Question

a marine biologist is preparing a deep - sea submersible for a dive. the sub stores breathing air under high pressure in a spherical air tank that measures 62.0 cm wide.
the biologist estimates she will need 1700 l of air for the dive. calculate the pressure to which this volume of air must be compressed in order to fit into the air tank. write your answer in atmospheres. be sure your answer has the correct number of significant digits.
atm

Explanation:

Step1: Calculate the volume of the spherical air tank

The formula for the volume of a sphere is \(V = \frac{4}{3}\pi r^{3}\). Given the diameter \(d = 62.0\ cm\), so the radius \(r=\frac{d}{2}=\frac{62.0}{2}=31.0\ cm\).

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Since \(1\ L = 1000\ cm^{3}\), then \(V = 124.9\ L\) (using unit conversion \(V=\frac{124900}{1000}\)).

Step2: Use Boyle's law (\(P_1V_1 = P_2V_2\))

Assume the initial pressure \(P_1 = 1\ atm\) (standard pressure for the air volume \(V_1 = 1700\ L\)) and the final volume \(V_2=124.9\ L\).
From \(P_1V_1 = P_2V_2\), we can solve for \(P_2\), \(P_2=\frac{P_1V_1}{V_2}\).
Substitute \(P_1 = 1\ atm\), \(V_1 = 1700\ L\) and \(V_2 = 124.9\ L\) into the formula:

$$ P_2=\frac{1\times1700}{124.9}\approx13.6\ atm $$

Answer:

\(13.6\ atm\)