Question

A vertical hollow cylinder of height $1.52m$ is fitted with a movable piston of negligible mass and thickness. The lower half of the cylinder contains an ideal gas and the upper half is filled with mercury. The cylinder is initially at $300K$. When the temperature is raised, half of the mercury comes out of the cylinder. Find this temperature, assuming the thermal expansion of mercury to be negligible.

Answer 1

Initial pressure of gas, $P_1=P_0+\frac{H}{2}dg,P_0$ is atmospheric pressure
Initial volume of gas, $V_1=\frac{V}{2}$
Initial temperature, $T_1=300K$
When half of the mercury comes out of the cylinder, the final pressure of gas $P_2=P_0+\frac{H}{4}dg$ and final volume of the gas

$V_2=\frac{V}{2}+\frac{V}{4}=\frac{3V}{4}$

We have $\frac{P_1{V}_1}{T_1}=\frac{P_2{V}_2}{T_2}$

$\Rightarrow T_2=\frac{P_2{V}_2}{P_1{V}_1}{T}_1=\frac{(P_0+\frac{H}{4}dg)\cdot \left(\frac{3V}{4}\right)\times 300}{(P_0+\frac{H}{2}dg)\cdot \frac{V}{2}}$

Putting the value $P_0=0.76dg$ (atmospheric pressure) we get
$T_2=337.5K$.