Change of state as a bubble rises in a liquid of constant volume

A liquid-filled container of constant volume is considered. A bubble of a perfect gas is allowed to rise from the bottom of the container to the top. Based on common assumptions, i.e. neglecting temperature changes and considering the liquid as incompressible, one obtains the unlikely result that, irrespective of the bubble’s size, the pressure in the bubble remains unchanged, while the pressure in the liquid increases by an amount equal to the hydrostatic pressure difference between the bottom and the top of the container. To resolve this paradox, the change of state of the thermodynamic system, consisting of a slightly compressible liquid, a bubble of a perfect gas, and the interface between them is determined on the basis of the energy balance. Solutions for the changes of pressure, temperature and bubble volume are given in closed form. Estimates of the orders of magnitude of non-dimensional parameters lead to conditions for neglecting surface tension and/or the temperature change of the liquid. Results of general nature, as well as for the particular case of an air bubble in water, are shown in diagrams, and several limiting cases, including the limiting case of constant pressure in the bubble, are discussed.