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Two containers $C_{1}$ and $C_{2}$ of volumes $V$ and $4 \,V$ respectively, hold the same ideal gas and are connected by a thin horizontal tube of negligible volume with a valve which is initially closed. The initial pressures of the gas in $C_{1}$ and $C_{2}$ are $p$ and $5 p$, respectively. Heat baths are employed to maintain the temperatures in the containers at $300 \,K$ and $400 \,K$, respectively. The valve is now opened. Select the correct statement.
The gas will flow from the hot container to the cold one and the process is reversible.
The gas will flow from one container to the other till the number of moles in two containers are equal.
A long time after the valve is opened, the pressure in both the containers will be $3 p$.
A long time af ter the valve is opened, number of moles of gas in the hot container will be thrice that of the cold one.
Solution

$(d)$ Let $n_{1}$ and $n_{2}$ are number of moles of gas present in container $C_{1}$ and $C_{2}$ respectively, before the value is opened.
Then, using $p V=n R T$.
$\text { and }$
$n_{1} =\frac{p V}{R(300)}$
$n_{2} =\frac{5 p(4 V)}{R(400)}$
When value is opened gas flows from $C_{2}$ to $C_{1}$ till pressure in $C_{1}$ and $C_{2}$ is equal. Let after equalisation of pressure in both $C_{1}$ and $C_{2}$, its value is $p_{0}$,
Then, using $p V=n R^{\prime} T$.
$p_{0} V=n_{1}^{\prime} R(300)\left(\right.$ Container $\left.C_{1}\right)$
and $p_{0} 4 V=n_{2}{ }^{\prime} R(400)$ (Container $C_{2}$ )
So, $\quad n_{1}^{\prime}=\frac{p_{0} V}{R(300)}$
and $\quad n_{2}^{\prime}=\frac{p_{0}(4 V)}{R(400)}$
As no gas is leaked from containers,
$n_{1}+n_{2}=n_{1}{ }^{\prime}: n_{2}{ }^{\prime}$
$\Rightarrow \quad \frac{p V}{R(300)}+\frac{20 p V}{R(400)}$ = $\quad \frac{p_{0} V}{R(300)}+\frac{4 p_{0} V}{R(400)}$
So, $p\left(\frac{1}{300}+\right.\left.\frac{20}{400}\right)$ $=p_{0}\left(\frac{1}{300}+\frac{4}{400}\right)$
$\Rightarrow \quad 4 p=p_{0}$
$\text { Now, } \frac{n_{2}{ }^{\prime}}{n_{1}{ }^{\prime}}=\frac{\frac{4 p_{0} V}{R(400)}}{\frac{p_{0} V}{R(300)}}=3$
Finally number of moles of $C_{2}$ is thrice of $C_{1}$.