The Physics
Opus in profectus


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theoretical definition

One has to be careful when defining temperature not to confuse it with heat. Heat is a form of energy. Temperature is something different. We could begin with a technical definition, but I would prefer to start with a question. How hot is it? The answer to this question (or a question like this) is a measurement of temperature. The hotter something is, the higher its temperature. Therefore I would like to propose the following informal definition — temperature is a measure of hotness.

Quantities in science are typically defined operationally (through the process by which they are measured) or theoretically (in terms of the theories of a specific discipline). We will begin with a theoretical definition of temperature and end with an operational definition.

Let's review what you should already know.

  1. A system possesses energy if it has the ability to do work.
  2. Energy comes in two basic forms: the kinetic energy of motion and the potential energy of position.
  3. Energy is conserved; which is to say, it cannot be created or destroyed. When one form of energy decreases another form must increase.

The archetypal example of this is the rock at the top of a hill. Due to its height above the bottom of the hill, it possesses gravitational potential energy. Give it a push and it will start to roll. If we assume the ideal situation of a closed system where no energy is lost on the way down, then the rock's initial potential energy will equal its final kinetic energy.

Now take the archetypal example one step further. Assume the rock crashes into a wall. Neither the rock nor the wall are elastic, so the rock comes to a halt. Now it appears as if we have violated the law of conservation of energy. The kinetic energy is lost and nothing has come along to replace it. Where has the energy gone?

Cartoon showing a boulder a the the top of a hill, halfway down, and at the bottom.

The answer to this question is inside the rock. The energy has been transformed from the external energy visible as the motion of the rock as a whole to the internal energy of the motion of the invisible parts that make up the rock. The two energies are identical in size, but different in appearance. External energy is visible because it is organized. The translational kinetic energy of a rock is due to coordinated motion. All the parts all move forward together. The rotational energy is also coordinated. The parts all rotate together around the center of mass. In contrast, the internal kinetic energy of a rock is invisible since the pieces are so small and numerous and their motion is completely uncoordinated. Their motions are statistically random with a mean value of zero making the energy largely invisible to us macroscopic beings.

Potential energy can also exist in external and internal forms. I won't provide you with an example here but I will say that external potential energy is relatively obvious. (Look, there's a rock on the top of that hill.) Internal potential energy is more obscure. (Look, there's an atom next to another atom.) Internal potential energy is responsible for latent heat, a topic discussed later in this book.

If you believe that objects can have internal energy, then it shouldn't be much of a stretch to believe that they can exchange this energy. This is known as thermal contact. The irreducible bits and pieces of objects responsible for carrying the internal energy are known as atoms — from the Greek "α τομή" [a tomi] meaning "can't be cut" — but belief in atoms is not a necessity. It just makes life easier. (Surprisingly, much of thermal physics and thermodynamics was worked out before atoms were generally considered real.) Since we are dealing with large numbers of atoms in uncoordinated motion, there will be times and places where the transfer of internal energy will run in one direction and different times and places where the transfer of internal energy will run in the opposite direction. Since the numbers are so unimaginably large, we really don't care about what happens to any one atom. All that we can observe in such cases is the net or overall transfer of internal energy. This is known as heat. If the net exchange of internal energy is zero; that is, if no heat flows from one region to another; then the whole system is said to be in thermal equilibrium. Heat, then, is the net transfer of internal energy from one region to another.

Nothing can be said to have heat or store heat. Instead we say heat flows from one place to another. The direction is indicated by the sign in front of the number. Use "+" when heat flows in and "−" when heat flows out. Heat can travel left, right, up, down, forward, or backward, but that's not usually the way it's described. Heat is a form of energy and energy is scalar, therefore specific directions and angles and all the rest of that vector stuff is irrelevant.

Heat is a form of energy and the unit of energy is the joule [J], therefore heat should be measured in joules. Before this was known, however, heat had its own special units; like the calorie and the British thermal unit [Btu]. These are still widely used in the United States for some reason — calorie for food energy (which is really a kilocalorie) and Btu for furnaces, air conditioners, stoves, and refrigerators. This units will be discussed in greater detail in a later section of this book.

Getting back to temperature. What is it?

Two regions in thermal contact have the same temperature when there is no net exchange of internal energy between them. Temperature, then, is what determines the direction of heat flow — out of the region with the higher temperature and in to the region with the lower temperature. In more concise terms, heat flows from hot to cold. That's the theoretical definition of temperature.

operational definition

Temperature is measured with a thermometer. The basic operating principle behind all thermometers is that there is some quantity, called a thermometric variable, that changes in response to changes in temperature. The relationship between temperature and the thermometric variable may be direct or inverse or it may be determined by a polynomial or power function. In any case, it is the thermometric variable that gets measured. There is no way to measure temperature directly.

Types of thermometers
type thermometric variable
liquid in glass volume
constant volume gas pressure
bimetallic strip coil pitch
electric resistor resistance
thermocouple voltage

Once we've settled on the thermometric variable to be measured, the next step is to decide on a temperature scale. Not because "units matter" (as every physics teacher says when they subtract points from students who forgot to write them on a test) but rather because temperature has no meaning without values defined as standard. In thermometry, what we need are fixed points: reproducible experiments based on natural phenomena that occur at a definite temperature under a proscribed set of conditions. Actually, we need at least two fixed points and a defined range of numbers (called a fundamental interval) between the lower fixed point and the upper fixed point. The other reason that the operational definition of temperature is so tightly bound with temperature scales is that the early science of thermometry is tied up with the invention and construction of thermometers.

The first thermometer was constructed in what is now northern Italy in the 17th century by either Sanctorius Sanctorius (1561–1636), the first physician to record vital signs like weight and body temperature; Galileo Galilei (1564–1642), the man who basically invented the scientific method; or Giovanni Francesco Sagredo (1571–1620), an instrument maker who is sometimes called a "disciple" of Galileo. All three men built what are known as liquid in glass thermometers, which consist of a glass reservoir of liquid attached to a narrow glass tube. When temperature increases, the liquid expands and rises up the tube. When temperature decreases, the liquid contracts and falls back down the tube. The height of the column is therefore related to the temperature in a simple linear fashion. Galileo did not put a scale on his device, so what he invented is better called a thermoscope since all it can do is show changes in temperature, not really measure them. Sanctorus added a scale to an air in glass thermoscope, and thus could be credited with inventing thermometer, but…. Air in glass devices respond to pressure changes as well as temperature changes and pressure was not something that was well understood at the time. Sagredo added a scale to his thermometer with 360 divisions imitating the Classical division of the circle. Ever since then, temperature units have been called "degrees" whether or not there were 360 of them in the fundamental interval.

Robert Hooke (1635–1703) of London was the first to suggest using the freezing point of water as a lower fixed point. Ole Rømer (1644–1710) of Copenhagen assigned a value of 7.5° to the freezing point and 60° to the boiling point of water so that normal body temperature would wind up as 22.5° or three times the freezing point. In the days when thermometers were graduated by hand, such tricks were commonly built into temperature scales.

In any case, normal body temperature is not the kind of fixed point that satisfies the needs of serious thermometry. There's just too much variation in the concept of "normal" as it applies to human beings. (The more meaningful term would be "average".) Different people can have different body temperatures and still be considered healthy and everyone's body temperature varies over the course of the day. We are coldest in the early morning and hottest in the middle of the afternoon. Such a variable number just doesn't cut it as a fixed point.

Some other failed ideas for fixed points include…


The longest lived of the temperature scales still in use is the work of Daniel Gabriel Fahrenheit (1686–1736). Fahrenheit was born to a German family living in Danzig, Prussia (now Gdansk, Poland). When he was 15 he lost both of his parents to mushroom poisoning and was apprenticed to a local merchant who later moved him to The Netherlands. Fahrenheit did not enjoy this arrangement and basically skipped out on his master. Apprenticeships are less like the internships modern college students deal with and are more like seven years of indentured servitude.

During his period as a runaway and for a few years after, Fahrenheit traveled throughout The Netherlands, Denmark, Germany, Sweden, and Poland; acquired technical skills like glassblowing and instrument making; and learned Dutch, French, English, and thermal physics.

When he was 28 he astounded the scientific community by constructing a pair of thermometers that gave consistently identical readings. What astounds me is that anyone would have found this act astounding, but apparently no one had ever done it before.

Sagredo's now historic 360 degree thermometer assigned 0° to a snow and salt mixture, 100° to snow, and 360° to the hottest summer day. Thermometers of the kind first built in northern Italy were calibrated to unreproduceable fixed points. This meant that thermometers manufactured in 1650 gave different results from those manufactured in 1651 and thermometers manufactured in Florence gave different results from those manufactured in Venice.

Fahrenheit settled on three fixed points, which he detailed in a paper presented before the Royal Society of London in 1724. (Emphasis has been added to certain keywords.)

Hujus scalæ divisio tribus nititur terminis fixis, qui arte sequentimodo parari possunt; primus illorum in informa parte vel initio scalæ reperitur, & commixtione glaciei, aquæ, & salis Armoniaci vel etiam maritimi acquiritur; huic mixturæ si thermometron imponitur, fluidum ejus usque ad gradum, qui zero notatur, descendit. Melius autem hyeme, quam æstate hoc experimentum succedit.   The division of the scale depends on three fixed points, which can be determined in the following manner. The first is found in the uncalibrated part or the beginning of the scale, and is determined by a mixture of ice, water and ammonium chloride or even sea salt. If the thermometer is placed in this mixture, its liquid descends as far as the degree that is marked with a zero. This experiment succeeds better in winter than in summer.
Secundus terminus obtinetur, si aqua & glacies absque memoratis salibus commiscentur, imposito thermometro huic mixturæ, fluidum ejus tricesimum secundum occupat gradum, & terminus initii congelationis a me vocatur; aquæ enim stagnantes tenuissima jam glacie obducuntur, quando hyeme liquor thermometri hunce gradum attingit.   The second point is obtained if water and ice are mixed without the aforementioned salts. When the thermometer is placed in this mixture, its liquid reaches the 32nd degree. I call this freezing point. For still waters are already covered with a very thin layer of ice when the liquid of the thermometer touches this point in winter.
Terminus tertius in nonagesimo sexto gradu reperitur; & spiritus usque ad hunc gradum dilatatur, dum thermometrum in ore vel sub axillis hominis in statu sano viventis tam diu tenetur donec perfectissime calorem corporis acquisivit.   The third point is situated at the 96th degree. Alcohol expands up to this point when it is held in the mouth or under the armpit of a living man in good health until it has completely acquired his body heat.
Daniel Gabriel Fahrenheit, 1724   Translation by J. Holland for

After Fahrenheit's death these fixed points were changed so that the scale bearing his name now has only two, more sensible fixed points. The normal freezing point of water stayed at the 32 °F but the saltwater and body heat points were dropped in favor of an upper fixed point of 212 °F at the normal boiling point of water. This divided the fundamental interval into 180 degrees, which was a tolerable number to work with. Dividing an interval up into halves or thirds (or powers of halves and thirds) is not that bad. It's fifths that are the real challenge. The factors of 96 are 2, 2, 2, 2, 2, 3; which is devoid of the dreaded fives. The factors of 180 are 2, 2, 3, 3, 5; which includes a five, but at least there's only one. The factors of 100 are 2, 2, 5, 5; which has twice as many fives as 180 and thus twice the dread.



René Réaumur (1683–1757) France. Anders Celsius (1701–1744) Sweden.

Since there are one hundred degrees between the two reference points, the the names degree centigrade and centesimal degree were used as well as the name degree Celsius. In 1948 these alternate names were dropped and degree Celsius was chosen as the official name. This was done to honor Celsius for his work in designing the original system and to avoid inconsistent use of the prefix centi. The name "centigrade" implies that there is a unit called the "grade".



William Thomson, Lord Kelvin (1824–1907) Ireland–Scotland suggests the first absolute temperature scale. Rudolf Clausius (1822–1888) Germany suggested that the scale be modified so that the size of one degree on Thomson's scale was the same as one centigrade degree.

International Temperature Scale (ITS)

Several fixed points.

temperature conversion

Most unit conversions are done by scaling. You take a number with a unit and multiply (or divide) by a conversion factor to get a new number with a new unit. The number by itself may be larger or smaller after the conversion, but the number with the unit is identical since the conversion factor is a ratio equal to one. Temperature units can't always be converted this way since not all temperature scales assign a value of zero to the same fixed point. Temperature conversions often require a translation to get the zeros to line up. You take a number with a unit and add (or subtract) a conversion factor with a number and a unit. You can do this before or after any scaling, depending on what you find convenient. A combination of scaling and translation is called a linear transformation (or a linear mapping).

Cartoon representation of the temperature scales using liquid in glass thermometers

The easiest temperature conversion is kelvin to degree Celsius. The size of the two units is identical by design. A temperature interval of 1 K is the same as 1 °C, therefore the scaling factor is 1 °C/1 K. A temperature of absolute zero is called 0 K on the kelvin scale and −273.15 °C on the Celsius scale, therefore a translation factor of −273 °C is needed. So we're basically multiplying by one, which is the same as doing nothing, and subtracting 273. The reverse conversion is equally simple.

  °C ← K  
T[°C] =  1 °C  T[K] − 273.15 °C
1 K
  °C = K − 273.15  
  K ← °C  
T[K] =  1 K  T[°C] + 273.15 K
1 °C
  °C = K + 273.15  

Let me tell ya somethin'. The last part of this section is really only useful for citizens and residents of the United States. There are 180 °F and 100 °C between the normal boiling and normal freezing points of water. This gives a scaling factor of 180100 when converting from degree Celsius to degree Fahrenheit, which reduces to 95. The zero of the Celsius scale is 32 degrees above the zero of the Fahrenheit scale, therefore a translation factor of +32 °F is needed.

The reverse conversion (degree Fahrenheit to degree Celsius) is, I think, best done in a slightly different way. Start by lining up the zero points by subtracting 32 °F, then use the scaling factor 100180 or 59.

  °F ← °C  
T[°F] =  180 °F  T[°C] + 32 °F
100 °C
°F =  9  °C + 32
 °C ← °F  
T[°C] = 
T[°F] − 32 °F
100 °C
180 °F
°C = 
°F − 32

For those of you who prefer your linear transformations in y = mx + b form, here's that last conversion again…

°C =  5  °F −  160
9 9

The only advantage to this notation is that it can be used to show that…

0 °F = −  160  °C

0 °F = −17.78 °C

Totally worth it.

Selected temperatures (fixed points in red)
device, event, phenomenon, process
~1032 planck temperature, upper limit of temperature
~1013 hottest laboratory experiment (LHC, 2012)
~1010 core of hottest stars
~107 core of the Sun
~107 nuclear explosion
~106 solar corona (the Sun's atmosphere)
25,000 surface of blue stars
24,000 lightning bolt
6500 D65 standard white hot (effective)
6000 center of Earth
5933 tungsten boils
5772 surface of the Sun
3683 tungsten melts
3500 surface of red stars
4900 2700 3000 incandescent light bulb
3100 1700 2000 typical flames
2200 1200 1500 fresh lava
1984.32 1084.62 1357.77 copper freezes
1947.52 1064.18 1337.33 gold freezes
1763.20 961.78 1234.93 silver freezes
1250 680 950 dull red hot
1220.58 660.323 933.473 aluminum freezes
930 500 770 incipient red heat
850 460 730 mean temperature on Venus
840 450 720 daytime temperature on Mercury
787.149 419.527 692.677 zinc freezes
674 357 630 mercury boils
621 327 600 lead melts
574.5875 301.4375 574.5875 fahrenheit and kelvin scales coincide
530 280 550 very hot home oven
451 233 506 paper burns, according to Ray Bradbury (paid link)
449.470 231.928 505.078 tin freezes
313.8773 156.5985 429.7485 indium freezes
252 122 395 upper limit for life under high pressure
212 100 373.15 water boils
134 56.7 329.817 hottest temperature on Earth (California, 1913)
106 41 314 New York City record high (Central Park, 1936)
100 37.778 310.928 nothing of importance
98.6 37.0 310.2 human body (traditional US)
98.2 36.8 309.9 human body (revised)
96 human body (according to Fahrenheit)
85.5763 29.7646 302.9146 gallium melts
80 27 300 numerically convenient "room temperature" (300 K)
68 20 293 numerically convenient "room temperature" (20 °C)
59 15 288 mean temperature on Earth
32.018 0.01 273.16 water triple point
32 0 273.15 water freezes
19 −7 266 optimal temperature of ice for skating
0 −17.8 255 ice-water-salt mixture (according to Fahrenheit)
−14.3 −25.7 247 New York City record low (Central Park, 1934)
−37.9019 −38.8344 234.3156 mercury triple point
−38 −39 234 mercury freezes
−40 −40 233 fahrenheit and celsius scales coincide
−56 −49 220 mean temperature on Mars
−108 −78 195 sublimation point of dry ice
−128.5 −89.2 183.95 coldest temperature on Earth (Antarctica, 1983)
−279.67 −173.15 100 nothing of importance
−300 −180 90 nighttime temperature on Mercury
−279 −183 90 oxygen liquefies
−308.8196 −189.3442 83.8058 argon triple point
−320 −196 77 nitrogen liquefies
63 nitrogen freezes
54.3584 oxygen triple point
50 mean temperature on Pluto
24.5561 neon triple point
20.3 hydrogen liquefies
13.8033 hydrogen triple point
4.22 helium liquefies
2.7260 cosmic microwave background
2.174 helium I/II λ point (0.050 atm)
~1 coldest point in space (Boomerang Nebula)
0.95 helium freezes (26 atm)
0.010 coldest cubic meter (CUORE, 2017)
10−8 stellar mass black hole
10−10 coldest laboratory experiment (Aalto University, 2000)
10−13~10−16 supermassive black hole
−459.67 −273.15 0 absolute zero