Ballast Volume Pressure

Two containers, I and II, filled with an ideal gas are connected by two small openings of the same area, A, through a ballast volume B (see Figure 1.1). The temperatures and pressures in the two containers are kept constant and equal to P, τ, and P, 2τ, respectively. The volume B is thermally isolated. Find the equilibrium pressure and temperature in the ballast volume, assuming the gas is in the Knudsen regime.

Figure 1.1

SOLUTION

The number of molecules per second entering the volume B from the left container I is proportional to the density of the molecules in I, n₁, the average velocity, (v₁), and the area of the opening, A. The constant of proportionality is unimportant for this problem. So, equating the rate of molecular flow in and out of volume B, we can write for flow rates  in equilibrium (see Figure 1.2)

(1)

or

(2)

Figure 1.2

The factor 2 appears for the flow rate since  there are two portals from region B. On the other hand, for an ideal gas, P = nτ, and therefore

(3)

We can rewrite (1) as

(4)

yielding

(5)

For the state of equilibrium, the energy in the volume B is constant. This means that the total rate of energy transfer out of volumes I and II should be equal to the rate of energy transfer out of volume B :

(6)

The average energy per particle is proportional to the temperature, so (6) becomes

(7)

We then have

(8)

Dividing (8) by (5), we can obtain

(9)

and

(10)