9.2 Relating Pressure, Volume, Amount, and Temperature: The Ideal Gas Law

The ideal gas law connects pressure, volume, moles, and temperature into a single equation. It's the go-to tool for predicting how gases behave under changing conditions, and most gas problems you'll encounter in this course come back to it.

Gas behavior makes more sense when you think about it at the particle level. Gas particles are constantly moving and colliding with each other and with container walls. When those particles move faster or collide more frequently, we observe changes in pressure, volume, and temperature at the macroscopic scale.

The Ideal Gas Law

Application of ideal gas law

The ideal gas law is expressed as:

$PV = nRT$

where:

• $P$ = pressure (in atm)
• $V$ = volume (in liters)
• $n$ = amount of gas (in moles)
• $R$ = ideal gas constant = 0.08206 L·atm/(mol·K)
• $T$ = temperature (in Kelvin)

You can rearrange this equation to solve for any variable. For example, to find volume: $V = \frac{nRT}{P}$

Unit consistency is critical. Before plugging anything in, make sure your units match the gas constant $R$:

1. Convert temperature from Celsius to Kelvin: $K = °C + 273.15$
2. Convert pressure to atmospheres if given in other units (1 atm = 760 mmHg = 101.325 kPa)
3. Make sure volume is in liters

Standard Temperature and Pressure (STP) is a common reference point: 0°C (273.15 K) and 1 atm. At STP, one mole of an ideal gas occupies 22.4 L.

Effects of variable changes on gases

The ideal gas law contains several simpler gas laws, each describing what happens when you hold certain variables constant.

Boyle's Law (constant temperature and amount):

$P_1V_1 = P_2V_2$

Pressure and volume are inversely proportional. If you compress a gas to half its volume, the pressure doubles.

Charles' Law (constant pressure and amount):

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

Volume and temperature are directly proportional. Heat a gas and it expands; cool it and it contracts. Temperature must be in Kelvin for this to work.

Gay-Lussac's Law (constant volume and amount):

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

Pressure and temperature are directly proportional. This is why a sealed container builds pressure when heated.

Avogadro's Law (constant pressure and temperature):

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

Volume and amount of gas are directly proportional. Add more gas to a flexible container and it expands.

Dalton's Law of Partial Pressures applies to gas mixtures: the total pressure equals the sum of each individual gas's partial pressure.

$P_{total} = P_1 + P_2 + P_3 + ...$

Each gas contributes to the total pressure as if the other gases weren't there.

Molecular interpretation of gas behavior

Kinetic molecular theory explains why gases behave the way the gas laws describe. The key ideas:

• Gas particles are in constant, random motion, colliding with each other and with container walls.
• Collisions are elastic, meaning no kinetic energy is lost during a collision.
• The average kinetic energy of gas particles is directly proportional to the absolute temperature (in Kelvin).

These ideas connect directly to the measurable properties of gases:

• Pressure comes from particles hitting the container walls. More particles or faster particles means more frequent and forceful collisions, which means higher pressure.
• Temperature is a measure of the average kinetic energy of the particles. Higher temperature means the particles are moving faster on average.
• Volume changes reflect changes in how spread out the particles are. In a larger container, particles travel farther between collisions with the walls, so collision frequency drops and pressure decreases (consistent with Boyle's Law).

Real gases and deviations from ideal behavior

The ideal gas law assumes that gas particles have no volume and don't attract or repel each other. Real gases don't perfectly follow these assumptions.

Deviations from ideal behavior become noticeable under two conditions:

• High pressure, where particles are squeezed close together and their actual volume matters
• Low temperature, where particles move slowly enough that attractive forces between them have a significant effect

The compressibility factor ($Z = \frac{PV}{nRT}$) quantifies how much a real gas deviates from ideal behavior. For an ideal gas, $Z = 1$. Values above or below 1 indicate the gas is behaving non-ideally.

For most problems in an intro course, the ideal gas law works well. Just keep in mind that it's an approximation, and it's most accurate at relatively high temperatures and low pressures, where gas particles are far apart and moving fast.