Two vessels A and B of equal volume V_{0} are connected by a narrow tube which can be closed by a valve. The vessels are fitted with pistons which can be moved to change the volumes. Initially, the valve is open and the vessels contain an ideal gas (C_{P}/C_{V} = γ) at atmospheric pressure p_{0} and atmospheric temperature T_{0}. The walls of the vessel A are diathermic and those of B are adiabatic. The valve is now closed and the pistons are slowly pulled out to increase the volumes of the vessels to double the original value.

(a) Find the temperatures and pressures in the two vessels.

(b) The valve is now opened for sufficient time so that the gases acquire a common temperature and pressure. Find the new values of the temperature and the pressure.

**Given:**

Two vessels A and B of equal volume V_{0} are connected by a narrow tube which can be closed by a valve.

Vessels contain an ideal gas () at atmospheric pressure p_{0} and atmospheric temperature T_{0}.

The walls of the vessel A are diathermic and those of B are adiabatic.

The pistons are slowly pulled out to increase the volumes of the vessels to double the original value.

(a) As the pistons are moved slowly to increase the volume, the expansion of gas in the diathermic vessel will be an isothermic process thus the temperature will be fixed at T_{0}. P,V and T represent the pressure, volume and temperature of the gasses and subscripts 1 and 2 denote initial and final state respectively.

Thus,

⇒

⇒

For the adiabatic vessel,

⇒

⇒

Again for ideal gasses,

⇒

⇒

Thus the temperature and pressure in the diathermic vessel will T_{0} and P_{0}/2 and in the adiabatic vessel, and .

(b) When the valve is open, the temperature will remain T_{0} throughout. Thus, there will be no change in temperature in the diathermic vessel so there will be change in pressure as well. For the gas in the diathermic vessel,

and for the adiabatic vessel

Therefore

Again,

Thus,

Thus the final temperature, when the valve is open will be T_{0} and the final pressure will be .

1