Chemical potential Totally Explained

Everything about Chemical Potential totally explained

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. Chemical potential is also referred to as partial molar Gibbs energy.

Example

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. 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, for example 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, for example "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:

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:
ť

delta (epsilon - Teta + P
u) = 0

when

delta

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

u

is the volume of the body.

Related terms

The precise meaning of the term chemical potential depends on the context in which it's 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. Depending on the experimental conditions, the characteristic thermodynamic state function is either: internal energy, enthalpy, Gibbs energy, or Helmholtz energy. 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. 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.

Thermodynamic chemical potential

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. If a system contains more than one species of particle, there's 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's conjugate to the particle number.
   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 chemical reactions, the equilibrium conditions are generally more complicated because more than two species are involved. In this case, the relation between the chemical potentials at equilibrium is given by the law of mass action.
   Since the chemical potential is a thermodynamic quantity, it's defined independently of the microscopic behavior of the system, for example 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. In electronic systems, the chemical potential is related to an effective electrical potential. 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,..., N_n:
ť $U=U(S,V,N_1,..N_n)$

By referring to U as the internal energy, it's 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 ť $mu_i = left(frac$ it follows that the temperature coefficient is equal to the negative molar entropy and the pressure coefficient is equal to the molar volume.
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. 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. 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's introduced. The application of chemical potential concepts for systems at absolute zero has significant appeal.
   For relativistic systems, for example, systems in which the rest mass is much smaller than the equivalent thermal energy, the chemical potential is related to symmetries and charges. Each conserved quantity is associated with a chemical potential.
   In a gas of photons in equilibrium with massive particles, the number of photons isn't conserved, and so in this case, the chemical potential is zero. Similarly, for a gas of phonons, there's also no chemical potential. 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. This would control the electric charge density of the system, and hence the excess of electrons over positrons, but not the number of photons. In the context in which one meets a phonon gas, temperatures high enough to pair produce other particles are seldom relevant. QCD matter is the prime example of a system in which many such chemical potentials appear.

Further Information
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