Einstein relations equation

We will perform a simple calculation in two parts. First, we will divide by the Avogadro number to obtain the energy per bond. Second, we convert the energy to a wavelength X with the Planck-Einstein relation (Equation (9.4)), E = hc/X. 

From the Stokes-Einstein relation, Equation (106), we then get... 

Solvent characteristics that influence the diffusion and extraction are found to be viscosity (t ) and polarity (e). For spherical solutes, the diffusion coefficient depends on the solvent according to the Stokes-Einstein relation (Equation 19). From this, it follows that the diffusion coefficient linearly increases with T/t. Hence, the permeability increases linearly with the reciprocal viscosity of the membrane solvent (12), Dozol et al. plotted the permeability P vs. and obtained a linear fit... 

Substituting the Nemst-Einstein relation Equation (15.6) into Eqnation (15.12) gives ... 

The peaks in Figure 3.13a (counted from the left) are ascribed to the following modes peak 3 (dominant) to cooperative or translational diffusion, peak 4 to selfdiffusion of the copolymer chain, and peak 5 to the diffusion of clusters of a hydrodynamic radius equal to about 120 nm, as estimated via the Stokes-Einstein relation (Equation 3.27). Peaks 1 and 2 correspond to thermal diffusion and to self-diffusion of solvent molecules, respectively. 

The Einstein relation can be rearranged to the following equation for relating Schmidt numbers at two temperatures ... 

For heterogeneous media composed of solvent and fibers, it was proposed to treat the fiber array as an effective medium, where the hydrodynamic drag is characterized by only one parameter, i.e., Darcy s permeability. This hydrodynamic parameter can be experimentally determined or estimated based upon the structural details of the network [297]. Using Brinkman s equation [49] to compute the drag on a sphere, and combining it with Einstein s equation relating the diffusion and friction coefficients, the following expression was obtained ... 

Einstein showed that mass and energy are equivalent. Energy can be converted into mass, and mass into energy. They are related by Einstein s equation ... 

Baxendale and Wardman (1973) note that the reaction of es with neutrals, such as acetone and CC14, in n-propanol is diffusion-controlled over the entire liquid phase. The values calculated from the Stokes-Einstein relation, k = 8jtRT/3jj, where 7] is the viscosity, agree well with measurement. Similarly, Fowles (1971) finds that the reaction of es with acid in alcohols is diffusion-controlled, given adequately by the Debye equation, which is not true in water. The activation energy of this reaction should be equal to that of the equivalent conductivity of es + ROH2+, which agrees well with the observation of Fowles (1971). 

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... 

Eimco High-Capacity thickener, 22 66 Einsteinium (Es), 1 463-491, 464t electronic configuration, l 474t ion type and color, l 477t metal properties of, l 482t Einstein relation, 22 238. See also Einstein s viscosity equation filled networks and, 22 571, 572 Einstein s coefficient, 14 662 Einstein s equation, 7 280 21 716 23 99 Einstein s law, 19 108 Einstein s viscosity equation, 22 54. 

These equations show that the ratio h/ Iu is proportional to the rate constant Hq for excimer formation. Assuming that the Stokes-Einstein relation (Eq. 4.12) is valid, ki is proportional to the ratio T /tj, tj being the viscosity of the medium. Application to the estimation of the fluidity of a medium will be discussed in Chapter 8. 

Various modifications of the Stokes-Einstein relation have been proposed to take into account the microscopic effects (shape, free volume, solvent-probe interactions, etc.). In particular, the diffusion of molecular probes being more rapid than predicted by the theory, the slip boundary condition can be introduced, and sometimes a mixture of stick and slip boundary conditions is assumed. Equation (8.3) can then be rewritten as... 

At low pressure, the only interactions of the ion with its surroundings are through the exchange of photons with the surrounding walls. This is described by the three processes of absorption, induced emission, and spontaneous emission (whose rates are related by the Einstein coefficient equations). In the circumstances of interest here, the radiation illuminating the ions is the blackbody spectrum at the temperature of the surrounding walls, whose intensity and spectral distribution are given by the Planck blackbody formula. At ordinary temperatures, this is almost entirely infrared radiation, and near room temperature the most intense radiation is near 1000 cm". ... 

Equation (2.158) is the Einstein relation relating the mobility and diffusivity tensors. 

Note that, in Equation 12.3, is a negative quantity since it is opposed to the direction x (see Fignre 12.5) and this explains the negative sign in the second member of Equation 12.3 where the absolnte qnantity It/ l is introduced. Equation 12.4 relates (4 to P- Useful alternative expressions for F are obtained by combining Equation 12.4 with Einstein s equation... 

Equation (6.41) is known as the Nernst-Einstein relation, originally deduced for the mobility of colloid particles in a liquid, but also valid for ionic solids. 

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 ... 

It should be noted that this derivation contains no assumptions about the shape of the particles. However, when the particles are assumed to be spherical, we can substitute Equation (8) for/, and the resulting equation for the diffusion coefficient is the well-known Stokes-Einstein relation. 

Equation (5.56) relates the correlation factor fA with the cross coefficient LAA . From the Nernst-Einstein relation we know that LAA = bA-cA = DAcA/R T. For a tracer experiment with a negligible fraction of A, the jump conservation requires that Da = Dv-Nv, so that instead of Eqn. (5.56) we have... 

Obviously, using the Einstein relation, Eq. (4.25) might have been written down right away as soon as the rotary mobility coefficient had been found. This is equally valid, of course, for both the Landau-Lifshitz and Gilbert representations of the magnetodynamic equation. Using formula (4.16) one finds... 

Equation (96) is known as the linear diffusion equation since the lowest-order field dependence is linear. Thus we have a microscopic derivation of the Einstein relation, eqn. (98). This relation is normally derived from quite different considerations based on setting the current equal to zero in the linear diffusion equation and comparing the concentration profile C (x) with that predicted by equilibrium thermodynamics. 

---