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 21

Chapter 17, Problem 21

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.

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Start by recalling the definitions of the coefficients of linear expansion (α) and volume expansion (β). The coefficient of linear expansion (α) describes how the length of a material changes with temperature, while the coefficient of volume expansion (β) describes how the volume of a material changes with temperature.

For a long narrow vessel, assume the fluid expands primarily in one direction (let's call it the x-direction). The volume of the fluid can be expressed as \( V = A \cdot L \), where \( A \) is the cross-sectional area and \( L \) is the length of the vessel.

When the temperature changes, the volume expansion can be expressed as \( \Delta V = \beta V \Delta T \), where \( \beta \) is the coefficient of volume expansion and \( \Delta T \) is the temperature change. Substituting \( V = A \cdot L \), we get \( \Delta V = \beta (A \cdot L) \Delta T \).

Now consider the linear expansion of the vessel. The length \( L \) changes according to \( \Delta L = \alpha L \Delta T \), where \( \alpha \) is the coefficient of linear expansion. Since the fluid expands primarily in one direction, the change in volume is approximately due to the change in length, so \( \Delta V \approx A \cdot \Delta L \).

Substitute \( \Delta L = \alpha L \Delta T \) into \( \Delta V \approx A \cdot \Delta L \), giving \( \Delta V \approx A \cdot (\alpha L \Delta T) \). Comparing this with \( \Delta V = \beta (A \cdot L) \Delta T \), we see that \( \alpha \approx \beta \), as the cross-sectional area \( A \) and length \( L \) cancel out.

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Key Concepts

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

Coefficient of Linear Expansion (α)

The coefficient of linear expansion (α) quantifies how much a material expands per unit length for each degree of temperature increase. It is defined as the change in length divided by the original length and the change in temperature. This concept is crucial for understanding how solids respond to temperature changes, particularly in one-dimensional expansion scenarios.

Coefficient of Volume Expansion (β)

The coefficient of volume expansion (β) measures how much a material's volume changes per unit volume for each degree of temperature increase. It is defined as the change in volume divided by the original volume and the change in temperature. This concept is essential for analyzing how fluids behave under temperature variations, especially in confined spaces.

Relationship Between Linear and Volume Expansion

In three-dimensional objects, the relationship between linear expansion and volume expansion is given by the equation β = 3α for isotropic materials. This means that if a material expands uniformly in all directions, its volume change is three times the linear change. In the context of a fluid in a narrow vessel, the effective linear expansion can be approximated to equal the volume expansion due to the constraints of the vessel allowing expansion primarily in one direction.

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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

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.

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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

Determine a formula for the change in surface area of a uniform solid sphere of radius r if its coefficient of linear expansion is α (assumed constant) and its temperature is changed by ∆T.

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Textbook Question

The pendulum in a grandfather clock is made of brass and keeps perfect time at 17°C. How much time is gained or lost in a year if the clock is kept at 26°C? (Assume the frequency dependence on length for a simple pendulum applies; see Chapter 14.)

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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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