§ C-4 Avogadro’s Law, Molar Volume & the Ideal Gas Equation (Grade 8–10 · Chemistry)
Molar volume Vₘ = 22.4 L/mol (STP), amount of substance ν & PV = νRT. Equal volumes of all ideal gases at the same temperature and pressure contain the exact same number of molecules: N = ν · N_A.
Theory Summary & Scientific Mechanism
In an ideal gas, molecular volume is negligible compared to container volume and intermolecular forces act only during elastic collisions. By Avogadro’s law, one mole of any ideal gas (ν = 1 mol, containing N_A = 6.022× 10²³ molecules) at the same temperature T and pressure P occupies the exact same molar volume Vₘ = (RT)/(P). At STP (T = 273.15 K, P = 101.325 kPa), Vₘ = 22.4 L/mol.
Combining Boyle’s, Charles’s, Gay-Lussac’s, and Avogadro’s laws yields the ideal gas equation of state (Mendeleev–Clapeyron equation): PV = ν RT = (m)/(M)RT. Rearranging gives gas density ρ = (m)/(V) = (PM)/(RT), demonstrating that at fixed P and T the macroscopic density of a gas is directly proportional to its molar mass M.
Microscopically, the mean translational kinetic energy per molecule langle Eₖrangle = (3)/(2)k_B T depends solely on absolute temperature T. While lighter molecules move faster (vᵣms = √((3RT)/(M)), so H₂ moves 4× faster than O₂), each individual collision carries less momentum; the higher collision frequency and lower momentum per impact cancel out so that equal molar amounts exert the exact same pressure P!
Everyday Analogy
Even though a CO₂ molecule is 22 times heavier than an H₂ molecule, Avogadro’s law guarantees that 1 mol of either gas inflates a balloon to the exact same volume (22.4 L at STP)! Lighter molecules hit the walls more often, while heavier ones hit harder per impact.
Interactive Question & Derivation
Does the volume V occupied by 1 mol of an ideal gas at fixed temperature T and pressure P depend on the chemical identity or molar mass of the gas (H₂, O₂, or CO₂)?
- A: No; by Avogadro’s law the molar volume Vₘ = RT/P is identical for all ideal gases (22.4 L/mol at STP), while molar mass M only affects gas density ρ = PM/(RT) and molecular speed
- B: Heavier CO₂ (M = 44 g/mol) occupies 22 times more volume per mole than light H₂ (M = 2 g/mol)
- C: Light H₂ gas exerts zero pressure on the container walls
Study Hint: From V = (ν R T)/(P), ideal gas volume is independent of molar mass M. Meanwhile, how does gas density ρ = (P M)/(R T) scale with molar mass M?
Proof & Explanation: Although light hydrogen molecules H₂ (M = 2 g/mol) move 4 times faster on average than oxygen O₂ (M = 32 g/mol), the average translational kinetic energy per molecule langle εₖ rangle = (3)/(2)k_B T depends solely on absolute temperature T. Thus 1 mol of any ideal gas occupies the same molar volume Vₘ = RT/P.
Key Equations & 3D Simulation Parameters
V = (ν R T)/(P), ρ = (P M)/(R T), vᵣms = √((3 R T)/(M))
- Amount of gas substance ν: 0.5 … 2.5 mol (default: 1 mol)
- Absolute temperature T: 200 … 500 K (default: 273.15 K)
- External pressure P: 50 … 200 kPa (default: 101.3 kPa)
- Gas molar mass M (2=H₂, 32=O₂, 44=CO₂): 2 … 44 g/mol (default: 32 g/mol)