Figure shows an adiabatic cylindrical tube of volume V_{0} divided in two parts by a frictionless adiabatic separator. Initially, the separator is kept in the middle, an ideal gas at pressure p_{1} and temperature T_{1} is injected into the left part and another ideal gas at pressure p_{2} and temperature T_{2} is injected into the right part. C_{P}/C_{V} = γ is the same for both the gases. The separator is slid slowly and is released at a position where it can stay in equilibrium. Find

(a) the volumes of the two parts,

(b) the heat given to the gas in the left part

(c) the final common pressure of the gases.

**Given:**

An adiabatic cylindrical tube of volume V_{0} is divided in two parts by a frictionless adiabatic separator.

An ideal gas at pressure p_{1} and temperature T_{1} is injected into the left part and another ideal gas at pressure p_{2} and temperature T_{2} is injected into the right part.

(a) When the piston is slowly moved to the equilibrium position, one side increases in volume when the other side decreases.

The processes will be adiabatic,

For the left part,

Where, subscript 1 and 2 represent the initial and the final state.

⇒ ……….(1)

And for the right part,

……….(2)

We are assuming P to be the common pressure.

Dividing (1) by (2) we get,

Again,

So,

⇒

⇒

Therefore,

The final volume of the left and the right side will be and respectively.

(b) The heat given will be zero as the whole process is taking place in an adiabatic surrounding.

(c) So putting the above result in (1) we get,

⇒

Thus the final common pressure of the gasses will be

1