Submarine rescue. When the U.S. submarine Squalus became...

Submarine rescue. When the U.S. submarine Squalus became disabled at a depth of 80 m, a cylindrical chamber was lowered from a ship to rescue the crew. The chamber had a radius of 1.00 m and a height of 3.50 m, was open at the bottom, and held two rescuers. It slid along a guide cable that a diver had attached to a hatch on the submarine. Once the chamber reached the hatch and clamped to the hull, the crew could escape into the chamber. During the descent, air was released from tanks to prevent water from flooding the chamber. Assume that the interior air pressure matched the water pressure at depth h as given by p0 rgh, where p0 1.000 atm is the surface pressure and r 1024 kg/m3 is the density of seawater. Assume a surface temperature of 20.0C and a submerged water temperature of 40.0C. (a) What is the air volume in the chamber at the surface? (b) If air had not been released from the tanks, what would have been the air volume in the chamber at depth h 80.0 m? (c) How many moles of air were needed to be released to maintain the original air volume in the chamber?

Submarine rescue. When the U.S. submarine Squalus became disabled at a depth of 80 m, a cylindrical chamber was lowered from a ship to rescue the crew. The chamber had a radius of
1.00 m and a height of 3.50 m, was open at the bottom, and held
two rescuers. It slid along a guide cable that a diver had attached
to a hatch on the submarine. Once the chamber reached the
hatch and clamped to the hull, the crew could escape into the
chamber. During the descent, air was released from tanks to prevent water from flooding the chamber. Assume that the interior
air pressure matched the water pressure at depth h as given by
p0 rgh, where p0 1.000 atm is the surface pressure and
r 1024 kg/m3 is the density of seawater. Assume a surface temperature of 20.0C and a submerged water temperature of
40.0C. (a) What is the air volume in the chamber at the surface? (b) If air had not been released from the tanks, what would
have been the air volume in the chamber at depth h 80.0 m?
(c) How many moles of air were needed to be released to maintain the original air volume in the chamber?

Key Concepts

Hydrostatic Pressure

Hydrostatic pressure describes how the pressure in a fluid increases with depth due to the weight of the overlying fluid. It is generally expressed by the equation p = p? + ?gh, where p? is the ambient pressure at the surface, ? the density of the fluid, g the acceleration due to gravity, and h the depth. This concept is essential in understanding the conditions encountered underwater and how external pressure affects enclosed gas volumes.

Ideal Gas Law

The ideal gas law, given by the equation pV = nRT, relates the pressure, volume, temperature, and number of moles of a gas. It provides a fundamental framework for predicting how gases will respond to changes in their thermodynamic conditions. In this context, it is used to determine how the internal air volume in the rescue chamber adjusts when it is subjected to the higher pressures found at depth.

Boyle's Law

Boyle's Law is a specific instance of the ideal gas law under isothermal conditions and states that the pressure of a gas is inversely proportional to its volume (p?V? = p?V?) when the temperature remains constant. This principle is critical for analyzing how the volume of gas in a confined space changes in response to an increase in pressure, such as what occurs when the rescue chamber descends underwater.

Cylindrical Volume Calculation

The calculation of the volume of a cylinder, using the formula V = ?r²h, is a key geometric concept needed to determine the initial air volume available in the rescue chamber. Understanding this is crucial when assessing how much gas is present or required to maintain the chamber under varying pressure conditions during the rescue operation.

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