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Q & A: meaning of temperature

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Most recent answer: 07/25/2012
Is temperature a microscopic or macroscopic concept?
- Anonymous
Itís more of a macroscopic concept. The definition of the absolute temperature is
T=dU/dS (taken at constant volume), where U is the total energy of some system and S is the entropy (a measure of how many quantum states are available to it) in conventional units. Now that derivative doesnít make any sense for really microscopic systems, at least at any particular time. In fact, if you specify what quantum state a system is in (a reasonable thing to do for an atom, for example), S=0 by definition. So T is really a meaningful quantity only for macroscopic systems. However, for a microscopic system exchanging energy with the outside over time one can describe the probabilities of the system being in its different states. If those probabilities follow the Boltzmann distribution, then one can speak of the T of the microscopic system. The reason for them to follow that distribution, however, is the energy exchange with the macroscopic environment. So something macroscopic ends up involved either way.

Mike W.

(published on 10/22/2007)

Follow-Up #1: meaning of temperature

More technically, is temperature really just a measure of the vibrational energy of a system? When the system vibrates, and if a sensor were place in it, the system now vibrates the sensor and via, either mercury, electricity, the sensor shows an increase. And then for humans, we assign a "temperature" to something based upon how we sense it's vibrations. A hot iron is said to have a high temperature because if we touch it, a high amount of vibrational energy migrates to our hand from the molecules and as it damages our hand by changing the chemical bonds (burning), we say it is "HOT!" (has a high temperature).
- Byron
Everything you describe is part of what's going on. However, T describes the distribution of lots of different kinds of energy, not just vibrational. For example, there are also thermal electromagnetic waves, including the glow you see from a hot coil. Also, large-scale organized vibrational energy, say the energy of a plucked string, is not part of the thermal energy.

Mike W.

(published on 03/07/2010)

Follow-Up #2: Energy, entropy, and temperature

Dear Mike W. First Q.: Does your A. intimate quality or quantity of energy in a system, as referring to your above definition of Absolute T, there appears that quality is involved? Second Q.: Regarding Entropy, could you please enlighten me about the way of how two independent systems could have the same Temp., yet having different Total Energies, (that which of course intimates quality of Energy). I read about this in P.W.Atkins' "The Second Law".ISBN 0-7167-6006-1. 1994.
- Francis F. Kish (age 89)
FLORE. ACT. Australia

I'm not sure I understand the first question. The definition of T involves the quantity of energy, but with the assumption that this energy has the quality of having being randomly distributed among the modes of the system.

It's of course easy for two systems to have different energies and the same T. Consider a little block of copper and a big block at the same T. Probably you're thinking of something less trivial. An example would be a kg of ice at 0°C and a kg of water at 0°C. There's more energy in the water but they're at the same T. This effect is possible at first-order phase transitions (like the liquid-solid transition here) where there's a coincidence in which the free-energies (which also include an entropy term) of two different forms happen to be equal.

Mike W.

(published on 07/25/2012)

Follow-up on this answer.