Question

A ring shaped tube contains two ideal gases with equal masses and molecular weight $m_1=32\text{g}$ and $m_2=28\text{g}$. The gases are separated by one fixed partition $P$ and another movable conducting partition $S$ which can move freely without friction inside the ring. The angle $\alpha$ as shown in the figure in equilibrium is:

• A. $\frac{7\pi }{8}$
• B. $\frac{8\pi }{7}$
• C. $\frac{25\pi }{16}$
• D. $\frac{16\pi }{15}$

Answer 1

The correct option is D $\frac{16\pi }{15}$

At the condition of equilibrium, the pressure of gas on both sides of movable partition $S$ will be equal.
$\Rightarrow P_1=P_2$

Also the partition $S$ is conducting hence it will maintain the thermal equilibrium of both gases by allowing exchange of heat between then.
Hence at the equilibrium position,
$T_1=T_2$

Considering the thermodynamic process to be reversible. because friction is absent,
From ideal gas equation,
$P_1=P_2$

$\frac{n_{1}R{T}_1}{V_1}=\frac{n_{2}R{T}_2}{V_2}$

$\frac{n_1}{V_1}=\frac{n_2}{V_2}(\because T_1=T_2).....\left(i\right)$
Equal mass of both the gases is present inside tube,
$n_1=\frac{m}{m_1}$$\&n_2=\frac{m}{m_2}$
Also the area of cross - section of tube is constant.
$\Rightarrow$ volume of the gas $=A\times l$
($l$ is length of arc)
$V\propto l$

$(\because l=r\theta )$
$V\propto \theta$

Let $V_1$ be Volume occupied by mass $m_1$ and $V_2$ be Volume occupied by mass $m_2$
$\frac{V_2}{V_1}=\frac{{\theta }_2}{{\theta }_1}$

$\frac{V_2}{V_1}=\frac{\alpha }{2\pi -\alpha }$

Substituting in Equation (i) we get,
$\frac{V_2}{V_1}=\frac{n_2}{n_1}$
$\frac{\alpha }{2\pi -\alpha }=\frac{\left(\frac{m}{m_2}\right)}{\left(\frac{m}{m_1}\right)}=\frac{m_1}{m_2}$


$\frac{\alpha }{2\pi -\alpha }=\frac{32}{28}=\frac{8}{7}$

$16\pi -8\alpha =7\alpha$

$\therefore \alpha =\frac{16\pi }{15}$
$\begin{array}{c}\text{Why this question}?\\ \text{Tip: In this problem the first crucial information is the}\\ \text{conducting nature of movable partition S, which gives}\\ \text{insight that it will enable exchange of heat between}\\ \text{both gases to maintain thermal equilibrium i.e.}{T}_2=T_2\\ \text{Trick : The angle subtended by are will give relation betwen}\\ \text{respective lengths of arc, thus volume of gas in each}\\ \text{section can be correlated.}\end{array}$