How does temperature become a number

An essential step to “understand” thermodynamics is to get to know temperature: how temperature comes down as an abstraction from empirical observations, and how it rises up as a consequence of the fundamental postulate.  I have just updated my notes on temperature.  The beginning paragraphs of the notes abstract temperature from empirical observations.  These paragraphs are posted here.

Thermal contact. Consider a glass of wine and a piece of cheese. They have been kept as two separate isolated systems for such a long time that each by itself is in equilibrium. However, the two isolated systems are not in equilibrium with each other. We now allow the two systems to interact in a particular way: energy from one system can go to the other system. Will energy go from the wine to the cheese, or the other way around?

This mode of interaction, which re-allocates energy between the two systems, is called thermal contact. The energy that goes from one system to the other is called heat. To focus on thermal contact, we block all other modes of interaction: the two systems do not exchange volume, molecules, etc. In reality, any contact of two systems does more than just re-allocating energy. For example, when two bodies touch each other, they may form atomic bonds. As another example, the presence of two bodies in space modifies the electromagnetic field. For the time being, we assume that such effects are negligible, and focus exclusively on a single mode of interaction: the two systems can only exchange energy.

Thermal equilibrium. When all other modes of interaction are blocked, two systems in thermal contact for a long time will cease to exchange energy, a condition known as thermal equilibrium. Our everyday experience with thermal contact and thermal equilibrium may be distilled in terms of several salient observations.

Observation 1: hotness is a property independent of system. If two systems are separately in thermal equilibrium with a third system, the two systems are in thermal equilibrium with each other. This observation is known as the zeroth law of thermodynamics.

This observation shows us that all systems in thermal equilibrium possess one property in common: hotness. The procedure to establish a level of hotness is empirical. We bring two systems into thermal contact, and check if they exchange energy. Two systems in thermal equilibrium are said to have the same level of hotness. Two systems not in thermal equilibrium are said to have different levels of hotness.

In thermodynamics, the word “hot” is used strictly within the context of thermal contact. It makes no thermodynamic sense to say that one movie is hotter than the other, because the two movies cannot exchange energy. The word hotness is synonymous to temperature.

Name a level of hotness. To facilitate communication, we give each level of hotness a name. As shown by the above empirical observation, levels of hotness are real: they exist in the experiment of thermal contact, regardless how we name them.

The situation is analogous to naming streets in a city. The streets are real: they exist regardless how we name them. We can name the streets by using names of presidents, or names of universities. We can use numbers. We can be inconsistent, naming streets in one part of the city by numbers, and those in another part of the city by names of presidents. We can even give the same street several names, using English, Chinese, and Spanish. Some naming schemes might be more convenient than others, but to call one scheme absolute is an abuse of language. We will give an example of such an abuse later. But for now, consider one specific naming scheme.

We can name each level of hotness after a physical event. For example, we can name a level of hotness after a substance that melts at this hotness. This practice is easily carried out because of the following empirical observation: at the melting point, a substance is a mixture of solid and liquid, and the hotness remains unchanged as the proportion of the solid and liquid changes. Thus, a system is said to be at the level of hotness WATER when the system is in thermal equilibrium with a mixture of ice and water at the melting point. Here are four distinct levels of hotness: WATER, LEAD, ALUMINUM, GOLD. We can similarly name other levels of hotness.

Observation 2: levels of hotness are ordered. When two systems of different levels of hotness are brought into thermal contact, energy goes only in one direction from one system to the other, but not in the opposite direction. This observation is known as the second law of thermodynamics.

This observation allows us to order any two levels of hotness. When two systems are brought into thermal contact, the system losing energy is said to have a higher level of hotness than the system gaining energy. For example, we say that hotness “LEAD” is lower than hotness “ALUMINUM” because, upon brining melting lead and melting aluminum into thermal contact, we observe that that the amount of solid lead decreases, while the amount solid aluminum increases. Similar experiments show us the order of the four levels of hotness as follows:

WATER, LEAD, ALUMINUM, GOLD.

Observation 3: levels of hotness are continuous. Between any two levels of hotness A and B there exists another level of hotness. The experiment may go like this. We have two systems at hotness A and B, respectively, where hotness A is lower than hotness B. We can always find another system, which loses energy when in thermal contact with A, but gains energy when in thermal contact with B.

This observation allows us to name all levels of hotness using a single real variable. Around 1720, Fahrenheit assigned the number 32 to the melting point of water, and the number 212 to the boiling point of water. What would he do for other levels of hotness? Mercury is a liquid within this range of hotness and beyond, sufficient for most purposes for everyday life. When energy is added to mercury, mercury expands. The various volumes of mercury could be used to name the levels of hotness.

What would he do for a high level of hotness when mercury is a vapor, or a low level of hotness when mercury is a solid? He could switch to materials other than mercury, or phenomena other than thermal expansion.

In naming levels of hotness using a real variable, in essence Fahrenheit chose a one-to-one function, whose domain was various levels of hotness, and whose range was a real number. Any choice of such a function is called a temperature scale. While the order of various levels of hotness is absolute, no absolute significance is attached to the number that names each level of hotness. In particular, using numbers to name levels of hotness does not authorize us to apply arithmetic rules: the addition of two levels of hotness has no more empirical significance than the addition of the numbers of two houses on a street.

As illustrated by the melting-point scale of hotness, a non-numerical naming scheme of hotness perfectly captures all we care about everyday experience of hotness. Naming levels of hotness by using numbers makes it easier to memorize that hotness 80 is hotter than hotness 60. Our preference to a numerical scale reveals more about the prejudice of our brains than the nature of hotness.

Once a scale of hotness is set up, any monotonically increasing function of the scale gives another scale of hotness. For example, in the Celsius scale, the freezing point of water is set to be 0C, and the boiling point of water 100C. We further set the Celsius (C) scale to be linear in the Fahrenheit (F) scale. These prescriptions give the transformation between the two scales:

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In general, however, the transformation from one scale of hotness to another need not be linear. Any increasing function will preserve the order of the levels of hotness.

We next wish to trace the above empirical observations of thermal contact to the fundamental postulate.  In doing so, we relate temperature to two other quantities:  energy and entropy.  This development is given in the notes on temperature.