Question

Two cylinders A and B of equal capacity are connected to each other via a stopcock. A contains a gas at standard temperature and pressure. B is completely evacuated. The entire system is thermally Insulated. The stopcock is now opened. Answer the following. 

1. What is the final pressure of the gas in A and B? 

2. What is the change in Internal energy of the gas? 

3. What is the change in temperature of the gas? 

4. Do the intermediate states of the system (before setting to the final equilibrium state) lie on its P-V-T surface?

Answer 1

1. The above process is a case of free expansion of gas. As soon as the stopcock is removed the gas expands to a total volume of twice its original value (as V_A = V_B).

From Boyle’s law, we have Pα₁/v

Since V doubles after the stopcock is removed. P reduces to one half the original value.

At STP, pressure of the gas is 100 kPa = 1 Bar

∴ Pressure after expansion = 50 kPa = 0.5 Bar.

2. From the first law of thermodynamics

∆ Q = ∆ U + ∆ W,

Since the process does not involve any work done by the gas such as moving a piston, and no heat is exchanged,

∆ Q = ∆ W = 0

∴ ∆ U = O

∴ There is no change in internal energy of the gas.

3. Since the internal energy of the gas is fixed for the given process, the temperature of the gas also does not change. 

ie, ∆ T = ∆ U and if ∆ U = 0, then ∆ T = 0

4. In case of free expansion of gas, the gas does not go through states of thermodynamic equilibrium before reaching the final state. There fore, the thermodynamic parameters such as pressure, volume and temperature are not well defined for these intermediate states and hence, they do not lie on the P-V-T surface for the gas.