Diffusion in liquids

The velocity can be determined from the solution of the hydrodynamic flow field. For a no-slip condihon at the interface of the diffusing spherical parhcles and the liquid, the force over the spherical particle is given by Stake s law as 

Combination of Equations 6.113 and 6.114 leads to the Stokes-Einstein equation as 

For slip condition at the interface of the diffusing particles and the liquid medium, the force over the particles is given as 

Another empirical correlation based on the Stokes-Einstein equation for low concentration of species i in liquid medium j is given by Wilke and Chang (Sherwood et al., 1975), 

Va = molar volume of solute at normal boiling point (cm /mol) molar volumes at normal boiling point are listed in Table 6.5 = viscosity of liquid (centipoise, cP) 

Whilst the diffusion of solution in a liquid is governed by the same equations as for the gas phase, the diffusion coefficient D is about two orders of magnitude smaller for a liquid than for a gas. Furthermore, the diffusion coefficient is a much more complex function of the molecular properties. 

For an ideal gas, the total molar concentration Cj is constant at a given total pressure P and temperature T. This approximation holds quite well for real gases and vapours, except at high pressures. For a liquid however, CT may show considerable variations as the concentrations of the components change and, in practice, the total mass concentration (density p of the mixture) is much more nearly constant. Thus for a mixture of ethanol and water for example, the mass density will range from about 790 to 1000 kg/m3 whereas the molar density will range from about 17 to 56 kmol/m3. For this reason the diffusion equations are frequently written in the form of a mass flux JA (mass/area x time) and the concentration gradients in terms of mass concentrations, such as cA. 

the diffusional process in a liquid gives rise to a situation where the components are being transferred at approximately equal and opposite mass (rather than molar) rates. 

Liquid densities exceed those of gases at normal atmospheric pressures by a factor of about 1000. These differences are reflected in the intermolecular distances that exist in the two phases. In gases xmder standard conditions these distances are some three orders of magnitude greater than the molecular dimensions. Liquid molecules are by contrast closely packed, with intermolecular distances of the same order as the molecular size. 

Gas molecules spend most of their time in transit between collisions and are only modestly affected by intermolecular forces. In liquids these forces are the dominant factor that determines the mobility of the molecules. They are notoriously difficult to quantify, and as a consequence, the prediction of liquid diffusivities has lagged behind theories describing the motion of gas molecules. 

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  

This relation, which is most successful for large molecules (1 500 ctvd/ mol), states that diffusivity varies inversely with the viscosity of the solvent and the molecular dimension of the diffusion molecule. This agrees with our intuitive grasp of the process. 

Diffusivities in molten salts and metals are even more difficult to predict and here we often resort to an Arrhenius-type relation to express the strong temperature dependence of the diffusion coefficienf, which is concealed in the viscosity of Equation 3.2 and Equation 3.3  

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Hwang L-P and Freed J H 1975 Dynamic effects of pair correlation functions on spin relaxation by translational diffusion in liquids J. Chem. Rhys. 63 4017-25... 

In the special case that A and B are similar in molecular weight, polarity, and so on, the self-diffusion coefficients of pure A and B will be approximately equal to the mutual diffusivity, D g. Second, when A and B are the less mobile and more mobile components, respectively, their self-diffusion coefficients can be used as rough lower and upper bounds of the mutual diffusion coefficient. That is, < D g < Dg g. Third, it is a common means for evaluating diffusion for gases at high pressure. Self-diffusion in liquids has been studied by many [Easteal AIChE]. 30, 641 (1984), Ertl and Dullien, AIChE J. 19, 1215 (1973), and Vadovic and Colver, AIChE J. 18, 1264 (1972)]. 

Few mechanisms of liquid/liquid reactions have been established, although some related work such as on droplet sizes and power input has been done. Small contents of surface-ac tive and other impurities in reactants of commercial quality can distort a reac tor s predicted performance. Diffusivities in liquids are comparatively low, a factor of 10 less than in gases, so it is probable in most industrial examples that they are diffusion controllech One consequence is that L/L reactions may not be as temperature sensitive as ordinary chemical reactions, although the effec t of temperature rise on viscosity and droplet size can result in substantial rate increases. L/L reac tions will exhibit behavior of homogeneous reactions only when they are very slow, nonionic reactions being the most likely ones. On the whole, in the present state of the art, the design of L/L reactors must depend on scale-up from laboratoiy or pilot plant work. 

In connection with the earlier consideration of diffusion in liquids using tire Stokes-Einstein equation, it can be concluded that the temperature dependence of the diffusion coefficient on the temperature should be T(exp(—Qvis/RT)) according to this equation, if the activation energy for viscous flow is included. 

In view of Swalin s treatment of diffusion in liquid metals. Are latter seems to be a better description. In binaty mixmres such as NaCl-KCl the equivalent conducAvities are a linear manner, but the KCl-CdCla mixture shows a marked negative departure from linear behaviour, probably because of the formation of the complex ion CdCh. ... 

Looking at translational diffusion in liquid systems, at least two elementary categories have to be taken into consideration self-diffusion and mutual diffusion [1, 2]. 

Following the general trend of looldng for a molecular description of the properties of matter, self-diffusion in liquids has become a key quantity for interpretation and modeling of transport in liquids [5]. Self-diffusion coefficients can be combined with other data, such as viscosities, electrical conductivities, densities, etc., in order to evaluate and improve solvodynamic models such as the Stokes-Einstein type [6-9]. From temperature-dependent measurements, activation energies can be calculated by the Arrhenius or the Vogel-Tamman-Fulcher equation (VTF), in order to evaluate models that treat the diffusion process similarly to diffusion in the solid state with jump or hole models [1, 2, 7]. 

H.J.V. Tyrrell and K.R. Harris, Diffusion in Liquids, Butterworths, London, 1984. 

Complete and Incomplete Ionic Dissociation. Brownian Motion in Liquids. The Mechanism of Electrical Conduction. Electrolytic Conduction. The Structure of Ice and Water. The Mutual Potential Energy of Dipoles. Substitutional and Interstitial Solutions. Diffusion in Liquids. 

Henry Eyring s research has been original and frequently unorthodox. He woj one of the first chemists to apply quantum mechanics in chemistry. He unleashed a revolution in the treatment of reaction rates by use of detailed thermodynamic reasoning. Having formulated the idea of the activated complex, Eyring proceeded to find a myriad of fruitful applications—to viscous flow of liquids, to diffusion in liquids, to conductance, to adsorption, to catalysis. 

Chandler D. Translational and rotational diffusion in liquids. I. Translational single-particle correlation functions. J. Chem. Phys. 60, 3500-507, (1974). Translational and rotational diffusion in liquids. II. Orientational single-particle correlation functions. J. Chem. Phys. 60, 3508-12 (1974). 

Maryott A. A., Farrar T. C., Malmberg M. S. 35C1 and 19F NMR spin-lattice relaxation time measurements and rotational diffusion in liquid CIO3F. J. Chem. Phys. 54, 64-71 (1971). 

Two solutes were used to study diffusion in liquids, methylbenzene, which is a small molecule that can be approximated as a sphere, and a liquid crystal that is long and rodlike. The two solutes were found to move and rotate in all directions to the same extent in benzene. In a liquid crystal solvent the methylbenzene again moved and rotated to the same extent in all directions, but the liquid crystal solute moved much more rapidly along the long axis of the molecule than it... 

Flayduk, W. and Minhas, B.S. (1982) Correlations for prediction of molecular diffusivities in liquids. Can.]. Chem. 

Diffusion in liquids is very slow. Turbulent transport or very narrow channels are necessary for good contact between the phases. The droplets must also be very small to minimize transport hmitations within the drops. An estimation of the time constant for diffusion in a 1-mm drop is (f (10-3)2... 

Tyrrell, HJV Harris, KR, Diffusion in Liquids Butterworths London, 1984. 

While the choice of a reference plane is usually simple for heat and electricity flow, this is not the case for diffusion in liquid mixtures. 

For diffusion in liquid electrolytes such as molten salts, two forces acting on an ion of interest should be taken into account the gradient of the chemical potential and the charge neutrality. Thus the electrochemical potential rather than the chemical potential should be the driving force for diffusion. 

DW McCall, DC Douglas, EW Anderson. Self diffusion in liquids Paraffin hydrocarbons. Phys Fluids 2 87-91, 1959. 

CN Satterfield, CK Colton, WH Pitcher. Restricted diffusion in liquids with fine pores. AIChE J 19 628-635, 1973. 

Diffusion constants are enhanced with the approximate inclusion of quantum effects. Changes in the ratio of diffusion constants for water and D2O with decreasing temperature are accurately reproduced with the QFF1 model. This ratio computed with the QFF1 model agrees well with the centroid molecular dynamics result at room temperature. Fully quantum path integral dynamical simulations of diffusion in liquid water are not presently possible. 

Poulsen, J. A. Nyman, G. Rossky, P. J., Quantum diffusion in liquid para-hydrogen an application of the Feynman-Kleinert linearized path integral approximation, J. Phys. Chem. B 2004,108, 19799-19808... 

The general equation of convective diffusion in liquids, equation (15), is a second-order, partial differential equation with variable coefficients. Its solution yields the spatial distribution of c, as a function of time, namely its transient behaviour. On an analytical level, solution of equation (15) into the transient c(t) is possible only for a number of relatively simple systems with well-defined geometry and flow properties. The problem is greatly simplified if the concentration function Cj(x,y,z) is essentially independent of time t, i.e. in the steady-state. Then equation (15) reduces to ... 

Let us assume that a sphere with radius a is immersed in a liquid of finite volume, e.g., a mineral in a hydrothermal fluid. Diffusion in liquids is normally fast compared to diffusion in solids, so that the liquid can be thought of as homogeneous. Similar conditions would apply to a sphere degassing into a finite enclosure, e.g., for radiogenic argon loss in a closed pore space. Given the diffusion equation with radial flux and constant diffusion coefficient... 

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