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. Temperature is operationally defined to be what we measure with a thermometer. (Many physical quantities are defined solely in terms of how they are measured. We shall see later how temperature is related to the kinetic energies of atoms and molecules, a more physical explanation.) Two accurate thermometers, one placed in hot water and the other in cold water, will show the hot water to have a higher temperature. If they are then placed in the tepid water, both will give identical readings (within measurement uncertainties). In this section, we discuss temperature, its measurement by thermometers, and its relationship to thermal equilibrium. Again, temperature is the quantity measured by a thermometer.

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 ([link]). Other properties used to measure temperature include electrical resistance and color, as shown in [link], and the emission of infrared radiation, as shown in [link].

# Temperature Scales

Thermometers are used to measure temperature according to well-defined scales of measurement, which use pre-defined 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 Celsius scale (which replaced the slightly different * centigrade* scale) has the freezing point of water at $0\text{\xba}\text{C}$ and the boiling point at $\text{100}\text{\xba}\text{C}$. Its unit is the degree Celsius$(\text{\xba}\text{C})$. On the Fahrenheit scale (still the most frequently used in the United States), the freezing point of water is at $\text{32}\text{\xba}\text{F}$ and the boiling point is at $\text{212}\text{\xba}\text{F}$. The unit of temperature on this scale is the degree Fahrenheit$(\text{\xba}\text{F})$. Note that a temperature difference of one degree Celsius is greater than a temperature difference of one degree Fahrenheit. Only 100 Celsius degrees span the same range as 180 Fahrenheit degrees, thus one degree on the Celsius scale is 1.8 times larger than one degree on the Fahrenheit scale $\text{180}/\text{100}=9/5\text{.}$

The Kelvin scale is the temperature scale that is commonly used in science. It is an *absolute temperature* scale defined to have 0 K at the lowest possible temperature, called absolute zero. The official temperature unit on this scale is the * kelvin*, which is abbreviated K, and is not accompanied by a degree sign. The freezing and boiling points of water are 273.15 K and 373.15 K, respectively. Thus, the magnitude of temperature differences is the same in units of kelvins and degrees Celsius. Unlike other temperature scales, the Kelvin scale is an absolute scale. It is used extensively in scientific work because a number of physical quantities, such as the volume of an ideal gas, are directly related to absolute temperature. The kelvin is the SI unit used in scientific work.

The relationships between the three common temperature scales is shown in [link]. Temperatures on these scales can be converted using the equations in [link].

To convert from . . . | Use this equation . . . | Also written as . . . |

Celsius to Fahrenheit | $$T\left(\text{\xba}\text{F}\right)=\frac{9}{5}T\left(\text{\xba}\text{C}\right)+\text{32}$$ | $${T}_{\text{\xba}\text{F}}=\frac{9}{5}{T}_{\text{\xba}\text{C}}+\text{32}$$ |

Fahrenheit to Celsius | $$T\left(\text{\xba}\text{C}\right)=\frac{5}{9}\left(T\left(\text{\xba}\text{F}\right)-\text{32}\right)$$ | $${T}_{\text{\xba}\text{C}}=\frac{5}{9}\left({T}_{\text{\xba}\text{F}}-\text{32}\right)$$ |

Celsius to Kelvin | $$T\left(\text{K}\right)=T\left(\text{\xba}\text{C}\right)+\text{273}\text{.}\text{15}$$ | $${T}_{\text{K}}={T}_{\text{\xba}\text{C}}+\text{273}\text{.}\text{15}$$ |

Kelvin to Celsius | $$T\left(\text{\xba}\text{C}\right)=T\left(\text{K}\right)-\text{273}\text{.}\text{15}$$ | $${T}_{\text{\xba}\text{C}}={T}_{\text{K}}-\text{273}\text{.}\text{15}$$ |

Fahrenheit to Kelvin | $$T\left(\text{K}\right)=\frac{5}{9}\left(T\left(\text{\xba}\text{F}\right)-\text{32}\right)+\text{273}\text{.}\text{15}$$ | $${T}_{\text{K}}=\frac{5}{9}\left({T}_{\text{\xba}\text{F}}-\text{32}\right)+\text{273}\text{.}\text{15}$$ |

Kelvin to Fahrenheit | $$T(\text{\xba}\text{F})=\frac{9}{5}\left(T\left(\text{K}\right)-\text{273}\text{.}\text{15}\right)+\text{32}$$ | $${T}_{\text{\xba}\text{F}}=\frac{9}{5}\left({T}_{\text{K}}-\text{273}\text{.}\text{15}\right)+\text{32}$$ |

Notice that the conversions between Fahrenheit and Kelvin look quite complicated. In fact, they are simple combinations of the conversions between Fahrenheit and Celsius, and the conversions between Celsius and Kelvin.

“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

2. Plug the known value into the equation and solve:

**Solution for (b)**

1. Choose the right equation. To convert from $\text{\xba}\text{C}$ to K, use the equation

2. Plug the known value into the equation and solve:

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 $0\text{\xba}\text{R}$ 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:

2. Plug the known value into the equation and solve:

# Temperature Ranges in the Universe

[link] 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 [link]).

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.

## Thermal Equilibrium and the Zeroth Law of Thermodynamics

Thermometers 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

*(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 thermal equilibrium, and no further changes will occur. The systems interact and change because their temperatures differ, and the changes stop once their temperatures are the same. Thus, if enough time is allowed for this transfer of heat to run its course, the temperature a thermometer registers*

*thermal contact**represent the system with which it is in thermal equilibrium. Thermal equilibrium is established when two bodies are in contact with each other and can freely exchange energy.*

*does*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 zeroth law of thermodynamics.

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.

## Section Summary

- Temperature is the quantity measured by a thermometer.
- Temperature is related to the average kinetic energy of atoms and molecules in a system.
- Absolute zero is the temperature at which there is no molecular motion.
- There are three main temperature scales: Celsius, Fahrenheit, and Kelvin.
- Temperatures on one scale can be converted to temperatures on another scale using the following equations:
${T}_{\text{\xba}\text{F}}=\frac{9}{5}{T}_{\text{\xba}\text{C}}+\text{32}$${T}_{\text{\xba}\text{C}}=\frac{5}{9}\left({T}_{\text{\xba}\text{F}}-\text{32}\right)$${T}_{\text{K}}={T}_{\text{\xba}\text{C}}+\text{273}\text{.}\text{15}$${T}_{\text{\xba}\text{C}}={T}_{\text{K}}-\text{273}\text{.}\text{15}$
- Systems are in thermal equilibrium when they have the same temperature.
- Thermal equilibrium occurs when two bodies are in contact with each other and can freely exchange energy.
- The zeroth law of thermodynamics states that when 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.

## Conceptual Questions

What does it mean to say that two systems are in thermal equilibrium?

Give an example of a physical property that varies with temperature and describe how it is used to measure temperature.

When a cold alcohol thermometer is placed in a hot liquid, the column of alcohol goes * down* slightly before going up. Explain why.

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

What is the Fahrenheit temperature of a person with a $\text{39}\text{.}0\text{\xba}\text{C}$ fever?

$\text{102}\text{\xba}\text{F}$

Frost 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?

To 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?

$\text{20}\text{.}0\text{\xba}\text{C}$ and $\text{25}\text{.}6\text{\xba}\text{C}$

A tungsten light bulb filament may operate at 2900 K. What is its Fahrenheit temperature? What is this on the Celsius scale?

The surface temperature of the Sun is about 5750 K. What is this temperature on the Fahrenheit scale?

$\text{9890}\text{\xba}\text{F}$

One 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?

(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 nine-fifths the change in Celsius degrees.

(a) $\text{22}\text{.}2\text{\xba}\text{C}$

(b) $\begin{array}{lll}\text{\Delta}T\left(\text{\xba}\text{F}\right)& =& {T}_{2}\left(\text{\xba}\text{F}\right)-{T}_{1}\left(\text{\xba}\text{F}\right)\\ & =& \frac{9}{5}{T}_{2}\left(\text{\xba}\text{C}\right)+\text{32}\text{.}0\text{\xba}-\left(\frac{9}{5}{T}_{1}\left(\text{\xba}\text{C}\right)+\text{32}\text{.}0\text{\xba}\right)\\ & =& \frac{9}{5}\left({T}_{2}\left(\text{\xba}\text{C}\right)-{T}_{1}\left(\text{\xba}\text{C}\right)\right)\text{}=\frac{9}{5}\text{\Delta}T\left(\text{\xba}\text{C}\right)\end{array}$

(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?

- College Physics
- Preface
- Introduction: The Nature of Science and Physics
- Kinematics
- Introduction to One-Dimensional Kinematics
- Displacement
- Vectors, Scalars, and Coordinate Systems
- Time, Velocity, and Speed
- Acceleration
- Motion Equations for Constant Acceleration in One Dimension
- Problem-Solving Basics for One-Dimensional Kinematics
- Falling Objects
- Graphical Analysis of One-Dimensional Motion

- Two-Dimensional Kinematics
- Dynamics: Force and Newton's Laws of Motion
- Introduction to Dynamics: Newton’s Laws of Motion
- Development of Force Concept
- Newton’s First Law of Motion: Inertia
- Newton’s Second Law of Motion: Concept of a System
- Newton’s Third Law of Motion: Symmetry in Forces
- Normal, Tension, and Other Examples of Forces
- Problem-Solving Strategies
- Further Applications of Newton’s Laws of Motion
- Extended Topic: The Four Basic Forces—An Introduction

- Further Applications of Newton's Laws: Friction, Drag, and Elasticity
- Uniform Circular Motion and Gravitation
- Work, Energy, and Energy Resources
- Linear Momentum and Collisions
- Statics and Torque
- Rotational Motion and Angular Momentum
- Introduction to Rotational Motion and Angular Momentum
- Angular Acceleration
- Kinematics of Rotational Motion
- Dynamics of Rotational Motion: Rotational Inertia
- Rotational Kinetic Energy: Work and Energy Revisited
- Angular Momentum and Its Conservation
- Collisions of Extended Bodies in Two Dimensions
- Gyroscopic Effects: Vector Aspects of Angular Momentum

- Fluid Statics
- Fluid Dynamics and Its Biological and Medical Applications
- Introduction to Fluid Dynamics and Its Biological and Medical Applications
- Flow Rate and Its Relation to Velocity
- Bernoulli’s Equation
- The Most General Applications of Bernoulli’s Equation
- Viscosity and Laminar Flow; Poiseuille’s Law
- The Onset of Turbulence
- Motion of an Object in a Viscous Fluid
- Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

- Temperature, Kinetic Theory, and the Gas Laws
- Heat and Heat Transfer Methods
- Thermodynamics
- Introduction to Thermodynamics
- The First Law of Thermodynamics
- The First Law of Thermodynamics and Some Simple Processes
- Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency
- Carnot’s Perfect Heat Engine: The Second Law of Thermodynamics Restated
- Applications of Thermodynamics: Heat Pumps and Refrigerators
- Entropy and the Second Law of Thermodynamics: Disorder and the Unavailability of Energy
- Statistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation

- Oscillatory Motion and Waves
- Introduction to Oscillatory Motion and Waves
- Hooke’s Law: Stress and Strain Revisited
- Period and Frequency in Oscillations
- Simple Harmonic Motion: A Special Periodic Motion
- The Simple Pendulum
- Energy and the Simple Harmonic Oscillator
- Uniform Circular Motion and Simple Harmonic Motion
- Damped Harmonic Motion
- Forced Oscillations and Resonance
- Waves
- Superposition and Interference
- Energy in Waves: Intensity

- Physics of Hearing
- Electric Charge and Electric Field
- Introduction to Electric Charge and Electric Field
- Static Electricity and Charge: Conservation of Charge
- Conductors and Insulators
- Coulomb’s Law
- Electric Field: Concept of a Field Revisited
- Electric Field Lines: Multiple Charges
- Electric Forces in Biology
- Conductors and Electric Fields in Static Equilibrium
- Applications of Electrostatics

- Electric Potential and Electric Field
- Electric Current, Resistance, and Ohm's Law
- Circuits, Bioelectricity, and DC Instruments
- Magnetism
- Introduction to Magnetism
- Magnets
- Ferromagnets and Electromagnets
- Magnetic Fields and Magnetic Field Lines
- Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field
- Force on a Moving Charge in a Magnetic Field: Examples and Applications
- The Hall Effect
- Magnetic Force on a Current-Carrying Conductor
- Torque on a Current Loop: Motors and Meters
- Magnetic Fields Produced by Currents: Ampere’s Law
- Magnetic Force between Two Parallel Conductors
- More Applications of Magnetism

- Electromagnetic Induction, AC Circuits, and Electrical Technologies
- Introduction to Electromagnetic Induction, AC Circuits and Electrical Technologies
- Induced Emf and Magnetic Flux
- Faraday’s Law of Induction: Lenz’s Law
- Motional Emf
- Eddy Currents and Magnetic Damping
- Electric Generators
- Back Emf
- Transformers
- Electrical Safety: Systems and Devices
- Inductance
- RL Circuits
- Reactance, Inductive and Capacitive
- RLC Series AC Circuits

- Electromagnetic Waves
- Geometric Optics
- Vision and Optical Instruments
- Wave Optics
- Introduction to Wave Optics
- The Wave Aspect of Light: Interference
- Huygens's Principle: Diffraction
- Young’s Double Slit Experiment
- Multiple Slit Diffraction
- Single Slit Diffraction
- Limits of Resolution: The Rayleigh Criterion
- Thin Film Interference
- Polarization
- *Extended Topic* Microscopy Enhanced by the Wave Characteristics of Light

- Special Relativity
- Introduction to Quantum Physics
- Atomic Physics
- Introduction to Atomic Physics
- Discovery of the Atom
- Discovery of the Parts of the Atom: Electrons and Nuclei
- Bohr’s Theory of the Hydrogen Atom
- X Rays: Atomic Origins and Applications
- Applications of Atomic Excitations and De-Excitations
- The Wave Nature of Matter Causes Quantization
- Patterns in Spectra Reveal More Quantization
- Quantum Numbers and Rules
- The Pauli Exclusion Principle

- Radioactivity and Nuclear Physics
- Medical Applications of Nuclear Physics
- Particle Physics
- Frontiers of Physics
- Atomic Masses
- Selected Radioactive Isotopes
- Useful Information
- Glossary of Key Symbols and Notation