Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law

Problem 55

Chapter 17, Problem 55

An air bubble at the bottom of a lake 32.0 m deep has a volume of 1.00 cm³ . If the temperature at the bottom is 5.5°C and at the top 18.5°C, what is the radius of the bubble just before it reaches the surface?

Verified step by step guidance

Start by identifying the relevant physical principles. This problem involves the Ideal Gas Law, which relates pressure, volume, and temperature:

PV = nRT

. Additionally, the relationship between the volume of a sphere and its radius is given by

V = \(\frac{4}{3}\)\(\pi\) r^3

Determine the pressure at the bottom of the lake. The pressure is the sum of atmospheric pressure and the pressure due to the water column. Use the formula

P_{bottom} = P_{atm} + \(\rho\) g h

, where

\(\rho\)

is the density of water (approximately 1000 kg/m³),

g

is the acceleration due to gravity (9.8 m/s²), and

h

is the depth of the lake (32.0 m).

Determine the pressure at the surface of the lake. At the surface, the pressure is equal to atmospheric pressure,

P_{atm}

, which is approximately 101,325 Pa (standard atmospheric pressure).

Apply the Ideal Gas Law to relate the initial and final states of the bubble. Since the number of moles of gas and the gas constant remain constant, the relationship simplifies to

\(\frac{P_{bottom}\) V_{bottom}}{T_{bottom}} = \(\frac{P_{surface}\) V_{surface}}{T_{surface}}

. Rearrange this equation to solve for the final volume of the bubble,

V_{surface} = V_{bottom} \(\cdot\) \(\frac{P_{bottom}\) T_{surface}}{P_{surface} T_{bottom}}

. Note that temperatures must be converted to Kelvin by adding 273.15 to the Celsius values.

Once the final volume

V_{surface}

is determined, use the formula for the volume of a sphere,

V = \(\frac{4}{3}\)\(\pi\) r^3

, to solve for the radius of the bubble at the surface. Rearrange to find

r = \(\sqrt\)[3]{\(\frac{3V}{4\pi}\)}

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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 and volume of a gas are inversely related when temperature is held constant. This means that as the pressure on a gas increases, its volume decreases, and vice versa. In the context of the air bubble rising in the lake, the pressure at the bottom is greater than at the surface, causing the bubble to expand as it ascends.

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 temperature of the air bubble increases from the bottom to the top of the lake, its volume will also increase. This relationship is crucial for determining the final volume of the bubble as it rises and warms up.

Ideal Gas Law

The Ideal Gas Law combines Boyle's Law and Charles's Law into a single equation: PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the ideal gas constant, and T is temperature in Kelvin. This law allows us to calculate the changes in volume and pressure of the air bubble as it rises through the water, taking into account both the depth and temperature variations.

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Related Practice

Textbook Question

A helium-filled balloon escapes a child’s hand at sea level and 20.0°C. When it reaches an altitude of 3600 m, where the temperature is 5.0°C and the pressure only 0.64 atm, how will its volume compare to that at sea level?

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A precise steel tape measure has been calibrated at 14°C. At 37°C, what will be the percentage error?

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A helium balloon rises because of a buoyant force. By what percentage does the buoyant force on a helium balloon change if the temperature of the helium is increased from 15°C to 25°C while the temperature of the surrounding air is unchanged? Assume that the pressure of the helium remains constant.

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Textbook Question

How many moles of water are there in 1.00 L at STP? How many molecules?

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Textbook Question

You buy an “airtight” potato chip bag packaged at sea level, and take the chips on an airplane flight. When you take the potato chip bag out of your “carry-on” bag, you notice it has noticeably “puffed up.” Airplane cabins are typically pressurized at 0.75 atm, and assuming the temperature inside an airplane is about the same as inside a potato chip processing plant, by what percentage has the bag “puffed up” in comparison to when it was packaged?

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Textbook Question

Use the ideal gas law to show that, for an ideal gas at constant pressure, the coefficient of volume expansion is equal to β = 1/ T, where T is the kelvin temperature. Compare to Table 17–1 for gases at T = 293 K.

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