Virtual particle

From Wikipedia, the free encyclopedia

In physics, a virtual particle is an explanatory conceptual entity that is found in mathematical calculations about quantum field theory. It refers to mathematical terms that have some appearance of representing particles inside a subatomic process such as a collision. Virtual particles, however, do not appear directly amongst the observable and detectable input and output quantities of those calculations, which refer only to actual, as distinct from virtual, particles. Virtual particle terms represent "particles" that are said to be 'off mass shell'. For example, they can progress backwards in time, can have apparent mass very different from their regular particle namesake's^{[dubious – discuss]}, and can travel faster than light. That is to say, when looked at individually, they appear to be able to violate basic laws of physics. Regular particles of course never do so. On the other hand, any particle that is actually observed never precisely satisfies the conditions theoretically imposed on regular particles. Virtual particles occur in combinations that mutually more or less nearly cancel from the actual output quantities, so that no actual violation of the laws of physics occurs in completed processes. Often the virtual-particle virtual "events" appear to occur close to one another in time, for example within the time scale of a collision, so that they are virtually and apparently "short-lived". If the mathematical terms that are interpreted as representing virtual particles are omitted from the calculations, the result is an approximation that may or may not be near the correct and accurate answer obtained from the proper full calculation.^[1][2][3]

Quantum theory is different from classical theory. The difference is in accounting for the inner workings of subatomic processes. Classical physics cannot account for such. It was pointed out by Heisenberg that what "actually" or "really" occurs inside such subatomic processes as collisions is not directly observable and no unique and physically definite visualization is available for it. Quantum mechanics has the specific merit of by-passing speculation about such inner workings. It restricts itself to what is actually observable and detectable. Virtual particles are conceptual devices that in a sense try to by-pass Heisenberg's insight, by offering putative or virtual explanatory visualizations for the inner workings of subatomic processes.

A virtual particle does not necessarily appear to carry the same mass as the corresponding real particle. This is because it appears as "short-lived" and "transient", so that the uncertainty principle allows it to appear not to conserve energy and momentum. The longer a virtual particle appears to "live", the closer its characteristics come to those of an actual particle.

Virtual particles appear in many processes, including particle scattering and Casimir forces. In quantum field theory, even classical forces — such as the electromagnetic repulsion or attraction between two charges — can be thought of as due to the exchange of many virtual photons between the charges.

Virtual particles appear in calculations of subatomic interactions, but never as asymptotic states or indices to the scattering matrix. A subatomic process involving virtual particles is schematically representable by a Feynman diagram in which they are represented by internal lines.

Antiparticles and quasiparticles should not be confused with virtual particles or virtual antiparticles.
Many physicists believe that, because of its intrinsically perturbative character, the concept of virtual particles is often confusing and misleading, and is thus best avoided.^[4][5]

Properties

The concept of virtual particles arises in the perturbation theory of quantum field theory, an approximation scheme in which interactions (in essence, forces) between actual particles are calculated in terms of exchanges of virtual particles. Such calculations are often performed using schematic representations known as Feynman diagrams, in which virtual particles appear as internal lines. By expressing the interaction in terms of the exchange of a virtual particle with four-momentum q, where q is given by the difference between the four-momenta of the particles entering and leaving the interaction vertex, both momentum and energy are conserved at the interaction vertices of the Feynman diagram.^[6]:119

A virtual particle does not precisely obey the energy–momentum relation m²c⁴ = E² − p²c². Its kinetic energy may not have the usual relationship to velocity–indeed, it can be negative.^[7]:110 This is expressed by the phrase off mass shell.^[6]:119 The probability amplitude for a virtual particle to exist tends to be canceled out by destructive interference over longer distances and times. Quantum tunnelling may be considered a manifestation of virtual particle exchanges.^[8]:235 The range of forces carried by virtual particles is limited by the uncertainty principle, which regards energy and time as conjugate variables; thus, virtual particles of larger mass have more limited range.^[9]

Written in the usual mathematical notations, in the equations of physics, there is no mark of the distinction between virtual and actual particles. The amplitude that a virtual particle exists interferes with the amplitude for its non-existence, whereas for an actual particle the cases of existence and non-existence cease to be coherent with each other and do not interfere any more. In the quantum field theory view, actual particles are viewed as being detectable excitations of underlying quantum fields. Virtual particles are also viewed as excitations of the underlying fields, but appear only as forces, not as detectable particles. They are "temporary" in the sense that they appear in calculations, but are not detected as single particles. Thus, in mathematical terms, they never appear as indices to the scattering matrix, which is to say, they never appear as the observable inputs and outputs of the physical process being modelled.

There are two principal ways in which the notion of virtual particles appears in modern physics. They appear as intermediate terms in Feynman diagrams; that is, as terms in a perturbative calculation. They also appear as an infinite set of states to be summed or integrated over in the calculation of a semi-non-perturbative effect. In the latter case, it is sometimes said that virtual particles contribute to a mechanism that mediates the effect, or that the effect occurs through the virtual particles.^[6]:118

Manifestations

There are many observable physical phenomena that arise in interactions involving virtual particles. For bosonic particles that exhibit rest mass when they are free and actual, virtual interactions are characterized by the relatively short range of the force interaction produced by particle exchange.^{[citation needed]} Examples of such short-range interactions are the strong and weak forces, and their associated field bosons. For the gravitational and electromagnetic forces, the zero rest-mass of the associated boson particle permits long-range forces to be mediated by virtual particles. However, in the case of photons, power and information transfer by virtual particles is a relatively short-range phenomenon (existing only within a few wavelengths of the field-disturbance, which carries information or transferred power), as for example seen in the characteristically short range of inductive and capacitative effects in the near field zone of coils and antennas.^{[citation needed]}

Some field interactions which may be seen in terms of virtual particles are:

• The Coulomb force (static electric force) between electric charges. It is caused by the exchange of virtual photons. In symmetric 3-dimensional space this exchange results in the inverse square law for electric force. Since the photon has no mass, the coulomb potential has an infinite range.
• The magnetic field between magnetic dipoles. It is caused by the exchange of virtual photons. In symmetric 3-dimensional space this exchange results in the inverse cube law for magnetic force. Since the photon has no mass, the magnetic potential has an infinite range.
• Electromagnetic induction. This phenomenon transfers energy to and from a magnetic coil via a changing (electro)magnetic field.
• The strong nuclear force between quarks is the result of interaction of virtual gluons. The residual of this force outside of quark triplets (neutron and proton) holds neutrons and protons together in nuclei, and is due to virtual mesons such as the pi meson and rho meson.
• The weak nuclear force - it is the result of exchange by virtual W and Z bosons.
• The spontaneous emission of a photon during the decay of an excited atom or excited nucleus; such a decay is prohibited by ordinary quantum mechanics and requires the quantization of the electromagnetic field for its explanation.
• The Casimir effect, where the ground state of the quantized electromagnetic field causes attraction between a pair of electrically neutral metal plates.
• The van der Waals force, which is partly due to the Casimir effect between two atoms.
• Vacuum polarization, which involves pair production or the decay of the vacuum, which is the spontaneous production of particle-antiparticle pairs (such as electron-positron).
• Lamb shift of positions of atomic levels.
• Hawking radiation, where the gravitational field is so strong that it causes the spontaneous production of photon pairs (with black body energy distribution) and even of particle pairs.
• Much of the so-called near-field of radio antennas, where the magnetic and electric effects of the changing current in the antenna wire and the charge effects of the wire's capacitive charge may be (and usually are) important contributors to the total EM field close to the source, but both of which effects are dipole effects that decay with increasing distance from the antenna much more quickly than do the influence of "conventional" electromagnetic waves that are "far" from the source. ["Far" in terms of ratio of antenna length or diameter, to wavelength]. These far-field waves, for which E is (in the limit of long distance) equal to cB, are composed of actual photons. It should be noted that actual and virtual photons are mixed near an antenna, with the virtual photons responsible only for the "extra" magnetic-inductive and transient electric-dipole effects, which cause any imbalance between E and cB. As distance from the antenna grows, the near-field effects (as dipole fields) die out more quickly, and only the "radiative" effects that are due to actual photons remain as important effects. Although virtual effects extend to infinity, they drop off in field strength as 1/r² rather than the field of EM waves composed of actual photons, which drop 1/r (the powers, respectively, decrease as 1/r⁴ and 1/r²). See near and far field for a more detailed discussion. See near field communication for practical communications applications of near fields.

Most of these have analogous effects in solid-state physics; indeed, one can often gain a better intuitive understanding by examining these cases. In semiconductors, the roles of electrons, positrons and photons in field theory are replaced by electrons in the conduction band, holes in the valence band, and phonons or vibrations of the crystal lattice. A virtual particle is in a virtual state where the probability amplitude is not conserved. Examples of macroscopic virtual phonons, photons, and electrons in the case of the tunneling process were presented by Günter Nimtz ^[10] and Alfons A. Stahlhofen.^[11]

History

Paul Dirac was the first to propose that empty space (a vacuum) can be visualized as consisting of a sea of electrons with negative energy, known as the Dirac sea. The Dirac sea has a direct analog to the electronic band structure in crystalline solids as described in solid state physics. Here, particles correspond to conduction electrons, and antiparticles to holes. A variety of interesting phenomena can be attributed to this structure. The development of quantum field theory (QFT) in the 1930s made it possible to reformulate the Dirac equation in a way that treats the positron as a "real" particle rather than the absence of a particle, and makes the vacuum the state in which no particles exist instead of an infinite sea of particles.

Feynman diagrams

One particle exchange scattering diagram

The calculation of scattering amplitudes in theoretical particle physics requires the use of some rather large and complicated integrals over a large number of variables. These integrals do, however, have a regular structure, and may be represented as Feynman diagrams. The appeal of the Feynman diagrams is strong, as it allows for a simple visual presentation of what would otherwise be a rather arcane and abstract formula. In particular, part of the appeal is that the outgoing legs of a Feynman diagram can be associated with actual, on-shell particles. Thus, it is natural to associate the other lines in the diagram with particles as well, called the "virtual particles". In mathematical terms, they correspond to the propagators appearing in the diagram.

In the image to the right, the solid lines correspond to actual particles (of momentum p₁ and so on), while the dotted line corresponds to a virtual particle carrying momentum k. For example, if the solid lines were to correspond to electrons interacting by means of the electromagnetic interaction, the dotted line would correspond to the exchange of a virtual photon. In the case of interacting nucleons, the dotted line would be a virtual pion. In the case of quarks interacting by means of the strong force, the dotted line would be a virtual gluon, and so on.

One-loop diagram with fermion propagator

Virtual particles may be mesons or vector bosons, as in the example above; they may also be fermions. However, in order to preserve quantum numbers, most simple diagrams involving fermion exchange are prohibited. The image to the right shows an allowed diagram, a one-loop diagram. The solid lines correspond to a fermion propagator, the wavy lines to bosons.

Vacuums

In formal terms, a particle is considered to be an eigenstate of the particle number operator a^†a, where a is the particle annihilation operator and a^† the particle creation operator (sometimes collectively called ladder operators). In many cases, the particle number operator does not commute with the Hamiltonian for the system. This implies the number of particles in an area of space is not a well-defined quantity but, like other quantum observables, is represented by a probability distribution. Since these particles do not have a permanent existence,^{[clarification needed]} they are called virtual particles or vacuum fluctuations of vacuum energy. In a certain sense, they can be understood to be a manifestation of the time-energy uncertainty principle in a vacuum.^[12]
An important example of the "presence" of virtual particles in a vacuum is the Casimir effect.^[13] Here, the explanation of the effect requires that the total energy of all of the virtual particles in a vacuum can be added together. Thus, although the virtual particles themselves are not directly observable in the laboratory, they do leave an observable effect: Their zero-point energy results in forces acting on suitably arranged metal plates or dielectrics.^[14] On the other hand, the Casimir effect can be interpreted as the relativistic van der Waals force.^[15]

Pair production

Virtual particles are often popularly described as coming in pairs, a particle and antiparticle, which can be of any kind. These pairs exist for an extremely short time, and then mutually annihilate. In some cases, however, it is possible to boost the pair apart using external energy so that they avoid annihilation and become actual particles.
This may occur in one of two ways. In an accelerating frame of reference, the virtual particles may appear to be actual to the accelerating observer; this is known as the Unruh effect. In short, the vacuum of a stationary frame appears, to the accelerated observer, to be a warm gas of actual particles in thermodynamic equilibrium.

Another example is pair production in very strong electric fields, sometimes called vacuum decay. If, for example, a pair of atomic nuclei are merged to very briefly form a nucleus with a charge greater than about 140, (that is, larger than about the inverse of the fine structure constant, which is a dimensionless quantity), the strength of the electric field will be such that it will be energetically favorable to create positron-electron pairs out of the vacuum or Dirac sea, with the electron attracted to the nucleus to annihilate the positive charge. This pair-creation amplitude was first calculated by Julian Schwinger in 1951.

Actual and virtual particles compared

As a consequence of quantum mechanical uncertainty, any object or process that exists for a limited time or in a limited volume cannot have a precisely defined energy or momentum. This is the reason that virtual particles — which exist only temporarily as they are exchanged between ordinary particles — do not necessarily obey the mass-shell relation. However, the longer a virtual particle exists, the more closely it adheres to the mass-shell relation. A "virtual" particle that exists for an arbitrarily long time is simply an ordinary particle.

However, all particles have a finite lifetime, as they are created and eventually destroyed by some processes. As such, there is no absolute distinction between "real" and "virtual" particles. In practice, the lifetime of "ordinary" particles is far longer than the lifetime of the virtual particles that contribute to processes in particle physics, and as such the distinction is useful to make.

Why people like Mike Huckabee are not legally qualified to hold any government position, let alone the presidency

It's quite simple, really.  Upon winning an election, the to be president has to take an oath  "to uphold, protect, and defend the Constitution ..."  If, however, people, like Huckabee, declare that the Bible (or Quran, or the Communist Manifesto) should be above the Constitution, then he cannot take that oath without committing perjury.  And since the oath is legally binding for as long as he is in power, he also cannot later adopt this position without violating that oath, for then he is committing treason against the country.  As either way, he is committing "high crimes and misdemeanors", he is subject to impeachment and prosecution.

 This is not a religious test for office.  A government official can hold any religious or other ideological ides as he pleases; he just can't, by his actions, impose them against the US and local constitutions.  He can even advocate his beliefs, and hold that the Constitution to be amended or rewritten to reflect them; he can't just act on them out of personal conviction or conscience.

Of course, I am alluding to Kim Davis as well, who is basically in the same position.  She isn't a county clerk simply she won the largest number of votes; she too must have taken an oath of office which mentioned both the state and federal constitutions.  Perjury cannot be proved here, because her views on gay marriage only came out recently.  But she is violating her oath by ignoring the Supreme and other courts, those determiners of what constitutions mean.  That to is treasonous, albeit on a much smaller scale.  It is a crime which should lead to her impeachment and prosecution.  And if impeachment is unobtainable, she still should be prosecuted and punished if found guilty.  Of course, the honorable thing to do would have been to resign, citing her conscience -- but that wouldn't have brought forth fame, legions of followers/supporters, and guaranteed royalties on the inevitable book (a ghost writer) publishes.

That, ironically, is something she does have the right to do.

Chemical equilibrium

From Wikipedia, the free encyclopedia

In a chemical reaction, chemical equilibrium is the state in which both reactants and products are present in concentrations which have no further tendency to change with time.^[1] Usually, 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 equal. Thus, there are no net changes in the concentrations of the reactant(s) and product(s). Such a state is known as dynamic equilibrium.^[2][3]

Historical introduction

Burette, a common laboratory apparatus for carrying out titration, an important experimental technique in equilibrium and analytical chemistry.

The concept of chemical equilibrium was developed after Berthollet (1803) found that some chemical reactions are reversible. For any reaction mixture to exist at equilibrium, the rates of the forward and backward (reverse) reactions are equal. In the following chemical equation with arrows pointing both ways to indicate equilibrium, A and B are reactant chemical species, S and T are product species, and α, β, σ, and τ are the stoichiometric coefficients of the respective reactants and products:

The equilibrium concentration position of a reaction is said to lie "far to the right" if, at equilibrium, nearly all the reactants are consumed. Conversely the equilibrium position is said to be "far to the left" if hardly any product is formed from the reactants.

Guldberg and Waage (1865), building on Berthollet’s ideas, proposed the law of mass action:

where A, B, S and T are active masses and k₊ and k₋ are rate constants. Since at equilibrium forward and backward rates are equal:

and the ratio of the rate constants is also a constant, now known as an equilibrium constant.

By convention the products form the numerator. However, the law of mass action is valid only for concerted one-step reactions that proceed through a single transition state and is not valid in general because rate equations do not, in general, follow the stoichiometry of the reaction as Guldberg and Waage had proposed (see, for example, nucleophilic aliphatic substitution by S_N1 or reaction of hydrogen and bromine to form hydrogen bromide). Equality of forward and backward reaction rates, however, is a necessary condition for chemical equilibrium, though it is not sufficient to explain why equilibrium occurs.

Despite the failure of this derivation, the equilibrium constant for a reaction is indeed a constant, independent of the activities of the various species involved, though it does depend on temperature as observed by the van 't Hoff equation. Adding a catalyst will affect both the forward reaction and the reverse reaction in the same way and will not have an effect on the equilibrium constant. The catalyst will speed up both reactions thereby increasing the speed at which equilibrium is reached.^[2][4]

Although the macroscopic equilibrium concentrations are constant in time, reactions do occur at the molecular level. For example, in the case of acetic acid dissolved in water and forming acetate and hydronium ions,

CH₃CO₂H + H₂O ⇌ CH₃CO₂⁻ + H₃O⁺

a proton may hop from one molecule of acetic acid on to a water molecule and then on to an acetate anion to form another molecule of acetic acid and leaving the number of acetic acid molecules unchanged. This is an example of dynamic equilibrium. Equilibria, like the rest of thermodynamics, are statistical phenomena, averages of microscopic behavior.

Le Chatelier's principle (1884) gives an idea of the behavior of an equilibrium system when changes to its reaction conditions occur. If a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to partially reverse the change. For example, adding more S from the outside will cause an excess of products, and the system will try to counteract this by increasing the reverse reaction and pushing the equilibrium point backward (though the equilibrium constant will stay the same).

If mineral acid is added to the acetic acid mixture, increasing the concentration of hydronium ion, the amount of dissociation must decrease as the reaction is driven to the left in accordance with this principle. This can also be deduced from the equilibrium constant expression for the reaction:

If {H₃O⁺} increases {CH₃CO₂H} must increase and {CH₃CO₂⁻} must decrease. The H₂O is left out, as it is the solvent and its concentration remains high and nearly constant.

A quantitative version is given by the reaction quotient.

J. W. Gibbs suggested in 1873 that equilibrium is attained when the Gibbs free energy of the system is at its minimum value (assuming the reaction is carried out at constant temperature and pressure). What this means is that the derivative of the Gibbs energy with respect to reaction coordinate (a measure of the extent of reaction that has occurred, ranging from zero for all reactants to a maximum for all products) vanishes, signalling a stationary point. This derivative is called the reaction Gibbs energy (or energy change) and corresponds to the difference between the chemical potentials of reactants and products at the composition of the reaction mixture.^[1] This criterion is both necessary and sufficient. If a mixture is not at equilibrium, the liberation of the excess Gibbs energy (or Helmholtz energy at constant volume reactions) is the “driving force” for the composition of the mixture to change until equilibrium is reached. The equilibrium constant can be related to the standard Gibbs free energy change for the reaction by the equation

where R is the universal gas constant and T the temperature.

When the reactants are dissolved in a medium of high ionic strength the quotient of activity coefficients may be taken to be constant. In that case the concentration quotient, K_c,

where [A] is the concentration of A, etc., is independent of the analytical concentration of the reactants. For this reason, equilibrium constants for solutions are usually determined in media of high ionic strength. K_c varies with ionic strength, temperature and pressure (or volume). Likewise K_p for gases depends on partial pressure. These constants are easier to measure and encountered in high-school chemistry courses.

Thermodynamics

At constant temperature and pressure, one must consider the Gibbs free energy, G, while at constant temperature and volume, one must consider the Helmholtz free energy: A, for the reaction; and at constant internal energy and volume, one must consider the entropy for the reaction: S.

The constant volume case is important in geochemistry and atmospheric chemistry where pressure variations are significant. Note that, if reactants and products were in standard state (completely pure), then there would be no reversibility and no equilibrium. Indeed, they would necessarily occupy disjoint volumes of space. The mixing of the products and reactants contributes a large entropy (known as entropy of mixing) to states containing equal mixture of products and reactants. The standard Gibbs energy change, together with the Gibbs energy of mixing, determine the equilibrium state.^[5][6]

In this article only the constant pressure case is considered. The relation between the Gibbs free energy and the equilibrium constant can be found by considering chemical potentials.^[1]

At constant temperature and pressure, the Gibbs free energy, G, for the reaction depends only on the extent of reaction: ξ (Greek letter xi), and can only decrease according to the second law of thermodynamics. It means that the derivative of G with ξ must be negative if the reaction happens; at the equilibrium the derivative being equal to zero.

: equilibrium

In general an equilibrium system is defined by writing an equilibrium equation for the reaction

In order to meet the thermodynamic condition for equilibrium, the Gibbs energy must be stationary, meaning that the derivative of G with respect to the extent of reaction: ξ, must be zero. It can be shown that in this case, the sum of chemical potentials of the products is equal to the sum of those corresponding to the reactants. Therefore, the sum of the Gibbs energies of the reactants must be the equal to the sum of the Gibbs energies of the products.

where μ is in this case a partial molar Gibbs energy, a chemical potential. The chemical potential of a reagent A is a function of the activity, {A} of that reagent.

, ( is the standard chemical potential ).

The definition of the Gibbs energy equation interacts with the fundamental thermodynamic relation to produce

.

Inserting into the above equation gives a Stoichiometric coefficient () and a differential that denotes the reaction occurring once (). At constant pressure and temperature the above equations can be written as

which is the "Gibbs free energy change for the reaction .

This results in:

.

By substituting the chemical potentials:

,

the relationship becomes:

: which is the standard Gibbs energy change for the reaction that can be calculated using thermodynamical tables.

The reaction quotient is defined as

Therefore

At equilibrium
leading to:

and

Obtaining the value of the standard Gibbs energy change, allows the calculation of the equilibrium constant

Addition of reactants or products

For a reactional system at equilibrium: ; .

If are modified activities of constituents, the value of the reaction quotient changes and becomes different from the equilibrium constant:

and

then

• If activity of a reagent increases

, the reaction quotient decreases.

then

and  : The reaction will shift to the right (i.e. in the forward direction, and thus more products will form).

• If activity of a product increases

then

and  : The reaction will shift to the left (i.e. in the reverse direction, and thus less products will form).

Note that activities and equilibrium constants are dimensionless numbers.

Treatment of activity

The expression for the equilibrium constant can be rewritten as the product of a concentration quotient, K_c and an activity coefficient quotient, Γ.

[A] is the concentration of reagent A, etc. It is possible in principle to obtain values of the activity coefficients, γ. For solutions, equations such as the Debye–Hückel equation or extensions such as Davies equation^[7] Specific ion interaction theory or Pitzer equations^[8] may be used.^{Software (below)}. However this is not always possible. It is common practice to assume that Γ is a constant, and to use the concentration quotient in place of the thermodynamic equilibrium constant. It is also general practice to use the term equilibrium constant instead of the more accurate concentration quotient. This practice will be followed here.

For reactions in the gas phase partial pressure is used in place of concentration and fugacity coefficient in place of activity coefficient. In the real world, for example, when making ammonia in industry, fugacity coefficients must be taken into account. Fugacity, f, is the product of partial pressure and fugacity coefficient. The chemical potential of a species in the gas phase is given by

so the general expression defining an equilibrium constant is valid for both solution and gas phases.

Concentration quotients

In aqueous solution, equilibrium constants are usually determined in the presence of an "inert" electrolyte such as sodium nitrate NaNO₃ or potassium perchlorate KClO₄. The ionic strength of a solution is given by

where c_i and z_i stands for the concentration and ionic charge of ion type i, and the sum is taken over all the N types of charged species in solution. When the concentration of dissolved salt is much higher than the analytical concentrations of the reagents, the ions originating from the dissolved salt determine the ionic strength, and the ionic strength is effectively constant. Since activity coefficients depend on ionic strength the activity coefficients of the species are effectively independent of concentration. Thus, the assumption that Γ is constant is justified. The concentration quotient is a simple multiple of the equilibrium constant.^[9]

However, K_c will vary with ionic strength. If it is measured at a series of different ionic strengths the value can be extrapolated to zero ionic strength.^[8] The concentration quotient obtained in this manner is known, paradoxically, as a thermodynamic equilibrium constant.

To use a published value of an equilibrium constant in conditions of ionic strength different from the conditions used in its determination, the value should be adjusted.

Metastable mixtures

A mixture may be appear to have no tendency to change, though it is not at equilibrium. For example, a mixture of SO₂ and O₂ is metastable as there is a kinetic barrier to formation of the product, SO₃.

2SO₂ + O₂ 2SO₃

The barrier can be overcome when a catalyst is also present in the mixture as in the contact process, but the catalyst does not affect the equilibrium concentrations.

Likewise, the formation of bicarbonate from carbon dioxide and water is very slow under normal conditions

CO₂ + 2H₂O HCO₃⁻ +H₃O⁺

but almost instantaneous in the presence of the catalytic enzyme carbonic anhydrase.

Pure substances

When pure substances (liquids or solids) are involved in equilibria their activities do not appear in the equilibrium constant^[10] because their numerical values are considered one.

Applying the general formula for an equilibrium constant to the specific case of a dilute solution of acetic acid in water one obtains

CH₃CO₂H + H₂O ⇌ CH₃CO₂⁻ + H₃O⁺

For all but very concentrated solutions, the water can be considered a "pure" liquid, and therefore it has an activity of one. The equilibrium constant expression is therefore usually written as

.

A particular case is the self-ionization of water itself

Because water is the solvent, and has an activity of one, the self-ionization constant of water is defined as




It is perfectly legitimate to write [H⁺] for the hydronium ion concentration, since the state of solvation of the proton is constant (in dilute solutions) and so does not affect the equilibrium concentrations. K_w varies with variation in ionic strength and/or temperature.

The concentrations of H⁺ and OH⁻ are not independent quantities. Most commonly [OH⁻] is replaced by K_w[H⁺]⁻¹ in equilibrium constant expressions which would otherwise include hydroxide ion.

Solids also do not appear in the equilibrium constant expression, if they are considered to be pure and thus their activities taken to be one. An example is the Boudouard reaction:^[10]

for which the equation (without solid carbon) is written as:

Multiple equilibria

Consider the case of a dibasic acid H₂A. When dissolved in water, the mixture will contain H₂A, HA⁻ and A²⁻. This equilibrium can be split into two steps in each of which one proton is liberated.

K₁ and K₂ are examples of stepwise equilibrium constants. The overall equilibrium constant,, is product of the stepwise constants.

Note that these constants are dissociation constants because the products on the right hand side of the equilibrium expression are dissociation products. In many systems, it is preferable to use association constants.

β₁ and β₂ are examples of association constants. Clearly β₁ = 1/K₂ and β₂ = 1/β_D; lg β₁ = pK₂ and lg β₂ = pK₂ + pK₁^[11] For multiple equilibrium systems, also see: theory of Response reactions.

Effect of temperature

The effect of changing temperature on an equilibrium constant is given by the van 't Hoff equation

Thus, for exothermic reactions, (ΔH is negative) K decreases with an increase in temperature, but, for endothermic reactions, (ΔH is positive) K increases with an increase temperature. An alternative formulation is

At first sight this appears to offer a means of obtaining the standard molar enthalpy of the reaction by studying the variation of K with temperature. In practice, however, the method is unreliable because error propagation almost always gives very large errors on the values calculated in this way.

Effect of electric and magnetic fields

The effect of electric field on equilibrium has been studied by Manfred Eigen^{[citation needed]} among others.

Types of equilibrium

1. In the gas phase. Rocket engines^[12]
2. The industrial synthesis such as ammonia in the Haber–Bosch process (depicted right) takes place through a succession of equilibrium steps including adsorption processes.
   
   

   Haber–Bosch process
3. atmospheric chemistry
4. Seawater and other natural waters: Chemical oceanography
5. Distribution between two phases
   1. LogD-Distribution coefficient: Important for pharmaceuticals where lipophilicity is a significant property of a drug
   2. Liquid–liquid extraction, Ion exchange, Chromatography
   3. Solubility product
   4. Uptake and release of oxygen by haemoglobin in blood
6. Acid/base equilibria: Acid dissociation constant, hydrolysis, buffer solutions, indicators, acid–base homeostasis
7. Metal-ligand complexation: sequestering agents, chelation therapy, MRI contrast reagents, Schlenk equilibrium
8. Adduct formation: Host–guest chemistry, supramolecular chemistry, molecular recognition, dinitrogen tetroxide
9. In certain oscillating reactions, the approach to equilibrium is not asymptotically but in the form of a damped oscillation .^[10]
10. The related Nernst equation in electrochemistry gives the difference in electrode potential as a function of redox concentrations.
11. When molecules on each side of the equilibrium are able to further react irreversibly in secondary reactions, the final product ratio is determined according to the Curtin–Hammett principle.

In these applications, terms such as stability constant, formation constant, binding constant, affinity constant, association/dissociation constant are used. In biochemistry, it is common to give units for binding constants, which serve to define the concentration units used when the constant’s value was determined.

Composition of a mixture

When the only equilibrium is that of the formation of a 1:1 adduct as the composition of a mixture, there are any number of ways that the composition of a mixture can be calculated. For example, see ICE table for a traditional method of calculating the pH of a solution of a weak acid.

There are three approaches to the general calculation of the composition of a mixture at equilibrium.

1. The most basic approach is to manipulate the various equilibrium constants until the desired concentrations are expressed in terms of measured equilibrium constants (equivalent to measuring chemical potentials) and initial conditions.
2. Minimize the Gibbs energy of the system.^[13]
3. Satisfy the equation of mass balance. The equations of mass balance are simply statements that demonstrate that the total concentration of each reactant must be constant by the law of conservation of mass.

Mass-balance equations

In general, the calculations are rather complicated or complex. For instance, in the case of a dibasic acid, H₂A dissolved in water the two reactants can be specified as the conjugate base, A²⁻, and the proton, H⁺. The following equations of mass-balance could apply equally well to a base such as 1,2-diaminoethane, in which case the base itself is designated as the reactant A:

With T_A the total concentration of species A. Note that it is customary to omit the ionic charges when writing and using these equations.

When the equilibrium constants are known and the total concentrations are specified there are two equations in two unknown "free concentrations" [A] and [H]. This follows from the fact that [HA]= β₁[A][H], [H₂A]= β₂[A][H]² and [OH] = K_w[H]⁻¹

so the concentrations of the "complexes" are calculated from the free concentrations and the equilibrium constants. General expressions applicable to all systems with two reagents, A and B would be

It is easy to see how this can be extended to three or more reagents.

Polybasic acids

Species concentrations during hydrolysis of the aluminium.

The composition of solutions containing reactants A and H is easy to calculate as a function of p[H]. When [H] is known, the free concentration [A] is calculated from the mass-balance equation in A.

The diagram alongside, shows an example of the hydrolysis of the aluminium Lewis acid Al³⁺_aq^[14] shows the species concentrations for a 5×10⁻⁶M solution of an aluminium salt as a function of pH. Each concentration is shown as a percentage of the total aluminium.

Solution and precipitation

The diagram above illustrates the point that a precipitate that is not one of the main species in the solution equilibrium may be formed. At pH just below 5.5 the main species present in a 5μM solution of Al³⁺ are aluminium hydroxides Al(OH)²⁺, Al(OH)₂⁺ and Al₁₃(OH)₃₂⁷⁺, but on raising the pH Al(OH)₃ precipitates from the solution. This occurs because Al(OH)₃ has a very large lattice energy. As the pH rises more and more Al(OH)₃ comes out of solution. This is an example of Le Chatelier's principle in action: Increasing the concentration of the hydroxide ion causes more aluminium hydroxide to precipitate, which removes hydroxide from the solution. When the hydroxide concentration becomes sufficiently high the soluble aluminate, Al(OH)₄⁻, is formed.

Another common instance where precipitation occurs is when a metal cation interacts with an anionic ligand to form an electrically neutral complex. If the complex is hydrophobic, it will precipitate out of water. This occurs with the nickel ion Ni²⁺ and dimethylglyoxime, (dmgH₂): in this case the lattice energy of the solid is not particularly large, but it greatly exceeds the energy of solvation of the molecule Ni(dmgH)₂.

Minimization of free energy

At equilibrium, G is at a minimum:

For a closed system, no particles may enter or leave, although they may combine in various ways. The total number of atoms of each element will remain constant. This means that the minimization above must be subjected to the constraints:

where is the number of atoms of element i in molecule j and b_i⁰ is the total number of atoms of element i, which is a constant, since the system is closed. If there are a total of k types of atoms in the system, then there will be k such equations.

This is a standard problem in optimisation, known as constrained minimisation. The most common method of solving it is using the method of Lagrange multipliers, also known as undetermined multipliers (though other methods may be used).

Define:

where the are the Lagrange multipliers, one for each element. This allows each of the to be treated independently, and it can be shown using the tools of multivariate calculus that the equilibrium condition is given by

    and    

This is a set of (m+k) equations in (m+k) unknowns (the and the ) and may, therefore, be solved for the equilibrium concentrations as long as the chemical potentials are known as functions of the concentrations at the given temperature and pressure. (See Thermodynamic databases for pure substances).
This method of calculating equilibrium chemical concentrations is useful for systems with a large number of different molecules. The use of k atomic element conservation equations for the mass constraint is straightforward, and replaces the use of the stoichiometric coefficient equations.^[12]