Definition of temperature: “absolute” temperature and linear relations between temperature and thermometric properties

Question

I'm confused about the definition of temperature, and I would like to make some points clearer since I did not find an explanation.

Firstly I have heard of three adjectives regarding temperature (scales): empirical, absolute and thermodynamical.

I think I understood the difference between empirical and thermodynamical, but what is exactly meant by absolute?

Is this the same as "indipendent from the particular system used to measure temperature", or does this adjective indicates that the scale can only have positive numbers, with the $0 K$ placed, indeed, at absolute zero?

For example, ideal gas constant volume thermometer gives absolute temperature, because the measurement is not affected by the use of different gases, as long as they are ideal? Or is it because in the definition $$T=\mathrm{lim}_{p_0->0} 273.16 \frac{p}{p_0} \, K$$

it is implicit that $T=0K$ is a temperature that can be reached only if $p=0 Pa$ (a limiting condition) and hence the scale has the zero in a very particular place, that cannot be reached?

---

My second question is about the ideal gas thermometer itself: in the previous definition it is used the fact that, "for an ideal gas $T \propto p$ at contant volume". That's how textbook introduces the ideal gas thermometer.

What I do not get is: how can we use the relation $T \propto p$ in ideal gases before even knowing what is temperature, i.e. to define (which means to measure) the temperature itself?

I do not get if the relation $T \propto p$ is imposed by the definition of temperature, for make things easier or if there is something behind it that guarantees that, before knowing how to even measure temperature, the temperature is proportional to pressure at constant volume. This last option does not makes a lot of sense, so if this is the case I surely misunderstood the reasoning behind the use of ideal gas thermometer.

And this holds also for any thermometric property $X$, where the relation to get the temperature $\theta$ from the property $X$ itself, say $\theta(X)$ is assumed to be linear, which means $$\theta(X)=C \, X$$

Is this really an assumption, in the sense that it is imposed to be true, or is in some way known before the definition of $\theta$?

Answer 1

Let me first organize your questions so everyone can navigate through this post more easily. If you happen to have more questions, or I misinterpreted your questions, please feel free to comment so I could edit.

QUESTIONS

1. What exactly is meant by absolute in an absolute temperature scale?
2. How can we use $T\propto p$ in order to construct thermometers if we don't have the definition of the temperature itself, or don't know what it means to measure temperature?
3. From where does the mentioned proportionality come from?
4. How do we deduce the types of various dependencies of thermodynamic parameters?

---

ANSWERS

I will start by answering question #2 first, since it seems to me that it is the most fundamental one.

2. Say I am a master blacksmith and I am sick and tired of not knowing how much do I need to heat up a material in order to melt it. There are no temperature scales, and none of that in my time and age but I do have an intuitive knowledge of what hot and cold is based on my sense of touch, and I also do know that for example boiling water is very hot, and ice is quite cold. So in order to know how much do I need to heat something up in order to melt it, I first have to come up with an idea of how to measure hotness. Hm, hotness sounds odd, let me call it temperature. So in the most basic and intuitive terms I have coined a noun that means "how something hot is" - temperature.
   $~~~~~~~~$Now, how am I going to describe this temperature thing so that when I say to my blacksmith friends how something hot is, they know exactly what I am talking about. Hm, let's try numbers. Okay so I know that ice is cold and boiling water is hot, let me attribute a value of 0 to the temperature of ice and 100 to the temperature of boiling water. Well okay, now I know that the temperature range of water when it goes from ice to vapor is 100, but how can I know how hot regular, liquid water is, after all I do feel the difference of cold and hot liquid water. Eureka! I have to measure it! Wait, how am I going to measure it? What does it mean to measure "how hot something is"? Let me consult my very smart friend. He tells me there exists this thing called pressure. He said that it is a resistance of a fluid, it is something that describes how much a fluid pushes onto you if you are in it. Yes, I remember that one time when I was swimming and went a bit deeper that I felt like something is pushing on me - it has to be pressure! But wait, he also said that he found out something. He found out that if you take a container and then fill the container up with water, and put a piston on top of the water, and then heat up the water, the piston eventually rises! The water starts pushing harder - pressure increases! Another Eureka!
   $~~~~~~~~$ Well I know how can I measure hotness, sorry temperature. By measuring pressure! I will measure pressure of water at 0 and at 100 and then I can find every other temperature in between or even less than 0 or higher than 100 indirectly by measuring pressure directly! I will use water for this. I have set up my apparatus, I start with water at 0 and I measure the pressure to obtain a value of $P_1$ then I have my workers make the fire that's heating the water slowly hotter - a greater temperature. As they are doing that I will read the pressure from the manometer and note it in my notebook. Okay, I'm done with the experiment, I have measured all pressures, and I note that if I draw the lines on a graph where y-axis is the measured pressure and x-axis is time, I see that I can draw an approximately straight line through the points, and since I know that the fire was slowly made hotter throughout the whole experiment, well then it surely means that temperature as a function of pressure is a linear function. I have values of pressures at 0 and 100 and I can draw a straight line, which means by interpolating the graph I can find the temperature just by looking at the pressure. And since I know that the slope of a linear function is constant, and my function has a slope of $P/T$ then that also is constant. Which leads me to $$P_1/T_1=P_2/T_2=const.$$

   1. The previous answer shows how you can go from knowing nothing at all about temperature to building a temperature scale and even a thermometer. Now, the problem that a temperature scale like the one that was described in answer one (Celsius temperature scale) is that there exists stuff that's colder than 0 degrees, and you have to use negative temperatures for this. Now, since it is more natural to have a temperature scale that doesn't use negative values Kelvin tried to come up with an absolute temperature scale, one that does not use negative values. He did that by using thermometers with various different fluids in them and found that when you graph all of those $P(T)$ lines for different curves they all intersect at one point which is $T=-273.15 ^\circ C=0~K$, basically saying that the only possible temperature where you can have zero pressure regardless of the fluid used in the thermometer is $0~K$ and based of the conversion function $T(K)=273.15+T(C)$ he set up the absolute thermodynamic scale.

   2. $T\propto p$ comes from Gay-Lussacs law or Amonton's Law. In terms of kinetic theory of gases it makes perfect sense, since in that theory temperature is defined as the parameter that describes the amount of movement of individual molecules, and in gases pressure is known to be caused by molecules piling up or bouncing off the walls of the container it is in, there fore if temperature is greater - then the molecules move around more (they have greater velocities and thus momenta) - as they hit the walls of the container with greater speeds - pressure increases.

   3. Thermodynamics is such a field where most of the functions that are used are obtained empirically. It's just a huge amount of molecules each acting in their own way, so it's really hard to come up with general equations especially from just a theoretical approach.

Ending note: empirical temperature scales are those such as Celsius or Fahrenheit, you assign a temperature value for some characteristic process and interpolate/extrapolate from there. Problem with these is they use negative temperatures. To avoid this you use absolute temperature scales such as Kelvin and Rankine. Problem with these is that they are obtained by measuring one temperature dependent property (like pressure, or volume) over the range of all temperatures, which is a problem since those dependencies aren't linear for the whole range of temperatures. To avoid this you use thermodynamic temperature scales that use correct thermometer types for appropriate temperature ranges and frequently also factor in systematic errors that occur in those thermometers (like vapors of the fluid inside the thermometer itself that bring on a certain deviation from the true value of pressure, if such a thermometer is being used, that in turn cause errors in temperature values).

Hope I helped.

Answer 2

You are quite right to worry about defining temperature in terms of the ideal gas law - there are two approaches that are more fundamental. I'm sorry this will only have to be a partial answer, I'm not well versed enough to give a complete one.

First we can talk about the classical thermodynamic definition of temperature. In order to define it, we need the Clausius statement of the 2nd law of thermodynamics: "No process is possible whose sole result is the transfer of heat from a cooler to a hotter body". This allows us to define 'cooler' and 'hotter' bodies in the following way: A is hotter than B if and only if there is no process whose sole result is the transfer of heat from B to A. If we combine this statement with the zeroth law of thermodynamics: "if two thermodynamic systems are each in thermal equilibrium with a third, then they are in thermal equilibrium with each other.", then we can start to talk about bodies possessing a property given by a real number that we might call the 'degree of hotness'. The degree of hotness is defined such that if A has greater degree of hotness than B, a process can transfer heat from A to B as its sole effect, and vice versa. If A has same degree of hotness as B, then the transfer can go both ways.

With this 'degree of hotness' in hand, we can go about trying to construct a measure of hotness which will turn out to be a function of this 'degree of hotness' and is in fact the Kelvin scale of temperature. To prove that this temperature has all the properties you expect (for example, to prove that it is the T in the ideal gas law), and to properly motivate its definition, is a lengthy task that I must leave out - it requires you to be familiar with arguments concerning heat engines. A good, concise exposition of it would be found in Enrico Fermi - Thermodynamics. Alternatively, you may try the chapter on classical thermodynamics in this text:

http://www.damtp.cam.ac.uk/user/tong/statphys/sp.pdf

The point I want to emphasise is that absolute/thermodynamic temperature has a deeper foundation than the ideal gas laws. It is something that can be rigorously defined and shown to have meaning from the laws of thermodynamics only.

An alternative approach for defining temperature is given in statistical mechanics. In that case temperature is defined as a tradeoff between energy and entropy. This definition can be shown using statistical arguments to be equivalent to the first - the argument is also rather lengthy. However I think the classical thermodynamic notion of temperature, grounded in the laws of thermodynamics, should be sufficient for you.

Answer 3

You need to put things in order. Temperature is first defined in text book (i.e. a thermostat with 0 degC for ice-water blend, 100 degC for boiling water and linear scale). By testing, people noticed that T is proportional to P for gas. This can be done by heating the gas of constant volume and measuring both T and P. Temperature is independent (not derived) to this ideal gas relationship.