Question

A vertical cylinder of height 100 cm contains air at a constant temperature. The top is closed by a frictionless light piston. The atmospheric pressure is equal to 75 cm of mercury. Mercury is slowly poured over the piston. Find the maximum height of the mercury column that can be put on the piston.

Answer 1

Infinite

Answer 2

The problem involves an isothermal compression of air. According to Boyle's Law, $P_1h_1 = P_2h_2$. Initially, $P_1 = P_{atm} = 75$ cm Hg and $h_1 = 100$ cm. When mercury of height $h_m$ is added, $P_2 = P_{atm} + h_m = 75 + h_m$. So, $75 \times 100 = (75 + h_m)h_2$. To maximize $h_m$, $h_2$ must be minimized. For an ideal gas, $h_2$ can approach zero, but never reach it. As $h_2 \to 0$, $P_2 \to \infty$, which implies $h_m \to \infty$. Thus, the maximum height of the mercury column is infinite.