Ideal Gas Laws and the Equation of State

The ideal gas law, pV = nRT = NkT, relates the pressure p, volume V, absolute temperature T and amount of gas n (R = 8.314 J/(mol·K) is the gas constant, k = 1.381 × 10⁻²³ J/K is the Boltzmann constant), and it combines the older gas laws of Boyle (pV = constant at constant T), Charles (V ∝ T at constant p) and Gay-Lussac (p ∝ T at constant V). In brief: temperature must be in kelvins and pressure in pascals (or use consistent units). One mole of an ideal gas at standard temperature and pressure (273.15 K, 101.3 kPa) occupies 22.4 liters. For a fixed amount of gas, p₁V₁/T₁ = p₂V₂/T₂. Real gases follow the law well at low pressure and high temperature.

Tires, balloons, scuba tanks and weather balloons all follow the gas laws. This guide states the individual laws and the ideal gas equation, shows how to use them for changes in state, discusses real gases and mixtures, compares the laws in a table, works through numerical examples and corrects common mistakes.

What are the individual gas laws?

• Boyle’s law (constant T and n): pV = constant, so p₁V₁ = p₂V₂. Halving the volume doubles the pressure. The graph of p against 1/V is a straight line.
• Charles’s law (constant p and n): V ∝ T (in kelvins), so V₁/T₁ = V₂/T₂. The volume extrapolates to zero at −273.15 °C (absolute zero).
• Gay-Lussac’s (pressure) law (constant V and n): p ∝ T, so p₁/T₁ = p₂/T₂. A sealed container heated gets a higher pressure.
• Avogadro’s law (constant p and T): V ∝ n. Equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.

What is the equation of state?

The combination of these laws gives the equation of state of the ideal gas: pV = nRT, where n is the number of moles. With the number of molecules N = nN_A and the Boltzmann constant k = R ÷ N_A: pV = NkT. For a fixed amount of gas, p₁V₁ ÷ T₁ = p₂V₂ ÷ T₂. The density of a gas is ρ = pM ÷ RT, so it decreases with temperature (hot air rises). For a mixture of gases, Dalton’s law of partial pressures states that the total pressure is the sum of the pressures that each gas would exert alone: p = p₁ + p₂ + …; the partial pressure of a gas is p_i = x_ip, where x_i is its mole fraction (oxygen, 21 percent of the air, has a partial pressure of about 21 kPa at sea level).

How do real gases differ?

Real molecules have a finite size and attract each other. At high pressure, the volume of the molecules becomes a significant part of the total volume, so the pressure is higher than predicted; at low temperature, attraction reduces the pressure and the gas can condense into a liquid. The van der Waals equation, (p + a n²/V²)(V − nb) = nRT, includes these corrections. For air at room temperature and atmospheric pressure, the ideal gas law is accurate to within about 0.1 percent. Gases deviate most near their liquefaction points.

Gas laws compared

Worked example: gas calculations

1. Find the pressure. 2.0 mol of gas at 300 K in a volume of 0.050 m³: p = nRT ÷ V = 2.0 × 8.314 × 300 ÷ 0.050 = 99,800 Pa = 99.8 kPa.
2. Boyle’s law. A bubble of 3.0 L at 100 kPa is compressed isothermally to 1.0 L: p₂ = 100 × 3.0 ÷ 1.0 = 300 kPa.
3. Scuba tank. A 12 L tank at an absolute pressure of 200 bar (2.0 × 10⁷ Pa) holds the equivalent of V = pV ÷ p_atm = 200 × 12 ÷ 1.01 = 2,380 L of air at atmospheric pressure (at the same temperature). A diver breathing 20 L/min (measured at atmospheric pressure) would use it up in 2,380 ÷ 20 = 119 minutes at the surface. At a depth of 20 m, where the absolute pressure is 3 bar, each breath of the same volume contains three times as many molecules, so the air is used three times as fast and lasts about 40 minutes.
4. Gay-Lussac. The air in a car tire has an absolute pressure of 321 kPa (220 kPa gauge) at 20 °C (293 K). After driving, its temperature is 50 °C (323 K), assuming a constant volume: p₂ = 321 × 323 ÷ 293 = 354 kPa absolute, which is 253 kPa gauge. The gauge pressure rises by about 33 kPa.
5. Combined law. A weather balloon with a volume of 2.0 m³ at the ground (101 kPa, 290 K) rises to a height where p = 20 kPa and T = 230 K: V₂ = V₁(p₁T₂) ÷ (p₂T₁) = 2.0 × 101 × 230 ÷ (20 × 290) = 8.0 m³.
6. Density. The density of dry air at 20 °C and 101.3 kPa: ρ = pM ÷ RT = 101,300 × 0.02897 ÷ (8.314 × 293.15) = 1.20 kg/m³.

How were the gas laws found?

Robert Boyle published his law in 1662, from experiments with a J-shaped tube of mercury (Robert Hooke built the apparatus). Jacques Charles discovered the proportionality of volume and temperature around 1787, and Joseph Gay-Lussac published it in 1802. Amedeo Avogadro proposed his hypothesis in 1811, and Émile Clapeyron combined the laws into the ideal gas equation in 1834. Johannes van der Waals published his equation for real gases in 1873 and won the Nobel Prize in 1910.

Which mistakes are common?

Mistake 1: Using Celsius temperatures

Gas laws require absolute temperature in kelvins.

Mistake 2: Using gauge pressure

Use absolute pressure: add atmospheric pressure to the gauge reading.

Mistake 3: Mixing units of R

With R = 8.314 J/(mol·K), use pascals and cubic meters. With R = 0.0821 L·atm/(mol·K), use liters and atmospheres.

Mistake 4: Applying the law to a gas near condensation

Near the boiling point or at high pressure, real gases deviate strongly from ideal behavior.