Question

A 1.6cm3 air bubble is released from the sandy bottom of a warm, shallow sea, where the gauge pressure is 2.0 atm . The bubble rises slowly enough that the air inside remains at the same constant temperature as the water.

What is the volume of the bubble as it reaches the surface?

Answer 1

Concepts and reason

The concepts used to solve this problem are variation of pressure with depth in a fluid and ideal gas equation.

First use relation of atmospheric pressure and static fluid pressure to calculate the pressure at the bottom.

Finally use the ideal gas equation to calculate the volume of the bubble as it reaches the surface.

Fundamentals

If a fluid is within a container then the depth of an object placed in that fluid can be measured.

Expression for the pressure at the depth is,

Here, is the pressure at the surface, is the atmospheric pressure, is the gauge pressure.

Gas inside the bubble behaves like an ideal gas.

Expression for the ideal gas law is,

Here, P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.

Pressure at the surface of the lake is equal to the atmospheric pressure.

Use the ideal gas equation to the bubble at bottom and surface at constant temperature.

Here, is the volume of the air bubble at depth and is the volume of the air bubble when it reaches the surface.

Rearrange the above equation to get the volume of the air bubble when it reaches the surface,

Expression for the pressure at the depth is,

Substitute for and for .

The expression for the volume of the air bubble when it reaches the surface is,

Substitute for , for , and for .

Ans:

The volume of the air bubble when it reaches the surface is.