B

The Balmer series is a sequence of emission lines in the spectrum of atomic hydrogen that terminate in the level with n_f = 2 and start in the levels with n_i = 3, 4,…. Many of the lines lie in the visible region. Their wavenumbers are proportional to 1/4 – 1/n_i ². See hydrogenic atom.

When each of N atoms of an element contributes one atomic orbital to the formation of a molecular orbital, the resulting N molecular orbitals form an almost continuous band of levels when N is large. The lowest energy molecular orbital of the band is fully bonding between all neighbours; the highest energy molecular orbital of the band is antibonding between all neighbours (Figure B.1). If the atomic orbitals are s orbitals, then they form an s-band. If the atomic orbitals are in p orbitals then they form a p-band. In a typical case there is so large an energy difference between the s and p atomic orbitals that the resulting s- and p-bands are separated by a band gap of width E_g. In certain cases, the symmetry of the crystal results in the formation of a band gap even though the s and p orbitals are so close in energy that the resulting bands are expected to overlap. In this case, the bands mix to form hybrid bands.

Electrons occupy orbitals in the band in accord with the Pauli principle. If insufficient electrons are available to fill a band, the result is a conduction band; the electrons at and close to the highest occupied level of the band are mobile and the solid is classified as a metallic conductor. The highest occupied orbital at T = 0 is called the Fermi level. Only the electrons close to the highest occupied level of the band contribute to the conduction; also, only they contribute to the heat capacity. If the band is full, then the electrons cannot transport the current and the solid is formally a semiconductor, its electrical conduction arising from electrons that have been thermally excited across the band gap. If the band gap is so large that few electrons populate an upper almost empty band, then conduction is negligible and the solid is regarded as an electrical insulator. The details of the temperature dependence of the population of the bands is expressed by the Fermi–Dirac distribution.

The unit of pressure 1 bar is defined as 10⁵ Pa exactly. Note that 1 atm = 1.101 325 bar exactly, so 1 bar is about 1 per cent smaller than 1 atm. The bar is used in the current definition of standard state.

An Arrhenius base is a compound that, when dissolved in water, generates OH⁻ ions. A Brønsted base is a proton acceptor. The conjugate base of an acid HA is the base A⁻ formed by proton loss from HA. A strong base is a Brønsted base that is effectively fully protonated in solution. A weak base is a Brønsted base that is only partly protonated at normal concentrations in solution. The stronger the base, the weaker is its conjugate acid. A base is weak if K_b < 1 and usually K_b ≪ 1. A Lewis base is an electron-pair donor. A hard base is a Lewis base that tends to bond strongly to a hard acid and a soft base is a Lewis base that tends to bond strongly to a soft acid. Hard acid–hard base combinations are largely ionic and soft acid–soft base combinations are largely covalent.

In molecular orbital theory a basis set is the set of atomic orbitals from which the molecular orbitals are constructed as a linear combination. In general, the larger the basis set, the more accurate is the molecular orbital. In group theory, a basis is the set of entities to which a symmetry transformation is applied when forming a representation of a group. In this context, the number of entities used is called the dimension of the representation.

The Bijvoet method is a technique in X-ray crystallography for the determination of the absolute configuration of a compound by making use of the phase shift in the scattered radiation as the X-rays approach an absorption frequency of the compound. The phase shifts are called anomalous scattering and result in different intensities in the diffraction pattern of different enantiomers.

Black-body radiation is electromagnetic radiation that is in thermal equilibrium with matter at a temperature T. A black body is a material that can absorb or emit without favouring any wavelength. The characteristics of black-body radiation are summarized by the following two laws:

• Wien displacement law: The temperature and wavelength of the radiation of maximum intensity satisfy the relation λT = constant, with the empirical value of the constant equal to 28 mm K.

• Stefan–Boltzmann law concerns the temperature dependence of the excitance, $M$ ⁠, the power emitted divided by the area, and states that $M=σ T 4$ ⁠, where σ is the Stefan–Boltzmann constant, with the empirical value 57 nW m⁻² K⁻⁴.

The Bohr frequency condition states the frequency, ν, of radiation emitted or absorbed when an atom makes a transition between two states that differ in energy by ΔE is hν = ΔE (more precisely, hν = |ΔE|). The basis of this expression is the conservation of energy when a photon of energy hv is generated or absorbed.

The energy of an electron with magnetic quantum number m_l in a magnetic field of flux density $B$ in the z-direction is $E m l = m l μ B B$ ⁠. See magnetogyric ratio.

The boiling temperature of a substance is the temperature at which its vapour pressure is equal to the ambient pressure. The normal boiling point is the temperature at which the vapour pressure is 1 atm. The standard boiling point is the temperature at which the vapour pressure is 1 bar. Boiling is characterized by by the formation of bubbles of vapour throughout the liquid.

Boltzmann’s constant, k, is the fundamental constant that relates the population of a state to the temperature by way of the Boltzmann distribution. It has the defined value k = 1.380 649 × 10⁻²³ J K⁻¹. The constant occurs wherever the Boltzmann distribution is implicitly present, which includes the statistical definition of entropy, the equipartition of energy, and the properties of a perfect gas. It often appears in disguise as the gas constant R = N_Ak, so R is effectively the molar Boltzmann’s constant. This relation accounts for the appearance of R in a number of expressions that have nothing to do with gases, such as the Nernst equation for the cell potential.

The bond dissociation energy, hcD̃₀ (with D̃₀ a wavenumber), is the minimum energy required to dissociate a bond from the vibrational ground state of a molecule. It differs from the energy minimum of the molecular potential energy curve by the zero-point energy (Figure B.4). It can be identified with the change in internal energy at T = 0. The closely related thermodynamic property is the bond dissociation enthalpy, ΔH(A−B), which is the molar enthalpy change accompanying the process A−B(g) → A(g) + B(g). This quantity differs from the dissociation energy itself in part because the process refers to a distribution over the available states at the stated temperature (ΔU_m(T) < ΔU_m(0) = hcD̃₀) and on account of the difference between enthalpy and internal energy. Thus, ΔH_m(T) = ΔU_m(T) + pΔV_m(T) ≈ ΔU_m(T) + RT. See mean bond enthalpy. The equilibrium bond length, R_e, is the distance between the nuclei of neighbouring atoms in a bond for the vibrational ground state.

The bond order, b, of a diatomic molecule or a diatomic fragment of a polyatomic molecule is defined as $b= 1 2 (n− n ⁎ )$ ⁠, where n is the number of electrons in bonding orbitals and n* the number in antibonding orbitals. The bond order correlates with a number of molecular properties. Thus, for a given pair of atoms, the bond length decreases, the bond dissociation energy increases, the force constant increases as the bond order between them increases.

A bonding orbital is a molecular orbital which, if occupied, lowers the energy of a molecule. It is often convenient to focus on pairs of neighbouring atoms in a molecule and to regard the orbital as bonding if there is no internuclear nodal plane and antibonding if there is an internuclear nodal plane (Figure B.5). The bonding character is commonly ascribed to the accumulation of electron density between two atoms as a result of the constructive overlap of atomic orbitals.

The Born–Haber cycle is a thermodynamic cycle in which one step is the formation of a solid from gas-phase ions. The cycle, as in the version depicted in Figure B.6, is most commonly used to determine the lattice enthalpy of a solid, with the steps corresponding to molar enthalpy changes.

The Born interpretation of a wavefunction ψ( r ) is that ψ*( r )ψ( r ) is proportional to the probability density for finding a particle at the location r , and therefore that ψ*( r )ψ( r )dτ is proportional to the probability of finding the particle in the infinitesimal region dτ at r . In each case, the proportionality becomes an equality if the wavefunction is normalized to 1. The implication is that ψ( r ) itself is a probability amplitude. The curve and the shading of the band in Figure B.7 depict the probability density in this instance.

According to the Born–Oppenheimer approximation, the distribution of electrons in molecule can be calculated by supposing that the nuclei are in fixed locations.

A boson is a particle with integral spin quantum number (including 0). Examples include the photon (spin 1), a helium-4 nucleus (spin 0), and a deuterium nucleus (spin 1). The Pauli principle requires their wavefunction to be unchanged under the exchange of identical pairs. As a result, the Pauli exclusion principle does not apply, and any number of bosons can occupy the same state. Bosons are the conveyors of forces between fermions, so matter can be regarded as fermions bound together by the exchange of bosons.

The general solution of a first-order differential equation has one undetermined constant; that of a second-order differential equation has two. These constants are determined by ensuring that the solutions have certain values at the boundaries of the system, such as at an impenetrable wall or at a great distance (Figure B.8). These values are the boundary conditions, and ensure that only some of the general solutions are acceptable. If one of the variables of a differential equation is time, the related constraint is called an initial condition, and sets the value of the solution at t = 0.

In chemistry, boundary conditions are imposed on solutions of the Schrödinger equation and the diffusion equation; initial conditions typically apply to the differential equations known as rate laws in chemical kinetics. The boundary conditions of the Schrödinger equation are typically related to the values of the wavefunction at the physical boundaries of the system (including the behaviour of wavefunctions at infinity), or, in certain systems, cyclic boundary conditions, the requirement that wavefunctions replicate themselves after a rotation of 2π (Figure B.9). Examples of the former are the requirement that ψ = 0 at the walls of a particle in a box, and examples of the latter at the requirement that ψ(ϕ + 2π) = ψ(ϕ) for a particle ion a ring. That boundary conditions lead to the selection of only a discrete set of solutions of the Schrödinger equation accounts for the quantization of its dynamical properties, specifically its energy and, for systems with circular symmetry, its angular momentum.

A boundary surface is a notional surface that captures a specified probability of finding an electron in an atom or molecule. An isodensity surface is a surface of constant probability density (Figure B.10). Each type of atomic orbital has a characteristic boundary surface (commonly represented as either an isodensity surface or more notionally by a shape that approximates to an isodensity surface), and those shapes, particularly their symmetry, is the basis of qualitative discussions of atomic and molecular structure.

All regular crystals can be modelled by attaching an asymmetric unit to one of the 14 Bravais lattices, depicted as a unit cell which when stacked together without rotation reproduces the crystal (Figure B.13). The lattices are classified according to the crystal system to which they correspond together with a specification of the location of the lattice points. A primitive lattice (P) has lattice points only at the vertices of the cell, a body-centred lattice (I) also has a lattice point at the centre, and a face-centred lattice (F) has lattice points at the vertices and at the centre of each face. In two cases (orthorhombic C and monoclinic C), there are points at the vertices and on only two opposite faces.

The continuous spread of frequencies in the X-ray region of the electromagnetic spectrum known as Bremsstrahlung (‘braking radiation’) is generated by fast electrons decelerating as they plunge into a metal.

A buffer solution is an aqueous solution that is resistant to changes in pH on the addition of a strong acid or strong base. It consists of a solution of a weak acid and its conjugate base or a weak base and its conjugate acid. The former is an acid buffer, and stabilizes solutions at pH < 7. The latter is a base buffer, and stabilizes at pH > 7. A buffer is typically most effective in the range pH = pK_a ± 1.

The building-up principle (or Aufbau principle) aims to generate the ground-state electron configuration of an atom. For an atom of atomic number Z, the Z electrons are allowed to occupy the available orbitals in order of increasing energy subject to the requirement of the Pauli exclusion principle that no more than two electrons can occupy any given orbital. When more that one orbital of the same energy is available, electrons enter each one separately and do so with parallel spins. Broadly speaking, the order of occupation of orbitals is the order of their energies after taking into account penetration and shielding. However, in some cases small adjustments are necessary in order to ensure that the final configuration is the one of lowest total energy. Thus, in the d-block, the lowest energy is obtained in certain cases by transferring an s-electron into a d orbital, especially if that results in a full or half-full d-subshell. Analogous adjustments occur in the f-block.

The electron configuration of a cation is obtained by removing an electron from the last-filled orbital of the parent atom: p-electrons are removed first, then s-electrons, and finally the requisite number of d-electrons. The configurations of chemically significant anions are obtained by completing the p-subshell.

The Butler–Volmer equation is often expressed in one of two limits. In the low overpotential limit, with ηf ≪ 1, j ≈ j₀fη. In this limit the current density is proportional to the overpotential, and the electrode is termed ohmic (as it behaves according to Ohm’s law). In the high overpotential limit, with ηf ≫ 1, ln j ≈ αfη. A Tafel plot is a plot of ln j against overpotential (Figure B.14). The slope of the plot is α and the extrapolated intercept at η = 0 gives ln j₀.