Conflicting definitions of temperature?

[Gerenuk]

Kelvin thus first defines an absolute temperature t = \int \mu (\theta) d\theta, which is now independent of the scale of \mu. It should be noted that t can indeed vary from (-\infty, +\infty) and has an arbitrary zero.

I didn't see it from the articles. Thanks for telling. Let's see...

What is \theta? How exactly do you measure to find \mu??
The expression \frac{V}{C_P}(T\alpha-1) contains temperature related properties, so they cannot be used before you know what temperature ist?!
Also you cannot know in advance that a process is isenthalpic?! For the Carnot definition one takes all reversible processes to be the isentropic, but here there is no way to pick only the right processes?

Also I don't see an argument why this definition should be material independent.

 

[Andy Resnick]

The idea that I have apparently failed to convey to you is that there are periods of time over which the net amount of energy radiated from a body is equal to the net amount of heat absorbed, and during such a steady state I may define a temperature as per the Onsager reciprocal relations. Do you still disagree?

Of course not- the Onsager relations, as is all of statistical mechanics, is valid for some phenomena. My point, which you are either ignoring or I am not being sufficiently clear in explaining- is that the statistical model of phenomena is *incomplete*. Using a statistical model of temperature (or any other physical quantity) is also *incomplete*. Furthermore, rather than simply giving up trying to develop a more general model that includes statistical mechanics as a limiting case, we should be trying to develop a more general ('better') model.

Which problem? The problem of developing a non-equilibrium statistical mechanics is certainly nowhere near solved, nor did I ever claim it was. I said the problem of defining "body temperature" was solved, as it is (effectively) a steady state problem and may be described by the Onsager reciprocal relations.

Again, how can you consider it solved when you are applying a result to conditions outside the region of validity? Onsager's relations are linear phenomenological relations. And please don't confuse my highlighting the assumptions inherent in statistical mechanics with questioning the validity of those assumptions.

 

[Andy Resnick]

I didn't see it from the articles. Thanks for telling. Let's see...

What is \theta? How exactly do you measure to find \mu??
The expression \frac{V}{C_P}(T\alpha-1) contains temperature related properties, so they cannot be used before you know what temperature ist?!
Also you cannot know in advance that a process is isenthalpic?! For the Carnot definition one takes all reversible processes to be the isentropic, but here there is no way to pick only the right processes?

Also I don't see an argument why this definition should be material independent.

This thread has been good- it's forced me to really dig down into some concepts I consider fundamental.

Ok- first, all those different symbols signifying 'temperature', which is fitting given the thread title. \theta is generally used to refer to the 'ideal gas temperature', and is measured by an ideal-gas themometer. 't' (or \tau when we need to use 't' for time) is Kelvin's first aboslute temeprate, and 'T' is Kelvin's second absolute temperature.

Next: the function \mu. Honestly, I do not have a clear understanding of what that is- the best derivation I have is from Truesdell's "The Tragicomical History of Thermodynamics 1822-1854". It's derived based on the heat generated though a (Carnot) cycle, and is thus material independent. The Carnot-Clapeyron theorem shows that

\mu \Lambda_{V} = \frac{\partial p}{\partial \theta}, where \Lambda_{V} is the latent heat at a specific volume- from this, one can generate the expression you presented. However, the original form of \mu is important because it is entirely *experimental*- measuring it allows a check on any theory regarding specific heats of real materials, the first law of thermodynamics, etc. etc.

In that context, there was quite a bit of experimental work by Clausius, Joule, Rankine, and Thompson to measure \mu for air, steam, etc. There's a lot of experimental results to sift through, Experimental measurements can be made isoenthalpic by simple insulation. One common criticism is that the measurements require an equation of state. However, by defining the various absolute temperatures the way they are, using different equations of state simply changes the relationship of \theta and T.

 

[Gerenuk]

Ok- first, all those different symbols signifying 'temperature', which is fitting given the thread title. \theta is generally used to refer to the 'ideal gas temperature', and is measured by an ideal-gas themometer.

That's not a good starting point, as it makes assumptions that you are able to find ideal gases. Also then the temperature cannot be applied to a deck of cards.

't' (or \tau when we need to use 't' for time) is Kelvin's first aboslute temeprate, and 'T' is Kelvin's second absolute temperature.

I have no idea what Kelvin did, but I played around with solely \mathrm{d}E=T\mathrm{d}S-p\mathrm{d}V and finally got the interesting equation
<br /> T(V,S)=T(V_0,S)\exp\left(-\int_{V_0,\text{const }S}^V \left(\frac{\partial p}{\partial E}\right)_V\mathrm{d}V\right)<br />
(and a similar for pressure) which can be used to determine the temperature for reversible processes.

\mu \Lambda_{V} = \frac{\partial p}{\partial \theta}, where \Lambda_{V} is the latent heat at a specific volume

I'm not sure how to derive that \Lambda_V from general assumptions.

One common criticism is that the measurements require an equation of state. However, by defining the various absolute temperatures the way they are, using different equations of state simply changes the relationship of \theta and T.

Oh, I see. With an equation of state I'd understand better and that would also be a big objection by me. I try to play around with equations a bit more to see if different equations of state just map the temperature to another function.

 

[Andy Resnick]

That's not a good starting point, as it makes assumptions that you are able to find ideal gases. Also then the temperature cannot be applied to a deck of cards.

Exactly! That said, the practical reality is that an air thermometer acts very much like an ideal gas thermometer for some range in temperature- recall the original goal of thermometry was to establish a method by which different thermometers could be compared. Not different air thermometers, but (say) an air thermometer and a mercury thermometer.

I'm not sure how to derive that \Lambda_V from general assumptions.

The 'fundamental' starting point of thermodynamics is that an object that absorbs a certain amount of heat Q experiences both a change in temperature and volume:

dQ/dt = \Lambda_{V} (V, \theta ) dV/dt + C_{V}(V, \theta ) d\theta /dt.

Where C_V is the specific heat at constant volume. Note that the latent heat, as written here, is not derived from assumptions regarding an equation of state. It's simply based on the observation that expanding or compressing a material involves the flow of heat, even at constant temperature. Note also that the above equation is a *dynamic* equation, in that time is explicit; much of 'thermodynamics' is really 'thermostatics' becasue the timedependence is supressed.