Appendix E Exercises
Chapter 1
1 Consider a model thermodynamic assembly in which the allowed (non-degenerate)
one-particle stateshave energies 0,ε, 2 ε,3ε, 4 ε,.... The assemblyhasfourdistin-
guishable (localized) particles and a total energyof 6 ε.Identifythenine possible
distributions,evaluateand work out theaveragedistribution of the four particles
in the energy states. (See also Chapter 5, question 1.)
Chapter 2
1 Verify (2.28), (a) by working through the outline derivation given in the text, and
(b)byusingittoderive an expressionforS[=−(∂F/∂T)V,N]whichisthe same
as that obtained bymethod 1 of section 2.5.
Chapter 3
1 Below what temperature will there be deviations of greater than 5 % from Curie’s
law (3.10), eveninanidealspin-^12 solid?
2 The magnetization of Pt nuclei (spin^12 )is commonlyused as a thermometric quan-
tity at very low temperatures. The measuring field is 10 mT, and the value ofμfor
Ptis 0. 60 μN. Estimate the usefulrangefor thethermometer, assuming, (a) thata
magnetization of less than 10−^4 of the maximum cannot be reliablymeasured, and
(b) that deviations from Curie’s law of greater than 5 % (see question 1) are unac-
ceptable. In practice NMR techniques are usedto single out the energysplitting;
what is the NMR frequency in this case?
3 Negative temperatures? Show thatfor thespin-^12 solidwiththermalenergyU(th),
the temperatureisgivenby 1 /T=(kkkB/ε)ln{[NNNε−U(th)]/U(th)}.Hence show
that negative temperatures would be reached ifU(th)>Nε/ 2.
(a) A madscientist,bowledover withthelogicof(3.7), suggests thatifshould be
possibletoachieve negative temperaturesinthespinsystembyan adiabatic
reversal of the applied field (i.e. demagnetization and remagnetization withB
reversed). Explainwhythis methodwillnot work.(Hints:Plot temperature
versus timefor the process; andFig. 3.7 should helpalso.)
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