Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

When the US submarine Squalus became disabled at a depth of , a cylindrical chamber was lowered from a ship to rescue the crew. The chamber had a radius of $1 m$ and a height of $4 m$ , 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 role="math" localid="1662369677002" $p_{0} +ρgh$ , where $p_{0} = 1.000 atm$
is the surface pressure and $ρ=$ $1024 \frac{kg}{m^{3}}$ is the density of sea water.
Assume a surface temperature of $20 ° C$ and a submerged water temperature of $- 30 ° C$ .
What is the air volume in the chamber at the surface?
If air had not been released from the tanks, what would have been the air volume in the chamber at depth h = $80.0 m$ ?
How many moles of air were needed to be released to maintain the original air volume in the chamber?

The air volume in the chamber at the surface is $12.6 m^{3} .$
The air volume in the chamber at depth $h = 80.0 m is 1.16 m^{3} .$
The number of moles of air needed to be released to maintain the original air volume in the chamberis. $5.1 \times 10^{3} mol$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Determine the concept

Find the volume of air in the chamber using the formula for volume of cylinder. Air volume at height h can be calculated from pressure and temperature at that height and pressure, volume, and temperature at the surface using gas law. Number of moles can be calculated by using gas law.

Formulae are as follow:

$P = P_{0} + ρ g h$

$P V = n R T$

Here, P is pressure, V is volume, T is temperature, is density, h is height, g is an acceleration due to gravity,R is universal gas constant and n is number of moles.

Step 3: (a) Determine the air volume in the chamber at the surface

Air volume in the chamber at the surface can be found as,

$V = π r^{2} h' V = 3.142 {(1)}^{2} (4)$

$V = 12.568 \approx 12.6 m^{3}$

Hence, the air volume in the chamber at the surface is $12.6 m^{3} .$

Step 4: (b) Determine the air volume in the chamber at depth

Pressure at depth hcan be found as,

$P = P_{0} + ρ g h$

$P = 1.01 \times 10^{5} + 1024 (9.8) (80)$

$P = 9.038 \times 10^{5} Pa$

According to the gas law,

$P V = n R T \frac{P_{1} V_{1}}{P_{2} V_{2}} = \frac{T_{1}}{T_{2}}$

Volume of the air at depth hcan be found as,

$∴ V_{2} = \frac{P_{1} V_{1}}{P_{2}} \frac{T_{2}}{T_{1}}$

$\frac{1.01 \times 10^{5} (12.568) (243)}{9.038 \times 10^{5} (293)} ∴ V_{2} = 1.16 m^{3}$

Hence, the air volume in the chamber at depth $h = 80.0 m is 1.16 m^{3} .$

Step 5: (c) Determine the number of moles of air needed to be released to maintain the original air volume in the chamber

The number of moles of air needed to be released to maintain the original air volume in the chamber is given by gas law as,

$n = \frac{P △ V}{R T} \frac{9.038 \times 10^{5} (12.6 - 1.16)}{8.314 \times 243} n = 5.1 \times 10^{3} mol$

Hence, the number of moles of air needed to be released to maintain the original air volume in the chamber is. $5.1 \times 10^{3} mol$

Therefore, the volume and the number of moles of air at height hcan be found from its pressure, volume, and temperature at the surface using gas law.

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

Step 1: Determine the concept

Step 3: (a) Determine the air volume in the chamber at the surface

Step 4: (b) Determine the air volume in the chamber at depth

Step 5: (c) Determine the number of moles of air needed to be released to maintain the original air volume in the chamber

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

Step 1: Determine the concept

Step 3: (a) Determine the air volume in the chamber at the surface

Step 4: (b) Determine the air volume in the chamber at depth

Step 5: (c) Determine the number of moles of air needed to be released to maintain the original air volume in the chamber

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