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k_B (Boltzmann constant) is the bridge between macroscopic (temperature T) and microscopic (molecular KE) domains. R = N_A × k_B where R = 8.314 J mol⁻¹K⁻¹ and N_A = 6.022 × 10²³ mol⁻¹.

Avogadro's hypothesis: Equal volumes of all gases at same T and P contain the same number of molecules. At STP, 22.4 L of any gas contains N_A = 6.022 × 10²³ molecules. This is consistent with the ideal gas equation P₁V₁/N₁T₁ = P₂V₂/N₂T₂ = k_B.

The factor 1/3 arises from isotropy — molecules move equally in all 3 directions, so ⟨vₓ²⟩ = ⟨vy²⟩ = ⟨vz²⟩ = (1/3)⟨v²⟩. This leads to PV = (2/3)E (translational KE).

This is the kinetic interpretation of temperature. Average translational KE per molecule = (3/2)k_BT regardless of the nature of the gas (mono, di, polyatomic). Temperature is a measure of average molecular KE.

Three important molecular speeds: v_mp < v_avg < v_rms. All ∝ √(T/M) — higher temperature or lower molar mass → faster molecules.

Law of equipartition: Energy is equally distributed among all available modes. A vibrational mode contributes 2 × (1/2)k_BT = k_BT because it has both kinetic AND potential energy quadratic terms.

Rotation along the molecular axis has negligible moment of inertia (quantum reasons) and doesn't contribute. So diatomic has only 2 (not 3) rotational DOF.

More degrees of freedom → higher Cv → lower γ. γ = (f+2)/f where f = number of DOF (translational + rotational only, at moderate temperatures).

Valid for non-reacting ideal gases. Each gas contributes its partial pressure independently. In kinetic theory terms: P = (n₁ + n₂ + ...)k_BT since all molecules at same T have equal average KE.

H₂ molecules move 4 times faster than O₂ at the same temperature. Lighter molecules move faster — at fixed T, both have same average KE but different speeds.

At STP for air (d ≈ 2Å): l ≈ 2.9 × 10⁻⁷ m ≈ 1500d. Mean free path increases with lower pressure (fewer molecules) and decreases with larger molecular size.

These values match experimental data for N₂, O₂, H₂ very well. Note: Cp − Cv = R always holds for any ideal gas (Mayer's relation).

Kinetic theory assumes perfectly elastic collisions — total kinetic energy and total momentum are both conserved. This ensures the distribution of molecular speeds remains steady (dynamic equilibrium). No energy is lost to heat (no internal molecular excitations).

This is the basis of uranium enrichment: ²³⁵UF₆ (M=349) diffuses 0.44% faster than ²³⁸UF₆ (M=352). H₂ diffuses 4× faster than O₂ at the same temperature.

Dulong-Petit law (C ≈ 3R ≈ 25 J/mol·K) works well for metals at room temperature. Carbon (diamond) is an exception — quantum effects make its C much lower at room temperature.

Avogadro's number N_A = 6.022 × 10²³. This is the number of molecules in one mole (22.4 L at STP) of any ideal gas. It is universal — same for He, O₂, CO₂, etc. (Avogadro's hypothesis + kinetic theory).

Key assumptions of kinetic theory: (1) Molecules are in incessant random motion. (2) Molecular volume is negligible compared to container. (3) Intermolecular forces negligible except during brief collisions. (4) Collisions are perfectly elastic. (5) Distribution of speeds follows Maxwell-Boltzmann distribution.

This relates macroscopic (P, V, T) and microscopic (KE, N) quantities. Internal energy of ideal gas depends only on temperature, not P or V.

Higher temperature → lower number density (gas expands at const P) → fewer collisions → longer mean free path. At constant volume (n fixed), l is unchanged.

Average translational KE per molecule = (3/2)k_BT for all gases, regardless of monatomic, diatomic or polyatomic nature. However, total KE per molecule differs: Ar has 3 DOF, O₂ has 5 DOF (rigid) → O₂ has higher total energy.

At low pressure: molecules are far apart → intermolecular forces negligible → behaves ideally. At high temperature: thermal KE >> intermolecular PE → interaction effects negligible. Real gases deviate at high pressure and low temperature (near liquefaction).

v_rms ∝ √T (not T). Doubling T → v_rms increases by factor √2 ≈ 1.414. For KE ∝ T, so doubling T doubles the average KE per molecule.

This is consistent with the ideal gas equation — a remarkable result showing that macroscopic pressure arises from molecular kinetic energy. The 2/3 factor comes from the 3D isotropy (only one direction contributes to wall pressure at a time).

Mean free path is inversely proportional to number density n and square of molecular diameter d. In a highly evacuated tube (small n), mean free path can be as large as the tube length itself.

About 517 m/s — close to the speed of sound in air (~340 m/s). This explains why sound can propagate through air effectively.

At stellar core temperatures (~10⁷ K), molecules have enough energy to overcome nuclear Coulomb barriers → nuclear fusion occurs!

The bubble expands to about 5.35 cm³ — more than 5 times its original volume as it rises due to decreasing pressure.

Helium atoms move about 3.16 times faster than argon atoms at the same temperature. This is why helium escapes Earth's atmosphere more readily than heavier gases.

Note: The 2:1 mass ratio doesn't affect the answer at all! Any mass ratio would give the same result, since KE and v_rms depend only on temperature and molecular mass.

Argon needs to be at ~2524 K for its atoms to have the same rms speed as helium at 253 K. Heavier atoms need much higher temperature to achieve the same speed.

Over 6 × 10²⁶ air molecules in a typical room — about 1000 times Avogadro's number! Despite their enormous number, they occupy negligible volume compared to the room.

Since volume is fixed (rigid cylinder), we use Cv. No work is done, all heat goes to internal energy.

Only 0.43% speed difference! This tiny difference requires the gas diffusion to be repeated many thousands of times to separate enough ²³⁵U for nuclear fuel. This is done in uranium enrichment plants.

ρ_Ne/ρ_O₂ = 0.947 — neon is slightly less dense than oxygen in this mixture despite more neon molecules, because neon atoms are lighter.