6.1 Gas Laws

Why Gas Laws Matter for the NAPT

Gas behavior is the mathematical backbone connecting chemistry to physics on the NAPT. A ship's high-pressure air (HP air) flasks, a diver's air tank, a steam plant's boiler, and a nuclear reactor's pressurizer all obey the same handful of equations covered in this section. The NAPT's chemistry content includes gas laws directly, and the same pressure-volume-temperature relationships reappear in Chapter 4's thermodynamics coverage and Chapter 7's reactor-pressure discussion — mastering the algebra here pays off twice.

Every gas law in this section relates the same four variables: pressure (P), volume (V), temperature (T), and the number of moles (n) of gas present. The trick to answering questions quickly is recognizing which variable is held constant and which equation applies.

Units and a Critical Rule: Always Use Kelvin

Before touching any gas-law equation, convert temperature to the Kelvin (K) scale using:

K = °C + 273

Gas laws are built on the relationship between molecular kinetic energy and absolute temperature — a scale that starts at absolute zero (0 K, or -273°C), where molecular motion theoretically stops. Plugging a Celsius value directly into Charles's Law or Gay-Lussac's Law is one of the most common errors on STEM screening tests. Pressure is commonly given in atmospheres (atm), kilopascals (kPa), or millimeters of mercury (mmHg), and volume in liters (L). Standard temperature and pressure (STP) is defined as 0°C (273 K) and 1 atm.

Boyle's Law: Pressure and Volume

Boyle's Law states that at constant temperature and a fixed amount of gas, pressure and volume are inversely proportional — squeeze a gas into a smaller space and its pressure rises; let it expand and pressure drops.

P1V1 = P2V2

Worked Example: A submarine's high-pressure air system transfers 4.0 L of compressed air from a 150 atm storage flask into a lower-pressure line regulated at 3.0 atm, at the same temperature. What volume does the air occupy at the lower pressure?

• P1 = 150 atm, V1 = 4.0 L, P2 = 3.0 atm
• V2 = P1V1 ÷ P2 = (150 atm × 4.0 L) ÷ 3.0 atm = 200 L

The same small volume of highly compressed air expands to 50 times its original volume once the pressure drops to 1/50th — exactly why HP air systems are treated as serious safety hazards aboard ship.

Charles's Law: Volume and Temperature

Charles's Law states that at constant pressure, volume and absolute temperature are directly proportional — heat a gas and it expands; cool it and it contracts.

V1/T1 = V2/T2

Worked Example: A gas sample confined in a piston (free to move, so pressure stays constant) occupies 2.00 L at 25°C. If heated to 100°C, what is the new volume?

• Convert: T1 = 25 + 273 = 298 K, T2 = 100 + 273 = 373 K
• V2 = V1 × (T2 ÷ T1) = 2.00 L × (373 K ÷ 298 K) = 2.50 L

Gay-Lussac's Law: Pressure and Temperature

Gay-Lussac's Law states that at constant volume, pressure and absolute temperature are directly proportional — heating a gas in a sealed, rigid container raises its pressure.

P1/T1 = P2/T2

Worked Example: A rigid cylinder of nitrogen gas reads 4.00 atm at 15°C in a cold storage space. A nearby fire warms it to 45°C. What pressure does the gauge reach (volume unchanged)?

• Convert: T1 = 15 + 273 = 288 K, T2 = 45 + 273 = 318 K
• P2 = P1 × (T2 ÷ T1) = 4.00 atm × (318 K ÷ 288 K) = 4.42 atm

This is precisely why sealed pressure vessels — air flasks, aerosol cans, boiler drums — carry warnings against exposure to open flame: a rigid container cannot relieve rising pressure by expanding.

The Combined Gas Law

When pressure, volume, and temperature all change simultaneously (moles held constant), use the Combined Gas Law, which folds Boyle's, Charles's, and Gay-Lussac's Laws into one equation:

P1V1/T1 = P2V2/T2

Worked Example: A gas sample has a volume of 6.00 L at 2.00 atm and 300 K. It is compressed to 3.00 L and cooled to 250 K. What is the new pressure?

• P2 = (P1 × V1 × T2) ÷ (T1 × V2) = (2.00 atm × 6.00 L × 250 K) ÷ (300 K × 3.00 L) = 3,000 ÷ 900 = 3.33 atm

The Ideal Gas Law: PV = nRT

The Ideal Gas Law adds the number of moles (n) and the universal gas constant (R) to link all four variables in a single equation:

PV = nRT

Use R = 0.0821 L·atm/(mol·K) when pressure is in atm and volume is in liters, or R = 8.314 J/(mol·K) when working in SI units (pascals and cubic meters).

Worked Example: How many liters does 2.50 mol of an ideal gas occupy at STP (273 K, 1.00 atm)?

• V = nRT ÷ P = (2.50 mol × 0.0821 L·atm/mol·K × 273 K) ÷ 1.00 atm = 56.0 L

This confirms the standard shortcut: 1 mole of any ideal gas occupies 22.4 L at STP (56.0 L ÷ 2.50 mol = 22.4 L/mol).

Summary Table: Which Law Applies?

Exam Strategy

1. Identify what's held constant first — that tells you which law to use.
2. Convert every temperature to Kelvin before plugging into an equation. This single step prevents the most common gas-law error.
3. Keep units consistent — if R = 0.0821 L·atm/(mol·K), pressure must be in atm and volume in L, not kPa or mL.
4. Sanity-check the direction of change — compressing a gas should never produce a smaller pressure; heating a rigid container should never produce a lower pressure. If your answer moves the wrong direction, recheck your algebra.