Editing Absolute zero (section)

==Thermodynamics near absolute zero==
At temperatures near {{convert|0|K|C F}}, nearly all molecular motion ceases and Δ''S''&nbsp;=&nbsp;0 for any [[adiabatic process]], where ''S'' is the [[entropy]]. In such a circumstance, pure substances can (ideally) form [[perfect crystal]]s with no structural imperfections as ''T'' → 0. [[Max Planck]]'s strong form of the [[third law of thermodynamics]] states the entropy of a perfect crystal vanishes at absolute zero. The original [[Walther Nernst|Nernst]] ''[[Nernst heat theorem|heat theorem]]'' makes the weaker and less controversial claim that the entropy change for any [[isothermal process]] approaches zero as ''T'' → 0:
:<math> \lim_{T \to 0} \Delta S = 0 </math>

The implication is that the entropy of a perfect crystal approaches a constant value. An adiabat is a state with constant entropy, typically represented on a graph as a curve in a manner similar to isotherms and isobars.

<blockquote>The [[Third law of thermodynamics|Nernst postulate]] identifies the [[isothermal process|isotherm]] T&nbsp;=&nbsp;0 as coincident with the [[adiabat]] S&nbsp;=&nbsp;0, although other isotherms and adiabats are distinct. As no two adiabats intersect, no other adiabat can [[Line–line intersection|intersect]] the T&nbsp;=&nbsp;0 isotherm. Consequently no adiabatic process initiated at nonzero temperature can lead to zero temperature (≈&nbsp;Callen, pp.&nbsp;189–190).</blockquote>

A perfect crystal is one in which the internal [[lattice (group)|lattice]] structure extends uninterrupted in all directions. The perfect order can be represented by translational [[symmetry]] along three (not usually [[orthogonality|orthogonal]]) [[Cartesian coordinate system|axes]]. Every lattice element of the structure is in its proper place, whether it is a single atom or a molecular grouping. For [[chemical substance|substances]] that exist in two (or more) stable crystalline forms, such as diamond and [[graphite]] for [[carbon]], there is a kind of ''chemical degeneracy''. The question remains whether both can have zero entropy at ''T''&nbsp;=&nbsp;0 even though each is perfectly ordered.

Perfect crystals never occur in practice; imperfections, and even entire amorphous material inclusions, can and do get "frozen in" at low temperatures, so transitions to more stable states do not occur.

Using the [[Debye model]], the [[specific heat capacity|specific heat]] and entropy of a pure crystal are proportional to ''T''<sup>&nbsp;3</sup>, while the [[enthalpy]] and [[chemical potential]] are proportional to ''T''<sup>&nbsp;4</sup> (Guggenheim, p.&nbsp;111). These quantities drop toward their ''T''&nbsp;=&nbsp;0 limiting values and approach with ''zero'' slopes. For the specific heats at least, the limiting value itself is definitely zero, as borne out by experiments to below 10&nbsp;K. Even the less detailed [[Einstein solid|Einstein model]] shows this curious drop in specific heats. In fact, all specific heats vanish at absolute zero, not just those of crystals. Likewise for the coefficient of [[thermal expansion]]. [[Maxwell relations|Maxwell's relations]] show that various other quantities also vanish. These phenomena were unanticipated.

Since the relation between changes in [[Gibbs free energy]] (''G''), the enthalpy (''H'') and the entropy is

:<math> \Delta G = \Delta H - T \Delta S \,</math>

thus, as ''T'' decreases, Δ''G'' and Δ''H'' approach each other (so long as Δ''S'' is bounded). Experimentally, it is found that all spontaneous processes (including [[chemical reaction]]s) result in a decrease in ''G'' as they proceed toward [[thermodynamic equilibrium|equilibrium]]. If Δ''S'' and/or ''T'' are small, the condition Δ''G''&nbsp;<&nbsp;0 may imply that Δ''H''&nbsp;<&nbsp;0, which would indicate an [[exothermic]] reaction. However, this is not required; [[endothermic]] reactions can proceed spontaneously if the ''T''Δ''S'' term is large enough.

Moreover, the slopes of the [[derivative]]s of Δ''G'' and Δ''H'' converge and are equal to zero at ''T''&nbsp;=&nbsp;0. This ensures that Δ''G'' and Δ''H'' are nearly the same over a considerable range of temperatures and justifies the approximate [[empiricism|empirical]] Principle of Thomsen and Berthelot, which states that ''the equilibrium state to which a system proceeds is the one that evolves the greatest amount of heat'', i.e., an actual process is the ''most exothermic one'' (Callen, pp.&nbsp;186–187).

One model that estimates the properties of an [[electron]] gas at absolute zero in metals is the [[Fermi gas]]. The electrons, being [[fermion]]s, must be in different quantum states, which leads the electrons to get very high typical [[velocities]], even at absolute zero. The maximum energy that electrons can have at absolute zero is called the [[Fermi energy]]. The Fermi temperature is defined as this maximum energy divided by the Boltzmann constant, and is on the order of 80,000 K for typical electron densities found in metals. For temperatures significantly below the Fermi temperature, the electrons behave in almost the same way as at absolute zero. This explains the failure of the classical [[equipartition theorem]] for metals that eluded classical physicists in the late 19th century.