Knudsen diffusion

In physics, Knudsen diffusion, named after Martin Knudsen, is a means of diffusion that occurs when the scale length of a system is comparable to or smaller than the mean free path of the particles involved. An example of this is in a long pore with a narrow diameter (2–50 nm) because molecules frequently collide with the pore wall. In physics, Knudsen diffusion, named after Martin Knudsen, is a means of diffusion that occurs when the scale length of a system is comparable to or smaller than the mean free path of the particles involved. An example of this is in a long pore with a narrow diameter (2–50 nm) because molecules frequently collide with the pore wall. Consider the diffusion of gas molecules through very small capillary pores. If the pore diameter is smaller than the mean free path of the diffusing gas molecules and the density of the gas is low, the gas molecules collide with the pore walls more frequently than with each other. This process is known as Knudsen flow or Knudsen diffusion. The Knudsen number is a good measure of the relative importance of Knudsen diffusion. A Knudsen number much greater than one indicates Knudsen diffusion is important. In practice, Knudsen diffusion applies only to gases because the mean free path for molecules in the liquid state is very small, typically near the diameter of the molecule itself. The diffusivity for Knudsen diffusion is obtained from the self-diffusion coefficient derived from the kinetic theory of gases: For Knudsen diffusion, path length λ is replaced with pore diameter d {displaystyle d} , as species A is now more likely to collide with the pore wall as opposed with another molecule. The Knudsen diffusivity for diffusing species A, D K A {displaystyle D_{KA}} is thus where R {displaystyle R} is the gas constant (8.3144 J/(mol·K) in SI units), molecular mass M A {displaystyle M_{A}} is expressed in units of kg/mol, and temperature T (in kelvins). Knudsen diffusivity D K A {displaystyle D_{KA}} thus depends on the pore diameter, species molecular mass and temperature. Expressed as a molecular flux, Knudsen diffusion follows the equation for Fick's first law of diffusion: Here, J K {displaystyle J_{K}} is the molcular flux in mol/m²·s, n {displaystyle n} is the molar concentration in m o l / m 3 {displaystyle { m {mol/m^{3}}}} . The diffusive flux is driven by a concentration gradient, which in most cases is embodied as a pressure gradient (i.e. n = P / R T {displaystyle n=P/RT} therefore ∇ n = Δ P R T l {displaystyle abla n={frac {Delta P}{RTl}}} where Δ P {displaystyle Delta P} is the pressure difference between both sides of the pore and l {displaystyle l} is the length of the pore). If we assume that Δ P {displaystyle Delta P} is much less than P a v e {displaystyle P_{ m {ave}}} , the average absolute pressure in the system (i.e. Δ P ≪ P a v e {displaystyle Delta Pll P_{ m {ave}}} ) then we can express the Knudsen flux as a volumetric flow rate as follows: where Q K {displaystyle Q_{K}} is the volumetric flowrate in m 3 / s {displaystyle { m {m^{3}/s}}} . If the pore is relatively short, entrance effects can significantly reduce to net flux through the pore. In this case, the law of effusion can be used to calculate the excess resistance due to entrance effects rather easily by substituting an effective length l e = l + 4 3 d {displaystyle l_{ m {e}}=l+{ frac {4}{3}}d} in for l {displaystyle l} . Generally, the Knudsen process is significant only at low pressure and small pore diameter. However there may be instances where both Knudsen diffusion and molecular diffusion D A B {displaystyle D_{AB}} are important. The effective diffusivity of species A in a binary mixture of A and B, D A e {displaystyle D_{Ae}} is determined by