What is meant by "molecular temperature"?

Hello,

I got into a debate with someone about the term "molecular temperature."

I said "You can't define the temperature of a single molecule. It doesn't make sense."

And they said. "But look at all these papers where people use the term molecular temperature. They must mean the temperature of a single molecule."

I said "I don't know what they mean, but they can't mean that."

So can anyone out there answer this?

So can someone please explain what any of these people mean when they invoke the phrase "molecular temperature"?

I seriously doubt that they mean the temperature of a single molecule.

In my opinion, the term "molecular temperature" does make intuitive sense. A single molecule can have excited electronic states, vibrational states and rotational states. The molecule may also lie in an external potential well, in which it is bouncing around. All of these excitations contribute to the molecule's "temperature".

It is possible to define temperature for any system that has multiple ways to store a given amount of energy.

The key missing point in your understanding is that you don't know wat temperature is.

Is it some kind of measurement of the average energy? No, and this is a common mistake.

In order to understand what temperature is, you must understand what entropy is. Entropy is the number of ways a system can arrange a given amount of energy.

Think about a cold object. If I add to it some energy, then its entropy goes up significantly because so many particles share so little energy (lots of ways to arrange)/.

Think about a hot object. If I add to it some energy, its entropy hardly increases because the entropy is already so large.

Entropy is the key to understanding temperature. Once you see how we can define molecular entropy it is easy to define molecular temperature.

You are confusing thermal with statistical temperature.

Thermal temperature is a old concept, it is derived from thermal entropy and it applicable only to macroscopic bodies.

Statistical entropy is a modern quantum generalization like statistical temperature. It is applicable to single molecules, atoms, or even single atomic nucleus.

In fact, from quantum thermodynamics one can predict phase transitions on nucleus. For more information on molecular, atomic, and mesoscopic temperatures can see CPS: physchem/0309002 and references therein on nuclear studies (a copy will be available online in www.canonicalscience.com in brief).

Note: In the macroscopic limit both temperatures agree.

If absolute zero is when particles come to a complete hault, then would the hottest temperature be when those particles are zooming around at light speed? If so, what temperature is that?

Ok, of course a single molecule can be in a particular vibrational, rotational, electronic, and rotational state. At any given instance it will be one energy state for each of these things. My understanding about temperature is that it represents an average of these things (including translational kinetic energy and such) of large numbers of bodies. ...

But a molecule is itself an assembly of smaller bodies (nuclei and electrons). Consider that, for example, a large protein molecule may contain many thousands of constituent atoms.

So are you saying that a molecule can be large enough to contain so many different energy states that one could possibly measure a temperature somehow from this molecule?

As in a strand of DNA or cellulose or something. It could be so long and wrapped up in itself and be of such a size as to have a single temperature?

So besides all that, it doesn't seem to me that the authors of the C70 matter beam paper(s) are using the term "molecular temperature" this way. Do you think they mean the temperature of single fullerene molecules in the sense that we have discussed above?

This interferometric setup is now complemented by a
heating stage about 1 m behind the oven and 7 cm in
front of the first grating. It serves to vary the internal
energy of the molecules by photon absorption

How can you have statistics on one molecule, atom, or nucleus?

What is temperature if not a thermal measurement?

Yes, that is what I mean, except that the temperature concept is even applicable to molecules much smaller than DNA.

Here is an except from one of the fullerene beam papers:

Originally Posted by arxiv.org/quant-ph/0412003 This interferometric setup is now complemented by a heating stage about 1 m behind the oven and 7 cm in front of the first grating. It serves to [B vary the internal energy [/b]of the molecules by photon absorption

They appear to mean "temperature" as the excitation of individual molecules. This is C70 they are talking about, so each one is really a little sheet of graphite curled in on itself. It's practically a macroscopic object.

It appears that you have read books in equilibrium statistical thermodynamics or kinetics and are misunderstanding statistical in a broad sense with statistics of a population of molecules or atoms. It also appears that you believe that thermal temperature and kinetic temperature are the same that statistical temperature.

One can perfectly do statistics of a single molecule, atom, or nucleus, taking the different quantum states of that system like a "population".

The general formula for the entropy of a system is



where is the density matrix of the single molecule, atom, or nucleus and is the Boltzman constant.

One also know the energy of a single molecule, atom, or nucleus, therefore

I think that you error is in believing that entropy is a macroscopic quantity applicable only to a great number of molecules, what is not true.

From above definition of temperature one can obtain kinetic temperature (average of kinetic motion) and thermal temperature (internal energy by unit of entropy for an Avogadro number of molecules) like special cases.

We change the internal temperature
of the molecules in a controlled way before they enter a
near-field interferometer, and observe the corresponding
reduction of the interference contrast.

Decoherence of the fullerene matter waves can be induced by heating the molecules with multiple laser beams (514.5 nm, 40 µm waist radius, 0 − 10 W) before they enter the interferometer. The resulting molecular temperature can be assessed by detecting the heating dependent fraction of fullerene ions using the electron multiplier D1 over the heating stage.

The laser heating increases the molecular temperature by 140 K per absorbed photon. We calculate that they reach up to 5000 K for very short times, but the re-emission of thermal photons is so efficient that even the hottest molecules are cooled to below∼ 3000 K when they enter the interferometer 7.2 cm behind the heating stage.

FIG. 3: Spectral photon emission rate R of C70 molecules, as used for the calculation of thermal decoherence. We use the published [25] absorption cross-section for (S0 !S1) and a heat capacity of CV = 202kB. The fall-off to short wavelengths is determined by the limited internal energy of the molecules, while the decrease at long wavelengths is due to the lack of accessible radiative transitions at energies below 1.5 eV. The figure shows that in the absence of cooling a single molecule at 2500 K travelling at 190 m s−1 (that is, with a transit time of 4 ms through the interferometer) would emit an integrated number of three visible photons. This is sufficient to determine the path of the molecule if the emission occurs close to the second grating.