Question

Two identical containers $A$ and $B$ with frictionless pistons contain the same ideal gas at the same temperature and same volume $V$. The mass of the gas in $A$ is $m_{A}$ and that in $B$ is $m_{B}$. The gas in each cylinder is now allowed to expand isothermally to the same final volume $2 V$. The changes in the pressure in $A$ and $B$ are found to be $\Delta P$ and $1.5 \Delta P$ respectively. Then

• A. $4 m_{A}=9 m_{B}$
• B. $2 m_{A}=3 m_{B}$
• C. $3 m_{A}=2 m_{B}$
• D. $9 m_{A}=4 m_{B}$

Answer 1

Answer: (C)

For isothermal change in $A$,
$P_{O A} V=P_{A}(2 V) \Rightarrow P_{A}=\frac{P_{O A}}{2}$
or $\Delta P=\left(\frac{-P_{O A}}{2}\right)$
For isothermal change in $B$
$P_{O B} V=P_{B}(2 V) \Rightarrow P_{B}=\frac{P_{O B}}{2}$
or $1.5 \Delta P=\left(\frac{-P_{O B}}{2}\right)$
$\Rightarrow \frac{1.5 \Delta P}{\Delta P}=\frac{P_{O B}}{P_{O A}}$
But $P_{O A}=\left(\frac{m_{A}}{M}\right) \frac{R T}{V}$
and $P_{O B}=\left(\frac{m_{B}}{M}\right) \frac{R T}{V}$
$\Rightarrow 1.5 \frac{\Delta P}{\Delta P}=\frac{m_{B}}{m_{A}}$
or $\frac{3}{2}=\frac{m_{B}}{m_{A}}$
or $3 m_{A}=2 m_{B}$