How to define/measure temperature?

[Gerenuk]

What is a general definition of temperature that can be applied to an arbitrary system?

My best guess so far was: Most systems in equilibrium will follow the Boltzmann distribution. If they do, then temperature is defined by P\propto e^{-E/T}

Unacceptable answers are:
Temperature is the mean kinetic energy. That is only correct for special systems where g(E)\propto E^c!
Temperature is \frac{\partial E}{\partial S}. This "definition" is not helpful as now entropy is an unknown.

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How can temperature be measured?

My best guess:
Examine the time-dependent or ensemble distributions of energies and fit the result to an exponential law e^{-E/T} to deduce the exponent and thus the temperature.
OR
Find an ideal gas where the allowed volumes are uncorrelated with energy and uniformly distributed. For such a system one can show that pV/T=\text{const}. Assuming constant pressure one can measure temperature by observing the change in volume.
OR
Find a system where you believe it has g(E)\propto E^c and measure the mean kinetic energy.

The last two methods rely on finding an ideal system that doesn't neccessarily exist.
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Bonus question:
How can the chemical potential be measured?

 

[Andy Resnick]

I'm not sure "temperature" can be applied to arbitrary systems. For example, systems not in thermal equilibrium (rapidly heated materials, possibly glassy states), or for nonthermal electromagnetic field distributions (i.e. laser light).

The most general definition I have for temperature is how 'hot' the object is. That's not very quantitative, but does allow for rank ordering, for example.

As for measuring temperature, that's what a thermometer does. Putting numbers on the side of the thermometer (or converting the voltages of a thermocouple) means defining a scale, and the Kelvin scale (absolute temperature) is material-independent.

The chemical potential (and the Gibbs free energy) are measured using calorimeters (sometimes differential scanning calorimeters, microcalorimeters...), but the data is obtained the same way- allowing a thermodynamic process to occur within a sealed, insulated chamber and measuring the resultant change in temperature of the insulating water bath.

 

[Gerenuk]

As you say, temperature cannot be applied to non-equilibrium system in a "basic" way, but abstract systems (which have nothing to do with a gas or solid) can be in equilibrium and have a temperature.

Saying how "hot" an object is, still doesn't explain what temperature is :) Define "hot".

If you put arbitrary numbers on the side of the thermometer, then there will be a discrepancy. And even if all people take the same thermometer, then there wouldn't be simple physical laws with temperature if its concept depends on an artificial measuring device that might be non-linear or whatever.

 

[flatmaster]

I was under the impression that the definition of temperature was determined by the location of the black-body peak. After all, this is what allows us to give the CMB a temperature, and it's not a physical object.

 

[Andy Resnick]

As you say, temperature cannot be applied to non-equilibrium system in a "basic" way, but abstract systems (which have nothing to do with a gas or solid) can be in equilibrium and have a temperature.

True enough, but your original question was for *arbitrary* systems, not specifically systems in equilibrium.

Saying how "hot" an object is, still doesn't explain what temperature is :) Define "hot".

Point taken, but that's not entirely fair- 'hotness' is a very intuitive concept, along with 'time' and 'space'. I see no reason why hotness can't be a primitive concept.

If you put arbitrary numbers on the side of the thermometer, then there will be a discrepancy. And even if all people take the same thermometer, then there wouldn't be simple physical laws with temperature if its concept depends on an artificial measuring device that might be non-linear or whatever.

Ahhh... but in order for a thermometer to be useful, the numbers *cannot* be arbitrary. The Joule-Thompson experiment was so important precisely becasue it demonstrated the existence of an absolute scale.

 

[Gerenuk]

I find your comment unscientific. I will try to answer you questions, but maybe there is no common ground for a discussion.

True enough, but your original question was for *arbitrary* systems, not specifically systems in equilibrium.

For an arbitrary system (stock markets, ...) as far as possible. It has to be in equilibirum, but that definition is still better, than restricting oneself to gases, fluid and solids only.

Point taken, but that's not entirely fair- 'hotness' is a very intuitive concept, along with 'time' and 'space'. I see no reason why hotness can't be a primitive concept.

Hotness is just a word. What is hotness? You are in the woods without a thermometer - how would you find a number signifying hotness. One has to be more specific.

Time and space is actually defined referring to universal reference processes and lengths. The question is still how temperature should be defined so that every scientist measures the same value.

Ahhh... but in order for a thermometer to be useful, the numbers *cannot* be arbitrary. The Joule-Thompson experiment was so important precisely becasue it demonstrated the existence of an absolute scale.

Please be more specific. How would you measure the temperature of some body in front of you without using an existing thermometer?

 

[Andy Resnick]

I find your comment unscientific. I will try to answer you questions, but maybe there is no common ground for a discussion.

For an arbitrary system (stock markets, ...) as far as possible. It has to be in equilibirum, but that definition is still better, than restricting oneself to gases, fluid and solids only.

Hotness is just a word. What is hotness? You are in the woods without a thermometer - how would you find a number signifying hotness. One has to be more specific.

Time and space is actually defined referring to universal reference processes and lengths. The question is still how temperature should be defined so that every scientist measures the same value.

Please be more specific. How would you measure the temperature of some body in front of you without using an existing thermometer?

I'm not really sure what I have to convince you *of*. I thought you were trying to gain a deeper understanding of temperature. Asking how one measures temperature without a thermometer is the same as asking how to measure distance without a ruler, or time without a stopwatch.

What's the temperature of a sandpile? Of a hard-sphere fluid, when the spheres are bowling balls in micro-gravity and the packing fraction near the crystallization point? The actin cytoskeleton of a filipodium of a cell? If you understand thermodynamics, you will understand why I mention these specific systems.

Kelvin wrote a paper back in 1848 which lay the foundations of thermometry, wherin he used Carnot's theory of motive power to establish an absolute interval of a degree. Perhaps you could read it...

 

[Dadface]

I think that the Zeroth law,which refers to thermal equilibrium, shows that temperature can be a well defined physical quantity.The establishment of a temperature scale is rather arbitary in that ,for example we can use the triple point of water and assign to this a temperature value.I suppose that water is chosen because of its easy availability,the ease with which it can be purified and so on.

 

[Gerenuk]

I'm not really sure what I have to convince you *of*. I thought you were trying to gain a deeper understanding of temperature. Asking how one measures temperature without a thermometer is the same as asking how to measure distance without a ruler, or time without a stopwatch.

Distance can be measure without a ruler. Just pick an object and count how much it fits into the object.

I'm convinced that you have an immense amount of knowledge, but I'm even more convinced that you don't understand much of it by yourself. It's probably good for teaching students, but not good for answering beyond textbook questions.

This is not getting anywhere, so I will wait for better answers from other people.

 

[Dadface]

The most accurate thermometer is the constant volume gas thermometer and uses the ideal gas equation.It works on the principle that as P tends to zero the gas approaches perfect ideal behaviour.P is measured at the triple point and at the unknown temperature and T calculated.The whole process is then repeated several times with increasingly reducing amounts of gas in the thermometer so that P reduces.The results are then extrapolated graphically to P= zero.It is a cumbersome instrument ,time consuming to use and used mainly for calibrating other thermometers.

 

[Gerenuk]

The most accurate thermometer is the constant volume gas thermometer and uses the ideal gas equation.It works on the principle that as P tends to zero the gas approaches perfect ideal behaviour.P is measured at the triple point and at the unknown temperature and T calculated.The whole process is then repeated several times with increasingly reducing amounts of gas in the thermometer so that P reduces.The results are then extrapolated graphically to P= zero.It is a cumbersome instrument ,time consuming to use and used mainly for calibrating other thermometers.

Thanks for this detailed answer. I didn't know of the complications.
My objection to using the ideal gas law was, that no-one really knows if the gas is really ideal. But using a gas with low pressure is probably the best one can do :)
So good to know how it's done for real.