Diffusion coefficient in liquids

Diffusion coefficients in liquids are about ten thousand times slower than those in dilute gases. To see what this means, we again calculate the penetration distance which was the distance we found central to unsteady diffusion. As an example, consider benzene diffusing into cyclohexane with a diffusion coefficient of about 2 10 cm /sec. At time zero, we bring the benzene and cyclohexane into contact. After 1 second, the diffusion has penetrated 0.004 cm, compared with 0.3 cm for gases after 1 minute, the penetration is 0.03 cm, compared with 4 cm after 1 hour, it is 0.3 cm, compared with 30 cm. 

The sloth characteristic liquid diffusion means that diffusion often limits the overall rate of processes occurring in liquids. In chemistry, diffusion limits the rate of acid-base reactions in physiology, diffusion limits the rate of digestion in metallurgy, diffusion can control the rate of surface corrosion in the chemical industry, diffusion is responsible for the rates of liquid-liquid extractions. Diffusion in liquids is important because it is slow. 

1 Liquid Diffusion Coefficients From the Stokes-Einstein Equation 

The most common basis for estimating diffusion coefficients in liquids is the Stokes-Einstein equation. Coefficients calculated from this equation are accurate to 

Note Known to very high accuracy, and so often used for calibration ci is in moles per liter. Source Data from Cussler (1976) and Poling et at. (2001). 

For an adequate description of transport phenomena the diffusion coefficient should not be left out. The performance of lithium-ion batteries is directly connected to the mass transport in the electrolyte, as studied by Sawai et al. [479]. He showed that the lithium-ion diffusion coefficient in the solution is even more important for the rate capability of graphite than the diffusion of lithium in the solid electrode. 

Although being one of the key determinants for the power output, very few diffusion data in liquid electrolytes are available. This has not changed since 1947, when Harned wrote There are few domains of physical science in which so much experimental effort over nearly a century has yielded so little accurate data as the field of diffusion in liquid systems [480]. 

For highly diluted systems an exact theory is modeled [481], but concentrated solutions bring complex problems, such as short-range forces or convection. Generally, diffusion can be described by Pick s first law [472]  

To estimate diffusion coefficients in liquid electrolytes, the Stokes-Einstein-relation [482] is generally used  

Measurements of diffusion coefficients are experimentally a difficult field. Cus-sler wrote about his work [475] ... that measurement of diffusion is a Holy Grail requiring noble knights who dedicate their lives to the quest.  

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Chemical Reaction Engineering Aspects of Homogeneous Hydrogenations Table 45.4 Estimation of the diffusion coefficient in liquid phase. 

Othmer, D.F. and M.S. Thakar (1953), Correlating diffusion coefficients in liquids, Industrial and... 

When one or more of the chemical reactions is sufficiently slow in comparison with the rate of diffusion to and away from the interface of the various species taking part in an extraction reaction, such that diffusion can be considered instantaneous, the solvent extraction kinetics occur in a kinetic regime. In this case, the extraction rate can be entirely described in terms of chemical reactions. This situation may occur either when the system is very efficiently stirred and when one or more of the chemical reactions proceeds slowly, or when the chemical reactions are moderately fast, but the diffusion coefficients of the transported species are very high and the thickness of the two diffusion films is close to zero. In practice the latter situation never occurs, as diffusion coefficients in liquids generally do not exceed 10 cm s, and the depth of the diffusion films apparently is never less than 10 cm. 

The molecular diffusion coefficients in liquid phase were estimated from the correlations Wilke and Chang (9) for organic solutions and Hayduk and Minhas (10) for aqueous solutions, respectively. The solvent viscosities needed in the correlations were obtained from the empirical equation based on the experimental data,... 

Although there are a lot of data in the literature regarding diffusion coefficients in liquids or then calculation from molecular properties (Appendix I, Section 1.2), it is not the case for diffusion coefficients in solids, where the phenomena appearing are more complex. In solids, the molecule may be forced to follow a longer and tortuous path due to the blocking of the cross-sectional area, and thus the diffusion is somehow impaired. Several models have been developed to take into consideration this effect in the estimation of diffusion coefficients, leading, however, to a variety of results. 

A number of useful empirical and semiempirical relations have been proposed for predicting diffusion coefficients in liquids a rather complete summary of these has been given by Treybal (T6, pp. 102 et seg.). Most of these are valid only for nonionic systems at very low concentrations. 

Equation (2) may be used for the rate constant k of a chemical reaction or applied to the diffusion coefficient in liquid or solid phases or to the fluidity of liquids (reciprocal of dynamic viscosity) or to the specific electrical conductivity of semiconductors. 

In chemical heterogeneous catalysis, it is common to use highly porous catalysts that come in particles of millimeter to centimeter size to increase the effective catalyst surface. In practical electrocatalysis, in particular applying electrocatalysis in fuel cells, it is also usual to use highly porous— although accounting for the low diffusion coefficients in liquid electrolytes compared to gases, 10 5 cm2/sec vs 1 cm2/sec, much smaller—catalyst particles. 

The sorption coefficient (K) in Equation (2.84) is the term linking the concentration of a component in the fluid phase with its concentration in the membrane polymer phase. Because sorption is an equilibrium term, conventional thermodynamics can be used to calculate solubilities of gases in polymers to within a factor of two or three. However, diffusion coefficients (D) are kinetic terms that reflect the effect of the surrounding environment on the molecular motion of permeating components. Calculation of diffusion coefficients in liquids and gases is possible, but calculation of diffusion coefficients in polymers is much more difficult. In the long term, the best hope for accurate predictions of diffusion in polymers is the molecular dynamics calculations described in an earlier section. However, this technique is still under development and is currently limited to calculations of the diffusion of small gas molecules in amorphous polymers the... 

Because of the low diffusion coefficients in liquids, the particle size for packed columns in LC needs to be very small (see section 7.2.1). For the same reason, all external volumes and diameters need to be minimized. This may easily be understood if we express the standard deviation in volume units (ctv) in the parameters that represent the dimensions of the column ... 

In relation 8.6 - rA is the reaction rate in kmol/(kg catalyst s), pc density of solid catalyst, R particle radius, I ) i diffusion coefficient in liquid and CAs reactant concentration at the catalyst surface. If CWp 1 there are no diffusion limitations, but if CWP 1 the catalyst effectiveness is severely affected. 

In general, diffusion coefficients in gases can be often be predicted accurately. Predictions of diffusion coefficients in liquids are also possible using the Stokes-Einstein equation or its empirical parallels. On the contrary in solids and polymers, models allow coefficients to be correlated but predictions are rarely possible. 

Othmer, D. F., and M. S. Thakar, Correlating Diffusion Coefficients in Liquids, Industrial and Engineering... 

Therefore, if the observed peak is allowed to have a volume 10% greater than the column peak volume, the extra-column peak volume should be one-half (ca. 46%) of the column peak volume. Table 1 lists column peak volumes and maximum extra-column peak volumes for various types of columns for LC because the largest contribution from this extra-column volume should be considered in liquid phase separations, the diffusion coefficients in liquids are very small. 

Diffusion coefficients in gases are usually about 1000 times larger than diffusion coefficients in liquids. 

Assumption 3 We consider the axial dispersion coefficient as constant. It has been shown that the dependence of diffusion coefficients in liquids on the pressure is not very important [20]. It has been reported that the HETP of insulin increases slightly with increasing column average pressure, from 50 to 250 bar. We neglect this ect here. The coefficient in Eqs. 2.1a to 2.1e accormts for the... 

The diffusion coefficient in a gas is proportional to -jP, the constant of proportionality being a rather slowly increasing function of temperature. The estimation of the diffusion coefficient in liquids is discussed briefly by Sherwood and Satterfield it is proportional to r// , where /z. is the viscosity. At atmospheric pressure and ordinary temperatures, the order of magnitude of D for a gas is 0.1-1 cm-/sec and for liquids it is smaller by a factor of about 10. ... 

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