Chapter 11 : Liquids, Solids, and Intermolecular Forces

Viscosity

Another manifestation of intermolecular forces is viscosity, the resistance of a liquid to flow. Motor oil, for example, is more viscous than gasoline, and maple syrup is more viscous than water. Viscosity is measured in a unit called the poise (P), defined as 1 g>cm # s. The viscosity of water at room temperature is approximately one centipoise (cP). Viscosity is greater in substances with stronger intermolecular forces because if molecules are more strongly attracted to each other they do not flow around each other as freely. Viscosity also depends on molecular shape; it increases in longer molecules that can interact over a greater area and possibly become entangled. Table 11.5 lists the viscosity of several hydrocarbons. Viscosity also depends on temperature because thermal energy partially overcomes the intermolecular forces, allowing molecules to flow past each other more easily. Table 11.6 lists the viscosity of water as a function of temperature. Nearly all liquids become less viscous as temperature increases. ◀ FIGure 11.15 Spherical Water Tiny water droplets are not distorted much by gravity and form nearly perfect spheres held together by intermolecular forces between water molecules.

Closest-Packed Structures

Another way to envision crystal structures, especially useful in metals where bonds are not usually directional, is to think of the atoms as stacking in layers, much as fruit is stacked at the grocery store. For example, we can envision the simple cubic structure as one layer of atoms arranged in a square pattern with the next layer stacking directly over the first, so that the atoms in one layer align exactly on top of the atoms in the layer beneath it, as shown to the right. As we saw previously, this crystal structure has a great deal of empty space—only 52% of the volume is occupied by the spheres, and the coordination number is 6. More space-efficient packing can be achieved by aligning neighboring rows of atoms within a layer not in a square pattern, but in a pattern with one row offset from the next by one-half a sphere, as shown to the left. In this way, the atoms pack more closely to each other in any one layer. We can further increase the packing efficiency by placing the next layer not directly on top of the first, but again offset so that any one atom actually sits in the indentation formed by three atoms in the layer beneath it, as shown here. This kind of packing leads to two different crystal structures called closest-packed structures, both of which have packing efficiencies of 74% and coordination numbers of 12. In the first of these two closest-packed structures—called hexagonal closest-packing—the third layer of atoms aligns exactly on top of the first, as shown at the bottom of this page (on the left). The pattern from one layer to the next is ABAB . . . with alternating layers aligning exactly on top of one another. Notice that the central atom in layer B of this structure is touching 6 atoms in its own layer, 3 atoms in the layer above it, and 3 atoms in the layer below, for a coordination number of 12. The unit cell for this crystal structure is not a cubic unit cell, but a hexagonal one. In the second of the two closest-packed structures—called cubic closest-packing— the third layer of atoms is offset from the first, as shown below and to the right. The pattern from one layer to the next is ABCABC . . . with every fourth layer aligning with the first. Although not easy to visualize, the unit cell for cubic closest-packing is the face-centered cubic unit cell, as shown in FIGure 11.40▲. Therefore the cubic closestpacked structure is identical to the face-centered cubic unit cell structure

The Clausius-Clapeyron Equation

As we can see from the graph in Figure 11.22, the vapor pressure of a liquid increases with increasing temperature. However, the relationship is not linear. In other words, doubling the temperature results in more than a doubling of the vapor pressure. The relationship between vapor pressure and temperature is exponential and can be expressed as follows: Pvap = b exp a -∆Hvap RT b In this expression Pvap is the vapor pressure, b is a constant that depends on the gas, ∆Hvap is the heat of vaporization, R is the gas constant (8.314 J>mol # K), and T is the temperature in kelvins. We can rearrange Equation 11.1 by taking the natural logarithm of both sides as follows: ln Pvap = lnc b exp a -∆Hvap RT b d Since ln AB = ln A + ln B, we can rearrange the right side of Equation 11.2. ln Pvap = ln b + lnc exp a -∆Hvap RT b d Since ln ex = x (see Appendix IB), we can simplify Equation 11.3. ln Pvap = ln b + -∆Hvap RT A slight additional rearrangement gives us the following important result: ln Pvap = -∆Hvap R a 1 T b + ln b y = m(x) + b Notice the parallel relationship between the Clausius-Clapeyron equation and the equation for a straight line. Just as a plot of y versus x yields a straight line with slope m and intercept b, so a plot of ln Pvap (equivalent to y) versus 1>T (equivalent to x) gives a straight line with slope -∆Hvap>R (equivalent to m) and y-intercept ln b (equivalent to b), as shown in FIGure 11.25▼. The Clausius-Clapeyron equation gives a linear relationship—not between the vapor pressure and the temperature (which have an exponential relationship)—but between the natural log of the vapor pressure and the inverse of temperature. This is a common technique in the analysis of chemical data. If two variables are not linearly related, it is often convenient to find ways to graph functions of those variables that are linearly related. The Clausius-Clapeyron equation leads to a convenient way to measure the heat of vaporization in the laboratory. We measure the vapor pressure of a liquid as a function of temperature and create a plot of the natural log of the vapor pressure versus the inverse of the temperature. We can then determine the slope of the line to find the heat of vaporization, The Clausius-Clapeyron equation can also be expressed in a two-point form that can be used with just two measurements of vapor pressure and temperature to determine the heat of vaporization. We can also use this form of the equation to predict the vapor pressure of a liquid at any temperature if we know the enthalpy of vaporization and the normal boiling point (or the vapor pressure at some other temperature)

Sublimation and Fusion

In Section 11.5, we examined a beaker of liquid water at room temperature from the molecular viewpoint. Now, let's examine a block of ice at -10 °C from the same molecular perspective, paying close attention to two common processes: sublimation and fusion. Sublimation Even though a block of ice is solid, the water molecules have thermal energy that causes each molecule to vibrate about a fixed point. The motion is much less than in a liquid, but significant nonetheless. Like the molecules in liquids, at any one instant, some molecules in a solid block of ice have more thermal energy than the average and some have less. The molecules with high enough thermal energy can break free from the ice surface—where molecules are held less tightly than in the interior due to fewer neighbor-neighbor interactions—and go directly into the gas state FIGure 11.27▶. This process is sublimation, the transition from solid to gas. Some of the water molecules in the gas state (those at the low end of the energy distribution curve for the gaseous molecules) collide with the surface of the ice and are captured by the intermolecular forces with other molecules. This process— the opposite of sublimation—is deposition, the transition from gas to solid. As is the case with liquids, the pressure of a gas in dynamic equilibrium with its solid is the vapor pressure of the solid. Although both sublimation and deposition are happening on the surface of an ice block open to the atmosphere at -10 °C, sublimation is usually happening at a greater rate than deposition because most of the newly sublimed molecules escape into the surrounding atmosphere and never come back. The result is a noticeable decrease in the size of the ice block over time (even though the temperature is below the melting point). If you live in a cold climate, you may have noticed the disappearance of ice and snow from the ground even when the temperature remains below 0 °C. Similarly, ice cubes left in the freezer for a long time slowly shrink, even though the freezer is always set below 0 °C. In both cases, the ice is subliming, turning directly into water vapor. Ice also sublimes out of frozen foods. You may have noticed, for example, the gradual growth of ice crystals on the inside of airtight plastic food-storage bags in your freezer. The ice crystals are composed of water that has sublimed out of the food and redeposited on the surface of the bag or on the surface of the food. For this reason, food that remains frozen for too long dries out. Such dehydration can be avoided to some degree by freezing foods to colder temperatures, a process called deep-freezing. The colder temperature lowers the vapor pressure of ice and preserves the food longer. Freezer burn on meats is another common manifestation of sublimation. When meat is improperly stored (that is, when its container is not airtight) sublimation continues unabated. The surface of the meat becomes dehydrated and discolored and loses flavor and texture. A substance commonly associated with sublimation is solid carbon dioxide or dry ice, which does not melt under atmospheric pressure no matter what the temperature. However, at -78 °C the CO2 molecules have enough energy to leave the surface of the dry ice and become gaseous through sublimation.

Crystalline Solids: Band Theory

In Section 9.11, we explored a model for bonding in metals called the electron sea model. We now turn to a model for bonding in solids that is both more sophisticated and more broadly applicable—it applies to both metallic solids and covalent solids. The band theory model is an extension of molecular orbital theory, first covered in Section 10.8. Recall that in molecular orbital theory, we combined the atomic orbitals of the atoms within a molecule to form molecular orbitals. These molecular orbitals are not localized on individual atoms, but delocalized over the entire molecule. Similarly, in band theory, we combine the atomic orbitals of the atoms within a solid crystal to form orbitals that are not localized on individual atoms, but delocalized over the entire crystal. In some sense then, the crystal is like a very large molecule and its valence electrons occupy the molecular orbitals formed from the atomic orbitals of each atom in the crystal. We begin our discussion of band theory by considering a series of molecules constructed from individual lithium atoms. The energy levels of the atomic orbitals and resulting molecular orbitals for Li, Li2, Li3, Li4, and LiN (where N is a large number on the order of 1023) are shown in FIGure 11.50▼. The lithium atom has a single electron in a single 2s atomic orbital. The Li2 molecule contains two electrons and two molecular orbitals. The electrons occupy the lower energy bonding orbital—the higher energy, or antibonding, molecular orbital is empty. The Li4 molecule contains four electrons and four molecular orbitals. The electrons occupy the two bonding molecular orbitals—the two antibonding orbitals are empty. The LiN molecule contains N electrons and N molecular orbitals. However, because there are so many molecular orbitals, the energy spacings between them are infinitesimally small; they are no longer discrete energy levels but instead form a band of energy levels. Half of the orbitals in the band (N>2) are bonding molecular orbitals and (at 0 K) contain the N valence electrons. The other N>2 molecular orbitals are antibonding and (at 0 K) are completely empty. If the atoms composing a solid have p orbitals available, then the same process leads to another band of orbitals at higher energies. In band theory, electrons become mobile when they make a transition from the highest occupied molecular orbital into higher energy empty molecular orbitals. For this reason, the occupied molecular orbitals are called the valence band and the unoccupied orbitals are called the conduction band. In lithium metal, the highest occupied molecular orbital lies in the middle of a band of orbitals, and the energy difference between it and the next higher energy orbital is infinitesimally small. Therefore, above 0 K, electrons can easily make the transition from the valence band to the conduction band. Since electrons in the conduction band are mobile, lithium, like all metals, is a good electrical conductor. Mobile electrons in the conduction band are also responsible for the thermal conductivity of metals. When a metal is heated, electrons are excited to higher energy molecular orbitals. These electrons can then quickly transport the thermal energy throughout the crystal lattice. In metals, the valence band and conduction band are always energetically continuous—the energy difference between the top of the valence band and the bottom of the conduction band is infinitesimally small. In semiconductors and insulators, however, an energy gap, called the band gap, exists between the valence band and conduction band as shown in FIGure 11.51▲. In insulators, the band gap is large, and electrons are not promoted into the conduction band at ordinary temperatures, resulting in no electrical conductivity. In semiconductors, the band gap is small, allowing some electrons to be promoted at ordinary temperatures and resulting in limited conductivity. However, the conductivity of semiconductors can be controlled by adding minute amounts of other substances to the semiconductor. These substances, called dopants, are minute impurities that result in additional electrons in the conduction band or electron holes in the valence band. The addition or subtraction of electrons affects the conductivity.

Water, No Gravity

In the space station there are no spills. When an astronaut squeezes a full water bottle, the water squirts out like it does on Earth, but instead of falling to the floor and forming a puddle, the water sticks together to form a floating, oscillating, blob of water. Over time, the blob stops oscillating and forms a nearly perfect sphere. Why? The reason is the main topic of this chapter: intermolecular forces, the attractive forces that exist among the particles that compose matter. The molecules that compose water are attracted to one another, much like a collection of small magnets are attracted to each other. These attractions hold the water together as a liquid (instead of a gas) at room temperature. They also cause samples of water to clump together into a blob; a behavior we can see most clearly in the absence of gravity. Over time, irregularities in the shape of the blob smooth out, and the blob becomes a sphere. The sphere is the geometrical shape with the lowest surface area to volume ratio. By forming a sphere, the water molecules maximize their interactions with one another, because the sphere minimizes the number of molecules at the surface of the liquid, where fewer interactions occur (compared to the interior of the liquid). Intermolecular forces exist, not only among water molecules, but among all particles that compose matter. We can see the effect of these attractive forces in the image at left, which shows an astronaut touching a floating blob of water in the absence of gravity. Notice how the water sticks to the astronaut's finger. The water molecules experience an attractive force to the molecules that compose skin. This attractive force deforms the entire blob of water. Intermolecular forces exist among all the particles that compose matter. Intermolecular forces are responsible for the very existence of the condensed states. The state of a sample of matter—solid, liquid, or gas—depends on the magnitude of intermolecular forces among the constituent particles relative to the amount of thermal energy in the sample. Recall from Chapter 6 that the molecules and atoms composing matter are in constant random motion that increases with increasing temperature. The energy associated with this motion is thermal energy. When thermal energy is high relative to intermolecular forces, matter tends to be gaseous. When thermal energy is low relative to intermolecular forces, matter tends to be liquid or solid.

Ionic Solids

Ionic solids are solids whose composite units are ions. Table salt (NaCl) and calcium fluoride (CaF2) are examples of ionic solids. Ionic solids are held together by the coulombic interactions that occur between the cations and anions occupying the lattice sites in the crystal. The coordination number of the unit cell for an ionic compound, therefore, represents the number of close cation-anion interactions. Since these interactions lower potential energy, the crystal structure of a particular ionic compound is the one that maximizes the coordination number, while accommodating both charge neutrality (each unit cell must be charge neutral) and the different sizes of the cations and anions that compose the particular compound. In general, the more similar the radii of the cation and the anion, the higher the coordination number. Cesium chloride (CsCl) is a good example of an ionic compound containing cations and anions of similar size (Cs+ radius = 167 pm; Cl- radius = 181 pm). In the cesium chloride structure, the chloride ions occupy the lattice sites of a simple cubic cell and one cesium ion lies in the very center of the cell, as shown in FIGure 11.42◀. The coordination number is 8; each cesium ion is in direct contact with eight chloride ions (and vice versa). Notice that the cesium chloride unit cell contains one chloride anion [(8 * 1 8) = 1] and one cesium cation (the cesium ion in the middle belongs entirely to the unit cell and complete chloride atoms are shown even though only a fraction of each is part of the single unit cell) for a ratio of Cs to Cl of 1:1, just as in the formula for the compound. Calcium sulfide (CaS) adopts the same structure as cesium chloride. The crystal structure of sodium chloride must accommodate the more disproportionate sizes of Na+(radius = 95 pm) and Cl-(radius = 181 pm). If ion size were the only consideration, the larger chloride anion could theoretically fit many of the smaller sodium cations around it, but charge neutrality requires that each sodium cation be surrounded by an equal number of chloride anions. Therefore, the coordination number is limited by the number of chloride anions that can fit around the relatively small sodium cation. The structure that minimizes the energy is shown in FIGure 11.43◀ and has a coordination number of 6 (each chloride anion is surrounded by six sodium cations and vice versa). We can visualize this structure, called the rock salt structure, as chloride anions occupying the lattice sites of a face-centered cubic structure with the smaller sodium cations occupying the holes between the anions. (Alternatively, we can visualize this structure as the sodium cations occupying the lattice sites of a face-centered cubic structure with the larger chloride anions occupying the spaces between the cations.) Each unit cell contains four chloride anions [(8 * 1 8) + (6 * 1 2) = 4] and four sodium cations [(12 * 1 4) + 1 = 4], resulting in a ratio of 1:1, just as in the formula of the compound. Other compounds that exhibit the sodium chloride structure include LiF, KCl, KBr, AgCl, MgO, and CaO. A greater disproportion between the sizes of the cations and anions in a compound makes a coordination number of even 6 physically impossible. For example, in ZnS (Zn2+ radius = 74 pm; S2- radius = 184 pm) the crystal structure, shown in FIGure 11.44◀, has a coordination number of only 4. We can visualize this structure, called the zinc blende structure, as sulfide anions occupying the lattice sites of a face-centered cubic structure with the smaller zinc cations occupying four of the eight tetrahedral holes located beneath each corner atom (towards the interior of the cell). A tetrahedral hole is the empty space that lies in the center of a tetrahedral arrangement of four atoms, as shown in the margin. Each unit cell contains four sulfide anions 318 * 1 8 2 + 16 * 1 2 2 = 44 and four zinc cations (each of the four zinc cations is completely contained within the unit cell), resulting in a ratio of 1:1, just as in the formula of the compound. Other compounds exhibiting the zinc blende structure include CuCl, AgI, and CdS. When the ratio of cations to anions is not 1:1, the crystal structure must accommodate the unequal number of cations and anions. Many compounds that contain a cation-to-anion ratio of 1:2 adopt the fluorite (CaF2) structure shown in FIGure 11.45▶. We can visualize this structure as calcium cations occupying the lattice sites of a face-centered cubic structure with the larger fluoride anions occupying all eight of the tetrahedral holes located beneath each corner atom. Each unit cell contains four calcium cations 318 * 1 8 2 + 16 * 1 2 2 = 44 and eight fluoride anions (each of the eight fluoride anions is completely contained within the unit cell), resulting in a cation-to-anion ratio of 1:2, just as in the formula of the compound. Other compounds exhibiting the fluorite structure include PbF2, SrF2, and BaCl2. Compounds with a cation-to-anion ratio of 2:1 often exhibit the antifluorite structure, in which the anions occupy the lattice sites of a face-centered cubic structure and the cations occupy the tetrahedral holes beneath each corner atom. Because the forces holding ionic solids together are strong coulombic forces (or ionic bonds), and because these forces are much stronger than the intermolecular forces discussed previously, ionic solids tend to have much higher melting points than molecular solids. For example, sodium chloride melts at 801 °C, while carbon disulfide (CS2) —a molecular solid with a higher molar mass—melts at -110 °C.

Fusion

Let's return to our ice block and examine what happens at the molecular level as we increase its temperature. The increasing thermal energy causes the water molecules to vibrate faster and faster. At the melting point (0 °C for water), the molecules have enough thermal energy to overcome the intermolecular forces that hold them at their stationary points, and the solid turns into a liquid. This process is melting or fusion, the transition from solid to liquid. The opposite of melting is freezing, the transition from liquid to solid. Once the melting point of a solid is reached, additional heating only causes more rapid melting; it does not raise the temperature of the solid above its melting point FIGure 11.28▼. Only after all of the ice has melted does additional heating raise the temperature of the liquid water past 0 °C. A mixture of water and ice always has a temperature of 0 °C (at 1 atm pressure).

Capillary Action

Medical technicians take advantage of capillary action—the ability of a liquid to flow against gravity up a narrow tube—when taking a blood sample. The technician pokes the patient's finger with a pin, squeezes some blood out of the puncture, and collects the blood with a thin tube. When the tube's tip comes into contact with the blood, the blood is drawn into the tube by capillary action. The same force plays a role in how plants draw water from the soil. Capillary action results from a combination of two forces: the attraction between molecules in a liquid, called cohesive forces, and the attraction between these molecules and the surface of the tube, called adhesive forces. The adhesive forces cause the liquid to spread out over the interior surface of the tube, while the cohesive forces cause the liquid to stay together. If the adhesive forces are greater than the cohesive forces (as is the case for the blood in a glass tube), the attraction to the surface draws the liquid up the tube while the cohesive forces pull along those molecules not in direct contact with the tube walls. The blood rises up the tube until the force of gravity balances the capillary action— the thinner the tube, the higher the rise. If the adhesive forces are smaller than the cohesive forces (as is the case for liquid mercury), the liquid does not rise up the tube at all. We can see the result of the differences in the relative magnitudes of cohesive and adhesive forces by comparing the meniscus of water to the meniscus of mercury FIGure 11.16◀. (The meniscus is the curved shape of a liquid surface within a tube.) The meniscus of water is concave because the adhesive forces are greater than the cohesive forces, causing the edges of the water to creep up the sides of the tube a bit, forming the familiar cupped shape. The meniscus of mercury is convex because the cohesive forces— due to metallic bonding between the atoms—are greater than the adhesive forces. The mercury atoms crowd toward the interior of the liquid to maximize their interactions with each other, resulting in the upward bulge at the center of the liquid surface. ▲ FIGure 11.16 Meniscuses of Water and Mercury The meniscus of water (dyed red for visibility at left) is concave because water molecules are more strongly attracted to the glass wall than to one another. The meniscus of mercury is convex because mercury atoms are more strongly attracted to one another than to the glass walls.

Changes between States

One state of matter can be transformed to another by changing the temperature, pressure, or both. For example, solid ice can be converted to liquid water by heating, and liquid water can be converted to solid ice by cooling. The diagram shown here illustrates the three states of matter and the changes in conditions that commonly induce transitions among them. Notice that transitions between the liquid and gas state can be achieved not only by heating and cooling, but also through changes in pressure. In general, increases in pressure favor the denser state, so increasing the pressure of a gas sample results in a transition to the liquid state. The most familiar example of this phenomenon is the LP (liquefied petroleum) gas used as a fuel for outdoor grills and lanterns. LP gas is mostly propane, which is a gas at room temperature and atmospheric pressure. However, it liquefies at pressures exceeding about 2.7 atm. The propane you buy in a tank is under pressure and therefore in the liquid form. When you open the tank, some of the propane escapes as a gas, lowering the pressure in the tank for a brief moment. Immediately, however, some of the liquid propane evaporates, replacing the gas that escaped. Storing gases like propane as liquids is efficient because, in their liquid form, they occupy much less space.

Hydrogen Bonding

Polar molecules containing hydrogen atoms bonded directly to small electronegative atoms—most importantly fluorine, oxygen, or nitrogen—exhibit an intermolecular force called hydrogen bonding. HF, NH3, and H2O all undergo hydrogen bonding. The hydrogen bond is a sort of super dipole-dipole force. The large electronegativity difference between hydrogen and these electronegative elements means that the H atom has a fairly large partial positive charge (d+) when bonded to F, O, or N, while the F, O, or N atom has a fairly large partial negative charge (d-). In addition, since these atoms are all quite small, they can approach one another very closely. The result is a strong attraction between the hydrogen in each of these molecules and the F, O, or N on its neighboring molecule, an attraction called a hydrogen bond. For example, in HF, the hydrogen is strongly attracted to the fluorine on neighboring molecules Figure 11.9▲. The electrostatic potential maps in Figure 11.9 illustrate the large differences in electron density that result in unusually large partial charges. Hydrogen bonds should not be confused with chemical bonds. Chemical bonds occur between individual atoms within a molecule, whereas hydrogen bonds—like dispersion forces and dipole-dipole forces—are intermolecular forces that occur between molecules. A typical hydrogen bond is only 2-5% as strong as a typical covalent chemical bond. Hydrogen bonds are, however, the strongest of the three intermolecular forces we have discussed so far. Substances composed of molecules that form hydrogen bonds have higher melting and boiling points than substances composed of molecules that do not form hydrogen bonds. For example, consider melting and boiling points of ethanol and dimethyl ether tabulated here: Since ethanol contains hydrogen bonded directly to oxygen, ethanol molecules form hydrogen bonds with each other as shown in Figure 11.10◀. Consequently, ethanol is a liquid at room temperature. Dimethyl ether, in contrast, has an identical molar mass but does not exhibit hydrogen bonding because the oxygen atom is not bonded directly to hydrogen, resulting in lower boiling and melting points. Dimethyl ether is a gas at room temperature. Water is another good example of a molecule with hydrogen bonding (Figure 11.11▶). Figure 11.12▶ shows the boiling points of the simple hydrogen compounds of the group 4A and group 6A elements. Notice that, in general, boiling points increase with increasing molar mass, as expected based on increasing dispersion forces. However, because of hydrogen bonding, the boiling point of water (100 °C) is much higher than expected based on its molar mass (18.0 g>mol). Without hydrogen bonding, all the water on our planet would likely be gaseous.

Crystalline Solids: Unit Cells and Basic Structures

Solids may be crystalline (having a well-ordered array of atoms or molecules) or amorphous (having no long-range order). Crystalline solids are composed of atoms or molecules arranged in structures with long-range order (see Section 11.2). If you have ever visited the mineral section of a natural history museum and seen crystals with smooth faces and well-defined angles between them, or if you have carefully observed the hexagonal shapes of snowflakes, you have witnessed some of the effects of the underlying order in crystalline solids. The often beautiful geometric shape that you see on the macroscopic scale is the result of a specific structural arrangement—called the crystalline lattice—on the molecular and atomic scale. The crystalline lattice of any solid is nature's way of aggregating the particles to minimize their energy. The crystalline lattice can be represented by a small collection of atoms, ions, or molecules called the unit cell. When the unit cell is repeated over and over—like the tiles of a floor or the pattern in a wallpaper design, but in three dimensions—the entire lattice is reproduced. For example, consider the two-dimensional crystalline lattice shown below. The unit cell for this lattice is the dark-colored square. Each circle represents a lattice point, a point in space occupied by an atom, ion, or molecule. By repeating and moving the pattern in the square throughout the two-dimensional space, we can generate the entire lattice. Unit cells are often classified by their symmetry, and many different unit cells exist. In this book, we focus primarily on cubic unit cells (although we look at one hexagonal unit cell). Cubic unit cells are characterized by equal edge lengths and 90° angles at their corners. The three cubic unit cells—simple cubic, body-centered cubic, and face-centered cubic—along with some of their basic characteristics, are shown in FIGure 11.35▼. The simple cubic unit cell FIGure 11.36▶ consists of a cube with one atom at each corner. The atoms touch along each edge of the cube, so the edge length is twice the radius of the atoms (l = 2r). Note that even though the unit cell may seem to contain eight atoms, it actually contains only one. Each corner atom is shared by eight other unit cells. Any single unit cell actually contains only one-eighth of each of the eight atoms at its corners, for a total of only one atom per unit cell. A characteristic feature of any unit cell is the coordination number, the number of atoms with which each atom is in direct contact. The coordination number represents the number of atoms with which a particular atom can have a strong interaction. The coordination number for the simple cubic unit cell is 6, because any one atom touches only six others, as we can see in Figure 11.36. A quantity closely related to the coordination number is the packing efficiency, the percentage of the volume of the unit cell occupied by the spheres. The higher the coordination number, the greater the packing efficiency. The simple cubic unit cell has a packing efficiency of 52%—there is a lot of empty space in the simple cubic unit cell. The body-centered cubic unit cell FIGure 11.37▼ consists of a cube with one atom at each corner and one atom of the same kind in the very center of the cube. Note that in the body-centered unit cell, the atoms do not touch along each edge of the cube, but rather touch along the diagonal line that runs from one corner, through the middle of the cube, to the opposite corner. The edge length in terms of the atomic radius is therefore l = 4r> 23, as shown on the next page. The body-centered unit cell contains two atoms per unit cell because the center atom is not shared with any other neighboring cells. The coordination number of the body-centered cubic unit cell is 8, which we can see by observing the atom in the very center of the cube, which touches the eight atoms at the corners. The packing efficiency is 68%, significantly higher than for the simple cubic unit cell. In this structure, any one atom strongly interacts with more atoms than in the simple cubic unit cell. The face-centered cubic unit cell FIGure 11.38▼ is characterized by a cube with one atom at each corner and one atom of the same kind in the center of each cube face. Note that in the face-centered unit cell (like the body-centered unit cell), the atoms do not touch along each edge of the cube. Instead, the atoms touch along the face diagonal. The edge length in terms of the atomic radius is therefore l = 222r, as shown at left. The face-centered unit cell contains four atoms per unit cell because the center atoms on each of the six faces are shared between two unit cells. So there are 1 2 * 6 = 3 face-centered atoms plus 1 8 * 8 = 1 corner atom, for a total of four atoms per unit cell. The coordination number of the face-centered cubic unit cell is 12, and its packing efficiency is 74%. In this structure, any one atom strongly interacts with more atoms than in either the simple cubic unit cell or the body-centered cubic unit cell. examPle 11.6 relating Density to Crystal Structure Aluminum crystallizes with a face-centered cubic unit cell. The radius of an aluminum atom is 143 pm. Calculate the density of solid crystalline aluminum in g>cm3. SORT You are given the radius of an aluminum atom and its crystal structure. You are asked to find the density of solid aluminum. STRATEGIZE The conceptual plan is based on the definition of density. Since the unit cell has the physical properties of the entire crystal, find the mass and volume of the unit cell and use these to calculate its density. SOLVE Begin by finding the mass of the unit cell. Obtain the mass of an aluminum atom from its molar mass. Since the facecentered cubic unit cell contains four atoms per unit cell, multiply the mass of aluminum by 4 to get the mass of a unit cell. Next, calculate the edge length (l) of the unit cell (in m) from the atomic radius of aluminum. For the face-centered cubic structure, l = 222r. Calculate the volume of the unit cell (in cm) by converting the edge length to cm and cubing the edge length. (Use centimeters to report the density in units of g>cm3). Finally, calculate the density by dividing the mass of the unit cell by the volume of the unit cell. CHECK The units of the answer are correct. The magnitude of the answer is reasonable because the density is greater than 1 g>cm3 (as you would expect for metals), but still not too high (aluminum is a low-density metal).

Atomic Solids

Solids whose composite units are individual atoms are atomic solids. Solid xenon (Xe), iron (Fe), and silicon dioxide (SiO2) are examples of atomic solids. As we saw in Figure 11.41, atomic solids can themselves be divided into three categories—nonbonding atomic solids, metallic atomic solids, and network covalent atomic solids—each held together by a different kind of force. Nonbonding atomic solids, a group that consists of only the noble gases in their solid form, are held together by relatively weak dispersion forces. In order to maximize these interactions, nonbonding atomic solids form closest-packed structures, maximizing their coordination numbers and minimizing the distance between atoms. Nonbonding atomic solids have very low melting points, which increase uniformly with molar mass. Argon, for example, has a melting point of -189 °C and xenon has a melting point of -112 °C. Metallic atomic solids, such as iron or gold, are held together by metallic bonds, which in the simplest model are represented by the interaction of metal cations with the "sea" of electrons that surrounds them, as described in Section 9.11 FIGure 11.46▶. Since metallic bonds are not directional, metals also tend to form closest-packed crystal structures. For example, nickel crystallizes in the cubic closest-packed structure, and zinc crystallizes in the hexagonal closest-packed structure FIGure 11.47▼. Metallic bonds are of varying strengths. Some metals, such as mercury, have melting points below room temperature; other metals, such as iron, have relatively high melting points (iron melts at 1538 °C). Network covalent atomic solids, such as diamond, graphite, and silicon dioxide, are held together by covalent bonds. The crystal structures of these solids are restricted by the geometrical constraints of the covalent bonds (which tend to be more directional than intermolecular forces, ionic bonds, or metallic bonds), so they do not tend to form closest-packed structures. In diamond FIGure 11.48(a)◀, each carbon atom forms four covalent bonds to four other carbon atoms in a tetrahedral geometry. This structure extends throughout the entire crystal, so that a diamond crystal can be thought of as a giant molecule, held together by these covalent bonds. Since covalent bonds are very strong, covalent atomic solids have high melting points. Diamond is estimated to melt at about 3550 °C. The electrons in diamond are confined to the covalent bonds and are not free to flow. Therefore diamond does not conduct electricity. In graphite FIGure 11.48(b)◀, carbon atoms are arranged in sheets. Within each sheet, carbon atoms are covalently bonded to each other by a network of sigma and pi bonds. The electrons within the pi bonds are delocalized over the entire sheet, making graphite a good electrical conductor along the sheets. The bond length between carbon atoms within a sheet is 142 pm. However, the bonding between sheets is much different. The separation between sheets is 341 pm. There are no covalent bonds between sheets, only relatively weak dispersion forces. Consequently, the sheets slide past each other relatively easily, which explains the slippery feel of graphite and its extensive use as a lubricant. The silicates (extended arrays of silicon and oxygen) are the most common network covalent atomic solids. Geologists estimate that 90% of Earth's crust is composed of silicates. The basic silicon oxygen compound is silica (SiO2), which in its most common crystalline form is called quartz. The structure of quartz consists of an array of SiO4 tetrahedra with shared oxygen atoms, as shown in FIGure 11.49(a)▼. The strong silicon-oxygen covalent bonds that hold quartz together result in its high melting point of about 1700 °C. Common glass is also composed of SiO2, but in its amorphous form FIGure 11.49(b)▼.

Heat of Vaporization

The amount of heat required to vaporize one mole of a liquid to gas is the heat (or enthalpy) of vaporization (∆Hvap). The heat of vaporization of water at its normal boiling point of 100 °C is +40.7 kJ>mol: H2O(l) S H2O(g) ∆Hvap = +40.7 kJ>mol The heat of vaporization is always positive because the process is endothermic—energy must be absorbed to vaporize a substance. The heat of vaporization is somewhat temperature dependent. For example, at 25 °C the heat of vaporization of water is +44.0 kJ>mol, slightly more than heat of vaporization at 100 °C because the water contains less thermal energy. Table 11.7 lists the heats of vaporization of several liquids at their boiling points and at 25 °C. When a substance condenses from a gas to a liquid, the same amount of heat is involved, but the heat is emitted rather than absorbed. H2O(g) S H2O(l) ∆H = -∆Hvap = -40.7 kJ (at 100 °C) When one mole of water condenses, it releases 40.7 kJ of heat. The sign of ∆H in this case is negative because the process is exothermic. We can use the heat of vaporization of a liquid to calculate the amount of heat energy required to vaporize a given mass of the liquid (or the amount of heat given off by the condensation of a given mass of liquid), using concepts similar to those covered in Section 6.5 (stoichiometry of ∆H). We essentially use the heat of vaporization as a conversion factor between number of moles of a liquid and the amount of heat required to vaporize it (or the amount of heat emitted when it condenses), as demonstrated in Example 11.3. EXAMPlE 11.3 Using the Heat of Vaporization in Calculations Calculate the mass of water (in g) that can be vaporized at its boiling point with 155 kJ of heat. SORT You are given a certain amount of heat in kilojoules and asked to find the mass of water that can be vaporized. STRATEGIZE The heat of vaporization gives the relationship between heat absorbed and moles of water vaporized. Begin with the given amount of heat (in kJ) and convert to moles of water that can be vaporized. Then use the molar mass as a conversion factor to convert from moles of water to mass of water. SOLVE Follow the conceptual plan to solve the problem.

Dipole-Dipole Force

The dipole-dipole force exists in all molecules that are polar. Polar molecules have electron-rich regions (which have a partial negative charge) and electron-deficient regions (which have a partial positive charge). For example, consider acetone, which is shown here: The image on the right is an electrostatic potential map of acetone; we introduced these maps in Section 10.5. Recall that the red-pink areas indicate electron-rich regions in the molecule and that the blue-green areas indicate electron-poor regions. Notice that acetone has an electron-rich region surrounding the oxygen atom (because oxygen is more electronegative than the rest of the molecule), and electron-poor regions surrounding the carbon and hydrogen atoms. As a result, acetone has a permanent dipole, which can interact with other acetone molecules as shown in Figure 11.6▶. Example 11.1 shows how to determine if a compound exhibits dipole-dipole forces. examPle 11.1 Dipole-Dipole Forces Determine whether each molecule has dipole-dipole forces. (a) CO2 (b) CH2Cl2 (c) CH4 SOLUTION A molecule has dipole-dipole forces if it is polar. To determine whether a molecule is polar, (1) determine whether the molecule contains polar bonds and (2) determine whether the polar bonds add together to form a net dipole moment (Section 10.5) (a) CO2 (1) The electronegativity of carbon is 2.5 and that of oxygen is 3.5 (Figure 9.7), so CO2 has polar bonds. (2) The geometry of CO2 is linear. Consequently, the dipoles of the polar bonds cancel, so the molecule is not polar and does not have dipole-dipole forces (b) CH2Cl2 (1) The electronegativity of C is 2.5, that of H is 2.1, and that of Cl is 3.5. Consequently, CH2Cl2 has two polar bonds (C¬Cl) and two bonds that are nearly nonpolar (C¬H). (2) The geometry of CH2Cl2 is tetrahedral. Since the C¬Cl bonds and the C¬H bonds are different, their dipoles do not cancel but sum to a net dipole moment. Therefore the molecule is polar and has dipole-dipole forces. (c) CH4 (1) The electronegativity of C is 2.5 and that of hydrogen is 2.1, so the C¬H bonds are nearly nonpolar. (2) In addition, since the geometry of the molecule is tetrahedral, any slight polarities that the bonds might have will cancel. CH4 is therefore nonpolar and does not have dipole-dipole forces. Polar molecules have higher melting and boiling points than nonpolar molecules of similar molar mass. Remember that all molecules (including polar ones) have dispersion forces. Polar molecules have, in addition, dipole-dipole forces. This additional attractive force raises the melting and boiling points of polar molecules relative to nonpolar molecules of similar molar mass. For example, consider the following two compounds: Formaldehyde is polar and therefore has a higher melting point and boiling point than nonpolar ethane, even though the two compounds have the same molar mass. shows the boiling points of a series of molecules with similar molar mass but progressively greater dipole moments. Notice that the boiling points increase with increasing dipole moment. The polarity of molecules composing liquids is also important in determining the miscibility—the ability to mix without separating into two phases—of liquids. In general, polar liquids are miscible with other polar liquids but are not miscible with nonpolar liquids. For example, water, a polar liquid, is not miscible with pentane (C5H12), a nonpolar liquid FIGure 11.8▶. Similarly, water and oil (nonpolar) do not mix. Consequently, oily hands or oily stains on clothes cannot be washed with plain water. The water does not mix with the oil.

Ion-Dipole Force

The ion-dipole force occurs when an ionic compound is mixed with a polar compound and is especially important in aqueous solutions of ionic compounds. For example, when sodium chloride is mixed with water, the sodium and chloride ions interact with water molecules via ion-dipole forces, as shown in FIGure 11.13◀. Notice that the positive sodium ions interact with the negative poles of water molecules, while the negative chloride ions interact with the positive poles. Ion-dipole forces are the strongest of the types of intermolecular forces that we have discussed, and these forces are responsible for the ability of ionic substances to form solutions with water. We will discuss aqueous solutions more thoroughly in Chapter 12. Table 11.4 summarizes the intermolecular forces we have discussed.

Energetics of Melting and Freezing

The most common way to cool a beverage quickly is to drop several ice cubes into it. As the ice melts, the drink cools because melting is endothermic—the melting ice absorbs heat from the liquid. The amount of heat required to melt 1 mol of a solid is the heat of fusion (∆Hfus). The heat of fusion for water is 6.02 kJ>mol: H2O(s) S H2O(l) ∆Hfus = 6.02 kJ>mol The heat of fusion is positive because melting is endothermic. Freezing, the opposite of melting, is exothermic—heat is released when a liquid freezes into a solid. For example, as water in a freezer turns into ice, it releases heat, which must be removed by the refrigeration system of the freezer. If the refrigeration system did not remove the heat, the water would not completely freeze into ice—the heat released as the water began to freeze would warm the freezer, preventing further freezing. The change in enthalpy for freezing has the same magnitude as the heat of fusion but the opposite sign. H2O(l) S H2O(s) ∆H = -∆Hfus = -6.02 kJ>mol Different substances have different heats of fusion. Table 11.8 lists some of these. In general, the heat of fusion is significantly less than the heat of vaporization, as shown in FIGure 11.29▶. We have already seen that the solid and liquid states are closer to each other in many ways than they are to the gas state. It takes less energy to melt 1 mol of ice into liquid than it does to vaporize 1 mol of liquid water into gas because vaporization requires complete separation of molecules from one another, which means that the intermolecular forces must be completely overcome. Melting, however, requires that intermolecular forces be only partially overcome; molecules in a liquid move around one another while still remaining in contact.

Intermolecular Forces in Action: Surface Tension, Viscosity, and Capillary Action

The most important manifestation of intermolecular forces is the very existence of liquids and solids. In liquids, we also observe several other manifestations of intermolecular forces including surface tension, viscosity, and capillary action. Surface Tension A fly fisherman delicately casts a small metal fly (a hook with a few feathers and strings attached to make it look like a fly) onto the surface of a moving stream. The fly floats on the surface of the water—even though the metal composing the hook is denser than water—and attracts trout. Why? The hook floats because of surface tension, which results from the tendency of liquids to minimize their surface area. We can understand surface tension by carefully examining FIGure 11.14▼, which depicts the intermolecular forces experienced by a molecule at the surface of the liquid compared to those experienced by a molecule in the interior. Notice that a molecule at the surface has relatively fewer neighbors with which to interact—and is therefore inherently less stable because it has higher potential energy—than one in the interior. (Remember that the attractive interactions with other molecules lower potential energy.) In order to increase the surface area of the liquid, molecules from the interior have to be moved to the surface, and, since molecules at the surface have a higher potential energy than those in the interior, this movement requires energy. Therefore, liquids tend to minimize their surface area. The surface tension of a liquid is the energy required to increase the surface area by a unit amount. For example, at room temperature, water has a surface tension of 72.8 mJ>m2 —it takes 72.8 mJ to increase the surface area of water by one square meter. Why does surface tension allow the fly fisherman's hook to float on water? The tendency for liquids to minimize their surface creates a kind of skin at the surface that resists penetration. For the hook to sink into the water, the water's surface area must increase slightly—an increase that is resisted by the surface tension. A fisherman's fly, even though it is denser than water, floats on the surface of the water. A slight tap on the fly will overcome the surface tension and cause it to sink. Surface tension decreases with decreasing intermolecular forces. We could not float a fisherman's fly on benzene (C6H6), for example, because the dispersion forces among the molecules composing benzene are significantly weaker than the hydrogen bonds among water molecules. The surface tension of benzene is only 28 mJ>m2 —just 40% that of water. Surface tension is also the reason for the behavior of water that we discussed in Section 11.1. We can see this behavior on Earth by looking at small water droplets (those not large enough to be distorted by gravity), on a leaf or spider's web FIGure 11.15▼. Just as larger samples of water form spheres in the space station, so smaller samples form spheres on Earth. Why? In the same way that gravity pulls the matter of a planet or star inward to form a sphere, intermolecular forces among collections of water molecules pull the water into a sphere. As we discussed in Section 11.1, the sphere is the geometrical shape with the smallest ratio of surface area to volume; therefore, the formation of a sphere minimizes the number of molecules at the surface, minimizing the potential energy of the system.

Dispersion Force

The one intermolecular force present in all molecules and atoms is the dispersion force (also called the London force). Dispersion forces are the result of fluctuations in the electron distribution within molecules or atoms. Since all atoms and molecules have electrons, they all exhibit dispersion forces. The electrons in an atom or molecule may, at any one instant, be unevenly distributed. For example, imagine a frame-by-frame movie of a helium atom in which each "frame" captures the position of the helium atom's two electrons. In any one frame, the electrons may not be symmetrically arranged around the nucleus. In frame 3, for example, helium's two electrons are on the left side of the helium atom. At that instant, the left side has a slightly negative charge (d-). The right side of the atom, which temporarily has no electrons, has a slightly positive charge (d+) because of the nucleus. We call this fleeting charge separation an instantaneous dipole or a temporary dipole. As shown in FIGure 11.3▶ (on the next page), an instantaneous dipole on one helium atom induces an instantaneous dipole on its neighboring atoms because the positive end of the instantaneous dipole attracts electrons in the neighboring atoms. The neighboring atoms then attract one another—the positive end of one instantaneous dipole attracting the negative end of another. This attraction is the dispersion force. The magnitude of the dispersion force depends on how easily the electrons in the atom or molecule can move or polarize in response to an instantaneous dipole, which in turn depends on the size (or volume) of the electron cloud. A larger electron cloud results in a greater dispersion force because the electrons are held less tightly by the nucleus and can therefore polarize more easily. If all other variables are constant, the dispersion force increases with increasing molar mass because molecules or atoms of higher molar mass generally have more electrons dispersed over a greater volume. Consider the boiling points of the noble gases in Table 11.3. As the molar masses and electron cloud volumes of the noble gases increase, the greater dispersion forces result in increasing boiling points. Molar mass alone, however, does not determine the magnitude of the dispersion force. For example, compare the molar masses and boiling points of n-pentane and neopentane, which are shown here: These molecules have identical molar masses, but n-pentane has a higher boiling point than neopentane. Why? Because the two molecules have different shapes. The n-pentane molecules are long and can interact with one another along their entire length FIGure 11.4(a)▼. In contrast, the bulky, round shape of neopentane molecules results in a smaller area of interaction between neighboring molecules FIGure 11.4(b)▼, and thus a lower boiling point. Although we must always consider molecular shape and other factors in determining the magnitude of dispersion forces, molar mass can serve as a guide when comparing dispersion forces within a family of similar elements or compounds, as shown in Figure 11.5▲ for some selected n-alkanes.

Intermolecular Forces: The Forces That Hold Condensed States Together

The structure of the particles that compose a substance determines the strength of the intermolecular forces that hold the substance together, which in turn determines if the substance is a solid, liquid, or gas at a given temperature. At room temperature, moderate-to-strong intermolecular forces tend to result in liquids and solids (high melting and boiling points) and weak intermolecular forces tend to result in gases (low melting and boiling points). Intermolecular forces originate from the interactions among charges, partial charges, and temporary charges on molecules (or atoms and ions), much as bonding forces originate from interactions among charged particles in atoms. Recall from Section 8.3 that according to Coulomb's law the potential energy (E) of two oppositely charged particles (with charges q1 and q2) decreases (becomes more negative) with increasing magnitude of charge and with decreasing separation (r) between them. E = 1 4pe0 q1q2 r (When q1 and q2 are opposite in sign, E is negative.) Therefore, as we have seen, protons and electrons are attracted to each other because their potential energy decreases as they get closer together. Similarly, molecules with partial or temporary charges are attracted to each other because their potential energy decreases as they get closer together. However, intermolecular forces, even the strongest ones, are generally much weaker than bonding forces. The reason for the relative weakness of intermolecular forces compared to bonding forces is also related to Coulomb's law. Bonding forces are the result of large charges (the charges of protons and electrons) interacting at very close distances. Intermolecular forces are the result of smaller charges (as we shall see in the following discussion) interacting at greater distances. For example, consider the interaction between two water molecules in liquid water illustrated here. The length of an O¬H bond in liquid water is 96 pm; however, the average distance between water molecules in liquid water is about 300 pm. The larger distances between molecules, as well as the smaller charges involved (partial charges on the hydrogen and oxygen atoms), result in weaker forces. To break the O¬H bonds in water, we have to heat the water to thousands of degrees Celsius. However, to completely overcome the intermolecular forces between water molecules, we have to heat water only to its boiling point, 100 °C. In this section, we examine several different types of intermolecular forces, including dispersion forces, dipole-dipole forces, hydrogen bonding, and ion-dipole forces. The first three of these can potentially occur in pure substances and mixtures; the last one is found only in mixtures.

Phase Diagrams

Throughout most of this chapter, we have examined how the state of a substance changes with temperature and pressure. We combine both the temperature dependence and pressure dependence of the state of a particular substance in a graph called a phase diagram. A phase diagram is a map of the state of a substance as a function of pressure (on the y-axis) and temperature (on the x-axis). In this section, we first examine the major features of a phase diagram, then we turn to navigating within a phase diagram. The Major Features of a Phase Diagram Consider the phase diagram of water FIGure 11.31▶. The y-axis displays the pressure in torr, and the x-axis shows the temperature in degrees Celsius. The main features of the phase diagram are categorized as regions, lines, and points. We examine each of these individually. Regions Each of the three main regions—solid, liquid, and gas—in the phase diagram represents conditions under which that particular state is stable. For example, under any of the temperatures and pressures within the liquid region in the phase diagram of water, the liquid is the stable state. Notice that the point 25 °C and 760 torr falls within the liquid region, as we know from everyday experience. In general, low temperature and high pressure favor the solid state; high temperature and low pressure favor the gas state; and intermediate conditions favor the liquid state. A sample of matter that is not in the state indicated by its phase diagram for a given set of conditions will convert to that state when those conditions are imposed. For example, steam that is cooled to room temperature at 1 atm condenses to liquid. Lines Each of the lines (curves) in the phase diagram represent a set of temperatures and pressures at which the substance is in equilibrium between the two states on either side of the line. For example, in the phase diagram for water, consider the curved line beginning just beyond 0 °C separating the liquid from the gas. This line is the vaporization curve (also called the vapor pressure curve) for water that we examined in Section 11.5. At any of the temperatures and pressures along this line, the liquid and gas states of water are equally stable and in equilibrium. For example, at 100 °C and 760 torr pressure, water and its vapor are in equilibrium—they are equally stable and will coexist. The other two major lines in a phase diagram are the sublimation curve (separating the solid and the gas) and the fusion curve (separating the solid and the liquid). The Triple Point The triple point in a phase diagram represents the unique set of conditions at which three states are equally stable and in equilibrium. In the phase diagram for water, the triple point occurs at 0.0098 °C and 4.58 torr. Under these unique conditions (and only under these conditions), the solid, liquid, and gas states of water are equally stable and coexist in equilibrium. The Critical Point As we learned in Section 11.5, at the critical temperature and pressure, the liquid and gas states coalesce into a supercritical fluid. The critical point in a phase diagram represents the temperature and pressure above which a supercritical fluid exists. Navigation within a Phase Diagram Changes in the temperature or pressure of a sample are represented by movement within the phase diagram. Consider the phase diagram of carbon dioxide (dry ice) shown in FIGure 11.32◀. An increase in the temperature of a block of solid carbon dioxide at 1 atm is indicated by horizontal movement in the phase diagram as shown by line A, which crosses the sublimation curve at -78.5 °C. At this temperature, the solid sublimes to a gas; you may have observed this behavior in dry ice. A change in pressure is indicated by vertical movement in the phase diagram. For example, a sample of gaseous carbon dioxide at 0 °C and 1 atm could be converted to a liquid by increasing the pressure, as shown by line B. Notice that the fusion curve for carbon dioxide has a positive slope—as the temperature increases the pressure also increases—in contrast to the fusion curve for water, which has a negative slope. The behavior of carbon dioxide is more typical than that of water. The fusion curve within the phase diagrams for most substances has a positive slope because increasing pressure favors the denser state, which for most substances is the solid state.

Solids, Liquids, and Gases: A Molecular Comparison

To begin to understand the differences among the three common states of matter, consider Table 11.1, which shows the density and molar volume of water in its three different states, along with molecular representations of each state. Notice that the densities of the solid and liquid states are much greater than the density of the gas state. Notice also that the solid and liquid states are more similar in density and molar volume to one another than they are to the gas state. The molecular representations show the reason for these differences. The molecules in liquid water and ice are in close contact with one another— essentially touching—while those in gaseous water are separated by large distances. The molecular representation of gaseous water in Table 11.1 is out of proportion—the water molecules in the figure should be much farther apart for their size. (Only a fraction of a water molecule could be included in the figure if it were drawn to scale.) From the molar volumes, we know that 18.0 mL of liquid water (slightly more than a tablespoon) occupies 30.5 L when converted to gas at 100 °C. The low density of gaseous water results from this large separation between molecules. Notice also that, for water, the solid is slightly less dense than the liquid. This is atypical behavior. Most solids are slightly denser than their corresponding liquids because the molecules move closer together upon freezing. As we will discuss in Section 11.9, ice is less dense than liquid water because the unique crystal structure of ice results in water molecules moving slightly farther apart upon freezing. From the molecular perspective, one important difference between liquids and solids is the freedom of movement of the constituent molecules or atoms. Even though the atoms or molecules in a liquid are in close contact, thermal energy can partially overcome the attractions among them, allowing them to move around one another. Not so in solids, where the atoms or molecules are virtually locked in their positions, only vibrating back and forth about a fixed point. Table 11.2 summarizes the properties of liquids and solids, as well as the properties of gases for comparison. Liquids assume the shape of their containers because the atoms or molecules that compose liquids are free to flow (or move around one another) FIGure 11.1▶. Liquids are not easily compressed because the molecules or atoms that compose them are in close contact—they cannot be pushed much closer together. The molecules in a gas, by contrast, have a great deal of space between them and are easily forced into a smaller volume by an increase in external pressure FIGure 11.2▼. Solids have a definite shape because, in contrast to liquids and gases, the molecules or atoms that compose solids are fixed in place. Like liquids, solids have a definite volume and generally cannot be compressed because the molecules or atoms composing them are already in close contact. Solids may be crystalline, in which case the atoms or molecules that compose them are arranged in a well-ordered three-dimensional array, or they may be amorphous, in which case the atoms or molecules that compose them have no long-range order

The Energetics of Vaporization

To understand the energetics of vaporization, consider again a beaker of water from the molecular point of view, except now the beaker is thermally insulated so that heat from the surroundings cannot enter the beaker. What happens to the temperature of the water left in the beaker as molecules evaporate? To answer this question, think about the energy distribution curve again (see Figure 11.18). The molecules that leave the beaker are the ones at the high end of the energy curve—the most energetic. If no additional heat enters the beaker, the average energy of the entire collection of molecules drops—much as the class average on an exam goes down if we eliminate the highest-scoring students—and as a result, vaporization slows. So vaporization is an endothermic process; it takes energy to vaporize the molecules in a liquid. Another way to understand the endothermicity of vaporization is to remember that vaporization involves overcoming the intermolecular forces that hold liquids together. Since energy must be absorbed to pull the molecules apart, the process is endothermic. Our bodies use the endothermic nature of vaporization for cooling. When you overheat, you sweat and your skin becomes covered with liquid water. As this water evaporates, it absorbs heat from your body, cooling your skin as the water and the heat leave your body. A fan makes you feel cooler because it blows newly vaporized water away from your skin, allowing more sweat to vaporize and causing even more cooling. High humidity, on the other hand, slows down the net rate of evaporation, preventing cooling. When the air already contains large amounts of water vapor, the sweat evaporates more slowly, making the body's cooling system less efficient. Condensation, the opposite of vaporization, is exothermic—heat is released when a gas condenses to a liquid. If you have ever accidentally put your hand above a steaming kettle, or opened a bag of microwaved popcorn too soon, you may have experienced a steam burn. As the steam condenses to a liquid on your skin, it releases a lot of heat, causing the burn. The condensation of water vapor is also the reason that winter overnight temperatures in coastal regions, where there is water vapor in the air, do not get as low as in deserts, which tend to have dry air. As the air temperature in a coastal area drops, water condenses out of the air, releasing heat and preventing the temperature from decreasing further. In deserts, there is little moisture in the air to condense, so the temperature drop is greater.

Water: An Extraordinary Substance

Water is the most common and important liquid on Earth. It fills our oceans, lakes, and streams. In its solid form, it caps our mountains, and in its gaseous form, it humidifies our air. We drink water, we sweat water, and we excrete bodily wastes dissolved in water. Indeed, the majority of our body mass is water. Life is impossible without water, and in most places on Earth where liquid water exists, life exists. Recent evidence for the past existence of water on Mars has fueled hopes of finding life or evidence of past life there. And, though not always obvious (because water is so familiar), water has many remarkable properties. Among liquids, water is unique. It has a low molar mass (18.02 g>mol), yet it is a liquid at room temperature. Other main-group hydrides have higher molar masses but lower boiling points, as shown in FIGure 11.33▶. No other substance of similar molar mass (except for HF) comes close to being a liquid at room temperature. We can understand the high boiling point of water (in spite of its low molar mass) by examining its molecular structure. The water molecule's bent geometry and the highly polar nature of its O¬H bonds result in a molecule with a significant dipole moment. Water's two O¬H bonds (hydrogen directly bonded to oxygen) allow a water molecule to form strong hydrogen bonds with four other water molecules FIGure 11.34▶, resulting in a relatively high boiling point. Water's high polarity also allows it to dissolve many other polar and ionic compounds, and even a number of nonpolar gases such as oxygen and carbon dioxide (by inducing a dipole moment in their molecules). Consequently, water is the main solvent within living organisms, transporting nutrients and other important compounds throughout the body. Water is also the main solvent in our environment, allowing aquatic animals to survive by breathing dissolved oxygen and allowing aquatic plants to survive by using dissolved carbon dioxide for photosynthesis. Recall from Section 6.4 that water has an exceptionally high specific heat capacity, which has a moderating effect on the climate of coastal cities. In some cities, such as San Francisco, for example, the daily fluctuation in temperature can be less than 10 °C. This same moderating effect occurs over the entire planet, two-thirds of which is covered by water. Without water, the daily temperature fluctuations on our planet might resemble those on Mars, where temperature fluctuates up to 63 °C (113 °F) between early morning and midday. Imagine awakening to below freezing temperatures, only to bake at summer desert temperatures in the afternoon! The presence of water on Earth and its uniquely high specific heat capacity are largely responsible for our planet's much smaller daily fluctuations. The way water freezes is also unique. Unlike other substances, which contract upon freezing, water expands upon freezing. Consequently, ice is less dense than liquid water, and it floats. This seemingly trivial property has significant consequences. The frozen layer of ice at the surface of a winter lake insulates the water in the lake from further freezing. If this ice layer sank, it would kill bottom-dwelling aquatic life and possibly allow the lake to freeze solid, eliminating virtually all life in the lake. The expansion of water upon freezing, however, is one reason that most organisms do not survive freezing. When the water within a cell freezes, it expands and often ruptures the cell, just as water freezing within a pipe bursts the pipe. Many foods, especially those with high water content, do not survive freezing very well either. Have you ever tried, for example, to freeze your own vegetables? If you put lettuce or spinach in the freezer, it will be limp and damaged when you defrost it.

Crystalline Solids: The Fundamental Types

We can categorize crystalline solids into three types—molecular, ionic, and atomic— based on the individual units that compose the solid. Atomic solids can themselves be divided into three categories—nonbonded, metallic, and network covalent—depending on the types of interactions between atoms within the solid. FIGure 11.41▼ shows the different categories of crystalline solids. Molecular Solids Molecular solids are solids whose composite units are molecules. The lattice sites in a crystalline molecular solid are therefore occupied by molecules. Ice (solid H2O) and dry ice (solid CO2) are examples of molecular solids. Molecular solids are held together by the kinds of intermolecular forces—dispersion forces, dipole-dipole forces, and hydrogen bonding—that we have discussed in this chapter. Molecular solids as a whole tend to have low to moderately low melting points. However, strong intermolecular forces (such as the hydrogen bonds in water) increase the melting points of some molecular solids.

Heating Curve for Water

We can combine and build on the concepts from Sections 11.5 and 11.6 by examining the heating curve for 1.00 mol of water at 1.00 atm pressure shown in FIGure 11.30▶. The y-axis of the heating curve represents the temperature of the water sample. The x-axis represents the amount of heat added (in kilojoules) during heating. As we can see from the diagram, we can divide the process into five segments: (1) ice warming; (2) ice melting into liquid water; (3) liquid water warming; (4) liquid water vaporizing into steam; and (5) steam warming. In two of these segments (2 and 4) the temperature is constant as heat is added because the added heat goes into producing the transition, not increasing the temperature. The two states are in equilibrium during the transition and the temperature remains constant. The amount of heat required to achieve the state change is given by q = n ∆H. In the other three segments (1, 3, and 5), temperature increases linearly. These segments represent the heating of a single state in which the deposited heat raises the temperature in accordance with the substance's specific heat capacity (q = mCs ∆T).

Vapor Pressure and Dynamic Equilibrium

We have already seen that if a container of water is left uncovered at room temperature, the water slowly evaporates away. But what happens if the container is sealed? Imagine a sealed evacuated flask—one from which the air has been removed—containing liquid water, as shown in FIGure 11.19▼. Initially, the water molecules evaporate, as they did in the open beaker. However, because of the seal, the evaporated molecules cannot escape into the atmosphere. As water molecules enter the gas state, some start condensing back into the liquid. When the concentration (or partial pressure) of gaseous water molecules increases, the rate of condensation also increases. However, as long as the water remains at a constant temperature, the rate of evaporation remains constant. Eventually the rate of condensation and the rate of vaporization become equal—dynamic equilibrium has been reached FIGure 11.20◀. Although condensation and vaporization continue at equal rates, the concentration of water vapor above the liquid is constant. The pressure of a gas in dynamic equilibrium with its liquid is the vapor pressure of the liquid. The vapor pressure of a particular liquid depends on the intermolecular forces present in the liquid and the temperature. Weak intermolecular forces result in volatile substances with high vapor pressures because the intermolecular forces are easily overcome by thermal energy. Strong intermolecular forces result in nonvolatile substances with low vapor pressures. A liquid in dynamic equilibrium with its vapor is a balanced system that tends to return to equilibrium if disturbed. For example, consider a sample of n-pentane (a component of gasoline) at 25 °C in a cylinder equipped with a moveable piston FIGure 11.21(a)▼. The cylinder contains no other gases except n-pentane vapor in dynamic equilibrium with the liquid. Since the vapor pressure of n-pentane at 25 °C is 510 mmHg, the pressure in the cylinder is 510 mmHg. Now, what happens when the piston is moved upward to expand the volume within the cylinder? Initially, the pressure in the cylinder drops below 510 mmHg in accordance with Boyle's law. Then, however, more liquid vaporizes until equilibrium is reached once again FIGure 11.21(b)▼. If the volume of the cylinder is expanded again, the same thing happens—the pressure initially drops and more n-pentane vaporizes to bring the system back into equilibrium. Further expansion causes the same result as long as some liquid n-pentane remains in the cylinder. Conversely, what happens if the piston is lowered, decreasing the volume in the cylinder? Initially, the pressure in the cylinder rises above 510 mmHg, but then some of the gas condenses into liquid until equilibrium is reached again FIGure 11.21(c)▼. We can describe the tendency of a system in dynamic equilibrium to return to equilibrium with the following general statement: When a system in dynamic equilibrium is disturbed, the system responds so as to minimize the disturbance and return to a state of equilibrium. If the pressure above the system is decreased, the pressure increases (some of the liquid evaporates) so that the system returns to equilibrium. If the pressure above a liquid-vapor system in equilibrium is increased, the pressure of the system drops (some of the gas condenses) so that the system returns to equilibrium. This basic principle—called Le Châtelier's principle—is applicable to any chemical system in equilibrium, as we will see in Chapter 14.

The Critical Point: The Transition to an Unusual State of Matter

We have considered the vaporization of a liquid in a container open to the atmosphere with and without heating, and the vaporization of a liquid in a sealed container without heating. We now examine the vaporization of a liquid in a sealed container while heating. Consider liquid n-pentane in equilibrium with its vapor in a sealed container initially at 25 °C. At this temperature, the vapor pressure of n-pentane is 0.67 atm. What happens if the liquid is heated? As the temperature rises, more n-pentane vaporizes and the pressure within the container increases. At 100 °C, the pressure is 5.5 atm, and at 190 °C the pressure is 29 atm. As more and more gaseous n-pentane is forced into the same amount of space, the density of the gas becomes higher and higher. At the same time, the increasing temperature causes the density of the liquid to become lower and lower. At 197 °C, the meniscus between the liquid and gaseous n-pentane disappears and the gas and liquid states commingle to form a supercritical fluid FIGure 11.26▲. For any substance, the temperature at which this transition occurs is the critical temperature (Tc)—it represents the temperature above which the liquid cannot exist (regardless of pressure). The pressure at which this transition occurs is the critical pressure (Pc)—it represents the pressure required to bring about a transition to a liquid at the critical temperature. ▲ FIGure 11.26 Critical Point Transition As n-pentane is heated in a sealed container, it undergoes a transition to a supercritical fluid. At the critical point, the meniscus separating the liquid and gas disappears, and the fluid becomes supercritical—neither a liquid nor a gas.

Vaporization and Vapor Pressure

We now turn our attention to vaporization, the process by which thermal energy can overcome intermolecular forces and produce a state change from liquid to gas. We first discuss the process of vaporization itself, then the energetics of vaporization, and finally the concepts of vapor pressure, dynamic equilibrium, and critical point. Vaporization is a common occurrence that we experience every day and even depend on to maintain proper body temperature. The Process of Vaporization Imagine the water molecules within a beaker of water sitting on a table at room temperature and open to the atmosphere FIGure 11.17▶. The molecules are in constant motion due to thermal energy. If we could actually see the molecules at the surface, we would witness Roald Hoffmann's "wild dance floor" (see the chapter-opening quote) because of all the vibrating, jostling, and molecular movement. The higher the temperature, the greater the average energy of the collection of molecules. However, at any one time, some molecules have more thermal energy than the average and some have less. The distributions of thermal energies for the molecules in a sample of water at two different temperatures are shown in FIGure 11.18▲. The molecules at the high end of the distribution curve have enough energy to break free from the surface—where molecules are held less tightly than in the interior due to fewer neighbor-neighbor interactions—and into the gas state. This process is vaporization, the state transition from liquid to gas. Some of the water molecules in the gas state, at the low end of the energy distribution curve for the gaseous molecules, plunge back into the water and are captured by intermolecular forces. This process—the opposite of vaporization—is condensation, the transition from gas to liquid. Although both evaporation and condensation occur in a beaker open to the atmosphere, under typical conditions (such as in relatively dry air at room temperature) evaporation takes place at a greater rate because most of the newly evaporated molecules escape into the surrounding atmosphere and never come back. The result is a noticeable decrease in the water level within an open beaker over time (usually several days). What happens if we increase the temperature of the water within the beaker? Because of the shift in the energy distribution to higher energies (see Figure 11.18), more molecules now have enough energy to break free and evaporate, so vaporization occurs more quickly. What happens if we spill the water on the table or floor? The same amount of water is now spread over a wider area, resulting in more molecules at the surface of the liquid. Since molecules at the surface have the greatest tendency to evaporate—because they are held less tightly—vaporization occurs more quickly. What happens if the liquid in the beaker is not water, but some other substance with weaker intermolecular forces, such as acetone? The weaker intermolecular forces allow more molecules to evaporate at a given temperature, again increasing the rate of vaporization. We call liquids that vaporize easily volatile and those that do not vaporize easily nonvolatile. Acetone is more volatile than water. Motor oil is virtually nonvolatile at room temperature. Summarizing the Process of Vaporization: ▶ The rate of vaporization increases with increasing temperature. ▶ The rate of vaporization increases with increasing surface area. ▶ The rate of vaporization increases with decreasing strength of intermolecular forces.

Temperature Dependence of Vapor Pressure and Boiling Point

When the temperature of a liquid increases, its vapor pressure rises because the higher thermal energy increases the number of molecules that have enough energy to vaporize (see Figure 11.18). Because of the shape of the energy distribution curve, however, a small change in temperature makes a large difference in the number of molecules that have enough energy to vaporize, which results in a large increase in vapor pressure. For example, the vapor pressure of water at 25 °C is 23.3 torr, while at 60 °C the vapor pressure is 149.4 torr. FIGure 11.22▶ shows the vapor pressure of water and several other liquids as a function of temperature. The boiling point of a liquid is the temperature at which its vapor pressure equals the external pressure. When a liquid reaches its boiling point, the thermal energy is enough for molecules in the interior of the liquid (not just those at the surface) to break free of their neighbors and enter the gas state FIGure 11.23▼. The bubbles in boiling water are pockets of gaseous water that have formed within the liquid water. The bubbles float to the surface and leave as gaseous water or steam. The normal boiling point of a liquid is the temperature at which its vapor pressure equals 1 atm. The normal boiling point of pure water is 100 °C. However, at a lower pressure, water boils at a lower temperature. In Denver, Colorado, where the altitude is around 1600 meters (5200 feet) above sea level, for example, the average atmospheric pressure is about 83% of what it is at sea level, and water boils at approximately 94 °C. For this reason, it takes slightly longer to cook food in boiling water in Denver than in San Francisco (which is at sea level). Once the boiling point of a liquid is reached, additional heating only causes more rapid boiling; it does not raise the temperature of the liquid above its boiling point, as shown in the heating curve in FIGure 11.24▼. Therefore, boiling water at 1 atm always has a temperature of 100 °C. As long as liquid water is present, the water's temperature cannot rise above its boiling point. However, after all the water has been converted to steam, the temperature of the steam can rise beyond 100 °C