AppendixE 195Chapter 7
1 Calculate the percentage contribution of vibration to the heat capacity of O 2 gas at
room temperature (293 K). Thecharacteristic(Einstein) temperaturefor vibration
inO 2 is22 00 K.
2 Work out the characteristic temperatures of rotation (7.7) for, (a) O 2 ,(b)H 2 ,(c)
D 2 , (d) HD. The masses of H, D and O are respectively 1, 2 and 1 6 timesMMP.
TheinternucleardistanceinO 2 is 1.2 0 × 10 −^10 m, andinthehydrogengasesis
0 .7 5 × 10 −^10 m.
3 Consider the properties ofD 2. (a) Show that theAspin states (i.e. para-deuterium)
are associatedwithodd-lrotationalstates,andtheSspin states (ortho-deuterium)
with even-lrotational states. (b) What is the ortho:para ratio in deuterium at room
temperature? (c) Whatisthe equilibrium composition atlow temperatures?
4 Suppose thepartition functionZofanMBgas depends onVandTasZ∝VxTy.
FindPandCV.
Chapter 8
1 Check the derivation of (8.5). Hence work out the Fermi energyand Fermi tem-
perature of the conduction electrons in metallic copper. You may assume that the
conduction electronsbehave as anidealgas offree electrons, one electron per
atom. Themolar volume ofcopperis7cm^3 .Showfromacalculation ofthe Fermi
velocity of the electrons (defined bykkkF=mvF)that relativistic corrections are
notimportant.
2 Calculate the Fermitemperatureforliquid^3 He, assumingtheidealgas model.The
molar volume of liquid^3 He is 3 5 cm^3.
3 Verify(8.8), that the average energyper particleinanidealFDgasis^35 μ.Prove
inaddition that the average speedofthegas particles (atT=0)is^34 vF,where the
Fermi velocityvFis defined bymvF=kkkF.
4 Asemiconductor can oftenbemodelledas anidealFD gas,but withan ‘effective
mass’ ofthe carriersdifferingfrom thefree electron massm.Aparticular sample
of InSb has a carrier density of 10^22 m−^3 ,with effective mass 0.01 4 m.You may
assume that the carrierdensitydoes not varysignificantlywithtemperature (more or
less valid for this material).(a)Findμ( 0 )andTTTF.(b) Show that at room temperature
(say 300 K) MB statistics may just be safely used. (c) Estimateμat 300 K.(Hint:
Itwill benegative–why? Note that the power series methodofAppendix C cannot
be applied accuratelyhere.) How doesμvarywithTabove 300 K?
Chapter 9
1 Workthroughthe stepsleading to (9.3) and(9.4).
2 Estimate the condensation temperatureTTTBfor anidealBEgas ofthe samedensity
of^4 He atoms as occurs in liquid^4 He(molar volume 27 cm^3 ).
3 TheStefan–Boltzmannlaw states that the energyflux radiatedfrom ablackbody
at temperatureTisσT^4 Jm−^2 s−^1. Derive thislawfromfirst principles, assuming