Worked cases: speeds, isotopes and a moving bat

Example 12.5: argon and chlorine (2 : 1 by mass) at 27 °C. Average kinetic energy per molecule depends only on T, so the ratio is 1 : 1.

Their rms speeds: (v_rms,Ar/v_rms,Cl)² = M_Cl/M_Ar = 70.9/39.9 = 1.77, so v_rms,Ar/v_rms,Cl = 1.33. The 2 : 1 mass ratio of the mixture plays no part.

Example 12.6: uranium hexafluoride made with ²³⁵U has molecular mass 235 + 6 × 19 = 349 u; with ²³⁸U it is 352 u. The lighter molecule is faster: v₃₄₉/v₃₅₂ = (352/349)^½ = 1.0044, a difference of 0.44%.

That small gap is used for enrichment. The gas is held inside a thick, narrow porous cylinder; molecules wander through the long pores one at a time, the faster ²³⁵U ones leak out slightly more, and the step is repeated many times (Fig. 12.5).

The same reasoning shows why the rate of diffusion of a gas is inversely proportional to the square root of its molecular mass.

Example 12.7: a ball approaching at speed u meets a massive bat moving towards it at V. Relative to the bat the ball arrives at V + u and leaves at V + u, so relative to the ground it leaves at 2V + u: it speeds up.

Map it onto a gas: piston → bat, cylinder → ground, molecule → ball. A piston pushed in makes molecules rebound faster, so compressing a gas raises its temperature; a piston moving out makes them rebound slower, so an expanding gas cools.

If the bat is not massive the rebound speed is less than u. For a cricketer, a heavier bat sends the ball back faster.