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

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

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

Problem 19

Chapter 17, Problem 19

It is observed that 55.50 mL of water at 20°C completely fills a container to the brim. When the container and the water are heated to 60°C, 0.35 g of water is lost.
(a) What is the coefficient of volume expansion of the container?
(b) What is the most likely material of the container? Density of water at 60°C is 0.98324 g/mL.

Verified step by step guidance

Determine the initial volume of water, which is given as 55.50 mL at 20°C. This is the volume of the container at 20°C since the container is completely filled to the brim.

Calculate the volume of water remaining in the container at 60°C. Use the mass of water lost (0.35 g) and the density of water at 60°C (0.98324 g/mL) to find the volume of water lost. Subtract this volume from the initial volume to find the remaining volume of water.

Determine the volume expansion of the water. Use the formula for volume expansion:

Δ V = V ₀ β Δ T

, where

Δ V

is the change in volume,

V ₀

is the initial volume,

β

is the coefficient of volume expansion for water, and

Δ T

is the temperature change. Use the known coefficient of volume expansion for water to calculate the expanded volume of water at 60°C.

Relate the expansion of the container to the expansion of the water. Since the container is filled to the brim at 20°C and some water is lost at 60°C, the container's volume expansion must be less than the water's volume expansion. Use the formula for volume expansion of the container:

Δ V = V ₀ α Δ T

, where

α

is the coefficient of volume expansion of the container. Solve for

α

using the known values.

Compare the calculated coefficient of volume expansion of the container to known values for various materials to identify the most likely material of the container. Common materials and their coefficients of volume expansion include glass, aluminum, and steel.

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Key Concepts

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

Coefficient of Volume Expansion

The coefficient of volume expansion is a measure of how much a material's volume changes with temperature. It is defined as the fractional change in volume per degree change in temperature. For solids and liquids, this coefficient helps predict how much a substance will expand or contract when subjected to temperature variations, which is crucial for understanding the behavior of materials in thermal processes.

Density and Mass Loss

Density is defined as mass per unit volume and is a critical property in understanding how substances behave under different conditions. In this scenario, the loss of mass (0.35 g of water) when the temperature increases indicates that the water expands and some of it spills out. The density of water at 60°C (0.98324 g/mL) is necessary to relate the mass loss to the volume change, allowing for calculations related to the container's expansion.

Thermal Expansion of Materials

Thermal expansion refers to the tendency of materials to change in shape, area, and volume in response to changes in temperature. Different materials expand at different rates, which is characterized by their specific coefficients of thermal expansion. Understanding this concept is essential for determining the material of the container, as it helps explain how the container's volume changes in relation to the water's expansion and the observed mass loss.

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Related Practice

Textbook Question

A brass plug is to be placed in a ring made of iron. At 15°C, the diameter of the plug is 8.756 cm and that of the inside of the ring is 8.742 cm. They must both be brought to what common temperature in order to fit?

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Textbook Question

A glass is filled to the brim with 450.0 mL of water, all at 100.0°C. If the temperature of glass and water is decreased to 20.0°C, how much water could be added to the glass?

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Textbook Question

A uniform rectangular plate of length ℓ and width ω has a coefficient of linear expansion α. Show that, if we neglect very small quantities, the change in area of the plate due to a temperature change ∆T is ∆A = 2αℓω ∆T. See Fig. 17–21.

Textbook Question

An aluminum sphere is 8.75 cm in diameter. What will be its % change in volume if it is heated from 30°C to 140°C?

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Textbook Question

If a fluid is contained in a long narrow vessel so it can expand in essentially one direction only, show that the effective coefficient of linear expansion α is approximately equal to the coefficient of volume expansion β.

Textbook Question

Wine bottles are never completely filled: a small volume of air is left in the glass bottle’s cylindrically shaped neck (inner diameter d = 18.5 mm) to allow for wine’s fairly large coefficient of thermal expansion. The distance H between the surface of the liquid contents and the bottom of the cork is called the “headspace height” (Fig. 17–22), and is typically H = 1.5 cm for a 750-mL bottle filled at 20°C. Due to its alcoholic content, wine’s coefficient of volume expansion is about double that of water; in comparison, the thermal expansion of glass can be neglected. Estimate H if the bottle is kept at 10°C.

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