Hole in Wall

A container is divided into two parts, I and II, by a partition with a small hole of diameter d. Helium gas in the two parts is held at temperatures T₁ = 150 K and T₂ = 300K, respectively, through heating of the walls (see Figure 1.1).

Figure 1.1

a) How does the diameter d determine the physical process by which the gases come to steady state?

b) What is the ratio of the mean free paths λ₁/λ₂ between the two parts when d << λ₁, d << λ₂?

c) What is the ratio λ₁/λ₂ when d >> λ₁, d >> λ₂?

SOLUTION

a) If the diameter of the hole is large compared to the mean free path in both gases, we have regular hydrodynamic flow of molecules in which the pressures are the same in both parts. If the diameter of the hole (and thickness of the partition) is small compared to the mean free path, there are practically no collisions within the hole and the molecules thermalize far from the hole (usually through collisions with the walls).

b) In case d << λ₁, d << λ₂, there are two independent effusive flows from I to II and from II to I. The number of particles N₁ and N₂ going through the hole from parts I and II are, respectively,

(1)

where A = πd²/4 is the area of the hole. At equilibrium, N₁ = N₂, so we have

(2)

The mean free path is related to the volume concentration of the molecules n by the formula

(3)

where σ is the effective cross section of the molecule, which depends only on the type of molecule, helium in both halves. Substituting (3) into (2) gives

or

c) When d >> λ₁, d >> λ₂, we have to satisfy the condition

or

which gives for the ratio of the mean free paths: