Einstein relations

But by the Einstein relation we know that the energy of a single photon on frequency oi is given by jod, and hence the total energy in tire field is... 

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by ... 

The relation between the microscopic friction acting on a molecule during its motion in a solvent enviromnent and macroscopic bulk solvent viscosity is a key problem affecting the rates of many reactions in condensed phase. The sequence of steps leading from friction to diflfiision coefficient to viscosity is based on the general validity of the Stokes-Einstein relation and the concept of describing friction by hydrodynamic as opposed to microscopic models involving local solvent structure. In the hydrodynamic limit the effect of solvent friction on, for example, rotational relaxation times of a solute molecule is [ ]... 

Supercritical Mixtures Dehenedetti-Reid showed that conven-tionaf correlations based on the Stokes-Einstein relation (for hquid phase) tend to overpredict diffusivities in the supercritical state. Nevertheless, they observed that the Stokes-Einstein group D g l/T was constant. Thus, although no general correlation ap es, only one data point is necessaiy to examine variations of fluid viscosity and/or temperature effects. They explored certain combinations of aromatic solids in SFg and COg. 

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

The oxide solid elecU olytes have elecuical conductivities ranging from lO Q cm to 10 cm at 1000°C and these can be converted into diffusion coefficient data, D, for die oxygen ions by the use of the Nernst-Einstein relation... 

Center-of-mass translational motion in MD simulations is often quantified in tenns of diffusion constants, D, computed from the Einstein relation. 

Analysis of neutron data in terms of models that include lipid center-of-mass diffusion in a cylinder has led to estimates of the amplitudes of the lateral and out-of-plane motion and their corresponding diffusion constants. It is important to keep in mind that these diffusion constants are not derived from a Brownian dynamics model and are therefore not comparable to diffusion constants computed from simulations via the Einstein relation. Our comparison in the previous section of the Lorentzian line widths from simulation and neutron data has provided a direct, model-independent assessment of the integrity of the time scales of the dynamic processes predicted by the simulation. We estimate the amplimdes within the cylindrical diffusion model, i.e., the length (twice the out-of-plane amplitude) L and the radius (in-plane amplitude) R of the cylinder, respectively, as follows ... 

For relatively higher volume fractions (<0.I5), a modification of the Einstein relation has been proposed [211 ... 

The flux by diffusion is described by the diffusivity Di and the migration by the conductivity cr-. The conductivity is proportional to the product of the mobility and the concentration of the mobile species. The diffusivity and mobility are related by the Nernst-Einstein relation [3J. The flux is in general given by... 

A number of bulk simulations have attempted to study the dynamic properties of liquid crystal phases. The simplest property to calculate is the translational diffusion coefficient D, that can be found through the Einstein relation, which applies at long times t ... 

Dunning, JW Angus, JC, Particle-Size Measurement by Doppler-Shifted Laser Light, a Test of the Stokes-Einstein Relation, Journal of Applied Physics 39, 2479, 1968. 

Since thermal agitation is the common origin of transport properties, it gives rise to several relationships among them, for example, the Nemst-Einstein relation between diffusion and conductivity, or the Stokes-Einstein relation between diffusion and viscosity. Although transport... 

Since the diffusion coefficient is the infinite-time integral of the velocity correlation function, we have the Einstein relation, D = kBT/Q. 

The mobility of eh was determined by measuring the equivalent conductance following pulse irradiation (Schmidt and Buck, 1966 Schmidt and Anbar, 1969). After correcting for the contribution of H30 and OH ions, they found the equivalent conductance of eh = 190 10 mho cm2. From this, these authors obtained the mobility p(eh) = 1.98 x 10"3 cm2/v.s. and the diffusion coefficient D(eh) = 4.9 x 10-5 cm2/s using the Nernst-Einstein relation, with about 5% uncertainty. The equivalent conductance of eh is the same as that for the OH - ion within experimental uncertainty. It is greater than that of the halide ion and smaller than that of eam... 

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). 

Williams (1964) derived the relation T = kBTrQV3De2, where T is the recombination time for a geminate e-ion pair at an initial separation of rg, is the dielectric constant of the medium, and the other symbols have their usual meanings. This r-cubed rule is based on the use of the Nernst-Einstein relation in a coulom-bic field with the assumption of instantaneous limiting velocity. Mozumder (1968) criticized the rule, as it connects initial distance and recombination time uniquely without allowance for diffusional broadening and without allowing for an escape probability. Nevertheless, the r-cubed rule was used extensively in earlier studies of geminate ion recombination kinetics. 

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... 

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. 

Next, we determine the wavelength, and start by rearranging the Planck-Einstein relation to make wavelength X the subject ... 

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