Diffusion, coefficients

Diffusion of an adsorbate molecule in the adsorbent particle occurs when there is concentration distribution in the particle. Since the mechanism of diffusion and the real driving force of diffusion may not 

Diffusion coefficient D C) is then a phenomenological coefficient and may be a function of concentration. The concentration may be a concentration in fluid phase or adsorbed phase. The physical meaning of (C) is dependent on the controlling mechanism in the diffusion concerned. Then only the mathematical problem of solving differential equations will be left. 

2 Diffusion Coefficient We learned about random walks in Section 1.2 to describe an ideal chain. The random walk on the cubic lattice (Fig. 3.6) shares stochastic nature of the whereabouts with the Brownian motion of the particle in solution. We can apply the results obtained for the ideal chains to the motion of the random walker. 

Let ti be the time of each step and b the displacement. The mean square displacement of N steps in time t = Mi, 

The last equality proves that the ratio is the same for the whole motion of N steps and for the single-step motion. To estimate D, we can use either an Mstep motion or a single-step motion. The results should be identical as long as the step motions are mutually independent (Markoffian). 

The diffusion coefficient of natural organics is related to the size of the molecules. Worch (1993) published a formula to relate MW with size for organic molecules (see Eqn. (2.1)). The size can then be related to diffusion coefficient by using the Stokes Einstein equation (see Appendix 5). While this method assumes spherical shapes, it allows the estimation as equivalent spheres. 

Cornel et al (1986) measured diffusion coefficients of HA in dilute solutions and found a decrease with increased molecular weight. This contradicts the above method and the anomalous behaviour was attributed to artifacts in the size measurements. Diffusivity increased with increasing ionic strength and decreasing pH. This could be attributed to the change in shape of the molecules as will be further explained in section 2.7. Diffusion coefficients were in the order of 1 to 2.5 lO mV for 50 to 100 kDa and 0.5 10 m-s for 1 kDa UF fractions, respectively. The diffusion coefficients for Suwannee River materials are shown in Table 2.1. 

The diffusion coefficient is a physical property that represents the speed of molecular diffusion. Recently it was shown that D changes during chemical reactions. Here, we describe the origin of the change in D. Intuitively, it may be easily understood that D decreases when molecular size increases because of the association reaction. In some cases, the relationship between D and the molecular size is well described by the Stokes-Einstein equation. The Stokes-Einstein equation is expressed by [9-13] 

Although the molecular size is certainly an important factor that determines D, D also depends on the intermolecular interaction between the molecule and the solvent. A clear example has been reported for chemical reaction of aromatic molecules [15-19]. It was found that the D values of organic radicals are much smaller than those of electronically closed shell molecules with similar sizes and shapes. This change was attributed to the enhanced intermolecular interaction between the radicals and solvent molecules. It was further reported that D of cytochrome c in its native form is much larger than that in the unfolded state [25-27], This difference was attributed to the larger intermolecular interaction between the protein and water due to the unfolded conformation of the a-helices. 

D has been sometimes expressed in terms of the hydrodynamic radius. However, we consider that hydrodynamic radius is not the proper term to describe the change of D, because this is not a well-defined radius such as the radius of gyration, which is clearly defined to show the molecular size. The hydrodynamic radius has just the same meaning as D as long as the Stoke-Einstein relation holds good. 

The diffusion coefficient, D, is a strong function of temperature. The temperature dependence arises because some energy (typically a few electron volts, eV) is required for an atom to jump from one atomic position to another. This energy is often called the activation energy, EA. The diffusion coefficient can be written 

The diffusion coefficient can be readily computed from the mean squared displacement of particles using the following expression  

In practice, the squared displacement of each particle is computed, the displacements of all particles are added and then divided by N to determine the mean. The diffusion coefficient can be calculated from the slope of the mean square displacement as a function of time. 

It is important to note that the diffusion coefficient is not accurately calculated when using the positions of particles after applying periodic boundary conditions. Periodic boundary conditions constrain the particles inside the original simulation box, and using updated positions results in significantly smaller diffusion coefficients. Instead, 

The rate of diffusion is governed by the diffusion coefficient in conjunction with Pick s laws of diffusion. Because diffusion is thermally activated, the diffusion coefficients follow the Arrhenius behavior and it is a common practice to express their thermal dependence in terms of a pre-exponential coefficient Do and an activation energy Q, 

To obtain a rough estimate of molecular radius and of Avogadro s number from a difiusion coefficient. 

If the diffusing molecules are lai e enough to obey Stokes law and are regarded as spheres, the coefficient of diffusion D is related to the viscosity q and the molecular radius a by the formula (Einsteia, Ann. Phys., Lpz. 1906, 19, 303) 

From the density and molecular weight we obtain the molar volume and thence an independent (approximate) value of By combining the values of ATa and Na we obtain estimates of N and a. 

MaTfing the simplest possible approximation, adequate for the present admittedly rough estimate, we treat the crystal as spherical molecules arranged on a simple cubic lattice. This leads to 

If instead of using the molar volume of solid sucrose we took the partial molar volume of sucrose in aqueous solution, the result would be effectively the same. The weakest point in this treatment is the assumption that Stokes law should be obeyed by such a small particle as a sucrose molecule. The approximation should be better for colloidal particles, but then there is the problem of determining their mass. Compare problem 13. 

As expressed in Pick s law (see Sec. 14.5), the diffusion coefficient, D, is a parameter that reiates the flux of a penetrant in a medium to its concentration gradient. A diffusion coefficient value for a polymer is always specific to a particuiar pene-trant-poiymer pair. For solid polymers, the diffusion coefficients of a large number 

In electrolyte systems the components diffiise as ionic species and the effictive ionic diffiisivities are related to the nature and concentration of the other ions present. Taylor and Krishna [1993] give as an exanple the diffiision of an aqueous solution of HCl and BaCl2. The infinite dilution diffusion coefficients for the individual ions in water are  

Charge carriers in thermal equilibrium with the liquid carry out an imdirected random motion. If an external electric field is applied the drift motion is superimposed on this random motion. At T = 300 K, the charge carriers have an average kinetic energy of 

The average thermal (random) velocity is then given as 

For electrons a random velocity of v = 1.210 cm/s is obtained. Atomic or molecular ions have masses 10 to 10 times larger, so their thermal velocities are in the range of 10 to 10 cm/s. 

The diffusion coefficient is related to the mobility of the charge carriers via the Nernst-Einstein equation (see Chapter 1, Equation 77), 

In the case of time of flight experiments (see Section 2.5.1), the random motion of the electrons parallel to the direction of the electric field leads to a broadening of the electrical drift signal. If an infinitesimal thin layer of Nq charge carriers starts at the cathode (x = 0) at t = 0, then their distribution at time t and at distance x is given as 

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The value of coefficient depends on the composition. As the mole fraction of component A approaches 0, approaches ZJ g the diffusion coefficient of component A in the solvent B at infinite dilution. The coefficient Z g can be estimated by the Wilke and Chang (1955) method ... 

For a binary mixture of two components A and B in the gas phase, the mutual diffusion coefficient such as defined in 4.3.2.3, does not depend on composition. It can be calculated by the Fuller (1966) method ... 

Umesi, N.O. (1980), Diffusion coefficients of dissoived gases in iiquids -Radius of gyration of solvent and solute . M.S. Thesis, The Pennsylvania State University, PA. 

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 ... 

Here (D is the diffusion coefficient and C is the concentration in the general bulk solution. For initial rates C can be neglected in comparison to C/ so that from Eqs. IV-59 and IV-60 we have... 

An alternative defining equation for surface diffusion coefficient Ds is that the surface flux Js is Js = - Ds dT/dx. Show what the dimensions of Js must be. 

It was commented that surface viscosities seem to correspond to anomalously high bulk liquid viscosities. Discuss whether the same comment applies to surface diffusion coefficients. 

The dififiision time gives the same general picture. The bulk self-diffusion coefficient of copper is 10"" cm /sec at 725°C [12] the Einstein equation... 

It is known that even condensed films must have surface diffusional mobility Rideal and Tadayon [64] found that stearic acid films transferred from one surface to another by a process that seemed to involve surface diffusion to the occasional points of contact between the solids. Such transfer, of course, is observed in actual friction experiments in that an uncoated rider quickly acquires a layer of boundary lubricant from the surface over which it is passed [46]. However, there is little quantitative information available about actual surface diffusion coefficients. One value that may be relevant is that of Ross and Good [65] for butane on Spheron 6, which, for a monolayer, was about 5 x 10 cm /sec. If the average junction is about 10 cm in size, this would also be about the average distance that a film molecule would have to migrate, and the time required would be about 10 sec. This rate of Junctions passing each other corresponds to a sliding speed of 100 cm/sec so that the usual speeds of 0.01 cm/sec should not be too fast for pressurized film formation. See Ref. 62 for a study of another mechanism for surface mobility, that of evaporative hopping. 

Reference 115 gives the diffusion coefficient of DTAB (dodecyltrimethylammo-nium bromide) as 1.07 x 10" cm /sec. Estimate the micelle radius (use the Einstein equation relating diffusion coefficient and friction factor and the Stokes equation for the friction factor of a sphere) and compare with the value given in the reference. Estimate also the number of monomer units in the micelle. Assume 25°C. 

The oil droplets in a certain benzene-water emulsion are nearly uniform in size and show a diffusion coefficient of 3.75 x 10 cm /sec at 25°C. Estimate the number of benzene molecules in each droplet. 

The state of an adsorbate is often described as mobile or localized, usually in connection with adsorption models and analyses of adsorption entropies (see Section XVII-3C). A more direct criterion is, in analogy to that of the fluidity of a bulk phase, the degree of mobility as reflected by the surface diffusion coefficient. This may be estimated from the dielectric relaxation time Resing [115] gives values of the diffusion coefficient for adsorbed water ranging from near bulk liquids values (lO cm /sec) to as low as 10 cm /sec. 

The rate of dissolving of a solid is determined by the rate of diffusion through a boundary layer of solution. Derive the equation for the net rate of dissolving. Take Co to be the saturation concentration and rf to be the effective thickness of the diffusion layer denote diffusion coefficient by . 

The viscosity, themial conductivity and diffusion coefficient of a monatomic gas at low pressure depend only on the pair potential but through a more involved sequence of integrations than the second virial coefficient. The transport properties can be expressed in temis of collision integrals defined [111] by... 

Sun Y-P and Saltiel J 1989 Application of the Kramers equation to stiibene photoisomerization in / -alkanes using translational diffusion coefficients to define microviscosity J. Phys. Chem. 93 8310-16... 

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

Three rotational diffusion coefficients, two polar angles 0 and T... 

Rotational diffusion coefficient, Dg, internal motion rate parameter, angle between the internal rotation axis and the internuclear axis... 

Tao T 1969 Time-dependent fluorescence depolarization and Brownian rotational diffusion coefficients of macromolecules Biopolymers 8 609-32... 

In dilute solutions, tire dependence of tire diffusion coefficient on tire molecular weight is different from tliat found in melts, eitlier entangled or not. This difference is due to tire presence of hydrodynamic interactions among tire solvent molecules. Such interactions arise from tire necessity to transfer solvent molecules from tire front to tire back of a moving particle. The motion of tire solvent gives rise to a flow field which couples all molecules over a... 

If tire diffusion coefficient is independent of tire concentration, equation (C2.1.22) reduces to tire usual fonn of Pick s second law. Analytical solutions to diffusion equations for several types of boundary conditions have been derived [M]- In tlie particular situation of a steady state, tire flux is constant. Using Henry s law (c = kp) to relate tire concentration on both sides of tire membrane to tire partial pressure, tire constant flux can be written as... 

Several ideas have been put forward to calculate tire diffusion coefficient of small molecules in polymers. Glasstone et al [M] proposed an expression based on transition-state tlieory... 

Otlier expressions for tire diffusion coefficient are based on tire concept of free volume [57], i.e. tire amount of volume in tire sample tliat is not occupied by tire polymer molecules. Computer simulations have also been used to quantify tire mobility of small molecules in polymers [58]. In a first approach, tire partition functions of tire ground... 

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