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A show on the weather channel said that as a water molecule ascends in the atmosphere it cools. Does it make sense to talk about the temperature of a single molecule?

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I think it makes sense to talk about the kinetic energy of a molecule, which is where the Kinetic Theory of Ideal Gases comes from: en.wikipedia.org/wiki/Kinetic_theory –  Greg May 24 '13 at 1:29
    
That in turn describes the temperature of a collection of molecules. –  Greg May 24 '13 at 1:34
    
The statistical mechanical definition of temperature is T = (∂E/∂S). Since entropy is directly related to the number of states, I suppose you could define a temperature for a molecule. Not sure how it'd be very useful though. –  Nick May 24 '13 at 4:09

4 Answers 4

I think it is a mistake, as often happens in popularizations of science.

A water or any molecule may lose kinetic energy and acquire potential energy, but it is the kinetic energy distribution that gives the temperature of an ensemble of molecules. The shape of the distribution shows that there will always be individual molecules at very high energy , in the ensemble, which they acquire from the random individual collisions.

From the link :

energy distr

and the shape shows that there always exist tails to high energies. The attribution of temperature labels to individual molecules is wrong.

shape

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Sure but if you have one molecule coupled to a thermostat and the system is ergodic then the distribution you are talking about can be thought as being a frequency with which each state is visited over a very long period of time. For once ergodicty is useful in this case –  gatsu May 24 '13 at 8:24

As the other answers have said, temperature is a collective property and can only be defined when you have an assemblage of particles. However by definition in a molecule you have an assemblage of atoms, and they have relative motions described by the vibrational excitations of the molecule.

So if you have a large enough molecule you can look at the excitations of its vibrational modes and use these to define a temperature. In effect what you're doing is saying that the excitation of the vibrational modes is the same as it would be if the molecule was in equilibrium with some enviroment of the defined temperature.

However I don't think this could usefully be applied to a water molecule. The vibrational excitations of water are of greater than thermal energy at ambient temperatures, and in any case there are only two normal modes. I suppose you could look at the rotation of the molecule, but this would give you only a rough guide to temperature.

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It makes sense if all you know about the molecule is its expected energy. Then you can show that it's energy distribution is the Boltzmann distribution $p(E) = e^{-E/kT}$ for some constant $T$, which is related to the expected energy.

So the question reduces to a philosophical view of probabilities. Does it make sense to assign probabilities to a deterministic system? If you accept probabilities as the reflection of your knowledge of the system rather than something intrinsic then it also makes sense to assign temperature to a single molecule.

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Thermodynamics makes sense when you have large numbers of particles. For example, the second law of thermodynamics has an extremely low probability of being violated when you have Avogadro's number's worth of particles. However, if you have a very small number of particles, the second law will frequently be violated.

This comes up in nuclear physics, where we routinely deal with nuclei consisting of 50 or 100 or 200 particles. Yes, we do talk about the temperature of an individual nucleus, and it does make sense.

However, a single water molecule is only 3 atoms, and in a system that size, it's nonsense to talk about temperature. In giant molecules, I can easily imagine that you could have enough atoms to talk about the temperature of an individual molecule.

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This sounds like an application of the "sorites problem"...as you add grains of sand to a collection one by one, when does it become a "heap". Similar issue here? –  Richardbernstein May 24 '13 at 13:21
    
@Richardbernstein: No, thermodynamics is just an approximation that becomes better and better as you have more particles. –  Ben Crowell May 25 '13 at 19:05

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