Diffusion in gases

In gases the diffusion rates are clearly dependent on the molecular speed, and consequently we should expect a dependence of the diffusion coefficient on temperature since the temperature indicates the average molecular speed. 

Gilliland [4] has proposed a semiempirical equation for the diffusion coefficient 

Equation (11-2) offers a convenient expression for calculating the diffusion coefficient for various compounds and mixtures, but it should not be used as a substitute for experimental values of the diffusion coefficient when they are available for a particular system. References 3 and 5 to 9 present more information on calculation of diffusion coefficients. An abbreviated table of diffusion coefficients is given in Appendix A. 

Calculate the diffusion coefficient for C02 in air at atmospheric pressure and 25°C using Eq. (I I-2), and compare this value with that in Table A-8. 

We realize from the discussion pertaining to Fig. 11-1 that the diffusion process is occurring in two ways at the same time i.e., gas A is diffusing into gas B at the same time that gas B is diffusing into gas A. We thus could refer to the diffusion coefficient for either of these processes. 

Mass Transfer and Separation Processes Principles and Applications 

A first simple expression for the diffusivity of gases was derived some 150 years ago based on the kinetic theory of gases. In this theory, the molecules are regarded as point entities that undergo elastic collisions with each other without the intrusion of intermolecular attractive or repulsive forces. This simple model led to the expression 

This first attempt at a prediction of D was followed by a series of more elaborate theories, which took account of the finite size of the gas molecules, as well as the effect of intermolecular forces. Probably the most popular among current prediction methods is that due to Fuller, Schettler, and Giddings, who proposed the following expression for the calculation of gas diffusivities  

Here D b is in units of square centimeter per second (cm /s), T is the absolute temperature (K), Pj is the total pressure in atmospheres, and V are the atomic and molecular volume contributions. These are empirical constants that correspond approximately to the molar volume of the substances in cubic centimeter per mole (cm /mol). They have been tabulated, and a partial list for use with organic molecules appears in Table 3.2. 

Cadmium vapor is a toxic substance whose diffusivity in air is not readily available in the literature. It is desired to calculate its diffusivity at its boiling point of 1038 K and a pressure of 1 atm. Because an empirical atomic volume is not available, we use the reported value for its liquid volume of 14 cm /mol for an atomic weight of 112.4. We obtain, using Equation 3.2, 

Diffusional mechanisms in (a) gases and (b) liquids and solids. TABLE 3.1 

At constant temperature the ratio DG2IDGI of the diffusion coefficients in a gas at two states 1 and 2 equals the ratio V2 Vt of the system volumes at the two states. Then together with the first assumption a starting point for modeling diffusion coefficients will be the relation DG2 = (V2IV,)Due p(qr) with the unit value Du = lm2/s and qr = iv/ for a perfect gas. By selecting V,- 105 V at pu = 1 Pa as a reference volume the following equation results for a self-diffusion coefficient, Do, in a perfect gas  

This equation is typical for ideal gases in which neither an interaction term in form of activation energy nor the volume of the diffusing particles are considered. At p = 1 bar and T=0°C for example, Dc = 1.36 cm2/s is obtained using Eq. (6-16) compared with 1.4 cm2/s measured in He (Landolt-Bornstein, 1969). 

For real gases two additional terms in the exponent of Eq. (6-16) must be introduced  

A molar activation energy EA of diffusion is defined as the product EA = wl eRTc, with the critical temperature, Tc, of the system. This definition takes into account the connection between the interaction terms of the model, w, and w/e and IS units, as shown in section 6.3.4. At the critical temperature, Tc, a pure translational amount, w,-w, e, of the relative energy density is responsible for the magnitude of Dc. 

Collecting the above results, the following final equation can be established for the self-diffusion coefficient in a macroscopic system in the gaseous state  

Imagine a gas in which a certain constant difference of concentration exists in the various fixed points (a nonequilibrium, but stationary system), whereas the temperature at any point in this system is the same and remains constant. Thus, there is a resulting flux of gas molecules referred to as stationary diffusion. We would like to emphasize that this flux is caused only by the chaotic movement of molecules and not for any other reasons. Another situation arises when the difference of concentration is not constant the system aspires to equalize concentration this will be nonstationary diffusion. 

By comparing eqs. (3.7.13) and (3.7.14), we can further derive the microscopic expression for the macroscopic diffusion coefficient D  

It follows from this expression that the physical sense of the diffusion coefficient/) consists of the fact that it shows the number of molecules that diffuse through a unit area in a unit time and at a nnit gradient of relative concentration. Data on key parameters of diffusion as well as other transport phenomena, i.e., of heat conductivity and viscosity, are shown in Table 3.3. 

If molecules differ considerably in their masses and the dimensions mentioned above, the calculations demand specification. More detailed examination shows that the process of diffusion is determined by the speed of the fastest (smallest) molecules, whereas for effective cross section determination, it is the larger molecules. 

For the nonstationary diffusion, it is possible to estimate the time t for which there is an alignment of concentration (reduced in e times) from a dimension consideration. In fact, t is defined only by the character of distribution of molecular masses in the initial instant of time and gas property. The initial state is defined by the dimension of heterogeneity area L. There is only one combination from D and L that has a dimension of time, namely 

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The Morse function which is given above was obtained from a study of bonding in gaseous systems, and dris part of Swalin s derivation should probably be replaced with a Lennard-Jones potential as a better approximation. The general idea of a variable diffusion step in liquids which is more nearly akin to diffusion in gases than the earlier treatment, which was based on the notion of vacant sites as in solids, remains as a valuable suggestion. 

Grew, K.E. and Ibbs, T.L. Thermal Diffusion in Gases (Cambridge University Press, Cambridge, 1952). 

The diffusivity in gases is about 4 orders of magnitude higher than that in liquids, and in gas-liquid reactions the mass transfer resistance is almost exclusively on the liquid side. High solubility of the gas-phase component in the liquid or very fast chemical reaction at the interface can change that somewhat. The Sh-number does not change very much with reactor design, and the gas-liquid contact area determines the mass transfer rate, that is, bubble size and gas holdup will determine reactor efficiency. 

Let us consider the typical mechanisms of spontaneous processes that decrease /. The direction and driving force of such mechanisms are determined by the laws of equilibrium thermodynamics, and the rate is proportional to diffusion in gases, viscosity in liquids, and transfer of atoms, vacancies, and other defects in solids. 

Cunningham, R. E. and R. J. J. Williams. 1980. Diffusion in Gases and Porous Media. Plenum Press, New York. 

Marrero and Mason [45] have reviewed the subject of diffusion in gases and give a comprehensive list of data. Diffusion coefficients of gases are inversely proportional to the pressure and vary with temperature according to a power of T between 1.5 and 2. If experimental values are not available, the Wilke—Lee method [46] predicts the diffusion coefficients of non-polar mixtures to within about 4% of their true value. 

The dimensionless time, t, for Sh to come within 100x% of the steady value indicates the duration of the unsteady state for Pe = 0, Tq.i == 31.8, and — 2.35. Diffusivities in gases are of order 10" times diffusivities in liquids hence, for particles with equal size and equal exposure, transient effects in a stagnant medium are much more significant in liquids. 

The molecules are generally much farther apart in gases, so the diffusivity of a compound in a gas is significantly larger than in a liquid. We will return to this comparison of diffusion in gases and hquids in Chapter 3. 

The Chapman-Enskog equation (see Chapman and Cowling, 1970) is semi-empirical because it uses equation (3.11) and adjusts it for errors in the observations of diffusivity in gases. It also includes a parameter, S2, to account for the elasticity of molecular collisions ... 

Drickamer contributed much to the knowledge of diffusion in gases and liquids. As an example, recent studies of the effect of pressure upon material transport in several inorganic systems have been made available... 

Barber, "D if fusion in and through Solids , Cambridge Univ press, NY(1952), 477 pp 16) K.E, Grew T.L. Ibbs, Thermal Diffusion in Gases , Cambridge Univ press,... 

The constant of proportionality, D, is the diffusion coefficient, and the negative sign is necessary because the net flux is from the region of high concentration to the region of low concentration. Table 23-1 shows that diffusion in liquids is 104 times slower than diffusion in gases. Macromole-cules such as ribonuclease and albumin diffuse 10 to 100 times slower than small molecules. 

The main observation from Table 2.1 is the enormous range of values of diffusion coefficients—from 10 1 to 10 30 cm2/s. Diffusion in gases is well understood and is treated in standard textbooks dealing with the kinetic theory of gases [24,25], Diffusion in metals and crystals is a topic of considerable interest to the semiconductor industry but not to membrane permeation. This book focuses principally on diffusion in liquids and polymers in which the diffusion coefficient can vary from about 10 5 to about 10-10 cm2/s. 

Typical values for p are between 0.3 and 0.6, and for tp between 2 and 5. So, a reasonable assumption for the effective diffusion De is that it is Vio of the diffusivity I). This diffusivity D can be calculated from the Knudsen (corresponding to collisions with the wall) and molecular diffusivity (intramolecular collisions). The molecular diffusivity was estimated at 10 5 m2/s, which is reasonable for the diffusion in gases. The Knudsen diffusivity depends on the pore diameter. The exact formulas for the molecular and Knudsen diffusion are given by Moulijn et at1. For zeolites, the determination of the diffusivity is more complicated. The microporous nature of zeolites strongly influences the diffusivity. Therefore, the diffusion... 

For multicomponent diffusion in gases at low density, the Maxwell-Stefan equations provide satisfactory approximations when species / diffuses in a homogeneous mixture... 

Fig. V1IL5. Illuwstrating mass diffusion in gases in one dimension.

Equations for diffusion through a layer of stagnant liquid can also be developed. The applicability of these equations is, however, limited because diffusivity in a liquid varies with concentration. In addition, unless the solutions are very dilute, the total molar concentration varies from point to point. These complications do not arise with diffusion in gases. 

Rates of mass transfer to the catalyst surface and pore diffusion can be calculated by the methods of Section 2.2.2 if the diffusion coefficients are known. However, the molecular theory of diffusion in liquids is relatively undeveloped and it is not yet possible to treat diffusion in liquids with the same rigour as diffusion in gases. The complicating factors are that the diffusion coefficient varies with concentration and that the mass density is usually more constant than the molar density of the solution. An empirical equation, due to Wilke and Chang, which applies in dilute solution, gives... 

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