3 Driving Forces for Diffusion

Consider a container (Figure 2a) of an ideal binary gas at uniform temperature, pressure and composition. On average, all of the molecules in the binary gas have the same kinetic energy because the temperature is uniform. Let us imagine a plane across the container so that it is divided into two sections as shown. Molecules of both components continuously pass back and forth across the plane due to their random thermally induced motion. We may also think of the dividing plane as a wall against which the molecules on either side collide. These collisions are responsible for exerting pressure on the wall. If the number of molecules per unit volume on both sides of the surface is the same, there is no pressure difference across the surface. The pressure created by molecules on one side is opposed by an equal pressure in the gas on the other. The contribution to total pressure made by an individual component is known as the partial pressure, pi, of that component. The partial pressure of each component is related to the component molar concentration by pi = CiRT.

Figure showing binary gas at uniform composition, pressure and temperature.

Figure 2a Binary gas at uniform composition, pressure and temperature.

Now suppose the composition of the binary gas is changed by removing from the left side some molecules of species A and replacing them with the same number of species-B molecules. The concentration of species-A molecules is now less on the left than on the right and the opposite is true for species B (Figure 2b). The total pressure on either side of our imaginary surface is unchanged because the total number of molecules is still the same on both sides of the surface. However, the number of A molecules impinging on the imaginary surface from the left is less than from the right and the opposite is true for B molecules. Thus, there is a net diffusion of species A from right to left and species B experiences a net diffusion from left to right. In both cases, diffusion occurs from high to low concentration or, equivalently, from high partial pressure to low partial pressure. Some authors employ differences in the thermodynamic quantity known as chemical potential (energy per mole) as the indicator of the direction of diffusion. However, concentration, partial pressure, and chemical potential are closely related quantities in isothermal ideal gases and each can be used to indicate the direction of component diffusion when the composition is not uniform.

Figure showing binary gas at uniform composition, pressure and temperature.

Figure 2b Binary gas with non-uniform composition. Molecules A diffuse to the left and molecules B diffuse to the right.

The gradient of concentration, partial pressure, or chemical potential may be used to express the driving force for the diffusion described in the previous paragraph. We choose to use the gradient of partial pressure for the moment because it has dimensions of force per unit volume and makes the discussion of momentum balances in the following section more intuitive and easier to grasp. It is important to note that the gradient of partial pressure is not an external force that acts on the fluid as a whole; rather, the gradient of partial pressure of a component is an internal force that drives diffusion of that component. The partial pressure of component i is related to the total gas pressure and the component mole fraction by pi = pxi. The gradient of partial pressure is expressed below in Equation 12.

\displaystyle \frac{dp_{i}}{dl}=p\frac{dx_{i}}{dl}+x_{i}\frac{dp}{dl} (12)

It follows that the driving force for diffusion of an individual component is affected by both a gradient of mole fraction and a pressure gradient in the gas as a whole. Of course, a pressure gradient also induces bulk gas flow, but the second term on the right side of Equation 12 derives from the gradient of partial pressure in the present context and is a force driving diffusion. Diffusion in response to the gradient of total gas pressure is often referred to as pressure diffusion.


Flux Equations for Gas Diffusion in Porous Media Copyright © 2021 by David B. McWhorter. All Rights Reserved.