C

The caesium-chloride structure (Figure C.1) is one of the simplest structures displayed by a number of ionic compounds, particularly when the ionic radii of the cations and anions are similar. It has (8,8) coordination.

A calorimeter is a device used to measure the energy transferred as heat due to a chemical reaction. A constant-volume calorimeter measures ΔU; a constant-pressure calorimeter measures ΔH. An adiabatic bomb calorimeter is a closed vessel that is thermally insulated from its surroundings; that isolation can be achieved if the walls are thermally conducting by ensuring that the surroundings are at the same temperature as the vessel itself. In an adiabatic bomb calorimeter the energy released as heat is monitored by noting the rise in temperature that accompanies reaction and interpreting that rise by using the calorimeter constant, C, in the form ΔU = CΔT. The calorimeter constant (essentially the heat capacity of the entire assembly) is determined by calibration either electrically or by the combustion of a compound of known heat output.

The Carnot cycle is the following sequence of reversible changes in a heat engine in which the working substance is a perfect gas with hot and cold reservoirs at temperatures T_hot and T_cold (Figure C.3).

1. Isothermal expansion at a temperature T_hot:

   $w 1 =−nR T hot ln V 2 V 1 , q 1 =nR T hot ln V 2 V 1 ,Δ S 1 =nRln V 2 V 1$
2. Adiabatic expansion from T_hot to T_cold:

   $w 2 =−n C V , m ( T hot − T cold ), q 2 =0,Δ S 2 =0$
3. Isothermal compression at T_cold:

   $w 3 =nR T cold ln V 4 V 3 , q 3 =−nR T cold ln V 4 V 3 ,Δ S 3 =−nRln V 4 V 3$
4. Adiabatic compression from T_cold to T_hot:

   $w 4 =n C V , m ( T hot − T cold ), q 4 =0,Δ S 4 =0$

The acronym CARS denotes coherent anti-Stokes Raman spectroscopy. The technique makes use of the mixing of two laser beams and adjusting their frequencies until they generate an intense narrow beam at one of the anti-Stokes transition frequencies.

In cathodic protection a relatively valuable metallic object is protected from corrosion (oxidation) by being connected to a more electropositive metal which undergoes preferential oxidation.

A cell reaction is the chemical reaction taking place in a galvanic cell. See electrochemical cell. It is formulated by expressing the reaction at the right-hand electrode (as specified by the cell diagram M|Red_L,Ox_L||Red_R,Ox_R|M) as a reduction, Ox_R + ν e⁻ → Red_R, and subtracting from it the reduction half-reaction at the left-hand electrode, Ox_L + ν e⁻ → Red_L. The overall cell reaction Ox_R + Red_L → Red_R + Ox_L is spontaneous in the direction written if the cell potential is positive. If the cell potential is negative, the reverse reaction is spontaneous under the prevailing conditions.

A centre of inversion, i, is the symmetry element corresponding to the operation of inversion in which all points of a body are projected through a single point to the same distance on the other side. A centre of inversion is equivalent to a two-fold axis of improper rotation: i = S₂ = σ_hC₂.

A chain reaction occurs when a reaction intermediate generated in one step attacks another species to produce another intermediate, and so on. The intermediate that propagates the chain is called the chain carrier. If the chain carrier is a radical, then the reaction is a radical chain reaction. A chain reaction typically has several characteristic types of steps:

1. Initiation: the initial formation of chain carriers. If they are produced by heat, then the step is a thermolysis. If they are produced by light, then the step is photolysis. A typical initial step might be A → B + R·.

2. Propagation, a reaction that gives rise to a new carrier, as in R· + A → B + R·.

3. Branching, the formation more than one chain carrier in a propagation step, as in R· + A → R· + R·. A branching step may result in a cascade of reaction and result in a chain-branching explosion.

4. Retardation, the attack of a chain carrier on a product molecule as in R· + P → A + R·.

5. Inhibition, the removal of a chain carrier by reaction with the walls of the vessel or with foreign radicals, as in R· + M· → B.

6. Termination, the combination of chain carriers, as in R· + R· → P.

1. The condition for an integral to be nonzero. An integral $I=∫ f 1 f 2 d τ$ may be nonzero only if the integrand has the symmetry species of the totally symmetric irreducible representation.

   To employ this rule, multiply the characters of the irreducible representation spanned by each function class by class, inspect the resulting set of characters, and see if it includes the characters of the totally symmetric irreducible representation of the group.

2. Selection rules. To decide whether the integral of the form $I=∫ f 1 f 2 f 3 d τ$ may be nonzero, as in the formulation of selection rules, proceed as above for all three functions.

3. Decomposition of a product. To identify the number of times, c_Г, that a particular irreducible representation with symmetry species Γ occurs in a product given the characters χ(C), form

   $c Γ = 1 h ∑ C g ( C ) χ ( Γ ) ( C ) χ ( C )$

   In the particular case of Γ being the totally symmetric irreducible representation, Γ⁽⁰⁾ with χ(C) = 1 for all classes C, which is needed for the two rules specified above,

   $c Γ ( 0 ) = 1 h ∑ C g ( C ) χ ( C )$
4. The formulation of a symmetry adapted linear combination. To construct a symmetry adapted linear combination from an arbitrary basis, use the following projection operator, an operator that projects out from that arbitrary basis a basis that spans a specific irreducible representation:

   $P Γ = 1 h ∑ R χ ( Γ ) ( R ) R$

   where R is an operation of the group.

The charge density, ρ( r ), in an atom or molecule is defined such that ρ( r )dτ is the electric charge in the infinitesimal volume element dτ. It is related to the wavefunction for an electron by ρ( r ) = −eψ*( r )ψ( r ).

Chemical exchange occurs when an atom or group of atoms is transferred between two species. An example is the chemical exchange of a proton (a hydrogen ion) between a molecule and the solvent. Chemical exchange is detected in nuclear magnetic resonance by loss of fine structure arising from the exchanged protons.

A group of nuclei are chemically equivalent if they are related by a symmetry operation of the molecule and have the same chemical shift. They are magnetically equivalent if as well as being chemically equivalent they have identical spin–spin interactions with any other magnetic nuclei in the molecule. Chemically equivalent nuclei couple together, but due to the selection rules that coupling does not appear in an NMR spectrum (Figure C.6).

Chemiluminescence is the production of light by a chemical reaction. It occurs when a chemical reaction leads to products in excited states, which then discard their excess energy as electromagnetic radiation. If the reaction leads to vibrationally excited states then the emitted radiation is in the infrared region and the phenomenon is termed infrared chemiluminescence.

A chiral molecule is a molecule that cannot be superimposed on its mirror image. Provided it is sufficiently long lived, a chiral molecule is optically active (see birefringence). A chiral molecule and its mirror image partner constitute an enantiomeric pair and rotate the plane of polarization of electromagnetic radiation in equal and opposite directions. A molecule is chiral if it does not possess an axis of improper rotation, S_n. Note that S₁ is equivalent to a mirror plane and S₂ is equivalent to a centre of inversion, so a molecule is achiral (not chiral) if it possesses either a mirror plane or a centre of inversion. Molecules with neither are also achiral if they possess an S₄ axis.

A cholesteric phase is a liquid crystalline mesophase in which molecules lie in a helical column (Figure C.7).

Chromatography is a technique for the separation of components of a mixture that makes use of the different chemical or physical properties of the components, particularly but not exclusively their abilities to adsorb to surfaces or to dissolve in liquids. The sample is carried in a mobile phase or eluent, which may be a liquid or a gas. The mobile phase passes through a stationary phase, which is a solid or a solid coated with a liquid. Liquid chromatography is employed in a variety of forms, including column chromatography, in which the stationary phase is packed into a column, and thin-layer chromatography, in which it forms a layer on a plate. In high performance liquid chromatography (HPLC), the solid phase consists of closely packed particles less than 10 μm in diameter. The eluent is pumped through the stationary phase at high pressure and at a precisely controlled rate. Detection devices make use of ultraviolet and visible absorption, redox properties, conductivity, fluorescence, or refractive index. In gas chromatography (GC) the stationary phase may be a liquid or a liquid coated on a solid. The latter technique is called gas–liquid chromatography (GLC). The stationary phase is packed inside a capillary tube but is maintained at an appropriate temperature in an oven. Detectors make use of thermal conduction, flame ionization (the detection of ion currents following pyrolysis), electron capture by halogen atoms and the resulting modification of the conductivity of a plasma of ions and electrons, or mass spectrometry.

A chromophore is a group of atoms that is responsible for specific optical absorption of molecules and is broadly transferable between them. Two common chromophores are C═C, in which the transition is π to π* and the carbonyl group, C═O, in which the transition is n to π*, where n denotes a nonbonding orbital on the O atom.

Circular dichroism (CD) is the differential absorption of left- and right-circularly polarized electromagnetic radiation. A CD spectrum is a record of ε_L – ε_R, where ε is a molar absorption coefficient, against the frequency of the incident radiation. The technique can be used to determine the absolute configuration of d-metal complexes.

Two symmetry elements S₁ and S₂ fall into the same class if they are related by a symmetry operation S₃ of the group in the sense that S₁ = S₃S₂S₃ ⁻¹. The number of irreducible representations of a group is equal to the number of classes in the group.

A close-packed structure is an arrangement of identical spheres with the greatest possible number density. It can be modelled by laying down a layer of spheres in which each sphere is surrounded and touched by six neighbours, to give layer A (Figure C.8). Then spheres are placed in the dips formed by each triangle of spheres in layer A to give a second layer, layer B. The process is continued to give a succession of layers. The relative locations of successive layers can produce a series of polytypes, structures that differ in one dimension. An ABAB… arrangement results in a hexagonally close packed (hcp) structure. An ABCABC… arrangement, in which the centres of the spheres in layer C lie above gaps in layer A, results in a cubic close-packed (ccp) structure. The corresponding Bravais lattices are hexagonal and face-centred cubic-F (fcc). Other polytypes (such as ABABCAB…) are also possible. The coordination number, the number of nearest neighbours, is 12 in all close-packed polytypes. The packing fraction, the fraction of space occupied by the sphere relative to the total volume, is $π / 3 2 ≈0.740.$

An atom has a closed shell if all its subshells are fully occupied by paired electrons. The orbital angular momentum of a closed shell is zero. The total orbital angular momentum of an atom is that of the electrons in the valence shell, which itself is closed for noble-gas atoms and ions with noble-gas configurations.

A colligative property is one that depends on the number of solute species present but not on their chemical identity. The principal colligative properties are the depression of vapour pressure, the elevation of boiling point, the depression of freezing point, and osmosis. In each case the property arises from the modification of the chemical potential of the liquid solvent by the solute and the consequent modification of the conditions, such as the pressure or the temperature, required to ensure the equality of the chemical potentials of the solvent in the solution and its vapour or solid phase.

The collisional lifetime, τ_col, of an excited state is the lifetime of the state due to deactivating collisions with other molecules or with walls of the container. The process of de-excitation by collision is called collisional deactivation and the resulting spectral linewidth is $ΔE=ℏ/ τ col$ ⁠. If each collision in the gas phase results in de-activation, then the collisional lifetime is the inverse of the collision frequency, z.

The relation of perceived colour to the frequency of electromagnetic radiation (and its vacuum wavelength and wavenumber) is set out in Table C.1.

Additive colouration occurs when the sample emits radiation of particular frequencies. Thus, emitted blue light is perceived as blue, emitted mixed red, blue, and green light is perceived as white. An incandescent black body emits all wavelengths (see black-body radiation) and at high temperatures is perceived as white. Subtractive colouration can have a variety of effects. If red and blue are removed from white light, the object appears green; it also appears green if only red is removed, indicating that red is the colour ‘complementary’ to green. Complementary colours, a pair of colours consisting of the colour itself and the colour that white light becomes when the colour is subtracted from it, are opposite each other on a colour wheel (Figure C.9).

The scattering of light, specifically its diffraction by structures with spacing comparable to the wavelength, gives rise to rays of different colour in different directions.

The combination principle states that any line in a spectrum can be expressed as the difference of two terms. A term is a spectroscopic state such as a vibrational or electronic state. The two terms are said to combine in a transition that gives rise to a spectral line. The combination principle is a consequence of the Bohr frequency condition.

An important consequence of the existence of a commutation relation is that a pair of physical observables represented by operators that do not commute are complementary in the sense that they cannot have simultaneously exactly specifiable values (see uncertainty principle).

See Figure C.10.

The electron configuration of an atom or molecule is the specification of its occupied atomic or molecular orbitals. A single configuration can give rise to a variety of terms (see term symbol). The ground-state electron configuration of atoms can be rationalized in terms of the building-up principle.

There are other meanings of ‘configuration’ in different contexts. The configuration of a macromolecule is the specification of the sequence of monomers and can be changed only by breaking and reforming chemical bonds. The absolute configuration of a molecule is the identification of the specific enantiomer. In statistical thermodynamics, the configuration of an ensemble is a specification of the distribution of molecules over the available states in a microcanonical ensemble or of the systems over a canonical ensemble.

The upper consolute temperature (or upper critical solution temperature) is the temperature at which a two-phase liquid mixture becomes a single phase (Figure C.11). The lower consolute temperature (the lower critical solution temperature) is the temperature below which a two-phase liquid mixture becomes a single phase. Some liquid mixtures have both upper and lower critical solution temperatures and so form partially miscible solutions only within a band of temperatures.

The (Fermi) contact interaction between an electron and the nucleus is a magnetic interaction that stems from the breakdown of the point-dipole approximation for the magnetic moment of the nucleus (as suggested in Figure C.12). Only s-electrons of an atom approach the central nucleus sufficiently closely for this breakdown to be relevant. The energy of the interaction is denoted aI.s , with a proportional to ψ(0)². The contact interaction plays a role in electron paramagnetic resonance, where it is responsible for the isotropic contribution to the hyperfine structure, and in nuclear magnetic resonance, where it contributes to the mechanism of spin–spin coupling.

Standard states can be defined for any specified temperature; however, it is common for tabulations of the data to be listed at the conventional temperature of 298.15 K (25.00 °C).

A cooling curve shows the temperature of a sample as it cools. A pause in the rate of cooling occurs at a transition temperature (Figure C.13). Therefore, a cooling curve can be used to identify phase transitions and to construct phase diagrams.

Straight lines are often obtained when log K for a reaction is plotted against log k_r, where K is the equilibrium constant for the reaction and k_r is its rate constant. The implication is that the standard reaction Gibbs energy of reaction is proportional to the activation Gibbs energy. The resulting relation is called a linear free energy relation (LFER).

A correlation diagram is a portrayal of the evolution of the state of a system when a parameter is varied. Correlation diagrams are constructed by taking note of the noncrossing rule of quantum mechanics, which forbids the crossing of states of the same symmetry (Figure C.14).

Correlation spectroscopy (COSY) is a pulse technique in nuclear magnetic resonance in which the basic pulse sequence is 90°-t₁-90°-acquire(t₂), where 90° denotes a pulse of sufficient strength and duration to rotate the magnetization vector through 90° around the x-axis. A series of acquisitions is taken with variable t₁. A double Fourier transform is then taken on the variable t₂ and the interferogram arising from t₁.

According to the correspondence principle, classical mechanical properties emerge from quantum mechanical formulations in the limit of high quantum numbers.

A covalent bond, A–B, is a chemical bond formed by a pair of shared electrons. The Lewis structure is A:B. According to valence bond theory, the bond has a wavefunction of the form {ψ_A(1)ψ_B(2) + ψ_A(2)ψ_B(1)}σ(1,2), where ψ_A and ψ_B are atomic orbitals on the bonded atoms and σ(1,2) = (1/2)^1/2{α(1)β(2) − α(2)β(1)}, denoting paired electron spins. According to molecular orbital theory, a covalent bond is ψ(1)ψ(2)σ(1,2), where ψ(i) = ψ_A(i) + ψ_B(i). In each description the strength of the bond stems largely from the accumulation of electron density in the internuclear region. The term dative bond or coordinate-covalent bond is still sometimes used to denote a covalent bond in which both electrons are provided by one atom. A covalent bond is polar if there are partial charges on the participating atoms. Typically, but not in every case, the more electronegative atom carries the negative partial charge. The limit of this polar character is an ionic bond, when one atom acquires the shared pair entirely. A single bond consists of one shared pair of electrons, a double bond two shared pairs (usually a σ bond + a π bond), and a triple bond three shared pairs (usually a σ bond + two π bonds). A covalent solid is a solid in which the atoms are held together by a network of covalent bonds (diamond is an example).

The critical isotherm of a substance is its isotherm at its critical temperature (see critical point). For a van der Waals equation of state, the critical isotherm has a flat inflexion at the critical pressure.

The critical micelle contribution (CMC) is the concentration above which micelles will form provided the temperature is above a minimum temperature, the Krafft temperature.

In cyclic voltammetry, the current through a working electrode is monitored as the applied potential difference is changed cyclically at a constant rate between preset limits (Figure C.15). As the potential approaches the standard potential E ^⦵(Ox/Red) for a solution that contains the reduced components (Red) current begins to flow as Red is oxidized then falls as Red is depleted near the electrode. A similar event occurs when the potential is reversed and Ox is reduced. The forward and reverse sweeps span E ^⦵(Ox/Red). If the redox reaction is rapid, the shape of the cyclic voltammogram is independent of the sweep rate and the reaction is ‘reversible’. If the reaction is irreversible (as a result of Ox reacting to form an inert material), the cyclic voltammogram is asymmetric because Ox becomes unavailable for reduction.