E

The acronym EELS stands for electronic energy loss spectroscopy. The technique relies on measuring the energy loss electrons undergo when they are reflected inelastically from a surface. The changes in energy are interpreted in terms of the vibrational spectrum of the adsorbate.

Effusion is the escape of a gas or vapour through a small hole in a container. According to Graham’s law, the rate of effusion is inversely proportional to the square root of the molar mass of the gas. That law is explained by Maxwell’s expression for the distribution of speeds, which implies that the mean speed of molecules is also inversely proportional to M ^1/2. Given that the superficial collision rate is Z_W = p/(2πmkT)^1/2 with m = M/N_A, the rate of effusion through a hole of area A is dN/dt = Z_WA.

According to the Ehrenfest classification of phase transitions, a first-order transition is one for which dμ/dT, where μ is the chemical potential, is discontinuous across the transition. The implication is that the enthalpy of transition is nonzero and the heat capacity is infinite at the transition temperature (Figure E.2). A second-order transition is one for which dμ/dT is continuous but d²μ/dT ² is not. The implication is that Δ_trsH = 0 and that the heat capacity is discontinuous but not infinite at the transition temperature.

A function f is an eigenfunction of an operator $Ω ˆ$ if $Ω ˆ$ f = ωf, where ω is a numerical factor, the eigenvalue of the operator corresponding to the eigenfunction f. If the operator is Hermitian, its eigenvalues are all real and its eigenfunctions are mutually orthogonal. The Schrödinger equation can be expressed as an eigenvalue equation, as $H ˆ$ ψ = Eψ, where $H ˆ$ is the Hamiltonian operator and E its eigenvalue corresponding to the wavefunction (the eigenfunction) ψ. The system is then said to be in an eigenstate of the Hamiltonian operator (and likewise for other physical observables). The outcome of any measurement of a system is an eigenvalue of the operator corresponding to the property of interest. A state can be a simultaneous eigenstate of two different operators only if they commute (see uncertainty principle, commutator).

The non-SI unit 1 einstein = 1 mole of photons.

In an elastic collision there is no transfer of energy from translational motion into the internal modes of motion (rotation and vibration) of the colliding species. The collision is inelastic if there is a transfer of energy between translation and internal modes.

An electric dipole consists of two equal and opposite charges Q and −Q separated by a distance L. The dipole moment is a vector of magnitude μ = QL directed from the negative charge to the positive charge. A point dipole is the dipole viewed from a distance r such that r ≫ L. The SI unit of dipole moment is coulomb metre (C m) but dipole moments are commonly reported in debye (D). A polar molecule is a molecule with a permanent nonzero electric dipole moment; a nonpolar molecule is a molecule with no permanent dipole moment.

A very approximate relation between the magnitude of the dipole moment and the electronegativity difference, Δχ, of a heteronuclear diatomic molecule or fragment is μ/D = Δχ. The negative partial charge is commonly on the more electronegative atom, but that might not be the case when antibonding orbitals are occupied (CO is an example). A molecule acquires a contribution to its electric dipole moment (in addition to one it might already possess) in an electric field: see polarizability.

An electric double layer consists of two layers of opposite charge. One of the layers may be diffuse. Three models of a double layer are used to discuss processes at surfaces. In the Helmholtz model (Figure E.6) it is supposed that the solvated ions are attracted to the surface but held away from it by their hydration spheres. In the Gouy–Chapman model, thermal disordering is taken into account by supposing that the outer plane is replaced by an ionic atmosphere, as in the Debye–Hückel model of an ionic solution. In the Stern model there is a rigid outer Helmholtz plane and beyond that there an ionic atmosphere.

Electrical conduction is the transport of electric charge. It may occur by the migration of electrons or ions. Electronic conductors are classified as metallic conductors if the conductivity decreases with increasing temperature (that is, if the resistance rises as the temperature is increased), as semiconductors if the conductivity increases with temperature, and as superconductors if the current flows with zero resistance.

Metallic conduction occurs when a solid has an incompletely filled band of levels (a ‘conduction band’, Figure E.7). Its decreasing conductivity with rising temperature is due to the disruption of electron flow by the increasingly vigorous vibrations of the lattice. Semiconduction arises when an empty band lies above but close to a filled band (a ‘valence band’); the increasing conductivity with temperature is due to the increasing population of the upper band as the temperature is raised (see Fermi–Dirac distribution). Many solids have such large band gaps between the valence band and an upper empty band that their ability to conduct an electric current remains very low at all temperatures and are considered to be electrical insulators.

Semiconduction can be tailored by doping the solid with elements that either remove electrons from the valence band (to give a p-type semiconductor, in which the conduction is effectively by ‘holes’ in the valence band), or with elements that provide electrons to populate the upper band (to give an n-type semiconductor). Low-temperature superconduction is due to the formation of Cooper pairs, in which distortion of the lattice by one electron captures a second, and the two linked electron travel through the lattice without being scattered. The mechanism of high-temperature superconduction remains obscure.

An electrochemical cell is device in which either a chemical reaction generates an electric current or in which an electric current is used to bring about a chemical reaction. A galvanic cell (or voltaic cell) is an electrochemical cell in which a spontaneous chemical reaction generates an electric current. An electrolytic cell is an electrochemical cell in which the passage of an electric current brings about a nonspontaneous chemical reaction. A primary cell is a galvanic cell in which the reactants are built in at manufacture. A secondary cell is a galvanic cell that must first be charged by acting as an electrolytic cell. A fuel cell is a galvanic cell in which the reactants are supplied externally. All types of cells consist of two metallic conductors in contact with an electrolyte, which may be an ionically conducting liquid or solid. The electrodes may share a common electrolyte, in which case the cell is designated M|electrolyte|M′ (Figure E.8, left). Alternatively, the cell may consist of two compartments. The cell may then be either M|electrolyte₁|electrolyte₂|M′ if the electrolytes are in direct contact, or M|electrolyte₁||electrolyte₂|M′ if the compartments are joined by a salt bridge to eliminate the junction potential (Figure E.8, right). If the compartments differ only in the concentrations of the same electrolyte, the cell is called a concentration cell.

The electrode at which reduction occurs is called the cathode (and is the point of entry of electrons into the cell from the external circuit of a working cell). The electrode at which oxidation occurs is called the anode (and is the electrode at which electrons leave the cell). Thus, the cathode is the positive electrode of a galvanic cell and the anode is the negative electrode. The cell potential, E_cell, is the potential difference of the cell (M′ – M for the cell as specified as above) when the progress of the chemical reaction is balanced by an externally applied potential difference and is therefore progressing reversibly. The Nernst equation expresses how the cell potential depends upon the composition of the reaction mixture.

The electrochemical series summarizes the relative reducing power of redox couples, with the more strongly reducing couples (couples with the more negative standard potentials) higher in the series.

An electrode is a metallic conductor that makes contact with an electrolyte in an electrochemical cell. If it is the site of reduction, it is termed a cathode; if it is the site of oxidation, then it is termed an anode. Examples of electrodes and their notation, where M is a metallic conductor and | denotes a phase boundary, are as follows:

1. Metal–metal ion electrode, M(s)|M⁺(aq). Example: Cu(s)|Cu²⁺(aq).

2. Metal–insoluble salt electrode, M(s)|MX(s)|X⁻(aq). Example: Ag(s)|AgCl(s)|Cl⁻(aq).

3. Gas electrode, M(s)|X₂(g)|X^+/−(aq). Example: Pt(s)|H₂(g)|H⁺(aq).

4. Redox electrode, M(s)|Ox(aq),Red(aq). Example: Pt(s)|Fe³⁺(aq),Fe²⁺(aq).

Each electrode is in its standard state when the reactants are in their standard states (a_J = 1, f_J = 1 bar). The standard hydrogen electrode (SHE) is an important example as it is the basis of the definition and compilation of standard potentials.

The electrokinetic potential, ζ (or zeta potential), is the electric potential at the radius of shear of a colloidal particle relative to the potential in the distant bulk medium (Figure E.9). The radius of shear is the radius of the sphere that surrounds a rigid layer of ions attached to the surface of a colloidal particle.

The term electrolyte is used in two slightly different ways. (1) An electrolyte is a substance that dissolves to give an ionically conducting solution. Such an electrolyte may be either strong or weak (see conductivity). (2) The electrolyte of an electrochemical cell is the ionically conducting medium in contact with each electrode.

The electromagnetic field is a distribution of oscillating electric and magnetic fields propagating in a vacuum at a common speed, c, the ‘speed of light’. Both component fields are perpendicular to the direction of propagation and are mutually perpendicular. The field is classified into different regions according to its frequency or its vacuum wavelength (Figure E.10). In its dual character, the field is treated as a collection of photons, each one of energy hν, the number of photons in a region determining its intensity there. The evidence for its particlelike character incudes the photoelectric effect; the evidence for its wavelike character is diffraction.

The field may be regarded as a collection of harmonic oscillators that span the entire and continuous frequency range. The excitation of an oscillator of frequency ν to its first excited state corresponds to the presence of one photon of energy hν, to its second excited state to two photons each of energy hν, and so on. This model leaves open the possibility that the oscillators have a zero-point energy and that the unexcited electromagnetic vacuum oscillates with electric and magnetic fields; that zero-point fluctuation accounts for the process of spontaneous emission (see Einstein coefficients).

In LEED, the use of low-energy electron beams means that only surface scattering events contribute to the diffraction. The technique is used to study the distribution of defects on the surface and the pattern of the attachment of an adsorbate to it.

An electronvolt (eV) is a non-SI unit of energy defined such at 1 eV is the difference in potential energy of an electron when it is moved between locations differing in electric potential by 1 V. Equivalently, it is the kinetic energy gained by an electron when it is accelerated through a potential difference of 1 V. Numerically, 1 eV = 1.602 176 634 × 10⁻¹⁹ J, corresponding to 96.485 kJ mol⁻¹. That the ionization energy of a hydrogen atom is 13.6 eV indicates that when an electron is lost from the ground state of the atom to infinity it experiences a change in electric potential of 13.6 V.

Electrophoresis is the motion of macromolecules under the influence of an electric field. In polyacrylamide gel electrophoresis (PAGE) the migration takes place through a crosslinked polyacrylamide gel. The technique is used to separate macromolecules and to determine their molar masses (after calibration).

An element is classified as electropositive if it has a strongly negative standard potential and hence is thermodynamically a powerful reducing agent. Note that ‘electropositive’ is not the opposite of ‘electronegative’: the former is a term in electrochemistry; the latter is a term relating to electron distributions in molecules.

An elementary reaction is a single step in a proposed reaction mechanism. It is classified as unimolecular if it involves a single reactant molecule (A → P) and as bimolecular if two reactant molecules are involved (A + B → P and A + A → P). A unimolecular elementary reaction has a first-order rate law; a bimolecular elementary reaction has a second-order rate law. The overall rate law implied by a proposed mechanism is formulated by combining the rate laws of the constituent elementary reactions, often by invoking the steady-state approximation or supposing that there is a pre-equilibrium.

The end point of an acid–base titration is the stage at which the acidic and basic forms of an indicator are in equal abundance (and similarly in other versions of titrations). Distinguish end point from stoichiometric point. A well-chosen indicator has an end point at the stoichiometric point of a titration.

The energy, E, of a system is its capacity to do work. The kinetic energy, E_k, of a body is the energy due to its motion; for a body of mass m and speed $v$ ⁠, $E k = 1 2 m v 2$ ⁠. The potential energy, E_p or V, of a body is the energy due to its position. The expression for potential energy depends on the type of field:

For a system free of external forces, the total energy, E_total = E_k + E_p (sometimes written, when the meaning of the symbols is clear, as E = T + V) is conserved (that is, remains unchanged as one form is converted into another).

Energy pooling is the process that occurs when two excited state molecules encounter one another and the excess energy of one is transferred to the other, so raising it to an even more highly excited state: A* + A* → A** + A.

An ensemble in statistical thermodynamics is a collection of imaginary replications of the actual system (Figure E.11). In a microcanonical ensemble, the number of molecules (N), volume (V), and energy (E), of each member of the ensemble are the same. In a canonical ensemble the number of molecules, the volume, and the temperature (T) of each member are the same and energy can be exchanged between the members; the total energy of the entire ensemble is constant. In a grand canonical ensemble the chemical potentials (μ) of the component, the volume, and the temperature are the same, with energy and particles able to move between the members, their totals being constant. The thermodynamic limit of a property is calculated as an average over the members of an ensemble as the total number of replications approaches infinity.

The enthalpy, H, of a system is defined as H = U + pV, where U is the internal energy of the system, p is its pressure, and V is its volume. Enthalpy is a state function. For a system of constant composition, the change in enthalpy accompanying a process is equal to the energy transferred as heat at constant pressure: ΔH = q_p. The isobaric (constant pressure) heat capacity is the derivative of the enthalpy of a closed system with respect to temperature at constant pressure: C_p = (∂H/∂T)_p,n.

The usefulness of the enthalpy is that changes in its value take into account automatically the work done on the system as it changes its volume (Figure E.12). An exenthalpic process is one for which ΔH < 0 and an endenthalpic process is one for which ΔH > 0. These processes are commonly referred to as exothermic and endothermic, respectively, but such terms should be retained for processes for q < 0 (energy leaving as heat) and q > 0 (energy entering as heat), respectively, regardless of the constraints on the system. In practice, there is rarely a distinction in the sense that most exothermic processes are exenthalpic. An isenthalpic process is one that occurs at constant enthalpy: see Joule–Thomson effect. The reversible isothermal expansion of a perfect gas is isenthalpic but endothermic.

The standard enthalpy of formation of a substance, Δ_fH ^⦵, at a specified temperature is the change in molar enthalpy accompanying its formation in its standard state from its elements in their reference states. The reference state of an element is its thermodynamically most stable state at that temperature. Thus, the reference state of carbon at room temperature and 1 bar is graphite and that of hydrogen at 1 bar is the gas. The one exception is white phosphorus, which is the most reproducible but not the most stable allotrope of the element. By implication, the standard enthalpy of formation of an element in its reference state is zero. Compounds are classified as exenthalpic or endenthalpic according to the sign of Δ_fH ^⦵ (Figure E.13). Standard enthalpies of formation are reported in kilojoules per mole of formula units of the compound and tabulated values commonly refer to the conventional temperature, 298.15 K (but any specified temperature can be used provided it is used consistently).

The problem with ions is that in solution cations are always accompanied by anions. To isolate the standard enthalpy of formation of ions in solution, the standard enthalpy of formation of hydrogen ions in water is defined as zero at all temperatures, Δ_fH ^⦵(H⁺,aq)=0, and values for other ions are reported on that basis.

At constant pressure, the enthalpy of reaction can be identified as the energy as heat required or released by the reaction. The standard value includes contributions from the mixing of the pure reactants, the reaction itself (the commonly dominant contribution, except for reactions involving ions), and the unmixing of the products into their pure forms. Reactions for which Δ_rH ^⦵ < 0 are classified as exenthalpic; those for which Δ_rH ^⦵ > 0 are classified as endenthalpic (although the less precise terms exothermic and endothermic are commonly used). An exenthalpic reaction at constant pressure in a diathermic container releases heat into the surroundings; in an adiabatic container it causes a rise in temperature. An endenthalpic reaction does the opposite.

The enthalpy of transition, Δ_trsH, is the change in molar enthalpy that accompanies a change in phase of a substance such as vaporization. It can be identified with the energy required as heat at constant pressure required or released as a result of the transition. For this reason, enthalpies of transition have in the past been called ‘latent heats’. The standard enthalpy of transition is the enthalpy change accompanying a transition when both phases are in their standard state (pure, 1 bar), Note that the enthalpy of transition α → β is the negative of the reverse transition α ← β (such as vaporization and condensation). As depicted in Figure E.14, the overall standard enthalpy of transition of α → β → γ is the sum of the individual standard enthalpies of transition (at the same temperature), as in $Δ sub H ⦵ T = Δ fus H ⦵ T + Δ vap H ⦵ (T)$ ⁠.

Entropy acts as a means of formulating the Second law of thermodynamics and expressing it quantitatively. Thus, the Second law states that the entropy of an isolated system increases in the course of a spontaneous change. Therefore, to decide whether a feasible change (one allowed by the First law) in an isolated system can occur spontaneously it is necessary to determine whether the entropy would increase if it took place. The ‘isolated system’ may consist of the system of interest and its immediate surroundings. The law opens the possibility that the entropy of the system of interest might decrease, for it is the sum of the entropy changes in it and its surroundings that matter, and the entropy of the system may decrease provided there is a compensating increase in the entropy of its surroundings.

The interpretation of the entropy as a measure of disorder is consistent with its definition in terms of the transfer of energy as heat. That transfer results in an increase in the thermal disorder of the system, but the extent of the change depends on the temperature. At high temperatures there is already a great deal of thermal disorder and a given transfer of energy results in only a small additional contribution to it. However, if the temperature is low then there is little thermal disorder present and the transfer of the same quantity of energy results in a large increase in disorder. (I have likened this difference to sneezing in a busy street or sneezing in a quiet library.)

An enzyme is a protein that acts as catalyst for a biochemical reaction. Enzymes are very specific and act on a specific substrate molecule. That specificity is often likened to a lock and key model or, increasingly commonly, to an induced-fit model in which the protein changes the shape of the active site in a specific way on the approach of the substrate. The kinetics of enzyme action are summarized by the Michaelis–Menten equation and its modifications. The action of enzymes may be inhibited by kinetic intention (a part of a natural biochemical process), accident (disease), or therapeutically. In competitive inhibition a poison competes for the active site. In noncompetitive inhibition the inhibitor attaches to another part of the molecule so distorting it and inhibiting attachment of the substrate to the active site. See inhibition.

The principle of equal a priori probabilities asserts that all states of the same energy are equally likely to be occupied at thermal equilibrium regardless of their nature (translational, rotational, and so on). That is, the only distinguishing feature of a state as far as its probability of occupation is concerned is its energy. The Boltzmann distribution is derived on this basis.

A system is at equilibrium if it has no tendency to change. The condition of equilibrium is a reversible state in the sense that an infinitesimal variation in the conditions results in a change in direction. An equilibrium state corresponds to a state of maximum total entropy. Providing the temperature and pressure are held constant, the system is at equilibrium if the Gibbs energy is at a minimum. Equilibria in chemistry are dynamic in the sense that although there is no net change, change continues at a molecular level.

A system is in mechanical equilibrium with its surroundings if an infinitesimal variation in the pressure results in a change in volume of the system. A system is in thermal equilibrium with the surroundings if an infinitesimal variation in the temperature results in a transfer of energy as heat. An open system is in chemical equilibrium if the infinitesimal change in the amount of the substance present results in a change in the extent of reaction.

A system produces maximum work if it passes through a series of steps in each of which it is in equilibrium with the surroundings; that is, maximum work is produced in a reversible change. Thus, maximum expansion work is done if the pressure of the surroundings is equal to the pressure of the system at all stages of the expansion. In that case, dw = −pdV. The maximum work that a system can produce is given by the Helmholtz energy, w_max = ΔA; the maximum nonexpansion work that a system can produce at constant temperature and pressure is given by the change in Gibbs energy: w_{nonexpansion,max} = ΔG.

According to the equipartition theorem, each quadratic contribution to the energy of a molecule in a system that is in thermal equilibrium with its surroundings at a temperature T is the same and equal to $1 2$  kT. A quadratic contribution means a contribution to the energy that is proportional to the square of a momentum or position variable. The kinetic energy of a particle of mass m in one dimension is p ²/2m, the average value of its energy is therefore $1 2$  kT. In three dimensions there are three such terms, so the average translational kinetic energy of the molecule is $3 2$  kT. The total energy of one-dimensional harmonic oscillator is p ²/2m +  $1 2$  k_fx ², the sum of two quadratic terms, so its average energy at a temperature T is kT. The theorem is a classical result and is invalid when the quantum effects are important.

The acronym ESCA stands for electron spectroscopy for chemical analysis. See photoelectron spectroscopy.

A eutectic mixture is a binary mixture that freezes without change of composition (Figure E.18). It has the lowest melting point of all such mixtures. When a series of liquid mixtures are allowed to cool the temperature remains constant longest for the mixture with the eutectic composition. This pause is known as the eutectic halt.

An excimer is a dimeric species of the form AA* that exists only in the excited state (denoted *). It is a special case of an exciplex, AB*, in which the two components are different.

An explosion is an abrupt increase in volume. In a thermal explosion the energy released by a reaction is unable to escape from the system. The resulting rise in temperature causes the reaction rate to increase, so releasing more energy, raising the temperature, resulting in a rapid growth in the reaction rate and a sudden increase in the volume occupied by the products. In a chain branching explosion, a branching step in a chain reaction results in the initiation of an additional chain; that chain branches, the resulting chain also branches, and so on, leading into a catastrophic increase in rate and volume of the products.

In a system that reacts by a chain mechanism there are regions of temperature and pressure in which the explosion occurs by one or other of these mechanisms and other regions in which reaction proceeds smoothly (Figure E.19). For a given temperature there is a pressure called the lower explosion limit below which reaction is smooth and above which it is explosive by chain branching until the upper explosion limit is reached. At that pressure the reaction becomes steady again. A further increase in pressure may take the reaction into the thermal explosion regime.

Consider a system that can be divided into fragments. If a property of the original system is the sum of the values of that property of all the fragments, then the property is extensive. Thus, mass and internal energy are extensive properties. If the property of the original system is not the sum of the values of that property of all the fragments, then the property is intensive. Thus, temperature and mass density are intensive properties. More loosely: an extensive property depends on the size of the system; an intensive property does not.