Einstein’s relation

Here, y is the friction coefficient of the solvent, in units of ps and Rj is the random force imparted to the solute atoms by the solvent. The friction coefficient is related to the diffusion constant D of the solvent by Einstein s relation y = kgT/mD. The random force is calculated as a random number, taken from a Gaussian distribu-... 

Self Diffusion Coefficients. Diffusion coefficients may be estimated by exploiting Einstein s relation ... 

The diffusion treatment of Brownian motion in general, Einstein s relation (Eq. (6)) in particular is valid only after the Brownian particle had multiple collisions with the surrounding atoms. For short times, before any collisions occur, there is no statistics we have determinacy. 

Even without solving this equation one can draw an important conclusion. It has the same form as the diffusion equation (IV.2.8) and in fact it is the diffusion equation for the Brownian particles in the fluid. Consequently a2 is identical with the phenomenological diffusion constant D. On the other hand, a2 is expressed in microscopic terms by (2.4) or by (1.6). This establishes Einstein s relation... 

Only at very low concentrations can the coils be considered as separate spheres. In that case Einstein s relation for the viscosity of suspensions can be applied ... 

Single-component diffusion under equilibrium conditions can be monitored either by labeling some of the molecules or by following their trajectories. Considering the diffusion flux of the labeled molecules, again a proportionality relation of the type of eq 2 may be established. The factor of proportionality is called the coefficient of self-diffusion (or tracer diffusion). In a completely equivalent way [2], the self-diffusion coefficient may be determined on the basis of Einstein s relation... 

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... 

From the coordinate data stored in the trajectory files, the mean-square displacements for all tiie movable atoms were calculated, and the diffusion coefficient, D sd, computed using Einstein s relation... 

These two core principles must be modified slightly for nuclear reactions, which occur at energies so high that matter and energy can be converted into one another through Einstein s relation, E = mc. The sum of mass and energy is conserved in nuclear reactions. 

Figure 1. Random walk traces recorded by Perrin [5] Three trajectories obtained by tracing a small grain of putty at intervals of 30sec. Using Einstein s relation between the macroscopic gas constant and the diffusion constant, Perrin found a quite accurate result for Avogadro s number. Refined results were successively obtained by Westgren and Kappler [7,8],...

Einstein s equation, its deduction, 449 Einstein s relation, 448, 451 Einstein-Schmolukowski theory, its mathematics, 583 the calculation of the fraction counted,... 

In the second place, the radius a of the electron has an actual physical significance by a we understand that length which satisfies Einstein s relation a. e ja = mc, where a is a numerical factor of the order of magnitude 1. We therefore have... 

In contrast to the traditional method, the nmr pulsed field gradient technique (4-10) is applied to samples under macroscopic equilibrium. By monitoring the precessional phases of the nuclear spins around an external magnetic field, this technique allows the determination of the displacements r(A> of the individual molecules of the sample during an observation time A of typically 1...100 ms. Using Einstein s relation... 

Taking into account that and using Einstein s relation Eq. (28) results... 

Estimating the mean exchange time between two positions at separation Az by using Einstein s relation... 

In Eq 7.25 ( ) is the maximum packing volume fraction. Thus, the magnitude of the shielding function My) depends on the reduced volume fraction, = ( ) / ( ). At low concentration, —> 0, the shielding factor vanishes and Einstein s relation is recovered. However, at high concentration, —> 1, the shielding function and relative viscosity both... 

Considering that the viscosity of a flxiid containing spherical particles dissolved in it, r s, increases with the volume fraction cp according to Einstein s relation. 

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