Pressure, temperature and molecular speed

Kinetic theory explains a gas's pressure and temperature from the motion of its molecules. Its picture of an ideal gas:

• Identical tiny molecules in fast, random motion in all directions.
• They collide with one another and with the walls elastically (no kinetic energy lost), and the collisions last almost no time.
• Between collisions there are no forces, so each molecule moves in a straight line at steady speed.

Why a gas pushes on its walls. Take a cube of side l. A molecule with x-velocity v_x hits a wall and bounces back with –v_x, so the wall receives momentum 2mv_x. It comes back after 2l/v_x, so the force from that one molecule is mv_x²/l.

• Add up all N molecules and divide by the wall's area l²: P = (mN/V)<v_x²>.
• Motion is random, so <v_x²> = <v_y²> = <v_z²> = <v²>/3.
• So P = (1/3)nm<v²> = (1/3)ρ<v²>, where n = N/V. Since the kinetic energy per unit volume is ½ρ<v²>, P = (2/3) × (kinetic energy per unit volume).

Compare this with PV = Nk_BT and temperature gets a meaning: ½m<v²> = (3/2)k_BT. Temperature measures the average kinetic energy of the molecules, and every gas at the same T has the same average kinetic energy per molecule. At 600 K it is 1.5 × 1.38 × 10⁻²³ × 600 ≈ 1.24 × 10⁻²⁰ J.

The rms speed is v_rms = √<v²> = √(3k_BT/m) = √(3RT/M) = √(3P/ρ). For oxygen at 27 °C, v_rms = √(3 × 8.31 × 300/0.032) ≈ 483 m/s. Take M in kg per mole (0.032, not 32): with 32 you get about 15 m/s, a sure sign of a unit slip.

• v_rms ∝ √T: to halve v_rms from 127 °C (400 K), cool the gas to 400/4 = 100 K, which is –173 °C.
• At T = 0 K the motion would stop, and there is nothing slower than stopped. That is why no temperature can go below absolute zero.
• Mix two lots of the same monatomic gas with no heat lost and their energies simply add: T = (N₁T₁ + N₂T₂)/(N₁ + N₂). 2 mol at 300 K and 1 mol at 600 K end at 400 K.

Not all molecules have the same speed, so there are three averages: v_rms = √(3RT/M), the mean speed √(8RT/πM) and the most probable speed √(2RT/M). The numbers under the root are about 3, 2.5 and 2, so v_rms > mean > most probable. For five molecules at 1, 2, 3, 4 and 5 m/s, the mean is 3 m/s but v_rms = √(55/5) ≈ 3.3 m/s.

At the same T, lighter molecules are faster, so they spread (diffuse) faster: the rate goes as 1/√M (Graham's law). Hydrogen diffuses √(32/2) = 4 times faster than oxygen. The same idea separates uranium-235 from the slightly heavier uranium-238.