-topically labeled molecules. As in the case of transport diffusion, the diffusivities are determined by comparing the measured curves of tracer exchange between the porous medium and the surroundings with the corresponding theoretical expressions. As a basic assumption of the isotopic tracer technique for studying self-diffusion, the isotopic forms are expected to have... 

The theoretical efficiency of a separation - as high as one million plates/metre in a column of length L - can be calculated from its effective length I and from the diffusion coefficient D (cm /s). This latest parameter is linked to the dispersion a and to the migration time via Einstein s law (cr = 2D Expression 8.9... 

Svedberg s primary focus as a physical chemist was the field of colloid chemistry. Colloids are mixtures of very small particles that when dispersed in solvents are not dissolved, but are held in suspension by various actions of the solvent. Svedberg and his collaborators studied the interaction of colloid suspensions with light and their sedimentation processes. These studies showed that the gas laws could be applied to colloidal systems. Svedberg s Ph.D. thesis on the diffusion of platinum colloidal particles elicited a response from Albert Einstein, since it supported Einstein s theory concerning the Brownian motions of colloidal particles. 

To determine the value of the diffusivity that connects the two approaches, we follow Einstein s thermodynamic arguments given in Section 5.2 for evaluating the translational Brownian diffusion coefficient. The basis for this is the random Brownian motion of the monomer units in the gel, which translates into the gel osmotic pressure. If, as above, the flow through the gel is assumed to follow Darcy s law (Eq. 4.7.7), then we may write the applied hydrodynamic force per mole of solution flowing through the gel as... 

The ability to use high-potential fields (100-900 V/cm) provides faster migration and flow rates, leading to rapid, highly efficient separations. Using Einstein s law of diffusion, the statistical equivalence of variance, and number of theoretical plates, the maximum separation efficiency (N) is given by Eq. (32) (N = -l- jUeo)U/2D), where D is the... 

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