Question

A container with insulating walls is divided into two equal parts by a partition fitted with a valve. One part is filled with an ideal gas at a pressure P and temperature T, whereas the other part is completely evacuated. If the valve is suddenly opened, the pressure and temperature of the gas respectively will be:

• A. $\frac{P}{2},T$
• B. $\frac{P}{2},\frac{T}{2}$
• C. $P,T$
• D. $P,\frac{T}{2}$

Answer 1

The correct option is A $\frac{P}{2},T$

Since the wall of container are insulating hence the process is adiabatic in nature i.e. dq = 0. Also the second part of valve is evacuated so when the valve is opened, there will be free expansion of ideal gas.
According to the first law of thermodynamics :
$dU=dq+w$
At adiabatic condition, dq = 0. hence,
$dU=w$
Since gas undergo free expansion, work done is 0.
$dU=0$
$=C_{v}dT=0$
This implies there is no change in temperature of gas during expansion. Also the given gas is ideal so we can apply Boyle's law to calculate the final pressure.
$P_1{V}_1=P_2{V}_2$
$P,V$ are initial pressure and volume and $P_2,2V$ are final pressure and volume of gas respectively
Thus,
$PV=P_2\times 2V$
$\Rightarrow P_2=\frac{P}{2}$