Einstein Brownian displacement equation

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. 

Equation 6.33 states that the root-mean-square displacement is proportional to the square root of the number of jumps. For very large values of n, the net displacement of any one atom is extremely small compared to the total distance it travels. It turns out, that the diffusion coefficient is related to this root-mean-square displacement. It was shown independently by Albert Einstein (1879-1955) and Marian von Smoluchowski (1872-1917) that, for Brownian motion of small particles suspended in a liquid, the root-mean-square displacement, is equal to V(2Dt), where t is the time... 

If the central particle is also in Brownian motion, the diffusion constant, D, should describe the relative motion of two particles. The relative displacement is given by Jt — xj, where a / and xj are the displacements of the two particles in the x direction measured from a given reference plane. The diffusion constant for the relative motion can be obtained from the Einstein equation for the diffusion coefficient (Chapter 2) ... 

The kinetic theory of the Brownian movement was developed by Einstein and Smoluchowski, giving equations for the mean square of the displacement in a given direction in a given time, and for the mean square of the angular velocity of rotation. These were verified by experiments. F. M. Exner had previously shown that the square of the velocity of the particles is approximately proportional to the temperature. By counting the number of particles in a given volume microscopically (or ultramicroscopically) it was found that... 

The second of (18.16) is similar to the Einstein equation for the mean square displacement of a Brownian particle in one dimension... 

Einstein interpreted diffusion as being a result of the random thermal motion of molecules. Such a random motion is caused by fluctuations in pressure in a liquid. Thus, diffusion is closely related to Brownian motion. The Brownian motion consists of zigzag motion in aU directions. It is a random walk, as discussed in Chapter 5, and is described by the parameter (x ), the square mean displacement. The equation of motion in one dimension for the Brownian motion of a particle in solution is given by... 

Brownian motion of a panicle is a result of the thermal motion of the molecular agitation of the liquid medium. Much stronger random displacement of a particle is usually observed in a less viscous liquid, smaller particle size, and higher temperature. A particle of size larger than 1 pm doesn t show a remarkable Brownian motion. There is much literature available on Brownian motion [7-9], and the Brownian motion is regarded as a diffusion process. For an isolated particle, i.e., there is no intcrparticlc action, the diffusion coefficient D , can be expressed as the Stokes-Einstein equation ... 

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