Chemical potential

Last updated

In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species in a mixture is defined as the rate of change of free energy of a thermodynamic system with respect to the change in the number of atoms or molecules of the species that are added to the system. Thus, it is the partial derivative of the free energy with respect to the amount of the species, all other species' concentrations in the mixture remaining constant. When both temperature and pressure are held constant, and the number of particles is expressed in moles, the chemical potential is the partial molar Gibbs free energy . [1] [2] At chemical equilibrium or in phase equilibrium, the total sum of the product of chemical potentials and stoichiometric coefficients is zero, as the free energy is at a minimum. [3] [4] [5] In a system in diffusion equilibrium, the chemical potential of any chemical species is uniformly the same everywhere throughout the system. [6]

Contents

In semiconductor physics, the chemical potential of a system of electrons at zero absolute temperature is known as the Fermi level. [7]

Overview

Particles tend to move from higher chemical potential to lower chemical potential because this reduces the free energy. In this way, chemical potential is a generalization of "potentials" in physics such as gravitational potential. When a ball rolls down a hill, it is moving from a higher gravitational potential (higher internal energy thus higher potential for work) to a lower gravitational potential (lower internal energy). In the same way, as molecules move, react, dissolve, melt, etc., they will always tend naturally to go from a higher chemical potential to a lower one, changing the particle number, which is the conjugate variable to chemical potential.

A simple example is a system of dilute molecules diffusing in a homogeneous environment. In this system, the molecules tend to move from areas with high concentration to low concentration, until eventually, the concentration is the same everywhere. The microscopic explanation for this is based on kinetic theory and the random motion of molecules. However, it is simpler to describe the process in terms of chemical potentials: For a given temperature, a molecule has a higher chemical potential in a higher-concentration area and a lower chemical potential in a low concentration area. Movement of molecules from higher chemical potential to lower chemical potential is accompanied by a release of free energy. Therefore, it is a spontaneous process.

Another example, not based on concentration but on phase, is an ice cube on a plate above 0 °C. An H2O molecule that is in the solid phase (ice) has a higher chemical potential than a water molecule that is in the liquid phase (water) above 0 °C. When some of the ice melts, H2O molecules convert from solid to the warmer liquid where their chemical potential is lower, so the ice cube shrinks. At the temperature of the melting point, 0 °C, the chemical potentials in water and ice are the same; the ice cube neither grows nor shrinks, and the system is in equilibrium.

A third example is illustrated by the chemical reaction of dissociation of a weak acid HA (such as acetic acid, A = CH3COO):

HA H+ + A

Vinegar contains acetic acid. When acid molecules dissociate, the concentration of the undissociated acid molecules (HA) decreases and the concentrations of the product ions (H+ and A) increase. Thus the chemical potential of HA decreases and the sum of the chemical potentials of H+ and A increases. When the sums of chemical potential of reactants and products are equal the system is at equilibrium and there is no tendency for the reaction to proceed in either the forward or backward direction. This explains why vinegar is acidic, because acetic acid dissociates to some extent, releasing hydrogen ions into the solution.

Chemical potentials are important in many aspects of multi-phase equilibrium chemistry, including melting, boiling, evaporation, solubility, osmosis, partition coefficient, liquid-liquid extraction and chromatography. In each case the chemical potential of a given species at equilibrium is the same in all phases of the system. [6]

In electrochemistry, ions do not always tend to go from higher to lower chemical potential, but they do always go from higher to lower electrochemical potential . The electrochemical potential completely characterizes all of the influences on an ion's motion, while the chemical potential includes everything except the electric force. (See below for more on this terminology.)

Thermodynamic definition

The chemical potential μi of species i (atomic, molecular or nuclear) is defined, as all intensive quantities are, by the phenomenological fundamental equation of thermodynamics. This holds for both reversible and irreversible infinitesimal processes: [8]

where dU is the infinitesimal change of internal energy U, dS the infinitesimal change of entropy S, dV is the infinitesimal change of volume V for a thermodynamic system in thermal equilibrium, and dNi is the infinitesimal change of particle number Ni of species i as particles are added or subtracted. T is absolute temperature, S is entropy, P is pressure, and V is volume. Other work terms, such as those involving electric, magnetic or gravitational fields may be added.

From the above equation, the chemical potential is given by

This is because the internal energy U is a state function, so if its differential exists, then the differential is an exact differential such as

for independent variables x1, x2, ... , xN of U.

This expression of the chemical potential as a partial derivative of U with respect to the corresponding species particle number is inconvenient for condensed-matter systems, such as chemical solutions, as it is hard to control the volume and entropy to be constant while particles are added. A more convenient expression may be obtained by making a Legendre transformation to another thermodynamic potential: the Gibbs free energy . From the differential (for and , the product rule is applied to) and using the above expression for , a differential relation for is obtained:

As a consequence, another expression for results:

and the change in Gibbs free energy of a system that is held at constant temperature and pressure is simply

In thermodynamic equilibrium, when the system concerned is at constant temperature and pressure but can exchange particles with its external environment, the Gibbs free energy is at its minimum for the system, that is . It follows that

Use of this equality provides the means to establish the equilibrium constant for a chemical reaction.

By making further Legendre transformations from U to other thermodynamic potentials like the enthalpy and Helmholtz free energy , expressions for the chemical potential may be obtained in terms of these:

These different forms for the chemical potential are all equivalent, meaning that they have the same physical content, and may be useful in different physical situations.

Applications

The Gibbs–Duhem equation is useful because it relates individual chemical potentials. For example, in a binary mixture, at constant temperature and pressure, the chemical potentials of the two participants A and B are related by

where is the number of moles of A and is the number of moles of B. Every instance of phase or chemical equilibrium is characterized by a constant. For instance, the melting of ice is characterized by a temperature, known as the melting point at which solid and liquid phases are in equilibrium with each other. Chemical potentials can be used to explain the slopes of lines on a phase diagram by using the Clapeyron equation, which in turn can be derived from the Gibbs–Duhem equation. [9] They are used to explain colligative properties such as melting-point depression by the application of pressure. [10] Henry's law for the solute can be derived from Raoult's law for the solvent using chemical potentials. [11] [12]

History

Chemical potential was first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. He defined it as follows:

If to any homogeneous mass in a state of hydrostatic stress we suppose an infinitesimal quantity of any substance to be added, the mass remaining homogeneous and its entropy and volume remaining unchanged, the increase of the energy of the mass divided by the quantity of the substance added is the potential for that substance in the mass considered.

Gibbs later noted[ citation needed ] also that for the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a substance, whether capable or not of existing by itself as a homogeneous body. This freedom to choose the boundary of the system allows the chemical potential to be applied to a huge range of systems. The term can be used in thermodynamics and physics for any system undergoing change. Chemical potential is also referred to as partial molar Gibbs energy (see also partial molar property). Chemical potential is measured in units of energy/particle or, equivalently, energy/mole.

In his 1873 paper A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces, Gibbs introduced the preliminary outline of the principles of his new equation able to predict or estimate the tendencies of various natural processes to ensue when bodies or systems are brought into contact. By studying the interactions of homogeneous substances in contact, i.e. bodies, being in composition part solid, part liquid, and part vapor, and by using a three-dimensional volumeentropyinternal energy graph, Gibbs was able to determine three states of equilibrium, i.e. "necessarily stable", "neutral", and "unstable", and whether or not changes will ensue. In 1876, Gibbs built on this framework by introducing the concept of chemical potential so to take into account chemical reactions and states of bodies that are chemically different from each other. In his own words from the aforementioned paper, Gibbs states:

If we wish to express in a single equation the necessary and sufficient condition of thermodynamic equilibrium for a substance when surrounded by a medium of constant pressure P and temperature T, this equation may be written:

Where δ refers to the variation produced by any variations in the state of the parts of the body, and (when different parts of the body are in different states) in the proportion in which the body is divided between the different states. The condition of stable equilibrium is that the value of the expression in the parenthesis shall be a minimum.

In this description, as used by Gibbs, ε refers to the internal energy of the body, η refers to the entropy of the body, and ν is the volume of the body.

Electrochemical, internal, external, and total chemical potential

The abstract definition of chemical potential given above—total change in free energy per extra mole of substance—is more specifically called total chemical potential. [13] [14] If two locations have different total chemical potentials for a species, some of it may be due to potentials associated with "external" force fields (electric potential energy, gravitational potential energy, etc.), while the rest would be due to "internal" factors (density, temperature, etc.) [13] Therefore, the total chemical potential can be split into internal chemical potential and external chemical potential:

where

i.e., the external potential is the sum of electric potential, gravitational potential, etc. (where q and m are the charge and mass of the species, Vele and h are the electric potential [15] and height of the container, respectively, and g is the acceleration due to gravity). The internal chemical potential includes everything else besides the external potentials, such as density, temperature, and enthalpy. This formalism can be understood by assuming that the total energy of a system, , is the sum of two parts: an internal energy, , and an external energy due to the interaction of each particle with an external field, . The definition of chemical potential applied to yields the above expression for .

The phrase "chemical potential" sometimes means "total chemical potential", but that is not universal. [13] In some fields, in particular electrochemistry, semiconductor physics, and solid-state physics, the term "chemical potential" means internal chemical potential, while the term electrochemical potential is used to mean total chemical potential. [16] [17] [18] [19] [20]

Systems of particles

Electrons in solids

Electrons in solids have a chemical potential, defined the same way as the chemical potential of a chemical species: The change in free energy when electrons are added or removed from the system. In the case of electrons, the chemical potential is usually expressed in energy per particle rather than energy per mole, and the energy per particle is conventionally given in units of electronvolt (eV).

Chemical potential plays an especially important role in solid-state physics and is closely related to the concepts of work function, Fermi energy, and Fermi level. For example, n-type silicon has a higher internal chemical potential of electrons than p-type silicon. In a p–n junction diode at equilibrium the chemical potential (internal chemical potential) varies from the p-type to the n-type side, while the total chemical potential (electrochemical potential, or, Fermi level) is constant throughout the diode.

As described above, when describing chemical potential, one has to say "relative to what". In the case of electrons in semiconductors, internal chemical potential is often specified relative to some convenient point in the band structure, e.g., to the bottom of the conduction band. It may also be specified "relative to vacuum", to yield a quantity known as work function, however, work function varies from surface to surface even on a completely homogeneous material. Total chemical potential, on the other hand, is usually specified relative to electrical ground.

In atomic physics, the chemical potential of the electrons in an atom is sometimes [21] said to be the negative of the atom's electronegativity. Likewise, the process of chemical potential equalization is sometimes referred to as the process of electronegativity equalization. This connection comes from the Mulliken electronegativity scale. By inserting the energetic definitions of the ionization potential and electron affinity into the Mulliken electronegativity, it is seen that the Mulliken chemical potential is a finite difference approximation of the electronic energy with respect to the number of electrons, i.e.,

Sub-nuclear particles

In recent years,[ when? ] thermal physics has applied the definition of chemical potential to systems in particle physics and its associated processes. For example, in a quark–gluon plasma or other QCD matter, at every point in space there is a chemical potential for photons, a chemical potential for electrons, a chemical potential for baryon number, electric charge, and so forth.

In the case of photons, photons are bosons and can very easily and rapidly appear or disappear. Therefore, at thermodynamic equilibrium, the chemical potential of photons is in most physical situations always and everywhere zero. The reason is, if the chemical potential somewhere was higher than zero, photons would spontaneously disappear from that area until the chemical potential went back to zero; likewise, if the chemical potential somewhere was less than zero, photons would spontaneously appear until the chemical potential went back to zero. Since this process occurs extremely rapidly - at least, it occurs rapidly in the presence of dense charged matter or also in the walls of the textbook example for a photon gas of blackbody radiation - it is safe to assume that the photon chemical potential here is never different from zero. A physical situation where the chemical potential for photons can differ from zero are material-filled optical microcavities, with spacings between cavity mirrors in the wavelength regime. In such two-dimensional cases, photon gases with tuneable chemical potential, much reminiscent to gases of material particles, can be observed. [22]

Electric charge is different because it is intrinsically conserved, i.e. it can be neither created nor destroyed. It can, however, diffuse. The "chemical potential of electric charge" controls this diffusion: Electric charge, like anything else, will tend to diffuse from areas of higher chemical potential to areas of lower chemical potential. [23] Other conserved quantities like baryon number are the same. In fact, each conserved quantity is associated with a chemical potential and a corresponding tendency to diffuse to equalize it out. [24]

In the case of electrons, the behaviour depends on temperature and context. At low temperatures, with no positrons present, electrons cannot be created or destroyed. Therefore, there is an electron chemical potential that might vary in space, causing diffusion. At very high temperatures, however, electrons and positrons can spontaneously appear out of the vacuum (pair production), so the chemical potential of electrons by themselves becomes a less useful quantity than the chemical potential of the conserved quantities like (electrons minus positrons).

The chemical potentials of bosons and fermions is related to the number of particles and the temperature by Bose–Einstein statistics and Fermi–Dirac statistics respectively.

Ideal vs. non-ideal solutions

The chemical potential of component i in solution for (left) ideal [incorrectly linearized] and (right) real solutions Chempotential.jpg
The chemical potential of component i in solution for (left) ideal [incorrectly linearized] and (right) real solutions

Generally the chemical potential is given as a sum of an ideal contribution and an excess contribution:

In an ideal solution, the chemical potential of species ii) is dependent on temperature and pressure.

μi0(TP) is defined as the chemical potential of pure species i. Given this definition, the chemical potential of species i in an ideal solution is

where R is the gas constant, and is the mole fraction of species i contained in the solution. The chemical potential becomes negative infinity when , but this does not lead to nonphysical results because means that species i is not present in the system.

This equation assumes that only depends on the mole fraction () contained in the solution. This neglects intermolecular interaction between species i with itself and other species [i–(ji)]. This can be corrected for by factoring in the coefficient of activity of species i, defined as γi. This correction yields

The plots above give a very rough picture of the ideal and non-ideal situation.

See also

Sources

Citations

  1. Atkins 2006 [ page needed ]
  2. Opacity, Walter F. Huebner, W. David Barfield, ISBN   1461487978, p. 105.
  3. Atkins 2002 , pp. 227, section 9.2
  4. Baierlein, Ralph (April 2001). "The elusive chemical potential" (PDF). American Journal of Physics. 69 (4): 423–434. Bibcode:2001AmJPh..69..423B. doi:10.1119/1.1336839.
  5. Job, G.; Herrmann, F. (February 2006). "Chemical potential–a quantity in search of recognition" (PDF). European Journal of Physics. 27 (2): 353–371. Bibcode:2006EJPh...27..353J. CiteSeerX   10.1.1.568.9205 . doi:10.1088/0143-0807/27/2/018. S2CID   16146320. Archived from the original (PDF) on 2015-09-24. Retrieved 2009-02-12.
  6. 1 2 Atkins 2002 , pp. 141, section 6.4
  7. Kittel 1980 , pp. 357
  8. Statistical Physics, F Mandl, (Wiley, London, 11971) ISBN   0 471 56658 6, page 88.
  9. Atkins 2006 , pp. 126, section 4.1
  10. Atkins 2006 , pp. 150–155, section 5.5
  11. McQuarrie, D. A.; Simon, J. D. Physical Chemistry – A Molecular Approach, p. 968, University Science Books, 1997.
  12. Atkins 2006 , pp. 143–145, section 5.3
  13. 1 2 3 Kittel 1980 , pp. 124
  14. Thermodynamics in Earth and Planetary Sciences by Jibamitra Ganguly, p. 240. This text uses "internal", "external", and "total chemical potential" as in this article.
  15. Mortimer, R. G. Physical Chemistry, 3rd ed., p. 352, Academic Press, 2008.
  16. Electrochemical Methods by Bard and Faulkner, 2nd edition, Section 2.2.4(a), 4–5.
  17. Electrochemistry at Metal and Semiconductor Electrodes, by Norio Sato, pages 4–5.
  18. Physics Of Transition Metal Oxides, by Sadamichi Maekawa, p. 323.
  19. The Physics of Solids: Essentials and Beyond, by Eleftherios N. Economou, page 140. In this text, total chemical potential is usually called "electrochemical potential", but sometimes just "chemical potential". The internal chemical potential is referred to by the unwieldy phrase "chemical potential in the absence of the [electric] field".
  20. Solid State Physics by Ashcroft and Mermin, page 257 note 36. Page 593 of the same book uses, instead, an unusual "flipped" definition, where "chemical potential" is the total chemical potential, which is constant in equilibrium, and "electrochemical potential" is the internal chemical potential; presumably this unusual terminology was an unintentional mistake.
  21. Morell, Christophe, Introduction to Density Functional Theory of Chemical Reactivity: The so-called Conceptual DFT Archived 2017-08-28 at the Wayback Machine , retrieved May 2016.
  22. J. Klaers; J. Schmitt; F. Vewinger & M. Weitz (2010). "Bose–Einstein condensation of photons in an optical microcavity". Nature. 468 (7323): 545–548. arXiv: 1007.4088 . Bibcode:2010Natur.468..545K. doi:10.1038/nature09567. PMID   21107426. S2CID   4349640.
  23. Baierlein, Ralph (2003). Thermal Physics . Cambridge University Press. ISBN   978-0-521-65838-6. OCLC   39633743.
  24. Hadrons and Quark-Gluon Plasma, by Jean Letessier, Johann Rafelski, p. 91.

Related Research Articles

In a chemical reaction, chemical equilibrium is the state in which both the reactants and products are present in concentrations which have no further tendency to change with time, so that there is no observable change in the properties of the system. This state results when the forward reaction proceeds at the same rate as the reverse reaction. The reaction rates of the forward and backward reactions are generally not zero, but they are equal. Thus, there are no net changes in the concentrations of the reactants and products. Such a state is known as dynamic equilibrium.

<span class="mw-page-title-main">Ideal gas</span> Mathematical model which approximates the behavior of real gases

An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics. The requirement of zero interaction can often be relaxed if, for example, the interaction is perfectly elastic or regarded as point-like collisions.

<span class="mw-page-title-main">Fermi level</span> Quantity in solid state thermodynamics

The Fermi level of a solid-state body is the thermodynamic work required to add one electron to the body. It is a thermodynamic quantity usually denoted by μ or EF for brevity. The Fermi level does not include the work required to remove the electron from wherever it came from. A precise understanding of the Fermi level—how it relates to electronic band structure in determining electronic properties; how it relates to the voltage and flow of charge in an electronic circuit—is essential to an understanding of solid-state physics.

<span class="mw-page-title-main">Fermi gas</span> Physical model of gases composed of many non-interacting identical fermions

A Fermi gas is an idealized model, an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density, temperature, and the set of available energy states. The model is named after the Italian physicist Enrico Fermi.

<span class="mw-page-title-main">Gibbs free energy</span> Type of thermodynamic potential

In thermodynamics, the Gibbs free energy is a thermodynamic potential that can be used to calculate the maximum amount of work, other than pressure-volume work, that may be performed by a thermodynamically closed system at constant temperature and pressure. It also provides a necessary condition for processes such as chemical reactions that may occur under these conditions. The Gibbs free energy is expressed as

<span class="mw-page-title-main">Thermodynamic potential</span> Scalar physical quantities representing system states

A thermodynamic potential is a scalar quantity used to represent the thermodynamic state of a system. Just as in mechanics, where potential energy is defined as capacity to do work, similarly different potentials have different meanings. The concept of thermodynamic potentials was introduced by Pierre Duhem in 1886. Josiah Willard Gibbs in his papers used the term fundamental functions.

<span class="mw-page-title-main">Helmholtz free energy</span> Thermodynamic potential

In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz energy during a process is equal to the maximum amount of work that the system can perform in a thermodynamic process in which temperature is held constant. At constant temperature, the Helmholtz free energy is minimized at equilibrium.

<span class="mw-page-title-main">Internal energy</span> Energy contained within a system

The internal energy of a thermodynamic system is the energy contained within it, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accounting for the gains and losses of energy due to changes in its internal state, including such quantities as magnetization. It excludes the kinetic energy of motion of the system as a whole and the potential energy of position of the system as a whole, with respect to its surroundings and external force fields. It includes the thermal energy, i.e., the constituent particles' kinetic energies of motion relative to the motion of the system as a whole. The internal energy of an isolated system cannot change, as expressed in the law of conservation of energy, a foundation of the first law of thermodynamics.

<span class="mw-page-title-main">Onsager reciprocal relations</span> Relations between flows and forces, or gradients, in thermodynamic systems

In thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists.

The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency towards further change. For a given set of reaction conditions, the equilibrium constant is independent of the initial analytical concentrations of the reactant and product species in the mixture. Thus, given the initial composition of a system, known equilibrium constant values can be used to determine the composition of the system at equilibrium. However, reaction parameters like temperature, solvent, and ionic strength may all influence the value of the equilibrium constant.

In statistical mechanics, the grand canonical ensemble is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium with a reservoir. The system is said to be open in the sense that the system can exchange energy and particles with a reservoir, so that various possible states of the system can differ in both their total energy and total number of particles. The system's volume, shape, and other external coordinates are kept the same in all possible states of the system.

<span class="mw-page-title-main">Thermodynamic equations</span> Equations in thermodynamics

Thermodynamics is expressed by a mathematical framework of thermodynamic equations which relate various thermodynamic quantities and physical properties measured in a laboratory or production process. Thermodynamics is based on a fundamental set of postulates, that became the laws of thermodynamics.

In physics, 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 Brownian motion. The more general form of the equation in the classical case is

<span class="mw-page-title-main">Conjugate variables (thermodynamics)</span> Pair of values which express a thermodynamic systems internal energy

In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature and entropy, pressure and volume, or chemical potential and particle number. In fact, all thermodynamic potentials are expressed in terms of conjugate pairs. The product of two quantities that are conjugate has units of energy or sometimes power.

The principle of minimum energy is essentially a restatement of the second law of thermodynamics. It states that for a closed system, with constant external parameters and entropy, the internal energy will decrease and approach a minimum value at equilibrium. External parameters generally means the volume, but may include other parameters which are specified externally, such as a constant magnetic field.

The grand potential or Landau potential or Landau free energy is a quantity used in statistical mechanics, especially for irreversible processes in open systems. The grand potential is the characteristic state function for the grand canonical ensemble.

The Gibbs adsorption isotherm for multicomponent systems is an equation used to relate the changes in concentration of a component in contact with a surface with changes in the surface tension, which results in a corresponding change in surface energy. For a binary system, the Gibbs adsorption equation in terms of surface excess is:

<span class="mw-page-title-main">Fundamental thermodynamic relation</span>

In thermodynamics, the fundamental thermodynamic relation are four fundamental equations which demonstrate how four important thermodynamic quantities depend on variables that can be controlled and measured experimentally. Thus, they are essentially equations of state, and using the fundamental equations, experimental data can be used to determine sought-after quantities like G or H (enthalpy). The relation is generally expressed as a microscopic change in internal energy in terms of microscopic changes in entropy, and volume for a closed system in thermal equilibrium in the following way.

<span class="mw-page-title-main">Gibbs–Duhem equation</span> Equation in thermodynamics

In thermodynamics, the Gibbs–Duhem equation describes the relationship between changes in chemical potential for components in a thermodynamic system:

<span class="mw-page-title-main">Diffusion</span> Transport of dissolved species from the highest to the lowest concentration region

Diffusion is the net movement of anything 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 potential. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, as in spinodal decomposition. Diffusion is a stochastic process due to the inherent randomness of the diffusing entity and can be used to model many real-life stochastic scenarios. Therefore, diffusion and the corresponding mathematical models are used in several fields beyond physics, such as statistics, probability theory, information theory, neural networks, finance, and marketing.

References