Question #43909

A vertical capillary tube of length $L$ and radius $r$ is closed at its top end. It initially contains air at atmospheric pressure $P_0$. The open bottom end is just brought in contact with the free surface of a liquid of density $\rho$ and surface tension $T$. The liquid rises rapidly in the tube to an equilibrium height $h$, compressing the trapped air. Assuming the contact angle is zero and the compression of the trapped air is perfectly adiabatic with an adiabatic exponent $\gamma$, which of the following expressions correctly gives the equilibrium height $h$? (Assume $h \ll L$ and acceleration due to gravity is $g$).