Stokes-Einstein and Free-Volume Theories

Stokes-Einstein and Free-Volume Theories The starting point for many correlations is the Stokes-Einstein equation. This equation is derived from continuum fluid mechanics and classical thermodynamics for the motion of large spherical particles in a liquid. For this case, the need for a molecular theory is cleverly avoided. The Stokes-Einstein equation is (Bird et al.) 

TABLE 5-13 Relationships for Diffusivities of Multicomponent Gas Mixtures at Low Pressure 

Another advance in the concepts of liquid-phase diffusion was provided by Hildebrand [Science, 174, 490 (1971)] who adapted a theory of viscosity to self-diffusivity. He postulated that DAA = B(V-V )/V , where DAA is the self-diffusion coefficient, V is the molar volume, and V is the molar volume at which fluidity is zero (i.e., the molar volume of the solid phase at the melting temperature). The difference (V -V ) can be thought of as the free volume, which increases with temperature and B is a proportionality constant. 

Ertl and Dullien (ibid.) found that Hildebrand s equation could not fit their data with B as a constant. They modified it by applying an empirical exponent n (a constant greater than unity) to the volumetric ratio. The new equation is not generally useful, however, since there is no means for predicting n. The theory does identify the free volume as an important physical variable, since n 1 for most liquids implies that diffusion is more strongly dependent on free volume than is viscosity. 

Dilute Binary Nonelectrolytes General Mixtures These correlations are outlined in Table 5-14. 

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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... 

Another microscopic approach to the viscosity problem was developed by Gierer and Wirtz (1953) and it is worthwhile describing the main aspects of this theory, which is of interest because it takes account of the finite thickness of the solvent layers and the existence of holes in the solvent (free volume). The Stokes-Einstein law can be modified using a microscopic friction coefficient ci micro... 

The conductivity of molten salts has been related to the existence of free volume in the melt [268] and it was argued that the Arrhenius activation energy Ba should be lower than the corresponding one for ion diffusion in the melt (see below), Bd as was in fact found. This would explain why the conductivity does not adhere to the Nemst-Einstein relation A = F D+ + D-)/RT for the diffusion or to Stokes law as mentioned above. The significant structure theory in this case [160] specifies that only the solid-like particles contribute to the conductivity. Their number per unit volume is where Fsd is the molar volume of the solid... 

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