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Temperature, Kinetic Theory, and the Gas Laws

93 Temperature

Learning Objectives

  • Define temperature.
  • Convert temperatures between the Celsius, Fahrenheit, and Kelvin scales.
  • Explain the concept of thermal equilibrium.
  • State the zeroth law of thermodynamics.

The idea of temperature originates from our everyday experiences of hot and cold. However, our biological senses can be misleading. For instance, if one hand is placed in cold water and the other in hot water, then both are placed in lukewarm water, the lukewarm water may feel hot to one hand and cold to the other. To avoid such subjective impressions, scientists define temperature as the quantity measured by a thermometer. This definition is operational—it is based on the reproducible physical changes in materials in response to thermal energy.

Misconception Alert: Human Sensation vs. Scientific Measurement

Objects at the same temperature may feel different to the touch. For example, metal feels colder than wood on a winter morning even though both are at the same temperature. This occurs because metal conducts heat away from your skin more rapidly than wood does. Similarly, on a humid day, your body may feel warmer because sweat does not evaporate as efficiently. This demonstrates that our sense of temperature is also affected by evaporation and thermal conductivity.

Many physical properties change predictably with temperature and can therefore be used to build thermometers. Examples include the expansion of liquids (as in alcohol thermometers), the bending of bimetallic strips (Figure 93.1), changes in electrical resistance, and emission of infrared radiation (Figure 93.2 and 93.3).

Thermometers based on these principles give consistent and reproducible readings, making them suitable for medical, laboratory, and industrial applications.

This figure has two parts, each of which shows a blue metallic strip attached lengthwise to a yellow metallic strip, thus forming a bimetallic strip. In part a, the bimetallic strip is straight and oriented vertically, and its temperature is given as T sub 0. In part b, the bimetallic strip is curving rightward away from the vertical, and its temperature is given as T, which is greater than T sub 0.
Figure 93.1: The curvature of a bimetallic strip depends on temperature. (a) The strip is straight at the starting temperature, where its two components have the same length. (b) At a higher temperature, this strip bends to the right, because the metal on the left has expanded more than the metal on the right.
A flat plastic thermometer used to measure forehead temperature; the thermometer can measure between ninety-five and one-hundred four degrees Fahrenheit, or between thirty-five and forty degrees Celsius.
Figure 93.2: Each of the six squares on this plastic (liquid crystal) thermometer contains a film of a different heat-sensitive liquid crystal material. Below [latex]\text{95}\text{º}\text{F}[/latex], all six squares are black. When the plastic thermometer is exposed to temperature that increases to [latex]\text{95}\text{º}\text{F}[/latex], the first liquid crystal square changes color. When the temperature increases above [latex]\text{96}\text{.}8\text{º}\text{F}[/latex] the second liquid crystal square also changes color, and so forth. (credit: Arkrishna, Wikimedia Commons)
A man holds a device that looks like a gun or a check-out scanner up toward an air vent. A red light emanates from the device and shines on the vent.
Figure 93.3: Fireman Jason Ormand uses a pyrometer to check the temperature of an aircraft carrier’s ventilation system. Infrared radiation (whose emission varies with temperature) from the vent is measured and a temperature readout is quickly produced. Infrared measurements are also frequently used as a measure of body temperature. These modern thermometers, placed in the ear canal, are more accurate than alcohol thermometers placed under the tongue or in the armpit. (credit: Lamel J. Hinton/U.S. Navy)

Temperature Scales

Temperature can be measured using various scales. The most common are the Celsius, Fahrenheit, and Kelvin scales.

  • The Celsius scale sets the freezing point of water at [latex]0^\circ \text{C}[/latex] and boiling at [latex]100^\circ \text{C}[/latex].
  • The Fahrenheit scale sets freezing at [latex]32^\circ \text{F}[/latex] and boiling at [latex]212^\circ \text{F}[/latex].
  • The Kelvin scale begins at absolute zero (the lowest possible temperature) and is used extensively in scientific work. Water freezes at 273.15 K and boils at 373.15 K.

Because 180 Fahrenheit degrees span the same range as 100 Celsius degrees, the relationship between the two is:

[latex]T^\circ_F = \frac{9}{5}T^\circ_C + 32[/latex]
Three temperature scales—Fahrenheit, Celsius, and Kelvin—are oriented horizontally, one below the other, and aligned to show how they relate to each other. Absolute zero is at negative four hundred fifty nine point six seven degrees F, negative two hundred seventy three point one five degrees C, and 0 K. Water freezes at thirty two degrees F, 0 degrees C, and two hundred seventy three point one five K. Water boils at two hundred twelve degrees F, one hundred degrees C, and three hundred seventy three point one five K. A temperature difference of 9 degrees F is the same as a temperature difference of 5 degrees C and 5 K.
Figure 93.4: Relationships between the Fahrenheit, Celsius, and Kelvin temperature scales, rounded to the nearest degree. The relative sizes of the scales are also shown.

Conversions between all three temperature scales are summarized below:

Table 93.1: Temperature Conversions
To convert from . . . Use this equation . . . Also written as . . .
Celsius to Fahrenheit [latex]T\left(\text{º}\text{F}\right)=\frac{9}{5}T\left(\text{º}\text{C}\right)+\text{32}[/latex] [latex]{T}_{\text{º}\text{F}}=\frac{9}{5}{T}_{\text{º}\text{C}}+\text{32}[/latex]
Fahrenheit to Celsius [latex]T\left(\text{º}\text{C}\right)=\frac{5}{9}\left(T\left(\text{º}\text{F}\right)-\text{32}\right)[/latex] [latex]{T}_{\text{º}\text{C}}=\frac{5}{9}\left({T}_{\text{º}\text{F}}-\text{32}\right)[/latex]
Celsius to Kelvin [latex]T\left(\text{K}\right)=T\left(\text{º}\text{C}\right)+\text{273}\text{.}\text{15}[/latex] [latex]{T}_{\text{K}}={T}_{\text{º}\text{C}}+\text{273}\text{.}\text{15}[/latex]
Kelvin to Celsius [latex]T\left(\text{º}\text{C}\right)=T\left(\text{K}\right)-\text{273}\text{.}\text{15}[/latex] [latex]{T}_{\text{º}\text{C}}={T}_{\text{K}}-\text{273}\text{.}\text{15}[/latex]
Fahrenheit to Kelvin [latex]T\left(\text{K}\right)=\frac{5}{9}\left(T\left(\text{º}\text{F}\right)-\text{32}\right)+\text{273}\text{.}\text{15}[/latex] [latex]{T}_{\text{K}}=\frac{5}{9}\left({T}_{\text{º}\text{F}}-\text{32}\right)+\text{273}\text{.}\text{15}[/latex]
Kelvin to Fahrenheit [latex]T\left(\text{º}\text{F}\right)=\frac{9}{5}\left(T\left(\text{K}\right)-\text{273}\text{.}\text{15}\right)+\text{32}[/latex] [latex]{T}_{\text{º}\text{F}}=\frac{9}{5}\left({T}_{\text{K}}-\text{273}\text{.}\text{15}\right)+\text{32}[/latex]

Converting between Temperature Scales: Room Temperature

“Room temperature” is generally defined to be [latex]\text{25}\text{º}\text{C}[/latex]. (a) What is room temperature in [latex]\text{º}\text{F}[/latex]? (b) What is it in K?

Strategy

To answer these questions, all we need to do is choose the correct conversion equations and plug in the known values.

Solution for (a)

  1. Choose the right equation. To convert from [latex]\text{º}\text{C}[/latex] to [latex]\text{º}\text{F}[/latex], use the equation
    [latex]{T}_{\text{º}\text{F}}=\frac{9}{5}{T}_{\text{º}\text{C}}+\text{32}\text{.}[/latex]
  2. Plug the known value into the equation and solve:
    [latex]{T}_{\text{º}\text{F}}=\frac{9}{5}\text{25}\text{º}\text{C}+\text{32}=\text{77}\text{º}\text{F}\text{.}[/latex]

Solution for (b)

  1. Choose the right equation. To convert from [latex]\text{º}\text{C}[/latex] to K, use the equation
    [latex]{T}_{\text{K}}={T}_{\text{º}\text{C}}+\text{273}\text{.}\text{15}\text{.}[/latex]
  2. Plug the known value into the equation and solve:
    [latex]{T}_{\text{K}}=\text{25}\text{º}\text{C}+\text{273}\text{.}\text{15}=\text{298}\phantom{\rule{0.15em}{0ex}}\text{K}\text{.}[/latex]

Converting between Temperature Scales: the Reaumur Scale

The Reaumur scale is a temperature scale that was used widely in Europe in the 18th and 19th centuries. On the Reaumur temperature scale, the freezing point of water is [latex]0\text{º}\text{R}[/latex] and the boiling temperature is [latex]\text{80}\text{º}\text{R}[/latex]. If “room temperature” is [latex]\text{25}\text{º}\text{C}[/latex] on the Celsius scale, what is it on the Reaumur scale?

Strategy

To answer this question, we must compare the Reaumur scale to the Celsius scale. The difference between the freezing point and boiling point of water on the Reaumur scale is [latex]\text{80}\text{º}\text{R}[/latex]. On the Celsius scale it is [latex]\text{100}\text{º}\text{C}[/latex]. Therefore [latex]\text{100}\text{º}\text{C}=\text{80}\text{º}\text{R}[/latex]. Both scales start at [latex]0\text{º}[/latex] for freezing, so we can derive a simple formula to convert between temperatures on the two scales.

Solution

  1. Derive a formula to convert from one scale to the other:
    [latex]{T}_{\text{º}\text{R}}=\frac{0\text{.}8\text{º}\text{R}}{\text{º}\text{C}}\phantom{\rule{0.15em}{0ex}}×\phantom{\rule{0.15em}{0ex}}{T}_{\text{º}\text{C}}\text{.}[/latex]
  2. Plug the known value into the equation and solve:
    [latex]{T}_{\text{º}\text{R}}=\frac{0\text{.}8\text{º}\text{R}}{\text{º}\text{C}}\phantom{\rule{0.15em}{0ex}}×\phantom{\rule{0.15em}{0ex}}\text{25}\text{º}\text{C}=\text{20}\text{º}\text{R}\text{.}[/latex]

Temperature Ranges in the Universe

Living organisms function within a narrow range of temperatures. Humans, for example, typically maintain a core body temperature around [latex]37.0^\circ \text{C}[/latex] ([latex]98.6^\circ \text{F}[/latex]). Survival is possible only within a limited range—from about [latex]24^\circ \text{C}[/latex] ([latex]75^\circ \text{F}[/latex]) to [latex]44^\circ \text{C}[/latex] ([latex]111^\circ \text{F}[/latex]). Deviations from the normal range may indicate medical conditions such as fever, infection, or circulatory issues (see Figure 93.5).

This figure consists of four different infrared thermographs of a person's head and neck, taken when the person's head was positioned at four different angles. The person's face and neck are mostly red and orange, with patches of white, green, and yellow. The red and white colors correspond to hot areas. The person's hair ranges in color from green to light blue to dark blue. The blue color corresponds to cold areas.
Figure 93.5: This image of radiation from a person’s body (an infrared thermograph) shows the location of temperature abnormalities in the upper body. Dark blue corresponds to cold areas and red to white corresponds to hot areas. An elevated temperature might be an indication of malignant tissue (a cancerous tumor in the breast, for example), while a depressed temperature might be due to a decline in blood flow from a clot. In this case, the abnormalities are caused by a condition called hyperhidrosis. (credit: Porcelina81, Wikimedia Commons)

The universe contains an extraordinary range of temperatures. In highly controlled laboratory conditions, temperatures as low as [latex]1.0 \times 10^{-10} \ \text{K}[/latex] have been achieved. In contrast, the coldest naturally occurring place in the known universe is the Boomerang Nebula, at about 1 K. On Earth, the coldest recorded surface temperature is 183 K (−89°C) in Vostok, Antarctica. These values are visualized in Figure 93.6.

The figure is a single vertical axis showing a wide range of temperatures on a logarithmic scale, measured in kelvin. The temperature range goes from the lowest temperature achieved at ten to the power of negative ten kelvin up to the temperature in experiments at the Relativistic Heavy Ion Collider at ten to the power of positive twelve kelvin.
Figure 93.6: presents this vast temperature scale on a logarithmic axis. Such scales are helpful in science and medicine when dealing with phenomena that span many orders of magnitude.

Making Connections: Absolute Zero

Absolute zero is the theoretical lowest possible temperature, defined as [latex]0 \ \text{K}[/latex] or [latex]-273.15^\circ \text{C}[/latex]. At absolute zero, all molecular motion would cease. This idea comes from the behavior of gases. As shown in Figure 93.7, the pressure of a gas decreases linearly with temperature and extrapolates to zero at absolute zero, assuming volume is held constant.

Although absolute zero can never be reached in practice—because gases condense or solidify first—it remains a fundamental concept in thermodynamics and plays a key role in scientific temperature scales like the Kelvin scale.

Line graph of pressure versus temperature of five gases. Each graph is linear with a positive slope. Each line extrapolates to a pressure of zero at a temperature of negative two hundred seventy three point one five degrees Celsius.
Figure 93.7: Graph of pressure versus temperature for various gases kept at a constant volume. Note that all of the graphs extrapolate to zero pressure at the same temperature.

Thermal Equilibrium and the Zeroth Law of Thermodynamics

A thermometer does not directly measure the temperature of another object—it measures its own. But if left in contact with an object long enough, it reaches the same temperature, assuming heat can flow between them. This state is called thermal equilibrium. At thermal equilibrium, there is no net heat flow between the systems, and both have the same temperature.

Thermal equilibrium is the basis for temperature measurement. If a thermometer reads the same after repeated contact with multiple systems, we can conclude that all systems are at the same temperature.

This leads to a fundamental principle known as the zeroth law of thermodynamics:

The Zeroth Law of Thermodynamics

If two systems (A and B) are in thermal equilibrium with a third system (C), then they are also in thermal equilibrium with each other. This law establishes temperature as a valid and measurable concept.

Although named later than the first and second laws of thermodynamics, the zeroth law provides the logical foundation for them. A real-world example involves infants in incubators. Despite wearing little clothing, the babies remain warm because the incubator maintains air and surface temperatures in equilibrium with the baby’s body.

Check Your Understanding

Question: Does the temperature of an object depend on its size?

Answer: No. Temperature is an intensive property—it does not depend on the amount of material. Each part of a system at equilibrium has the same temperature, regardless of size.

Section Summary

  • Temperature is a measurable quantity that reflects the average kinetic energy of the particles in a system.
  • Absolute zero is the lowest possible temperature, at which all molecular motion ceases.
  • Temperature can be measured in Celsius, Fahrenheit, or Kelvin. Kelvin is the SI unit and is commonly used in science.
  • Temperatures can be converted using the following equations:
[latex]{T}_{\text{F}} = \frac{9}{5}{T}_{\text{C}} + 32[/latex]
[latex]{T}_{\text{C}} = \frac{5}{9}({T}_{\text{F}} - 32)[/latex]
[latex]{T}_{\text{K}} = {T}_{\text{C}} + 273.15[/latex]
[latex]{T}_{\text{C}} = {T}_{\text{K}} - 273.15[/latex]
  • Thermal equilibrium occurs when two systems in contact have the same temperature and no net heat flows between them.
  • The zeroth law of thermodynamics establishes that temperature is a transitive property between systems in equilibrium.

 Conceptual Questions

  1. What does it mean to say that two systems are in thermal equilibrium?
  2. Give an example of a physical property that varies with temperature and describe how it is used to measure temperature.
  3. When a cold alcohol thermometer is placed in a hot liquid, the column of alcohol goes down slightly before going up. Explain why.
  4. If you add boiling water to a cup at room temperature, what would you expect the final equilibrium temperature of the unit to be? You will need to include the surroundings as part of the system. Consider the zeroth law of thermodynamics.

Problems & Exercises

  1. What is the Fahrenheit temperature of a person with a [latex]\text{39}\text{.}0\text{º}\text{C}[/latex] fever?
  2. Frost damage to most plants occurs at temperatures of [latex]\text{28}\text{.}0\text{º}\text{F}[/latex] or lower. What is this temperature on the Kelvin scale?
  3. To conserve energy, room temperatures are kept at [latex]\text{68}\text{.}0\text{º}\text{F}[/latex] in the winter and [latex]\text{78}\text{.}0\text{º}\text{F}[/latex] in the summer. What are these temperatures on the Celsius scale?
  4. A tungsten light bulb filament may operate at 2900 K. What is its Fahrenheit temperature? What is this on the Celsius scale?
  5. The surface temperature of the Sun is about 5750 K. What is this temperature on the Fahrenheit scale?
  6. One of the hottest temperatures ever recorded on the surface of Earth was [latex]\text{134}\text{º}\text{F}[/latex] in Death Valley, CA. What is this temperature in Celsius degrees? What is this temperature in Kelvin?
  7. (a) Suppose a cold front blows into your locale and drops the temperature by 40.0 Fahrenheit degrees. How many degrees Celsius does the temperature decrease when there is a [latex]\text{40}\text{.}0\text{º}\text{F}[/latex] decrease in temperature? (b) Show that any change in temperature in Fahrenheit degrees is nine-fifths the change in Celsius degrees.
  8. (a) At what temperature do the Fahrenheit and Celsius scales have the same numerical value? (b) At what temperature do the Fahrenheit and Kelvin scales have the same numerical value?

Glossary

temperature
the quantity measured by a thermometer
Celsius scale
temperature scale in which the freezing point of water is [latex]0\text{º}\text{C}[/latex] and the boiling point of water is [latex]\text{100}\text{º}\text{C}[/latex]
degree Celsius
unit on the Celsius temperature scale
Fahrenheit scale
temperature scale in which the freezing point of water is [latex]\text{32}\text{º}\text{F}[/latex] and the boiling point of water is [latex]\text{212}\text{º}\text{F}[/latex]
degree Fahrenheit
unit on the Fahrenheit temperature scale
Kelvin scale
temperature scale in which 0 K is the lowest possible temperature, representing absolute zero
absolute zero
the lowest possible temperature; the temperature at which all molecular motion ceases
thermal equilibrium
the condition in which heat no longer flows between two objects that are in contact; the two objects have the same temperature
zeroth law of thermodynamics
law that states that if two objects are in thermal equilibrium, and a third object is in thermal equilibrium with one of those objects, it is also in thermal equilibrium with the other object
definition

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