Liquid diffusion coefficients and

Some Pressure Effects on Liquid Diffusion Coefficients and Equilibrium... 

A requirement for modeling gas-liquid reactors is that the gas and liquid diffusion coefficients and mass transfer coefficients are known. For an estimation of the diffusion coefficients in the gas and liquid phases, see Appendices 4 and 6, respectively. Estimation of mass transfer coefficients is considered in Appendices 5 and 7, and methods for calculating the gas solubilities are discussed in Appendix 8. 

Flemings estimates lo by starting with the number of critical nuclei n in equilibrium with N atoms n = N exp( AG VI b F). He then takes the number of atoms adjacent to a critical nucleus, 4Tr(rcriticai)and estimates the nucleation rate as the rate at which these atoms join the critical nucleus to push it over the energy barrier. The rate at which the adjacent atoms join a critical nucleus is given by Di ja, where Dl is the liquid diffusion coefficient and flo is an atomic radius. If one takes N lO atoms cm, r nticaxho... 

There is also a traffic between the surface region and the adjacent layers of liquid. For most liquids, diffusion coefficients at room temperature are on the order of 10 cm /sec, and the diffusion coefficient is related to the time r for a net displacement jc by an equation due to Einstein ... 

It is important to recognize that the effects of temperature on the liquid-phase diffusion coefficients and viscosities can be veiy large and therefore must be carefully accounted for when using /cl or data. For liquids the mass-transfer coefficient /cl is correlated in terms of design variables by relations of the form... 

Higher diffusion coefficients and lower viscosities compared with liquids. Rapid diffusion of GO2 through condensed phases, e.g. polymers. 

Examples of this procedure for dilute solutions of copper, silicon and aluminium shows the widely different behaviour of these elements. The vapour pressures of the pure metals are 1.14 x 10, 8.63 x 10 and 1.51 x 10 amios at 1873 K, and the activity coefficients in solution in liquid iron are 8.0, 7 X 10 and 3 X 10 respectively. There are therefore two elements of relatively high and similar vapour pressures, Cu and Al, and two elements of approximately equal activity coefficients but widely differing vapour pressures. Si and Al. The right-hand side of the depletion equation has the values 1.89, 1.88 X 10- , and 1.44 X 10 respectively, and we may conclude that there will be depletion of copper only, widr insignificant evaporation of silicon and aluminium. The data for the boundaty layer were taken as 5 x lO cm s for the diffusion coefficient, and 10 cm for the boundary layer thickness in liquid iron. 

It is seen from equation (26) that the optimum velocity is determined by the magnitude of the diffusion coefficient and is inversely related to the particle diameter. Unfortunately, in LC (where the mobile phase is a liquid as opposed to a gas), the diffusivity is four to five orders of magnitude less than in GC. Thus, to achieve comparable performance, the particle diameter must also be reduced (c./., 3-5 p)... 

With regard to the enantioselective transport through the membrane, one advantage of liquid membrane separation is the fact that the diffusion coefficient of a solute in a liquid is orders of magnitude higher as compared to the diffusion coefficient in a solid. The flux through the membrane depends linearly on the diffusion coefficient and concentration of the solute, and inversely on the thickness of the membrane [7]. 

The behavior of ionic liquids as electrolytes is strongly influenced by the transport properties of their ionic constituents. These transport properties relate to the rate of ion movement and to the manner in which the ions move (as individual ions, ion-pairs, or ion aggregates). Conductivity, for example, depends on the number and mobility of charge carriers. If an ionic liquid is dominated by highly mobile but neutral ion-pairs it will have a small number of available charge carriers and thus a low conductivity. The two quantities often used to evaluate the transport properties of electrolytes are the ion-diffusion coefficients and the ion-transport numbers. The diffusion coefficient is a measure of the rate of movement of an ion in a solution, and the transport number is a measure of the fraction of charge carried by that ion in the presence of an electric field. 

Systems that are near to ideality can be described satisfactorily with Equation 4.4-4, but the equation does not work very well in systems that are far from thermodynamic ideality, even if the self-diffusion coefficients and activities are known. Since systems with ionic liquids show strong intermolecular forces, there is a need... 

The mass transfer coefficient is calculated for a given diffusivity coefficient and reaction rate constant at the equilibrium concentration of oxygen. When oxygen is continuously transported and removed from the liquid phase we may write ... 

Calderbank and Moo-Young (C5) have studied gas-liquid mass transfer in systems characterized by high viscosities and high diffusion coefficients, and have on the basis of data obtained in this and other studies developed correlations for the mass-transfer coefficients. 

Table 10.7. Diffusivities (diffusion coefficients) and Schmidt numbers, in liquids at 293 K 41...

We have seen that particle sizes of 100-150 y in classical LC lead to slow and inefficient columns. The reason for this is related to the slow diffusion of molecules in the liquid state (e.g., gaseous diffusion coefficient 10 liquid diffusion coefficients). 

While electrical conductivity, diffusion coefficients, and shear viscosity are determined by weak perturbations of the fundamental diffu-sional motions, thermal conductivity is dominated by the vibrational motions of ions. Heat can be transmitted through material substances without any bulk flow or long-range diffusion occurring, simply by the exchange of momentum via collisions of particles. It is for this reason that in liquids in which the rate constants for viscous flow and electrical conductivity are highly temperature dependent, the thermal conductivity remains essentially the same at lower as at much higher temperatures and more fluid conditions. 

The gas A must transfer from the gas phase to the liquid phase. Equation (1) describes the specific (per m2) molar flow (JA) of A through the gas-liquid interface. Considering only limitations in the liquid phase, this molar flow notably depends on the liquid molecular diffusion coefficient DAL (m2 s ). Based on the liquid state theories, DA L can be calculated using the Stokes-Einstein expression, and many correlations have been developed in order to estimate the liquid diffusion coefficients. The best-known example is the Wilke and Chang (W-C) relationship, but many others have been established and compared (Table 45.4) [28-33]. 

The coefficients are defined for infinitely dilute solution of solute in the solvent L. However, they are assumed to be valid even for concentrations of solute of 5 to 10 mol.%. The relationships are available for pure solvent, and could be used for mixture of solvents composed of molecules of close size and shape. They all refer to the solvent viscosity which can be estimated or measured. Pressure has a negligible influence on liquid viscosity, which decreases with temperature. As a consequence, pressure has a weak influence on liquid diffusion coefficient conversely, diffusivity increases significantly with temperature (Table 45.4). For mixtures of liquids, an averaged value for the viscosity should be employed. 

The Hl value is reduced by an increase in the viscosity of the solvent or by a decrease in the temperature. Longitudinal diffusion can thus be reduced by decreasing the diffusion coefficient and increasing the flow rate however, these two actions are counter-effective in liquid chromatography because of the mass transport term. 

The second major difference found in vapor-liquid extraction of polymeric solutions is related to the low values of the diffusion coefficients and the strong dependence of these coefficients on the concentration of solvent or monomer in a polymeric solution or melt. Figure 2, which illustrates how the diffusion coefficient can vary with concentration for a polymeric solution, shows a variation of more than three orders of magnitude in the diffusion coefficient when the concentration varies from about 10% to less than 1%. From a mathematical viewpoint the dependence of the diffusion coefficient on concentration can introduce complications in solving the diffusion equations to obtain concentration profiles, particularly when this dependence is nonlinear. On a physiced basis, the low diffiisivities result in low mass-transfer rates, which means larger extraction equipment. 

The book by Reid et al. [9] is an excellent source of information on properties such as thermal conductivities, diffusion coefficients and viscosities of gases and liquids. Not only are there extensive tables of data, but many estimation methods and correlations are critically reviewed. 

Because the theory of the liquid state is not nearly so well developed as the kinetic theory of gases, estimation methods for liquid diffusion coefficients are not as reliable as those used for gases. For dilute solutions of non-electrolytes, one widely used correlation is that due to Wilke and Chang[48]... 

J is the flux of particles crossing a 1-cm2 plane in 1 s (i.e., number cm 2 s ). The constant D is known as the diffusion coefficient and is simply the proportionality constant relating the flux to the concentration gradient. (Fick s first law applies, of course, not only to particles but also to gas and liquid molecules.)... 

Typical values of self-diffusion coefficients and mutual diffusion coefficients in aqueous solutions and in molten salt systems such as (K,Ag)N03 are of the order of 10 m s and the coefficients do not usually vary by more than a factor of 10 over the whole composition range [1, 2, 15]. From measurements in pure ionic liquids we have learned that their self-diffusion coefficients are only of the order of 10 m s From this point of view it is interesting to investigate systems of ordinary and ionic liquids. Figure 4.4-3 shows the results of first measurements in the methanol/[BMIM][PF6] system, which can be seen as a prototype for a system in which an organic and an ionic liquid are mixed. 

The physical properties of solvents greatly influence the choice of solvent for a particular application. The solvent should be liquid under the temperature and pressure conditions at which it is employed. Its thermodynamic properties, such as the density and vapor pressure, temperature and pressure coefficients, as well as the heat capacity and surface tension, and transport properties, such as viscosity, diffusion coefficient, and thermal conductivity, also need to be considered. Electrical, optical, and magnetic properties, such as the dipole moment, dielectric constant, refractive index, magnetic susceptibility, and electrical conductance are relevant, too. Furthermore, molecular... 

The model parameters q and ML can be estimated from experimental data for radius of gyration, intrinsic viscosity, sedimentation coefficient, diffusion coefficient and so on in dilute solutions. The typical methods are expounded in several recent articles and books [20-22], Here we refer only to the results of the application to representative liquid-crystalline polymers (See Table 1). 

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