Diffusion

# Diffusion

Overview
Molecular diffusion, often called simply diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) of the particles. Diffusion explains the net flux
Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...

of molecules from a region of higher concentration to one of lower concentration, but it is important to note that diffusion also occurs when there is no concentration gradient.
Discussion

Encyclopedia
Molecular diffusion, often called simply diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) of the particles. Diffusion explains the net flux
Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...

of molecules from a region of higher concentration to one of lower concentration, but it is important to note that diffusion also occurs when there is no concentration gradient. The result of diffusion is a gradual mixing of material. In a phase
Phase (matter)
In the physical sciences, a phase is a region of space , throughout which all physical properties of a material are essentially uniform. Examples of physical properties include density, index of refraction, and chemical composition...

with uniform temperature, absent external net forces acting on the particles, the diffusion process will eventually result in complete mixing.

Diffusive equilibrium is reached when the concentration
Concentration
In chemistry, concentration is defined as the abundance of a constituent divided by the total volume of a mixture. Four types can be distinguished: mass concentration, molar concentration, number concentration, and volume concentration...

s of the diffusing substance
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...

in the two compartments becomes equal.

Consider two systems; S1 and S2 at the same temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

and capable of exchanging particles
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...

. If there is a change in the potential energy
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

of a system; for example μ12 (μ is Chemical potential
Chemical potential
Chemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...

) an energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

flow will occur from S1 to S2, because nature always prefers low energy and maximum entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

.

Though the different systems are at equilibrium, there is still water passing through the semipermeable membrane. So if food coloring is put in system A, eventually it would be of equal color to system B.

Molecular diffusion is typically described mathematically using Fick's laws of diffusion.

## Applications

Diffusion is of fundamental importance in many disciplines of physics, chemistry, and biology. Some example applications of diffusion:
• Sintering
Sintering
Sintering is a method used to create objects from powders. It is based on atomic diffusion. Diffusion occurs in any material above absolute zero, but it occurs much faster at higher temperatures. In most sintering processes, the powdered material is held in a mold and then heated to a temperature...

to produce solid materials (powder metallurgy
Powder metallurgy
Powder metallurgy is the process of blending fine powdered materials, pressing them into a desired shape , and then heating the compressed material in a controlled atmosphere to bond the material . The powder metallurgy process generally consists of four basic steps: powder manufacture, powder...

, production of ceramic
Ceramic
A ceramic is an inorganic, nonmetallic solid prepared by the action of heat and subsequent cooling. Ceramic materials may have a crystalline or partly crystalline structure, or may be amorphous...

s)
• Chemical reactor
Chemical reactor
In chemical engineering, chemical reactors are vessels designed to contain chemical reactions. The design of a chemical reactor deals with multiple aspects of chemical engineering. Chemical engineers design reactors to maximize net present value for the given reaction...

design
• Catalyst
Catalysis
Catalysis is the change in rate of a chemical reaction due to the participation of a substance called a catalyst. Unlike other reagents that participate in the chemical reaction, a catalyst is not consumed by the reaction itself. A catalyst may participate in multiple chemical transformations....

• Steel
Steel
Steel is an alloy that consists mostly of iron and has a carbon content between 0.2% and 2.1% by weight, depending on the grade. Carbon is the most common alloying material for iron, but various other alloying elements are used, such as manganese, chromium, vanadium, and tungsten...

can be diffused (e.g., with carbon or nitrogen) to modify its properties
• Doping
Doping (semiconductor)
In semiconductor production, doping intentionally introduces impurities into an extremely pure semiconductor for the purpose of modulating its electrical properties. The impurities are dependent upon the type of semiconductor. Lightly and moderately doped semiconductors are referred to as extrinsic...

during production of semiconductor
Semiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...

s.

## Significance

Diffusion is part of the transport phenomena
Transport Phenomena
Transport Phenomena is the first textbook that is about transport phenomena. It is specifically designed for chemical engineering students...

. Of mass transport mechanisms, molecular diffusion is known as a slower one.

### Biology

In cell biology
Cell biology
Cell biology is a scientific discipline that studies cells – their physiological properties, their structure, the organelles they contain, interactions with their environment, their life cycle, division and death. This is done both on a microscopic and molecular level...

, diffusion is a main form of transport for necessary materials such as amino acid
Amino acid
Amino acids are molecules containing an amine group, a carboxylic acid group and a side-chain that varies between different amino acids. The key elements of an amino acid are carbon, hydrogen, oxygen, and nitrogen...

s within cells. Diffusion of water (H2O) through a partially permeable membrane is classified as osmosis
Osmosis
Osmosis is the movement of solvent molecules through a selectively permeable membrane into a region of higher solute concentration, aiming to equalize the solute concentrations on the two sides...

.

Metabolism
Metabolism
Metabolism is the set of chemical reactions that happen in the cells of living organisms to sustain life. These processes allow organisms to grow and reproduce, maintain their structures, and respond to their environments. Metabolism is usually divided into two categories...

and respiration
Respiration (physiology)
'In physiology, respiration is defined as the transport of oxygen from the outside air to the cells within tissues, and the transport of carbon dioxide in the opposite direction...

rely in part upon diffusion in addition to bulk or active processes. For example, in the alveoli
Pulmonary alveolus
An alveolus is an anatomical structure that has the form of a hollow cavity. Found in the lung parenchyma, the pulmonary alveoli are the dead ends of the respiratory tree, which outcrop from either alveolar sacs or alveolar ducts, which are both sites of gas exchange with the blood as well...

of mammal
Mammal
Mammals are members of a class of air-breathing vertebrate animals characterised by the possession of endothermy, hair, three middle ear bones, and mammary glands functional in mothers with young...

ian lung
Lung
The lung is the essential respiration organ in many air-breathing animals, including most tetrapods, a few fish and a few snails. In mammals and the more complex life forms, the two lungs are located near the backbone on either side of the heart...

s, due to differences in partial pressures across the alveolar-capillary membrane, oxygen
Oxygen
Oxygen is the element with atomic number 8 and represented by the symbol O. Its name derives from the Greek roots ὀξύς and -γενής , because at the time of naming, it was mistakenly thought that all acids required oxygen in their composition...

diffuses into the blood and carbon dioxide
Carbon dioxide
Carbon dioxide is a naturally occurring chemical compound composed of two oxygen atoms covalently bonded to a single carbon atom...

diffuses out. Lungs contain a large surface area to facilitate this gas exchange process.

## Tracer, self- and chemical diffusion

Fundamentally, two types of diffusion are distinguished:
• Tracer diffusion and Self-diffusion, which is a spontaneous mixing of molecules taking place in the absence of concentration (or chemical potential) gradient. This type of diffusion can be followed using isotopic tracers
Isotopic labeling
Isotopic labeling is a technique for tracking the passage of a sample of substance through a system. The substance is 'labeled' by including unusual isotopes in its chemical composition...

, hence the name. The tracer diffusion is usually assumed to be identical to self-diffusion
Self-diffusion
According to IUPAC definition, self-diffusion coefficient is the diffusion coefficient D_i^* of species i when the chemical potential gradient equals zero. It is linked to the diffusion coefficient D_i by the equation:...

(assuming no significant isotopic effect
Kinetic isotope effect
The kinetic isotope effect is the ratio of reaction rates of two different isotopically labeled molecules in a chemical reaction. It is also called "isotope fractionation," although this term is somewhat broader in meaning...

). This diffusion can take place under equilibrium. An excellent method for the measurement of self-diffusion
Self-diffusion
According to IUPAC definition, self-diffusion coefficient is the diffusion coefficient D_i^* of species i when the chemical potential gradient equals zero. It is linked to the diffusion coefficient D_i by the equation:...

A pulsed field gradient is a short, timed pulse with spatial-dependent field intensity. Any gradient is identified by four characteristics: axis, strength, shape and duration....

(PFG) NMR
NMR
NMR may refer to:Applications of Nuclear Magnetic Resonance:* Nuclear magnetic resonance* NMR spectroscopy* Solid-state nuclear magnetic resonance* Protein nuclear magnetic resonance spectroscopy* Proton NMR* Carbon-13 NMR...

, where no isotopic tracers are needed. In a so-called NMR spin echo
Spin echo
In magnetic resonance, a spin echo is the refocusing of precessing spin magnetisation by a pulse of resonant radiation. Modern nuclear magnetic resonance and magnetic resonance imaging rely heavily on this effect....

experiment this technique uses the nuclear spin precession phase, allowing to distinguish chemically and physically completely identical species e.g. in the liquid phase, as for example water molecules within liquid water. The self-diffusion coefficient of water has been experimentally determined with high accuracy and thus serves often as a reference value for measurments on other liquids. The self-diffusion coefficient of neat water is: 2.299·10−9 m²·s−1 at 25 °C and 1.261·10−9 m²·s−1 at 4 °C.

• Chemical diffusion occurs in a presence of concentration (or chemical potential) gradient and it results in net transport of mass. This is the process described by the diffusion equation. This diffusion is always a non-equilibrium process, increases the system entropy, and brings the system closer to equilibrium.

The diffusion coefficients for these two types of diffusion are generally different because the diffusion coefficient for chemical diffusion is binary and it includes the effects due to the correlation of the movement of the different diffusing species.

## Non-equilibrium system

Because chemical diffusion is a net transport process, the system in which it takes place is not an equilibrium
Chemical equilibrium
In a chemical reaction, chemical equilibrium is the state in which the concentrations of the reactants and products have not yet changed with time. It occurs only in reversible reactions, and not in irreversible reactions. Usually, this state results when the forward reaction proceeds at the same...

system (i.e. it is not at rest yet). Many results in classical thermodynamics are not easily applied to non-equilibrium systems. However, there sometimes occur so-called quasi-steady states, where the diffusion process does not change in time, where classical results may locally apply. As the name suggests, this process is a not a true equilibrium since the system is still evolving.

Non-equilibrium fluid systems can be successfully modeled with Landau-Lifshitz fluctuating hydrodynamics. In this theoretical framework, diffusion is due to fluctuations whose dimensions range from the molecular scale to the macroscopic scale.

Chemical diffusion increases the entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

of a system, i.e. diffusion is a spontaneous and irreversible process. Particles can spread out by diffusion, but will not spontaneously re-order themselves (absent changes to the system, assuming no creation of new chemical bonds, and absent external forces acting on the particle).

## Concentration dependent "collective" diffusion

Collective diffusion is the diffusion of a large number of particles, most often within a solvent
Solvent
A solvent is a liquid, solid, or gas that dissolves another solid, liquid, or gaseous solute, resulting in a solution that is soluble in a certain volume of solvent at a specified temperature...

.

Contrary to brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

, which is the diffusion of a single particle, interactions between particles may have to be considered, unless the particles form an ideal mix with their solvent (ideal mix conditions correspond to the case where the interactions between the solvent and particles are identical to the interactions between particles and the interactions between solvent molecules; in this case, the particles do not interact when inside the solvent).

In case of an ideal mix, the particle diffusion equation holds true and the diffusion coefficient D the speed of diffusion
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...

in the particle diffusion equation is independent of particle concentration. In other cases, resulting interactions between particles within the solvent will account for the following effects:
• the diffusion coefficient D in the particle diffusion equation becomes dependent of concentration. For an attractive interaction between particles, the diffusion coefficient tends to decrease as concentration increases. For a repulsive interaction between particles, the diffusion coefficient tends to increase as concentration increases.
• In the case of an attractive interaction between particles, particles exhibit a tendency to coalesce and form clusters if their concentration
Concentration
In chemistry, concentration is defined as the abundance of a constituent divided by the total volume of a mixture. Four types can be distinguished: mass concentration, molar concentration, number concentration, and volume concentration...

lies above a certain threshold. This is equivalent to a precipitation
Precipitation (chemistry)
Precipitation is the formation of a solid in a solution or inside anothersolid during a chemical reaction or by diffusion in a solid. When the reaction occurs in a liquid, the solid formed is called the precipitate, or when compacted by a centrifuge, a pellet. The liquid remaining above the solid...

chemical reaction (and if the considered diffusing particles are chemical molecules in solution, then it is a precipitation
Precipitation (chemistry)
Precipitation is the formation of a solid in a solution or inside anothersolid during a chemical reaction or by diffusion in a solid. When the reaction occurs in a liquid, the solid formed is called the precipitate, or when compacted by a centrifuge, a pellet. The liquid remaining above the solid...

).

## Molecular Diffusion of Gases

Transport of material in stagnant fluid or across streamlines of a fluid in a laminar flow occurs by molecular diffusion.Two adjacent compartments, separated by partition containing pure gases A or B may be envisaged. Random movement of all molecules occurs so that after a period molecules are found remote from their original positions. If the partition is removed, some molecules of A move towards the region occupied by B, their number depends on the number of molecules at the point considered. Concurrently, molecules of B diffuse toward regimens formerly occupied by pure A.
Finally, complete mixing occurs. Before this point in time, a gradual variation in the concentration of A occurs along an axis, designated x, which joins the original compartments. This variation, expressed mathematically -dCA/dx, where CA is the concentration of A. The negative sign arises because the concentration of A decreases as the distance x increases. Similarly, the variation in the concentration of gas B is -dCB/dx. The rate of diffusion of A, NA, depend on concentration gradient and the average velocity with which the molecules of A moves in the x direction. This relationship is expressed by Fick's Law
"only applicable for no bulk motion"

where D is the Diffusivity of A through B, proportional to the average molecular velocity and, therefore depend on the temperature and pressure of gases. The rate of Diffusion NA,is usually expressed as the number of moles diffusing across unit area in unit time. As with the basic equation of heat transfer, indicates that the rate of force is directly proportional to the driving force, which is the concentration gradient.

This basic equation applied to a number of situations. Restricting discussion exclusively to steady state conditions, in which neither dCA/dx or dCB/dx change with time, equimolecular counterdiffusion is considered first.

## Equimolecular Counterdiffusion

If no bulk flow occurs in an element of length dx, the rates of diffusion of two gases A and B must be equal and opposite, that is NA=-NB.

The partial pressure of A changes by dPA over the distance dx. Similarly, the partial pressure of B changes dPB. As there is no difference in total pressure across the element (no bulk flow), dPA/dx must equal -dPB/dx. For an ideal gas the partial pressure is related to the molar concentration by the relation

where nA is the number of moles of gas A in a volume V. As the molar concentration CA is equal to nA/ V therefore

Consequently, for gas A,

where DAB is the diffusivity of A in B. Similarly,

It therefore allows that DAB=DBA=D. If the partial pressure of A at x1 is PA1 and x2 is PA2, integration of above equation,

A similar equation may be derived for the counterdiffusion of gas B.