Question

A vertical hollow cylinder of height 152 cm is fitted with a weightless piston. The lower half is filled with an ideal gas and the upper half with mercury. The cylinder is now heated at 300 K, so that $\frac{1}{4}$th of the mercury comes out. Find the temperature at which it will happen assuming thermal expansion of Hg to be negligible.

Answer 1

The total pressure on the gas would be the sum of the atmospheric pressure plus the pressure due to the mercury column. Atmospheric pressure is equivalent to 76 cm of Hg, and there is another 76 cm of Hg due to the mercury on top of the gas. We can use the ideal gas law and express pressures in cm of Hg to solve this question.
So, before heating,
$P_1=76+76=152cm$ of Hg
$V_1=\frac{V}{2}$
$T_1=300K$
After heating,
Since $\frac{1}{4}$th of the mercury goes out, the final pressure would be the atmospheric pressure plus that exerted by the remaining mercury.
$P_2=76+76\times \left(\frac{3}{4}\right)=133cm$ of Hg
The increase in volume after heating is: $V_2=\frac{V}{2}+\frac{V}{8}=\frac{5}{8}V$
Let, the required temperature be $T_2$
We know, $\frac{P_1\times V_1}{T_1}=\frac{P_2\times V_2}{T_2}$
$T_2=328.125K$