Ch 18: A Macroscopic Description of Matter

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 18: A Macroscopic Description of Matter

Problem 59

Chapter 18, Problem 59

A diver 50 m deep in 10°C fresh water exhales a 1.0-cm-diameter bubble. What is the bubble's diameter just as it reaches the surface of the lake, where the water temperature is 20°C?

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Identify the key principles involved: This problem involves the Ideal Gas Law and the relationship between pressure, volume, and temperature. The bubble's volume will change as it rises due to changes in pressure and temperature.

Write the Ideal Gas Law: $ PV = nRT $. Since the number of moles $ n $ and the gas constant $ R $ are constant, the relationship between pressure, volume, and temperature can be expressed as $ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} $.

Determine the initial and final conditions: At 50 m depth, the pressure $ P_1 $ is the sum of atmospheric pressure ($ P_{atm} = 1.0 \; \text{atm} $) and the water pressure ($ \rho g h $, where $ \rho $ is the density of water, $ g $ is the acceleration due to gravity, and $ h $ is the depth). The temperature $ T_1 $ is 10°C (convert to Kelvin: $ T_1 = 10 + 273.15 $). At the surface, $ P_2 = P_{atm} $ and $ T_2 = 20 + 273.15 $.

Relate the bubble's volume to its diameter: The volume of a sphere is $ V = \frac{4}{3} \pi r^3 $, where $ r $ is the radius. Since the diameter $ d $ is twice the radius, $ V \propto d^3 $. Use this relationship to express the change in diameter: $ \frac{d_2}{d_1} = \left( \frac{V_2}{V_1} \right)^{1/3} $.

Substitute the known values into the equations: Use $ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} $ to find $ \frac{V_2}{V_1} $, then calculate $ \frac{d_2}{d_1} $. Finally, multiply $ d_1 = 1.0 \; \text{cm} $ by $ \frac{d_2}{d_1} $ to find the final diameter of the bubble.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Boyle's Law

Boyle's Law states that the pressure of a gas is inversely proportional to its volume when temperature is held constant. In the context of the diver's bubble, as the bubble rises from the high pressure at 50 m depth to the lower pressure at the surface, its volume will increase, causing its diameter to expand.

Charles's Law

Charles's Law describes how the volume of a gas is directly proportional to its temperature when pressure is constant. As the bubble rises and the temperature increases from 10°C to 20°C, this law indicates that the bubble will also expand due to the increase in temperature, further affecting its final diameter.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. At a depth of 50 m, the diver experiences significant hydrostatic pressure, which compresses the bubble. Understanding how this pressure changes as the bubble ascends is crucial for calculating the final diameter at the surface.

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