Sutherland-Einstein equation

The proportionality factor F is called friction factor and is identical for diffusion and sedimentation. Using the Einstein Sutherland equation... 

According to the Einstein-Sutherland equation, the diffusion coefficient D is related to the molecular frictional coefficientfo. ... 

The electrophoretic mobility p is defined as the velocity of movement induced by the effect of an electric field of 1 V/cm. On inserting into the Einstein-Sutherland Equation (7-17), one obtains, with Equation (7-26),... 

Here D is the diffusion coefficient of the solute, R is the gas constant, Tis the temperature, and y is the activity coefficient of the solute. The second expression employs the Einstein-Sutherland equation, which may be derived from the thermodynamics of irreversible processes. 

Figure 3-35 Comparison of calailated diffusivity and experimental diffusivity of noble gas elements in water. Noble gas radius from Zhang and Xu (1995). Molecular diffusivity data are from Jahne et al. (1987) except for Ar (Cussler, 1997). A different symbol for Ar is used because different sources for diffusion data may not be consistent. The solid curve is calculated from the Einstein equation, and the dashed curve is calculated from the Sutherland equation. The curve from Glasstone et al. (1941) is outside the scale.

By the classical Stokes-Einstein (SE) (or Einstein-Sutherland) relation we will describe our experimental results between the individual ion diffusion coefficients in Fig. 10 and the viscosity in Fig. 11. The SE equation is given as... 

If E is measured as volts per meter, then dx/dt is called the mobility of the ion for the chosen experimental conditions. The coefficient / has a theoretical foundation in hydrodynamics, and a functional relationship between / and the coordinates of the particle can be derived for a few regular shapes (for a sphere it is known as the Stokes equation, but there are mathematical solutions for ellipsoids and cylinders). It is also known that / is inversely proportional to the randomizing effects of diffusion of a large number of ions (Einstein-Sutherland relationship). Essentially, Eq. (6) describes the movement of a single ion under the influence of an electrical field. Rarely, if ever, can one ion be studied experimentally, because at finite concentrations of ions there are > 10 ions per liter (a 10 ° M solution of 0.1 /xg liter for an ion of relative mass 10 contains 10 ions per liter). Diffusion of this population of ions spreads the boundary about an elec-trophoretically transported point called the centroid, and it is the velocity of this point that is described by Eq. (6) for experimental situations. The centroid or first moment (x) can be evaluated from Eq. (7) using a set of rectangular coordinates determined experimentally over an elec-trophoresing boundary. 

Diffusion coefficients may be estimated using the Wilke-Chang equation (Danckwerts, 1970), the Sutherland-Einstein equation (Gobas et al., 1986), or the Hayduk-Laudie equation (Tucker and Nelken, 1982), which state that Dw values decrease with the molar volume (Vm) to the power 0.3 to 0.6. Alternatively, the semi-empirical Worch relation may be used (Worch, 1993), which predicts diffusion coefficients to decrease with increasing molar mass to the power of 0.53. These four equations yield very similar D estimates (factor of 1.2 difference). Using the estimates from the most commonly used Hayduk-Laudie equation... 

This important equation is commonly known as the Einstein or the Stokes-Einstein equation, though it was obtained independently by Sutherland. 

The Sutherland-Einstein equation has very often been used to obtain information about the radii of the diffusing molecules from the observed values of their diffusion coefficients. In the case of translational free diffusion of neutral molecules the driving force per mole is the negative gradient of the chemical potential. Thus,... 

The Stokes-Einstein equation, one of tire earliest theoretical expressions for liquid diffusivities, viewed the diffusion process as a hydrodynamic phenomenon in which the thermal motion of the molecules is resisted by a Stokesian drag force. This theory, along with subsequent modification by Sutherland and Eyring, established the following proportionality for the diffusion coefficient ... 

As the standard, the Stokes-Einstein equation has often been extended and adapted. Two adaptations deserve special mention. The first is for small solutes. For this case, the factor 6n in Eq. 5.2-1 is often replaced by a factor of 47t or of two. The substitution of 4jt can be rationalized on mechanical grounds as signifying solvent shpping past the surface of the solute molecule (Sutherland, 1905). The factor of two can be supported with the theory of absolute reaction rates (Glasstone et al., 1941). Neither substitution always works. 

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