Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

Question: An air bubble of volume 20 cm³ is at the bottom of a lake 40 m deep, where the temperature is 4. 0 ⁰C . The bubble rises to the surface, which is at a temperature of . Take the temperature of 20 ⁰C the bubble’s air to be the same as that of the surrounding water. Just as the bubble reaches the surface, what is its volume?

Answer

The volume of the bubble as it reaches the surface is $1.02 \times 10^{2} {cm}^{3}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given

Volume of the air bubble at the bottom of the lake, $V_{1} = 20 {cm}^{3} = 20 \times 10^{- 6} m^{3} .$
The depth of the lake,h = 40 m
Temperature of the air bubble at the bottom of the lake, $T_{1} = 4.0 ° C = 277 K .$

Temperature of the air bubble at the surface of the lake, $T_{0} = 20 ° C = 293 K .$

Step 2: Determine the concept

Find the number of moles using the gas law. Using this value of number of moles, find the volume of the bubble as it reaches the surface by applying the gas law to the air bubble at the surface.

Formula is as follow:

$p_{i} v_{i} = n R T_{i}$

Here, p is pressure, v is volume, T is temperature, R is universal gas constant and n is number of moles.

Step 3: Determine the volume of the bubble as it reaches the surface

According to the gas law,

pV = n RT

Pressure at the bottom of the lake is given by,

, $p_{1} = p_{0} + ρ g h$

Where, $p_{0}$ is the atmospheric pressure and $ρ$ is the density of water.

Thus,

$n = \frac{(p_{0} + ρ g h) V_{1}}{R T_{1}} n = \frac{(1.013 \times 10^{5} + 998 (9.8) (40)) (20 \times 10^{- 6})}{8.314 \times 277} n = 4.277 \times 10^{- 3} moles$

Consider the bubble reaches the surface the gas, the law becomes:

$p_{0} V_{0} = n R T_{0} V_{0} = \frac{n R T_{0}}{P_{0}}$

Substitute the values and solve as:

$V_{0} = \frac{4.277 \times 10^{- 3} (8.314) (293)}{1.013 \times 10^{5}} V_{0} = 1.02 \times 10^{- 4} m^{3} V_{0} = 1.02 \times 10^{2} {cm}^{3}$

Hence, the volume of the bubble as it reaches the surface is $1.02 \times 10^{2} {cm}^{3}$ .

Therefore, the volume of the bubble as it reaches the surface can be found using the gas law.

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