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Chapter 12: Problem 97

When \(4200 \mathrm{J}\) of heat are added to a \(0.15-\mathrm{m}\) -long silver bar, its length increases by \(4.3 \times 10^{-3} \mathrm{m} .\) What is the mass of the bar?

Short Answer

Expert verified
The mass of the silver bar is approximately 0.118 kg.

Step by step solution

01

Understanding the Problem

We need to find the mass of a silver bar given the heat added, initial length, and change in length. The initial given are: heat added \( Q = 4200 \mathrm{J} \), initial length \( L = 0.15 \mathrm{m} \), and the change in length \( \Delta L = 4.3 \times 10^{-3} \mathrm{m} \).
02

Recall the Formula for Linear Expansion

The formula for linear expansion is \( \Delta L = \alpha L \Delta T \). Here, \( \Delta L \) is the change in length, \( \alpha \) is the coefficient of linear expansion for silver, and \( \Delta T \) is the change in temperature.
03

Find the Coefficient of Linear Expansion

The typical coefficient of linear expansion for silver is \( \alpha = 19 \times 10^{-6} \mathrm{掳C}^{-1} \). We will use this to find \( \Delta T \).
04

Calculate the Change in Temperature

Rearrange the linear expansion formula to solve for \( \Delta T \): \( \Delta T = \frac{\Delta L}{\alpha L} = \frac{4.3 \times 10^{-3}}{19 \times 10^{-6} \times 0.15} \approx 150.88 \mathrm{掳C} \).
05

Relate Heat, Temperature Change, and Mass

Use the formula \( Q = mc\Delta T \) where \( c \) is the specific heat capacity of silver \( (c = 235 \mathrm{J/kg掳C}) \). Here, \( Q \) is the heat added, \( m \) is the mass, and \( \Delta T \) is the temperature change.
06

Solve for the Mass of the Bar

Rearrange the formula \( Q = mc\Delta T \) to solve for \( m \): \( m = \frac{Q}{c \Delta T} = \frac{4200}{235 \times 150.88} \approx 0.1181 \mathrm{kg} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coefficient of Linear Expansion
The coefficient of linear expansion, denoted by \( \alpha \), is a measure of how much a material expands per degree change in temperature. It is specific to each material and indicates the fractional change in length per degree Celsius. For example, silver has a coefficient of linear expansion of \( 19 \times 10^{-6} \, \text{掳C}^{-1} \). This means that for every degree Celsius increase in temperature, a silver object will expand by \( 19 \times 10^{-6} \) times its original length. It is crucial in applications where temperature changes are expected, as it informs engineers how the size will change, ensuring structures can accommodate these expansions and contractions.
Specific Heat Capacity
Specific heat capacity, denoted by \( c \), is the amount of heat required to change the temperature of one kilogram of a substance by one degree Celsius. It tells us how much energy a material can store and how it reacts to heat changes.
  • Silver's specific heat capacity is \( 235 \, \text{J/kg掳C} \), meaning it requires \( 235 \text{ J} \) to raise the temperature of \( 1 \text{ kg} \) of silver by \( 1 \text{掳C} \).
  • Knowing the specific heat capacity is essential for calculating temperature changes and energy requirements in a system.
This concept helps in determining the energy needed for temperature modifications without changing the state of the material.
Thermal Expansion
Thermal expansion refers to the tendency of materials to change in volume in response to a change in temperature. It encompasses both linear and volumetric expansion, though linear expansion is most often discussed regarding solids, such as metals.
  • In the case of the silver bar, the linear expansion is observed as the bar increases in length when heated.
  • The degree of expansion depends on the material's coefficient of linear expansion, the initial length, and the temperature change.
Thermal expansion is a key consideration in many engineering fields, affecting how components fit and move under temperature changes.
Heat and Temperature Change
The relationship between heat and temperature change can be quantified using the formula \( Q = mc\Delta T \), where \( Q \) is the heat added, \( m \) is the mass of the material, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature. This formula allows us to calculate one of these variables if the others are known.
  • In the context of the exercise, \( 4200 \text{ J} \) of heat causes the temperature of the silver bar to change by \( 150.88 \text{掳C} \).
  • This temperature change, in turn, causes the bar to expand due to its coefficient of linear expansion.
Understanding how heat transfer affects temperature and material properties is vital in designing systems that operate efficiently under various thermal conditions.

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Most popular questions from this chapter

A 42 -kg block of ice at \(0^{\circ} \mathrm{C}\) is sliding on a horizontal surface. The initial speed of the ice is \(7.3 \mathrm{m} / \mathrm{s}\) and the final speed is \(3.5 \mathrm{m} / \mathrm{s} .\) Assume that the part of the block that melts has a very small mass and that all the heat generated by kinetic friction goes into the block of ice. Determine the mass of ice that melts into water at \(0^{\circ} \mathrm{C}\).

A steel bicycle wheel (without the rubber tire) is rotating freely with an angular speed of \(18.00 \mathrm{rad} / \mathrm{s}\). The temperature of the wheel changes from \(-100.0 \mathrm{to}+300.0^{\circ} \mathrm{C} .\) No net external torque acts on the wheel, and the mass of the spokes is negligible. (a) Does the angular speed increase or decrease as the wheel heats up? Why? (b) What is the angular speed at the higher temperature?

You are sick, and your temperature is 312.0 kelvins. Convert this temperature to the Fahrenheit scale.

A person eats a container of strawberry yogurt. The Nutritional Facts label states that it contains 240 Calories ( 1 Calorie \(=4186\) J). What mass of perspiration would one have to lose to get rid of this energy? At body temperature, the latent heat of vaporization of water is \(2.42 \times 10^{6} \mathrm{J} / \mathrm{kg} .\)

Occasionally, huge icebergs are found floating on the ocean's currents. Suppose one such iceberg is \(120 \mathrm{km}\) long, \(35 \mathrm{km}\) wide, and \(230 \mathrm{m}\) thick. (a) How much heat would be required to melt this iceberg (assumed to be at \(0^{\circ} \mathrm{C}\) ) into liquid water at \(0^{\circ} \mathrm{C}\) ? The density of ice is \(917 \mathrm{kg} / \mathrm{m}^{3}\). (b) The annual energy consumption by the United States is about \(1.1 \times 10^{20}\) J. If this energy were delivered to the iceberg every year, how many years would it take before the ice melted?
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