# Puzzled: why is this even linear? Must it be? Looking for clarification

And as for "I can create my own scale (Z, the zgozvrm unit) that works just fine. I can define the freezing point of water as -23 deg Z and the boiling point of water as +69.5 deg Z if I want to. And, I can divide that scale into as many parts as I want to."
No you can't, by definition you'll have 92.5 scales going from freezing to boiling; what you can (and must) do is still give meaning to -22 deg, -21 deg, etc
If we can have
$${}^\circ F = {}^\circ C \times \frac{9}{5} + 32$$

$${}^\circ F = K \times \frac{9}{5} - 459.67$$
or

$${}^\circ F = {}^\circ R - 459.67$$

Then we can also have

$${}^\circ Z = {}^\circ C \times \frac{37}{40} - 23$$
or
$${}^\circ F = ({}^\circ Z + 23 ) \times \frac{72}{37} +32$$

As long as we define [itex]-23^\circ Z[/tex] as the freezing point of water and [itex]69.5^\circ Z[/tex] as the boiling point of water.

In fact, I can build such a thermometer if I want to.
I could explain it's scale to someone who's never used it before and, after getting used to it, they will realize that [itex]-2.4^\circ Z[/tex] is a nice, comfortable air temperature to have in the summer, whereas [itex]18.1^\circ Z[/tex] would be a very hot, desert-like temperature.

I wasn't saying it did follow. I meant that once you'd found some set of things that were 'regular with hotness', i.e. things whose change against our current temperature definition was linear, exponential (or any strictly increasing function in the mathematical sense), then you would prefer to chose the one that acted linearly, because it 'makes sense' to us that combining temperatures is linear.
Hm, I'm trying to understand, but I'm getting a bit confused:
I've used the word "linear" in two ways so far:
1) Celsius, Fahrenheit, ... are linear with respect to each other
2) The defining of temperature by making it linear to the expansion of a certain liquid (principle of a thermometer)
Now you bring in a 3rd notion of linearity, if I'm correct:
3) Two identical objects at a different temperature equilibrate to a temperature that is the mean of both initial temperatures.

Now I'm trying to get if this linearity has a tie with the other 2 (well we can ignore the first one, I suppose). But anyway, if we had no temperature and we wanted 2 to be fulfilled for a certain liquid, we could easily define temperature that way. Now how would you define a temperature starting from nothing that has linear property 3? (I don't really see the connection with our current temperature measure which I thought to be historically based on number 2 for some liquid or gas)

D H - I'm not asking why they were stupid at all, just trying to understand the issue phenomenologically.

D H
Staff Emeritus
Now I'm trying to get if this linearity has a tie with the other 2
After the fact, yes. Once again, this additional characteristic of temperature could not have been discovered without having a quantitative definition of temperature already in place.

I'm not asking why they were stupid at all, just trying to understand the issue phenomenologically.
It's pretty simple: The definition of temperature as the quantity that thermometers measure turned out to be very useful. Other concepts regarding heat from the same time frame were less useful. You probably have never heard of phlogiston (1670s), caloric (1770s), or frigoric (1780s) because those concepts turned out to be not that useful (in fact they turned out to be wrong). Science tends to hold on to concepts that are useful but discard concepts that turn out to be wrong or of minimal value. Harping on what should now be obvious, the concept of temperature turned out to be very useful. If it wasn't it would have been discarded.

Mathematics and science moves two steps forward, one step back. After the fact those backward steps are often edited out of the picture to present what appears to be a smoother progression of thought than actually took place.

"After the fact, yes. Once again, this additional characteristic of temperature could not have been discovered without having a quantitative definition of temperature already in place."

Is it logical that number 2 and 3 are related? I can't see it, at all. Does the nr 3 linearity only hold for materials for which the nr 2 linearity holds? (for a certain defined temperature)

D H
Staff Emeritus
Is it logical that number 2 and 3 are related? I can't see it, at all.
Yes, so long as both thermometers use the same two temperature reference points and the same liquid.

Suppose we have two thermometers with different temperature scales, call them A and B, and two easily verifiable temperature reference points, call them 1 and 2. In scale A we arbitrarily assign temperature values A1 and A2 (0 and 100 for example) to these reference temperatures while in scale B we arbitrarily call these points B1 and B2 (32 and 212, for example). Now we define temperature to be proportional to the height above (below) reference point 1 of the top of the liquid in the tube such that the temperature of the two reference points are our arbitrary values (i.e., A1 or B1 when the temperature is known to be at reference point 1, A2 or B2 when the temperature is known to be at reference point 2). The transformation from one scale to another will the same affine relationship everywhere by definition.

Now suppose we use different pairs of temperature reference points such as (A) an ice/brine mix and Fahrenheit's wife's armpit versus (B) the freezing and boiling points of pure water at standard atmospheric pressure. In this case a simple affine relationship between the temperature scales is not a given. That this was the case showed the utility of 'temperature' as a meaningful concept.

"The transformation from one scale to another will the same affine relationship everywhere by definition."

Why? What if we defined one of those scales with a liquid whose volume rises exponentially for every degree celsius?

D H
Staff Emeritus
Irrelevant. All that matters is that whatever is being measured is a monotonic function of state only (as opposed to the path taken to arrive at that state).

epenguin
Homework Helper
Gold Member
I think the OP is right in his questionings and probably a lot of students are shortchanged by not having these things explained properly.

The expansion of liquids with temperature is a complex phenomenon for which the theory is unlikely to be complete. There might be a rough theory to suggest why roughly most liquids would increase in volume roughly proportionally to another. But I am sure no two liquids would expand exactly proportionately to each other. When they invented the mercury thermometer there was not even a glimmering of a theory. But mercury would be more convenient than others, and you might think the temperature measured by the length of a mercury thread might mean something even if you didn't know what. BTW you don't need to use length of liquid columns - many physical measurements might do, e,g, electrical resistance of some substance or other. But if you didn't have a physical understanding what use was it?

At first standardisation and discrimination. Early scientists had only a qualitative description of temperature - terms like 'dung heat' or 'furnace heat'. There were only at most half a dozen different temperatures! Even without a theory it is quite useful to be told that a chemical reaction goes well between 60 and 89 deg. And for the human body just a few degrees turn out very important for health and disease and diagnosis, so an instrument to measure within a fraction of a degree instead of judging the degree of fever is very useful.

(In one way on the other hand temperature measurement was ahead of others. All measurements are comparisons. When you report a measurement in units you are comparing something with something else in some other laboratory where the instrument was directly or indirectly calibrated against some standard. A meter and a gram were originally blocks of stuff in some laboratory in Paris. But for temperature you could make your own standard where you were agreeing with others because the freezing and boiling points of water are the same everywhere (simplifying). These days they try to get standards like this that can be reproduced everywhere. When last I heard they had got this for time and length but not yet mass.)

Then came along the kinetic theory of gasses and their volume at constant pressure was proportional to the average kinetic energy of the molecules so this volume resembled the mercury thermometer but reflected something fundamental and understandable. So That could be the measure of temperature. Well, you still need a substance that is near enough to what the theory is. OK - dilute noble gases. I remember that there was a gas thermometer made of platinum filled with argon in some standards laboratory somewhere. So all decent thermometers were calibrated, no doubt mostly indirectly against this. You cannot check experimentally that the volume of this argon is linear with temperature of course - that would be circular. But you can check experimentally that other predictions of the theory like Boyle's law are well obeyed.

So through this chain of comparisons we are able, maybe without thinking about it, to measure e.g. rates of a chemical reaction, varying with our temperature measured with our (ultimately calibrated) and explain the dependency quantitatively in terms of an activation energy because we know an energy is essentially what our thermometer is reporting to us.

epenguin
Homework Helper
Gold Member
Let me give some examples of students being fed stuff that is if not meaningless, not meaning what they are told.

An outstanding example is when they are told that Galileo discovered the law of the simple pendulum by noticing the period of the pendant lamp in the Cathedral of Florence was constant and independent of amplitude. How did he time that pendulum? He measured its frequency with his own heartbeats. So we could say that he used a poor clock to check on the regularity of a better one, or that he experimentally proved that his heartbeats were rather regular that day.

Or I remember at school we did experimentally 'prove' Charles' laws, pressure and volume of air were proportional to temperature. Again using a so-so instrument to check up on a better one! What we had proved of course that mercury column length was reasonably proportional to temperature - a useful but not very fundamental fact.

Or a niece not long ago went through with me some school lab work. They had made some measurement of current and voltage and shown they were proportional. Proved Ohm's law! Current proportional to voltage! Wurll I said, do you know how these meters work? The voltmeter is just an ammeter with high resistance. So you have proved that a current distributes itself to flow in a constant proportion independent of current between two resistors in parallel. Not that you have actually measured current, you have measured the magnetic effects of two currents. So it looks like depends on a Faraday law of the magnetic effect of currents. But maybe it is better than that. Maybe it didn't depend on that current-force law applying in practice in a rather complicated instrumental arrangement, maybe the ammeters were just calibrated against a better measure of current, like electrochemical silver deposition which is something rather fundamental and understandable and more easily made a precise measurement I think. But I got the impression this was confusing and that at school it was better to believe they had proved voltage ∝ current. *

So there are these examples all the time comparing one instrument reading with another without understanding. No wonder it's a bit boring or mistrusted for some of the students.

*What Ohm actually did was questioned here on a thread, I don't think we did find out.

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D H
Staff Emeritus
Let me give some examples of students being fed stuff that is if not meaningless, not meaning what they are told.

An outstanding example is when they are told that Galileo discovered the law of the simple pendulum by noticing the period of the pendant lamp in the Cathedral of Florence was constant and independent of amplitude. How did he time that pendulum? He measured its frequency with his own heartbeats. So we could say that he used a poor clock to check on the regularity of a better one, or that he experimentally proved that his heartbeats were rather regular that day.
That is an outstanding example, but not of students being fed stuff that is meaningless. It is an outstanding example of modern science at work during its very infancy.

You have the advantage of hindsight and 400 years of developments in physics to know that a pendulum is a good timekeeper. Galileo did not have that hindsight, nor did he have calculus, nor Newtonian mechanics. There weren't any good clocks in Galileo's day. He used the best clock he had -- his heartbeat. Galileo's study of pendulums was one of the motivating factors that led Huygens to develop a pendulum clock 50 years after Galileo's studies.

epenguin
Homework Helper
Gold Member
That is an outstanding example, but not of students being fed stuff that is meaningless. It is an outstanding example of modern science at work during its very infancy.
They are not explained that either. Aha, thank you epenguin, that helps.

So if I understand well, the underlying key understanding is that the ideal gas was very understandable concept and it made the natural suggestion to define temperature as to make it linear with its expansion. Because the ideal gas is also an (exteme) idealized model for most gases and fluids, its linearity became a good general approximation.
Is the fact that "if we have two identical objects at different temperatures, they equilibrate at the temperature that is the mathematical mean of the two initial temperatures" (if the temp.differences are not too big) also a result of the same principle? Namely that this happens to be true for an ideal gas and that the ideal gas is a basic rudimentary approximation for gases and fluids? Ah yes I see that these two characteristics of the ideal gas are logically equivalent: if temperature is a linear measure for energy content and if two identical boxes at different T's equilibrate, the energy content of one of either boxes will be the mathematical mean :)

Hm, afterwards we noticed we could define a temperature $$\theta$$ independent of any material using the 2nd law (and 0th and 1st) by choosing an arbitrary reference temperature and put it in $$f(\theta,\theta_0) = Q/Q_0$$ if Q is the heat absorbed at $$\theta$$ and Q_0 the heat deposited at $$\theta_0$$ by a reversible cycle. The funny thing is that... the most simple choice for a function, namely $$\theta / \theta_0$$ makes this general definition coincide with that of the ideal gas temperature T... Is this truly a coincidence?

epenguin
Homework Helper
Gold Member
I was about to log off, here it's very late, I cannot concentrate.

For your first part I'd roughly agree. Temperature is a bit like time. At first the rotation of the earth was the standard of time, the clock, your mercury thermometer. But like as you first said, different thermometers didn't necessarily agree nor do different clocks. If the rotation of the earth is the clock, then it makes no sense to say the earth is slowing down or varying in any way. You can only say that if there is a better clock. How do you know another clock is better? Well because you have a theory, like for gases, that makes you think another clock has simpler, better understood physics, is less influenced e.g. by environment outside the clock. The rotating earth on second approximation is quite complicated.

I have not followed you last para today. Perhaps you will re-examine it. You talk of heat - but to measure heat e.g. in a calorimeter requires you measure temperature.