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 suddenly opened. Answer the following :
$(a)$ What is the final pressure of the gas in $A$ and $B$?
$(b)$ What is the change in internal energy of the gas?
$(c)$ What is the change in the temperature of the gas?
$(d)$ Do the intermediate states of the system (before settling to the final equilibrium state) lie on its $P-V-T$ surface?

(A) $0.5 \text{ atm}$: The volume available to the gas is doubled as soon as the stopcock between cylinders $A$ and $B$ is opened. Since volume is inversely proportional to pressure at constant temperature,the pressure will decrease to one-half of the original value. Since the initial pressure of the gas is $1 \text{ atm}$,the final pressure in each cylinder will be $0.5 \text{ atm}$.
$(b)$ Zero: The internal energy of an ideal gas depends only on its temperature. Since this is a case of free expansion in a thermally insulated system,no work is done $(W = 0)$ and no heat is exchanged $(Q = 0)$. According to the first law of thermodynamics,$\Delta U = Q - W = 0$. Thus,the internal energy does not change.
$(c)$ Zero: Since the internal energy of an ideal gas is a function of temperature only and $\Delta U = 0$,the temperature of the gas remains constant.
$(d)$ No: The given process is a case of free expansion. It is rapid and uncontrolled. The intermediate states are non-equilibrium states and do not satisfy the ideal gas equation $PV = nRT$; therefore,they do not lie on the $P-V-T$ surface.