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 24a

Chapter 17, Problem 24a

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.

Verified step by step guidance

Start by recalling the formula for the surface area of a sphere: \( A = 4\pi r^2 \), where \( r \) is the radius of the sphere.

The coefficient of linear expansion \( \alpha \) relates the change in length (or radius in this case) to the change in temperature \( \Delta T \) by the formula: \( \Delta r = \alpha r \Delta T \).

Substitute the new radius \( r + \Delta r \) into the surface area formula to find the new surface area: \( A' = 4\pi (r + \Delta r)^2 \).

Expand \( (r + \Delta r)^2 \) using the binomial theorem: \( A' = 4\pi (r^2 + 2r\Delta r + (\Delta r)^2) \).

Subtract the original surface area \( A = 4\pi r^2 \) from \( A' \) to find the change in surface area \( \Delta A \), and simplify using \( \Delta r = \alpha r \Delta T \). The final formula for \( \Delta A \) will be expressed in terms of \( \alpha \), \( r \), and \( \Delta T \).

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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

The coefficient of linear expansion (α) quantifies how much a material expands per unit length for each degree of temperature increase. It is a material-specific property that indicates how sensitive a material is to temperature changes. For solids, this expansion is typically uniform in all directions, which is crucial for understanding how dimensions change with temperature.

Surface Area of a Sphere

The surface area (A) of a sphere is calculated using the formula A = 4πr², where r is the radius. When the radius changes due to thermal expansion, the surface area will also change. Understanding how the radius varies with temperature is essential for determining the change in surface area as the sphere is heated or cooled.

Thermal Expansion of Solids

Thermal expansion refers to the increase in size of a solid material when its temperature rises. For a uniform solid sphere, the change in radius (Δr) can be expressed as Δr = αrΔT, where ΔT is the change in temperature. This relationship allows us to derive how the surface area changes as the temperature varies, linking thermal expansion to geometric properties.

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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

An aluminum bar has the desired length when at 12°C. How much stress is required to keep it at this length if the temperature increases to 38°C? [See Table 12–1.]

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Textbook Question

Water’s coefficient of volume expansion in the temperature range from 0°C to about 20°C is given approximately by β = α + bT + cT² , with α = - 6.43 x 10⁻⁵ (C°)⁻¹ , b = 1.70 x 10⁻⁵ (C°)⁻² , and c = -2.02 x 10⁻⁷ ((C°)⁻³. Using the formula for density from Problem 22, show that water has its greatest density at approximately 4.0°C.

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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

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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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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