In a hot summer day, a spherical air bubble that has a volume of 1.20 cm3 is released at temperature 17.0 °C by a scuba diver 25.0 m below the surface of ocean. Calculate the radius of the spherical bubble when it reaches the surface at temperature 30 °C? Assume that the number of air molecules in the bubble remain the same (rhosalt water = 1.027 g/cm3 ).

The radius of the bubble when it reaches the surface at 30 ºC is 1.015 centimeters.

Explanation:

Let suppose that air bubble behaves as ideal gas, whose equation of state is:

(Eq. 1)

Where:

- Pressure of the bubble, measured in kilopascals.

- Volume of the bubble, measured in cubic meters.

- Molar amount of the bubble, measured in kilomoles.

- Temperature, measured in Kelvin.

- Ideal gas constant, measured in kilopascal-cubic meter per kilomole-Kelvin.

Then, we eliminate the molar amount and the ideal gas constant by constructing the following relationship:

(Eq. 2)

Where:

, - Pressure of the bubble at bottom and surface, measured in kilopascals.

, - Volume of the bubble at bottom and surface, measured in cubic meters.

, - Temperature of the bubble at bottom and surface, measured in Kelvin.

The pressure experimented by the bubble at bottom and surface are, respectively:

If we know that , , , and , then the volume of the bubble at surface is:

And the volume of the air bubble is determined by this formula:

(Eq. 3)

Where is the radius of the air bubble, measured in centimeters.

If we know that , then the radius of the air bubble is:

The radius of the bubble when it reaches the surface at 30 ºC is 1.015 centimeters.

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