Question

An air bubble at the bottom of a lake 38.5 m deep has a volume of 1.00 cm³. If the temperature at the bottom is 5.5∘ C and at the top 20.1∘ C, what is the radius of the bubble just before it reaches the surface? Express your answer to two significant figures and include the appropriate units.

Answer 1

The radius of the bubble just before it reaches the surface is approximately 0.50 cm.

Step-by-step explanation:

To calculate the radius of the bubble just before it reaches the surface, we can use the combined gas law equation:

P1V1/T1 = P2V2/T2

Where P1, V1, and T1 are the pressure, volume, and temperature at the bottom of the lake, and P2, V2, and T2 are the pressure, volume, and temperature at the top of the lake.

Plugging in the given values:

(1 atm)(1.00 cm³)/(5.5°C + 273.15) = (1 atm)(V2)/(20.1°C + 273.15)

Solving for V2:

V2 = (1.00 cm³ * (20.1°C + 273.15))/(5.5°C + 273.15)

V2 = 1.05 cm³

Finally, we can find the radius of the bubble using the formula for the volume of a sphere:

V = (4/3)πr³

Plugging in the value for V2:

1.05 cm³ = (4/3)πr³

Solving for r:

r³ = (3 * 1.05 cm³)/(4π)

r = ∛((3 * 1.05 cm³)/(4π))

r ≈ 0.50 cm