6.5 About Tem­per­a­ture

The pre­vi­ous sec­tion dis­cussed the wave func­tion for a macro­scopic sys­tem of bosons in its ground state. How­ever, that is re­ally a very the­o­ret­i­cal ex­er­cise.

A macro­scopic sys­tem of par­ti­cles is only in its ground state at what is called ab­solute zero tem­per­a­ture. Ab­solute zero tem­per­a­ture is $\vphantom0\raisebox{1.5pt}{$-$}$273.15 $\POW9,{\circ}$C in de­grees Cel­sius (Centi­grade) or $\vphantom0\raisebox{1.5pt}{$-$}$459.67 $\POW9,{\circ}$F in de­grees Fahren­heit. It is the cold­est that a sta­ble sys­tem could ever be.

Of course, you would hardly think some­thing spe­cial was go­ing on from the fact that it is $\vphantom0\raisebox{1.5pt}{$-$}$273.15 $\POW9,{\circ}$C or $\vphantom0\raisebox{1.5pt}{$-$}$459.67 $\POW9,{\circ}$F. That is why physi­cists have de­fined a more mean­ing­ful tem­per­a­ture scale than Centi­grade or Fahren­heit; the Kelvin scale. The Kelvin scale takes ab­solute zero tem­per­a­ture to be 0 K, zero de­grees Kelvin. A one de­gree tem­per­a­ture dif­fer­ence in Kelvin is still the same as in Centi­grade. So 1 K is the same as $\vphantom0\raisebox{1.5pt}{$-$}$272.15 $\POW9,{\circ}$C; both are one de­gree above ab­solute zero. Nor­mal am­bi­ent tem­per­a­tures are near 300 K. More pre­cisely, 300 K is equal to 27.15 $\POW9,{\circ}$C or 80.6 $\POW9,{\circ}$F.

A tem­per­a­ture mea­sured from ab­solute zero, like a tem­per­a­ture ex­pressed in Kelvin, is called an “ab­solute tem­per­a­ture.” Any the­o­ret­i­cal com­pu­ta­tion that you do re­quires the use of ab­solute tem­per­a­tures. (How­ever, there are some em­pir­i­cal re­la­tions and ta­bles that are mis­tak­enly phrased in terms of Cel­sius or Fahren­heit in­stead of in Kelvin.)

Ab­solute zero tem­per­a­ture is im­pos­si­ble to achieve ex­per­i­men­tally. Even get­ting close to it is very dif­fi­cult. There­fore, real macro­scopic sys­tems, even very cold ones, have an en­ergy no­tice­ably higher than their ground state. So they have a tem­per­a­ture above ab­solute zero.

But what ex­actly is that tem­per­a­ture? Con­sider the clas­si­cal pic­ture of a sub­stance, in which the mol­e­cules that it con­sists of are in con­stant chaotic ther­mal mo­tion. Tem­per­a­ture is of­ten de­scribed as a mea­sure of the trans­la­tional ki­netic en­ergy of this chaotic mo­tion. The higher the tem­per­a­ture, the larger the ther­mal mo­tion. In par­tic­u­lar, clas­si­cal sta­tis­ti­cal physics would say that the av­er­age ther­mal ki­netic en­ergy per par­ti­cle is equal to $\frac32{k_{\rm B}}T$, with $k_{\rm B}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1.38 10$\POW9,{-23}$ J/K the Boltz­mann con­stant and $T$ the ab­solute tem­per­a­ture in de­grees Kelvin.

Un­for­tu­nately, this story is only true for the trans­la­tional ki­netic en­ergy of the mol­e­cules in an ideal gas. For any other kind of sub­stance, or any other kind of ki­netic en­ergy, the quan­tum ef­fects are much too large to be ig­nored. Con­sider, for ex­am­ple, that the elec­tron in a hy­dro­gen atom has 13.6 eV worth of ki­netic en­ergy even at ab­solute zero tem­per­a­ture. (The bind­ing en­ergy also hap­pens to be 13.6 eV, {A.17}, even though phys­i­cally it is not the same thing.) Clas­si­cally that ki­netic en­ergy would cor­re­spond to a gi­gan­tic tem­per­a­ture of about 100 000 K. Not to 0 K. More gen­er­ally, the Heisen­berg un­cer­tainty prin­ci­ple says that par­ti­cles that are in any way con­fined must have ki­netic en­ergy even in the ground state. Only for an ideal gas is the con­tain­ing box big enough that it does not make a dif­fer­ence. Even then that is only true for the trans­la­tional de­grees of free­dom of the ideal gas mol­e­cules. Don’t look at their elec­trons or ro­ta­tional or vi­bra­tional mo­tion.

The truth is that tem­per­a­ture is not a mea­sure of ki­netic en­ergy. In­stead the tem­per­a­ture of a sys­tem is a mea­sure of its ca­pa­bil­ity to trans­fer ther­mal en­ergy to other sys­tems. By de­f­i­n­i­tion, if two sys­tems have the same tem­per­a­ture, nei­ther is able to trans­fer net ther­mal en­ergy to the other. It is said that the two sys­tems are in ther­mal equi­lib­rium with each other. If how­ever one sys­tem is hot­ter than the other, then if they are put in ther­mal con­tact, en­ergy will flow from the hot­ter sys­tem to the colder one. That will con­tinue un­til the tem­per­a­tures be­come equal. Trans­ferred ther­mal en­ergy is re­ferred to as “heat,” so it is said that heat flows from the hot­ter sys­tem to the colder.

The sim­plest ex­am­ple is for sys­tems in their ground state. If two sys­tems in their ground state are brought to­gether, no heat will trans­fer be­tween them. By de­f­i­n­i­tion the ground state is the state of low­est pos­si­ble en­ergy. There­fore nei­ther sys­tem has any spare en­ergy avail­able to trans­fer to the other sys­tem. It fol­lows that all sys­tems in their ground state have the same tem­per­a­ture. This tem­per­a­ture is sim­ply de­fined to be ab­solute zero tem­per­a­ture, 0 K. Sys­tems at ab­solute zero have zero ca­pa­bil­ity of trans­fer­ring heat to other sys­tems.

Sys­tems not in their ground state are not at zero tem­per­a­ture. Be­sides that, ba­si­cally all that can be said is that they still have the same tem­per­a­ture as any other sys­tem that they are in ther­mal equi­lib­rium with. But of course, this only de­fines equal­ity of tem­per­a­tures. It does not say what the value of that tem­per­a­ture is.

For iden­ti­fi­ca­tion and com­pu­ta­tional pur­poses, you would like to have a spe­cific nu­mer­i­cal value for the tem­per­a­ture of a given sys­tem. To get it, look at an ideal gas that the sys­tem is in ther­mal equi­lib­rium with. A nu­mer­i­cal value of the tem­per­a­ture can sim­ply be de­fined by de­mand­ing that the av­er­age trans­la­tional ki­netic en­ergy of the ideal gas mol­e­cules is equal to $\frac32{k_{\rm B}}T$, where $k_{\rm B}$ is the Boltz­mann con­stant, 1.380 65 10$\POW9,{-23}$ J/K. That ki­netic en­ergy can be de­duced from such eas­ily mea­sur­able quan­ti­ties as the pres­sure, vol­ume, and mass of the ideal gas.


Key Points
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A macro­scopic sys­tem is in its ground state if the ab­solute tem­per­a­ture is zero.

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Ab­solute zero tem­per­a­ture means 0 K (Kelvin), which is equal to $\vphantom0\raisebox{1.5pt}{$-$}$273.15 $\POW9,{\circ}$C (Centi­grade) or $\vphantom0\raisebox{1.5pt}{$-$}$459.67 $\POW9,{\circ}$F (Fahren­heit).

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Ab­solute zero tem­per­a­ture can never be fully achieved.

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If the tem­per­a­ture is greater than ab­solute zero, the sys­tem will have an en­ergy greater than that of the ground state.

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Tem­per­a­ture is not a mea­sure of the ther­mal ki­netic en­ergy of a sys­tem, ex­cept un­der very lim­ited con­di­tions in which there are no quan­tum ef­fects.

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In­stead the defin­ing prop­erty of tem­per­a­ture is that it is the same for sys­tems that are in ther­mal equi­lib­rium with each other.

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For sys­tems that are not in their ground state, a nu­mer­i­cal value for their tem­per­a­ture can be de­fined us­ing an ideal gas at the same tem­per­a­ture.