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DOI: 10.1093/acprof:oso/9780198729945.001.0001

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Preamble

Chapter:
(p.1) 1 Preamble
Source:
Bonding, Structure and Solid-State Chemistry
Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198729945.003.0001

Abstract and Keywords

Chapter 1 introduces material which is developed in later chapters as the main substance of the book. It includes discussions of the atomic nature of matter, states of matter, crystalline and amorphous solids, isomorphism and polymorphism, solid-state transitions and liquid crystals. An outline of bonding in covalent, molecular, ionic, metallic and hydrogen-bonded solids is provided, together with a classification of solids in terms of these sub-divisions.

“Wherever we look, the work of the chemist has raised the level of civilization and has increased the productive capacity of our nation.”

President Calvin Coolidge

1.1 Introduction

It has been suggested that a contribution to the failure of Napoleon’s 1812 Russian campaign was linked to buttons, tin buttons that fastened the greatcoats and trousers of Napoleon’s officers and foot soldiers. When the temperature falls below $13∘C$, ‘white’ β‎‎-tin begins to transform to a crumbling powder of brittle, ‘grey’ α‎‎-tin, a process that proceeds rapidly at $−40∘C$, a temperature not unknown in a Russian winter:

(1.1)
$Display mathematics$

Were the soldiers of the Grande Armée fatally weakened by cold, because the buttons of their uniforms degraded to powder and their clothing fell apart (Fig. 1.1)? (p.2) How different might the world have been if tin did not disintegrate at low temperatures and the French had continued their eastward expansion—but the veracity of this intriguing story of failure is not without challenge.

Fig. 1.1 Napoleon’s retreat from Moscow, October 1812.

The crystal structures of these two allotropes of tin are illustrated by the stereoviews of Figs. 1.2 and 1.3. The silvery β‎‎-tin is the better known and exhibits the properties of a metal, whereas α‎‎-tin is a grey, powdery, non-metallic substance. The crystal structure of α‎‎-tin is similar in type to that of the diamond form of carbon, a cubic crystal with coordination number four, whereas β‎‎-tin forms an irregular six-coordinated solid; the transformation Eq. (1.1) will be considered again later. Appendix A1 provides information on stereoviewers and stereoviewing.

Fig. 1.2 Stereoview of the unit cell and environs of the crystal structure of β‎‎-tin, showing the distorted octahedral arrangement of six tin atoms about any other tin atom.

Fig. 1.3 Stereoview of the unit cell and environs of the crystal structure of α‎‎-tin, showing the regular tetrahedral arrangement of four tin atoms about any other tin atom; the cubic structure type is the same as that of the diamond form of carbon.

1.2 Atomic nature of matter

Notwithstanding many subatomic particles are now known, the fundamental particles of interest here are electrons, protons and neutrons. Atoms may be regarded as spherical in shape, with diameters of the order of 0.1 nm. The atomic nucleus is 1–10 fm in size1 and contains positively charged protons and uncharged (p.3) neutrons. These particles constitute most of the mass of an atom, being approximately $1.67×10−27kg$ each. The nucleus is surrounded by negatively charged electrons, each of mass approximately $9.11×10−31kg$; the number of electrons is equal to that of the protons, the atomic number, thus ensuring electrical neutrality of the atom as a whole.

Electrons may be considered as particles located mostly within regions of space surrounding the atom known as atomic orbitals; they are associated with specific energy states. In terms of an electron density around the atom, the term atomic orbital implies a one-electron wavefunction ψ‎‎ describing the electron. The product $|ψψ∗|$ may be regarded as an electron density,2 such that $|ψψ∗|dτ$ is a measure of the probability of finding an electron in a volume element dτ‎‎ (see Appendix A6) in a region around the nucleus, as will be discussed further in the next chapter. The chemistry of a substance is determined by its electrons, particularly those that are furthest from the nucleus.

Atoms in combination exist in four different states.3 Sodium chloride, for example, is encountered usually as a colourless, crystalline, ionic solid. If pure, it is a non-conductor of electricity, but at 1077 K it melts to form a colourless liquid that conducts electricity by ion transport. At the higher temperature of 1686 K the liquid boils and the vapour4 consists almost entirely of NaCl molecules in which the atoms are mainly covalently bonded, in strong contrast to the ionic nature of the substance in the solid state.

The changes $solid→liquid→gas$ occur for most substances, and at widely different ranges of temperature.

(1.2)
$Display mathematics$

In the first of these transition sequences, the covalency of the water molecule is preserved in all three phases.

1.3 States of matter

The state, solid, liquid or gas, of any substance at a given temperature is determined by the result of a competition between the interatomic forces acting on its components and their thermal energy. At equilibrium, there exists a balance between forces of attraction and repulsion that is dependent upon temperature.

Gases may be characterized by their large volume changes consequent upon variations in the temperature or pressure to which they are subjected, and by their ability to flow into the space available to them. Gases are fixed in neither shape nor volume; they are miscible one with the other in all proportions. They (p.4) differ significantly from liquids and solids, in that many of their properties are independent of their nature and respond to general laws.

A liquid is fixed in volume at a given temperature but, like a gas, has no definite form: it assumes the shape of its bounding container. Unlike a gas, the molecular species in a liquid are close enough one to the other for interatomic forces to influence their relative spatial distribution and cause local small clusters of molecules to form.

The radial distribution function g(r) of a liquid measures the probability of finding molecules of the liquid within a spherical shell of width δ‎‎r at a distance r from a reference molecule acting as an origin (see also Section 3.9.2). The following diagram, obtained by x-ray diffraction from liquid water, shows a strong peak at approximately 0.28 nm. It derives from multiple polar interactions at this distance arising from intermolecular hydrogen bonding; an average intermolecular distance in the absence of polarity is ca. 0.37 nm. However, the thermal energy of a liquid is commensurate with its intermolecular bonding energy, and order does not exist over more than a few atomic dimensions, often termed short-range order.

Radial distribution function for water based on x-ray diffraction measurements

A solid is a material of fixed volume and shape at a given temperature. In atomic terms, the mean, or equilibrium, positions of the atoms are invariant with time at a given temperature. However, the atoms are not static; they are vibrating about their mean positions, and their vibrational energy makes a major contribution to the heat capacity of the solid. The vibrations themselves are anharmonic: Fig. 1.4 illustrates the variation of the energy U of a pair of atoms as a function of their interatomic distance r. At absolute zero, the theoretical potential (dissociation) energy De at the equilibrium distance re is represented by the point A. As the temperature is increased from T1 to T2, the potential energy of the system decreases (becomes less negative) in moving from U1 to U2. Owing to the anharmonicity of the atomic vibrations, the potential energy variation is asymmetric, and the mean interatomic distance moves along the curved path $A→B$ as the temperature is increased.

Fig. 1.4 Variation in potential energy U with interatomic distance r for a pair of atoms. The curve is anharmonic and U rises steeply as r decreases below the equilibrium value re. The energies U1 and U2 may be taken to correspond with temperatures T1 and T2.

(p.5) Anharmonicity arises because at any given value of the energy U, a movement $±Δr$ from the equilibrium value re produces the steeper energy change for $−Δr$ because of electron–electron repulsion; hence, solids have very small values of compressibility.

In an assembly of atoms there are added complications. The problem must be considered in terms of free energy, which introduces an entropy factor, and it may be shown [2] that even if the atomic vibrations were harmonic an increase in free energy of the system would lead to an increase in its volume. Nevertheless, the simple potential energy curve provides a useful qualitative picture that assists in understanding physical properties of solids.

The invariance of mean atomic position with time needs further consideration. In some solids, certain groups of atoms may behave as though their symmetry were higher than that apparent from their structure.

Potassium cyanide at room temperature has the sodium chloride structure type (Fig. 1.5); although the cyanide ion is linear, it behaves as though it possessed spherical symmetry. This situation may arise in one of two ways. On the one hand the (C–N) group may exhibit dynamic disorder, performing free rotation about its mean position so that, although linear, its envelope of motion over a period of time is spherical in shape. On the other hand the group may exhibit static, or orientational, disorder, with the multiplicity of the positional distributions of the anionic groups over many unit cells again simulating a sphere. In the case of potassium cyanide experimental evidence supports the orientationally-disordered model, with the axes of the cyanide ions lying mostly normal to planes of the types (100) and (111) in the cubic unit cell.5 Thus, the mean positions averaged over time are constant at a given temperature.

Fig. 1.5 Stereoview of the unit cell and environs of the face-centred sodium chloride crystal structure type for potassium cyanide at ambient temperature; circles in decreasing order of size represent the (CN) and K+ species. Random orientations of the cyanide anions about their mean positions account for the effective spherical symmetry of the anions in the cubic unit cell.

At 233 K, potassium cyanide transforms to a polymorphic structure (Fig. 1.6) in which the cyanide ions are orientated regularly in the unit cell, lending support for the preference of orientational disorder at the lower temperatures.

Fig. 1.6 Stereoview of the unit cell and environs of the body-centred orthorhombic crystal structure of potassium cyanide below 233 K; circles in decreasing order of size represent the K, C and N species.

(p.6) 1.4 Crystalline and amorphous solids

Crystals, sometimes referred to now as classical crystals, are characterized by a three-dimensional periodicity of their structural units, atoms or groups of atoms, a feature termed long-range order. The regular appearance of a well-formed crystal (Fig. 1.7) is a manifestation of its ordered internal arrangement.

Fig. 1.7 Crystal of the hydrated silicate mineral hemimorphite, Zn4Si2O7(OH)2.H2O. The crystallographic right-handed reference axes are shown in their conventional orientation, with y and z in the plane of the diagram and x directed forwards.

Lower degrees of crystallinity are recognized: certain stretched polymer sheets exhibit two-dimensional order in the planes of the sheets, and many natural and synthetic fibres exhibit long-range order in only one dimension, that of the fibre direction. Mammalian hair contains the fibrous protein α‎‎-keratin, the structural unit of which is illustrated here:

Repeating unit of the α‎‎-keratin fibrous protein

The Ri groups are amino acid residues and other chemical groups. α‎‎-Keratin shows a periodicity to x-rays of approximately 0.52 nm along its fibre axis, but if the hair is extended in steam by 100% of its length, β‎‎-keratin is obtained, with a periodicity of 0.35 nm along its fibre axis:

Repeating unit of the β‎‎-keratin fibrous protein

(p.7) In steam the > C=O- - -H–N< hydrogen bonds in α‎‎-keratin are broken, allowing the polymer chain to extend and so develop the smaller periodicity. The $α→β$ transformation is reversible, but if the stretched hair is held in the steam for some time the reversibility is lost, a feature that is involved in the ‘permanent waving’ of hair.

Most crystalline materials are anisotropic in their physical properties, that is, the magnitude of a physical property varies with direction in the material. Stretched fibres often exhibit optical anisotropy under polarized light. However, this may not indicate crystallinity because optical anisotropy can be induced by stress in an unannealed, drawn fibre; further evidence must be sought by x-ray diffraction. A fibre of ethylene–polypropylene–diene (co)polymer when subjected to x-ray

A repeating unit of ethylene–polypropylene–diene (co)polymer

irradiation produced the diffraction diagrams in Fig. 1.8. They indicate an increase in the crystallinity of the polymer from 10% in (a) to 70% in (b), corresponding to an increase in the extension of the fibre.

Fig. 1.8 X-ray diffraction patterns from an ethylene–polypropylene-diene (co)polymer: (a) zero extension, 10% crystallinity; (b) 220% extension, 70% crystallinity. The increasing crystallinity from (a) to (b) is indicated by a beginning of the development of a spot pattern, a characteristic of all crystalline materials

[Dr EJ Wheeler. Personal communication, 1979.]

Amorphous solids, like liquids, exhibit short-range order; Fig. 1.9 shows a schematic diagram for experimental x-ray diffraction from a powdered material. The sharply defined rings characteristic of a crystalline material, of which only two are shown on the diagram, arise from an aluminium sample. The diffuse (p.8) band is from the amorphous Sellotape that is used to enclose the powder in the specimen holder. From Fig. 1.9,

(1.3)
$Display mathematics$

Fig. 1.9 Schematic diagram for the diffraction of x-rays from a powder specimen: X, incident x-ray beam; S, powder specimen; F, photographic film; L, lead trap for transmitted beam. The two sharp diffraction rings arise from the (111) and (200) planes in aluminium, and the diffuse band is the strongest diffraction spectrum from a Sellotape film enclosing the powder. The dashed lines are the traces in the plane of the 200 ‘cone’ of diffracted rays. In practice, a strip of film encompassing the central portion of the film F is sufficient, as shown in Fig. 5.35.

where $θhkl$ is the Bragg angle for dffraction from (hkl) planes in the aluminium crystals; Appendix A3 shows that

(1.4)
$Display mathematics$

where λ‎‎ is the wavelength of the x-rays and a is the dimensions of the cubic unit cell, of aluminium in this discussion. Hence,

(1.5)
$Display mathematics$

(p.9) In an actual experiment using Cu $Kα$ x-radiation of wavelength 0.15418 nm and with $R=30.00mm$, measurements on the film gave $r200=29.5mm$; thus, from Eq. (1.5), the unit cell dimension a evaluates as 0.4070 nm.

The diffuse band cannot be examined in this manner; it is the strongest part of the scattered radiation from the Sellotape, a part of its radial distribution, and it arises from a prominent interatomic spacing D given by [3]:

(1.6)
$Display mathematics$

By experiment, the mean radius of the diffuse ring was 11.5 mm; hence, from Eqs. (1.3) and (1.6), $D=0.61×(0.15418nm)$/sin{$tan−1$[(11.5 mm)/(30.00 mm)]}, which evaluates to 0.517 nm. Sellotape is a polysaccharide and a model of a 1,4-glucosidic monomer fragment, constructed from standard molecular geometry, shows an overall length of the major structural unit of 0.52 nm. Discussions on crystal geometry and x-ray diffraction may be found in the literature [4].

Repeating unit of a 1,4-glucoside monomer

As well as their different responses to x-ray diffraction, crystalline solids may be differentiated from amorphous solids by their behaviour on melting. Pure crystalline solids generally have a clearly defined melting point, with a melting range of 0.5–1 degree, whereas amorphous solids soften over a wider range of temperature before becoming fully liquid.

(p.10) The materials normally understood as crystals (classical crystals) possess an ideally infinite three-dimensional periodic arrangement of atoms, and exhibit rotational symmetry degrees R of 1, 2, 3, 4 or 6: a rotation of $(360/R)∘$ about the rotation axis brings the crystal into a position that is indistinguishable from that prior to the rotation.

In 1982, Shechtman, experimenting with an alloy of composition $Al6Mn$, obtained transmission electron microscopy photographs (Section 2.5.3) showing apparent tenfold symmetry (Fig. 1.10); however, x-ray diffraction introduces a centre of symmetry into an x-ray spectral record. Further experimentation established unequivocally the presence of fivefold symmetry in the crystalline material [5]. The crystal structure of the alloy material exhibits long-range order and is space-filling in an aperiodic manner, that is, without the three-dimensional periodicity that is characteristic of normal crystals. The aluminium–manganese alloy is one of a class of substances termed quasicrystals [6].

Fig. 1.10 Transmission electron micrograph of an $Al6Mn$ alloy surface; the apparent tenfold symmetry pattern indicates a crystalline nature for the specimen, but without the periodicity normally associated with a crystal. The true symmetry is fivefold; x-ray diffraction introduces a centre of symmetry into the pattern [4].

[Professor Daniel Shechtman, personal communication, 2013.]

The concept of aperiodic space-filling material was introduced by Schrödinger in 1944 [7]. He sought to explain how hereditary information is stored. Molecules were deemed to be too small, and amorphous solids were plainly chaotic; so it had to be a kind of crystal. Since the periodic structure of a crystal could not encode information it had to be pattern of another type, a new type of periodicity, namely aperiodicity.

Schrödinger proposed that the blueprint of life would be found in a compound with components arranged in a long and irregular sequence, but which carried information in the form of a genetic code embedded within its chemical structure. A protein was the obvious candidate for an aperiodic crystal, with the amino acid sequence providing the code. Later, the structure of DNA (deoxyribonucleic acid) was discovered and shown to possess properties similar to those predicted by Schrödinger—a regular but aperiodic structure. An on-line account of the DNA story [8] summarizes briefly events over the period 1869, when Miescher extracted DNA from white blood cell material, to 1953, when the structure of DNA was finally determined.

1.5 Isomorphism and polymorphism

The correspondence between similar crystal shape and chemical composition was demonstrated first by Mitscherlich in 1819. His law of isomorphism may be illustrated with data on the axial ratios a/b and c/b of sulphates and selenates of a number of singly-charged cations listed in Table 1.1. The results show that the components of the different crystals pack with Rb+ taking the place of the K+ cation, Se2− taking the place of the S2− anion and so forth.

Table 1.1 Axial ratios of selected isomorphous sulphates and selenates

a/b

b

c/b

K2SO4

0.573

1

0.742

Rb2SO4

0.572

1

0.749

Cs2SO4

0.571

1

0.753

Tl2SO4

0.564

1

0.732

K2SeO4

0.573

1

0.732

Rb2SeO4

0.571

1

0.739

Cs2SeO4

0.570

1

0.742

Tl2SeO4

0.555

1

0.724

On the basis of the totality of structural information now available, a wider definition of isomorphism is recognized. Chemical resemblance is not wholly essential; more significant is a similarity in the nature, size and shape of atoms and groups of atoms comprising a structure. Thus, KCl and RbCl are isomorphous in (p.11) the sodium chloride structure type (Fig. 1.5), but CsCl has a different structure (Fig. 1.11), on account of the larger radius of the Cs+ cation.

Fig. 1.11 Stereoview of the unit cell and environs of the cesium chloride crystal structure type; circles in decreasing order of size represent the Cs and Cl species. The unit cell is primitive, not body-centred.

Again, KNO3 and NaNO3 are not isomorphous, but NaNO3 is isomorphous with the calcite form of CaCO3 (Fig. 1.12). Here, the similarity of ionic radii is the controlling factor:

Fig. 1.12 Stereoview of the unit cell and environs of the calcite CaCO3 crystal structure type; circles in decreasing order of size represent the O, Ca and C species; this structure is isomorphous with that of sodium nitrate NaNO3.

 Ion Na+ Ca2+ K+ r/nm 0.112 0.118 0.144

In these examples, the shape and size of the structural units in the crystals are important factors, as well as the ionic radii themselves (see Table 4.9); both the (p.12) $(NO3)−$ and $(CO3)2−$ anions are trigonal planar in shape, with bond lengths $N−O≈C−O≈0.13nm$. The potassium cation is too large to be accommodated in the calcite structure type, but potassium nitrate is isomorphous with calcium carbonate in the aragonite structure type (Fig. 1.13). An octahedral coordination is present in the crystal structures of both KNO3 and NaNO3, but energetics determine the patterns of crystallization.

Solids that possess more than one structure type are said to exhibit polymorphism, which is often referred to as allotropy when the species involved are elements. The different structures may involve quite different interatomic forces and, therefore, show quite different physical properties, as with elemental tin (Section 1.1). Another example of allotropy is afforded by the graphite and diamond structures of carbon (see Figs. 2.49 and 2.50); graphene, a two-dimensional allotropic form carbon, is discussed in Section 6.3.4ff.

Fig. 1.13 Stereoview of the unit cell and environs of the aragonite CaCO3 crystal structure type; circles in decreasing order of size represent the O, Ca and C species; this structure is isomorphous with that of potassium nitrate, KNO3.

1.6 Solid-state transitions

Generally, polymorphs are stable over a particular range of temperature and pressure and transformations between them may take place rapidly or over a period of time. For example, at atmospheric pressure α‎‎-tin is stable below 286 K and β‎‎-tin is stable above this temperature; the $β→α$ transformation becomes more rapid as the temperature is further decreased below the transition temperature of $13∘C$.

Much of the available data on the solid state refers to ambient conditions, 293–298 K and 1 atm.6 The temperature range of availability of solids is from $−272.2K$ (He, s) to 4488 K (tantalum hafnium carbide, $Ta4HfC5$), above which latter temperature all known solids melt, vaporize or decompose.

Conditions of temperature and pressure are of importance in the study of polymorphic transitions. For example, above 500 K potassium nitrate transforms from the aragonite structure type (Fig. 1.13) to that of calcite (Fig. 1.12), so that potassium nitrate and sodium nitrate then become isomorphous under these conditions. Transitions may be classed as sharp or gradual and each type possesses specific characteristics.

(p.13) 1.6.1 Sharp transitions

A sharp transition occurs at a precise temperature and pressure, and the Clapeyron equation (1.7) is obeyed. This equation expresses the rate of variation of vapour pressure p with temperature T in terms of the changes in both enthalpy $ΔHt$ and volume $ΔVt$ at the transition temperature [9]:

(1.7)
$Display mathematics$

Many physical properties, such as density, heat capacity and entropy, exhibit a discontinuity at the transition point that is characteristic of a first-order transition, one in which the discontinuity is in the parameter itself. The variation of the density of white phosphorus with temperature is shown in Fig. 1.14; a sharp, first-order transition occurs at 317 K, when the density falls markedly but then continues to decrease at approximately the same rate as before the transition.

Fig. 1.14 Variation of density with temperature for white phosphorus P4; the sharp transition at 317 K is accompanied by a discontinuity in the density.

(p.14) When a solid undergoes a transition, the packing of its component species changes significantly and abruptly. The resultant molar volume, at constant mass, is reflected in the change in density. Generally, the solid phase of a substance is denser than its liquid form; ice is a notable exception.

In gradual transitions, the temperature of transformation is not always clearly defined and the change may take place over a considerable range of temperature. Discontinuities in physical properties are not observed, but maxima or minima occur in the temperature or pressure variations of properties such as heat capacity and compressibility. The Clapeyron equation is not obeyed, and hysteresis may occur in a physical property, such as that in the variation of heat capacity with temperature for cesium (Fig. 1.15). The gradual transition is of second order, that is, one in which the discontinuity is in the first derivative of the parameter.

Fig. 1.15 Variation of molar heat capacity with temperature for elemental cesium; hysteresis occurs between 110 K and 170 K, the lower curve indicating the path followed on cooling. The area of the hysteresis loop represents the heat content retained on cooling. Similar effects in other substances have been attributed to crystal imperfections and to surface entropy changes arising from migration of surface atoms from lattice sites to a random distribution.

(p.15) The change in heat capacity with temperature for ammonium chloride exhibits a Λ‎‎-point transition (Fig. 1.16), so-called because of the mnemonic resemblance of the heat capacity versus temperature graph to the Greek letter lambda, Λ‎‎.

Fig. 1.16 Variation of molar heat capacity with temperature for ammonium chloride; a Λ‎‎-type transition occurs at 242.5 K, from a non-centrosymmetric cubic crystal structure to the cesium chloride structure type, Fig 1.12; there is further transition to the sodium chloride structure type at 457.5 K.

Ammonim chloride has the cesium chloride structure type (Fig. 1.11); at low temperatures, the N–H bonds in the $(NH4)+$ ion are directed tetrahedrally to the same four corners of the unit cells throughout the crystal. As the temperature is increased a new structure arises, now of twice the size in order to accommodate the tetrahedral ammonium cation in both possible orientations in the unit cells. Further heating leads to the Λ‎‎-point transition in the heat capacity at 242 K. A new structure is formed having the volume of the low temperature form, but with the $(NH4)+$ ions now in twofold orientational disorder. The two possible positions of the cation are distributed randomly throughout the structure, so that the cation assumes statistically the spherical envelope required by the cesium chloride structure type; the thermodynamic properties of this transition are discussed in Section 1.6.3.

Another well-known example of a Λ‎‎-point transition is in crystalline quartz, for which the low temperature form α‎‎-quartz transforms at 846 K to the high temperature β‎‎-quartz (see Fig. 2.53). The $α→β$ transition is reversible and is accompanied by a linear expansion of ca. 0.5%; on account of this type of change, silica-containing ceramic ware must be cooled very slowly after firing in order to avoid cracking of the material.

(p.16) 1.6.3 Entropy of transition and the Boltzmann equation

The molar entropy change of ammonium chloride at the transition point is known from experiment to be $4.6J K−1mol−1$. Entropy S may be defined statistically as a measure of the disorder of a system: for example, a liquid has a higher value of entropy (greater disorder) than that of its solid phase because the components of the liquid phase exhibit only short-range order.

Entropy is related to disorder through the Boltzmann equation [9]:

(1.8)
$Display mathematics$

where W is a measure of the disorder, or probability of the system. A molar entropy change between states 1 and 2 is then

(1.9)
$Display mathematics$

For ammonium chloride $(W2/W1)$ is 2 for the disordering of the structure;7 the nitrogen atom is the central species in the diagrams of the two structures of ammonium chloride:

Disordered forms of $NH4Cl$

Hence, $ΔSm=(8.3145J K−1mol−1×0.6931/1)$, or $5.76J K−1mol−1$. The difference of ca. 1 $J K−1mol−1$ from the experimental value indicates that the disorder is probably not total throughout the crystal.

The Boltzmann distribution of energies in a system of molecules at equilibrium relates the number $nε$ of molecules of energy greater than a given value ε‎‎ to the energy ε‎‎ by the equation

(1.10)
$Display mathematics$

where n0 is a constant, equal to the number of molecules in the lowest energy state. In molar terms, Eq. (1.10) may be written as

(1.11)
$Display mathematics$

A simple derivation of this important distribution equation, used in later chapters, is given in Appendix A4.

(p.17) As the temperature of any solid is increased, the thermal motion imparted to its components increases. The consequent changes in the structure and properties of the substance depend upon the balance between this thermal energy and the internal bonding energy. Increased thermal vibrations may bring about polymorphic transformations: melting, in KCN at 233 K and in NaCN at 371 K; sublimation, in I2 at 458 K;8 decomposition, in $Pb(NO3)2$ at 743 K. In each example the entropy change $ΔS$ for the process is positive, indicating an increase in the degree of disorder in the substance arising from the transformation.

According to the third law of thermodynamics, the entropy of an infinite crystal of a pure substance is zero at 0 K, indicating a state of perfect order. This law does not preclude complete order at a higher temperature. Indeed, it is possible that it exists in a superconductor as, for example, in elemental molybdenum at 0.9 K. The ordering process near absolute zero may be more complex than that simply of progressive reduction in entropy by cooling based on the third law and may also involve magnetic interactions.

As the temperature of a solid is decreased, its atomic vibrations decrease in amplitude and rotational motion ceases until, finally at absolute zero, the species remains with only the zero-point energy of vibrational motion. The changes in atomic vibrations are accompanied by a decrease in heat capacity which, from the third law of thermodynamics, tends to zero as the temperature approaches 0 K. The loss of vibrational modes can be detected by a decrease in intensity of infrared spectra, and finally its absence.

Electrical resistivities of solids change markedly with changes in temperature: the resistivities of ionic solids tend to high values whereas those of metals tend to zero at 0 K. These effects will be discussed further after a study of ionic and metallic solids in the later chapters.

The effect of pressure on solids, though important, is often less dramatic than that of temperature. Generally, an increase in pressure on a solid produces a change in its physical properties akin to that occasioned by a decrease in temperature. On application of external pressure, a crystal may undergo a polymorphic transition that produces a closer packed structure. This process is usually accompanied by both a change in the coordination number, and a weakening in the intensity of infrared spectra occasioned by the decreased atomic vibrations. Cesium chloride, for example (Fig. 1.11), transforms to the sodium chloride structure type (Fig. 1.5) at 743 K but also at ambient temperature under a pressure of 10 GPa.

The variations in electrical resistivity differ considerably among solids within the four classifications chosen; Figs. 1.17 and 1.18 illustrate the changes in resistivity for cesium and selenium respectively as a function of applied pressure. The discontinuity in the curve for cesium occurs at a pressure of ca. 24 kbar (2.4 GPa) and corresponds to a structural change from the body-centred cubic structure, coordination number 8, to the face-centred cubic structure, coordination (p.18) number 12. In the case of selenium, a metallic structure develops at a pressure of ca.120 kbar.

Fig. 1.17 Variation of relative electrical resistivity with pressure for elemental cesium at ambient pressure; $ρ0$ is the resistivity at zero pressure. A discontinuity accompanies the polymorphic transition occurring at a pressure of 23.4 × 103 atm (23.7 kbar).

Fig. 1.18 Variation of relative electrical resistivity with pressure for elemental selenium at ambient temperature; $ρ0$ is the resistivity at zero pressure. The discontinuity at ca. 120,000 atm initiates metallic characteristics for selenium.

1.6.4 Thermodynamic properties at transition points

Thermodynamic data on a selection of substances at their transition points are listed in Table 1.2. All $ΔHt$ values are positive: liquids are in a state of higher enthalpy (and energy) than are the corresponding solids at their melting points; gases are of higher enthalpy than the vapours of their liquids at the boiling points, as Table 1.2 shows.

Table 1.2 Selected thermodynamic data at transition points

$ΔHt$/kJ mol−1

$ΔSt$/J K−1 mol−1

Mp/K

Bp/K

Fusion

Vaporization

Fusion

Vaporizationa

NaCl

1074

1738

28.5

171

26.5

98

KCl

1043

1680

25.5

162

24.4

96

BeCl2

678

793

12.6

105

18.6

132

MgCl2

985

1691

43.1

137

43.9

81

H2O

273

373

5.86

47.3

21.5

127

CH4

91

112

0.96

9.20

10.5

82

Hg

234

630

2.43

64.9

10.4

103

Ge

1210

3103

34.7

285

28.7

92

Si

1683

2953

46.4

297

27.6

101

Na

371

1165

2.64

103

7.1

88

H2

14.0

20.0

0.13

0.92

9.3

46

He

1.0b

4.2

0.021

0.084

6.3

21

(a) From Trouton’s rule.

(b) At 26 atm.

Entropy changes at transition points are positive; the degree of randomness increases in the order solid → liquid → gas. With the exception of helium, enthalpies of fusion range from ca. 0.1 to $46kJmol−1$, and the corresponding entropy changes from ca. 6 to 44 J$K−1mol−1$. For the vaporization transitions, the enthalpy change ranges from ca. 1 to $300kJmol−1$, but the $ΔSt$ values cluster around $100JK−1mol−1$ to within 20%, excluding hydrogen and helium.

(p.19) The thermodynamic free energy change $ΔG$ expresses the tendency for a reaction to take place spontaneously:

(1.12)
$Display mathematics$

A negative value for $ΔG$ in a given reaction $A→B$ indicates a spontaneous reaction in the forward direction; a free energy change represents a compromise between the normally opposing enthalpic and entropic terms in the equation. The term ‘spontaneous’ in this context does not mean necessarily that the reaction is immediate; kinetic and catalytic factors may control the rates of reactions. For example, a mixture of hydrogen and oxygen confined in darkness will not react although the free energy change is $−237.1kJmol−1$ for the reaction:

$Display mathematics$

However, in the presence of sunlight or other intense radiation the reaction takes place at an explosive rate.

In the liquid → gas transition, because of the much greater randomness of a gas compared to its liquid state, the entropy change of the reaction is dominated by the entropy of the gas. From the Avogadro law, gases have approximately equal molar volumes; hence, the entropy of vaporization tends to a constant value. This result is embodied in Trouton’s rule, which states that the entropy of vaporization at the transition point, given by

(1.13)
$Display mathematics$

is approximately $88J K−1mol−1$ for a range of different liquids. However, it is evident from Table 1.2 that Trouton’s rule is only approximate; other factors, such as hydrogen bonding or dimerization, modify the Trouton value because they influence the degree of randomness in the liquid and vapour phases.

Hydrogen-bonded acetic acid dimer: Trouton constant $62.4J K−1mol−1$

1.7 Liquid crystals

Certain organic substances when heated pass into a state that is intermediate between those of solid and liquid. Physically, they flow like liquids, yet have some properties of crystalline solids. Liquid crystals were discovered in 1888 by the Austrian botanist Friedrich Reinitzer. If cholesteryl benzoate $C34H50O2$ is heated, it melts sharply at 419 K forming a cholesteric type, opaque liquid crystal, and at 452 K there is a sudden change to an isotopic liquid.

Cholesteryl benzoate

(p.20) Liquid crystals may be considered as crystalline substances that have lost some or all of their positional order, while maintaining full orientational order. They consist usually of large elongated molecules, at least 1.3 nm in length, not severely branched or angular and possessed of polar groups such as $−NH2$, –OH or > CO and low melting points. The best materials tend to be structurally rigid and anisotropic in shape, so that the more useful liquid crystals are frequently based on benzene derivatives.

In the crystalline state, the molecules of liquid crystals are aligned parallel one to the other and bonding takes place between the polar groups and also through van der Waals (London) forces of attraction. On heating, the weaker van der Waals forces are overcome first by the thermal energy supplied to the crystal, and relative movement of the molecules can then occur. Further heating breaks the dipolar linkages and the substance passes into the true liquid state.

Several phases (mesophases) of liquid crystal are recognized (Fig. 1.19), but not all liquid crystals necessarily exhibit each phase. As well as the cholesteric phase, there is the nematic phase, taking ammonium oleate $CH3(CH2)7CH=CH(CH2)7CO2−NH4+$ as an example,

Ammonium oleate

Fig. 1.19 Liquid crystal mesophases; T1, T2 and T3 are transition temperatures; not all liquid crystals exhibit all mesophases.

in which elongated molecules are arranged parallel one to the other but without periodicity, rather like an army of descending parachutists. The smectic phase, exemplified by p-azoxyanisole $C14H14N2O3$, has its molecules arranged on equally spaced planes but without lateral periodicity,

p-Azoxyanisole

(p.21) rather like a crowd of shoppers in a department store. The transitions between mesophases are reversible and occur at definite temperatures which vary according to the Clapeyron equation (1.7).

• Liquid crystals have numerous applications in science, technology and engineering on account of both their thermochromic sensitivity and their reactions to electrical and magnetic fields. Applications of these materials, which are still being developed, have already provided effective solutions to many different problems.

• Liquid crystal displays (LCDs) find uses in watches, calculators and computer screens. An LCD consists of an array of tiny segments, or pixels, of a material that can be manipulated to present information to an observer or a detector device (LCDs are discussed further in a later chapter).

• Liquid crystal thermometers using nematic or cholesteric mesophases that reflect light with a colour that depends upon temperature find applications in thermometric devices over a wide range of temperatures.

• Optical imaging and recording technology make use of a liquid crystal between two layers of photoconducting material. On irradiation by light the conductivity of the material is increased, which develops an electric field in the crystal of a magnitude dependent on the light intensity. The electric pattern is then converted to an image which can be digitized and recorded.

• Other applications of liquid crystals include inter alia non-destructive stress analysis of materials, medical applications, computer-aided design and even the somewhat less scientific mood ring (Fig. 1.20), which changes colour in response to the body temperature and is alleged to indicate the emotional state of the wearer.

Fig. 1.20 Mood rings: a ‘stone’ is a glass or quartz shell containing a strip of thermochromic liquid crystal material. A change in temperature causes the crystalline material to deform and so bring about a change in structure which modifies the wavelength of light interacting with the crystal.

[Reproduced by courtesy of Bestmoodrings.com]

A recent innovative application in the field of nematic liquid-crystal technology incorporates strongly-emitting inorganic cluster species into a nematic liquid, in particular, the $Cs2Mo6Br14$ cluster combined with a polyanionic crown ether (CE) species. The $Cs+$ cation and the CE species are brought together to form a nematic mesophase ($2CE9⋅Cs)2Mo6Br14$. It preserves the strong red–NIR luminescent properties of the functional inorganic cluster species and the resulting liquid crystal material has important applications in temperature sensors and liquid crystal displays [10].

(p.22) 1.8 Classification of solids

No scheme of classification of solids or structures is free from some degree of ambiguity, yet it is desirable that the vast body of available structural information be discussed over a framework that groups solids according to a few chosen parameters. The method selected herein is based on the type of bonding that is mainly responsible for cohesion in the solid state. Four classes evolve; they are introduced briefly in the following subsections and will be discussed in detail in the following chapters. The subject of nanoscience has been accorded a chapter outside this classification since the materials involved therein embrace all four classes, and belong to a rapidly developing subject of academic and technological importance.

1.8.1 Covalent solids

Covalent bonding involves an electron sharing mechanism by overlap of the atomic orbitals of atoms whereby a stable electron configuration is obtained. By virtue of the nature of atomic orbitals the covalent bond is strongly directional. A necessary repulsion energy, balancing the attraction derived from electron sharing, arises mainly from the inner electron shells. A pair of electrons shared by two atoms may remain localized to the orbitals of these atoms or they may be delocalized over the complete molecular entity.

The diamond form of carbon is an excellent example of a covalent solid. Crystals of organic molecules in which the atoms are bonded covalently are sometimes quoted incorrectly as covalent solids, but the bonding responsible for their cohesion derives from dipolar and van der Waals forces, and so they belong to the ‘molecular’ class.

1.8.2 Molecular solids

While van der Waals forces make a small contribution to bonding in most solids, they are solely responsible for the cohesion between electrically neutral species, such as the inert gases, methane, benzene and other non-polar compounds in the solid state, the molecular solids. The cohesive energy in these substances arises through dipolar or induced dipolar interactions. The close-packed cubic structure of krypton, typical of the inert gases in the solid state, is illustrated by Fig. 1.21. Its unit cell dimension a is 0.5648 nm at 4 K; since the krypton atoms are in close contact along the face diagonal of the cubic unit cell of length $a2,r=a2/4$, or $0.1997nm$, which is a measure of the van der Waals radius for krypton at 4 K.

Fig. 1.21 Unit cell and environs of the crystal structure of krypton; the krypton atoms form a close-packed cubic structure which is representative of the solid inert gases and many elemental metals; eight unit cells are shown in the figure.

[Reproduced from http://www.webelements.com/krypton/crystal_structure.html by courtesy of Professor Mark Winter, University of Sheffield, UK.]

1.8.2.1 Hydrogen-bonded solids

Hydrogen bonding is discussed conveniently within the molecular group of solids; it is an important interaction in the solid and liquid states. Bonded hydrogen (p.23) atoms are able to form additional links between two atoms, most strongly with fluorine, oxygen and nitrogen, which enhance the total bonding energy. Hydrogen bonding can be either intramolecular, between atoms of one and the same molecule, or intermolecular, between atoms in different, adjacent molecules in a condensed phase. Clear evidence of hydrogen bonding can be found among the hydrides of groups 15, 16 and 17 of the periodic table, as may be judged from the example of Table 1.3. Normally, melting temperatures increase with an increase in molecular mass; exceptions may be seen along the first row of the table. Hydrogen bonds increase the attractive energy in both ionic and molecular (van der Waals) compounds. In gypsum $CaSO4.2H2O$, for example, the cohesion in one direction is governed by hydrogen bonds (Fig. 1.22). As well as hydrogen-bonded compounds, it will be convenient to include clathrate compounds, charge transfer structures, π‎‎-electron overlap compounds as well as other structure-types in Chapter 3.

Fig. 1.22 Stereoview of the unit cell and environs of gypsum CaSO4.2H2O; circles in decreasing order of size represent the O, Ca, S and H species. The hydrogen bonds, shown by double lines, are responsible for cohesion along one direction in the crystal.

Table 1.3 Melting temperature/K of hydrides of periodic table groupsa 15 (V A), 16 (VI A) and 17 (VII A)

 H3N 195 H2O 273 HF 190 H3P 140 H2S 190 HCl 159 H3As 157 H2Se 207 HBr 186 H3Sb 185 H2Te 224 HI 222

(a) Earlier group notation is given in parentheses.

1.8.3 Ionic solids

Ionic bonding involves an electron donor–acceptor mechanism among the participating atoms. A structure is formed in which ions are attracted one to the other by forces that are predominantly coulombic in nature. They are balanced by repulsion forces that increase markedly with a decrease in the interionic distance below the equilibrium value for the solid. Potassium fluoride, which has the sodium chloride structure type (Fig. 1.5), is a good example of an ionically bonded solid.

(p.24) 1.8.4 Metallic solids

In a molecule such as benzene, the valence electrons are delocalized, or distributed, over the whole molecule. In metals this delocalization is extended to the complete crystal. A regular array of positive ions is bound by an all-pervading distribution of electron density, sometimes referred to as a ‘sea’ of electrons. Because of the mobility of the outermost, conduction electrons, metals have high electrical and thermal conductivities. Gold is a good example of a metallic solid and it has the same close-packed cubic structure as that shown by Fig. 1.21.

1.8.5 Comments on the classification of solids

A schematic illustration of the four types of bonding by which solids have been classified is shown by Fig. 1.23ad. With the exception of the crystalline forms of the inert gases, solids cohere by an amalgam of bonding forces, frequently with a given compound exhibiting properties that are associated mainly with one type of idealized bonding. (p.25)

Fig. 1.23 Pictorial representations of the four main bonding forces: (a) covalent, (b) molecular, (c) ionic and (d) metallic.

[Crystal Structures: Lattices and Solids in Stereoview, 1999, Ellis Horwood Limited, UK; reproduced by courtesy of Woodhead Publishing, UK.]

(p.26) In general, it may be considered that a given bond type is disposed, conceptually, somewhere within a tetrahedron (Fig. 1.24), with each apex indicating a good example of its principal bond type. An actual bond can have the character of more than one of the four extreme types while remaining a unique chemical bond; Table 1.4 summarizes, with examples, the classification described. Fuller discussions appear in the following chapters of the book which will reveal more clearly the nature of interatomic bonding in the chosen classes.

Fig. 1.24 Schematic representation of bond type, with typical representatives in parentheses.

Table 1.4 Structural and physical properties of solids

Bonding class

COVALENT

MOLECULAR

Structure class

Ia

II

I

II

Structural units

Atoms or groups bonded covalently in three dimensions.

As for structure class I

Atoms.

Molecules.

Close packed or nearly close packed

Diamond, Si, Ge; compounds of groups 14–16 elements among themselves. Generally, hard, high mp crystals; low thermal and electrical conductivity.

Compounds of groups 14–16 metals with P, As; NiAs; pyrite.

Inert gases. Lowmp; poor thermal and electrical conductivity.

Molecular gases, e.g. N2, O2, HCl; S; organic compounds, e.g. methane, sucrose, phenol; low to moderately high mp.

Chain

Rubber, cellulose; fibrous proteins.

Layer

Graphite, graphene.

Framework

Quartz, cristobalite.

Globular proteins.

Bonding class

IONIC

METALLIC

Structure class

I

EPSb $<|zX|/2$

II

$EBS=|zX|/2$

III

$EBS>|zX|/2$

I

II

Structural units

Positive and negative ions or groups, and covalently-bonded groups carrying + or − charge.

Atoms.

Atoms.

Close packed or nearly close packed

Halides andoxides ofmetals: MX,MX2 types; spinels; perovskites.

Borates, silicates, germanates.

Salts ofinorganicoxy-acids.

True metals andtheir alloys.

Zn, Cd, Sn; alloys ofmore metallic group14–16 elements onewith the other.

Hard, brittle crystals. Poor electrical and thermal conductivity in solid; conduct electricity in the melt.

Soft to hard crystals; low to very high mp; excellent thermal and electrical conductivity.

Chain

Pyroxenes, amphiboes.

Na, Mg, Al

Se, Te, Sb2S3

Layer

Mica

Gypsum

As, Sb, Bi; MoS2

Framework

Felspars, zeolites.

Interstitial compounds.

(a) I = Class bond type predominates; II = Overlap with other areas of the classification.

(b) EBS (electrostatic bond strength) is the oxidation number of a species divided by the coordination number, and $zX$ is the change on an anionic species: NaCl, $EBS(Na−Cl)$=1/6; $|zx|=12$.

References 1

Bibliography references:

[1] Eliezer S and Eliezer Y. The Fourth State of Matter: An Introduction to Plasma Science, 2nd ed. Taylor and Francis, 2001.

[2] Fowler RH and Guggenheim EA. Statistical Mechanics, 2nd ed. Cambridge University Press, 1960.

[3] James RW. The Crystalline State. Vol. 2: The Optical Principles of the Diffraction of X-rays from Crystals. G. Bell and Sons Limited, 1948.

[4] Ladd M and Palmer R. Structure Determination by X-ray Crystallography, 5th ed. Springer Science+Business Media, 2013.

[5] Shechtman D et al. Phys. Rev. Lett. 1984; 53: 1951.

[6] Senechal M. Quasicrystals and Geometry. Cambridge University Press, 1996.

[7] Schrödinger E. What Is Life. Cambridge University Press, 1944.

[8] The DNA story. Chem. World, 2003; http://www.rsc.org/chemistryworld/Issues/2003/April/story.asp

[9] Atkins PW and de Paula J. Atkins’ Physical Chemistry, 9th ed. Oxford University Press, 2009.

[10] Nayak SK et al. Chem. Commun. 2015; 51: 3774,

[11] Wyckoff RWG. Crystal Structures, Vol. I, 2nd ed. Wiley, 1963.

[12] Wyckoff RWG. The Structure of Crystals, Reinhold, 1935; on-line as The Structure of Crystals. https://archive.org/details/structureofcryst030914mbp

Problems 1

1. 1.1. At 420 K, ammonium nitrate has been found to exhibit the cesium chloride structure type. What structural implication may be associated with this observation?

2. 1.2. In an x-ray diffraction experiment, similar to that described in Section 1.4, but with ammonium nitrate at 420 K, the first two sharp diffraction rings had diameters of 22.1 mm and 32.7 mm respectively. The indices of these spectral rings were 100 and 110, and the specimen to film distance was (p.27) 30.00 mm. If the x-ray wavelength was 0.15418 nm, calculate an average value for the unit cell dimension a.

3. 1.3. Which of the following nine pairs of substances illustrate structural isomorphism?

 NaCl KCl BaSO4 PbSeO4 CaF2 $β−PbF2$ RbCl CsCl SrSO4 CaSO4 NaBr MgO KNO3 CaCO3 (calcite) CaF2 MgF2 CaO BeO

Publications by R. W. G. Wyckoff may be helpful in this problem [11; 12].

4. 1.4. The entropy of vaporization of BeCl2 is 36 J K−1 mol−1 greater than that for KCl. What does this result suggest about the liquid state of BeCl2?

5. 1.5. The enthalpies of combustion of orthorhombic α‎‎-sulphur (the thermodynamically stable form) and monoclinic β‎‎-sulphur to form SO2 at 298.15 K are $−297.0kJmol−1$ and $−297.3kJmol−1$ respectively. Draw an enthalpy-level diagram to illustrate the changes involved. Assuming that the enthalpies of α‎‎-sulphur and β‎‎-sulphur do not change between 298 K and 386 K, calculate the entropy of β‎‎-sulphur at the transition temperature of 386 K given that $Sα=31.73J K−1mol−1$.

6. 1.6. Classify the following 12 substances according to the principal type of bonding type responsible for cohesion in the solid state.

 RbF CO2 C6H6 Cu3Au P4 AlN Ne Na2SO4 P2Cl10 Pb KClO3 SiC

7. 1.7. What are the Miller indices of crystal planes that make the following intercepts on the x, y and z crystallographic axes?

 (i) a/2 b –c/3 (ii) a ‖ to b 2c/3 (iii) –a/3 ‖ to b ‖ to c (iv) a $−2b/3$ $3c/4$ (v) ‖ to a $b/2$ $−3c/4$ (vi) ‖ to a –b 2c

8. (p.28) 1.8. Consider the close-packed, cubic structure of aluminium (the same structure type as that in Fig. 1.21). Using the information in Section 1.4, what is the diameter of the first sharp diffraction ring on Fig. 1.9?

9. 1.9. Iron transforms from a body-centred cubic structure to a face-centred cubic structure at 812 °C. If the atoms of iron behave as hard spheres of radius r, what is the percentage change in packing fraction accompanying the transition? (The packing fraction is the volume of an atom divided by the volume occupied by that atom in the structure.)

10. 1.10. The unit cell dimension for krypton (Fig. 1.21) at 4 K is 0.5648 nm. What is its density at 4 K?

Notes:

(1) 1 fm = 10−15 m.

(2) ψ‎‎* is the complex conjugate of ψ‎‎; $ψψ∗≡|ψ|2$ if the electron density function is real.

(3) Plasma, the fourth state of matter, is not discussed herein [1].

(4) A vapour is a gas below its critical temperature.

(5) Miller indices (hkl) are discussed in Appendix A2.

(6) 1 atm = 101,325 Pa (Nm−2).

(7) W1 represents the state of no disorder.

(8) At a partial pressure less than 90 mmHg (ca. 12,000 Pa).