# Temperature Scales

The most widely adopted temperature scale is the Celsius or centigrade scale. Nowadays, we use these terms as synonyms, but, in fact, they were not such, originally.

The most widely adopted temperature scale is the **Celsius** or **centigrade** scale. Nowadays, we use these terms as synonyms, but, in fact, they were not such, originally. The Celsius temperature scale was invented by Anders Celsius that, in 1742, proposed its use by publishing “Observationer om twänne beständiga grader på en thermometer” (Observations of two constant gradations on a thermometer). In his paper, Celsius reported the observations, according to which the temperature of melting ice (marked as C) and that of boiling water (marked as D) appeared to be constant, if measured with a mercury thermometer.

He then proposed, on page 179, to divide the length between these two points into 100 parts and assigned the value 100 to point C (melting ice), saving the value 0 for point D (boiling water).

At item 3 of the above figure (taken from archive.org), one can read

delas distancen CD uti 100 lika delar eller Grader

that can be translated with the aid of an automatic translator like Google translate, as

the distance CD is divided into 100 equal parts or Degrees.

followed by

så at 0 grad svarar emot kokande vatns: punkten D, och 100 emot vatnfrusnings: punkten C. När samma grader kontinueras sedan utföra under C ånda till A, så är termometern fardig

whose meaning is approximately

let's say (så) that 0 degree corresponds (svarar) to boiling water (kokande vatns): point D, and 100 to the freezing water (vatnfrusnings): point C. Continuing the same degrees from C to A [the lowest point in the thermometer, ed], then the thermometer is finished.

In “Sur l’invention du thermomètre centigrade à mercure, faite à Lyon par M. Christin”, by M. J. Fournet, one can read a history of thermometers from the point of view of a French scientist. Apparently, during the same year 1742, a French physicist, Jean-Pierre Christin, made, independently, the same observations and proposed the centigrade scale, in use today. Indeed, at page 260 we can read

Ce fut Stroemer qui effectua la modification par laquelle le thermomètre de Celsius put concorder avec celui de M. Christin; mais les observations qu'il fit avec son instrument à Upsal, ne datant que de 1744

The two scales, then, were reconciled two years later.

In fact, the first commercially available thermometers were built by Daniel Gabriel Fahrenheit, starting from 1717. The scale he invented to mark temperatures on his thermometers is still popular among anglo-saxons countries, and is the cause of many headaches among physics students. The conversion from degrees Fahrenheit and degrees Celsius, in fact, is significantly more complex than any other unit conversion. In fact, the Fahrenheit scale is such that 0 °F corresponds to -18 °C, while 100 °F corresponds to 37.8 °C. The conversion between the two scales can thus be obtained by means of the following formulas:

I heard about many different explanations about such a difference. According to the weirdest one, Fahrenheit used the lowest temperature he could achieve in his laboratory as T=0° and the one of his horse’s blood as T=100°.

Searching for original documents, as made above in the case of Celsius and Christin, is always good, and, indeed, it could be fun. This applies, in fact, to every information you get from any source (websites, books, etc.): **being able to find the correct, original information is a key competence**, especially nowadays.

In 1724, Fahrenheit himself explained how he defined the scale in a paper entitled “Experimenta et Observationes de Congelatione aquae in vacuo factae” published in the Royal Society’s *Philosophical Transactions* (London), vol. **33**, no. 382, page 78.

The following part explains how the scale was determined (it starts as the penultimate sentence in the image above): “The scale of thermometers used for meteorological observations begins with zero and ends at 96 degrees. The division of the scale depends on three fixed points, which can be determined as follows. The first is found at the bottom of the scale, and is determined by a mixture of ice, water and either ammonium chloride or sea salt. When the thermometer is submerged in this mixture, its fluid [alcohol, ed] descends to a degree marked with a zero. This experiment succeeds better in winter than in summer. The second point is obtained if water and ice are mixed. Placing the thermometer in this mixture, its fluid reaches the 32ⁿᵈ degree. This point corresponds to the start of freezing. Such a point is reached by the thermometer when still waters are covered with a thin layer of ice in winter. The third point is fount 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.”

It is worth noting that 96 °F corresponds, in fact, to 36 °C. The question, thus, becomes: how the starting and final points were chosen? Choosing zero for a given, low temperature, can be natural; the point at 0 °F can be obtained mixing snow and salt in equal parts. What about 96?

Ernst Cohen and his wife Wilhelma Abramina Titia de Meester, in 1936, published¹ a Fahrenheit’s letter to this Herman Boerhaave, in which the inventor of the scale said that “As [Ole Rømer, ed] graduation is inconvenient and awkward because of the fractions, I decided to alter the scale, and to use 96 instead of 22½ or 90”. Indeed, at that time, units were always expressed in base-12, as they still are in imperial units (miles, inches, feet, etc.). Base 12 is useful because it has more divisors (1, 2, 3, 4, 6, 12) than 10 (1, 2, 5) and measurements are easy to express as fractions.

Most probably, Fahrenheit used a scale invented by Rømer that, however, he found to be inconvenient, because the interval between the fixed points (the mixture of ice and salts, and the human body temperature) cannot be divided into a multiple of 12. From what is reported in the letter it would appear that in Rømer’s original scale, the human body temperature corresponded to 22.5 degrees. Having only a bit more than twenty divisions of the interval, made it difficult to avoid fractions, so Fahrenheit decided to multiply the scale by 4: 22.5⨉4=90 that, however, is not a multiple of 12. Hence, he redefined the scale such that the interval was divided into 96 degrees. In this scale, the temperature of melting ice is actually 32 °F.

The current internationally agreed unit for temperature is the **Kelvin** (not the “degree Kelvin”), abbreviated **K** (not °K), after William Thomson, also known as Lord Kelvin. It is based on an absolute scale, deriving from the observation (due to Boltzmann) that temperature is nothing but a measurement of the average kinetic energy of some body’s constituents. Kinetic energy cannot be negative, then the minimum possible temperature is 0 K and that point correspond to the case in which constituents are “at rest”. Such a condition, in fact, cannot be reached, in practice, however, we can approach it indefinitely. Nowadays, we are able to “routinely” reach temperatures as low as 1.9 K (corresponding to about -271°C) in the CERN’s LHC accelerator that is the macroscopic (LHC is 27 km long) coldest object in the Universe, as far as we know.

In fact, lower temperatures can be achieved in several laboratories where, however, the cold volume is orders of magnitude lower (in LHC we use about 100 tons of liquid helium at about 2K). In particular², the CUORE experiment at Gran Sasso, has a 1 cubic meter cold volume at less than 10 mK.

A first hint about the fact that temperatures cannot be lower than a certain value comes, in fact, from gas laws. In fact, it can be shown that the product of the pressure *p* times the volume *V* of a gas, *obeys* the following relationship: *pV=aT+b, *where *a* and *b* are constants. If *T*, the temperature, is measured in degrees Celsius, for one mole of gas, it si found experimentally that *a=8.314* in SI units and *b=2 269.722*, still in SI units (J). Manifestly, the volume *V* of a gas cannot be negative, then, if the product *pV* is proportional to the temperature, there must exist a temperature such that *pV=0* that cannot be exceeded. Imposing *pV=0*, we find that *T=-b/a=-273* in some unit coherent with those of *pV *(actually, Kelvins).

¹ Ernst Cohen and W. A. T. Cohen-De Meester. *Chemisch Weekblad*, volume **33**(*1936*), pages 374–393.

² I am indebted to Mirco Milletarì for noticing this initially missing piece of information.

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