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What substance will have the largest specific heat capacity integrated from T=0 to, say, room temperature? In other words, given a finite amount of mass, what object or collection of objects has the largest number of degrees of freedom that can be excited as it absorbs energy starting from T=0? Would it be a complicated molecular polymer that can be tangled in all sorts of ways, or some kind of gas of low-mass particles, or maybe a spin lattice of some sort? Is there some kind of fundamental limit in the universe of the number of quantum degrees of freedom per mass or perhaps per volume that is allowed?

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(1) If you allow a very long time for the substance to warm up, go with a kilogram of $\nu_1$, the lightest neutrino.

(2) If you require that the substance heat up using electromagnetism (i.e. photons), then go with a kilogram of electrons.

(3) If you also require that the substance be electrically neutral and not decay, then a kilogram of hydrogen will fit the bill.

(4) If you further require that the substance be chemically stable, then molecular hydrogen is almost as good.

The reason these are the answers is because they have the lightest weights. So they have more degrees of freedom per kilogram.

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Dear Carl, +1, a good answer. – Luboš Motl Jan 24 '11 at 7:40
""So they have more degrees of freedom per kilogram."" Freedom per kilogram? Strange value. – Georg Jan 24 '11 at 11:33
A kilogramm of electrons were a Fermi gas? Wouldn't they? The electrons in metals contibute almost nothing to specific heat at RT. – Georg Jan 24 '11 at 11:43
@Georg: Carl's unit is degrees-of-freedom per unit mass. Assuming an ideal gas, count the accessible modes (i.e. 3 degrees of freedom for each independent particle in a gas, two rotation modes and 1 vibrational mode for each diatomic molecule (assuming the temperature is high enough)....). By the equipartition theorem each mode takes (on average) a equal fraction of the energy. One way of looking at the temperature is as measure of the mean energy a single mode, so the specific heat scales by 1/(mass * number of modes per particle). But this doesn't prove that gasses are the "best" phase. – dmckee Jan 25 '11 at 2:36
If light particles are such great reservoirs of 'cold' then why do I put ethylene glycol in my car's radiator instead of hydrogen gas? That's a joke, but the point is that this still nags at me - I guess it would be more interesting to specify a fixed volume rather than a fixed mass. It seems like there might be some kind of fundamental limit on the 'potential cooling' power of a passive macroscopic object. – user1552 Jan 25 '11 at 4:08

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