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Q & A: Can a water molecule have a temperature?

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Most recent answer: 06/17/2013
Can a water molecule have a temperature?
- Amy (age 16)

Not really, at least not at any particular time. A temperature can be thought of in several ways. One is as a measure of the ratio of how likely the system is to be in high energy states rather than low energy states. (The ratio is given by the Boltzmann factor exp(-ΔE/kT), where ΔE is the energy difference between the states, k is Boltzmann's factor, and T is the absolute temperature. Now in any snapshot of a small thing like a water molecule, no quantum state can look like the sort of distribution you get at some temperature. You need either a big collection of little parts or many different samples of the same little part over time to build up the sort of probability distribution needed. In principle, temperature doesn't quite become precisely defined except for infinite systems, but that formal limitation has no practical significance. A collection of maybe a couple dozen water molecules, each of which can have energy in several different forms, can start to get close to being well-described by a temperature. 

You may be familiar with another definition of T: the inverse of the derivative dS/dU, where S is entropy and U is internal energy. This is basically equivalent to the probability-based definition, and likewise only becomes well defined for large systems.

Mike W.

(published on 06/17/2013)

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