What is a gas made of?
A gas is made of tiny particles (atoms or molecules) with lots of empty space between them. They move fast, in straight lines, and bounce off the walls. Every bounce pushes the wall a little. Pressure is the total push on each square metre of wall. Its SI unit is the pascal (Pa); 1 atm = 101.325 kPa = 1.01325 bar.
Four things describe a gas: pressure P, volume V, temperature T (in kelvin) and amount n (in moles).
The three simple gas laws
Boyle's law (T and n fixed)
P ∝ 1/V, so P₁V₁ = P₂V₂. Squeeze a gas to half its volume and its pressure doubles.
Charles's law (P and n fixed)
V ∝ T, so V₁/T₁ = V₂/T₂. T must be in kelvin: T(K) = t(°C) + 273.15. At 0 K the volume of an ideal gas would shrink to zero, which is why 0 K is called absolute zero.
Gay-Lussac's (pressure) law (V and n fixed)
P ∝ T. A closed can heated in a fire can burst.
Avogadro's law (P and T fixed)
V ∝ n. Equal volumes of all gases at the same T and P hold equal numbers of molecules. At 273.15 K and 1 bar, one mole of an ideal gas takes 22.7 L (22.4 L at 1 atm).
The ideal gas equation PV = nRT
Joining the laws: V ∝ nT/P, so PV = nRT. R is the gas constant: 8.314 J mol⁻¹ K⁻¹ (= 8.314 Pa m³ mol⁻¹ K⁻¹ = 0.0821 L atm mol⁻¹ K⁻¹ = 0.0831 L bar mol⁻¹ K⁻¹).
The combined gas law for a fixed amount: P₁V₁/T₁ = P₂V₂/T₂.
Because n = m/M, the equation also gives molar mass and density: M = mRT/(PV) and d = PM/(RT).
Dalton's law of partial pressures: in a mixture that does not react, total P = p₁ + p₂ + …, and each p = (mole fraction) × P.
Deviation from ideal behaviour
An ideal gas is a model: particles have no size and do not attract each other. Real gases are close to this when the pressure is low and the temperature is high (particles far apart and fast).
The compressibility factor Z = PV/(nRT) shows how far a gas is from ideal. Z = 1 for an ideal gas.
- Z < 1: attractions pull particles together, so the gas is easier to squeeze (for example CO₂ or NH₃ at moderate pressure).
- Z > 1: at very high pressure the particles' own volume matters, so the gas is harder to squeeze (H₂ and He show this at most pressures).
The van der Waals equation fixes the model: (P + an²/V²)(V − nb) = nRT, where a measures attraction and b the particles' own volume. Gases with strong attractions can be liquefied below their critical temperature (CO₂: 31 °C).
Key formulas and definitions
- PV = nRT
- R = 8.314 J mol⁻¹ K⁻¹ = 0.0821 L atm mol⁻¹ K⁻¹
- P₁V₁/T₁ = P₂V₂/T₂
- M = mRT/(PV); d = PM/(RT)
- Z = PV/(nRT)
- (P + an²/V²)(V − nb) = nRT
Worked examples
1. A gas occupies 6.0 L at 100 kPa. It is squeezed to 2.0 L at the same temperature. New pressure?
Boyle: P₁V₁ = P₂V₂ → 100 × 6.0 = P₂ × 2.0 → P₂ = 300 kPa.
2. A balloon holds 3.0 L of air at 27 °C. What is its volume at 127 °C (same pressure)?
Change to kelvin: 300 K and 400 K. V₂ = V₁ × T₂/T₁ = 3.0 × 400/300 = 4.0 L.
3. What pressure do 2.0 mol of gas exert in a 10.0 L (0.0100 m³) container at 300 K?
P = nRT/V = 2.0 × 8.314 × 300 / 0.0100 = 498 840 Pa ≈ 499 kPa.
4. How many moles of gas are in a 24.6 L cylinder at 1.00 atm and 300 K?
n = PV/(RT) = 1.00 × 24.6 / (0.0821 × 300) = 24.6 / 24.63 ≈ 1.00 mol.
5. 0.64 g of a gas fills 0.50 L at 1.0 atm and 300 K. Find its molar mass.
M = mRT/(PV) = 0.64 × 0.0821 × 300 / (1.0 × 0.50) = 15.76/0.50 ≈ 31.5 ≈ 32 g/mol (oxygen, O₂).
6. A gas at 2.0 bar, 5.0 L and 27 °C is changed to 4.0 bar and 127 °C. What is the new volume?
P₁V₁/T₁ = P₂V₂/T₂ → V₂ = P₁V₁T₂/(T₁P₂) = 2.0 × 5.0 × 400 / (300 × 4.0) = 4000/1200 ≈ 3.3 L.
7. At 300 K and 100 bar, one mole of a gas has V = 0.22 L. Is it easier or harder to compress than an ideal gas? (R = 0.0831 L bar mol⁻¹ K⁻¹)
Z = PV/(nRT) = 100 × 0.22 / (1 × 0.0831 × 300) = 22 / 24.93 ≈ 0.88. Z < 1, so attractions win: it is easier to compress than an ideal gas.
Common mistakes
- Using °C in PV = nRT. Always convert: T(K) = t(°C) + 273.
- Mixing units of R: with P in atm and V in L use 0.0821; with SI (Pa, m³) use 8.314.
- Forgetting 1 L = 10⁻³ m³ when using P in pascals.
- Thinking real gases always have Z < 1. At very high pressure Z becomes more than 1 because the particles have their own volume.