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 86

Chapter 17, Problem 86

A brass lid screws tightly onto a glass jar at 15°C. To help open the jar, it can be placed into a bath of hot water. After this treatment, the temperatures of the lid and the jar are both 55°C. The inside diameter of the lid is 7.0 cm. Find the size of the gap (difference in radius) that develops by this procedure.

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Determine the formula for linear thermal expansion: ΔL = α × L₀ × ΔT, where ΔL is the change in length, α is the coefficient of linear expansion, L₀ is the original length, and ΔT is the change in temperature.

Identify the coefficients of linear expansion for brass and glass. For brass, α_brass ≈ 19 × 10⁻⁶ /°C, and for glass, α_glass ≈ 9 × 10⁻⁶ /°C.

Calculate the change in radius for the brass lid using the formula ΔL = α × L₀ × ΔT. Here, L₀ is the original radius of the lid (7.0 cm / 2 = 3.5 cm), and ΔT = 55°C - 15°C = 40°C.

Similarly, calculate the change in radius for the glass jar using the same formula, with the same ΔT and the same initial radius (3.5 cm). Use the coefficient of linear expansion for glass.

Find the size of the gap by subtracting the change in radius of the glass jar from the change in radius of the brass lid: Gap = ΔR_brass - ΔR_glass.

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Key Concepts

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

Thermal Expansion

Thermal expansion refers to the increase in size of materials as they are heated. Different materials expand at different rates, characterized by their coefficient of linear expansion. In this scenario, the brass lid will expand more than the glass jar when both are heated, which is crucial for understanding how the gap develops.

Coefficient of Linear Expansion

The coefficient of linear expansion is a material-specific value that quantifies how much a material expands per degree of temperature increase. For brass, this coefficient is higher than that of glass, meaning that for the same temperature change, brass will expand more than glass, leading to a change in the fit between the lid and the jar.

Gap Calculation

To find the size of the gap that develops between the lid and the jar, one must calculate the change in radius for both the lid and the jar due to thermal expansion. This involves using the formula for linear expansion, which is ΔL = αL0ΔT, where ΔL is the change in length (or radius), α is the coefficient of linear expansion, L0 is the original length (or radius), and ΔT is the change in temperature.

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