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A primitive diving bell consists of a cylindrical tank with one end open and one end closed. The tank is lowered into a freshwater lake, open end downward. Water rises into the tank, compressing the trapped air, whose temperature remains constant during the descent. The tank is brought to a halt when the distance between the surface of the water in the tank and the surface of the lake is 46.6 m. Atmospheric pressure at the surface of the lake is 1.01 x 105 Pa. Find the fraction of the tank's volume that is filled with air.

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Answer:

0.18095

Explanation:

[tex]P_1[/tex] = Atmospheric pressure = [tex]1.01\times 10^5\ Pa[/tex]

[tex]\rho[/tex] = Density of water = 1000 kg/m³

[tex]V_1[/tex] = Volume of air

[tex]V_2[/tex] = Volume of water

At the given depth the absolute pressure is

[tex]P_2=P_1+\rho gh\\\Rightarrow P_2=1.01\times 10^5+1000\times 9.81\times 46.6\\\Rightarrow P_2=558146\ Pa[/tex]

From Boyle's law

[tex]P_1V_1=P_2V_2\\\Rightarrow \frac{V_2}{V_1}=\frac{P_1}{P_2}\\\Rightarrow \frac{V_2}{V_1}=\frac{1.01\times 10^5}{558146}\\\Rightarrow \frac{V_2}{V_1}=0.18095\\\Rightarrow V_2=0.18095V_1[/tex]

The fraction is 0.18095

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