Introduction to Solid State Chemistry

By James E. House

Introduction to Solid State Chemistry provides a strong background to the structures of solids and factors that determine this structure. The content presented will also stress transformations of solids both in physical forms and chemical composition. In so doing, topics such as phase transitions, sintering, reactions of coordination compounds, photovoltaic compounds are described, whilst kinetics and mechanisms of solid state reactions are covered in depth.

There are currently few books that deal with solid state chemistry, where a considerable number instead deal with solid state physics and materials science/engineering. This book provides someone needing or wishing to learn about the chemistry of solids a comprehensive resource that describes structures of solids, the behaviour of solids under applied stresses, the types of reactions that solids undergo, and the phenomenological aspects of reactions in solids. Kinetics of reactions in solids is very seldom covered in current literature and an understanding of the mechanisms of reactions in solids is necessary for many applications.

James E. House provides a balanced treatment of structure, dynamics, and behaviour of solids at a level commensurate with upper-level undergraduates or beginning graduate students who wish to obtain an introduction and overview to solid state chemistry.

• Provides a?fundamental introduction and entry point to solid state chemistry, acting as a useful prerequisite for further learning in the area
• Presents a balanced approach that not only emphasizes structures of solids but also provides information on reactions of solids and how they occur
• Gives much-needed focus to the kinetics of reactions of solids and their mechanisms where existing literature covers little of this
• Explores crucial solid state chemistry topics such as solar energy conversion, reactions of solid coordination compounds, diffusion, sintering, and other transformations of solids
• Features accessible and well-written examples and case studies featuring many new and bespoke supporting illustrations, offering an excellent framework that will help students to understand reaction mechanisms

• Language: English
• Publisher: Elsevier Science
• Release date: Feb 21, 2024
• ISBN: 9780443134296

James E. House

J.E. House is Scholar in Residence, Illinois Wesleyan University, and Emeritus Professor of Chemistry, Illinois State University. He received BS and MA degrees from Southern Illinois University and the PhD from the University of Illinois, Urbana. In his 32 years at Illinois State, he taught a variety of courses in inorganic and physical chemistry. He has authored almost 150 publications in chemistry journals, many dealing with reactions in solid materials, as well as books on chemical kinetics, quantum mechanics, and inorganic chemistry. He was elected Professor of the Year in 2011 by the student body at Illinois Wesleyan University. He has also been elected to the Southern Illinois University Chemistry Alumni Hall of Fame. He is the Series Editor for Elsevier's Developments in Physical & Theoretical Chemistry series, and a member of the editorial board of The Chemical Educator.

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Preface

Chemistry is the study of matter and the changes it undergoes. Therefore, solid state chemistry should be the study of solids and the changes they undergo. In this book, emphasis will be placed on such topics. However, in the study of properties and reactions of solids, any separation of the fields of chemistry, physics, and materials science is essentially arbitrary. For example, the behavior of solids exposed to light may involve chemical composition, the physical action of electrons, or the materials science related to structure or dopants. The field of solid state chemistry is vast, and there are many interesting subareas and topics. Therefore, an attempt has been made to strike a balance of information that shows some of the relationships.

Reactions of solids have been important since very early times. For example, lime that is used in mortar, concrete, steel production, glass, and other uses is obtained by the decomposition of calcium carbonate. Alloys have been used since at least the Bronze Age. The processes of diffusion and sintering are vital to industries involved in powder metallurgy and the production of ceramics. However, courses in undergraduate chemistry curricula do not, in general, include discussions of these topics and numerous others related to the behavior of solids. Accordingly, the purpose of this book is to provide a teaching resource for students and teachers who wish to obtain an understanding of many aspects of solid state chemistry. It is not intended to be a review of the enormous body of literature on solid state chemistry. Rather, it is intended to present an overview of some of the basic principles of solid state chemistry and to amplify those topics with illustrations from the literature.

Although a great deal of original literature is cited, this book is intended to serve as a teaching tool. Most chapters begin with a discussion of basic principles related to the topic, followed by more specific case studies showing applications. In some areas, those applications have involved the use of simple equipment and techniques that reveal a great deal about solid state processes. Some of that work was done by undergraduate students, and it is to be hoped that users of this book may be inspired to try simple techniques in the study of solids. Moreover, the development of numerous topics is presented in such a way as to develop intuitive approaches to making correlations and predictions.

It should be a pleasant experience to write a book. For this author it is, and in writing this book, the pleasure has been afforded once again. In large measure, the pleasure is due to working with Charles Bath and Lena Sparks of Elsevier. Their constant encouragement and expert guidance are acknowledged and greatly appreciated. In the preface to the 6th edition of his monumental book, Introduction to Solid State Physics (Wiley, New York, 1986), Professor Charles Kittel has the statement, I wrote parts of this edition in the country with a word processor powered by photovoltaic panels, as appropriate to the solid state revolution. Writing is an art form, and many sections of this book were first prepared the old-fashioned way with a classic fountain pen and ink in campgrounds in Wyoming and Montana. As stated by Alessandra Elia, Writing is such an intimate gesture….. [Alessandra Elia, Director of Writing Culture at Montblanc, https://www.youtube.com/watch?v=aYUMQ9l7PHM&t=610s (Accessed July 23, 2023).] Writing as I did brought back memories and provided new ones while using some of my pen collection, which has been expanded thanks to Leena Shrestha Menon at The Pen Boutique and Brian Anderson of Anderson's Pens. Finally, the author would like to acknowledge the support and patience shown by his wife, Kathleen, during the preparation of this book, without which the experience would not have been so pleasant. Her careful reading of the manuscript resulted in numerous suggestions for improvement and helped to eradicate many errors that resulted from a rebellious keyboard refusing to follow the script.

A note on units

In publications that deal with various aspects of solid state topics, a variety of units are used. If the event is associated with an atomic or molecular species, the electron volt (eV) is commonly used. For chemical reactions, energy changes are often expressed as kJ mol-1 or kcal mol-1. A great deal of information is included in this book from original publications. In presenting that information, the units included are those in which the data were originally published. This situation will be encountered by people who consult the original literature.

Chapter 1 Energy and space factors in ionic crystals

Abstract

Solid-state chemistry begins with the structure of solids. Predictions of crystal structures are illustrated based on ionic radii and energy. The significance of the lattice energy is described, and the relationship of lattice energy to properties such as solubility, proton affinity, and hardness is described. Important structures such as sodium chloride, cesium chloride, fluorite, rutile, wurtzite, zinc blend, and others are described. The concept of electrostatic bond character is used to predict the number of nearest neighbors in crystal lattices. Hardness of solids is considered as a consequence of the bonding modes.

Keywords

Lattice energy; Born–Mayer equation; Kapustinskii equation; Madelung constant; Solubility; Heat of solution; Hardness; Electrostatic bond character

Although numerous types of solids will be discussed throughout this book, many compounds exist in the form of ionic crystals, so this is an appropriate place to begin the discussion of solids. Ionic crystals are held together by electrostatic forces between ions. To form ions, electrons must be removed from atoms, which is a process that requires energy. The energies required to remove electrons from gaseous atoms are referred to as ionization potentials. Even to get an atom into the gaseous state requires energy, but other energies are involved in forming an ionic compound. In spite of these factors, many ionic crystals are very stable, rigid, and solid structures. This chapter will be devoted to an examination of the energies involved in the formation and properties of ionic crystals and how the sizes of the ions are related to the structures of the solids.

In general chemistry courses, students are taught that solids have rigid structures that give them fixed shape and volume. However, that is only partially true because some solids do change shape or structure, at least at the atomic or ionic level such as sulfur changing from a monoclinic to a rhombic structure (a phase change). Such changes also occur for some ionic solids. Generally, phase changes are the result of changing conditions such as temperature and/or pressure. This chapter is concerned with the basic principles that relate to energy and size factors of ionic crystals.

1.1 Bonds between atoms

In general chemistry and other courses, the concept of chemical bonding is explained in terms of what happens to electrons when atoms combine. In simplest terms, ions result when electrons are transferred from one atom to another resulting in charged species or ions. Covalent bonds are described as cases in which electrons are shared by the two atoms. There are cases in which electrons are shared equally as in the case of homonuclear diatomic molecules such as F2, H2, and N2. In these cases, the two atoms in the molecule are identical and have the same attraction for electrons. This is not the case when the two atoms are different. The ability of an atom in a molecule to attract electrons is known as its electronegativity. This property of atoms and how it varies with position in the periodic table should be familiar to readers of this book, but if not, a general chemistry text can be consulted for a review. There are several numerical scales that represent the electronegativity of atoms, the most commonly encountered being that known as the Pauling scale in which the electronegativity of fluorine has the value 3.98. In that way, the electronegativities of all atoms are positive values between 0 and 4. The values are based on bond energies that have both covalent and ionic character that are estimated from heat of formation data. Generally, electronegativities of atoms decrease in a fairly predictable way. The values increase for atoms to the right in a given period (Be, 0.98 and F, 3.98) and decrease lower in a given group (F, 3.97 and I, 2.66).

Bonding between atoms can be considered to be partially ionic and partially covalent depending the degree to which electrons are transferred and shared. That ratio will depend on the electronegativities of the atoms and the greater the difference, the more nearly the bond approaches one that is covalent. The percent of ionic character to a bond between two atoms A and B is commonly approximated by the empirical equation:

Equation

   (1.1)

in which χ represents the electronegativity value. Thus, when the atoms are identical, the difference in electronegativity is zero, and the bond is considered to be covalent. The percent ionic character corresponding to a difference in electronegativity of 2.1 corresponds to a bond that is almost exactly 50% ionic and 50% covalent in character. On the other hand, bonding between Fr (0.78) and F (3.98) would be slightly over 90% ionic. Fig. 1.1 shows a general relationship between percent ionic character and the difference in electronegativity between atoms.

Fig. 1.1

Fig. 1.1 The relationship of percent ionic character to electronegativity difference. Note that for electronegativity differences from approximately 1 to 2.5, the relationship is almost linear.

The gradual transition in bond character can be represented in terms of a triangle that is sometimes known as a bond or Ketelaar–Van Arkel triangle. This graphic representation is shown in Fig. 1.2. There are, however, many other forms of such triangles, some of which are based properties other than electronegativity (1–3).

Fig. 1.2

Fig. 1.2 A qualitative diagram showing the transition between bonding types. The positions of formulas are only qualitative to show the transition between the limiting types of covalent, ionic, and metallic.

This chapter is devoted to a discussion of bonds that are predominantly ionic in character and the structures that result from arranging them. However, as will be shown later, there is a substantial covalent contribution to the bonding in crystals such as silver halides. Metals generally have low electronegativities, so the bonding is covalent. However, the bonds in metals involve many electrons in energy bands or levels that span many atoms. The bonding and structure of metals will be described in Chapter 3.

1.2 Energy considerations and Madelung constants

The formation of an ionic bond as in the case of forming NaCl from sodium and chlorine is not as simple as removing an electron from a sodium atom and placing it on a chlorine atom to produce ions. However, an important part of the process involves the electrostatic interaction of the ions after they are produced. Even though the formation of NaCl(s) from the elements is energetically favorable,

Equation

   (1.2)

an electron does not simply jump from a sodium atom to a chlorine atom. Eq. (1.2) does not give any information about how the reaction occurs.

The interaction of charged ions is governed by Coulomb's law,

Equation    (1.3)

where F is the force between the ions, q1 and q2 are the charges on the ions, r is the distance of separation between the ions, and ɛ is the dielectric constant of the medium separating the ions. For a vacuum (the so-called free space), the value of ɛ is considered to be 1. The internuclear distance between Na+ and Cl− in sodium chloride is 279 pm (2.79 Å). If a mole of Na+ and Cl− ions form a mole of ion pairs having this distance of separation, the energy released is −439 kJ.

Important questions to be answered are: how does the reaction occur and what energies are involved? To produce the Na+ ion, an electron must be removed, and the energy necessary to remove an electron from an isolated, gaseous atom is the ionization energy. To produce the Cl− ion, an electron must be added to a chlorine atom, and the energy released when an electron is added to a gaseous atom is the electron addition energy (usually negative), but the energy to remove the electron after it is added is known as the electron affinity. Note that the energies for electron transfer from Na to Cl are for the gaseous atoms. In order to make use of the known values for these thermodynamic quantities, the atoms involved must be in the gaseous state. A thermodynamic cycle, known as a Born–Haber cycle, can be constructed as shown in Fig. 1.3.

Fig. 1.3

Fig. 1.3 A Born–Haber cycle for the formation of NaCl(s) from the elements. In this cycle, S is the sublimation enthalpy of sodium, D is the dissociation enthalpy of Cl 2 , I Na is the ionization enthalpy of sodium, and EA Cl is the electron affinity of chlorine. Note that the steps that lead to the formation of NaCl(s) are steps involving atoms and molecules for which the energies are known. Note that the solid sodium must be vaporized, and the diatomic chlorine molecules must be separated into atoms.

For a thermodynamic cycle, the enthalpy change is a thermodynamic state function so the overall change is the same regardless of the path if the initial and final states are the same. In the Born–Haber cycle, metallic sodium and gaseous chlorine are first converted into gaseous atoms. This requires the sublimation energy, S, for sodium and one-half the dissociation energy for molecular chlorine to produce a mole of atoms. Next, an electron is removed from the sodium atom (which requires addition of the ionization energy, I) and the electron is added to the chlorine atom (which releases the energy equivalent to the electron affinity, EA). At this point, the gaseous Na+ and Cl− ions are produced, but at the expenditure of energies represented as S, ½ D, I, and EA the values of which are 109, 121, 496, and −349 kJ mol−1, respectively. The overall enthalpy change to produce the ions is +315 kJ mol−1. The enthalpy of formation of NaCl(s) is −411 kJ mol−1 as a result of the ions interacting after they are formed. The energy necessary to separate a mole of crystal into the gaseous ions is the lattice energy for the crystal, which is represented as U in the Born–Haber cycle. Thus, from the cycle shown in Fig. 1.3 we can write

Equation

   (1.4)

From this equation, a value of 788 kJ mol−1 is obtained for the lattice energy for sodium chloride. Note that the formation of NaCl from the elements is energetically feasible only as a result of the interaction of the ions that are produced. The simple transfer of an electron from one atom to another is not energetically favorable even for Cs (I = 374 kJ mol−1) and Cl (EA = 349 kJ mol−1).

The question now arises with regard to how the reaction between sodium and chlorine takes place. The Born–Haber cycle is constructed to make use of atomic properties for which the enthalpies are known, but it does not necessarily indicate that these are the actual steps followed in the formation of NaCl(s). However, it does not invalidate the calculated value for the lattice energy because by Hess’ law if a process takes place in a series of steps, the overall enthalpy change is the sum of the enthalpies for the individual steps, and because a cycle is completed, the enthalpy change is the same regardless of the path. The reason for considering the steps indicated is because those are steps for which the energies are known. It would not be useful to devise a scheme involving steps for which the energies are unknown quantities. Tables 1.1 and 1.2 show atomic properties that can be used for calculations dealing with Group I halides.

Table 1.1

a Ionization potential of the metal.

b The heat of sublimation of the metal.

Table 1.2

Although Born–Haber cycles can be used to determine lattice energies when the required thermodynamic quantities are known, there is another important use of similar cycles. Lattice energies have been determined for many compounds by experimental techniques, but electron affinities of atoms are often not determined experimentally, but rather they are calculated. Thus, in such cases, the unknown quantity in Eq. (1.4) becomes the electron affinity, and some values reported for electron affinities are known only as calculated values.

Because the formation of solid ionic compounds is favorable only when the ions form a lattice, it is important to understand how the energy released is related to crystal structure. A quantity known as the Madelung constant takes into account the interaction of ions in a crystal based on their arrangement, charges, and distances of separation (4).

In order to show the basis for how a Madelung constant is involved, and to show how it is determined, a simple example will be used. This example is not based on an actual crystal structure, but it suffices to illustrate the procedure, which is much more complicated for real crystals (5).

It will be shown later that if the formation of a mole of ion pairs is considered, the energy released would −439 kJ mol−1. However, instead of forming ion pairs, let us assume that a mole of Na+ and Cl− ions form a chain as shown in Fig. 1.4.

Fig. 1.4

Fig. 1.4 A hypothetical chain of sodium and chloride ions. The test ion to be considered as the test charge in the electrostatic field produced by all the other ions is indicated with a star.

For a charge, q, interacting with an electrostatic field of strength V, the energy of interaction is related by the product of the charges, in this case −Vq. To determine the energy for the assembly of ions shown in Fig. 1.4, the energy can be determined by calculating the field strength at the ion identified with a star (the test ion which has a charge of +e) and multiplying the charge on that ion by the electric field strength at the point where it is located. To calculate the electric field strength at the starred ion, we begin there and work outward considering that ɛ = 1. It can be seen that there are two chloride ions at a distance of r from the test ion that contribute −2e/r to the field strength. Working farther outward from the test ion, two positive ions are encountered at a distance of 2r, and they contribute +2e/2r to the field strength at the starred ion. Next, there are two negative ions at a distance of 3r from the test ion that contribute −2e/3r to the field strength. Continuing in this way, the total field strength at the test ion can be shown to be represented by the series

Equation

   (1.5)

When the quantity –e/r is factored from each term in the series, the result is

Equation

   (1.6)

The series inside the parentheses is a converging series that gives a sum that is equal to 2 ln 2 or 1.38629. This would be the Madelung constant for a linear chain of Na+ and Cl−. Earlier it was shown that the formation of a mole of ion pairs is energetically favorable with the value being −Noe²/r = −497 kJ mol−1, but the interaction of a chain of ions would give an energy of −1.38629Noe²/r. Thus, the chain is more stable than isolated ion pairs by a factor of 1.38629. However, sodium chloride exists in a three-dimensional lattice so that must represent a lower energy than does a chain. However, determining the contribution of ions in a three-dimensional lattice to the electric field strength at a particular point is a difficult process. The simple model described illustrates a general approach to determining Madelung constants for arrangements of ions.

The series of terms in the electrostatic potential for the linear example illustrated above leads to a recognizable sum. However, in general that would not be the case so that a summation process would be needed. The sum of first term is 2, and the sum of the first two terms is 2–1 = 1. The sum of the first three terms is 2–1 + 2/3 = 1.667, etc., and the sum of the partial sums of the first five terms can written as shown in Table 1.3 in which A1,2,3… represent the terms in the series, and A1,2 represents the sum of the first two terms, A1,2,3 represents the sum of the first three terms, etc. In the averaging process to arrive at an approximate sum, averages are obtained between adjacent partial sums, then those averages are averaged, etc., until only one value remains. That value is an approximate value for the sum of the series.

Table 1.3

Using the values for the first five terms in the series, a value of 1.3896 is obtained as the sum of the series. The actual value for 2 ln 2 is 1.38629, so the series has converged rapidly. It must be emphasized that the process in this case is simple, but for a three-dimensional crystal, it is by no means as simple.

As mentioned previously, the situation is much more complicated for a three-dimensional crystal of sodium chloride. The problem is to determine the electric field strength at the test ion. Although procedures have been developed, they are beyond the scope of this discussion (6).

Consider one layer of ions that constitutes a two-dimensional layer in a crystal as shown in Fig. 1.5.

Fig. 1.5

Fig. 1.5 A layer of ions arranged in the sodium chloride pattern. The starred ion is considered to be the test ion at which the field strength is to be determined.

The layers above and below the layer shown will contain ions of opposite charges so that directly above and below the starred cation will reside chloride ions. The nearest neighbors to the starred ion are six chloride ions at a distance of r. These six ions contribute −6e/r to the electrostatic potential. Working outward from the starred ion are four cations in the plane at a distance of r√2 from the starred ion. However, there will be four others above and below the nearest neighbor anions. Therefore, there will be 12 cations at a distance of r√2 from the starred ion that contribute 12e/r√2. The layers above and below the one shown each have four anions that are at r√3 from the starred ion, and they contribute −8e/r√3 to the field strength. In the two layers above and below the one shown, the arrangement of ions are like that in Fig. 1.5. Therefore, there are six cations at a distance of 2r from the starred ion. Working outward from the starred ion leads to an electrostatic potential that can be written as

Equation

   (1.7)

Removing the factor –e/r from each term leads to

Equation

   (1.8)

Unlike the series that resulted for a linear chain of ions, the series shown in the parentheses in Eq. (1.8) does not converge very quickly, nor does it converge to any identifiable numerical sum. Although the procedure will not be illustrated here, the series converges to a value of 1.74756 (7).

This value is approximately equal to the ratio of the energy released (equal to the lattice energy in magnitude) when a mole crystal is formed to that when a mole of ion pairs form. In fact, the Madelung constant is precisely that ratio, and it takes into account all of the interactions in the three-dimensional crystal.

Madelung constants have been determined for numerous types of crystal structures most of which will be shown later in this chapter. However, the values are difficult to calculate, so they will be tabulated here without any attempt being made to determine the values. Madelung constants for several common crystal types are summarized in Table 1.4.

Table 1.4

a Rutile, TiO2, and fluorite, CaF2, have twice as many anions as cations. The factor of 2 is not included in the numerical value shown. As a result, some sources give the values for these crystals as twice the value shown.

The interaction of ions having charges of +1 and −1 results in an electrostatic interaction energy that can be expressed as

Equation    (1.9)

In this equation, e is the charge of an electron, r is the distance of separation, and ɛ is assumed to be 1. For Avogadro's number of ions of each type, the energy released will be No times as great or

Equation    (1.10)

The charge on an electron (the charges on the ions) is 4.8 × 10−10 esu and 1 esu = 1 g½ cm³/² s−1. If the Na+ and Cl− ions have the same distance of separation as in the crystal, 2.79 Å (279 pm), the energy of interaction of a mole of ion pairs is given by

Equation

   (1.11)

From the Born–Haber cycle a value of 788 kJ mol−1 was obtained for the lattice energy of crystalline NaCl. Therefore, the ratio of the energy released when a mole of crystal of NaCl forms from gaseous ions to that when a mole of ion pairs form is 788/497 = 1.59. The ratio should be equal to the Madelung constant, 1.74756, so other factors are at work.

However, if the value for energy released when Na+ and Cl− ion pairs form is multiplied by 1.74756, the Madelung constant for the sodium chloride lattice, the result is 869 kJ mol−1 rather than the actual lattice energy of 788 kJ mol−1. We need to explain why the actual lattice energy is lower than that calculated in this way. The answer lies in the fact that although the equation

Equation    (1.12)

should give the energy released when a mole of ion pairs form, the interacting species are ions rather than point charges.

There are 10 electrons remaining in a sodium ion, and a chloride ion contains 18 electrons. Therefore, as the ions approach each other, the attractive force is the dominant force, but as the ions get close together, the electrons in the ions repel each other with a force that increases as the ions get closer together. Therefore, the equation for the calculation of lattice energy should include a term that takes repulsion into account. The repulsion in inversely related to the distance of separation, and it increases rapidly as the distance between ions gets smaller. A term that represents such repulsion, R, can be written as

Equation    (1.13)

where n and B are constants, and r is the distance of separation of the ions. This equation shows the inverse relationship of repulsion to R, but the value for n depends on the number of electrons in the ions. It has been found that values of n can be assigned that are approximately 5, 7, 9, 10, or 12 for simple ions having the electron configurations of He, Ne, Ar, Kr, or Xe, respectively. For a crystal composed of Li+ (He configuration) and F− (Ne configuration), an average value for n of 6 can be used. Including the repulsion term, the equation for lattice energy can be written as the sum of the attraction and repulsion terms as

Equation    (1.14)

The quantities represented in this equation are thus known except for B. However, the attractive force greatly exceeds the repulsive force in crystals so the potential energy becomes more negative as the value of r decreases. At some most favorable distance between the ions, the sum of the attractive and repulsive forces reaches a minimum as the distance decreases. This situation is illustrated by the potential energy diagram shown in Fig. 1.6.

Fig. 1.6

Fig. 1.6 A potential energy curve for the interaction of positive and negative ions as a function of distance. At the minimum in the curve, the attraction energy is n times the repulsion.

A curve such as that shown in Fig. 1.6 has an inflection point at the minimum where the slope changes sign. At that point, the slope of the curve represents the minimum in U. Therefore, to find the energy at which this occurs, the derivative dU/dr is obtained and set equal to zero. The result is shown in Eq. (1.15).

Equation    (1.15)

It is now possible to solve this equation for B to obtain

Equation    (1.16)

Although a negative value for the energy is obtained when calculated from this standpoint, the lattice energy is defined as the energy required to separate a mole of crystal into the gaseous ions (a positive quantity). The equation obtained after substituting the value of B in Eq. (1.14) and simplifying is

Equation    (1.17)

This equation, known as the Born–Landè equation, can be used to calculate lattice energies in many cases for which the Madelung constant, r, and n are known. For cases in which the ions are multiply charged, factors Zc and Za must be used for the charges on the cation and anion, respectively, instead of simply representing the charges as e. Although it is sometimes stated that at the equilibrium distance the forces of attraction and repulsion are equal, that is not true. It can be seen from Eq. (1.17) that the attraction is actually n times the repulsion. In Section 1.4, the topic of lattice energies will be revisited.

1.3 Ionic sizes and crystal environments

To the uninitiated, it may sometimes seem that crystals are arranged with haphazard structures of ions. However, this is not the case, and in 1929, Linus Pauling published an article that gives a set of very important rules that pertain to crystal structures (8). The first of these rules has to do with the stable arrangement of ions based on the relative sizes of the anions and cations as discussed here. In a crystal, if the ions touch each other, attraction will arise from the interaction of the positive and negative ions. If the anions are larger (as is usually the case), they may touch before they make contact with the positive ions unless they are arranged in a specific pattern. Consider the arrangement of ions shown in Fig. 1.7.

Fig. 1.7

Fig. 1.7 A cation surrounded by four anions in a plane with another above and below the plane giving an octahedral arrangement of anions around the cation. The anions touch each other as they make contact with the cation. In this case, the cation resides in an interstitial space referred to as an octahedral hole.

A stable arrangement of the ions shown in Fig. 1.7 is determined when the anions all contact the cation. By considering the geometry of the ions shown in Fig. 1.7, it is possible to calculate how large the cation must be relative to the anions for the arrangement to be stable with the anions all contacting the cation. In this approach, the ions are considered to be hard spheres. From the figure it can be seen that the four anions shown lead to a value of θ = 45° between the line showing the sum of the ionic radii, S, and the line showing the radius of only one anion. Therefore,

Equation

   (1.18)

This equation can be solved for rc/ra to obtain a value of 0.414 as the minimum size of the cation relative to that of the anion (known as the radius ratio) in order for the anions to make contact with the cation. If the cation is smaller than 0.414 ra, the anions make contact with each other but not with the cation, and the arrangement should not be stable.

Similar calculations can be performed with a cation surrounded by other geometrical arrangements of anions. For example, if a cation resides in a hole generated by four anions in a tetrahedral arrangement, it can be shown that the radius of the cation must be at least 0.225 that of the anion for the anions to make contact with the cation. Otherwise, the anions make contact only with each other. From the results of these two examples, it is apparent that for 0.225 < rc/ra < 0.414, a tetrahedral arrangement of anions around a cation should be stable. The results of these and other arrangements are shown in Table 1.5.

Table 1.5

afcc and hcp are face-centered cubic and hexagonal close packing of spheres (see Section 3.1)

A word of caution is in order with regard to predicting crystal structures based on ionic radii. First, ions do not behave exactly as hard spheres, and they may be subject to some distortion or polarization that results from interactions with other ions in the crystal. Ionic radii also depend to some extent on the number of nearest neighbors. In general, there is a slight increase in apparent ionic radius as the coordination number increases. Moreover, some ions are not spherical even though radii have been determined by means of the Kapustinskii equation (discussed in Section 1.4). For example, the NO3− is planar even though we know an effective or thermochemical radius. Using a value for the radius of such ions may not always predict the actual arrangement of ions in a crystal. As shown in Table 1.5, an octahedral arrangement of anions around a cation is predicted when rc/ra is in the range from 0.414 to 0.732. Disagreements between predictions and observations generally occur when the radius ratio is close to the limits for predicting environments. For example, if rc/ra differs only slightly from 0.414 or 0.732, other factors may cause the structure to be different from that predicted of having an octahedral arrangement around the cation. On the other hand, if rc/ra is somewhere toward the middle of the range, say 0.52 or 0.60, the predicted structure will almost always be correct.

Because ionic radii are important in determining crystal structures and other properties of solids, the radii of many ions are shown in Table 1.6.

Table 1.6

1.4 The Kapustinskii equation

In order to make use of the Born–Landé equation, information about the crystal structure must be known including the value of the Madelung constant. Also, there is no way to include information about the ions other than their radii. This may not be possible because not all ions are spherical. Ions such as NO3− and CO3²− are planar, but SO4²− and NH4+ ions are tetrahedral. As a result, the distance of separation of ions may not be the same in all directions. As shown in Eq. (1.19), a different approach to the calculation of lattice energy has been provided by Kapustinskii (9,10).

Equation

   (1.19)

The Kapustinskii equation does not make use of the Madelung constant but requires only the charges on the cation and anion, Zc and Za, the sum of the ionic radii, rc + ra, and the number of ions in the formula, m. Thus, it is not necessary to know the structure of the crystal or the radii of the individual ions. The equation gives results that are quite accurate, but it is based on ionic interactions, and the results obtained for lattice energies of crystals that have substantial covalent character are not as accurate. For example, the calculated lattice energies for alkali halides are quite close to the actual values. However, for crystals such as AgI, CuBr, or TlCl where the degree of covalency is substantial, the calculated lattice energies are generally low because the contribution of van der Waals forces between polarizable ions is not included. This is shown by the lattice energies for silver halides (shown as calculated/actual values in kJ mol−1): AgF, 816/912; AgCl, 715/858; AgBr, 690/845; AgI, 653/833. Note that the disagreement between the value calculated assuming an ionic crystal and the actual lattice energy becomes larger as the halide ion increases in size. This is expected on the basis of the larger −1 ion being more polarizable so that the bonds assume more covalent character in the case of AgI than for AgF. Although lattice energies of silver halides were considered in the illustration, it must be remembered that there is some degree of covalence between other types of ions. Such effects normally depend on the difference in electronegativity of the