Ensemble (mathematical physics)

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https://en.wikipedia.org/wiki/Ensemble_(mathematical_physics)

In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a possible state that the real system might be in. In other words, a statistical ensemble is a set of systems of particles used in statistical mechanics to describe a single system. The concept of an ensemble was introduced by J. Willard Gibbs in 1902.

A thermodynamic ensemble is a specific variety of statistical ensemble that, among other properties, is in statistical equilibrium (defined below), and is used to derive the properties of thermodynamic systems from the laws of classical or quantum mechanics.

Physical considerations

The ensemble formalises the notion that an experimenter repeating an experiment again and again under the same macroscopic conditions, but unable to control the microscopic details, may expect to observe a range of different outcomes.

The notional size of ensembles in thermodynamics, statistical mechanics and quantum statistical mechanics can be very large, including every possible microscopic state the system could be in, consistent with its observed macroscopic properties. For many important physical cases, it is possible to calculate averages directly over the whole of the thermodynamic ensemble, to obtain explicit formulas for many of the thermodynamic quantities of interest, often in terms of the appropriate partition function.

The concept of an equilibrium or stationary ensemble is crucial to many applications of statistical ensembles. Although a mechanical system certainly evolves over time, the ensemble does not necessarily have to evolve. In fact, the ensemble will not evolve if it contains all past and future phases of the system. Such a statistical ensemble, one that does not change over time, is called stationary and can be said to be in statistical equilibrium.

Terminology

• The word "ensemble" is also used for a smaller set of possibilities sampled from the full set of possible states. For example, a collection of walkers in a Markov chain Monte Carlo iteration is called an ensemble in some of the literature.
• The term "ensemble" is often used in physics and the physics-influenced literature. In probability theory, the term probability space is more prevalent.

Main types

Visual representation of five statistical ensembles (from left to right): microcanonical ensemble, canonical ensemble, grand canonical ensemble, isobaric-isothermal ensemble, isoenthalpic-isobaric ensemble

The study of thermodynamics is concerned with systems that appear to human perception to be "static" (despite the motion of their internal parts), and which can be described simply by a set of macroscopically observable variables. These systems can be described by statistical ensembles that depend on a few observable parameters, and which are in statistical equilibrium. Gibbs noted that different macroscopic constraints lead to different types of ensembles, with particular statistical characteristics.

"We may imagine a great number of systems of the same nature, but differing in the configurations and velocities which they have at a given instant, and differing in not merely infinitesimally, but it may be so as to embrace every conceivable combination of configuration and velocities..." J. W. Gibbs (1903)

Three important thermodynamic ensembles were defined by Gibbs:

• Microcanonical ensemble (or NVE ensemble) —a statistical ensemble where the total energy of the system and the number of particles in the system are each fixed to particular values; each of the members of the ensemble are required to have the same total energy and particle number. The system must remain totally isolated (unable to exchange energy or particles with its environment) in order to stay in statistical equilibrium.
• Canonical ensemble (or NVT ensemble)—a statistical ensemble where the energy is not known exactly but the number of particles is fixed. In place of the energy, the temperature is specified. The canonical ensemble is appropriate for describing a closed system which is in, or has been in, weak thermal contact with a heat bath. In order to be in statistical equilibrium, the system must remain totally closed (unable to exchange particles with its environment) and may come into weak thermal contact with other systems that are described by ensembles with the same temperature.
• Grand canonical ensemble (or μVT ensemble)—a statistical ensemble where neither the energy nor particle number are fixed. In their place, the temperature and chemical potential are specified. The grand canonical ensemble is appropriate for describing an open system: one which is in, or has been in, weak contact with a reservoir (thermal contact, chemical contact, radiative contact, electrical contact, etc.). The ensemble remains in statistical equilibrium if the system comes into weak contact with other systems that are described by ensembles with the same temperature and chemical potential.

The calculations that can be made using each of these ensembles are explored further in their respective articles. Other thermodynamic ensembles can be also defined, corresponding to different physical requirements, for which analogous formulae can often similarly be derived. For example, in the reaction ensemble, particle number fluctuations are only allowed to occur according to the stoichiometry of the chemical reactions which are present in the system.

Representations

The precise mathematical expression for a statistical ensemble has a distinct form depending on the type of mechanics under consideration (quantum or classical). In the classical case, the ensemble is a probability distribution over the microstates. In quantum mechanics, this notion, due to von Neumann, is a way of assigning a probability distribution over the results of each complete set of commuting observables. In classical mechanics, the ensemble is instead written as a probability distribution in phase space; the microstates are the result of partitioning phase space into equal-sized units, although the size of these units can be chosen somewhat arbitrarily.

Requirements for representations

Putting aside for the moment the question of how statistical ensembles are generated operationally, we should be able to perform the following two operations on ensembles A, B of the same system:

• Test whether A, B are statistically equivalent.
• If p is a real number such that 0 < p < 1, then produce a new ensemble by probabilistic sampling from A with probability p and from B with probability 1 – p.

Under certain conditions, therefore, equivalence classes of statistical ensembles have the structure of a convex set.

Quantum mechanical

Main article: Density matrix

A statistical ensemble in quantum mechanics (also known as a mixed state) is most often represented by a density matrix, denoted by ${\displaystyle \hat{\rho }}$. The density matrix provides a fully general tool that can incorporate both quantum uncertainties (present even if the state of the system were completely known) and classical uncertainties (due to a lack of knowledge) in a unified manner. Any physical observable X in quantum mechanics can be written as an operator, X̂. The expectation value of this operator on the statistical ensemble ${\displaystyle \rho }$ is given by the following trace:

${\displaystyle ⟨X⟩=\mathrm{Tr}(\hat{X}\rho ).}$

This can be used to evaluate averages (operator X̂), variances (using operator X̂ ²), covariances (using operator X̂Ŷ), etc. The density matrix must always have a trace of 1: ${\displaystyle \mathrm{Tr}\hat{\rho }=1}$ (this essentially is the condition that the probabilities must add up to one).

In general, the ensemble evolves over time according to the von Neumann equation.

Equilibrium ensembles (those that do not evolve over time, ${\displaystyle d\hat{\rho }/dt=0}$) can be written solely as a function of conserved variables. For example, the microcanonical ensemble and canonical ensemble are strictly functions of the total energy, which is measured by the total energy operator Ĥ (Hamiltonian). The grand canonical ensemble is additionally a function of the particle number, measured by the total particle number operator N̂. Such equilibrium ensembles are a diagonal matrix in the orthogonal basis of states that simultaneously diagonalize each conserved variable. In bra–ket notation, the density matrix is

${\displaystyle \hat{\rho }=\sum _i{P}_i|{\psi }_i⟩⟨{\psi }_i|}$

where the |ψ_i⟩, indexed by i, are the elements of a complete and orthogonal basis. (Note that in other bases, the density matrix is not necessarily diagonal.)

Classical mechanical

Evolution of an ensemble of classical systems in phase space (top). Each system consists of one massive particle in a one-dimensional potential well (red curve, lower figure). The initially compact ensemble becomes swirled up over time.

In classical mechanics, an ensemble is represented by a probability density function defined over the system's phase space. While an individual system evolves according to Hamilton's equations, the density function (the ensemble) evolves over time according to Liouville's equation.

In a mechanical system with a defined number of parts, the phase space has n generalized coordinates called q₁, ... q_n, and n associated canonical momenta called p₁, ... p_n. The ensemble is then represented by a joint probability density function ρ(p₁, ... p_n, q₁, ... q_n).

If the number of parts in the system is allowed to vary among the systems in the ensemble (as in a grand ensemble where the number of particles is a random quantity), then it is a probability distribution over an extended phase space that includes further variables such as particle numbers N₁ (first kind of particle), N₂ (second kind of particle), and so on up to N_s (the last kind of particle; s is how many different kinds of particles there are). The ensemble is then represented by a joint probability density function ρ(N₁, ... N_s, p₁, ... p_n, q₁, ... q_n). The number of coordinates n varies with the numbers of particles.

Any mechanical quantity X can be written as a function of the system's phase. The expectation value of any such quantity is given by an integral over the entire phase space of this quantity weighted by ρ:

${\displaystyle ⟨X⟩=\sum _{N_1=0}^{\mathrm{\infty }}\dots \sum _{N_s=0}^{\mathrm{\infty }}\int \dots \int \rho Xd{p}_1\dots d{q}_n.}$

The condition of probability normalization applies, requiring

${\displaystyle \sum _{N_1=0}^{\mathrm{\infty }}\dots \sum _{N_s=0}^{\mathrm{\infty }}\int \dots \int \rho d{p}_1\dots d{q}_n=1.}$

Phase space is a continuous space containing an infinite number of distinct physical states within any small region. In order to connect the probability density in phase space to a probability distribution over microstates, it is necessary to somehow partition the phase space into blocks that are distributed representing the different states of the system in a fair way. It turns out that the correct way to do this simply results in equal-sized blocks of canonical phase space, and so a microstate in classical mechanics is an extended region in the phase space of canonical coordinates that has a particular volume. In particular, the probability density function in phase space, ρ, is related to the probability distribution over microstates, P by a factor

${\displaystyle \rho =\frac{1}{h^{n}C}P,}$

where

• h is an arbitrary but predetermined constant with the units of energy×time, setting the extent of the microstate and providing correct dimensions to ρ.
• C is an overcounting correction factor (see below), generally dependent on the number of particles and similar concerns.

Since h can be chosen arbitrarily, the notional size of a microstate is also arbitrary. Still, the value of h influences the offsets of quantities such as entropy and chemical potential, and so it is important to be consistent with the value of h when comparing different systems.

Correcting overcounting in phase space

Typically, the phase space contains duplicates of the same physical state in multiple distinct locations. This is a consequence of the way that a physical state is encoded into mathematical coordinates; the simplest choice of coordinate system often allows a state to be encoded in multiple ways. An example of this is a gas of identical particles whose state is written in terms of the particles' individual positions and momenta: when two particles are exchanged, the resulting point in phase space is different, and yet it corresponds to an identical physical state of the system. It is important in statistical mechanics (a theory about physical states) to recognize that the phase space is just a mathematical construction, and to not naively overcount actual physical states when integrating over phase space. Overcounting can cause serious problems:

• Dependence of derived quantities (such as entropy and chemical potential) on the choice of coordinate system, since one coordinate system might show more or less overcounting than another.
• Erroneous conclusions that are inconsistent with physical experience, as in the mixing paradox.
• Foundational issues in defining the chemical potential and the grand canonical ensemble.

It is in general difficult to find a coordinate system that uniquely encodes each physical state. As a result, it is usually necessary to use a coordinate system with multiple copies of each state, and then to recognize and remove the overcounting.

A crude way to remove the overcounting would be to manually define a subregion of phase space that includes each physical state only once and then exclude all other parts of phase space. In a gas, for example, one could include only those phases where the particles' x coordinates are sorted in ascending order. While this would solve the problem, the resulting integral over phase space would be tedious to perform due to its unusual boundary shape. (In this case, the factor C introduced above would be set to C = 1, and the integral would be restricted to the selected subregion of phase space.)

A simpler way to correct the overcounting is to integrate over all of phase space but to reduce the weight of each phase in order to exactly compensate the overcounting. This is accomplished by the factor C introduced above, which is a whole number that represents how many ways a physical state can be represented in phase space. Its value does not vary with the continuous canonical coordinates, so overcounting can be corrected simply by integrating over the full range of canonical coordinates, then dividing the result by the overcounting factor. However, C does vary strongly with discrete variables such as numbers of particles, and so it must be applied before summing over particle numbers.

As mentioned above, the classic example of this overcounting is for a fluid system containing various kinds of particles, where any two particles of the same kind are indistinguishable and exchangeable. When the state is written in terms of the particles' individual positions and momenta, then the overcounting related to the exchange of identical particles is corrected by using

${\displaystyle C=N_1!N_2!\dots N_s!.}$

This is known as "correct Boltzmann counting".

Ensembles in statistics

Main articles: Principle of maximum entropy and Markov random field

The formulation of statistical ensembles used in physics has now been widely adopted in other fields, in part because it has been recognized that the canonical ensemble or Gibbs measure serves to maximize the entropy of a system, subject to a set of constraints: this is the principle of maximum entropy. This principle has now been widely applied to problems in linguistics, robotics, and the like.

In addition, statistical ensembles in physics are often built on a principle of locality: that all interactions are only between neighboring atoms or nearby molecules. Thus, for example, lattice models, such as the Ising model, model ferromagnetic materials by means of nearest-neighbor interactions between spins. The statistical formulation of the principle of locality is now seen to be a form of the Markov property in the broad sense; nearest neighbors are now Markov blankets. Thus, the general notion of a statistical ensemble with nearest-neighbor interactions leads to Markov random fields, which again find broad applicability; for example in Hopfield networks.

Ensemble average

In statistical mechanics, the ensemble average is defined as the mean of a quantity that is a function of the microstate of a system, according to the distribution of the system on its micro-states in this ensemble.

Since the ensemble average is dependent on the ensemble chosen, its mathematical expression varies from ensemble to ensemble. However, the mean obtained for a given physical quantity does not depend on the ensemble chosen at the thermodynamic limit. The grand canonical ensemble is an example of an open system.

Classical statistical mechanics

For a classical system in thermal equilibrium with its environment, the ensemble average takes the form of an integral over the phase space of the system:

${\displaystyle \overline{A}=\frac{\int A{e}^{-\beta H(q_1,q_2,...q_M,p_1,p_2,...p_N)}d\tau }{\int e^{-\beta H(q_1,q_2,...q_M,p_1,p_2,...p_N)}d\tau }}$

where:

${\displaystyle \overline{A}}$ is the ensemble average of the system property A,

${\displaystyle \beta }$ is ${\displaystyle \frac{1}{kT}}$, known as thermodynamic beta,

H is the Hamiltonian of the classical system in terms of the set of coordinates ${\displaystyle q_i}$ and their conjugate generalized momenta ${\displaystyle p_i}$, and

${\displaystyle d\tau }$ is the volume element of the classical phase space of interest.

The denominator in this expression is known as the partition function, and is denoted by the letter Z.

Quantum statistical mechanics

In quantum statistical mechanics, for a quantum system in thermal equilibrium with its environment, the weighted average takes the form of a sum over quantum energy states, rather than a continuous integral:

${\displaystyle \overline{A}=\frac{\sum _i{A}_i{e}^{-\beta E_i}}{\sum _i{e}^{-\beta E_i}}}$

Canonical ensemble average

The generalized version of the partition function provides the complete framework for working with ensemble averages in thermodynamics, information theory, statistical mechanics and quantum mechanics.

The microcanonical ensemble represents an isolated system in which energy (E), volume (V) and the number of particles (N) are all constant. The canonical ensemble represents a closed system which can exchange energy (E) with its surroundings (usually a heat bath), but the volume (V) and the number of particles (N) are all constant. The grand canonical ensemble represents an open system which can exchange energy (E) as well as particles with its surroundings but the volume (V) is kept constant.

Operational interpretation

In the discussion given so far, while rigorous, we have taken for granted that the notion of an ensemble is valid a priori, as is commonly done in physical context. What has not been shown is that the ensemble itself (not the consequent results) is a precisely defined object mathematically. For instance,

• It is not clear where this very large set of systems exists (for example, is it a gas of particles inside a container?)
• It is not clear how to physically generate an ensemble.

In this section, we attempt to partially answer this question.

Suppose we have a preparation procedure for a system in a physics lab: For example, the procedure might involve a physical apparatus and some protocols for manipulating the apparatus. As a result of this preparation procedure, some system is produced and maintained in isolation for some small period of time. By repeating this laboratory preparation procedure we obtain a sequence of systems X₁, X₂, ....,X_k, which in our mathematical idealization, we assume is an infinite sequence of systems. The systems are similar in that they were all produced in the same way. This infinite sequence is an ensemble.

In a laboratory setting, each one of these prepped systems might be used as input for one subsequent testing procedure. Again, the testing procedure involves a physical apparatus and some protocols; as a result of the testing procedure we obtain a yes or no answer. Given a testing procedure E applied to each prepared system, we obtain a sequence of values Meas (E, X₁), Meas (E, X₂), ...., Meas (E, X_k). Each one of these values is a 0 (or no) or a 1 (yes).

Assume the following time average exists:

${\displaystyle \sigma (E)=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{N}\sum _{k=1}^N\mathrm{Meas}(E,X_k)}$

For quantum mechanical systems, an important assumption made in the quantum logic approach to quantum mechanics is the identification of yes-no questions to the lattice of closed subspaces of a Hilbert space. With some additional technical assumptions one can then infer that states are given by density operators S so that:

${\displaystyle \sigma (E)=\mathrm{Tr}(ES).}$

We see this reflects the definition of quantum states in general: A quantum state is a mapping from the observables to their expectation values.

Lewis acids and bases

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Lewis_acids_and_bases

 

Diagram of some Lewis bases and acids

A Lewis acid (named for the American physical chemist Gilbert N. Lewis) is a chemical species that contains an empty orbital which is capable of accepting an electron pair from a Lewis base to form a Lewis adduct. A Lewis base, then, is any species that has a filled orbital containing an electron pair which is not involved in bonding but may form a dative bond with a Lewis acid to form a Lewis adduct. For example, NH₃ is a Lewis base, because it can donate its lone pair of electrons. Trimethylborane (Me₃B) is a Lewis acid as it is capable of accepting a lone pair. In a Lewis adduct, the Lewis acid and base share an electron pair furnished by the Lewis base, forming a dative bond. In the context of a specific chemical reaction between NH₃ and Me₃B, a lone pair from NH₃ will form a dative bond with the empty orbital of Me₃B to form an adduct NH₃•BMe₃. The terminology refers to the contributions of Gilbert N. Lewis.

The terms nucleophile and electrophile are more or less interchangeable with Lewis base and Lewis acid, respectively. However, these terms, especially their abstract noun forms nucleophilicity and electrophilicity, emphasize the kinetic aspect of reactivity, while the Lewis basicity and Lewis acidity emphasize the thermodynamic aspect of Lewis adduct formation.

Depicting adducts

In many cases, the interaction between the Lewis base and Lewis acid in a complex is indicated by an arrow indicating the Lewis base donating electrons toward the Lewis acid using the notation of a dative bond — for example, Me₃B←NH₃. Some sources indicate the Lewis base with a pair of dots (the explicit electrons being donated), which allows consistent representation of the transition from the base itself to the complex with the acid:

Me₃B + :NH₃ → Me₃B:NH₃

A center dot may also be used to represent a Lewis adduct, such as Me₃B·NH₃. Another example is boron trifluoride diethyl etherate, BF₃·Et₂O. In a slightly different usage, the center dot is also used to represent hydrate coordination in various crystals, as in MgSO₄·7H₂O for hydrated magnesium sulfate, irrespective of whether the water forms a dative bond with the metal.

Although there have been attempts to use computational and experimental energetic criteria to distinguish dative bonding from non-dative covalent bonds, for the most part, the distinction merely makes note of the source of the electron pair, and dative bonds, once formed, behave simply as other covalent bonds do, though they typically have considerable polar character. Moreover, in some cases (e.g., sulfoxides and amine oxides as R₂S → O and R₃N → O), the use of the dative bond arrow is just a notational convenience for avoiding the drawing of formal charges. In general, however, the donor–acceptor bond is viewed as simply somewhere along a continuum between idealized covalent bonding and ionic bonding.

Lewis acids

Major structural changes accompany binding of the Lewis base to the coordinatively unsaturated, planar Lewis acid BF₃

Lewis acids are diverse and the term is used loosely. Simplest are those that react directly with the Lewis base, such as boron trihalides and the pentahalides of phosphorus, arsenic, and antimony.

In the same vein, CH₃⁺ can be considered to be the Lewis acid in methylation reactions. However, the methyl cation never occurs as a free species in the condensed phase, and methylation reactions by reagents like CH₃I take place through the simultaneous formation of a bond from the nucleophile to the carbon and cleavage of the bond between carbon and iodine (S_N2 reaction). Textbooks disagree on this point: some asserting that alkyl halides are electrophiles but not Lewis acids, while others describe alkyl halides (e.g. CH₃Br) as a type of Lewis acid. The IUPAC states that Lewis acids and Lewis bases react to form Lewis adducts, and defines electrophile as Lewis acids.

Simple Lewis acids

Some of the most studied examples of such Lewis acids are the boron trihalides and organoboranes:

BF₃ + F⁻ → BF₄⁻

In this adduct, all four fluoride centres (or more accurately, ligands) are equivalent.

BF₃ + OMe₂ → BF₃OMe₂

Both BF₄⁻ and BF₃OMe₂ are Lewis base adducts of boron trifluoride.

Many adducts violate the octet rule, such as the triiodide anion:

I₂ + I⁻ → I₃⁻

The variability of the colors of iodine solutions reflects the variable abilities of the solvent to form adducts with the Lewis acid I₂.

Some Lewis acids bind with two Lewis bases, a famous example being the formation of hexafluorosilicate:

SiF₄ + 2 F⁻ → SiF₆²⁻

Complex Lewis acids

Most compounds considered to be Lewis acids require an activation step prior to formation of the adduct with the Lewis base. Complex compounds such as Et₃Al₂Cl₃ and AlCl₃ are treated as trigonal planar Lewis acids but exist as aggregates and polymers that must be degraded by the Lewis base. A simpler case is the formation of adducts of borane. Monomeric BH₃ does not exist appreciably, so the adducts of borane are generated by degradation of diborane:

B₂H₆ + 2 H⁻ → 2 BH₄⁻

In this case, an intermediate B₂H₇⁻ can be isolated.

Many metal complexes serve as Lewis acids, but usually only after dissociating a more weakly bound Lewis base, often water.

[Mg(H₂O)₆]²⁺ + 6 NH₃ → [Mg(NH₃)₆]²⁺ + 6 H₂O

H⁺ as Lewis acid

The proton (H⁺) is one of the strongest but is also one of the most complicated Lewis acids. It is convention to ignore the fact that a proton is heavily solvated (bound to solvent). With this simplification in mind, acid-base reactions can be viewed as the formation of adducts:

• H⁺ + NH₃ → NH₄⁺
• H⁺ + OH⁻ → H₂O

Applications of Lewis acids

A typical example of a Lewis acid in action is in the Friedel–Crafts alkylation reaction. The key step is the acceptance by AlCl₃ of a chloride ion lone-pair, forming AlCl₄⁻ and creating the strongly acidic, that is, electrophilic, carbonium ion.

RCl +AlCl₃ → R⁺ + AlCl₄⁻

Lewis bases

A Lewis base is an atomic or molecular species where the highest occupied molecular orbital (HOMO) is highly localized. Typical Lewis bases are conventional amines such as ammonia and alkyl amines. Other common Lewis bases include pyridine and its derivatives. Some of the main classes of Lewis bases are

• amines of the formula NH_3−xR_x where R = alkyl or aryl. Related to these are pyridine and its derivatives.
• phosphines of the formula PR_3−xA_x, where R = alkyl, A = aryl.
• compounds of O, S, Se and Te in oxidation state -2, including water, ethers, ketones

The most common Lewis bases are anions. The strength of Lewis basicity correlates with the pK_a of the parent acid: acids with high pK_a's give good Lewis bases. As usual, a weaker acid has a stronger conjugate base.

• Examples of Lewis bases based on the general definition of electron pair donor include:
  • simple anions, such as H⁻ and F⁻
  • other lone-pair-containing species, such as H₂O, NH₃, HO⁻, and CH₃⁻
  • complex anions, such as sulfate
  • electron-rich π-system Lewis bases, such as ethyne, ethene, and benzene

The strength of Lewis bases have been evaluated for various Lewis acids, such as I₂, SbCl₅, and BF₃.

Heats of binding of various bases to BF₃

Lewis base	Donor atom	Enthalpy of complexation (kJ/mol)
Quinuclidine	N	150
Et3N	N	135
Pyridine	N	128
Acetonitrile	N	60
DMA	O	112
DMSO	O	105
THF	O	90.4
Et2O	O	78.8
Acetone	O	76.0
EtOAc	O	75.5
Trimethylphosphine	P	97.3
Tetrahydrothiophene	S	51.6

Applications of Lewis bases

Main article: Homogeneous catalysis

Nearly all electron pair donors that form compounds by binding transition elements can be viewed as a collections of the Lewis bases—or ligands. Thus a large application of Lewis bases is to modify the activity and selectivity of metal catalysts. Chiral Lewis bases thus confer chirality on a catalyst, enabling asymmetric catalysis, which is useful for the production of pharmaceuticals.

Many Lewis bases are "multidentate," that is they can form several bonds to the Lewis acid. These multidentate Lewis bases are called chelating agents.

Hard and soft classification

Main article: HSAB theory

Lewis acids and bases are commonly classified according to their hardness or softness. In this context hard implies small and nonpolarizable and soft indicates larger atoms that are more polarizable.

• typical hard acids: H⁺, alkali/alkaline earth metal cations, boranes, Zn²⁺
• typical soft acids: Ag⁺, Mo(0), Ni(0), Pt²⁺
• typical hard bases: ammonia and amines, water, carboxylates, fluoride and chloride
• typical soft bases: organophosphines, thioethers, carbon monoxide, iodide

For example, an amine will displace phosphine from the adduct with the acid BF₃. In the same way, bases could be classified. For example, bases donating a lone pair from an oxygen atom are harder than bases donating through a nitrogen atom. Although the classification was never quantified it proved to be very useful in predicting the strength of adduct formation, using the key concepts that hard acid—hard base and soft acid—soft base interactions are stronger than hard acid—soft base or soft acid—hard base interactions. Later investigation of the thermodynamics of the interaction suggested that hard—hard interactions are enthalpy favored, whereas soft—soft are entropy favored.

Quantifying Lewis acidity

Many methods have been devised to evaluate and predict Lewis acidity. Many are based on spectroscopic signatures such as shifts NMR signals or IR bands e.g. the Gutmann-Beckett method and the Childs method.

The ECW model is a quantitative model that describes and predicts the strength of Lewis acid base interactions, −ΔH. The model assigned E and C parameters to many Lewis acids and bases. Each acid is characterized by an E_A and a C_A. Each base is likewise characterized by its own E_B and C_B. The E and C parameters refer, respectively, to the electrostatic and covalent contributions to the strength of the bonds that the acid and base will form. The equation is

−ΔH = E_AE_B + C_AC_B + W

The W term represents a constant energy contribution for acid–base reaction such as the cleavage of a dimeric acid or base. The equation predicts reversal of acids and base strengths. The graphical presentations of the equation show that there is no single order of Lewis base strengths or Lewis acid strengths and that single property scales are limited to a smaller range of acids or bases.

History

MO diagram depicting the formation of a dative covalent bond between two atoms

The concept originated with Gilbert N. Lewis who studied chemical bonding. In 1923, Lewis wrote An acid substance is one which can employ an electron lone pair from another molecule in completing the stable group of one of its own atoms. The Brønsted–Lowry acid–base theory was published in the same year. The two theories are distinct but complementary. A Lewis base is also a Brønsted–Lowry base, but a Lewis acid doesn't need to be a Brønsted–Lowry acid. The classification into hard and soft acids and bases (HSAB theory) followed in 1963. The strength of Lewis acid-base interactions, as measured by the standard enthalpy of formation of an adduct can be predicted by the Drago–Wayland two-parameter equation.

Reformulation of Lewis theory

Lewis had suggested in 1916 that two atoms are held together in a chemical bond by sharing a pair of electrons. When each atom contributed one electron to the bond, it was called a covalent bond. When both electrons come from one of the atoms, it was called a dative covalent bond or coordinate bond. The distinction is not very clear-cut. For example, in the formation of an ammonium ion from ammonia and hydrogen the ammonia molecule donates a pair of electrons to the proton; the identity of the electrons is lost in the ammonium ion that is formed. Nevertheless, Lewis suggested that an electron-pair donor be classified as a base and an electron-pair acceptor be classified as acid.

A more modern definition of a Lewis acid is an atomic or molecular species with a localized empty atomic or molecular orbital of low energy. This lowest-energy molecular orbital (LUMO) can accommodate a pair of electrons.

Comparison with Brønsted–Lowry theory

A Lewis base is often a Brønsted–Lowry base as it can donate a pair of electrons to H⁺; the proton is a Lewis acid as it can accept a pair of electrons. The conjugate base of a Brønsted–Lowry acid is also a Lewis base as loss of H⁺ from the acid leaves those electrons which were used for the A—H bond as a lone pair on the conjugate base. However, a Lewis base can be very difficult to protonate, yet still react with a Lewis acid. For example, carbon monoxide is a very weak Brønsted–Lowry base but it forms a strong adduct with BF₃.

In another comparison of Lewis and Brønsted–Lowry acidity by Brown and Kanner, 2,6-di-t-butylpyridine reacts to form the hydrochloride salt with HCl but does not react with BF₃. This example demonstrates that steric factors, in addition to electron configuration factors, play a role in determining the strength of the interaction between the bulky di-t-butylpyridine and tiny proton.

Electrophile

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Electrophile

In chemistry, an electrophile is a chemical species that forms bonds with nucleophiles by accepting an electron pair. Because electrophiles accept electrons, they are Lewis acids. Most electrophiles are positively charged, have an atom that carries a partial positive charge, or have an atom that does not have an octet of electrons.

Electrophiles mainly interact with nucleophiles through addition and substitution reactions. Frequently seen electrophiles in organic syntheses include cations such as H⁺ and NO⁺, polarized neutral molecules such as HCl, alkyl halides, acyl halides, and carbonyl compounds, polarizable neutral molecules such as Cl₂ and Br₂, oxidizing agents such as organic peracids, chemical species that do not satisfy the octet rule such as carbenes and radicals, and some Lewis acids such as BH₃ and DIBAL.

Organic chemistry

Addition of halogens

These occur between alkenes and electrophiles, often halogens as in halogen addition reactions. Common reactions include use of bromine water to titrate against a sample to deduce the number of double bonds present. For example, ethene + bromine → 1,2-dibromoethane:

C₂H₄ + Br₂ → BrCH₂CH₂Br

This takes the form of 3 main steps shown below;

1. Forming of a π-complex

   The electrophilic Br-Br molecule interacts with electron-rich alkene molecule to form a π-complex 1.

2. Forming of a three-membered bromonium ion

   The alkene is working as an electron donor and bromine as an electrophile. The three-membered bromonium ion 2 consisted of two carbon atoms and a bromine atom forms with a release of Br⁻.

3. Attacking of bromide ion

   The bromonium ion is opened by the attack of Br⁻ from the back side. This yields the vicinal dibromide with an antiperiplanar configuration. When other nucleophiles such as water or alcohol are existing, these may attack 2 to give an alcohol or an ether.

This process is called Ad_E2 mechanism ("addition, electrophilic, second-order"). Iodine (I₂), chlorine (Cl₂), sulfenyl ion (RS⁺), mercury cation (Hg²⁺), and dichlorocarbene (:CCl₂) also react through similar pathways. The direct conversion of 1 to 3 will appear when the Br⁻ is large excess in the reaction medium. A β-bromo carbenium ion intermediate may be predominant instead of 3 if the alkene has a cation-stabilizing substituent like phenyl group. There is an example of the isolation of the bromonium ion 2.

Addition of hydrogen halides

Hydrogen halides such as hydrogen chloride (HCl) adds to alkenes to give alkyl halides in hydrohalogenation. For example, the reaction of HCl with ethylene furnishes chloroethane. The reaction proceeds with a cation intermediate, being different from the above halogen addition. An example is shown below:

1. Proton (H⁺) adds (by working as an electrophile) to one of the carbon atoms on the alkene to form cation 1.
2. Chloride ion (Cl⁻) combines with the cation 1 to form the adducts 2 and 3.

In this manner, the stereoselectivity of the product, that is, from which side Cl⁻ will attack relies on the types of alkenes applied and conditions of the reaction. At least, which of the two carbon atoms will be attacked by H⁺ is usually decided by Markovnikov's rule. Thus, H⁺ attacks the carbon atom that carries fewer substituents so as the more stabilized carbocation (with the more stabilizing substituents) will form.

This is another example of an Ad_E2 mechanism. Hydrogen fluoride (HF) and hydrogen iodide (HI) react with alkenes in a similar manner, and Markovnikov-type products will be given. Hydrogen bromide (HBr) also takes this pathway, but sometimes a radical process competes and a mixture of isomers may form. Although introductory textbooks seldom mentions this alternative, the Ad_E2 mechanism is generally competitive with the Ad_E3 mechanism (described in more detail for alkynes, below), in which transfer of the proton and nucleophilic addition occur in a concerted manner. The extent to which each pathway contributes depends on the several factors like the nature of the solvent (e.g., polarity), nucleophilicity of the halide ion, stability of the carbocation, and steric effects. As brief examples, the formation of a sterically unencumbered, stabilized carbocation favors the Ad_E2 pathway, while a more nucleophilic bromide ion favors the Ad_E3 pathway to a greater extent compared to reactions involving the chloride ion.

In the case of dialkyl-substituted alkynes (e.g., 3-hexyne), the intermediate vinyl cation that would result from this process is highly unstable. In such cases, the simultaneous protonation (by HCl) and attack of the alkyne by the nucleophile (Cl⁻) is believed to take place. This mechanistic pathway is known by the Ingold label Ad_E3 ("addition, electrophilic, third-order"). Because the simultaneous collision of three chemical species in a reactive orientation is improbable, the termolecular transition state is believed to be reached when the nucleophile attacks a reversibly-formed weak association of the alkyne and HCl. Such a mechanism is consistent with the predominantly anti addition (>15:1 anti:syn for the example shown) of the hydrochlorination product and the termolecular rate law, Rate = k[alkyne][HCl]². In support of the proposed alkyne-HCl association, a T-shaped complex of an alkyne and HCl has been characterized crystallographically.

In contrast, phenylpropyne reacts by the Ad_E2_ip ("addition, electrophilic, second-order, ion pair") mechanism to give predominantly the syn product (~10:1 syn:anti). In this case, the intermediate vinyl cation is formed by addition of HCl because it is resonance-stabilized by the phenyl group. Nevertheless, the lifetime of this high energy species is short, and the resulting vinyl cation-chloride anion ion pair immediately collapses, before the chloride ion has a chance to leave the solvent shell, to give the vinyl chloride. The proximity of the anion to the side of the vinyl cation where the proton was added is used to rationalize the observed predominance of syn addition.

Hydration

One of the more complex hydration reactions utilises sulfuric acid as a catalyst. This reaction occurs in a similar way to the addition reaction but has an extra step in which the OSO₃H group is replaced by an OH group, forming an alcohol:

C₂H₄ + H₂O → C₂H₅OH

As can be seen, the H₂SO₄ does take part in the overall reaction, however it remains unchanged so is classified as a catalyst.

This is the reaction in more detail:

1. The H–OSO₃H molecule has a δ+ charge on the initial H atom. This is attracted to and reacts with the double bond in the same way as before.
2. The remaining (negatively charged) ⁻OSO₃H ion then attaches to the carbocation, forming ethyl hydrogensulphate (upper way on the above scheme).
3. When water (H₂O) is added and the mixture heated, ethanol (C₂H₅OH) is produced. The "spare" hydrogen atom from the water goes into "replacing" the "lost" hydrogen and, thus, reproduces sulfuric acid. Another pathway in which water molecule combines directly to the intermediate carbocation (lower way) is also possible. This pathway become predominant when aqueous sulfuric acid is used.

Overall, this process adds a molecule of water to a molecule of ethene.

This is an important reaction in industry, as it produces ethanol, whose purposes include fuels and starting material for other chemicals.

Chiral derivatives

Many electrophiles are chiral and optically stable. Typically chiral electrophiles are also optically pure.

One such reagent is the fructose-derived organocatalyst used in the Shi epoxidation. The catalyst can accomplish highly enantioselective epoxidations of trans-disubstituted and trisubstituted alkenes. The Shi catalyst, a ketone, is oxidized by stoichiometric oxone to the active dioxirane form before proceeding in the catalytic cycle.

Use of a chiral oxaziridine for asymmetric synthesis.

Oxaziridines such as chiral N-sulfonyloxaziridines effect enantioselective ketone alpha oxidation en route to the AB-ring segments of various natural products, including γ-rhodomycionone and α-citromycinone.

Polymer-bound chiral selenium electrophiles effect asymmetric selenenylation reactions. The reagents are aryl selenenyl bromides, and they were first developed for solution phase chemistry and then modified for solid phase bead attachment via an aryloxy moiety. The solid-phase reagents were applied toward the selenenylation of various alkenes with good enantioselectivities. The products can be cleaved from the solid support using organotin hydride reducing agents. Solid-supported reagents offers advantages over solution phase chemistry due to the ease of workup and purification.

Electrophilicity scale

Electrophilicity index

Fluorine	3.86
Chlorine	3.67
Bromine	3.40
Iodine	3.09
Hypochlorite	2.52
Sulfur dioxide	2.01
Carbon disulfide	1.64
Benzene	1.45
Sodium	0.88
Some selected values  (no dimensions)

Several methods exist to rank electrophiles in order of reactivity and one of them is devised by Robert Parr with the electrophilicity index ω given as:

${\displaystyle \omega =\frac{{\chi }^2}{2\eta }}$

with ${\displaystyle \chi }$ the electronegativity and ${\displaystyle \eta }$ chemical hardness. This equation is related to the classical equation for electrical power:

${\displaystyle P=\frac{V^2}{R}}$

where ${\displaystyle R}$ is the resistance (Ohm or Ω) and ${\displaystyle V}$ is voltage. In this sense the electrophilicity index is a kind of electrophilic power. Correlations have been found between electrophilicity of various chemical compounds and reaction rates in biochemical systems and such phenomena as allergic contact dermititis.

An electrophilicity index also exists for free radicals. Strongly electrophilic radicals such as the halogens react with electron-rich reaction sites, and strongly nucleophilic radicals such as the 2-hydroxypropyl-2-yl and tert-butyl radical react with a preference for electron-poor reaction sites.

Superelectrophiles

Superelectrophiles are defined as cationic electrophilic reagents with greatly enhanced reactivities in the presence of superacids. These compounds were first described by George A. Olah. Superelectrophiles form as a doubly electron deficient superelectrophile by protosolvation of a cationic electrophile. As observed by Olah, a mixture of acetic acid and boron trifluoride is able to remove a hydride ion from isobutane when combined with hydrofluoric acid via the formation of a superacid from BF₃ and HF. The responsible reactive intermediate is the [CH₃CO₂H₃]²⁺ dication. Likewise, methane can be nitrated to nitromethane with nitronium tetrafluoroborate NO⁺
₂BF⁻
₄ only in presence of a strong acid like fluorosulfuric acid via the protonated nitronium dication.

In gitionic (gitonic) superelectrophiles, charged centers are separated by no more than one atom, for example, the protonitronium ion O=N⁺=O⁺—H (a protonated nitronium ion). And, in distonic superelectrophiles, they are separated by 2 or more atoms, for example, in the fluorination reagent F-TEDA-BF₄.