Kinetic Theory and the Gaseous State: Maxwell Distribution, Molecular Speeds, Mean Free Path and Real Gases

5.1Gas laws and the ideal gas

• Ideal gas: point molecules with no forces between them; the volume of the gas is essentially the volume of the container, and its pressure comes from collisions with the walls. Real gases approach it at low pressure and high temperature.
• Absolute zero: the ideal-gas temperature scale is defined through the limit (pV_m)_p→0 = RT, which becomes zero at T = 0, i.e. −273.15 °C, the natural zero of temperature.
• Units: 1 bar = 10⁵ Pa; 1 atm = 101 325 Pa = 760 torr; 1 L bar = 100 J; 1 dm³ = 1 L = 10⁻³ m³. Always use kelvin.

5.2Kinetic theory and kinetic energy

Postulates

• A gas is made of a very large number of tiny particles whose own volume is negligible compared with that of the container.
• They move randomly in straight lines (no force field) and collide only now and then with one another and with the walls.
• Collisions are perfectly elastic: the total kinetic energy of the colliding pair is conserved, though it may be shared differently afterwards.
• There are no attractive or repulsive forces between the particles except during collisions.
• The average kinetic energy depends only on the absolute temperature.

• The kinetic gas equation follows from the momentum change of molecules hitting the walls. Combining it with pV = nRT gives the energy result.
• Average translational kinetic energy depends only on T, not on the nature or mass of the gas: H₂, He, O₂ at the same temperature have the same average kinetic energy per molecule (ratio 1 : 1). Heavier molecules therefore move more slowly.
• KE ∝ T (energy); speeds ∝ √T. At 300 K, (3/2)kT = 6.2 × 10⁻²¹ J per molecule and (3/2)RT = 3.74 kJ per mole; 2 mol have 7.5 kJ.
• For one mole, E = ½Mu_rms², so u_rms = √(2E/M).
• When molecules exchange energy by collisions (a sample starting with equal energies relaxes to the Maxwell–Boltzmann distribution), the total energy is unchanged, so the temperature is set by the original average energy: T = 2Ē/3k.

5.3Maxwell–Boltzmann distribution of speeds

Molecules of a gas do not all move at one speed. The fraction with speeds between u and u + du is

• Near the origin the exponential is about 1, so the curve rises as u² (parabolic); at high speeds the exponential wins and the curve falls with a long tail. Very few molecules have zero or very high speed.
• The peak (where the first derivative is zero) is at the most probable speed u_mp; the molecules cluster around u_mp, not around the average speed.
• The area under the curve is 1 (the whole sample) at every temperature.
• Higher temperature: the peak moves to higher speed, becomes lower and the curve broader; more molecules have high speeds. So in a set of curves the one peaking furthest right is the hottest.
• Lighter gas at the same temperature behaves like a hotter gas: broader curve, peak at higher speed. Heavier gases (CO₂) give narrow, tall curves at low speed; CH₄ gives a broad curve at high speed. At 300 K, u_mp: CO₂ 337, O₂ 395, N₂ 422, CH₄ 558 m s⁻¹.
• M and T enter only as the ratio M/T: O₂ at T and SO₂ (M = 64) at 2T have identical distributions.