Question

A closed hollow insulated cylinder is filled with a gas at $0^{\circ }\text{C}$ and also contains an insulated piston of negligible weight and negligible thickness at the middle point. The gas on one side of the piston is heated to ${100}^{\circ }\text{C}$. If the piston moves by $5\text{cm}$, the length of the hollow cylinder is

• A. $13.65\text{cm}$
• B. $27.3\text{cm}$
• C. $38.6\text{cm}$
• D. $64.6\text{cm}$

Answer 1

The correct option is D $64.6\text{cm}$


Pressure on both sides of the piston will be the same throughout [for equilibrium of piston]
We know that for a constant pressure, $\frac{V}{T}=\text{constant}$
$\Rightarrow \frac{V_1}{T_1}=\frac{V_2}{T_2}$
$\Rightarrow \frac{A_1{l}_1}{273+100}=\frac{A_2{l}_2}{(273+0)}....\left(1\right)$
$\because$ Cross sectional area will be same.

As the piston moves by $5\text{cm}$, the length of one side will be $(\frac{l}{2}+5)$ and the other side $(\frac{l}{2}-5)$ as shown in the figure.
On using this in $\left(1\right)$, we get
$\frac{\frac{l}{2}+5}{373}=\frac{\frac{l}{2}-5}{273}$ [$\because A_1=A_2$]
$\Rightarrow 373\times \frac{l}{2}-273\times \frac{l}{2}=(273+373)5$
$\Rightarrow 100\times \frac{l}{2}=3230$
$\Rightarrow l=64.6\text{cm}$
Thus, option (d) is the correct answer.