Lab Idea from Jaime Skiba: Volume of a Rising Air Bubble

rising bubbles from alaska-in-pictures.com

New York physics teacher Jaime Skiba posed a novel (to me) idea for a thermodynamics laboratory exercise.  She teaches chemistry as well as physics, so she and her colleagues are even more heavily invested in teaching gas laws than we physicists.  Her colleague, she says, long ago took excellent picture s of a gas bubble rising in a long column of liquid.  The diameter of the bubble increases as the bubble rises, because the pressure decreases while the temperature remains essentially constant.

Jaime proposes having students create some rising bubbles in a long, transparent graduated cylinders just by plunging an air hose to the bottom.  Smartphones can created video of the rising bubble; then (if you have the knowing of the software) the frames can be analyzed on a computer.  The diameter of the bubble in a frame can be measured by knowing the diameter of the cylinder itself, and scaling proportionally.

Students would make a graph of bubble diameter d on the vertical axis vs. depth h on the horizontal axis.  A straight line plot can be created by plotting the diameter cubed vs. the reciprocal of the depth.  Why?

Start with the ideal gas law, PV = nRT.  Pressure in the column is ρgh, where ρ is the density of the water.   The volume is the volume of a sphere.  By solving for the cube of diameter, I get 

Using this equation and the slope of the d-cubed vs. 1/h graph, you could solve for the number of moles in the gas bubble.