Chapter 12

One way to describe ionic crystal structures is in terms of cations filling voids among closely packed anions. Show that in order for cations to fill the tetrahedral voids in a close packed arrangement of anions, the radius ratio of cation, \(r_{\mathrm{c}},\) to anion, \(r_{\mathrm{a}},\) must fall between the following limits \(0.225 < r_{\mathrm{c}}: r_{\mathrm{a}} < 0.414\)

Show that the formation of \(\mathrm{NaCl}_{2}(\mathrm{s})\) is very unfavorable; that is, \(\Delta \mathrm{H}_{\mathrm{f}}^{\circ}\left[\mathrm{NaCl}_{2}(\mathrm{s})\right]\) is a large positive quantity. To do this, use data from Section \(12-7\) and assume that the lattice energy for \(\mathrm{NaCl}_{2}\) would be about the same as that of \(\mathrm{MgCl}_{2},-2.5 \times 10^{3} \mathrm{kJ} \mathrm{mol}^{-1}\)

A crystalline solid contains three types of ions, \(\mathrm{Na}^{+}, \mathrm{O}^{2-},\) and \(\mathrm{Cl}^{-}\). The solid is made up of cubic unit cells that have \(\mathrm{O}^{2-}\) ions at each corner, \(\mathrm{Na}^{+}\) ions at the center of each face, and \(\mathrm{Cl}^{-}\) ions at the center of the cells. What is the chemical formula of the compound? What are the coordination numbers for the \(\mathrm{O}^{2-}\) and \(\mathrm{Cl}^{-}\) ions? If the length of one edge of the unit cell is \(a,\) what is the shortest distance from the center of a \(\mathrm{Na}^{+}\) ion to the center of an \(\mathrm{O}^{2-}\) ion? Similarly, what is the shortest distance from the center of a \(\mathrm{Cl}^{-}\) ion to the center of an \(\mathrm{O}^{2-}\) ion?

A certain mineral has a cubic unit cell with calcium at each corner, oxygen at the center of each face, and titanium at its body center. What is the formula of the mineral? An alternate way of drawing the unit cell has calcium at the center of each cubic unit cell. What are the positions of titanium and oxygen in such a representation of the unit cell? How many

In some barbecue grills the electric lighter consists of a small hammer-like device striking a small crystal, which generates voltage and causes a spark between wires that are attached to opposite surfaces of the crystal. The phenomenon of causing an electric potential through mechanical stress is known as the piezoelectric effect. One type of crystal that exhibits the piezoelectric effect is lead zirconate titanate. In this perovskite crystal structure, a titanium(IV) ion sits in the middle of a tetragonal unit cell with dimensions of \(0.403 \mathrm{nm} \times 0.398 \mathrm{nm} \times 0.398 \mathrm{nm} .\) At each corner is a lead(II) ion, and at the center of each face is an oxygen anion. Some of the Ti(IV) are replaced by Zr(IV). This substitution, along with \(\mathrm{Pb}(\mathrm{II}),\) results in the piezoelectic behavior. (a) How many oxygen ions are in the unit cell? (b) How many lead(II) ions are in the unit cell? (c) How many titanium(IV) ions are in the unit cell? (d) What is the density of the unit cell?

We have learned that the enthalpy of vaporization of a liquid is generally a function of temperature. If we wish to take this temperature variation into account, we cannot use the Clausius-Clapeyron equation in the form given in the text (that is, equation 12.2 ). Instead, we must go back to the differential equation upon which the Clausius-Clapeyron equation is based and reintegrate it into a new expression. Our starting point is the following equation describing the rate of change of vapor pressure with temperature in terms of the enthalpy of vaporization, the difference in molar volumes of the vapor \(\left(V_{g}\right),\) and liquid \(\left(V_{1}\right),\) and the temperature. $$\frac{d P}{d T}=\frac{\Delta H_{\mathrm{vap}}}{T\left(V_{\mathrm{g}}-V_{1}\right)}$$ Because in most cases the volume of one mole of vapor greatly exceeds the molar volume of liquid, we can treat the \(V_{1}\) term as if it were zero. Also, unless the vapor pressure is unusually high, we can treat the vapor as if it were an ideal gas; that is, for one mole of vapor, \(P V=R T\). Make appropriate substitutions into the above expression, and separate the \(P\) and \(d P\) terms from the \(T\) and \(d T\) terms. The appropriate substitution for \(\Delta H_{\text {vap }}\) means expressing it as a function of temperature. Finally, integrate the two sides of the equation between the limits \(P_{1}\) and \(P_{2}\) on one side and \(T_{1}\) and \(T_{2}\) on the other. (a) Derive an equation for the vapor pressure of \(\mathrm{C}_{2} \mathrm{H}_{4}(\mathrm{l})\) as a function of temperature, if \(\Delta H_{\mathrm{vap}}=\) \(15,971+14.55 T-0.160 T^{2}\left(\text { in } J m o l^{-1}\right)\) (b) Use the equation derived in (a), together with the fact that the vapor pressure of \(\mathrm{C}_{2} \mathrm{H}_{4}(1)\) at \(120 \mathrm{K}\) is 10.16 Torr, to determine the normal boiling point of ethylene.

All solids contain defects or imperfections of structure or composition. Defects are important because they influence properties, such as mechanical strength. Two common types of defects are a missing ion in an otherwise perfect lattice, and the slipping of an ion from its normal site to a hole in the lattice. The holes discussed in this chapter are often called interstitial sites, since the holes are in fact interstices in the array of spheres. The two types of defects described here are called point de kcts because they occur within specific sites. In the 1930 s, two solidstate physicists, W. Schottky and J. Fraenkel, studied the two types of point defects: A Schottky defect corresponds to a missing ion in a lattice, while a Fraenkel defect corresponds to an ion that is displaced into an interstitial site. (a) An example of a Schottky defect is the absence of a \(\mathrm{Na}^{+}\) ion in the NaCl structure. The absence of a \(\mathrm{Na}^{+}\) ion means that a \(\mathrm{Cl}^{-}\) ion must also be absent to preserve electrical neutrality. If one NaCl unit is missing per unit cell, does the overall stoichiometry change, and what is the change in density? (b) An example of a Fraenkel defect is the movement of \(a \mathrm{Ag}^{+}\) ion to a tetrahedral interstitial site from its normal octahedral site in \(\mathrm{AgCl}\), which has a structure like \(\mathrm{NaCl}\). Does the overall stoichiometry of the compound change, and do you expect the density to change? (c) Titanium monoxide (TiO) has a sodium chloridelike structure. X-ray diffraction data show that the edge length of the unit cell is \(418 \mathrm{pm}\). The density of the crystal is \(4.92 \mathrm{g} / \mathrm{cm}^{3}\) Do the data indicate the presence of vacancies? If so, what type of vacancies?

In an ionic crystal lattice each cation will be attracted by anions next to it and repulsed by cations near it. Consequently the coulomb potential leading to the lattice energy depends on the type of crystal. To get the total lattice energy you must sum all of the electrostatic interactions on a given ion. The general form of the electrostatic potential is $$V=\frac{Q_{1} Q_{2} e^{2}}{d_{12}}$$ where \(Q_{1}\) and \(Q_{2}\) are the charges on ions 1 and \(2, d_{12}\) is the distance between them in the crystal lattice. and \(e\) is the charge on the electron. (a) Consider the linear "crystal" shown below. The distance between the centers of adjacent spheres is \(R .\) Assume that the blue sphere and the green spheres are cations and that the red spheres are anions. Show that the total electrostatic energy is $$V=-\frac{Q^{2} e^{2}}{d} \times \ln 2$$ (b) In general, the electrostatic potential in a crystal can be written as $$V=-k_{M} \frac{Q^{2} e^{2}}{R}$$ where \(k_{M}\) is a geometric constant, called the Madelung constant, for a particular crystal system under consideration. Now consider the NaCl crystal structure and let \(R\) be the distance between the centers of sodium and chloride ions. Show that by considering three layers of nearest neighbors to a central chloride ion, \(k_{M}\) is given by $$k_{M}=\left(6-\frac{12}{\sqrt{2}}+\frac{8}{\sqrt{3}}-\frac{6}{\sqrt{4}} \cdots\right)$$ (c) Carry out the same calculation for the CsCl structure. Are the Madelung constants the same?

In your own words, define or explain the following terms or symbols: (a) \(\Delta H_{\text {vap }} ;\) (b) \(T_{c} ;\) (c) instantaneous dipole; (d) coordination number; (e) unit cell.

Briefly describe each of the following phenomena or methods: (a) capillary action; (b) polymorphism; (c) sublimation; (d) supercooling; (e) determining the freezing point of a liquid from a cooling curve.

Explain the important distinctions between each pair of terms: (a) adhesive and cohesive forces; (b) vaporization and condensation; (c) triple point and critical point; (d) face-centered and body-centered cubic unit cell; (e) tetrahedral and octahedral hole.

The magnitude of one of the following properties must always increase with temperature; that one is (a) surface tension; (b) density; (c) vapor pressure; (d) \(\Delta H_{\text {vap }}\)

Of the compounds \(\mathrm{HF}, \mathrm{CH}_{4}, \mathrm{CH}_{3} \mathrm{OH}, \mathrm{N}_{2} \mathrm{H}_{4},\) and \(\mathrm{CHCl}_{3},\) hydrogen bonding is an important intermolecular force in (a) none of these; (b) two of these; (c) three of these; (d) all but one of these; (e) all of these.

A metal that crystallizes in the body-centered cubic (bcc) structure has a crystal coordination number of (a) \(6 ;\) (b) \(8 ;\) (c) \(12 ;\) (d) any even number between 4 and 12

A unit cell of an ionic crystal (a) shares some ions with other unit cells; (b) is the same as the formula unit; (c) is any portion of the crystal that has a cubic shape; (d) must contain the same number of cations and anions.

If the triple point pressure of a substance is greater than 1 atm, which two of the following conclusions are valid? (a) The solid and liquid states of the substance cannot coexist at equilibrium. (b) The melting point and boiling point of the substance are identical. (c) The liquid state of the substance cannot exist. (d) The liquid state cannot be maintained in a beaker open to air at 1 atm pressure. (e) The melting point of the solid must be greater than \(0^{\circ} \mathrm{C}\) (f) The gaseous state at 1 atm pressure cannot be condensed to the solid at the triple point temperature.

In each of the following pairs, which would you expect to have the higher boiling point? (a) \(\mathrm{C}_{7} \mathrm{H}_{16}\) or \(\mathrm{C}_{10} \mathrm{H}_{22} ;\) (b) \(\mathrm{C}_{3} \mathrm{H}_{8}\) or \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{O} ;\) (c) \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{SH}\) or \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\)

Arrange the following substances in the expected order of increasing melting point: \(\mathrm{KI}\), \(\mathrm{Ne}, \mathrm{K}_{2} \mathrm{SO}_{4}\) \(\mathrm{C}_{3} \mathrm{H}_{8}, \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}, \mathrm{MgO}, \mathrm{CH}_{2} \mathrm{OHCHOHCH}_{2} \mathrm{OH}\)

Is it possible to obtain a sample of ice from liquid water without ever putting the water in a freezer or other enclosure at a temperature below \(0^{\circ} \mathrm{C} ?\) If \(\mathrm{so}\) how might this be done?

The following data are given for \(\mathrm{CCl}_{4}\). Normal melting point, \(-23^{\circ} \mathrm{C} ;\) normal boiling point, \(77^{\circ} \mathrm{C} ;\) density of liquid \(1.59 \mathrm{g} / \mathrm{mL} ; \Delta H_{\text {fus }}=3.28 \mathrm{kJ} \mathrm{mol}^{-1} ;\) vapor pressure at \(25^{\circ} \mathrm{C}, 110\) Torr. (a) What phases-solid, liquid, and/or gas-are present if \(3.50 \mathrm{g} \mathrm{CCl}_{4}\) is placed in a closed \(8.21 \mathrm{L}\) container at \(25^{\circ} \mathrm{C} ?\) (b) How much heat is required to vaporize 2.00 L of \(\mathrm{CCl}_{4}(\mathrm{l})\) at its normal boiling point?

Of the following liquids at \(20^{\circ} \mathrm{C}\), which has the smallest viscosity? (a) Dodecane, \(\mathrm{C}_{12} \mathrm{H}_{26} ;\) (b) n-nonane, \(\mathrm{C}_{9} \mathrm{H}_{20} ;\) (c) n-heptane \(\mathrm{C}_{7} \mathrm{H}_{16} ;\) (d) n-pentane \(\mathrm{C}_{5} \mathrm{H}_{12}\)

Would you expect an ionic solid or a network covalent solid to have the higher melting point?

In the lithium iodide crystal, the Li-I distance is \(3.02 \AA\) Calculate the iodide radius, assuming that the iodide ions are in contact.

Construct a concept map representing the different types of intermolecular forces and their origin.

Construct a concept map using the ideas of packing of spheres and the structure of metal and ionic crystals.

Construct a concept map showing the ideas contained in a phase diagram.