Bell Jar

A vessel with a small hole of diameter in it is placed inside a high-vacuum chamber (see Figure 1.1). The pressure is so low that the mean free path λ >> d. The temperature of the gas in the chamber is T₀, and the pressure is P₀. The temperature in the vessel is kept at a constant T₁ = 4T₀. What is the pressure inside the vessel when steady state is reached?

Figure 1.1

SOLUTION

The pressure inside the vessel

(1)

where n₁ is the concentration of the molecules inside the vessel and n₀ is the concentration of the molecules in the chamber. Disregarding the thickness of the walls of the vessel, we can write the condition that the number of molecules entering the vessel per second is equal to the number of molecules leaving it:

(2)

where A is the area of the hole, and we used the result for the number of molecules striking a unit area per second. Actually, the only important point here is that this number is proportional to the product of concentration and average velocity. Therefore,

(3)

The average velocity   So, from (3), we have

(4)

Substituting (4) into (1), we obtain

(5)