In a thermally isolated system, two boxes filled with an ideal gas are connected by a valve. When the valve is in closed position, states of the box 1 and 2 , respectively, are (1 atm, V, T) and (0.5 atm, $4 \mathrm{~V}, \mathrm{~T}$ ). When the valve is opened, the final pressure of the system is approximately

• A

  $0.5$ atm
• Correct answer
  B

  $0.6 \mathrm{~atm}$
• C

  $0.75 \mathrm{~atm}$
• D

  $1.0 \mathrm{~atm}$

Solution

After opening of at equilibrium temperature and pressure of whole gas is $\mathrm{T}_{1}$ and $\mathrm{P}_{1}$ $\begin{array}{l} n_{1}=\frac{1 \times V}{R T}, n_{2}=\frac{0.5 \times V \times 4}{R T} \\ n_{1}+n_{2}=n \\ \frac{V}{R T}+\frac{V \times 4}{2 R T}=\frac{5 V P_{1}}{R T_{1}} \\ \frac{3 V}{R T}=\frac{5 V P_{1}}{R T_{1}} \\ \frac{P_{1}}{T_{1}}=\frac{0.6}{T} \\ \Delta Q=0, \quad \Delta W=0 \\ \therefore \Delta U=0 \\ n_{1} C_{V} T+n_{2} C_{V} T=\left(n_{1}+n_{2}\right) C_{V} T_{1} \\ T_{1}=T \\ \frac{P_{1}}{T}=\frac{0.6}{T} \\ P_{1}=0.6 \text { atm } \end{array}$

Hence, the correct answer is option: 'B'.