11.1 Tem­per­a­ture

This book fre­quently uses the word tem­per­a­ture, but what does that re­ally mean? It is of­ten said that tem­per­a­ture is some mea­sure of the ki­netic en­ergy of the mol­e­cules, but that is a du­bi­ous state­ment. It is OK for a thin no­ble gas, where the ki­netic en­ergy per atom is $\frac32{k_{\rm B}}T$ with $k_{\rm B}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1.380 65 10$\POW9,{-23}$ J/K the Boltz­mann con­stant and $T$ the (ab­solute) tem­per­a­ture in de­grees Kelvin. But the va­lence elec­trons in an metal typ­i­cally have ki­netic en­er­gies many times greater than $\frac32{k_{\rm B}}T$. And when the ab­solute tem­per­a­ture be­comes zero, the ki­netic en­ergy of a sys­tem of par­ti­cles does not nor­mally be­come zero, since the un­cer­tainty prin­ci­ple does not al­low that.

In re­al­ity, the tem­per­a­ture of a sys­tem is not a mea­sure of its ther­mal ki­netic en­ergy, but of its hot­ness. So, to un­der­stand tem­per­a­ture, you first have to un­der­stand hot­ness. A sys­tem A is hot­ter than a sys­tem B, (and B is colder than A,) if heat en­ergy flows from A to B if they are brought into ther­mal con­tact. If no heat flows, A and B are equally hot. Tem­per­a­ture is a nu­mer­i­cal value de­fined so that, if two sys­tems A and B are equally hot, they have the same value for the tem­per­a­ture.

The so-called “ze­roth law of ther­mo­dy­nam­ics” en­sures that this de­f­i­n­i­tion makes sense. It says that if sys­tems A and B have the same tem­per­a­ture, and sys­tems B and C have the same tem­per­a­ture, then sys­tems A and C have the same tem­per­a­ture. Oth­er­wise sys­tem B would have two tem­per­a­tures: A and C would have dif­fer­ent tem­per­a­tures, and B would have the same tem­per­a­ture as each of them.

The sys­tems are sup­posed to be in ther­mal equi­lib­rium. For ex­am­ple, a solid chunk of mat­ter that is hot­ter on its in­side than its out­side sim­ply does not have a (sin­gle) tem­per­a­ture, so there is no point in talk­ing about it.

The re­quire­ment that sys­tems that are equally hot must have the same value of the tem­per­a­ture does not say any­thing about what that value must be. De­f­i­n­i­tions of the ac­tual val­ues have his­tor­i­cally var­ied. A good one is to com­pute the tem­per­a­ture of a sys­tem A us­ing an ideal gas B at equal tem­per­a­ture as sys­tem A. Then $\frac32{k_{\rm B}}T$ can sim­ply be de­fined to be the mean trans­la­tional ki­netic en­ergy of the mol­e­cules of ideal gas B. That ki­netic en­ergy, in turn, can be com­puted from the pres­sure and den­sity of the gas. With this de­f­i­n­i­tion of the tem­per­a­ture scale, the tem­per­a­ture is zero in the ground state of ideal gas B. The rea­son is that a highly ac­cu­rate ideal gas means very few atoms or mol­e­cules in a very roomy box. With the vast un­cer­tainty in po­si­tion that the roomy box pro­vides to the ground-state, the un­cer­tainty-de­manded ki­netic en­ergy is van­ish­ingly small. So ${k_{\rm B}}T$ will be zero.

It then fol­lows that all ground states are at ab­solute zero tem­per­a­ture, re­gard­less how large their ki­netic en­ergy. The rea­son is that all ground states must have the same tem­per­a­ture: if two sys­tems in their ground states are brought in ther­mal con­tact, no heat can flow: nei­ther ground state can sac­ri­fice any more en­ergy, the ground state en­ergy can­not be re­duced.

How­ever, the ideal gas ther­mome­ter is lim­ited by the fact that the tem­per­a­tures it can de­scribe must be pos­i­tive. There are some un­sta­ble sys­tems that in a tech­ni­cal and ap­prox­i­mate, but mean­ing­ful, sense have neg­a­tive ab­solute tem­per­a­tures [4]. Un­like what you might ex­pect, (aren’t neg­a­tive num­bers less than pos­i­tive ones?) such sys­tems are hot­ter than any nor­mal sys­tem. Sys­tems of neg­a­tive tem­per­a­ture will give off heat re­gard­less of how sear­ingly hot the nor­mal sys­tem that they are in con­tact with is.

In this chap­ter a de­f­i­n­i­tion of tem­per­a­ture scale will be given based on the quan­tum treat­ment. Var­i­ous equiv­a­lent de­f­i­n­i­tions will pop up. Even­tu­ally, sec­tion 11.14.4 will es­tab­lish it is the same as the ideal gas tem­per­a­ture scale.

You might won­der why the laws of ther­mo­dy­nam­ics are num­bered from zero. The rea­son is his­tor­i­cal; the first, sec­ond, and third laws were al­ready firmly es­tab­lished be­fore in the early twen­ti­eth cen­tury it was be­lat­edly rec­og­nized that an ex­plicit state­ment of the ze­roth law was re­ally needed. If you are al­ready fa­mil­iar with the sec­ond law, you might think it im­plies the ze­roth, but things are not quite that sim­ple.

What about these other laws? The “first law of ther­mo­dy­nam­ics” is sim­ply stolen from gen­eral physics; it states that en­ergy is con­served. The sec­ond and third laws will be de­scribed in sec­tions 11.8 through 11.10.