Modified Joule–Thomson

Figure 1.1 shows container A of variable volume V controlled by a frictionless piston, immersed in a bath at temperature τ. This container is connected by a pipe with a porous plug to another container, B, of fixed volume V'. Container A is initially occupied by an ideal gas at pressure P while container B is initially evacuated. The gas is allowed to flow through the plug, and the pressure on the piston is maintained at the constant value P. When the pressure of the gas in B reaches P, the experiment is terminated. Neglecting any heat conduction through the plug, show that the final temperature of the gas in B is τ₁ = (C_P/C_V) τ, where C_p and C_V are the molar heats at constant pressure and volume of the gas.

Figure 1.1

SOLUTION

The work done by the piston goes into changing the internal energy of the part of the gas of volume dV that enters the plug and into the work done by the gas to enter container B occupying volume dV'. So we may write

(1)

where c_v is the constant-volume heat capacity for one molecule and dN is the number of molecules in the volume dV. On the other hand, before and after the plug, we have, respectively,

(2)

(3)

Substituting dV from (2) and from (3) into (1), we have

(4)

So,

(5)

When P = P' (5) becomes

(6)