H

Heat is the mode of transfer of energy that is due to a temperature difference between a system and its surroundings. The quantity of energy transferred to the system in this way is denoted q, or dq of the quantity is infinitesimal. The resulting change in internal energy, provided no other types of transfer take place, is dU = dq. Specifically, if the volume of a closed (constant composition) system is constant during the transfer, and no other processes are taking place, then ΔU = q_V, the subscript denoting constant volume. If the pressure on a closed system is constant, and no other processes are taking place, then ΔH = q_p.

At a molecular level, heat corresponds to the transfer of energy that makes use of or generates disorderly motion in the surroundings (Figure H.9). As such, it is distinguished from work, the transfer of energy that makes use of orderly motion in the surroundings. The term ‘heat’ should be distinguished from thermal motion, the disorderly motion of molecules, which is commonly used as a synonym.

At a molecular level, the heat capacity is large if there are many energy levels thermally accessible to the system, in which case populations can be transferred readily between them. At room temperature, many monatomic nonmetallic solids follow, approximately at least, Dulong and Petit’s law, that C_V,m ≈ 3R. This law can be justified on the basis of the equipartition theorem and the oscillation of atoms in three dimensions. The heat capacities of all substance tend to zero as T → 0 because fewer energy levels become accessible. At low temperatures, nonmetallic solids obey the Debye-T ³ law, that C_V,m ≈ aT ³, where a is a constant characteristic of the solid. Metallic solids show an additional contribution stemming from the possibility of exciting electrons near the top of the conduction band. For them, C_V,m ≈ aT ³ + bT.

A heat engine is a cyclic device for the conversion of heat into work (Figure H.12). That is, it continuously draws energy as heat from a hot source, transfers some of that energy to the surroundings as work, and deposits the rest of the energy as heat into a cold sink. See Carnot cycle and efficiency. A heat engine in reverse is a refrigerator.

The Hermann–Mauguin system (or International system) is used to label symmetry groups, more commonly for crystals than for individual molecules. In the system, which is set out in Table H.1,

• n denotes the presence of an n-fold axis

• m denotes a mirror plane

• / indicates that the mirror plane is perpendicular to the symmetry axis

• – (a bar over a number) indicates that the element is combined with an inversion

The Hückel approximation is a primitive scheme for calculating the π orbitals and energies of conjugated polyenes. The Hamiltonian matrix is constructed by setting all diagonal elements equal to α – E, all off-diagonal elements between neighbours equal to β, all other elements to zero; overlap is ignored. The eigenvalues and eigenfunctions of the matrix are the energy levels and linear combination coefficients (of the carbon atom π orbitals), respectively, and are best found by using mathematical software. Figure H.13 shows two typical outcomes.

Hund’s rules are a guide to the lowest energy term of an atom and can be extended to molecules.

1. Electrons occupy orbitals singly before occupying them doubly.

2. Electrons occupy orbitals that result in the maximum value of the total orbital angular momentum quantum number, L.

3. Electrons adopt the maximum number of parallel spins compatible with their orbital occupation. This is the maximum multiplicity rule, with atoms achieving the maximum value of the total spin angular momentum, S.

4. If the shell is less than half full, minimum J lies lowest; if it is more than half full, maximum J lies lowest.

The maximum multiplicity rule stems from the quantum mechanical property of spin correlation in which electrons with the same spin tend to stay apart from one another. This mutual avoidance permits the atom to shrink slightly with the result that electrons experience a more favourable nuclear attraction.

Hybridization is the mixing of two or more atomic orbitals on the same atom. It is a feature of valence bond theory aimed at providing equivalent, local bonds that conform to the shape of the molecule. Three characteristic hybridization schemes and their mutually orthogonal orbital compositions are:

Tetrahedral and trigonal hybrids are shown in Figure H.14.

Note that N atomic orbitals give rise to N mutually orthogonal hybrid orbitals. Other symmetrical schemes are listed in Table H.2. Hybrid orbitals can be constructed for a variety of geometries by selecting combinations that give rise to hybrids that match the geometry and are mutually orthogonal.

A hydrogen bond is a link between atoms of two electronegative elements, A and B, one of which (B) has a lone pair of electrons, bridged by a hydrogen atom and denoted A–H⋯B. Typically, A and B are any of N, O, and F, but anions may also participate. A variety of theories have been given for its formation; all depend on the small size of the proton and its ability to lie between two nearby atoms. In the molecular orbital explanation (Figure H.15), the atoms A, H, and B each supply one atomic orbital and form a bonding, an almost nonbonding, and an antibonding molecular orbital. The first two are occupied by the four electrons provided by the A–H bond and the B lone pair. As the antibonding orbital remains unoccupied, this arrangement has a lower energy than the separate AH and B components.

The strength of a hydrogen bond is approximately 20 kJ mol⁻¹. Its formation accounts for many of the anomalous properties of water (including its low vapour pressure and the relative mass densities of the liquid and solid phases). It is of major importance in determining the secondary structure of biological macromolecules.

The description of the bond in the hydrogen molecule, H₂, can be regarded as the starting point of the valence bond description of the chemical bond, and that of the hydrogen molecule-ion, H₂ ⁺, the starting point of the molecular orbital theory of bonding. Valence bond theory ascribes the bond to spin pairing of the electrons supplied by the two atoms, which enables (via the Pauli principle) the two electrons to be described by the spatial wavefunction ψ_A(1)ψ_B(2) + ψ_A(2)ψ_B(1). In that combination there is constructive interference in the internuclear region and a consequent lowering of energy of the two paired electrons. More complex molecules are described by localized bonds formed similarly. In molecular orbital theory, bonding and antibonding molecular orbitals, ψ_A ± ψ_B are constructed from the H1s atomic orbitals. The occupation of the bonding combination by even a single electron lowers the energy, once again largely due to constructive interference in the internuclear region. A second electron can join the first to strengthen the bond provided its spin is opposite to that of the first electron. The theory is extended to more complex molecules by including more atomic orbitals in the linear combinations. The Schrödinger equation for the electron in H₂ ⁺ can be solved analytically, but the resulting wavefunctions (in spheroidal coordinates) are of too great a complexity to be useful.

All orbitals of the same principal quantum number, n, have the same energy and belong to the same shell of the atom. A shell is sometimes designated by the letters K, L, M,… for n = 1, 2, 3,…. The members of a shell with the same value of the orbital angular momentum quantum number l form a subshell of the atom. There are n subshells in a shell with quantum number n. They are commonly designated by the letters s, p, d, f,… for l = 0, 1, 2, 3,…. A subshell with quantum number l consists of 2l + 1 orbitals that are distinguished by the magnetic quantum number, m_l. It follows that there are n ² degenerate orbitals in a shell with quantum number n. See quantum number for a further description of their significance and the entries for each type of orbital.

The ground state of a hydrogenic atom has the configuration 1s¹ and term symbol ²S_1/2. Its ionization energy is hcR̃_NZ ² ≈ 13.59Z ² eV.

The hyperfine structure of a spectrum is the splitting of absorption lines that arises from the interaction of an electron with a nucleus other than by the Coulombic interaction. Two common sources are magnetic interaction of electrons and nuclei and the interaction of the nuclear quadrupole moment with the electric field gradient at the nucleus.