Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
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4.4. DEGENERATE GAS. 79<br />
<br />
<br />
n<br />
F = NkBT log( ) − 1 + Fint(T, N) (4.49)<br />
nQ(T )<br />
Fint(T, N) = −NkBT log Zint(T ) (4.50)<br />
Experimental access to the internal degrees of freedom.<br />
For the entropy and internal energy we also find formulas th<strong>at</strong> show th<strong>at</strong><br />
these quantities are the sum of the entropy/energy for the motion of the center<br />
of mass plus a term pertaining to the internal degrees of freedom. Since we<br />
have assumed th<strong>at</strong> the energy of the internal motion does not depend on other<br />
particles, Fint does not depend on volume and the ideal gas law pV = NkT<br />
remains valid! The he<strong>at</strong> capacities do change, however, since the internal energy<br />
is different. Therefore, the value of the r<strong>at</strong>io γ is different. As we will<br />
see l<strong>at</strong>er, the value of this r<strong>at</strong>io is a direct measure of the number of active<br />
degrees of freedom. Hence by measuring the pressure-volume rel<strong>at</strong>ion in adiab<strong>at</strong>ic<br />
expansion gives us direct inform<strong>at</strong>ion about the internal structure of the<br />
molecules!<br />
Do we have them all?<br />
The entropy is the sum of the entropy associ<strong>at</strong>ed with the center of mass<br />
motion and the entropy of the internal st<strong>at</strong>e of the molecules. Hence the entropy<br />
is now larger than the entropy of mono-<strong>at</strong>omic molecules. We need more<br />
inform<strong>at</strong>ion to describe the st<strong>at</strong>e of large molecules, and hence we have more<br />
”un-knowledge”, which transl<strong>at</strong>es into a larger entropy.<br />
How do we know if we have all internal degrees of freedom accounted for?<br />
In principle, we do not know, because there could always be hidden degrees of<br />
freedom. For example, do we have to separ<strong>at</strong>e electrons and nucleï? Neutrons<br />
and protons? Quarks? Hence the real value of the entropy could even be larger<br />
as wh<strong>at</strong> we calcul<strong>at</strong>ed here. We can also ask the question from an experimental<br />
point of view. In th<strong>at</strong> case, the answer is simple, cool the system down to a low<br />
temper<strong>at</strong>ure. But wh<strong>at</strong> is low? Sometimes we see phase transitions <strong>at</strong> very low<br />
temper<strong>at</strong>ures, and we are never sure if we went low enough. For example, the<br />
nuclear spins could order below a certain temper<strong>at</strong>ure, and the values of such<br />
temper<strong>at</strong>ures are small indeed.<br />
4.4 Degener<strong>at</strong>e gas.<br />
The entropy of an ideal or Boltzmann gas is given by the Sackur-Tetrode formula.<br />
The temper<strong>at</strong>ure enters this equ<strong>at</strong>ion through the quantum concentr<strong>at</strong>ion.<br />
In the limit T → 0, the entropy is approxim<strong>at</strong>ely 3<br />
2NkB log(T ) and approaches<br />
−∞, which is obviously incorrect. The Sackur-Tetrode formula is not