Diffusion coefficients of a mixture

NMR spectroscopic measurements provide support for these conclusions. Specifically, the diffusion coefficient of a mixture of 40 and 43 was first measured at low monomer concentration (total monomer concentration =1.4 mM). Under these conditions, a relatively large diffusion coefficient is obtained (cf. Fig. 12.30e). On the other hand, at a lOx higher monomer concentration (total mraiomer concentration = 14 mM), a diffusion coefficient of roughly 0.33 of that recorded at low concentration is obtained (cf. Fig. 12.30e). Such a finding is consistent with the formation of larger aggregates ([40 43] ) in solution, which are less free to diffuse in solution. 

Hence, the behavior of the mass-diffusion coefficient of a mixture in the critical region is controlled by the behavior of (dn/dx)T,p = x As pointed out in Section 6.1, in binary mixtures the osmotic susceptibility x diverges near a consolute point and leads to the critical slowing down of the mass diffusion. Near a vapor-liquid critical point, on the other hand, where the thermodynamic properties undeigo a crossover from pure-fluid-like behavior before they display their asymptotic mixture behavior (Jin et al. 1993), the osmotic susceptibility does not exhibit a critical behavior except at temperatures very close to the plait-point temperature (for the system mentioned above, the reduced temperature has to be smaller than 5 x 10 ). Therefore, not too close to the critical point, the mutual diffiisivity is dominated by its background value d/(px)> and the critical slowing down that follows the Stokes-Einstein diffusion law is not seen in the mass diffusion coefficient. 

The binary diffusion coefficient of a mixture can be estimated by using an expression given by Fuller et al. [6]. For two gases (A and B), the adequate expressions are as follows ... 

The dimensionless parameter Dpc / is called the Lewis number, which is the ratio of the diffusion coefficient of a gas through the mixture divided by the thermal diffusion coefficient of the gas mixture. 

Fig. 16. Panorama of values in the literature for diffusion coefficients of hydrogen in silicon and for other diffusion-related descriptors. Black symbols represent what can plausibly be argued to be diffusion coefficients of a single species or of a mixture of species appropriate to intrinsic conditions. Other points are effective diffusion coefficients dependent on doping and hydrogenation conditions polygons represent values inferred from passivation profiles [i.e., similar to the Dapp = L2/t of Eq. (95) and the ensuing discussion] pluses and crosses represent other quantities that have been called diffusion coefficients. The full line is a rough estimation for H+, drawn assuming the top points to refer mainly to this species otherwise the line should be higher at this end. The dashed line is drawn parallel a factor 2 lower to illustrate a plausible order of magnitude of the difference between 2H and H.

Mixture-Averaged Diffusion Coefficient The ordinary diffusion coefficient of a species k into a mixture may be evaluated as... 

At moderate pressures the diffusion coefficient of a binary gas mixture of molecules i and j is well described by the Chapman-Enskog theory, discussed in Section 12.4 ... 

The formal description of thermodiffusion in the critical region has been discussed in detail by Luettmer-Strathmann [79], The diffusion coefficient of a critical mixture in the long wavelength limit contains a mobility factor, the Onsager coefficient a = ab + Aa, and a thermodynamic contribution, the static structure factor S(0) [7, 79] ... 

In Table 6-5 the mutual diffusion coefficients of a binary mixture of n-heptane and n-hexadecane at 25 °C are calculated for different molar fractions of the solutes and compared with experimental values (Landolt-Bornstein, 1969). 

In Equation 7.67 DAf (Ps) and Dm (Ps) are the Maxwell diffusion coefficients of A in a binary mixture respectively of P and Dt at surface pressure. Neither D, (Ps) nor DAn (P,) depend on the gas composition or pressure. DM is the Knudsen diffusion coefficient of A in the catalyst pores, which is also independent of the gas composition and the pressure. The term is the mole fraction of Dj that would have been obtained... 

In these equations JA is the mole flux of A in moles of A per second per square metre flowing through surface area of the catalyst pores. This is not the same as the mole flux in moles of A per second per square metre flowing through surface area of the catalyst pellet. This is elaborated in Appendix E. The term DAP is the Maxwell diffusion coefficient of A in a binary mixture with P and DM is the Knudsen diffusion coefficient of A inside the catalyst pores. ka is the mole fraction of A. J, Dpdp Dn, kp etc. are similarly defined. VA is a factor that accounts for viscous flow inside the pores. If VA is much smaller than one, viscous flow can be neglected. We will neglect viscous flow for all components and substitute... 

In a binary ideal gas mixture of species A and B, the diffusion coefficient of A in B is equal to the diffusion coefficient of S in A, and both increase with temperature... 

It is possible to use alternative formulations considering mole fractions rather than mass fractions. For most cases, mass fraction formulations will be adequate. An estimation of the diffusion coefficient (of component k) in a multicomponent mixture Dkm) however, is not straightforward. For mixtures of ideal gases, the diffusion coefficient in a mixture can be estimated as (Hines and Maddox, 1985)... 

THE PROGRAM USES THE WILKE-CHANG METHOD TO CALCULATE THE DIFFUSION COEFFICIENT OF A BINARY MIXTURE OF SOLUTE A IN SOLVENT B. 

For an ideal-liquid mixture, this equation shows a linear plot of In D vs. molar concentration. The term is the liquid-phase activity coefficient, and the term d InjJdXi is the slope of the conventional plot of the logarithm of the activity coefficient versus mole fraction (see Figures 12.7 and 12.9 in Chapter 12, or see Chapter 4). Thus, for the value of the liquid diffusion coefficient of a concentrated binary mixture i-j, Equations (8.5) or (8.6) should be used, depending on ideality of the solution. 

P1I-8b A device for measuring the diffusion coefficient of a gas mixture (Figure PI l-8)consis[s of two chambers connected by a small tube. Initially the chambers contain different proportions of two gases. A and B. The total pressure is the same in each chamber. 

Thus, if ti is in a time range accessible to autocorrelation (roughly 1—10—7 sec), fluorescence fluctuations may be used to measure macromolecular translational diffusion coefficients. The presence of a fluorescent label enables this method to measure the translational diffusion coefficient of a molecule in a complex mixture. Such a measurement would be very difficult in an ordinary light-scattering experiment because all components of the mixture contribute.17 The advantage of fluorescent probes is that they allow particular species to be labeled and thereby separately studied. For or of the order of 10 4 cm and for a particle with a diffusion coefficient of the order 10 5 cm2/sec, tt = 10 3 sec, well within our ability to measure. This leads to the interesting possibility of measuring diffusion coefficients of labeled molecules in membranes, and in cells in vivo. 

In (5.51), r stands for the intrinsic reaction rate at liquid bulk conditions. For worst-case-estimations, one should use a highest rate value possible in the considered RD column. In this respect it should be kept in mind that the reaction rates under RD conditions strongly depends on the operating pressure that influences the boiling temperatures, that is the reaction temperature. D g/ represents the effective diffusion coefficient of a selected reaction component inside the catalyst particles. One should use the component with the lowest mole fraction Xj in the liquid bulk mixture as key component [35]. Its effective diffusion coefficient can be estimated from the diffusion coefficient at infinite dilution Dg((/ = (sfr)D with the total porosity e and the tortuosity r of the applied catalyst. Based on (5.51) one can say that intraparticle diffusion resistances will be negligible, if 1. 

The mechanism operative in the unstable part of the phase diagram can be understood if we revisit equation (4.4.11) for the mutual diffusion coefficient of a polymer mixture, which we can rewrite in a simplified form as... 

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