Question

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

Answer 1

The force exerted by a substance per unit area on another substance is called pressure. In case of gas, the pressure is defined as the force exerted by the gas molecules on the walls of its container. Pressure is the property which simply defines the direction in which mass flows.

Complete Step by Step Solution
According to the question, the lower half of a vertical hollow cylinder is filled with ideal gas and the upper half is filled with mercury. With the given initial temperature of $300K$ . On increasing the temperature half the amount of mercury comes out of the cylinder. We have to find the temperature at which mercury comes out of the cylinder.
So, at initial stage:
Pressure of gas = Pressure of mercury + Pressure of atmospheric air
Let pressure of gas at initial stage be ${P_i}$
\ {P_i} = (76 + 76)cm \\\ \Rightarrow 152cm \\\ \
Now, given temperature be $T = 300K$
Volume = half of the volume of cylinder
That is $V = \dfrac{1}{2}{V_1}$
Now, at final stage after heating:
Pressure of gas = Pressure of mercury + Pressure of atmospheric air
Let the pressure of gas at final stage be ${P_f}$
\ {P_f} = (38 + 76)cm \\\ \Rightarrow {P_f} = 114cm \\\ \
Volume of gas = $\dfrac{3}{4}$ of volume of cylinder
That is, $V = \dfrac{3}{4}{V_1}$
Now, by applying gas equation:
\ \dfrac{{152 \times {V_1}}}{{2 \times 300}} = \dfrac{{114 \times \dfrac{3}{4}{V_1}}}{T} \\\ \Rightarrow T = \dfrac{{114 \times 3 \times 2 \times 300}}{{4 \times 152}}K \\\ \Rightarrow T = 337.5K \\\ \
So, the temperature at which mercury comes out of the container is $337.5K$ .

Note
Kinetic molecular theory of gases: The rapid motion and collisions of molecules with the walls of the container causes pressure (force on a unit area). Pressure is proportional to the number of molecular collisions and the force of the collisions in a particular area. The more collisions of gas molecules with the walls, the higher the pressure.