🧪 Contents · Interactive Lab Chapter 10 · Liquids and Solids
Chemistry · Chapter 10

Liquids and Solids

A gas fills any room you put it in. A liquid and a solid do not. Pour water into a glass and it stays in the glass. Set an ice cube on a plate and it keeps its shape. Something holds those particles close together. This chapter is about that something. You will meet the weak pulls that act between molecules. You will see how they set boiling points, thickness and crystal shape. You will walk a sample from ice to steam and count the heat each step takes. Then you will read the map of the three states and find the odd one out, which turns out to be water.

By the end of this chapter you can

  • Name the four main pulls between particles and rank their strength.
  • Predict which of two liquids boils at the higher temperature, and say why.
  • Explain thickness, surface tension and capillary rise in terms of those pulls.
  • Find the heat a phase change takes, using heat of fusion and heat of vaporization.
  • Read a heating curve and say what each flat stretch means.
  • Use vapor pressure to find a boiling point at any outside pressure.
  • Read a phase diagram, point to its triple point and critical point, and count the atoms in a unit cell.

Section 1Intermolecular forces

Inside a molecule, and between molecules

Bonds hold the atoms inside one molecule. Weaker pulls act between whole molecules. Those weaker pulls are the intermolecular forces, and they run this chapter. Break the bonds in water and you get hydrogen and oxygen. Break the forces between water molecules and you get steam. It is still water. The molecules are just spread out. That is why boiling a pot of water takes far less energy than splitting it would.

Dispersion forces

Every molecule holds electrons, and electrons move. For an instant, more of them crowd one side. That side turns slightly negative and the far side turns slightly positive. The molecule now has a brief dipole, which is a split of charge. It tugs on the electrons next door and sets up a matching dipole there. The two then attract. This is the dispersion force, and every substance has it.

A big, loose electron cloud is easy to push out of shape. We say such a molecule is more polarizable. So dispersion grows as the electron count grows. The noble gases show it cleanly. They boil at 4 K, 27 K, 87 K, 120 K and 165 K going from helium down to xenon. Shape counts too. A long thin molecule lies against its neighbor along its whole length, so it sticks better than a round one of the same mass.

Dipole-dipole forces

Some molecules are lopsided all the time. One atom pulls the shared electrons harder, so one end stays slightly negative. These molecules line up plus to minus and hold on. That pull is the dipole-dipole force. It adds to dispersion. It never replaces it.

Hydrogen bonds

One case is so strong that it gets its own name. Tie an H atom to N, O or F and the H is left very exposed. Its one electron is pulled away, and no inner shell hides it. That bare H can reach a lone pair on an N, O or F atom next door. The link is a hydrogen bond. In water it is worth about 20 kJ/mol. The O–H bond inside the molecule is about 464 kJ/mol, so a hydrogen bond is nowhere near a real bond. It is still several times stronger than a plain dipole pull of the same size.

Water shows the payoff. H2O boils at 100 °C. H2S has the same bent shape and almost twice the mass, and it boils at −60 °C. Sulfur is not electronegative enough to make hydrogen bonds. That one missing force is worth 160 °C.

Ion-dipole forces

Stir table salt into water and the ions come loose. Each Na+ gathers the negative ends of water molecules around itself. Each Cl− gathers the positive ends. That pull is the ion-dipole force, and it runs from 40 to 600 kJ/mol. It is the strongest of the four. It is also what pays the bill for taking a crystal apart.

Dispersion, dipole-dipole and hydrogen bonding compared DISPERSION every substance · 0.05 to 40 kJ/mol δ− δ+ δ− δ+ a flicker on the left forces a match on the right DIPOLE–DIPOLE polar molecules · 5 to 25 kJ/mol H Cl δ+ δ− H Cl δ+ δ− the minus end of one meets the plus end of the next HYDROGEN BOND H on N, O or F · about 20 kJ/mol in water O H H O H H O to O: about 2.8 Å a bare H reaches the lone pair on the next O
Figure 10.1 The three forces between neutral molecules, drawn to the same idea. Dispersion is a flicker that spreads. A dipole-dipole pull is permanent. A hydrogen bond is a special, stronger case of that pull, and it only happens when H sits on N, O or F.

Ranking them

Compare like with like. For molecules of about the same size, the order is clear. Hydrogen bonding beats dipole-dipole, which beats dispersion alone. But size wins when the mismatch is large. I2 has no dipole at all and boils at 184 °C. HCl is polar and boils at −85 °C. Iodine has far more electrons, so its dispersion forces swamp the dipoles in HCl.

Worked example · Rank three liquids by boiling point

Put CH4, CH3F and CH3OH in order from lowest boiling point to highest.

Step 1 · name the forcesCH4 is a symmetric nonpolar molecule, so it has dispersion only. CH3F is polar, so it has dispersion plus dipole-dipole. CH3OH has an H on an O, so it adds hydrogen bonding on top of both.

Step 2 · check the massesThe molar masses are 16.04, 34.03 and 32.04 g/mol. The last two are nearly a tie, so their dispersion forces are close. Mass will not decide between them.

Step 3 · rankMore forces and stronger forces mean a higher boiling point. So CH4 < CH3F < CH3OH.

Step 4 · check itThe measured values are −161 °C, −78 °C and 65 °C. The order holds, and the jump to the alcohol is the biggest one.

Key points
  • Intermolecular forces act between molecules. Bonds act inside them.
  • Dispersion is in every substance and grows with the electron count.
  • Dipole-dipole needs a permanent lopsided molecule.
  • A hydrogen bond needs H joined to N, O or F, and a lone pair to reach.
  • Ion-dipole is the strongest of the four and drives salts into water.
  • Rank by comparing molecules of similar size. Otherwise dispersion can win on mass alone.
A water molecule can make four hydrogen bonds. It gives two through its H atoms and takes two through the lone pairs on its O. In liquid water the average is a bit under four at any instant. That dense web is why water holds so much heat and boils so high for a molecule of mass 18.02 g/mol.
Interactive Compare dispersion, dipole-dipole and hydrogen bonding. Open full ↑

Section 2Properties of liquids

Thickness, in a word: viscosity

Viscosity measures how much a liquid resists flowing. Water pours. Honey crawls. Two things raise viscosity. Stronger forces between molecules make them harder to slide past each other. Long molecules also tangle like cooked spaghetti. Glycerol has three OH groups on a three-carbon chain, so it does both at once. At 25 °C its viscosity is about 934 mPa·s. Water at the same temperature is 0.890 mPa·s. Glycerol is more than a thousand times thicker.

Heat thins a liquid. Faster molecules spend less time in each other's grip. Water drops from 1.79 mPa·s at 0 °C to 0.890 at 25 °C, and down to 0.282 at 100 °C. That is why warm syrup pours and cold syrup will not.

Surface tension

A molecule deep in a liquid is pulled on from every side. The pulls cancel. A molecule at the top surface has no neighbors above it, so the leftover pull aims sideways and down. The surface behaves like a stretched skin. Surface tension is the energy it costs to make new surface area. We report it in mN/m, which is the same as mJ/m2.

Water comes in at 72.0 mN/m at 25 °C, which is high for a liquid. Ethanol is only 22.1 mN/m, since it makes fewer hydrogen bonds per molecule. Mercury reaches 486 mN/m, because metal atoms hold each other with a sea of shared electrons. High surface tension pulls a drop into a bead and lets a water strider stand on a pond without sinking.

Cohesion, adhesion and the meniscus

Cohesive forces pull a liquid toward itself. Adhesive forces pull it toward the walls of its container. Which one wins sets the curve at the edge, called the meniscus. Water sticks to glass better than to itself, so it climbs the wall and the curve dips in the middle. Read the bottom of that dip. Mercury does the reverse. It holds itself far better than it holds glass, so the curve humps up. Read the top.

Capillary action

In a narrow tube the climb does not stop at the rim of the meniscus. The whole column rises. That lift is capillary action. The surface pulls up on the ring where liquid meets glass, and the weight of the raised column pulls back down. The two balance at one height.

h = 2T cos θ / (r ρ g)

Here T is surface tension, θ is the contact angle, r is the tube radius, ρ is density and g is 9.81 m/s2. The radius sits on the bottom, so a thinner tube lifts the liquid higher. Paper towels and plant stems are packed with very thin channels, and that is the whole trick.

Meniscus shape and capillary rise WATER IN GLASS sticking beats self-pull read here ↓ curve dips in the middle MERCURY IN GLASS self-pull beats sticking read here ↑ curve humps in the middle CAPILLARY RISE IN WATER halve the radius and the column doubles open dish 2.9 cm r = 0.50 mm 5.9 cm r = 0.25 mm
Figure 10.2 Water wets glass, so its meniscus curves up at the wall and the column rises in a thin tube. Mercury does not wet glass, so its meniscus humps and the column is pushed down instead. The two rise heights shown are the worked example below.

Worked example · How high does water climb?

Find the rise of water at 25 °C in a clean glass tube of radius 0.25 mm.

Step 1 · collect the valuesT = 72.0 mN/m = 0.0720 N/m. Water wets clean glass, so θ is near 0 and cos θ = 1. ρ = 1.00 × 103 kg/m3. r = 0.25 mm = 2.5 × 10−4 m.

Step 2 · build the bottom rρg = (2.5 × 10−4)(1.00 × 103)(9.81) = 2.45 N/m2 per m.

Step 3 · divideh = 2(0.0720)(1) / 2.45 = 0.144 / 2.45 = 0.0587 m, which is 5.9 cm.

Step 4 · test the patternDouble the radius to 0.50 mm and the bottom doubles to 4.90. Then h = 0.144 / 4.90 = 0.0294 m, or 2.9 cm. Twice the width, half the climb.

Key points
  • Stronger intermolecular forces mean higher viscosity and higher surface tension.
  • Heating a liquid always lowers its viscosity.
  • Cohesion is the pull to itself. Adhesion is the pull to the wall.
  • Water wets glass and curves down in the middle. Mercury does not wet glass and humps up.
  • Capillary rise goes as 1/r, so thin tubes lift liquid the furthest.
Soap drops water's surface tension from about 72 mN/m to near 30 mN/m. The soap molecules crowd the surface and get in the way of the hydrogen bonds. Low surface tension lets water spread into cloth and creep into the gaps in dirt instead of sitting on top in beads.
Interactive Test viscosity, surface tension and capillary action. Open full ↑

Section 3Phase changes and heating curves

Six changes, three in each direction

Matter moves between states in six named ways. Melting, vaporizing and subliming all take heat in. Freezing, condensing and depositing all give that heat back. Sublimation is the jump from solid straight to gas, with no liquid in between. Dry ice does it in a warm room. So does old snow on a cold dry day, which shrinks without ever making a puddle.

Heat that does not raise the temperature

Here is the surprise of this section. Keep heating ice that has reached 0 °C and the thermometer stops moving. The energy is still going in. It is being spent to pull molecules out of their fixed spots, not to speed them up. The reading holds at 0 °C until the last crystal is gone.

The heat of fusion is the energy to melt a set amount at the melting point. For water it is 334 J/g, which is 6.01 kJ/mol. The heat of vaporization is the energy to boil a set amount at the boiling point. For water that is 2,260 J/g, or 40.7 kJ/mol. Freezing and condensing return the same amounts with the sign flipped.

Look at the gap. Boiling water takes almost seven times the energy of melting it. Melting only loosens the grip. The molecules stay packed and touching. Boiling has to break the hold completely and send each molecule off alone. That is the expensive step.

Reading a heating curve

Put a cold sample on a steady flame. Plot temperature up and heat added across. You get five stretches: three slopes and two flats. On a slope only the temperature changes, and the heat is q = mcΔT. On a flat only the state changes, and the heat is the mass times the heat of fusion or vaporization. The specific heats for water differ by state: ice is 2.09 J/g·°C, liquid water is 4.184, and steam is 1.84.

The flat at 0 °C is melting. The flat at 100 °C is boiling, and it is far wider on the page. Width on this plot means energy, so the widest stretch is the most expensive change. A cooling curve is the same picture run backward, with heat leaving instead of entering.

Heating curve for water, with both plateaus to scale °C 0 100 melting, 0 °C 8,350 J boiling, 100 °C · 56,500 J, the widest stretch ice warms 784 J water warms 10,460 J steam warms 690 J heat added → total 76,784 J for 25.0 g Flat stretches are drawn to scale, so you can see that boiling costs almost three quarters of the whole bill.
Figure 10.3 A heating curve for 25.0 g of water, from ice at −15 °C to steam at 115 °C. Each stretch is labeled with its own energy. The two flats are phase changes, where heat goes in and the temperature holds still.

Worked example · Ice at −15 °C to steam at 115 °C

How much heat does 25.0 g of water need for the whole trip? Use 2.09, 4.184 and 1.84 J/g·°C for ice, liquid and steam.

Step 1 · warm the ice q = (25.0 g)(2.09 J/g·°C)(15.0 °C) = 784 J.

Step 2 · melt it q = (25.0 g)(334 J/g) = 8,350 J.

Step 3 · warm the water q = (25.0 g)(4.184 J/g·°C)(100.0 °C) = 10,460 J.

Step 4 · boil it q = (25.0 g)(2,260 J/g) = 56,500 J.

Step 5 · warm the steam q = (25.0 g)(1.84 J/g·°C)(15.0 °C) = 690 J.

Step 6 · add and look 784 + 8,350 + 10,460 + 56,500 + 690 = 76,784 J, or 76.8 kJ. Step 4 alone is 73.6% of that. Phase changes, not temperature changes, own the energy budget.

Key points
  • During a phase change the temperature holds still while heat keeps flowing.
  • On a slope use q = mcΔT. On a flat use mass times the heat of fusion or vaporization.
  • Water: 334 J/g to melt and 2,260 J/g to boil. That is 6.01 and 40.7 kJ/mol.
  • Boiling costs almost seven times what melting costs for the same mass.
  • Freezing and condensing release the same heat they took in.
This is why steam burns are so much worse than hot water burns. A gram of steam at 100 °C that condenses on your skin dumps 2,260 J before its temperature has dropped at all. Add the heat of fusion to the heat of vaporization and you get the heat of sublimation, near 47 kJ/mol. The measured value for ice is about 51 kJ/mol, because vaporizing takes more energy at 0 °C than it does at 100 °C.
Interactive Add heat steadily and watch temperature stall at each change. Open full ↑

Section 4Vapor pressure and boiling

Where vapor pressure comes from

Molecules in a liquid do not all move at the same speed. A few at the top surface are fast enough to break free. In a sealed flask those escapees build up in the space above the liquid. Some of them hit the surface and fall back in. Escape and return speed up until they match. At that point the gas pressure above the liquid stops climbing and holds. That steady value is the vapor pressure.

Vapor pressure depends on just two things: which liquid it is, and how warm it is. Raise the temperature and more molecules clear the bar, so the pressure climbs fast. Water sits at 23.8 torr at 25 °C, 92.5 torr at 50 °C, 355 torr at 80 °C and 760 torr at 100 °C. Note the shape of that run. It nearly quadruples from 25 to 50 °C, then nearly quadruples again by 80 °C.

Weak forces mean a high vapor pressure at any given temperature. A liquid like that is called volatile, and you can smell it from across the room. Diethyl ether reaches 533 torr at 25 °C, which is more than twenty times water's value at the same temperature.

Boiling

Warming a liquid in an open beaker does not make bubbles until one condition is met. The vapor pressure must match the pressure pressing down from above. Only then can a bubble inside the liquid hold its own shape instead of being squeezed flat. So boiling is a tug of war that the vapor wins at one exact temperature.

The normal boiling point is the temperature where vapor pressure reaches 1 atm, which is 760 torr. For water that is 100 °C. Drop the outside pressure and the liquid boils sooner. Denver sits near 625 torr, and water there boils at about 95 °C. On the summit of Everest the pressure is near 253 torr and water boils at about 72 °C. Food takes longer to cook up there, because the water bathing it is cooler. A pressure cooker runs the trade the other way. At 2 atm water will not boil until 121 °C, and the hotter water cooks beans in a third of the time.

Vapor pressure against temperature for three liquids torr 760 torr = 1 atm 400 200 ether 34.6 °C ethanol 78.4 °C water 100 °C 0 25 50 75 100 temperature (°C) Where a curve meets the gold line, that liquid boils in open air at sea level.
Figure 10.4 Vapor pressure rises steeply with temperature. The three curves come from measured data for diethyl ether, ethanol and water. Each one crosses 760 torr at its own normal boiling point. Lower the gold line and every crossing slides left.

The Clausius-Clapeyron idea

Those curves look hopeless to work with, but one trick straightens them. Plot the natural log of vapor pressure against 1 over the Kelvin temperature. The points fall on a line. Its slope is −ΔHvap/R. Two points on that line give the useful form:

ln(P2/P1) = −(ΔHvap/R)(1/T2 − 1/T1)

A steep line means a large heat of vaporization, which means strong forces. So the slope of this plot is a direct read on the forces from section 1. The equation also runs backward. Give it a target pressure and it hands you the boiling point at that pressure.

Worked example · Predict a vapor pressure

Water's vapor pressure at 60 °C is 149 torr. Estimate it at 80 °C. Use ΔHvap = 40.7 kJ/mol and R = 8.314 J/mol·K.

Step 1 · go to KelvinT1 = 333.15 K and T2 = 353.15 K.

Step 2 · the bracket1/353.15 − 1/333.15 = 0.0028317 − 0.0030017 = −1.700 × 10−4 K−1.

Step 3 · the front factorΔHvap/R = 40,700 / 8.314 = 4,895 K.

Step 4 · multiplyln(P2/149) = −(4,895)(−1.700 × 10−4) = 0.832. So P2/149 = e0.832 = 2.30.

Step 5 · finishP2 = 149 × 2.30 = 343 torr.

Step 6 · judge itThe measured value is 355 torr, so the estimate is 3% low. The equation assumes the heat of vaporization never changes. Near 70 °C water's real value is about 42.4 kJ/mol, and that number gives 355 torr. Short jumps in temperature stay accurate. Long ones drift.

Key points
  • Vapor pressure is the pressure of vapor sitting over its own liquid at balance.
  • It depends only on the liquid and the temperature, not on the amount of liquid.
  • Boiling happens when vapor pressure equals the outside pressure.
  • Lower outside pressure gives a lower boiling point. Higher pressure raises it.
  • ln P against 1/T is a straight line with slope −ΔHvap/R.
Vapor pressure does not care how much liquid you have. A teaspoon of water and a full bathtub both sit at 23.8 torr at 25 °C. Surface area changes how fast balance arrives, not where it lands. Add a wider dish and you get there sooner, at the very same pressure.
Interactive Change temperature and altitude and find the boiling point. Open full ↑

Section 5Phase diagrams

A map of the three states

A phase diagram plots pressure up the side and temperature across the bottom. Every point on it is one set of conditions. The region a point lands in names the state that is stable there. Three lines divide the map. The solid-liquid line is melting and freezing. The liquid-gas line is boiling and condensing, and it is exactly the vapor pressure curve from section 4. The solid-gas line is sublimation. Crossing a line means changing state.

The triple point

The three lines meet at one point and only one. At the triple point solid, liquid and gas all sit together in balance. Ice, water and vapor coexist, and none of them is winning. For water it is 0.01 °C and 0.006 atm, which is 4.58 torr. Below that pressure liquid water cannot exist at any temperature. Warm ice at 0.001 atm and it goes straight to vapor. Freeze drying lives in that gap. Pump a sealed chamber down below 0.006 atm and the ice in frozen food leaves as vapor, with no soggy liquid stage on the way.

The critical point

Now follow the liquid-gas line up and to the right. It does not run forever. It stops dead at the critical point, which for water is 374 °C and 218 atm. Why would a line just end? Because at that temperature the hot, dense gas and the hot, thin liquid have the same density. There is nothing left to tell them apart, and no surface divides them. Past that point the substance is a supercritical fluid. It flows and spreads like a gas but dissolves things like a liquid. Supercritical CO2 is how coffee beans get decaffeinated without a drop of solvent left behind.

Water leans the wrong way

Look at the solid-liquid line on almost any other substance and it tilts to the right. Squeeze that substance and you help it freeze. Water's line tilts left instead. Press harder on ice held at 0 °C and it melts.

The reason is density. Ice is 0.917 g/cm3 and liquid water is 1.00 g/cm3, so ice floats. The hydrogen bond network in ice locks molecules into an open cage with gaps in it. Melting collapses part of that cage, and the pieces pack tighter. Pressure always favors whichever phase takes up less room, so here pressure favors the liquid. The effect is real but small. It takes roughly 135 atm to pull water's melting point down by a single degree.

Phase diagram of water, with carbon dioxide for contrast pressure (atm) 0.0001 0.01 1 100 1000 how almost every other substance leans SOLID LIQUID GAS (vapor) triple point 0.01 °C, 0.006 atm critical point 374 °C, 218 atm supercritical 0 °C 100 °C the 1 atm line, where everyday melting and boiling happen −80 0 100 200 300 400 temperature (°C) · water, pressure on a log scale CO₂ for contrast 1 atm SOL LIQ GAS triple point 5.11 atm sits above the 1 atm line, so no liquid. Sketch only.
Figure 10.5 Water's phase diagram, drawn from measured data with pressure on a log scale. The solid-liquid line really does lean left, though only slightly. The grey dashed line shows the right-leaning direction nearly every other substance takes. The small sketch at the right shows why CO2 has no liquid state at 1 atm.

CO2, the useful contrast

Carbon dioxide's triple point is −56.6 °C at 5.11 atm. One atmosphere is below that pressure, so the 1 atm line never touches the liquid region at all. Solid CO2 sublimes at −78.5 °C, which is why dry ice smokes away and leaves the floor dry. Its critical point is 31.0 °C and 72.9 atm. A CO2 cylinder in a cool room holds liquid and gas together near 57 atm. Let the room warm past 31 °C and the liquid surface vanishes into a supercritical fluid.

Worked example · Walk two paths on the map

Start with ice at −10 °C. Warm it to 50 °C at steady pressure. Do it once at 0.002 atm and once at 1 atm.

Step 1 · place the first path0.002 atm is below the triple point pressure of 0.006 atm. The path runs under the triple point for its whole length.

Step 2 · count the crossingsIt leaves the solid region and crosses the solid-gas line only. The ice sublimes. No liquid forms at any moment.

Step 3 · place the second path1 atm is above 0.006 atm, so the path runs over the triple point.

Step 4 · count againIt crosses the solid-liquid line at 0 °C, then sits in the liquid region. At 50 °C it is still liquid, since boiling waits until 100 °C. So the answer is ice to water, and the same sample took two different routes for one reason: pressure.

Key points
  • A phase diagram names the stable state for every pressure and temperature pair.
  • At the triple point all three states sit in balance at once.
  • Below the triple point pressure, a solid sublimes and the liquid never appears.
  • The liquid-gas line ends at the critical point, where liquid and gas become one fluid.
  • Water's solid-liquid line leans left because ice is less dense than water.
  • CO2 has a triple point at 5.11 atm, above 1 atm, so it has no liquid in open air.
The triple point of water is so reproducible that it was used to define the kelvin for decades. The scale was set by calling that one point exactly 273.16 K. The 2019 redefinition moved the anchor to the Boltzmann constant, but the triple point still sits at 273.16 K to within a hair.
Interactive Move a point around a phase diagram across every boundary. Open full ↑

Section 6Solids and crystal lattices

Four families of solid

Solids differ by what sits at the lattice points and what holds those pieces together. Ionic solids stack positive and negative ions. The pulls are strong in every direction, so they melt high and shatter rather than bend. NaCl melts at 801 °C. Metallic solids park metal atoms in a shared pool of loose electrons, which is why they conduct heat and current and can be hammered flat.

Covalent network solids join every atom to its neighbors with real bonds, so melting one means breaking bonds. Diamond does not melt below 3,500 °C. Molecular solids hold whole molecules in place using only the forces from section 1. They melt low. Ice goes at 0 °C and solid CO2 never melts at 1 atm at all.

One more split cuts across all four. A crystalline solid repeats one pattern over long distances and melts at a sharp temperature. An amorphous solid has order only in small patches. Glass is the classic case. It softens over a range of temperatures instead of melting at a single one.

The unit cell

The smallest repeating box in a crystal is the unit cell. Stack copies of it in all three directions and you rebuild the whole crystal. Three cubic cells cover a great many metals, so they are worth knowing cold.

Counting atoms in a cell takes one rule: share them. An atom sitting on a corner is shared by the eight cells that meet there, so this cell owns 1/8 of it. An atom on a face is shared by two cells, so this cell owns 1/2. An atom fully inside belongs to this cell alone.

Where the spheres touch sets a link between the edge length a and the radius r. In a simple cubic cell they touch along the edge, so a = 2r. In a body-centered cell they touch along the body diagonal, giving 4r = √3 a. In a face-centered cell they touch along the face diagonal, giving 4r = √2 a.

The three cubic unit cells and how their atoms are counted SIMPLE CUBIC 8 corners × 1/8 = 1 atom touches 6 · fills 52% · a = 2r example: polonium BODY-CENTERED 8 × 1/8 + 1 = 2 atoms touches 8 · fills 68% · 4r = √3 a example: iron, tungsten, sodium FACE-CENTERED 8 × 1/8 + 6 × 1/2 = 4 atoms touches 12 · fills 74% · 4r = √2 a example: copper, aluminum, nickel
Figure 10.6 The three cubic unit cells. Teal spheres sit on corners and each one is shared by eight cells. The gold sphere in the middle cell belongs to that cell alone. The gold spheres on the right sit on faces and are split between two cells each.

From a cell to a density

Here is the payoff. X-rays measure the edge length of a unit cell. From that one number you can predict the density of the bulk metal, and the match is a real test of the model. Density is mass over volume, so take one cell. Its mass is the Z atoms it owns, each weighing M/NA. Its volume is a3.

ρ = Z M / (NA a3)

Worked example · The density of copper from one length

Copper is face-centered cubic with a = 361.5 pm and M = 63.55 g/mol. Find its density and the radius of a copper atom.

Step 1 · count the atomsFace-centered means Z = 4.

Step 2 · get the volume in cm3 a = 361.5 pm = 3.615 × 10−8 cm. a3 = 4.724 × 10−23 cm3.

Step 3 · get the mass of one cell (4)(63.55 g/mol) / (6.022 × 1023 /mol) = 4.221 × 10−22 g.

Step 4 · divideρ = 4.221 × 10−22 g / 4.724 × 10−23 cm3 = 8.94 g/cm3. The handbook value is 8.96 g/cm3.

Step 5 · find the radiusSpheres touch along the face diagonal, so 4r = √2 a and r = a/2√2 = 361.5/2.828 = 127.8 pm. The listed radius of copper is 128 pm.

Key points
  • Ionic, metallic, covalent network and molecular solids differ in what holds them together.
  • Crystalline solids melt sharply. Amorphous solids soften over a range.
  • A corner atom counts as 1/8, a face atom as 1/2, a body atom as 1.
  • Simple cubic holds 1 atom, body-centered holds 2, face-centered holds 4.
  • ρ = ZM/(NAa3) turns one measured edge into a density.
Table salt is two face-centered lattices laid on top of each other, one of Na+ and one of Cl−, offset by half a cell edge. Count each one and you get 4 ions of each kind per cell. The ratio is 4 to 4, which is the 1 to 1 you already write as NaCl.
Interactive Count atoms in simple, body-centered and face-centered unit cells. Open full ↑

The chapter in one breath

  • Dispersion forces act in every substance and grow with the number of electrons.
  • Dipole-dipole needs a lopsided molecule, and a hydrogen bond needs H on N, O or F.
  • Ion-dipole is the strongest of the four, and it is what pulls salts into water.
  • Stronger forces mean higher boiling point, higher viscosity and higher surface tension.
  • A phase change takes heat without changing temperature: 334 J/g to melt water, 2,260 J/g to boil it.
  • Vapor pressure rises steeply with temperature, and boiling comes when it matches the outside pressure.
  • ln P against 1/T is a line whose slope gives the heat of vaporization.
  • A phase diagram holds a triple point and a critical point, and water's melting line leans left.
  • Unit cells hold 1, 2 or 4 atoms, and one edge length gives you the density.

ReferenceWords worth knowing

These are the words this chapter leans on. Each one is defined the way it is used here.

adhesive force
The attraction between a liquid and the surface it touches. It competes with cohesion to set the shape of the meniscus.
body-centered cubic
A cubic cell with atoms on the 8 corners plus 1 in the middle. It owns 2 atoms and fills 68% of space.
capillary action
The rise of a liquid in a narrow tube, driven by adhesion to the walls. The height goes as 1 over the radius.
Clausius-Clapeyron equation
The relation ln(P2/P1) = −(ΔHvap/R)(1/T2 − 1/T1), linking vapor pressure to temperature.
critical point
The end of the liquid-gas line. Above it liquid and gas have the same density and no surface separates them. Water's is 374 °C and 218 atm.
dipole-dipole attraction
The pull between permanently lopsided molecules lining up plus to minus. Usually 5 to 25 kJ/mol.
dispersion force
The pull from a brief, flickering split of charge in an electron cloud. Present in all matter and stronger for bigger clouds.
face-centered cubic
A cubic cell with atoms on the 8 corners plus 1 on each of the 6 faces. It owns 4 atoms and fills 74% of space.
heat of fusion
The energy a set amount of solid needs to melt at its melting point. Water needs 334 J/g, or 6.01 kJ/mol.
heat of vaporization
The energy a set amount of liquid needs to boil at its boiling point. Water needs 2,260 J/g, or 40.7 kJ/mol.
hydrogen bond
A strong dipole pull between an H atom bonded to N, O or F and a lone pair on a nearby N, O or F. About 20 kJ/mol in water.
intermolecular force
Any attraction acting between whole molecules rather than inside one. It sets melting point, boiling point and most liquid behavior.
ion-dipole attraction
The pull between an ion and a polar molecule. At 40 to 600 kJ/mol it is the strongest force in this chapter, and it drives dissolving.
phase diagram
A plot of pressure against temperature that names the stable state at every point, with lines marking where states change.
sublimation
The change from solid straight to gas, with no liquid stage. Dry ice does it at room temperature and 1 atm.
surface tension
The energy cost of making new surface, in mN/m. Water is 72.0 mN/m at 25 °C and mercury is 486 mN/m.
triple point
The one pressure and temperature where solid, liquid and gas all sit in balance. Water's is 0.01 °C and 0.006 atm.
unit cell
The smallest repeating box of a crystal. Stacking it in three directions rebuilds the whole solid.
vapor pressure
The pressure of vapor over its own liquid once escape and return are balanced. It depends on the liquid and the temperature only.
viscosity
A liquid's resistance to flow. It rises with stronger forces and with molecule length, and it falls as temperature rises.

Check yourself

1. Water’s heat of fusion is 6.01 kJ/mol. Show that this is 334 J/g.
334 J/g. 6.01 kJ/mol is 6,010 J/mol. Divide by the molar mass: 6,010 J/mol ÷ 18.02 g/mol = 333.5 J/g, which rounds to 334 J/g. The same trick on 40.7 kJ/mol gives 40,700 ÷ 18.02 = 2,259 J/g, the 2,260 J/g used in this chapter.
2. How much heat does 45.0 g of ice at 0 °C need to melt completely?
15.0 kJ. Melting is a flat stretch, so use mass times the heat of fusion. q = (45.0 g)(334 J/g) = 15,030 J = 15.0 kJ. The temperature never leaves 0 °C while this happens.
3. Find the heat needed to take 50.0 g of liquid water from 20.0 °C to steam at 100.0 °C.
1.30 × 105 J, or 130 kJ. Two steps. Warming: q = (50.0)(4.184)(80.0) = 16,736 J. Boiling: q = (50.0)(2,260) = 113,000 J. Total = 129,736 J = 130 kJ. Boiling is 87% of the bill even though the warming covers 80 degrees.
4. At what temperature does water boil when the outside pressure is 608 torr (0.800 atm)? Use 760 torr at 373.15 K and ΔHvap = 40.7 kJ/mol.
About 93.8 °C. ln(608/760) = ln(0.800) = −0.2231. And ΔHvap/R = 40,700/8.314 = 4,895 K. So −0.2231 = −4,895(1/T2 − 1/373.15). That gives 1/T2 − 0.0026799 = 4.558 × 10−5, so 1/T2 = 0.0027255 and T2 = 366.9 K = 93.8 °C. Steam tables give 93.9 °C, so the estimate lands right on it.
5. Iron is body-centered cubic with an edge length of 286.6 pm and a molar mass of 55.85 g/mol. Find its density.
7.88 g/cm3. Body-centered means Z = 2. a = 2.866 × 10−8 cm, so a3 = 2.354 × 10−23 cm3. Cell mass = (2)(55.85)/(6.022 × 1023) = 1.855 × 10−22 g. Divide: 1.855 × 10−22 / 2.354 × 10−23 = 7.88 g/cm3. The handbook value is 7.87 g/cm3.
6. How high does water climb in a clean glass tube of radius 0.10 mm? Use T = 0.0720 N/m, cos θ = 1 and ρ = 1.00 × 103 kg/m3.
About 15 cm. h = 2Tcosθ/(rρg). The bottom is (1.0 × 10−4)(1.00 × 103)(9.81) = 0.981. The top is 2(0.0720) = 0.144. So h = 0.144/0.981 = 0.147 m = 14.7 cm. That is 2.5 times the 5.9 cm from the worked example, because the radius is 2.5 times smaller.
7. Count the atoms in a face-centered cubic unit cell, showing the shares.
4 atoms. The 8 corner atoms each give 1/8, for 8 × 1/8 = 1. The 6 face atoms each give 1/2, for 6 × 1/2 = 3. Add them: 1 + 3 = 4 atoms per cell. Compare simple cubic at 1 and body-centered at 2.
8. Rank argon, HCl and water from lowest boiling point to highest, and name the deciding force.
Ar < HCl < H2O. Argon has dispersion only and boils at −186 °C. HCl adds dipole-dipole and boils at −85 °C. Water adds hydrogen bonding and boils at 100 °C. All three have similar electron counts, so the type of force decides, not the mass.
9. Why can you not pour liquid CO2 out of a bottle in an open room?
Its triple point pressure is above 1 atm. CO2’s triple point is −56.6 °C at 5.11 atm. At 1 atm the sample never enters the liquid region, no matter the temperature. Solid CO2 sublimes at −78.5 °C instead.
10. Why does water’s solid-liquid line lean left when nearly every other substance leans right?
Because ice is less dense than liquid water. Pressure pushes a sample toward whichever phase takes less room. Ice is 0.917 g/cm3 and liquid water is 1.00 g/cm3, so squeezing favors the liquid and the melting point drops. For most substances the solid is denser, so squeezing favors freezing.
11. Glycerol is about 1,000 times more viscous than water at 25 °C. Give two reasons, and say what warming it does.
Three OH groups and a longer chain, and warming thins it. Glycerol makes several hydrogen bonds per molecule instead of water’s few, and its longer chain tangles with its neighbors. Heating gives molecules enough energy to slip out of each grip sooner, so viscosity always falls as temperature rises.
12. Freeze drying holds frozen food below 0.006 atm. Name the phase change it uses and say why the low pressure matters.
Sublimation, below the triple point pressure. Water’s triple point is 0.01 °C and 0.006 atm. Below that pressure the liquid region is gone from the map entirely. Ice warmed there turns straight to vapor, so the food dries without ever getting wet.