132. TemperatureLearning Objectives
The concept of temperature has evolved from the common concepts of hot and cold. Human perception of what feels hot or cold is a relative one. For example, if you place one hand in hot water and the other in cold water, and then place both hands in tepid water, the tepid water will feel cool to the hand that was in hot water, and warm to the one that was in cold water. The scientific definition of temperature is less ambiguous than your senses of hot and cold. Misconception Alert: Human Perception vs. Reality:On a cold winter morning, the wood on a porch feels warmer than the metal of your bike. The wood and bicycle are in thermal equilibrium with the outside air, and are thus the same temperature. They feel different because of the difference in the way that they conduct heat away from your skin. The metal conducts heat away from your body faster than the wood does (see more about conductivity in Conduction). This is just one example demonstrating that the human sense of hot and cold is not determined by temperature alone. Another factor that affects our perception of temperature is humidity. Most people feel much hotter on hot, humid days than on hot, dry days. This is because on humid days, sweat does not evaporate from the skin as efficiently as it does on dry days. It is the evaporation of sweat (or water from a sprinkler or pool) that cools us off. Any physical property that depends on temperature, and whose response to temperature is reproducible, can be used as the basis of a thermometer. Because many physical properties depend on temperature, the variety of thermometers is remarkable. For example, volume increases with temperature for most substances. This property is the basis for the common alcohol thermometer, the old mercury thermometer, and the bimetallic strip (Figure 1). Other properties used to measure temperature include electrical resistance and color, as shown in Figure 2, and the emission of infrared radiation, as shown in Figure 3. Temperature ScalesThermometers are used to measure temperature according to welldefined scales of measurement, which use predefined reference points to help compare quantities. The three most common temperature scales are the Fahrenheit, Celsius, and Kelvin scales. A temperature scale can be created by identifying two easily reproducible temperatures. The freezing and boiling temperatures of water at standard atmospheric pressure are commonly used. The The The relationships between the three common temperature scales is shown in Figure 4. Temperatures on these scales can be converted using the equations in Table 1. Example 1: Converting between Temperature Scales: Room Temperature“Room temperature” is generally defined to be $\text{25}\text{\xba}\text{C}$. (a) What is room temperature in $\text{\xba}\text{F}$? (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 $\text{\xba}\text{C}$ to $\text{\xba}\text{F}$, use the equation $${T}_{\text{\xba}\text{F}}=\frac{9}{5}{T}_{\text{\xba}\text{C}}+\text{32}\text{.}$$2. Plug the known value into the equation and solve: $${T}_{\text{\xba}\text{F}}=\frac{9}{5}\text{25}\text{\xba}\text{C}+\text{32}=\text{77}\text{\xba}\text{F}\text{.}$$Solution for (b) 1. Choose the right equation. To convert from $\text{\xba}\text{C}$ to K, use the equation $${T}_{\text{K}}={T}_{\text{\xba}\text{C}}+\text{273}\text{.}\text{15}\text{.}$$2. Plug the known value into the equation and solve: $${T}_{\text{K}}=\text{25}\text{\xba}\text{C}+\text{273}\text{.}\text{15}=\text{298}\phantom{\rule{0.25em}{0ex}}\text{K}\text{.}$$Example 2: Converting between Temperature Scales: the Reaumur ScaleThe 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 $0\text{\xba}\text{R}$ and the boiling temperature is $\text{80}\text{\xba}\text{R}$. If “room temperature” is $\text{25}\text{\xba}\text{C}$ 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 $\text{80}\text{\xba}\text{R}$. On the Celsius scale it is $\text{100}\text{\xba}\text{C}$. Therefore $\text{100}\text{\xba}\text{C}=\text{80}\text{\xba}\text{R}$. Both scales start at $0\text{\xba}$ 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: $${T}_{\text{\xba}\text{R}}=\frac{0\text{.}8\text{\xba}\text{R}}{\text{\xba}\text{C}}\phantom{\rule{0.25em}{0ex}}\times \phantom{\rule{0.25em}{0ex}}{T}_{\text{\xba}\text{C}}\text{.}$$2. Plug the known value into the equation and solve: $${T}_{\text{\xba}\text{R}}=\frac{0\text{.}8\text{\xba}\text{R}}{\text{\xba}\text{C}}\phantom{\rule{0.25em}{0ex}}\times \phantom{\rule{0.25em}{0ex}}\text{25}\text{\xba}\text{C}=\text{20}\text{\xba}\text{R}\text{.}$$Temperature Ranges in the UniverseFigure 6 shows the wide range of temperatures found in the universe. Human beings have been known to survive with body temperatures within a small range, from $\text{24}\text{\xba}\text{C}$ to $\text{44}\text{\xba}\text{C}$ $(\text{75}\text{\xba}\text{F}$ to $\text{111}\text{\xba}\text{F}$). The average normal body temperature is usually given as $\text{37}\text{.}0\text{\xba}\text{C}$ ($\text{98}\text{.}6\text{\xba}\text{F}$), and variations in this temperature can indicate a medical condition: a fever, an infection, a tumor, or circulatory problems (see Figure 5). The lowest temperatures ever recorded have been measured during laboratory experiments: $4\text{.}5\times {\text{10}}^{\u2013\text{10}}\phantom{\rule{0.25em}{0ex}}\text{K}$ at the Massachusetts Institute of Technology (USA), and $1\text{.}0\times {\text{10}}^{\u2013\text{10}}\phantom{\rule{0.25em}{0ex}}\text{K}$ at Helsinki University of Technology (Finland). In comparison, the coldest recorded place on Earth’s surface is Vostok, Antarctica at 183 K $(\u2013\text{89}\text{\xba}\text{C})$, and the coldest place (outside the lab) known in the universe is the Boomerang Nebula, with a temperature of 1 K. Making Connections: Absolute Zero:What is absolute zero? Absolute zero is the temperature at which all molecular motion has ceased. The concept of absolute zero arises from the behavior of gases. Figure 7 shows how the pressure of gases at a constant volume decreases as temperature decreases. Various scientists have noted that the pressures of gases extrapolate to zero at the same temperature, $\u2013\text{273}\text{.}\text{15}\text{\xba}\text{C}$. This extrapolation implies that there is a lowest temperature. This temperature is called absolute zero. Today we know that most gases first liquefy and then freeze, and it is not actually possible to reach absolute zero. The numerical value of absolute zero temperature is $\u2013\text{273}\text{.}\text{15}\text{\xba}\text{C}$ or 0 K. Thermal Equilibrium and the Zeroth Law of ThermodynamicsThermometers actually take their own temperature, not the temperature of the object they are measuring. This raises the question of how we can be certain that a thermometer measures the temperature of the object with which it is in contact. It is based on the fact that any two systems placed in thermal contact (meaning heat transfer can occur between them) will reach the same temperature. That is, heat will flow from the hotter object to the cooler one until they have exactly the same temperature. The objects are then in Furthermore, experimentation has shown that if two systems, A and B, are in thermal equilibrium with each another, and B is in thermal equilibrium with a third system C, then A is also in thermal equilibrium with C. This conclusion may seem obvious, because all three have the same temperature, but it is basic to thermodynamics. It is called the The Zeroth Law of Thermodynamics:If two systems, A and B, are in thermal equilibrium with each other, and B is in thermal equilibrium with a third system, C, then A is also in thermal equilibrium with C. This law was postulated in the 1930s, after the first and second laws of thermodynamics had been developed and named. It is called the zeroth law because it comes logically before the first and second laws (discussed in Thermodynamics). An example of this law in action is seen in babies in incubators: babies in incubators normally have very few clothes on, so to an observer they look as if they may not be warm enough. However, the temperature of the air, the cot, and the baby is the same, because they are in thermal equilibrium, which is accomplished by maintaining air temperature to keep the baby comfortable. Check Your UnderstandingDoes the temperature of a body depend on its size? Show/Hide Solution SolutionNo, the system can be divided into smaller parts each of which is at the same temperature. We say that the temperature is an intensive quantity. Intensive quantities are independent of size. Section Summary
Conceptual QuestionsExercise 1What does it mean to say that two systems are in thermal equilibrium? Exercise 2Give an example of a physical property that varies with temperature and describe how it is used to measure temperature. Exercise 3When a cold alcohol thermometer is placed in a hot liquid, the column of alcohol goes down slightly before going up. Explain why. Exercise 4If 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 & ExercisesExercise 1What is the Fahrenheit temperature of a person with a $\text{39}\text{.}0\text{\xba}\text{C}$ fever? Show/Hide Solution Solution$\text{102}\text{\xba}\text{F}$ Exercise 2Frost damage to most plants occurs at temperatures of $\text{28}\text{.}0\text{\xba}\text{F}$ or lower. What is this temperature on the Kelvin scale? Exercise 3To conserve energy, room temperatures are kept at $\text{68}\text{.}0\text{\xba}\text{F}$ in the winter and $\text{78}\text{.}0\text{\xba}\text{F}$ in the summer. What are these temperatures on the Celsius scale? Show/Hide Solution Solution$\text{20}\text{.}0\text{\xba}\text{C}$ and $\text{25}\text{.}6\text{\xba}\text{C}$ Exercise 4A tungsten light bulb filament may operate at 2900 K. What is its Fahrenheit temperature? What is this on the Celsius scale? Exercise 5The surface temperature of the Sun is about 5750 K. What is this temperature on the Fahrenheit scale? Show/Hide Solution Solution$\text{9890}\text{\xba}\text{F}$ Exercise 6One of the hottest temperatures ever recorded on the surface of Earth was $\text{134}\text{\xba}\text{F}$ in Death Valley, CA. What is this temperature in Celsius degrees? What is this temperature in Kelvin? Exercise 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 $\text{40}\text{.}0\text{\xba}\text{F}$ decrease in temperature? (b) Show that any change in temperature in Fahrenheit degrees is ninefifths the change in Celsius degrees. Show/Hide Solution Solution(a) $\text{22}\text{.}2\text{\xba}\text{C}$ (b) $\begin{array}{lll}\text{\Delta}T\left(\text{\xba}\text{F}\right)& =& {T}_{2}\left(\text{\xba}\text{F}\right){T}_{1}\left(\text{\xba}\text{F}\right)\\ & =& \frac{9}{5}{T}_{2}\left(\text{\xba}\text{C}\right)+\text{32}\text{.}0\text{\xba}\left(\frac{9}{5}{T}_{1}\left(\text{\xba}\text{C}\right)+\text{32}\text{.}0\text{\xba}\right)\\ & =& \frac{9}{5}\left({T}_{2}\left(\text{\xba}\text{C}\right){T}_{1}\left(\text{\xba}\text{C}\right)\right)\text{}=\frac{9}{5}\text{\Delta}T\left(\text{\xba}\text{C}\right)\end{array}$ Exercise 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?
