Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

When the temperature of a metal cylinder is raised from $0 . 0^{o} C to 100^{o} C$ , its length increases by 0.23%
(a) Find the percent change in density.
(b) What is the metal? Use Table.

The percent change in density is -0.69%
The coefficient of expansion is $23 \times 10^{- 6} / {}^{o}C$ . The given metal is Aluminum.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: The given data

Temperature of the cylinder changes from 0^oC to 100^oC and its length increases by 0.23%.

Step 2: Understanding the concept of thermal expansion

When an object's temperature changes, it expands and grows larger, a process known as thermal expansion. By using the formula for density, we can find the percent change in density. Also, we can use the concept of linear expansion to find the linear expansion coefficient.

Formula:

The density of a body, $ρ = \frac{m}{V}$ …(i)

The linear expansion of a body, data-custom-editor="chemistry" $∆ L = Lα ∆ T$ …(ii)

Where data-custom-editor="chemistry" $α$ is the coefficient of linear expansion of body, L is length, data-custom-editor="chemistry" $∆ T$ is change in temperature, m is mass and V is volume

Step 3: (a) Calculation of percent change in density

By differentiating equation (i), we can get

$\begin{array}{rcl} d ρ & = & - (\frac{m}{V^{2}}) d V \\ ∆ ρ & = & - (\frac{m}{V^{2}}) ∆ V \\ = & - \frac{m ∆ V}{V^{2}} \\ = & - \frac{ρ ∆ V}{V^{2}} . . . . . . . . . . . . . . . . (a) (from equation (i)) \end{array}$

But we know the relation of expansion of volume and length, that is:

$\frac{∆ V}{V} = 3 \frac{∆ L}{L}$

Therefore, substituting the above value in equation (a), we get

$∆ ρ = - \frac{3 ρ ∆ L}{L} \frac{∆ ρ}{ρ} = - \frac{3 ∆ L}{L}$

Where $\frac{∆ L}{L}$ is the percent change in length which is 0.23%

So, the percent change in density is given as:

$\begin{array}{rcl} \frac{∆ ρ}{ρ} & = & - 3 \times 0.23 \\ = & - 0.69 % \end{array}$

Hence, the value of percent change in density is -0.69%

Step 4: (b) identifying the metal of thermal expansion

The coefficient of thermal expansion can tell us which metal it is. So, we have to find the coefficient of thermal expansion from equation (ii) as follows:

$\begin{array}{rcl} α & = & \frac{∆ L}{L ∆ T} \\ = & \frac{0.23 \times 10^{- 2}}{{(100 - 0)}^{o} C} (∵ \frac{∆ L}{L} = 0.23 % = 0.23 \times 10^{- 2}) \\ = & 23 \times 10^{- 6} / {}^{o}C \end{array}$

From this value of the coefficient of thermal expansion, we can conclude that the given metal is aluminum.

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Fundamentals Of Physics

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Short Answer

When the temperature of a metal cylinder is raised from $0 . 0^{o} C to 100^{o} C$ , its length increases by 0.23%
(a) Find the percent change in density.
(b) What is the metal? Use Table.

Step by Step Solution

Step 1: The given data

Step 2: Understanding the concept of thermal expansion

Step 3: (a) Calculation of percent change in density

Step 4: (b) identifying the metal of thermal expansion

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Short Answer

When the temperature of a metal cylinder is raised from 0.0oC to 100oC, its length increases by 0.23% (a) Find the percent change in density. (b) What is the metal? Use Table.

Step by Step Solution

Step 1: The given data

Step 2: Understanding the concept of thermal expansion

Step 3: (a) Calculation of percent change in density

Step 4: (b) identifying the metal of thermal expansion

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When the temperature of a metal cylinder is raised from $0 . 0^{o} C to 100^{o} C$ , its length increases by 0.23%
(a) Find the percent change in density.
(b) What is the metal? Use Table.