Pyrex glass is almost impermeable to all gases but helium. For example, the diffusivity of He through pyrex (D_hp)is about 25 times the diffusivity of H2 through pyrex, hydrogen being the closest competitor in the diffusion process. This fact suggests that a method for separating helium from natural gas could be based on the relative diffusion rates through pyrex. Suppose a natural gas mixture is contained in a pyrex tube with length L, inner radius R_i , and outer radius R_o.

a) Derive the equation governing the concentration profile in the wall of the tube.

b) Solve for the concentration profile in the tube. The interfacial concentrations of the helium are C_he(R_i) = C_i and C_he(R_o) = C_o.

c) Obtain an expression for the rate at which helium will leak out of the tube, in terms of the diffusivity of helium through the pyrex, the interfacial concentrations of the helium in the pyrex, and the dimensions of the tube.