In thermodynamics and chemistry, chemical potential, symbolized by μ, is a term introduced in by the American mathematical physicist Willard Gibbs, which he defined as follows:

Gibbs noted 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. A chemical element is a type of Atom that is distinguished by its Atomic number; that is by the number of Protons in its nucleus. Chemical potential is also referred to as partial molar Gibbs energy. In Thermodynamics, the Gibbs free energy ( IUPAC recommended name Gibbs energy or Gibbs function) is a Thermodynamic potential which

Example

Example of a plot of chemical potential. Grayscale shades indicate a value; black is the lowest number and white is the highest.

Suppose the chemical potential function defined over a 2D region shown in the figure. Particles will tend to move from regions of high chemical potential (shown as lighter shades in plot) to regions of low chemical potential (shown as darker shades in plot).

Various thermodynamic properties define what the chemical potential is. Here is a partial list of thermodynamic properties of Fluids T Temperature *\rho Density For example, consider charged particles in a fluid. The pressure gradient in a fluid may push particles in one direction, and the electric potential gradient may push the particles in another. The chemical potential would take both the pressure and electric effects into account and describe a potential distribution, which particles will tend to move down.

History

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 volume-entropy-internal energy graph, Gibbs was able to determine three states of equilibrium, i. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy In Thermodynamics, the internal energy of a Thermodynamic system, or a body with well-defined boundaries, denoted by  U, or sometimes  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 which are chemically different from each other. In his own words, to summarize his results in 1873, Gibbs states:

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. In Thermodynamics, the internal energy of a Thermodynamic system, or a body with well-defined boundaries, denoted by  U, or sometimes  In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically

Related terms

The precise meaning of the term chemical potential depends on the context in which it is used.

• When speaking of thermodynamic systems, chemical potential refers to the thermodynamic chemical potential. In this context, the chemical potential is the change in a characteristic thermodynamic state function per change in the number of molecules. In Thermodynamics, a state function, state quantity, or a function of state, is a property of a system that depends only on the current Depending on the experimental conditions, the characteristic thermodynamic state function is either: internal energy, enthalpy, Gibbs energy, or Helmholtz energy. In Thermodynamics, the internal energy of a Thermodynamic system, or a body with well-defined boundaries, denoted by  U, or sometimes  In Thermodynamics and molecular chemistry, the enthalpy (denoted as H, h, or rarely as χ) is a quotient or description of In Thermodynamics, the Gibbs free energy ( IUPAC recommended name Gibbs energy or Gibbs function) is a Thermodynamic potential which In Thermodynamics, the Helmholtz free energy is a Thermodynamic potential which measures the “useful” work obtainable from a closed thermodynamic This particular usage is most widely used by experimental chemists, physicists, and chemical engineers.

• Theoretical chemists and physicists often use the term chemical potential in reference to the electronic chemical potential, which is related to the functional derivative of the density functional, sometimes called the energy functional, found in Density Functional Theory. Density functional theory (DFT is a quantum mechanical theory used in Physics and Chemistry to investigate the Electronic structure (principally This particular usage of the term is widely used in the field of electronic structure theory.

• Physicists sometimes use the term chemical potential in the description of relativistic systems of fundamental particles. In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made

Thermodynamic chemical potential

Main article: Thermodynamic potentials

The chemical potential of a thermodynamic system is the amount by which the energy of the system would change if an additional particle were introduced, with the entropy and volume held fixed. The particle number, N, is the number of so called ' Elementary particles (or elementary constituents in a thermodynamical system. In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning " In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically If a system contains more than one species of particle, there is a separate chemical potential associated with each species, defined as the change in energy when the number of particles of that species is increased by one. The chemical potential is a fundamental parameter in thermodynamics and it is conjugate to the particle number. In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning " In Thermodynamics, the Internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature/entropy or pressure/volume The particle number, N, is the number of so called ' Elementary particles (or elementary constituents in a thermodynamical system.

The chemical potential is particularly important when studying systems of reacting particles. Consider the simplest case of two species, where a particle of species 1 can transform into a particle of species 2 and vice versa. An example of such a system is a supersaturated mixture of water liquid (species 1) and water vapor (species 2). If the system is at equilibrium, the chemical potentials of the two species must be equal. Otherwise, any increase in one chemical potential would result in an irreversible net release of energy of the system in the form of heat (see second law of thermodynamics) when that species of increased potential transformed into the other species, or a net gain of energy (again in the form of heat) if the reverse transformation took place. In Physics, heat, symbolized by Q, is Energy transferred from one body or system to another due to a difference in Temperature The second law of Thermodynamics is an expression of the universal law of increasing Entropy, stating that the entropy of an Isolated system which In chemical reactions, the equilibrium conditions are generally more complicated because more than two species are involved. A chemical reaction is a process that always results in the interconversion of Chemical substances The substance or substances initially involved in a chemical reaction are called In this case, the relation between the chemical potentials at equilibrium is given by the law of mass action. In chemistry Law of Mass Action has two aspects 1 the equilibrium aspect concerning the composition of a reaction mixture at equilibrium and 2 the kinetic

Since the chemical potential is a thermodynamic quantity, it is defined independently of the microscopic behavior of the system, i. e. the properties of the constituent particles. However, some systems contain important variables that are equivalent to the chemical potential. In Fermi gases and Fermi liquids, the chemical potential at zero temperature is equivalent to the Fermi energy. A Fermi gas, or Free electron gas, is a collection of non-interacting Fermions. Fermi liquid is a generic term for a quantum mechanical Liquid of Fermions that arises under certain physical conditions when the Temperature Absolute zero is the point at which molecules do not move (relative to the rest of the body more than they are required to by a quantum mechanical effect called Zero-point The Fermi energy is a concept in Quantum mechanics usually referring to the energy of the highest occupied Quantum state in a system of Fermions at In electronic systems, the chemical potential is related to an effective electrical potential. Electronics refers to the flow of charge (moving Electrons through Nonmetal conductors (mainly Semiconductors, whereas electrical At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field

A way to understand the chemical potential is to consider one mole of methane and 2 moles of oxygen. If a flame is brought near this mixture, the following reaction will occur: CH₄ + 2 O₂ --> CO₂ + 2 H₂O and energy (heat) will be released. This energy comes from the difference in chemical potential between CH4 and O2 on one hand (higher potential) and CO2 and H2O on the other hand (lower). The whole energy that will be released will be given by µ(CH₄) + 2 µ(O₂) - µ(CO₂) - 2 µ(H₂O)

Similar examples can be found within batteries where chemical energy is converted into electrical energy.

Precise definition

Consider a thermodynamic system containing n constituent species. Its total internal energy U is postulated to be a function of the entropy S, the volume V, and the number of particles of each species N₁,. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject . . , N_n:

U = U(S,V,N₁,. . N_n)

By referring to U as the internal energy, it is emphasized that the energy contributions resulting from the interactions between the system and external objects are excluded. For example, the gravitational potential energy of the system with the Earth are not included in U.

The chemical potential of the i-th species, μ_i is defined as the partial derivative

where the subscripts simply emphasize that the entropy, volume, and the other particle numbers are to be kept constant. In Mathematics, a partial derivative of a function of several variables is its Derivative with respect to one of those variables with the others held constant

In real systems, it is usually difficult to hold the entropy fixed, since this involves good thermal insulation. The term thermal insulation can refer to materials used to reduce the rate of Heat transfer, or the methods and processes used to reduce heat transfer It is therefore more convenient to define the Helmholtz free energy A, which is a function of the temperature T, volume, and particle numbers:

A = A(T,V,N₁,. In Thermodynamics, the Helmholtz free energy is a Thermodynamic potential which measures the “useful” work obtainable from a closed thermodynamic Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature . N_n)

In terms of the Helmholtz free energy, the chemical potential is

Laboratory experiments are often performed under conditions of constant temperature and pressure. Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface Under these conditions, the chemical potential is the partial derivative of the Gibbs free energy with respect to number of particles

A similar expression for the chemical potential can be written in terms of partial derivative of the enthalpy (under conditions of constant entropy and pressure). In Thermodynamics, the Gibbs free energy ( IUPAC recommended name Gibbs energy or Gibbs function) is a Thermodynamic potential which In Thermodynamics and molecular chemistry, the enthalpy (denoted as H, h, or rarely as χ) is a quotient or description of

Here, the chemical potential has been defined as the «energy» per molecule. A variant of this definition is to define the chemical potential as the «energy» per mole.

Electronic chemical potential

The electronic chemical potential is the functional derivative of the density functional with respect to the electron density. In Mathematics and theoretical Physics, the functional derivative is a generalization of the Directional derivative. Electron density is the measure of the Probability of an Electron being present at a specific location

Formally, a functional derivative yields many functions, but is a particular function when evaluated about a reference electron density - just as a derivate yields a function, but is a particular number when evaluated about a reference point. The density functional is written as

where is the external potential, e. g. , the electrostatic potential of the nuclei and applied fields, and F is the Universal functional, which describes the electron-electron interactions, e. At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field g. , electron Coulomb repulsion, kinetic energy, and the non-classical effects of exchange and correlation. The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 Electronic correlation refers to the interaction between Electrons in a quantum system whose Electronic structure is being considered With this general definition of the density functional, the chemical potential is written as

Thus, the electronic chemical potential is the effective electrostatic potential experienced by the electron density.

The ground state electron density is determined by a constrained variational optimization of the electronic energy. A variational principle is a principle in Physics which is expressed in terms of the Calculus of variations. The Lagrange multiplier enforcing the density normalization constraint is also called the chemical potential, i. In mathematical optimization problems the method of Lagrange multipliers, named after Joseph Louis Lagrange, is a method for finding the extrema of e. ,

where N is the number of electrons in the system and μ is the Lagrange multiplier enforcing the constraint. When this variational statement is satisfied, the terms within the curly brackets obey the property

where the reference density is the density that minimizes the energy. This expression simplifies to

The Lagrange multiplier enforcing the constraint is, by construction, a constant; however, the functional derivative is, formally, a function. Therefore, when the density minimizes the electronic energy, the chemical potential has the same value at every point in space. The gradient of the chemical potential is an effective electric field. In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can An electric field describes the force per unit charge as a function of space. In Physics, a force is whatever can cause an object with Mass to Accelerate. Therefore, when the density is the ground state density, the electron density is stationary, because the gradient of the chemical potential (which is invariant with respect to position) is zero everywhere, i. e. , all forces are balanced. As the density undergoes a change from a non-ground state density to the ground state density, it is said to undergo a process of chemical potential equalization.

The chemical potential of an atom is sometimes said to be the negative of the atom's electronegativity. " Electronegativity " is the opposite of " Electropositivity," which describes an element's ability to donate electrons Similarly the process of chemical potential equalization is sometimes referred to as the process of electronegativity equalization. This connection comes from the Mulliken definition of electronegativity. " Electronegativity " is the opposite of " Electropositivity," which describes an element's ability to donate electrons By inserting the energetic definitions of the ionization potential and electron affinity into the Mulliken electronegativity, it is possible to show that the Mulliken chemical potential is a finite difference approximation of the electronic energy with respect to the number of electrons. The ionization potential, ionization energy or EI of an Atom or Molecule is the Energy required to remove an Electron The electron affinity, E ea of an Atom or Molecule is the energy required to detach an electron from a singly charged negative , i. e. ,

where IP and EA are the ionization potential and electron affinity of the atom, respectively.

The values of the chemical potential

For standard conditions (T = 298. In Physical sciences standard conditions for temperature and pressure are Standard sets of conditions for experimental measurements to allow comparisons to be made 15 K; p = 101,325 Pa) the values of the chemical potential are tabulated, see under "Weblinks". If the chemical potential is known in a certain state (e. g. for standard conditions), then it can be calculated in linear approximation for pressures and temperatures in the vicinity of this state:
μ(T) = μ(T₀) + α(T – T₀)
and
μ(p) = μ(p₀) + β(p – p₀)
Here

is the temperature coefficient and

is the pressure coefficient.
With the Maxwell relations

and

it follows that the temperature coefficient is equal to the negative molar entropy and the pressure coefficient is equal to the molar volume. Maxwell's relations are a set of equations in Thermodynamics which are derivable from the definitions of the Thermodynamic potentials.

Fundamental particle chemical potential

In recent years, thermal physics has applied the definition of chemical potential to systems in particle physics and its associated processes. Thermal physics is the combined study of Thermodynamics, Statistical mechanics, and Kinetic theory. Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them In general, chemical potential measures the tendency of particles to diffuse. This characterization focuses on the chemical potential as a function of spatial location. Particles tend to diffuse from regions of high chemical potential to those of low chemical potential. ^[1] Being a function of internal energy, chemical potential applies equally to both fermion and boson particles, That is, in theory, any fundamental particle can be assigned a value of chemical potential, depending upon how it changes the internal energy of the system into which it is introduced. In Thermodynamics, the internal energy of a Thermodynamic system, or a body with well-defined boundaries, denoted by  U, or sometimes  In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made The application of chemical potential concepts for systems at absolute zero has significant appeal. Absolute zero is the point at which molecules do not move (relative to the rest of the body more than they are required to by a quantum mechanical effect called Zero-point

For relativistic systems, i. This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism. e. , systems in which the rest mass is much smaller than the equivalent thermal energy, the chemical potential is related to symmetries and charges. In Physics, mass–energy equivalence is the concept that for particles slower than light any Mass has an associated Energy and vice versa. A thermal column (or thermal) is a column of rising Air in the lower altitudes of the Earth's atmosphere. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. Each conserved quantity is associated with a chemical potential.

In a gas of photons in equilibrium with massive particles, the number of photons is not conserved, and so in this case, the chemical potential is zero. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena Similarly, for a gas of phonons, there is also no chemical potential. In Physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the Atomic lattice of a Solid However, if the temperature of such a system were to rise above the threshold for pair production of electrons, then it might be sensible to add a chemical potential for the electrical charge. See also Electron-positron annihilation Meitner–Hupfeld effect Pair instability supernova The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J This would control the electric charge density of the system, and hence the excess of electrons over positrons, but not the number of photons. Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The positrons or antielectron is the Antiparticle or the Antimatter counterpart of the Electron. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena In the context in which one meets a phonon gas, temperatures high enough to pair produce other particles are seldom relevant. In Physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the Atomic lattice of a Solid QCD matter is the prime example of a system in which many such chemical potentials appear. Quark matter or QCD matter (see QCD) refers to any of a number of theorized phases of matter whose degrees of freedom include Quarks and Gluons

See also

• Chemical equilibrium
• Electrochemical potential
• Thermodynamic equilibrium
• Activity (chemistry)
• Excess chemical potential

References

• Baierlein, Ralph (April 2001). "The elusive chemical potential". American Journal of Physics 69 (4): 423-434. doi:10.1119/1.1336839. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.  

• Kaplan, T. A. (March 2006). "The Chemical Potential". Journal of Statistical Physics 122 (6): 1237-1260. doi:10.1007/s10955-005-8067-x. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.  

• Job, G. ; Herrmann, F. (February 2006). "Chemical potential–a quantity in search of recognition" (PDF). European Journal of Physics 27: 353-371. doi:10.1088/0143-0807/27/2/018. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.  

External links

• Chemical Potential
• Chemical Potentials
• Values of the chemical potential of 1300 substances
• Chemical potential in experiments: Demonstration experiments "dissolution of marble", "ammonia fountain", "carbide lamp" (instructions and videos)