Statistical Physics, Second Revised and Enlarged Edition

112 Entropyin other situations

The calculationofSfor thissituationgoes asfollows:

S=kkkBln definition(1.5)

=kkkB

(

lnN!−

∑

L

lnNNL!

)

from(10.1)

=kkkB

(

NlnN−

∑

L

NNLlnNNL

)

Stirling’s approx., also

∑

NNNL=N

=−kkkB

[

∑

L

NNL(lnNNL−lnN)

]

puttingN=

∑

NNNL

=−NkkkB

∑

L

PLlnPL (10.2)

The answer (10.2) is a nice simple one. For instance if we were to have a 50–5 0

mixture of two isotopes, it wouldgive an ‘entropyof mixing’ofS=NkkkBln2,anot

unexpected result, analogous to tossing pennies (Appendix B).

Whether this entropyisobservableis another matter. Infact ourblockofcopper

does not separate out undergravityto have all the^65 Cu at the bottom and all the

(^63) Cu at the top, however cold it is made! Rather this disorder is frozen in, and a cold
piece ofcopperisin a metastable state whichcontains thisfixedamount ofdisorder.
Therefore the entropy of isotopic disorder is usually omitted from consideration,
sinceithas noinfluence on thethermalproperties. Metastable states ofthis sortdo
not violate thethirdlaw ofthermodynamics, since no entropychangesoccur near the
absolute zero.
Actually partialisotopic separationdoes occurinjust one case, that ofliquid helium.
A liquidmixture of^3 Heand^4 Heis entirelyrandom above 0.8 K. However, when
it is cooled below this temperature, a phase separation occurs to give a solution
ofalmost pure^3 He floating on top of a dilute solution of about 6 %^3 Hein^4 He.
Thisphase separationisimportant tolow-temperature physicists, notleastbecause
the difference in thermal properties of the^3 He between the two phases forms the
basis of a ‘dilution refrigerator’, the work-horse cooling method for reaching 5 mK
nowadays.
1 0.1.2 Localized particles
Asimilar calculation can be made for the entropyof an assemblyofNlocalizedpar-
ticles in thermal equilibrium, the topic of Chapter 2. We have seen (equation (2.4))
that,the totalnumber ofmicrostates, canbe replacedbyt∗,the number of
microstates associated with the most probable distribution. Hence the entropy is
givenbyS=kkkBlnt∗,witht∗=N!/nn∗j (equation (2.3)). The evaluation ofS