Mass Diffusivity
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Diffusivity, mass diffusivity or diffusion coefficient is a proportionality constant between the molar flux due to molecular
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 p ...
and the gradient in the concentration of the species (or the driving force for diffusion). Diffusivity is encountered in
Fick's law Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion eq ...
and numerous other equations of
physical chemistry Physical chemistry is the study of macroscopic and microscopic phenomena in chemical systems in terms of the principles, practices, and concepts of physics such as motion, energy, force, time, thermodynamics, quantum chemistry, statistical mecha ...
. The diffusivity is generally prescribed for a given pair of species and pairwise for a multi-species system. The higher the diffusivity (of one substance with respect to another), the faster they diffuse into each other. Typically, a compound's diffusion coefficient is ~10,000× as great in air as in water. Carbon dioxide in air has a diffusion coefficient of 16 mm2/s, and in water its diffusion coefficient is 0.0016 mm2/s.Diffusion
/ref> Diffusivity has dimensions of length2 / time, or m2/s in
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
and cm2/s in CGS units.


Temperature dependence of the diffusion coefficient


Solids

The diffusion coefficient in solids at different temperatures is generally found to be well predicted by the
Arrhenius equation In physical chemistry, the Arrhenius equation is a formula for the temperature dependence of reaction rates. The equation was proposed by Svante Arrhenius in 1889, based on the work of Dutch chemist Jacobus Henricus van 't Hoff who had noted in 18 ...
: D = D_0 \exp\left(-\frac\right) where *''D'' is the diffusion coefficient (in m2/s), *''D''0 is the maximal diffusion coefficient (at infinite temperature; in m2/s), *''E''A is the
activation energy In chemistry and physics, activation energy is the minimum amount of energy that must be provided for compounds to result in a chemical reaction. The activation energy (''E''a) of a reaction is measured in joules per mole (J/mol), kilojoules pe ...
for diffusion (in J/mol), *''T'' is the absolute temperature (in K), *''R'' ≈ 8.31446J/(mol⋅K) is the
universal 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 ...
.


Liquids

An approximate dependence of the diffusion coefficient on temperature in liquids can often be found using
Stokes–Einstein equation In physics (specifically, the kinetic theory of gases), the Einstein relation is a previously unexpected connection revealed independently by William Sutherland in 1904, Albert Einstein in 1905, and by Marian Smoluchowski in 1906 in their works on ...
, which predicts that \frac = \frac \frac , where * ''D'' is the diffusion coefficient, * ''T''1 and ''T''2 are the corresponding absolute temperatures, * ''μ'' is the
dynamic viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...
of the solvent.


Gases

The dependence of the diffusion coefficient on temperature for gases can be expressed using
Chapman–Enskog theory Chapman–Enskog theory provides a framework in which equations of hydrodynamics for a gas can be derived from the Boltzmann equation. The technique justifies the otherwise phenomenological constitutive relations appearing in hydrodynamical descri ...
(predictions accurate on average to about 8%): D = \frac\sqrt, where * ''D'' is the diffusion coefficient (cm2/s), * ''A'' is an empirical coefficient equal to 1.859 \times 10^ \mathrm * 1 and 2 index the two kinds of molecules present in the gaseous mixture, * ''T'' is the absolute temperature (K), * ''M'' is the molar mass (g/mol), * ''p'' is the pressure (atm), * \sigma_ = \frac(\sigma_1 + \sigma_2) is the average collision diameter (the values are tabulated page 545) (Å), * Ω is a temperature-dependent collision integral (the values are tabulated but usually of order 1) (dimensionless).


Pressure dependence of the diffusion coefficient

For self-diffusion in gases at two different pressures (but the same temperature), the following empirical equation has been suggested: \frac = \frac , where * ''D'' is the diffusion coefficient, * ''ρ'' is the gas mass density, * ''P''1 and ''P''2 are the corresponding pressures.


Population dynamics: dependence of the diffusion coefficient on fitness

In population dynamics, kinesis is the change of the diffusion coefficient in response to the change of conditions. In models of purposeful kinesis, diffusion coefficient depends on fitness (or reproduction coefficient) ''r'': D = D_0 e ^, where D_0 is constant and ''r'' depends on population densities and abiotic characteristics of the living conditions. This dependence is a formalisation of the simple rule: Animals stay longer in good conditions and leave quicker bad conditions (the "Let well enough alone" model).


Effective diffusivity in porous media

The effective diffusion coefficient describes diffusion through the pore space of
porous media A porous medium or a porous material is a material containing pores (voids). The skeletal portion of the material is often called the "matrix" or "frame". The pores are typically filled with a fluid (liquid or gas). The skeletal material is usu ...
. It is
macroscopic The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic. Overview When applied to physical phenomena an ...
in nature, because it is not individual pores but the entire pore space that needs to be considered. The effective diffusion coefficient for transport through the pores, ''D''e, is estimated as follows: D_\text = \frac, where *''D'' is the diffusion coefficient in gas or liquid filling the pores, *''εt'' is the
porosity Porosity or void fraction is a measure of the void (i.e. "empty") spaces in a material, and is a fraction of the volume of voids over the total volume, between 0 and 1, or as a percentage between 0% and 100%. Strictly speaking, some tests measure ...
available for the transport (dimensionless), *''δ'' is the
constrictivity Constrictivity is a dimensionless parameter used to describe transport processes (often molecular diffusion) in porous media. Constrictivity is viewed to depend on the ratio of the diameter of the diffusing particle to the pore diameter. The val ...
(dimensionless), *''τ'' is the
tortuosity Tortuosity is widely used as a critical parameter to predict transport properties of porous media, such as rocks and soils. But unlike other standard microstructural properties, the concept of tortuosity is vague with multiple definitions and vari ...
(dimensionless). The transport-available
porosity Porosity or void fraction is a measure of the void (i.e. "empty") spaces in a material, and is a fraction of the volume of voids over the total volume, between 0 and 1, or as a percentage between 0% and 100%. Strictly speaking, some tests measure ...
equals the total porosity less the pores which, due to their size, are not accessible to the diffusing particles, and less dead-end and blind pores (i.e., pores without being connected to the rest of the pore system). The constrictivity describes the slowing down of diffusion by increasing the
viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...
in narrow pores as a result of greater proximity to the average pore wall. It is a function of pore diameter and the size of the diffusing particles.


Example values

Gases at 1 atm., solutes in liquid at infinite dilution. Legend: (s) – solid; (l) – liquid; (g) – gas; (dis) – dissolved.


See also

*
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 ...
*
Effective diffusion coefficient The effective diffusion coefficient of a in atomic diffusion of solid polycrystalline materials like metal alloys is often represented as a weighted average of the grain boundary diffusion coefficient and the lattice diffusion coefficient.P. Hei ...
*
Lattice diffusion coefficient Lattice diffusion (also called bulk or volume diffusion) refers to atomic diffusion within a crystalline lattice.P. Heitjans, J. Karger, Ed, “Diffusion in condensed matter: Methods, Materials, Models,” 2nd edition, Birkhauser, 2005, pp. 1-96 ...
*
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 w ...


References

{{DEFAULTSORT:Mass Diffusivity Transport phenomena Diffusion