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

Hello,

When the Celsius temperature was first defined, it was defined by saying that 0°C was the freezing point and 100°C the boiling point. Now, this is far (infinitely far, actually) from a definition of temperature (this was before the 2nd law or Stat. Mech.; they are not important for our discussion). One might intuitively say "well you got two points: two points define a line so now you can extrapolate the celsius temperature for any other system", but surely this is completely wrong? What kind of graph is one thinking of? The x-axis temperature, okay; and the y-axis? (if my last two lines confuse you: ignore them and skip on to the next paragraph)

The thing I'm trying to get is why all our temperature scales (Fahrenheit, Celsius, Kelvin, ...) are linear to each other, cause there is no reason at all why they should be, or is there? Anyway, Celsius then defined his scale by saying: if V is the expansion of water from 0°C to 100°C, then if the water rises by V/100, the temperature has gone up by one degree Celsius. So Celsius defined his temperature to be linear with the expansion of water: this is arbitrary, right? If he had made it linear with the expansion of mercury, then it wouldn't have been linear with the expansion of water? I'm basing this last line on following quote from the Feynman Lectures

At one time "the temperature" was defined arbitrarily by dividing the expansion of water into even degrees of certain size. But when one then measures temperature with a mercury thermometer, one finds that the degrees are no longer even.
I'm not sure what he means by "even", but I assume he means in correlation to the volumetric expansion of the fluid? So with our current definition of T, water and ideal gases expand linearly with the degree, and mercury doesn't? (I suppose the difference is due to... ehm, chemistry? Well one would expect that water and mercury would ressemble each other more chemically than water and an ideal gas...)

N.B.: so is the linearity of Celsius, Fahrenheit, Kelvin, ... with respect to each other simply because they were all (arbitrarily) made to be linear with the expansion of water?

Thank you very much!

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D H
Staff Emeritus
mr. vodka,

First a minor correction: Celsius' temperature scale had zero as the boiling point of water at sea level pressure and 100 as the freezing point. Linnaeus later reversed this scale so that the freezing point was zero and the boiling point was 100.

People had been developing the concept of 'temperature' based on the expansion of liquids in a column for over a 100 years prior to Celsius. Regarding the use of water as the medium: Not a good idea, and that certainly is not what was used by Fahrenheit or Celsius or Linnaeus or any of the other 18th century physicists who worked on refining the concept of temperature.

Beyond the obvious reason that the freezing point of water is a bit too high (a water-based thermometer would be of little use in winter), there is an even bigger problem with using water as the liquid in a thermometer: Water does not undergo a linear expansion from 0 to 100 C. Ice floats.

The equal division of the temperature scale was derived from the ideal gas equation and therefore (at least in theory) "really equal".
AFAIK, when they measured temperature to high precision in the 19th century, a thing called constant volume gas thermometer was used. (Ideal gas equation , you know). And they knew that mercury and alcohol have different temperature dependency of expansion coefficients, so they corrected for it. (very good instruments came with correction tables).
I don´t know of "water thermometers" in scientific applications.

jtbell
Mentor
The equal division of the temperature scale was derived from the ideal gas equation and therefore (at least in theory) "really equal".
AFAIK, when they measured temperature to high precision in the 19th century, a thing called constant volume gas thermometer was used.
This was long after Fahrenheit and Celsius did their work and invented their temperature scales. They simply assumed, whether they consciously realized it or not, that the volume of their thermometric fluids varies linearly with temperature. They and the scientists who followed them were fortunate that this is a "good enough" assumption for most practical purposes, for the fluids that they used and the temperature range that they were measuring.

D H
Staff Emeritus
They simply assumed, whether they consciously realized it or not, that the volume of their thermometric fluids varies linearly with temperature.
I don't think they assumed that. Linnaeus' first thermometer (he contracted out the development to a scientific instrument manufacturer) looks remarkably modern. People had investigated various liquids as the basis for a thermometer for quite about hundred years before Linnaeus' time. Some of those liquid-based thermometers agreed quite nicely with one another, but others (e.g. water) did not. Water is obviously non-linear. It increases in density from 0 to 4 C and then decreases in density from 4 to 100 C.

One might intuitively say "well you got two points: two points define a line so now you can extrapolate the celsius temperature for any other system", but surely this is completely wrong? What kind of graph is one thinking of? The x-axis temperature, okay; and the y-axis?
Not all lines exist in 2 dimensions. In this case, think of the line being on the x-axis (or a number line) only.

EDIT: Actually, I should have stated this as, "Lines don't exist only in 2 dimensions."

Then you can assume that the origin is 0 deg C and 100 on the number line correlates to 100 deg C. Any other temperature scale can be shown on this same number line such that the freezing and boiling points of water coincide between scales.

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The Centigrade scale is not based on the expansion of water... water certainly doesn't increase linearly between its freezing point to its boiling point, and it actually shrinks going from 0 deg to about 4 deg.

"At one time "the temperature" was defined arbitrarily by dividing the expansion of water into even degrees of certain size"

This doesn't actually say that is how Celsius defined it, just that it once was defined that way.

The centigrade system simply makes 0 degrees the temperature at which water freezes and 100 deg the temp at which water boils, and 1 degree is defined as 1/100th of the temperature difference between those two points.
This is similar to the way Fahrenheit is defined and hence why they are linear to one another.

Thank you all for your responses!

Oh of course, water can't expand linearly especially with its weird behavior!

Anyway, I really don't get:
The centigrade system simply makes 0 degrees the temperature at which water freezes and 100 deg the temp at which water boils, and 1 degree is defined as 1/100th of the temperature difference between those two points.
How does this define anything? So if you have some water: what temperature is it at? The definition only tells you how to know if it is 0 or 100 degrees, but the degrees in between have to be given some physical significance, otherwise how do you even know when you've reached 1 degree? Am I really just missing something fundamental here? Saying "1 degree is defined as 1/100th of the temperature difference between those two points" just has no meaning what so ever for as far as I can see.

I'll keep this reply short, because the above paragraph stresses my main confusion: when you've pointed out the 0°C and 100°C, you still have to physically determine what corresponds to 1°C, 2°C etc, right? (again without using 2nd law or Stat. Mech) How could they have even done that? They must've used some sort of fluid and defined a temp that was linear with its expansion? (and if Fahrenheit used the same fluid as Celsius, then that would explain their linearity between each other, due to the transitivity of linearity)

maimonides suggests they used gases that were as ideal as possible and then defined temperature as to be linear with PV? (assuming N = constant) Does everyone agree?

So if you have some water: what temperature is it at? The definition only tells you how to know if it is 0 or 100 degrees, but the degrees in between have to be given some physical significance, otherwise how do you even know when you've reached 1 degree?
I'm not sure why this is so hard to understand:

If you took an unmarked (no scale) alcohol or mercury-based thermometer, for example, you could make marks at 0 and 100 deg C easily enough. You could then divide the space between those marks into 100 segments and call each division 1 deg C. Now, when you measure something that lies between 0 and 100 deg C, you simply read the scale on the thermometer.

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I'm not sure why this is so hard to understand:

If you took an unmarked (no scale) alcohol or mercury-based thermometer, for example, you could make marks at 0 and 100 deg C easily enough. You could then divide the space between those marks into 100 segments and call each division 1 deg C. Now, when you measure something that lies between 0 and 100 deg C, you simply read the scale on the thermometer.
This, essentially, but of course assuming whatever liquid you happen to be using does in fact increase in volume linearly with temperature.

D H
Staff Emeritus
So if you have some water: what temperature is it at? The definition only tells you how to know if it is 0 or 100 degrees, but the degrees in between have to be given some physical significance, otherwise how do you even know when you've reached 1 degree?
Simple. Stick a closed-bulb thermometer in the water. The level of the surface of the liquid in the thermometer will be somewhere between the rather arbitrary 0° and 100° markers. The 'temperature' is simply the 100 times the ratio of the height of the surface above the 0° mark to the distance between the 0° and 100° marks.

That is essentially all they were measuring 250+ years ago. Even the (now discarded) caloric theory (late 1700s) had not yet been developed in the time of Rømer, Fahrenheit, Celsius, and Linnaeus (late 1600s-mid 1700s). Thermodynamics was a development of the last half of the 19th century.

Don't fault these early scientists for not knowing concepts that weren't developed until a hundred years after they died!

This, essentially, but of course assuming whatever liquid you happen to be using does in fact increase in volume linearly with temperature.
If you're referring to my mention of alcohol and/or mercury, these liquids have already been shown to increase their volume linearly with temperature, which is why we use them in thermometers today.

Simple. Stick a closed-bulb thermometer in the water. The level of the surface of the liquid in the thermometer will be somewhere between the rather arbitrary 0° and 100° markers. The 'temperature' is simply the 100 times the ratio of the height of the surface above the 0° mark to the distance between the 0° and 100° marks.
This is essentially the same thing that I said, stated a slightly different way...

zgozvrm: do we have any reason to expect the scale you'd make with the alcohol and the mercury would be equal? (They might be shown to be so afterwards) I'm kind of bothered (not personally! I mean content-wise) by the reply of leakeg as it confuses me and seems wrong, but nobody seems to pick up on it, implying it's me that's wrong: "This, essentially, but of course assuming whatever liquid you happen to be using does in fact increase in volume linearly with temperature." But this doesn't make sense, it's assuming temperature exists before you define it, whilst it definition is totally arbitrary. There is no "volume that expands linearly with temperature" unless you've defined temperature that way for that volume.

D H - so you choose a certain liquid in your thermometer and you basically decide to define temperature as to make it linear with its volumetric expansion, right? I mean different fluids could result into different scales, maybe there is even a fluid A that expand exponentially when using a Celsius scale, but my point is: the person using Celsius would have no right in saying "the temperature based on fluid A rises exponentially with my temperature" no more than the person using the liquid A scale has in saying "the temperature based on (whatever Celsius used) rises logarithmically with my temperature".

The ideal gas definition was the answer to the "physical significance" question, once the concept of ideal gas had been developed. As jtbell pointed out, my answer was historically not correct. (Kelvin and the ideal gas were in the OP, so please forgive me.)

zgozvrm: do we have any reason to expect the scale you'd make with the alcohol and the mercury would be equal? (They might be shown to be so afterwards) I'm kind of bothered (not personally! I mean content-wise) by the reply of leakeg as it confuses me and seems wrong, but nobody seems to pick up on it, implying it's me that's wrong: "This, essentially, but of course assuming whatever liquid you happen to be using does in fact increase in volume linearly with temperature." But this doesn't make sense, it's assuming temperature exists before you define it, whilst it definition is totally arbitrary. There is no "volume that expands linearly with temperature" unless you've defined temperature that way for that volume.

D H - so you choose a certain liquid in your thermometer and you basically decide to define temperature as to make it linear with its volumetric expansion, right? I mean different fluids could result into different scales, maybe there is even a fluid A that expand exponentially when using a Celsius scale, but my point is: the person using Celsius would have no right in saying "the temperature based on fluid A rises exponentially with my temperature" no more than the person using the liquid A scale has in saying "the temperature based on (whatever Celsius used) rises logarithmically with my temperature".
Now you're deviating from your original question. You were trying to determine why all temperature scales are "linear to each other" and how you could define "other" temperatures between freezing and boiling.

I answered that question, as did D H.

Nobody said the scales between an alcohol-based and mercury-based thermometer would be equal. That is, you couldn't take 2 empty glass bulb thermometers and fill them with equal volumes of whatever alcohol/methanol mixture they use in thermometers and mercury and expect the scales on each to be of the same length or to "begin" at the same point along the container. Simply, I was stating that if you used one the known type of thermometers (there may be more, but for the sake of your original question, that is irrelevant), you could utilize the processes described by D H and myself to come up with a thermometer that works.

Yes, the different liquids probably expand at different (linear) rates and, therefore, you couldn't take one type of thermometer (say alcohol-based), empty it, and fill it with another type of liquid (say mercury) and expect it to read accurately.

Also, one cake may call for a teaspoon of vanilla extract and another may ask for a teaspoon of lemon juice. Obviously the cakes are different, but they're both still cakes, and you can cut them both and eat them both.

zgozvrm, you're not getting my point: you seem to assume all liquids expand linearly, as if there was a predefined linearity. This has much to do with my original question, because if there existed a predefiend linearity as you assume, then the linearity of Celsius, Fahrenheit etc with respect to each other would be evident. However, I would say there is no such thing, meaning their respective linearity is a great coincidence, unless there was some obvious liquid everybody used

alxm
IMO, the answer here is that they neither knew or assumed it was linear, they defined it to be linear.

To begin with: What is temperature? The best answer I've heard to that is the deceptively simple: "Temperature is something you measure with a thermometer."

The physical phenomenon we're talking about here is heat. We know from everyday experience all about heat and hot-vs-cold. Heat, being a form of energy, is something that exists. Temperature does not exist, it's an abstract measure of thermal energy which was defined into existence by the invention of the thermometer. Assume you know nothing about our current idea of 'temperature', and you're looking for a way to quantify heat. How'd you go about it? The answer is you'd need to find some object, some physical thing, which changed with temperature, and did so as regularly as possible. I.e. consistently got larger or smaller when the 'hotness' or 'coldness' we perceive went up or down. For reasons explained, the expansion of water, for instance, doesn't fit the bill for this.

Uncoincidentally the things that did were often behaving linearly with what we now call temperature (because they were exhibiting some form of ideal-gas-like behavior). That wasn't an arbitrary choice. We humans strongly prefer linear scales. Probably the most important property we wanted our temperature scale to have, was that if you mixed two equal amounts of the same liquid or gas at two different temperatures, the resulting mixture would have the average of the two temperatures. The temperature scale we use has that property.

In that respect, it's a useful scale. But from the modern, theoretical point of view, there's no real reason to say "temperature is linear". There aren't actually that many things that are linear in T. Most of the time you're seeing T accompanied by k in some exponential expression. A logarithmic temperature scale wouldn't make any less sense from the theoretical perspective.

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zgozvrm, you're not getting my point: you seem to assume all liquids expand linearly, as if there was a predefined linearity. This has much to do with my original question, because if there existed a predefiend linearity as you assume, then the linearity of Celsius, Fahrenheit etc with respect to each other would be evident. However, I would say there is no such thing, meaning their respective linearity is a great coincidence, unless there was some obvious liquid everybody used
No, you're missing my point! I never said that all liquids expand linearly. I merely suggested 2 types that have been shown to do so (which is why we use them in thermometers today).

I'm sure that in building the first thermometers, they first determined that this was the case.

And, the fact that different scales of temperature have any relationship to each other is not of great coincidence; they are all based on known values such as absolute zero, the boiling point of water, the freezing point of water, etc.

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.

IMO, the answer here is that they neither knew or assumed it was linear, they defined it to be linear.
Exactly! That is the way it was defined. It is based on 2 easily-reproducible temperatures.
We could have created some exponential scale to delineate the values between freezing and boiling, but most people don't think in those terms. It's much easier to think in linear terms.

D H
Staff Emeritus
do we have any reason to expect the scale you'd make with the alcohol and the mercury would be equal?
After the fact, yes.

Back in the early 1700s, by experimentation. People were experimenting with different liquids, and different liquids well away from their freezing points and boiling points do expand nearly linearly with temperature.

Having $0 US means having no money. Having$1 US means having an amount of money that can buy items that cost 1 dollar in the United States (based on the inflation rate, but let's not get into that).
We decided to divide the dollar into 100 cents and call the individual cent a penny.
How do we know that 48 pennies is actually equal to 48 cents?
We defined it that way.

We could have divide the dollar into 58 parts and called them "squipples."
It would then take 58 squipples to make one dollar.
Half a dollar would be 29 squipples.

We could even convert between squipples and cents: 1 cent = 0.58 squipples,
or 48 cents is equal to 27.84 squipples

Banks do the same thing ever day when they convert between one currency and another. All currencies are linear because we defined them that way.

Temperature scales are linear because we defined them that way.

With the Centigrade scale, we decide to call the freezing point of water 0 deg and the boiling point of water 100 deg
We also decide to divide it into 100 parts such that each part represents 1 deg C.

Again, I don't see why that is so hard to understand.

zgozvrm - Let me put it antoher way: what do you mean when you say "this certain liquid B expands linearly" before you've defined temperature?
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

alxm - Indeed, most of what you're saying is exactly what I was thinking.
Two things:
"Probably the most important property we wanted our temperature scale to have, was that if you mixed two equal amounts of the same liquid or gas at two different temperatures, the resulting mixture would have the average of the two temperatures. The temperature scale we use has that property." I agree we'd like that property, but how would that follow?

"A logarithmic temperature scale wouldn't make any less sense from the theoretical perspective." I suppose you that when you say "logarithmic temperature", you simply mean "with respect to our temperature", right?

D H - Hm, so could one say we basically looked at what kind of liquids behaved similarly, as in that when in contact with each other but with rising temperature their expansion was equal in percentages and we then partition all liquids into "linear-with-respect-to-each-other" groups and then take the biggest such group and define temperature according to its expansion?

EDIT (to the post above me): because with money it makes a unique sense to say "This cup costs as much as that cup" as this corresponds to being able to buy the new cup twice instead of the other cup once, but it has no meaning to say "This cup is twice as hot as that cup" (even using Kelvin, and I'm using hot as meaning temperature), after all what would it mean (not using the kinetic energy interpretation that came afterwards). Somebody might define temperature so that he would say about the same two cups "this cup is three times as hot as that cup" and you could bring nothing against it, while you could do so if someone had a certain currency and said "this cup costs three times as much as that cup": this would make no sense! This is because going from 0 to 1 dollar is the same as going from 100 to 101 dollars, while this has no meaning for temperature.

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alxm
"Probably the most important property we wanted our temperature scale to have, was that if you mixed two equal amounts of the same liquid or gas at two different temperatures, the resulting mixture would have the average of the two temperatures. The temperature scale we use has that property." I agree we'd like that property, but how would that follow?
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.

I suppose you that when you say "logarithmic temperature", you simply mean "with respect to our temperature", right?
Right.

D H
Staff Emeritus
IMO, the answer here is that they neither knew or assumed it was linear, they defined it to be linear.
Exactly.

To begin with: What is temperature? The best answer I've heard to that is the deceptively simple: "Temperature is something you measure with a thermometer."
That is such a good answer that it bears repeating:
To begin with: What is temperature? The best answer I've heard to that is the deceptively simple: "Temperature is something you measure with a thermometer."

After the fact, physicists found that this simple definition of temperature was very, very useful.
After the fact, physicists found that this simple definition of temperature had a very strong theoretical basis.
After the fact, physicists found that this simple definition needed some slight refinements to make it better match theory.

Before the fact, you are essentially asking why early 18th century physicists were so stupid. All of those after-the-fact events could not have taken place without some quantitative definition of temperature.