12

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?

1 Answers1

7

(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.

Carl Brannen
  • 12,811
  • Dear Carl, +1, a good answer. – Luboš Motl Jan 24 '11 at 07:40
  • ""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 --- ex-moderator kitten Jan 25 '11 at 02:36
  • 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. –  Jan 25 '11 at 04:08
  • 1
    @BrianC: At least part of the answer is that for a heat transfer system you want heat capacity per unit volume. And transfer is generally more efficient in a dense material as well. – dmckee --- ex-moderator kitten Jan 25 '11 at 05:45
  • @dmckee, I am beginning to realize that this seems to be some wording I do not know. 1st, I am used to do thermodynamics on the molar scale, not per particle or mass. 2nd Im not used to assign degrees of freedom to any particle and sum up afterwards and then to call the sum "degrees of freedom". I am not aware of the correct English expressions in that case, I assume something like "Zustandssumme" is meant. I'll try not to shiver, when reading that wording in future. – Georg Jan 25 '11 at 13:16
  • BrianC, indeed, an example of hydrogen being used as coolant is in the space shuttle main engines: http://en.wikipedia.org/wiki/Space_Shuttle_main_engine . And dmckee, re "degrees of freedom" in the context of statistical mechanics see the section with that title in: http://en.wikipedia.org/wiki/Heat_capacity – Carl Brannen Jan 25 '11 at 23:45
  • Zustanssumme is partition function in English which is related to, but not the same thing, as degrees of freedom. Assuming you speak German, Freiheitsgrad is a translation for degree of freedom (according to wikipedia, the infallible source of information) –  Dec 09 '13 at 20:46
  • Might be worth noting that this answer is based on the equipartition theorem which says that if each particle has f degrees of freedom to store energy, then the total internal energy U of a system of N identical particles at temp. T is $U=(f/2)NkT$. So, if you add heat Q to this system, Q is related to $\Delta T$ by $Q = (f/2)Nk\Delta T$, and heat capacity at constant volume is $C_V = Q/\Delta T$ so this gives $C_V = (f/2)Nk$. And specific heat capacity is just $C_V$ divided by the system's mass M, or (f/2)(N/M)k, so to maximize it you should maximize N/M, number of particles per unit mass. – Hypnosifl Jan 09 '15 at 20:51
  • Also, wouldn't molecular Hydrogen be better than individual Hydrogen atoms even if you didn't care about chemical stability? A monoatomic molecule has 3 degrees of freedom so two individual Hydrogen would have 6, whereas a diatomic molecule like H2 has 7 degrees of freedom (3 based on translational kinetic energy, 2 based on rotational kinetic energy and 2 based on kinetic and potential energy due to vibrations which change the distance between atoms), as mentioned on p. 2 of this pdf. I also wonder about plasma, with free protons and electrons... – Hypnosifl Jan 09 '15 at 20:53