Unit 15: Temperature, Heat, and Expansion

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Additional Reading

Heat, Temperature, and Internal Energy

Even though heat and temperature are closely related, they are two distinct physical quantities. Heat flow describes a change in the internal energy content of a substance, whereas temperature is a measure of the average kinetic energy of the atoms (or molecules) in the substance. Quantitatively, heat is measured in joules (J) or calories (cal), whereas temperature is measured in degrees Celsius (°C), Kelvin (K), or degrees Fahrenheit (°F).

Internal energy is the total (kinetic plus potential) microscopic energy in an object. Heat does not necessarily flow from an object of higher internal energy content to an object of lower internal energy content. However, heat flows spontaneously from an object of higher temperature to an object of lower temperature until the temperature of both objects become equal.

Consider two blocks of steel at room temperature, say 20°C, where one block has a mass of 1 kg and the other is 2 kg. Suppose you supply enough heat (110 cal) to raise the temperature of the larger block by 0.5°C so its temperature becomes 20.5°C. If you supply the same amount of heat to the smaller block, it will reach a final temperature of 21°C. Note that although both blocks received equal amounts of heat, the temperature increase of the smaller block is double that of the larger block.

Specific Heat Capacity

Consider a block of steel and a block of sliver, each initially at room temperature (20°C) and each having a mass of 1 kg. If you supply 110 cal to the steel block, its temperature increases to 21°C, as mentioned earlier. However, the same amount of heat will raise the temperature of the silver block to 22°C. This is because steel has a greater specific heat capacity than silver.

The specific heat capacity of a substance is the quantity of heat required to raise the temperature of a unit mass of the substance by 1°C. For example, you need 1 cal (4.186 J) to increase the temperature of 1 g of water from 14.5°C to 15.5°C. Therefore, the specific heat capacity of water is equal to $1\,\text{cal}/{\text{g}\cdot\text{C}^\circ}$.

Temperature Scales

The first modern thermometers were designed by Daniel Gabriel Fahrenheit (1686–1736), a Dutch manufacturer of meteorological instruments, who experimented with using alcohol and mercury in thermometers. On his scale, the temperature of melting ice is 32°F and that of boiling water is 212°F. This scale, known as the Fahrenheit scale, is still used in some countries, including the United States.

The 180-degree difference between the freezing and boiling point of water was not aesthetically pleasing to many scientists, and they sought to establish a scale based on a difference of 100 degrees between the two fixed points. Anders Celsius (1701–1744), a Swedish astronomer, was the first to use such a scale, with the freezing temperature of water set at 0°C and the boiling temperature set at 100°C.

Scientists, however, prefer to use the absolute scale of temperature known as the Kelvin scale. Absolute zero on the Kelvin scale ($0\,\text{K} = -273^\circ\text{C}$) corresponds to the temperature at which all atomic and molecular motion comes to a halt.

Thermal Expansion

One of the earliest discoveries in the study of the thermal properties of matter was that most substances expand with the application of heat. This phenomenon is called thermal expansion. For example, when heated, a long, thin wire will have a noticeable increase in length that is proportional to its original length and to the increase in temperature. A solid object also expands in all three dimensions (i.e., swells in volume) as it becomes warmer.

You may have noticed that railroad tracks are not continuous: there are gaps between successive pieces of track to accommodate expected thermal expansion. Without the gaps, on a hot day, the tracks would experience a huge thermal stress and would buckle. Engineers must make allowances for thermal stresses when they are designing buildings, bridges, and, of course, railroad tracks.

The Ideal Gas Law

Suppose you put a certain amount of nitrogen gas in a cylinder with an adjustable piston. The piston, which is assumed to be airtight, can be moved to change the volume inside the cylinder. The gas exerts a pressure ($P$) on the walls of the cylinder and on the inside surface of the piston.

Suppose you push the piston so the volume of the container—and, therefore, the volume of the gas ($V$)—is now half of the original volume. Note that the cylinder still contains the same number of nitrogen molecules ($N$). If you make sure the temperature of the gas does not change during the procedure, you will find that the pressure exerted by the gas increases to twice the original pressure. This property—that pressure exerted by a gas is inversely proportional to its volume when the temperature is constant—is known as Boyle’s law.

Now suppose you increase of the temperature of the gas while holding the piston to maintain a constant volume. You will notice that the pressure exerted by the gas increases linearly with the increasing temperature ($T$), measured in kelvins. If you adjust the piston in a controlled way to ensure that the pressure remains constant, you will find that the volume increases linearly with temperature, which is known as Charles’s law.

An important relation that combines all the variables mentioned above is referred to as the ideal gas law, and it can be expressed in the following form: \begin{equation} P V = N k\, T \label{unit15_PVNKT} \end{equation} In this equation, $k$ is called the Boltzmann constant.

Questions

The following questions are selected from the end of Chapter 15 of the eText. It is important to your learning that you try to answer each question independently before you read through the answer and explanation given.

For questions that ask you to explain, defend, or discuss your answer, the response revealed by the Answer button would earn you only partial marks on a quiz or exam in this course. Use the Answer to help you formulate a complete answer before you select the Explanation button to check your work.

Note that some questions have been adapted to suit the format of this course.

Exercises

Spend some time completing the following exercises to test your understanding of the main concepts in Chapter 15 and increase your efficiency in answering exam questions.