In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, Knudsen diffusion, named after
Martin Knudsen, is a means of
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical ...
that occurs when the scale length of a system is comparable to or smaller than the
mean free path
In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as ...
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
The Knudsen number (Kn) is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale. This length scale could be, for example, the radius of a body in a fluid. The number is name ...
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
In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as ...
for molecules in the liquid state is very small, typically near the diameter of the molecule itself.
Mathematical description
The diffusivity for Knudsen diffusion is obtained from the self-diffusion coefficient derived from the
kinetic theory of gases
Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to:
* Kinetic theory, describing a gas as particles in random motion
* Kinetic energy, the energy of an object that it possesses due to its motion
Art and ent ...
:
:
For Knudsen diffusion, path length λ is replaced with pore diameter
, 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'',
is thus
:
where
is the
gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
(8.3144 J/(mol·K) in SI units), molar mass
is expressed in units of kg/mol, and temperature ''T'' (in
kelvins). Knudsen diffusivity
thus depends on the pore diameter, species molar mass and temperature. Expressed as a molecular flux, Knudsen diffusion follows the equation for
Fick's first law of diffusion:
:
Here,
is the molecular flux in mol/m²·s,
is the molar concentration in
. The diffusive flux is driven by a concentration gradient, which in most cases is embodied as a pressure gradient (
i.e. therefore
where
is the pressure difference between both sides of the pore and
is the length of the pore).
If we assume that
is much less than
, the average absolute pressure in the system (
i.e. ) then we can express the Knudsen flux as a volumetric flow rate as follows:
:
,
where
is the volumetric flow rate in
. 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
in for
. 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
are important. The effective diffusivity of species ''A'' in a binary mixture of ''A'' and ''B'',
is determined by
:
where
and
is the flux of component ''i''.
For cases where α = 0 (
, i.e. countercurrent diffusion) or where
is close to zero, the equation reduces to
:
Knudsen self diffusion
In the Knudsen diffusion regime, the molecules do not interact with one another, so that they move in straight lines between points on the pore channel surface. Self-diffusivity is a measure of the translational mobility of individual molecules. Under conditions of
thermodynamic equilibrium
Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In the ...
, a molecule is tagged and its trajectory followed over a long time. If the motion is diffusive, and in a medium without long-range correlations, the squared displacement of the molecule from its original position will eventually grow linearly with time (
Einstein’s equation). To reduce statistical errors in simulations, the self-diffusivity,
, of a species is defined from ensemble averaging Einstein’s equation over a large enough number of molecules ''N''.
See also
*
Knudsen flow
*
Knudsen equation
In fluid dynamics, the Knudsen equation is used to describe how gas flows through a tube in free molecular flow. When the mean free path of the molecule
A molecule is a group of two or more atoms held together by attractive forces known ...
*
Atomic diffusion
Atomic may refer to:
* Of or relating to the atom, the smallest particle of a chemical element that retains its chemical properties
* Atomic physics, the study of the atom
* Atomic Age, also known as the "Atomic Era"
* Atomic scale, distances com ...
*
Mass diffusivity
Diffusivity, mass diffusivity or diffusion coefficient is a proportionality constant between the molar flux due to molecular diffusion and the gradient in the concentration of the species (or the driving force for diffusion). Diffusivity is enc ...
References
{{reflist
External links
Knudsen number and diffusivity calculators
Diffusion