Diffusion Coefficients in Binary Mixtures

1 Relationship Between Fick and Maxwell-Stefan Diffusion Coefficients 

The Maxwell-Stefan equation for diffusion in a two component system is Eq. 2.2.11. 

We see that, for a binary system, the Fick diffusivity D and the Maxwell-Stefan diffusivity 67 

The correlation and prediction of Fick and Maxwell-Stefan diffusion coefficients is discussed in the sections that follow. The Fick D incorporates two aspects (1) the significance of an inverse drag ( )) and (2) thermodynamic nonideality (F). Consequently, the physical interpretation of the Fick D is less transparent than for the Maxwell-Stefan diffusivity. 

For ideal systems F is unity and the Fick D and the Maxwell-Stefan D are identical. 

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The Stokes-Einstein equation for liquid-phase ordinary molecular diffusion coefficients in binary mixtures suggests that the product of Hab and the solvent viscosity /u-b should scale linearly with temperature T. Cite references (i.e., equations) from the literature and evaluate the product of Hab and /xb in terms of its scaling-law dependence on temperature for low-density gases. In other words ... 

The reversed-flow method for measurement of gas diffusion coefficients in binary mixtures can also be extended to simultaneous determination of effective diffusion coefficients for each substance in a multicomponent gas mix-ture. This extension of the method is achieved by filling the column section / (Fig. 1) with a chromatographic material, which can affect the separation of all components of the gas mixture. Effective diffusion coefficients of various mixtures of gaseous hydrocarbons into the carrier gases N2, H2, and He, determined by RF-GC, with the aid of Eq. 12, can be found in the literature. 

The procedure to calculate diffusion coefficients in binary mixtures, as presented in the literature by Satterfield, will be given here. The Lennard-Jones parameter, (Tab, is calculated by the following equation ... 

Bulk diffusion coefficients in binary gas mixture are almost independent of the ratio of components of the mixture. Therefore, it was supposed that if diffusion in the measurements described above is of the bulk type, i.e., the free path of molecules is much lesser than the diameter of pores, then the first gas diffuses into the second gas at the same rate as the second gas diffuses into the first. 

The explanation of Graham s law given by Hoogschagen is not complete, as subsequent authors (17,18) stated. However, the attempts of these authors to give a more complete explanation for the law are not convincing. It is known that at conventional measurements of diffusion coefficients in binary gas mixtures using wide capillaries, equal velocities of counterdiffusion of the components are observed. From the considerations developed... 

The description of diffusion may be complex in mixtures with more than two components. Diffusion coefficients in multicomponent mixtures are usually unknown, although sufficient experimental and theoretical information on binary systems is available. The Maxwell-Stefan diffusivities can be estimated for dilute monatomic gases from D k Dkl when the Fick diffusivities are available. The Maxwell diflfusivity is independent of the concentration for ideal gases, and almost independent of the concentration for ideal liquid mixtures. The Maxwell-Stefan diffusivities can be calculated from... 

The Stefan tube, depicted schematically in Figure 2.4, is a simple device sometimes used for measuring diffusion coefficients in binary vapor mixtures. In the bottom of the tube is a pool of quiescent liquid. The vapor that evaporates from this pool diffuses to the top of the tube. A stream of gas across the top of the tube keeps the mole fraction of diffusing vapor there to essentially nothing. The mole fraction of the vapor at the vapor-liquid interface is its equilibrium value. 

Diffusion coefficients in binary liquid mixtures are of the order 10 m /s. Unlike the diffusion coefficients in ideal gas mixtures, those for liquid mixtures can be strong functions of concentration. We defer illustration of this fact until Chapter 4 where we also consider models for the correlation and prediction of binary diffusion coefficients in gases and liquids. 

For process engineering calculations it is almost inevitable that experimental values of D or f), even if available in the literature, will not cover the entire range of temperature, pressure, and concentration that is of interest in any particular application. It is, therefore, important that we be able to predict these coefficients from fundamental physical and chemical data, such as molecular weights, critical properties, and so on. Estimation of gaseous diffusion coefficients at low pressures is the subject of Section 4.1.1, the correlation and prediction of binary diffusion coefficients in liquid mixtures is covered in Sections 4.1.3-4.1.5. We do not intend to provide a comprehensive review of prediction methods since such are available elsewhere (Reid et al., 1987 Ertl et al., 1974 Danner and Daubert, 1983) rather, it is our purpose to present a selection of methods that may be useful in engineering calculations. 

Diffusion coefficients in binary liquid mixtures can be strong functions of composition. To illustrate this fact we have plotted experimental data for a few systems in Figure 4.1. The Maxwell-Stefan coefficient > also is shown in Figure 4.1. To obtain the Maxwell-Stefan coefficients we have divided the Fick D by the thermodynamic factor F... 

The two bulb diffusion cell is a simple device that can be used to measure diffusion coefficients in binary gas mixtures. Figure 5.3 is a schematic of the apparatus. Two vessels containing gas mixtures with different compositions are connected by a capillary tube. At the start of the experiment (at t = 0), the valve is opened and the gases in the two bulbs allowed to diffuse along the capillary tube. Samples from each bulb are taken after some time and this information is used to calculate the binary diffusion coefficient. 

Carry out a review of methods for estimating infinite dilution diffusion coefficients in binary liquid mixtures. Your review should include calculations to test the accuracy of the methods that have been proposed. Fundamental data for computing the coefficients should be obtained from a single source as far as is possible we recommend the compilation by Daubert and Danner (1985). 

The Stefan tube, depicted schematically in Figure 1.8, is a simple device sometimes used for measuring diffusion coefficients in binary vapor mixtures. In the bottom of the tube is a pool of quiescent liquid. The vapor that evaporates from this pool diffuses to the top of the tube. A stream of gas across the top of the tube keeps the mole fraction of the diffusing vapors there to essentially zero. The compositon of the vapor at the vapor-liquid interface is its equilibrium value. Carty and Schrodt (1975) evaporated a binary liquid mixture of acetone (1) and methanol (2) in a Stefan tube. Air (3) was used as the carrier gas. In one of their experiments the composition of the vapor at the liquid interface was yx - 0.319, y2 - 0.528, and y3 = 0.153. The pressure and temperature in the gas phase were 99.4 kPa and 328.5 K, respectively. The length of the diffusion path was 0.24 m. The MS diffusion coefficients of the three binary pairs are ... 

Zistler, M., Wachter, P., Wasserscheid, P., Gerhard, D., Hinsch, A., Sastrawan, R., and Gores, H.J. (2006) Comparison of electrochemical methods for triiodide diffusion coefficient measurements and observation of non-Stokesian diffusion behaviour in binary mixtures of two ionic liquids. Electrochim. Acta, 52, 161-169. 

An injection port is added at the closed end of the diffusion column when diffusion coefficients in binary gas mixtures are measured. 

Fig. 1 Schematic representation of columns and gas connections for studying (a) diffusion coefficients in binary gas mixtures, (b) interaction between gases and liquids, and (c) interaction between gases and solids.

Note that for extremely low pressures the mean free path becomes the order of magnitude of the vessel diameter, which is then limiting and has to be used instead of A in Eqs. (3.1.68)-(3.1.70). For air (at 20 °C), a pressure of less than 10 mbar is needed to obtain a mean free path of the order of magnitude of a cm [Eq. (3.1.72)] Xg is then proportional to p, and thus this effect is used for superinsulations by highly evacuated casings. In addition, note that the diffusivities given in Table 3.1.7 are only valid for pure gases (self-diffusion coefficients). In binary gas mixtures, the binary coefficient D g g has to be used (Table 3.1.8). Note that in a binary gas mixture the diffusion coefficient is independent of the content of both components and that the diffusion coefficient of A in B is equal to the diffusion coefficient of B in A. 

The preceding sections have shown how it is possible to use the available kinetic theory of low-density reacting binary mixtures to calculate the viscosity and thermal conductivity of the alkali metal vapors. The model can also be applied to the binary diffusion coefficient in the mixture. It can be easily shown that, for this model, the ratio [Avid] 1 / [Avi] i = This result says that the density and temperature depen-... 

The diffusion coefficient in a mixture is obviously affected by the other species present, and to account for this a binary diffusion coefficient f>i2 is specifically related to the diffusivity of species 1 into 2 can be used. Since diffusion of species 1 into 2 is identical to species 2 into 1, D12 = f>2i. In cases where the density is low and diffusion flux is primarily a result of self-interaction, the species self-diffusion coefficient (Djj = D) is used, and the effect of interaction with other species is not included. Later in this chapter methods for calculation of diffusion coefficients are given. 

The generalized Stefan-Maxwell equations using binary diffusion coefficients are not easily applicable to hquids since the coefficients are so dependent on conditions. That is, in hquids, each Dy can be strongly composition dependent in binary mixtures and, moreover, the binaiy is strongly affected in a multicomponent mixture. Thus, the convenience of writing multicomponent flux equations in terms of binary coefficients is lost. Conversely, they apply to gas mixtures because each is practically independent of composition by itself and in a multicomponent mixture (see Taylor and Krishna for details). 

The ail viscosity can be found in Table 1.2, Appendix I, where we find that the viscosity at 644 K is about 3.1 X 10 5 kg/m s. The Fuller—Schettler—Giddings equation is proposed for the determination of the diffusion coefficient of nonpolar gases in binary mixtures at... 

Several other empirical relations for diffusion coefficients have been suggested Olson and Walton (01) have devised a means for estimating diffusion coefficients of organic liquids in water solution from surface-tension measurements. Hill (H5) has proposed a method based on Andrade s theory of liquids which allows for the concentration dependence of the diffusion coefficient in a binary liquid mixture. The formula of Arnold (A2, T6, p. 102) does not seem generally useful inasmuch as it contains two constants ( abnormality factors ) characteristic of the solute and of the solvent. 

Although there has not been much theoretical work other than a quantitative study by Hynes et al [58], there are some computer simulation studies of the mass dependence of diffusion which provide valuable insight to this problem (see Refs. 96-105). Alder et al. [96, 97] have studied the mass dependence of a solute diffusion at an infinite solute dilution in binary isotopic hard-sphere mixtures. The mass effect and its influence on the concentration dependence of the self-diffusion coefficient in a binary isotopic Lennard-Jones mixture up to solute-solvent mass ratio 5 was studied by Ebbsjo et al. [98]. Later on, Bearman and Jolly [99, 100] studied the mass dependence of diffusion in binary mixtures by varying the solute-solvent mass ratio from 1 to 16, and recently Kerl and Willeke [101] have reported a study for binary and ternary isotopic mixtures. Also, by varying the size of the tagged molecule the mass dependence of diffusion for a binary Lennard-Jones mixture has been studied by Ould-Kaddour and Barrat by performing MD simulations [102]. There have also been some experimental studies of mass diffusion [106-109]. 

The dec8y rate of the order-parameter fluctuations is proportional to the thermal diffusivity in case of pure gases near the vapor-liquid critical point and is proportional to the binary diffusion coefficient in case of liquid mixtures near the critical mixing point (6). Recently, we reported (7) single-exponential decay rate of the order-parameter fluctuations in dilute sugercritical solutions of liquid hydrocarbons in CO for T - T 10 C. This implied that the time scales associated with thermal diffusion and mass diffusion are similar in these systems. 

The Maxwell-Stefan diffusion coefficients represent binary diffusivities for ideal and many nonideal mixtures, they are independent of the concentration of the species in the multicomponent mixtures. 

The behavior of the Fick diffusion coefficient in nonideal systems may be complicated, while the Maxwell-Stefan diffusion coefficients behave quite well, and are always positive for binary systems. In nonideal binary systems, the Fick diffusivity varies with concentration. As seen in Figure 6.1, water-acetone and water-ethanol systems exhibit a minimum diffusivity at intermediate concentrations. Table 6.1 displays the dependency of binary diffusivity coefficients on concentration for selected alkenes in chloroform at 30°C and 1 atm. As the nonideality increases, mixture may split into two liquid phases at certain composition and temperature. 

Many more correlations are available for diffusion coefficients in the liquid phase than for the gas phase. Most, however, are restricted to binary diffusion at infinite dilution or to self-diffusivity DA A. This reflects the much greater complexity of liquids on a molecular level. For example, gas-phase diffusion exhibits negligible composition effects and deviations from thermodynamic ideality. Conversely, liquid-phase diffusion almost always involves volumetric and thermodynamic effects due to composition variations. For concentrations greater than a few mole percent of A and B, corrections are needed to obtain the true diffusivity. Furthermore, there are many conditions that do not fit any of the correlations presented here. Thus, careful consideration is needed to produce a reasonable estimate. Again, if diffusivity data are available at the conditions of interest, then they are strongly preferred over the predictions of any correlations. Experimental values for liquid mixtures are listed in Table 2-325. 

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