States of Matter and the Gas Laws

Zoom in far enough and the three everyday states look like this: particles in a solid are tightly packed, often in a regular pattern; in a liquid they are close together with no regular arrangement; in a gas they are far apart with no regular arrangement.^[6]

The motion differs too. Particles in a solid vibrate about fixed positions. In a liquid they slide past each other but stay in nearly constant contact, which is why a liquid flows and takes the shape of its container yet keeps its volume. In a gas they move independently except when they collide,^[6] so a gas spreads to fill whatever container it is in and is easy to squeeze.

Which state a substance is in depends on a contest between the attractions holding particles together and the kinetic energy pulling them apart. Heat a solid and its particles shake harder until they break out of their fixed places; heat a liquid and they eventually escape one another completely.

The kinetic molecular theory (KMT) is a model of an ideal gas built on five postulates:^[4]

1. Gas molecules are in continuous motion in straight lines, changing direction only when they collide with each other or with the walls.
2. The molecules are negligibly small compared to the distances between them.
3. Pressure comes from collisions of the molecules with the container walls.
4. The molecules neither attract nor repel each other, so their collisions are elastic: no energy is lost.
5. The average kinetic energy of the molecules is proportional to the kelvin temperature.

Keep these five ideas handy, because each gas law below grows out of them. Push the same molecules into a smaller box and they hit each wall more often, so pressure goes up. Heat them and they move faster and hit harder, so pressure goes up again, unless the box is allowed to grow.

Melting, boiling and sublimation (solid straight to gas) all need energy. Freezing, condensation and deposition (gas straight to solid) release it. Watch the thermometer, because it does something odd: while a substance is changing state, its temperature stays constant.^[7] The heat goes into pulling particles apart, not into making them move faster.

For water, melting takes ΔH_fus = 6.01 kJ/mol and boiling takes ΔH_vap = 40.67 kJ/mol at the boiling point.^[7] Warming takes 2.09 J/g·°C for ice, 4.18 J/g·°C for liquid water and 1.86 J/g·°C for steam.^[7] A liquid boils when its vapor pressure equals the pressure of the air above it,^[7] which is why water boils below 100 °C high in the mountains.

Worked example: heat to take 100 g of ice at −20 °C to steam at 120 °C

100 g ÷ 18.02 g/mol = 5.55 mol.

1. Warm the ice: 100 g × 2.09 J/g·°C × 20 °C = 4.18 kJ
2. Melt it: 5.55 mol × 6.01 kJ/mol = 33.4 kJ
3. Warm the water: 100 g × 4.18 J/g·°C × 100 °C = 41.8 kJ
4. Boil it: 5.55 mol × 40.67 kJ/mol = 226 kJ
5. Warm the steam: 100 g × 1.86 J/g·°C × 20 °C = 3.72 kJ

Total ≈ 309 kJ. Boiling alone is 73% of it: separating molecules completely takes far more energy than loosening them.

A phase diagram shows which state is stable at each temperature and pressure. The lines are where two phases coexist. Where all three lines meet, ice, liquid water and water vapor coexist: the triple point.^[8] For water it is at 0.01 °C and 0.6 kPa.^[8] At pressures below the triple point water cannot be a liquid at any temperature: ice turns straight to vapor.^[8] The boiling curve stops at the critical point, 374 °C and 218 atm; above that temperature there is no boundary between liquid and gas.^[8]

Water's melting line leans slightly to the left, so squeezing ice lowers its melting point a little. Most substances lean the other way.

Pressure is force divided by the area it acts on: P = F/A.^[1] In a gas, that force comes from countless molecules striking the walls. At sea level, the air above you presses with 1 atmosphere, enough to hold up a column of mercury about 760 mm high.^[1]

Worked example: a weather report says 98.0 kPa. What is that in atm and mmHg?

1. 98.0 kPa × (1 atm / 101.325 kPa) = 0.967 atm
2. 0.967 atm × 760 mmHg/atm = 735 mmHg

Kelvin first

If a gas is cooled at constant volume, its pressure falls in a straight line that reaches zero at −273 °C: absolute zero, the lowest possible temperature and 0 on the kelvin scale.^[2] Celsius puts its zero at the freezing point of water, so it has negative values and doubling a Celsius number means nothing physically. Temperatures must be in kelvin for every gas law calculation:^[2] K = °C + 273.15 (or + 273 for 3 significant figures).

Boyle's law: pressure and volume

At constant temperature, the volume of a given amount of gas is inversely proportional to its pressure.^[2] Halve the volume and the pressure doubles.

Charles's law: volume and temperature

At constant pressure, the volume of a given amount of gas is directly proportional to its kelvin temperature.^[2] A balloon taken from a warm room into a freezer shrinks.

Gay-Lussac's law (Amontons's law): pressure and temperature

At constant volume, pressure is directly proportional to kelvin temperature. Guillaume Amontons first found this around 1700, and Joseph Louis Gay-Lussac measured it more precisely around 1800.^[2] It is why a sealed can is dangerous in a fire.

Avogadro's law: volume and amount

At the same temperature and pressure, equal volumes of gases contain the same number of molecules,^[2] so volume is proportional to moles.

Put the first four together, holding only the amount fixed, and you get the combined gas law.^[2]

Worked example (Boyle): 2.00 L of gas at 1.00 atm is allowed to expand until the pressure is 0.500 atm

V₂ = P₁V₁/P₂ = (1.00 atm × 2.00 L) / 0.500 atm = 4.00 L

Worked example (Charles): 3.00 L of gas at 27 °C is heated to 127 °C at constant pressure

1. Convert: 27 °C = 300. K, 127 °C = 400. K
2. V₂ = V₁ × T₂/T₁ = 3.00 L × 400. K / 300. K = 4.00 L

In Celsius the temperature looked like it nearly quintupled; the volume grew by only a third.

Worked example (Gay-Lussac): a tire at 2.20 atm and 20 °C warms to 50 °C

P₂ = 2.20 atm × 323 K / 293 K = 2.43 atm

Worked example (combined): 5.00 L at 1.00 atm and 300. K goes to 2.00 atm and 400. K

V₂ = V₁ × (P₁/P₂) × (T₂/T₁) = 5.00 L × (1.00/2.00) × (400./300.) = 3.33 L

PV = nRT

All four relationships fit in one equation. R is the gas constant; the values you will use most are 0.08206 L·atm/(mol·K) and 8.314 kPa·L/(mol·K).^[2] Pick the one whose units match your data.

STP. To compare gases, chemists use standard temperature and pressure: 273.15 K (0 °C) and 1 atm (101.325 kPa). At STP one mole of an ideal gas takes up about 22.4 L, the standard molar volume.^[2] This is the same STP as the Stoichiometry unit. In 1982 IUPAC changed standard pressure to 1 bar (100 kPa), but the older 1 atm definition is still widely used.^[2] At 0 °C and 1 bar the molar volume works out to 8.314 × 273.15 / 100 = 22.7 L, so check which definition a problem uses.

Worked example: pressure of 0.500 mol of gas in 5.00 L at 25 °C

1. T = 25 + 273 = 298 K
2. P = nRT/V = 0.500 mol × 0.08206 L·atm/(mol·K) × 298 K / 5.00 L = 2.45 atm

Worked example: how many moles are in a 10.0 L balloon at STP?

n = PV/RT = (1.00 atm × 10.0 L) / (0.08206 × 273.15 K) = 0.446 mol. Check: 10.0 L ÷ 22.4 L/mol = 0.446 mol.

Unless gases react, each gas in a mixture exerts the same pressure it would if it were alone in the container. The total pressure is the sum of these partial pressures:^[3]

P_total = P_A + P_B + P_C + …

Worked example: a tank holds N₂ at 0.60 atm, O₂ at 0.30 atm and Ar at 0.10 atm

P_total = 0.60 + 0.30 + 0.10 = 1.00 atm. The fraction of the molecules that are O₂ equals its share of the pressure: 0.30/1.00 = 30%.

The ideal gas law works well near everyday pressures, up to about 1 or 2 atm for most gases.^[4] It does poorly at relatively high pressures,^[5] where the molecules are crowded so close that their own volume matters, and the error grows at low temperatures, where slow molecules feel each other's attractions more.^[5] In 1879 Johannes van der Waals added two corrections to PV = nRT, one for molecular volume and one for attractions.^[5]

• Gay-Lussac's law in action: heat a sealed can and its pressure rises. Cosmetic aerosols sold in Canada must carry “CONTAINER MAY EXPLODE IF HEATED” and warn against puncturing, burning, or storing them above 50 °C.^[11]
• Never put a sealed container, including a capped bottle, in a fire, an oven or hot water.
• Gas cylinders in a lab are chained upright. Don't move or open one without your teacher.
• When heating anything in a test tube, leave it open and point it away from people.