Molecular Diffusion

Molecular diffusion of matter is analogous to the conductive transfer of heat. In heat conduction, energy moves from a region of high temperature to a region of low temperature due to the random motion of gas or liquid molecules or the vibration of solid molecules. Similarly, in molecular diffusion, matter moves from a region of high concentration to a region of low concentration due to the random motion of the molecules of that substance. Therefore, molecular diffusion can be defined as the net transport of matter on a molecular scale due to a concentration or partial pressure gradient through a medium, which is either stagnant or has a laminar flow with a direction perpendicular to that of the concentration gradient. For example, water will evaporate from an open surface into still surrounding air, and a piece of sugar will dissolve in a cup of coffee and spread without stirring.

Fick's First Law

Fick first recognized the analogy between heat conduction and mass diffusion in 1855. Following Fourier's equation for heat conduction, Fick quantitatively expressed the rate of diffusion of a substance, through an isotropic medium with a unit surface area, as being proportional to the concentration gradient measured as a vector normal to the surface:

dC dy

where j is the diffusion rate, C is the concentration of the substance being transferred, y is the distance, and D is a measure of the resistance against the diffusion of matter, usually called the diffusivity or diffusion coefficient. The rate j is usually expressed in mol/m2 • s; C, in mol/m3; y, in m; and D, in m2/s. The negative sign in Fick's law indicates that the flow of matter is in the direction of decreasing concentration. It should be noted that equation 1 applies equally well to diffusion in gases, liquids, and solids (assuming isotropic properties).

Diffusion in Gases

From the standpoint of molecular kinetics, gas molecules contain energy and are in a state of continuous random motion. During such movement, the molecules bounce against each other or against other surfaces, continuously changing their direction. Consequently, in a region of constant concentration, the probability of an individual molecule moving in any direction is the same. This phenomenon is called the random walk. When considering a homogeneous mixture of two gases, for example, and imaging a certain number of molecules traveling in one di rection, there must be an equal number of molecules traveling in the opposite direction and no net mass transfer is taking place. If a concentration gradient present were, however, then the number of molecules traveling from the high to the low concentration region would be larger than that traveling in the opposite direction. Therefore, net mass transfer would take place from the high to the low concentration region.

Assuming that a gas A follows the ideal gas law, the concentration term in Fick's law can be expressed as:

where nA is the amount of gas A in moles, V is the total volume, PA is the partial pressure of A, R is the gas constant, and T is the absolute temperature. Thus, Fick's law becomes

In a more general case, the medium is moving in a laminar flow with a bulk speed vb while the component is diffusing through the medium. NA, the total flux of component A, is the sum of diffusion and bulk flow of the mixture

where vd is the velocity of A due to true diffusion and CAvd is the number of molecules being diffused. The term CAvd can be expressed by Fick's law and is equivalent to jA. Hence the total flux of A is

For a binary mixture of gases A and B, the total mass flux is the sum of the fluxes of the two gases

where C is the total concentration of the mixture (A + B). By substituting equation 7 into equation 5, and using CA = Pa/RT, and C = P/RT, gives

For ideal gases where CA/C = PA/P, Fick's law gives the following general equation for diffusion in gases

Applications of equation 9 in two specific cases will be discussed in the following section.

Equimolecular Gas Counterdiffusion. Consider two gases A and B in two jars connected by a tube as shown in Figure 1. When the total pressure in each jar is the same, a molecular diffusion through a stationary medium will occur due to the presence of partial pressure difference between each individual component. Thus

As long as the partial pressure PA1 > PA2 and PB2 > Pbi, diffusion will continue to occur. The number of molecules of A diffusing from jar 1 to jar 2 is equal to the number of molecules of B diffusing in the opposite direction (to maintain the equal total pressure in both jars)

Na = ~Nb or ja = -jB Setting Na = —Nb in equation 9, we have

Dab dPA RT dy

Dba dpg RT dy

For steady-state conditions, integration of equation 10 gives and

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