Engineering Chemistry: Solids (PDF)

CHAPTER THREE SOLIDS 3.1 STRUCTURES OF SOLIDS Solids can be either crystalline or amorphous (noncrystalline). A crystalline solid is a solid whose atoms, ions, or molecules are ordered in well-defined arrangements. These solids usually have flat surfaces or...

CHAPTER THREE SOLIDS 3.1 STRUCTURES OF SOLIDS Solids can be either crystalline or amorphous (noncrystalline). A crystalline solid is a solid whose atoms, ions, or molecules are ordered in well-defined arrangements. These solids usually have flat surfaces or faces that make definite angles with one another. The orderly stacks of particles that produce these faces also cause the solids to have highly regular shapes like quartz and diamond. An amorphous solid is a solid whose particles have no orderly structure. These solids lack well-defined faces and shapes. Many amorphous solids are mixtures of molecules that do not stack together well. Most others are composed of large, complicated molecules like rubber and glass. Quartz, SiO2, is a crystalline solid with a three-dimensional structure like that shown in Figure 3.1(a). When quartz melts (at about 1600°C), it becomes a viscous, tacky liquid. Although the silicon-oxygen network remains largely intact, many Si O bonds are broken. If the melt is rapidly cooled, the atoms are unable to return to an orderly arrangement. An amorphous solid known as quartz glass or silica glass results [Figure 3.1(b)]. FIGURE 3.1 Schematic comparisons of (a) crystalline SiO2 (quartz) and (b) amorphous SiO2 (quartz glass). The structure is actually three-dimensional and not planar as drawn. The unit shown as the basic building block (silicon and three oxygens) actually has four oxygens, the fourth coming out of the plane of the paper and capable of bonding to other silicon atoms. Because the particles of an amorphous solid lack any long-range order, intermolecular forces vary in strength throughout a sample. Thus, amorphous solids do not melt at specific temperatures. Instead, they soften over a temperature range as intermolecular 76 forces of various strengths are overcome. A crystalline solid, in contrast, melts at a specific temperature. Unit Cells The repeating unit of a solid, the crystalline is known as the unit cell. A simple two- dimensional example appears in the sheet shown in Figure 3.2. There are several ways of choosing the repeat pattern, or unit cell, of the design, but the choice is usually the smallest one that shows clearly the symmetry characteristic of the entire pattern. FIGURE 3.2 Wallpaper design showing a characteristic repeat pattern. Each dashed denotes a unit cell of the repeat pattern. The unit cell could equally well be selected at the corners. A crystalline solid can be represented by a three-dimensional array of points. Such an array of points is called a crystal lattice. We can imagine forming the entire crystal structure by arranging the contents of the unit cell repeatedly on a network of points. Figure 3.3 shows a crystal lattice and its associated unit cell. In general, unit cells are parallelepipeds (six-sided figures whose faces are parallelograms). Each unit cell can be described in terms of the lengths of the edges of the cell and by the angles between these edges. The lattices of all crystalline compounds can be described in terms of seven basic types of unit cells. The simplest of these is the cubic unit cell, in which all the sides are equal in length and all the angles are 90°. FIGURE 3.3 Simple crystal lattice and its associated unit cell. 77 There are three kinds of cubic unit cells, as illustrated in Figure 3.4. When lattice points are at the corners only, the unit cell is described as primitive cubic. When a lattice point also occurs at the center of the unit cell, the cell is known as body- centered cubic. A third type of cubic cell has lattice points at the center of each face, as well as at each corner, an arrangement known as face-centered cubic. FIGURE 3.4 The three types of unit cells found in cubic lattices. The simplest crystal structures are cubic unit cells with only one atom centered at each lattice point. For example, nickel has a face-centered cubic unit cell, whereas sodium has a body-centered cubic one. Figure 3.5 shows how atoms fill the cubic unit cells. Notice that the atoms on the corners and faces do not lie wholly within the unit cell. Instead, these atoms are shared between unit cells. Table 3.1 summarizes the fraction of an atom that occupies a unit cell when atoms are shared between unit cells. FIGURE 3.5 Space-filling views of cubic unit cells. Only the portion of each atom that belongs to the unit cell is shown. 3.2 BONDING IN SOLIDS The physical properties of crystalline solids, such as their melting points and hardness, depend both on the arrangements of particles and on the attractive forces 78 between them. Table 3.2 classifies solids according to the types of forces between particles in solids. 3.3 METALLIC SOLIDS Metallic solids consist entirely of metal atoms. Metallic solids usually have hexagonal close-packed, cubic close-packed (face-centered-cubic), or body-centered- cubic structures. Thus, each atom typically has 8 or 12 adjacent atoms. The bonding in metals is too strong to be due to London dispersion forces, and yet there are not enough valence electrons for ordinary covalent bonds between atoms. The bonding is due to valence electrons that are delocalized throughout the entire solid. In fact, we can visualize the metal as an array of positive ions immersed in a sea of delocalized valence electrons, as shown in Figure 3.6. Metals vary greatly in the strength of their bonding, as is evidenced by their wide range of physical properties such as hardness and melting point. In general, however, the strength of the bonding increases as the number of electrons available for bonding increases. Thus, sodium, which has only one valence electron per atom, melts at 97.5°C, whereas chromium, with six electrons beyond the noble-gas core, melts at 79 1890°C. The mobility of the electrons explains why metals are good conductors of heat and electricity. FIGURE 3.6 A cross-section of a metal. Each sphere represents the nucleus and inner-core electrons of a metal atom. The surrounding colored "fog" represents the mobile sea of electrons that binds the atoms together. To maximize the bonding in a metal it makes sense to pack as many atoms around each other as possible, maximize the number of nearest neighbors (called the "coordination number") and minimize the volume. Close packed layers of atoms If we treat the atoms as spheres and consider all the atoms in the solid to be of equal size (as is the case for elemental metals), the most efficient form of packing is the close packed layer. This is illustrated below where it is clear that close-packing of spheres is more efficient than, for example, square packing. Below on the left is a square packed array compared to the more densely packed close packed array. Within the square packed layer the coordination # of each atom is 4, in the close packed layer it is 6. 80  Close Packed Crystal Structures. To build our 3-dimensional metal structures we now need to stack the close packed layers on top of each other. There are several ways of doing this. The most efficient space saving way is to have the spheres in one layer fit into the "holes" of the layer below. If we call the first layer "A", then the second layer ("B") is positioned as shown on the left of the diagram below. The third layer can then be added in two ways. In the first way the third layer fits into the holes of the B layer such that the atoms lie above those in layer A. By repeating this arrangement one obtains ABABAB... stacking or exagonal close packing. Hexagonal Close Packing. (HCP) The {ABAB... } type stacking of close packed layers is called Hexagonal Close Packing (hcp) because the smallest lattice repeat is a Primitive Hexagonal unit cell. A more extended side view of the packing is shown below: 81 In the primitive hexagonal cell we have 1 atom at each of the corners of the cell (each is "worth" 1/8) and 1 atom within the cell giving us 2 atoms/unit cell. The coordination number of the atoms in this structure is 12. They have 6 nearest neighbors in the same close packed layer, 3 in the layer above and 3 in the layer below. This is one of the most efficient methods of packing spheres (the other that is equally efficient is cubic close packing, see below). In both cases the spheres fill 74% of the available space. HCP is a very common type of structure for elemental metals. Examples include Be, Mg, Ti, Zr, etc. 82  Cubic Close Packing (CCP) While for the HCP structure the third close packed layer was positioned above the first , an alternate method of stacking is to place the third layer such that it lies in an unique position, in this way an "ABCABC..." close packed layer sequence can be created, see below. This method of stacking is call Cubic Close Packing (ccp) The reason this is called cubic close packing is because the smallest unit cell that can describe this arrangement is face centered cubic. Two sides of the unit cell are apparent in the figure below. By reorienting the structure the entire face centered cubic cell and its contents are apparent. 83 Because the ccp structure is still close packed it is as efficient in its packing as the hcp structure (74%), and the coordination number of the atoms is still 12. How many atoms are there in the fcc unit cell ? 8 at the corners (8x1/8 = 1), 6 in the faces (6x1/2=3), giving a total of 4 per unit cell. Again there are many examples of ccp (fcc) (ABCABC) metal structures, e.g. Al, Ni, Cu, Ag, Pt. Density Calculations The density of solids depends on its crystal structure. The density of the unit cell is calculated if we know the volume of the unit cell and the number of atome contained in it. Beginning with the general definition of density as mass per unit volume, the density of unit cell can be defined as follows: dunit cell = m unit cell/ Vunit cell [ 3.1 ] where m is the total mass of the atoms in the unit cell and V is the volume of the unit cell. The mass of the atoms in the unit cell in grams is: m unit cell = Z M / NA [ 3.2 ] where Z is the number of atoms per per unit cell, M is the molar mass, and NA is the Avogadro,s number. This mass can be substituted into the equation for dunit cell , d = ZM / NAV [ 3. 3 ] For a cubic unit cell where a is the length of the edge of the unit cell and V = a3, d = ZM / NAa3 [ 3.4 ] Sample Exercise 3.1: 84 X-ray diffraction experiments show that copper crystallizes in an FCC unit cell 3.608oA along an edge. Density measurements give the value 8.92 g/cm3. What is the molar mass of copper? Comment Solve for M by rearranging the preceding equation for calculating density, M = dNAa3 / Z The number of atoms per unit cell Z is determined in exactly : Z = [8 corners / cell ] X [1 atom / 8corner] + [6 faces/ cell]X[1 atom / 2 faces]= 4 atoms / cell Solution: M = dNAa3 / Z = (8.92 g/cm3) (6.022x1023 atoms/mole) (3.608 x 10-8 cm)3 / 4 atoms = 63.1 g/mol Using a model for the structure of a metal of hard spheres in direct contact, one can calculate the relationship between the size of the spheres and the size of the unit cell. The radius r of the sphere corresponds to the atomic radius r. Consider the FCC unit cell of a metal, corner atoms do not touch each other. However, atoms on the corners touch atoms in the centers of faces, and the face diagonal is equal to four atomic radii. Thus, in a metal that crystallizes in an FCC lattice, the atomic radius r can be calculated: r = a√2 / 4 In the BCC structure of metal the corner atoms do not touch. However, atoms on the corners touch the atom in the center of the unit cell, and the body diagonal is equal to four atomic radii. From which the atomic radius r can be calculated: r = a√3 / 4 Sample exercise 3. 2 Al has a ccp arrangement of atoms. The radius of Al = 1.423Å ( = 143.2pm). Calculate the lattice parameter of the unit cell and the density of solid Al (atomic weight = 26.98). Solution: Because Al is ccp we have an fcc unit cell. Cell contents: 4 atoms/cell [8 at corners (each 1/8), 6 in faces (each 1/2)] Lattice parameter: atoms in contact along face diagonal, therefore 4rAl = a(2)1/2 a = 4(1.432Å)/(2)1/2 = 4.050Å. Density (= Al) = Mass/Volume = Mass per unit cell/Volume per unit cell g/cm3 85 Mass of unit cell = mass 4 Al atoms = (26.98)(g/mol)(1mol/6.022x1023atoms)(4 atoms/unit cell) = 1.792 x 10-22 g/unit cell Volume unit cell = a3 = (4.05x10-8cm)3 = 66.43x10-24 cm3/unit cell Therefore Al = {1.792x10-22g/unit cell}/{66.43x10-24 cm3/unit cell} = 2.698 g/cm3 Other common types of metal structures 1. Body Centered Cubic (BCC) Not close packed - atoms at corners and body center of cube. # atoms/unit cell = 2. Coordination number = 8  less efficient packing (68%) The atoms are only in contact along the body diagonal. For a unit cell edge length a, length body diagonal = a(3)1/2. Therefore 4r = a(3)1/2 Examples of BCC structures include one form of Fe, V, Cr, Mo, W. 2. Simple Cubic 86 Again not close packed - primitive or simple cubic cell with atoms only at the corners. # atoms/unit cell = 1. Coordination number = 6  least efficient method of packing (52%) The atoms are in contact along the cell edge. Therefore a = 2r. A very rare packing arrangement for metals, one example is a form of Polonium (Po) Conduction properties Solids can exhibit a wide range of conduction properties; these can be rationalized and interpreted by considering the character of their bonding and structure. Summary of Packing types for Metallic Structures: Benzene and Graphite. 87 The similarity between the bonding in methane and diamond allowed us to interpret the insulating properties of diamond; in a similar way it is possible to interpret the electronic properties of graphite. The properties of diamond and graphite are very different, diamond is hard, colorless, and an insulator; graphite is a well-known lubricant, black, and a conductor. These differences arise from the different bonding and structure. The structure of graphite is based on layers of connected six-member rings of carbon atoms - the structure of the layers is very similar to that of benzene, see below. While the bonding in the layers is very strong, only weak Van Der Waals forces bond the layers together (thus the effectiveness of graphite as a lubricant). Benzene Layered structure of Graphite The conduction of electrons within the graphite layers can be rationalized using M.O. theory. Consider benzene: each carbon atom forms sp2 hybrids that overlap to form  and * M.O.'s. The remaining pz orbitals overlap to form 6  molecular orbitals - the nature of the overlap is shown below. The  M.O.'s have 4 discrete energy levels. The separation of the levels is small compared to the  * separation. In graphite each of the levels is broadened into bands and the  orbitals overlap to form a continuous band. This band is half filled and graphite exhibits metallic conduction within the layers. 3.4 IONIC SOLIDS General Bonding considerations For compound (A)n(B)m we can expect ionic bonding to predominate when atom A has low electronegativity and atom B has a high electronegativity. In this case 88 electron transfer from one atom to another leads to the formation of A+B-. For the main group elements the electron transfer continues until the ions have closed shell configurations. For ionic compounds the bonding forces are electrostatic and therefore omni- directional. The bonding forces should be maximized by packing as many cations around each anion, and as many cations around each anion as is possible. The number of nearest neighbor ions of opposite charge is called the coordination number. We must realize however that the coordination numbers are constrained by the stoichiometry of the compound and by the sizes of the atoms. e.g. For sodium chloride, Na+Cl-, there are 6 anions around each cation (coordination number Na = 6); because of the 1:1 stoichiometry there must also be 6 Na cations around each Cl anion. For Zr4+O2-2 there are 8 anions around each cation, therefore there must be only 4 cations around each anion (Figure 4.6). The strength of an ionic bond depends greatly on the charges of the ions. Thus, NaCl, in which the ions have charges of 1+ and 1-, has a melting point of 801°C, whereas MgO, in which the charges are 2+ and 2-, melts at 2852°C. The structures of simple ionic solids can be classified as a few basic types. The NaCl structure is a representative example of one type. Other compounds that possess this same structure include LiF, KCl, AgCl, and CaO. FIGURE 3.6 Unit cells of sodium chloride 89 NON CLOSE-PACKED STRUCTURES FIGURE 3.7 UNIT CELLS OF CSCL, LATTICE: CUBIC - P (N.B. PRIMITIVE!) The structure adopted by an ionic solid depends largely on the charges and relative sizes of the ions. In the NaCl structure, for example, the Na+ ions have a coordination number of 6: Each Na+ ion is surrounded by 6 Cl- ion nearest neighbors. In the CsCl structure [Figure 4.7], by comparison, each Cs+ ion is surrounded by 8 Cl- ions. The increase in the coordination number as the alkali metal ion is changed from Na+ to Cs+ is a consequence of the larger size of Cs+ compared to Na+. In the zinc blende, ZnS, structure [Figure 4.8], the S2- ions adopt a face-centered-cubic arrangement, with the smaller Zn2+ ions arranged so they are each surrounded tetrahedrally by four S2- ions. CuCl also adopts this structure. In the fluorite, CaF2, structure [Figure 4.9] the Ca2+ ions are shown in a face-centered- cubic arrangement. As required by the chemical formula of the substance, there are twice as many F- ions in the unit cell as there are Ca2+ ions. Other compounds that have the fluorite structure include BaCl2 and PbF2. 90  FIGURE 3.8 UNIT CELLS OF ZNS ZINC BLENDE (SPHALERITE) FIGURE 3.9 UNIT CELLS OCAF2 FLUORITE / {NA2O ANTI-FLUORITE} 3.5 COVALENT SOLIDS In covalent compounds both atoms have similar and fairly high electronegativities. The nature of the bonding therefore relies upon electron-sharing through orbital overlap. In contrast to ionic and metallic solids, the bonding in extended covalent solids is highly directional, this leads to low coordination numbers (e.g. 4). Covalent-network solids consist of atoms held together in large networks or chains by covalent bonds. Because covalent bonds are much stronger than intermolecular forces, these solids are much harder and have higher melting points than molecular solids. Two of the most familiar examples of covalent-network solids are diamond and graphite, two allotropes of carbon. Other examples include quartz, SiO2, silicon carbide, SiC, and boron nitride, BN. 91 Diamond, analogy to methane In methane the bonding network of the carbon atoms is terminated by the H atoms. Because the H atoms can only form one covalent bond, which is confined within the molecule, there is no possibility for extended covalent bonding between different molecules. The inter-molecular bonds must therefore rely on weaker dispersion forces for their cohesive forces. In diamond, which can be constructed by replacing the H atoms in methane by other C atoms, the bonding is not terminated and the  sp3 bonds between the C atoms are extended throughout the solid. Each carbon atom is bonded to four other carbon atoms as shown in Figure 4.10. This interconnected three-dimensional array of strong carbon-carbon single bonds contributes to diamond's unusual hardness. Industrial- grade diamonds are employed in the blades of saws for the most demanding cutting jobs. In keeping with its structure and bonding, diamond also has a high melting point, 3550°C. The crystal strucutre of diamond is comprised of tetrahedrally coordinated C atoms. The unit cell is fcc, with 2 carbon atoms per lattice point, and 8 atoms per cell. The diamond structure is also adopted by several well known semi-conductors such as Si, Ge (and also gray Sn). Methane Crystal Structure of Diamond FIGURE 3.10 Structures of diamond In graphite the carbon atoms are arranged in layers of interconnected hexagonal rings as shown in Figure 4.11. Each carbon atom is bonded to three others in the layer. The distance between adjacent carbon atoms in the plane, 1.42 Å, is very close to the C C distance in benzene, 1.395 Å. In fact, the bonding resembles that of benzene, with delocalized bonds extending over the layers. Electrons move freely through the delocalized orbitals, making graphite a good conductor of electricity along the layers. 92 (If you have ever taken apart a flashlight battery, you know that the central electrode in the battery is made of graphite.) The layers, which are separated by 3.41 Å, are held together by weak dispersion forces. The layers readily slide past one another when rubbed, giving the substance a greasy feel. Graphite is used as a lubricant and in making the "lead" in pencils. FIGURE 4.11 Structures of graphite. 3.6 MOLECULAR SOLIDS Molecular solids consist of atoms or molecules held together by intermolecular forces (dipole-dipole forces, London dispersion forces, and hydrogen bonds). Because these forces are weak, molecular solids are soft. Furthermore, they normally have relatively low melting points (usually below 200°C). Most substances that are gases or liquids at room temperature form molecular solids at low temperature. Examples include Ar, H2O, and CO2. The properties of molecular solids depend not only on the strengths of the forces that operate between molecules but also on the abilities of the molecules to pack efficiently in three dimensions. For example, benzene, C6H6, is a highly symmetrical planar molecule. It has a higher melting point than toluene, a compound in which one of the hydrogen atoms of benzene has been replaced by a CH3 group (Figure 4.12). The lower symmetry of toluene molecules prevents them from packing as efficiently as benzene molecules. As a result, the intermolecular forces that depend on close 93 contact are not as effective, and the melting point is lower. In contrast, the boiling point of toluene is higher than that of benzene, indicating that the intermolecular attractive forces are larger in liquid toluene than in liquid benzene. Figure 11.41 also presents the melting and boiling points of another substituted-benzene compound, phenol. Both the melting and boiling points of phenol are higher than those of benzene because of the hydrogen-bonding ability of the OH group of phenol. FIGURE 4.12 Comparative melting and boiling points for benzene, toluene, and phenol. 3.7 ORDERLINESS OF LIQUID CRYSTALS: liquid crystals are an itermediate form of matter, displaying the flow properties ofliquids and the light scattering properties of so;ids. That combination of properties has taken on great scientific and technological significance in such diverse fields as computer electronics and automative products. The liquid crystal state dependes on the presence of molecules with rigid, rod-like structures. At higher temperatures, these rods are oriented in a random fashion, resulting in typical properties of a disordered liquid (Fig. 4.13 A). However as the compound cools, some ordering of the random structure is possible. Ordering is observed because the molecules have preferred orientations. The simplest of these is the nematic phase ( Fig. 4.13 b), in which the molecules are all parallel to each other, but the cernters of the molecules are distributed at random. Shown in Figure 4.13 c is the sematic phase, which is more ordered than the nematic phase because the molecules not only have the same preferred direction, but their centers are located in planes thjat run through the structure.Finelly the full ordering of a molecular crystal is shown in Figure 4.13 d, completing the transition from liquid to crystalline solid. 94  Figure 3.13 Molecular orders in nematic and smectic liquid crystals. In the liquid phase of any substance, the molecules are arranged randomly, whereas in the liquid crystalline phases the molecules are arranged in a partially ordered way. The orientation of the molecules in a liquid crystal is very sensitive to electric and magnetic fields. This is useful in liquid crystal devices, particularly liquid crystal displays, or LCDs. Problems The element iron crystallizes in a form called -iron, which has a body-centered- cubic unit cell. How many iron atoms are in the unit cell? Answer: two The body-centered cubic unit cell of a particular crystalline form of iron is 2.8664 Å on each side. Calculate the density of this form of iron. Answer: 7.8753 g/cm3 Compare and contrast the structures of the following pairs of solids. Include descriptions of the structures and particularly discuss the reasons for their adoption. (a) NaCl & CsCl. (e) C (Diamond) & C (Graphite). Silver crystallizes with a fcc structure and a lattice parameter a=4.086 Å. What is its theoretical density? Al has a ccp arrangement of atoms. The radius of Al = 1.423Å. Calculate the lattice parameter of the unit cell and the density of solid Al (atomic weight = 26.98). Vanadium has a BCC unit cell with a unit cell edge of 3.011oA. Its density is 5.96 g/cm3. Calculate its molar mass? Calcium has a Fcp lattice with a lattice constant a =5.56oA. Determine the atomic radius of calcium? Calculate the fraction of empty space in an FCP structure of a metal, assuming the metal atoms are hard spheres that just touch along the face diagonal? 95