3.4 Gas Laws

Gas Laws

Gas laws describe how gases respond to changes in pressure, volume, temperature, and amount. They give you a set of predictable relationships so that if one variable changes, you can figure out what happens to the others.

These relationships show up everywhere: weather systems, car engines, scuba diving, even the way a bag of chips puffs up on an airplane. Mastering these laws also sets you up for stoichiometry problems involving gases later on.

Gas Laws

Pressure-Volume Relationship: Boyle's Law

Boyle's Law captures a simple idea: if you squeeze a gas into a smaller space (and the temperature stays the same), the pressure goes up. If you give it more room, the pressure drops. Pressure and volume move in opposite directions. This is called an inverse relationship.

The equation is:

$P_1V_1 = P_2V_2$

where $P$ is pressure and $V$ is volume. This only works when temperature and the amount of gas stay constant.

• Think of a bicycle pump. As you push the handle down, you decrease the volume inside the pump, and the air pressure increases enough to force air into the tire.
• A balloon rising in the atmosphere expands because the outside air pressure drops, so the gas inside spreads out into a larger volume.
• On a pressure-vs-volume graph, Boyle's Law produces a curved line (a hyperbola), not a straight line. That curve reflects the inverse relationship.

Solving a Boyle's Law problem:

1. Identify which two variables are changing (pressure and volume) and confirm temperature is constant.
2. Label your known values as $P_1$, $V_1$, and whichever of $P_2$ or $V_2$ you're given.
3. Plug into $P_1V_1 = P_2V_2$ and solve for the unknown.
4. Check that your answer makes sense: if pressure went up, volume should have gone down (and vice versa).

Quick example: A gas occupies 4.0 L at 2.0 atm. What's the volume at 8.0 atm?

$P_1V_1 = P_2V_2$ $(2.0)(4.0) = (8.0)(V_2)$ $V_2 = 1.0 \text{ L}$

Pressure quadrupled, so volume dropped to one-quarter. That checks out.

Temperature-Volume Relationship: Charles's Law

Charles's Law says that when pressure is held constant, a gas expands as it heats up and shrinks as it cools down. Volume and temperature are directly proportional.

The equation is:

$\frac{V_1}{T_1} = \frac{V_2}{T_2}$

where $V$ is volume and $T$ is temperature in Kelvin. You must use Kelvin here. If you plug in Celsius, your answer will be wrong.

To convert: $T(K) = T(°C) + 273.15$

• Hot air balloons work because heating the air inside the balloon increases its volume, making the air inside less dense than the cooler air outside.
• Tires can appear to lose pressure in cold weather because the air inside contracts as temperature drops.
• On a volume-vs-temperature (Kelvin) graph, Charles's Law gives a straight line that, if extended, would pass through the origin at 0 K.

Pressure-Temperature Relationship: Gay-Lussac's Law

Gay-Lussac's Law connects pressure and temperature when volume is held constant. Heat a gas in a rigid container, and the pressure rises. Cool it down, and the pressure falls. Like Charles's Law, this is a direct proportion.

The equation is:

$\frac{P_1}{T_1} = \frac{P_2}{T_2}$

where $P$ is pressure and $T$ is temperature in Kelvin (always Kelvin for gas laws).

• On a hot day, the air molecules inside your car tires move faster and hit the walls harder, raising the tire pressure. That's why tire pressure readings are best taken when tires are cool.
• Pressure cookers seal in steam at constant volume. As the temperature rises, pressure builds, which raises the boiling point of water and cooks food faster.
• The graph of pressure vs. temperature in Kelvin is a straight line, just like Charles's Law.

Comprehensive Gas Behavior: Combined Gas Law and Ideal Gas Law

When more than one variable changes at the same time, the individual laws aren't enough on their own. The Combined Gas Law merges Boyle's, Charles's, and Gay-Lussac's Laws into one equation:

$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$

This works whenever the amount of gas (moles) stays constant. If one of the three variables happens to stay constant too, the equation simplifies back to whichever individual law applies. For example, if temperature doesn't change, the $T$ values cancel and you're left with Boyle's Law.

Solving a Combined Gas Law problem:

1. List all six possible values: $P_1$, $V_1$, $T_1$, $P_2$, $V_2$, $T_2$.
2. Convert all temperatures to Kelvin.
3. Identify the one unknown.
4. Plug in and solve.
5. Sanity-check the result against what you'd expect from the individual laws.

The Ideal Gas Law goes one step further by including the amount of gas:

$PV = nRT$

• $P$ = pressure
• $V$ = volume
• $n$ = number of moles
• $R$ = the universal gas constant ($8.314$ J/(mol·K), or $0.0821$ L·atm/(mol·K) depending on your units)
• $T$ = temperature in Kelvin

Which value of $R$ you use depends on your pressure and volume units. If pressure is in atm and volume is in liters, use $0.0821$. If you're working in SI units (Pascals and cubic meters), use $8.314$.

This equation assumes an ideal gas, meaning the gas particles take up no space themselves and don't attract or repel each other. Most real gases behave close to ideal under normal conditions (moderate temperatures, moderate pressures). At very high pressures or very low temperatures, real gases start to deviate because particles get forced close together and intermolecular forces become significant.

Molar Volume: Avogadro's Law

Avogadro's Law states that the volume of a gas is directly proportional to the number of moles, as long as temperature and pressure stay constant.

$\frac{V_1}{n_1} = \frac{V_2}{n_2}$

It doesn't matter what the gas is. One mole of oxygen, one mole of helium, and one mole of carbon dioxide all occupy the same volume under the same conditions.

• At standard temperature and pressure (STP), defined as 0°C (273.15 K) and 1 atm, one mole of any ideal gas occupies 22.4 L. This value, called the molar volume, is worth memorizing.
• Two balloons containing equal numbers of molecules of different gases (say, helium and nitrogen) will have the same volume at the same temperature and pressure, even though they have very different masses.
• Avogadro's Law is the foundation for gas stoichiometry. In balanced chemical equations involving gases, the mole ratios are also volume ratios (at the same temperature and pressure).

Gas Properties

Fundamental Gas Characteristics: Pressure and Temperature

Pressure is the force that gas particles exert on the walls of their container, per unit area. Every time a gas molecule bounces off a wall, it pushes on that wall a tiny bit. Trillions of these collisions per second add up to a measurable pressure.

Common pressure units and their conversions:

• 1 atm = 101,325 Pa = 760 mmHg = 760 torr

Atmospheric pressure is the weight of all the air above you pressing down. At sea level, that's about 1 atm. At higher elevations, there's less air above, so atmospheric pressure is lower.

Temperature measures the average kinetic energy of the gas particles. Hotter gas means faster-moving particles on average.

• For everyday use, you'll see Celsius (°C) or Fahrenheit (°F).
• For all gas law calculations, you need Kelvin (K). The Kelvin scale starts at absolute zero (0 K, or $-273.15°C$), the theoretical point where all particle motion stops. There are no negative Kelvin values, which is exactly why the scale works in gas law equations: a negative temperature would break the proportional relationships.

Spatial and Quantitative Measures: Volume and Moles

Volume is the three-dimensional space a gas fills. Gases expand to fill whatever container they're in, so the volume of a gas equals the volume of its container.

• Common units: liters (L), milliliters (mL), cubic meters ($m^3$)
• Volume changes whenever pressure, temperature, or the amount of gas changes.

Moles tell you how much gas you have in terms of particle count. One mole equals $6.022 \times 10^{23}$ particles (Avogadro's number). This bridges the gap between the invisible world of individual molecules and the quantities you can actually measure in a lab, like mass in grams or volume in liters.

Physical Properties: Density and Kinetic Energy

Density is mass per unit volume:

$\rho = \frac{m}{V}$

Gas density is much lower than liquid or solid density because gas particles are spread far apart. For gases, density changes with conditions:

• Increasing pressure pushes particles closer together, raising density.
• Increasing temperature makes the gas expand (at constant pressure), lowering density.

This is why hot air rises: it's less dense than the cooler air around it.

Kinetic energy of gas particles is directly proportional to temperature:

$KE = \frac{3}{2}kT$

where $k$ is Boltzmann's constant ($1.38 \times 10^{-23}$ J/K) and $T$ is temperature in Kelvin. If you double the Kelvin temperature, you double the average kinetic energy of the particles.

This molecular motion explains two important processes:

• Diffusion: gas particles spread out from areas of high concentration to low concentration (like a perfume scent filling a room). This happens gradually because gas molecules constantly collide with each other, following zigzag paths rather than straight lines.
• Effusion: gas escapes through a tiny hole in a container. Lighter gas molecules move faster at the same temperature, so they effuse more quickly than heavier ones. For example, helium (molar mass 4 g/mol) effuses much faster than oxygen (molar mass 32 g/mol), which is why helium balloons deflate relatively quickly.