Leaky Balloon

Sometimes helium gas in a low-temperature physics lab is kept temporarily in a large rubber bag at essentially atmospheric pressure. A physicist left a 40-L bag filled with He floating near the ceiling before leaving on vacation. When she returned, all the helium was gone (diffused through the walls of the bag). Find the entropy change of the gas. Assume that the atmospheric helium concentration is approximately 5 ×10^-4%. What is the minimum work needed to collect the helium back into the bag?

SOLUTION

Let us consider the bag as part of a very large system (the atmosphere) which initially has N molecules of air, which we consider as one gas, and molecules N₁ of helium. The bag has volume V₀, and the number of helium molecules is N₀. Omitting all the temperature-dependent terms, we may write for the initial entropy of the system

(1)

When the helium has diffused out, we have

(2)

We wish to find  in the limit where V >> V₀. Then

(3)

We then obtain

(4)

where n₀/n₁ ~ 1/(5 × 10^-6) is the concentration ratio of helium molecules in the bag to their concentration in the air. In regular units

(5)

Substituting the standard pressure and temperature into (5) gives

(6)

The minimum work necessary to separate the helium at constant temperature is

(7)

∆S₁ = -∆S, since after we separate the helium molecules from the rest of the air, the total entropy of that system would decrease. So