12.11

Concepts & Calculations

This section contains examples that discuss one or more conceptual questions, followed by a related quantitative problem. Example 18 provides insight on the variables involved when the length and volume of an object change due to a temperature change. Example 19 discusses how different factors affect the temperature change of an object to which heat is being added.

| Concepts & Calculations Example 18 | | | Linear and Volume Thermal Expansion

Figure 12-40 shows three rectangular blocks made from the same material. The initial dimensions of each are expressed as multiples of D, where D = 2.00 cm. They are heated and their temperatures increase by 35.0 C°. The coefficients of linear and volume expansion are and , respectively. Determine the change in their (a) vertical heights and (b) volumes.

Figure 12-40 The temperatures of the three blocks are raised by the same amount. Which one(s) experience the greatest change in height and which the greatest change in volume?

Concept Questions and Answers Does the change in the vertical height of a block depend only on its height, or does it also depend on its width and depth? Without doing any calculations, rank the blocks according to their change in height, largest first.

Answer According to Equation 12.2, , the change in the height of an object depends on its original height L0, the change in temperature , and the coefficient of linear expansion, . It does not depend on the width or depth of the object. Since blocks B and C have twice the height of A, their heights will increase by twice as much as that of A. The heights of B and C, however, will increase by the same amount, even though C is twice as wide.

Does the change in the volume of a block depend only on its height, or does it also depend on its width and depth? Without doing any calculations, rank the blocks according to their greatest change in volume, largest first.

Answer According to Equation 12.3, , the change in the volume of an object depends on its original volume V0 , the change in temperature , and the coefficient of volume expansion, . Thus, the change in volume depends on height, width, and depth, because the original volume is the product of these three dimensions. The initial volumes of A, B, and C, are, respectively, , , and . Blocks A and C have equal volumes, which are greater than that of B. Thus, we expect A and C to exhibit the greatest increase in volume, while B exhibits the smallest increase.

Solution

a. | The change in the height of each block is given by Equation 12.2 as

As expected, the heights of B and C increase more than that of A.
b. | The change in the volume of each block is given by Equation 12.3 as

As discussed earlier, the greatest change in volume occurs with A and C followed by B.

| Concepts & Calculations Example 19 | | | Heat and Temperature Changes

Objects A and B in Figure 12-41 are made from copper, but the mass of B is three times that of A. Object C is made from glass and has the same mass as B. The same amount of heat Q is supplied to each one: . Determine the rise in temperature for each.

Figure 12-41 Which block has the greatest change in temperature when the wame heat is supplied to each?

Concept Questions and Answers Which object, A or B, experiences the greater rise in temperature?

Answer Consider an extreme example. Suppose a cup and a swimming pool are filled with water. For the same heat input, would you intuitively expect the temperature of the cup to rise more than that of the pool? Yes, because the cup has less mass. Another way to arrive at this conclusion is to solve Equation 12.4 for the change in temperature: . Since the heat Q and the specific heat capacity c are the same for A and B, is inversely proportional to the mass m. So smaller masses give rise to larger temperature changes. Therefore, A has a greater temperature change than B.

Which object, B or C, experiences the greater rise in temperature?

Answer Objects B and C are made from different materials. To see how the rise in temperature depends on the type of material, let’s again use Equation 12.4: . Since the heat Q and the mass m are the same for B and C, is inversely proportional to the specific heat capacity c. So smaller specific heat capacities give rise to larger temperature changes. Table 12-2 indicates that the specific heat capacities of copper and glass are 387 and 840 J/(kg·C°), respectively. Since B is made from copper, which has the smaller specific heat capacity, it has the greater temperature change.

Solution According to Equation 12.4, the temperature change for each object is

As anticipated, is greater than , and is greater than .

At the end of the problem set for this chapter, you will find homework problems that contain both conceptual and quantitative parts. These problems are grouped under the heading Concepts & Calculations.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.