Answer:
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:
[tex]P\cdot V = n\cdot R_{u}\cdot T[/tex] (Eq. 1)
Where:
[tex]P[/tex] - Pressure of the bubble, measured in kilopascals.
[tex]V[/tex] - Volume of the bubble, measured in cubic meters.
[tex]n[/tex] - Molar amount of the bubble, measured in kilomoles.
[tex]T[/tex] - Temperature, measured in Kelvin.
[tex]R_{u}[/tex] - 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:
[tex]\frac{P_{A}\cdot V_{A}}{T_{A}} = \frac{P_{B}\cdot V_{B}}{T_{B}}[/tex] (Eq. 2)
Where:
[tex]P_{A}[/tex], [tex]P_{B}[/tex] - Pressure of the bubble at bottom and surface, measured in kilopascals.
[tex]V_{A}[/tex], [tex]V_{B}[/tex] - Volume of the bubble at bottom and surface, measured in cubic meters.
[tex]T_{A}[/tex], [tex]T_{B}[/tex] - Temperature of the bubble at bottom and surface, measured in Kelvin.
The pressure experimented by the bubble at bottom and surface are, respectively:
[tex]P_{A} = 101.325\,kPa+\left(1027\,\frac{kg}{m^{3}} \right)\cdot \left(9.807\,\frac{kg}{m^{3}} \right)\cdot (25\,m)\cdot \left(\frac{1}{1000}\,\frac{kPa}{Pa} \right)[/tex]
[tex]P_{A} = 353.120\,kPa[/tex]
[tex]P_{B} = 101.325\,kPa[/tex]
If we know that [tex]P_{A} = 353.120\,kPa[/tex], [tex]P_{B} = 101.325\,kPa[/tex], [tex]V_{A} = 1.20\times 10^{-6}\,m^{3}[/tex], [tex]T_{A} = 290.15\,K[/tex] and [tex]T_{B} = 303.15\,K[/tex], then the volume of the bubble at surface is:
[tex]\frac{(353.120\,kPa)\cdot (1.20\times 10^{-6}\,m^{3})}{290.15\,K} = \frac{(101.325\,kPa)\cdot V_{B}}{303.15\,K}[/tex]
[tex]1.460\times 10^{-6} = 0.334\cdot V_{B}[/tex]
[tex]V_{B} = 4.372\times 10^{-6}\,m^{3}[/tex]
[tex]V_{B} = 4.372\,cm^{3}[/tex]
And the volume of the air bubble is determined by this formula:
[tex]V_{B} = \frac{4\pi\cdot R^{3}}{3}[/tex] (Eq. 3)
Where [tex]R[/tex] is the radius of the air bubble, measured in centimeters.
If we know that [tex]V_{B} = 4.372\,cm^{3}[/tex], then the radius of the air bubble is:
[tex]4.372 = \frac{4\pi\cdot R^{3}}{3}[/tex]
[tex]R^{3} = 1.044[/tex]
[tex]R \approx 1.015\,cm[/tex]
The radius of the bubble when it reaches the surface at 30 ºC is 1.015 centimeters.
