Venting

A thermally insulated chamber is pumped down to a very low pressure. At some point, the chamber is vented so that it is filled with air up to atmospheric pressure, whereupon the valve is closed. The temperature of the air surrounding the chamber is T₀ = 300K What is the temperature T of the gas in the chamber immediately after venting?

SOLUTION

The air surrounding the chamber may be thought of as a very large reservoir of gas at a constant pressure P₀ and temperature T₀. The process of venting is adiabatic, so we can assume that there is no energy dissipation. We then find that the energy of the gas admitted to the chamber equals the sum of its energy E₀ in the reservoir plus the work done by the gas of the reservoir at P₀ to expel the gas into the chamber. This may be calculated by considering the process of filling a cylinder by pushing a piston back, where the piston offers a resistant force of P₀A, A being the cross section of the cylinder. The total energy E is then given by

(1)

Figure 1.1

where V₀ is the volume of the gas needed to fill the volume of the chamber V (note that V does not coincide with V₀ because the temperature of the gas in the chamber T presumably is not the same as T₀, see Figure 1.1). On the other hand.

(2)

where C_V is the heat capacity of the gas, c_v is the heat capacity per molecule, and N₀ is the number of molecules. From (1) and (2), we have

(3)

Using the ideal gas law

(4)

we have

(5)

So

(6)

The air is mostly nitrogen and oxygen (78% nitrogen and 21% oxygen = 99%), diatomic gases, so that

and therefore T = 240 K. Thus, the temperature of the gas in the chamber will increase. Note that the result does not depend on the outside pressure P₀, the volume of the chamber V, or whether it is filled to P₀.