University of Illinois at Chicago Department of Physics. Thermodynamics & Statistical Mechanics Qualifying Examination

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1 Univrsity of Illinois at Chicago Dpartmnt of hysics hrmodynamics & tatistical Mchanics Qualifying Eamination January 9, am 1:00 pm Full crdit can b achivd from compltly corrct answrs to 4 qustions. If th studnt attmpts all 5 qustions, all of th answrs will b gradd, and th top 4 scors will b countd toward th am s total scor.

2 roblm 1 A vrtical cylindr contains n mols of an idal gas, and is closd off by a piston of mass M and ara A. h acclration du to gravity is g. h molar spcific hat Cv (at constant volum) of th gas is a constant indpndnt of tmpratur. h hat capacitis of th piston and th cylindr ar ngligibly small, and any frictional forcs btwn th piston and th cylindr walls can b nglctd. h whol systm is thrmally isolatd. Initially th piston is clampd in position so that th gas has a volum o and a tmpratur o. h piston is now rlasd and, aftr som oscillations, coms to rst in a final quilibrium stat corrsponding to a largr volum of th gas. Assum that th trnal air prssur is ngligibl. (a) Dos th tmpratur of th gas incras, dcras, or stay th sam. Eplain your answr. (b) Dos th ntropy of th gas incras, dcras, or rmain th sam? Eplain your answr. (c) Eprss th ratio f / f in trms of M, g, A, n and th gas constant R, whr f and f ar th final tmpratur and volum, rspctivly, of th gas. (d) What is th nt work don by th gas in this procss? Eprss your answr in trms of o, f, M, g, and A. () What is th final tmpratur f? Eprss your answr in trms of o, o, M, g, A, C v, n and R. roblm h diagram shows th nrgy flow in a hat ngin. (a) tat th dfinition of th fficincy of this hat ngin in trms of th hat Q h absorbd from th hot rsrvoir, and th hat Q c rjctd to th cold rsrvoir. Do not assum that th ngin is rvrsibl. (b) asd upon ntropy considrations, driv an inquality btwn th fficincy of th ngin, and th tmpraturs of th rsrvoirs. Hot Rsrvoir, h Q h Engin Q c Cold Rsrvoir, c W For parts (c)-(), rplac th rsrvoirs by two idntical but finit bodis, ach charactrizd by a hat capacity at constant prssur C, which is indpndnt of tmpratur. h bodis rmain at constant prssur and undrgo no chang of phas. Initially, thir tmpraturs ar 1 and, rspctivly finally, as a rsult of th opration of th hat ngin, th bodis will attain a common final tmpratur f.

3 (c) What is th total amount of work W don by th ngin? Do not assum that th ngin is opratd rvrsibly. Eprss your answr in trms of C, 1,, and f. (d) As in part (b), us ntropy considrations to driv an inquality rlating f to th initial tmpraturs 1 and. () For givn initial tmpraturs 1 and, what is th maimum amount of work obtainabl by oprating an ngin btwn ths two bodis? roblm 3 Considr a paramagntic matrial consisting of N non-intracting spin-1 particls (with r µ r magntic dipol momnt µ, z mh, m 0, ± 1). uppos that ths spins rsid h in (and ar in thrmal quilibrium with) a crystal lattic, which is in good thrmal contact with a rsrvoir at tmpratur. W apply a magntic fild of magnitud in th z- dirction. (a) Writ th partition function of th paramagntic systm in trms of k and µ. (b) How dos th avrag nrgy of th N spins vary with tmpratur? (c) Calculat th spin contribution to th ntropy of th crystal and show that it is only a function of µ / k. (d) What ar th limiting valus of th ntropy as 0 and as? How would your rsults chang if th systm consistd of spin-1/ particls? () h crystal is initially in contact with a rsrvoir at 1 and th magntic fild has a magnitud 1. h crystal is now thrmally isolatd, and th magntic fild strngth is slowly turnd down to a valu. Calculat th final tmpratur of th spins (and thrfor th tmpratur of th crystal lattic). roblm 4 (a) Considr a spinlss quantum mchanical particl of mass m in a simpl harmonic oscillator potntial in on dimnsion of frquncy ν and at tmpratur. h nrgy lvls for this particl ar givn by E n ( n +1/ ) hν. Writ down th partition function, Z 1, of a singl such particl in th oscillator. Eprss your rsult in trms of th dimnsionlss quantity α hν / k (b) Considr N spinlss, distinguishabl particls placd in this oscillator. Writ down th partition function Z N for N such particls. 3

4 (c) Now considr two idntical spinlss particls placd in this oscillator. Writ down th partition function Z for th two particls at tmpratur as an pansion in α 4 ξ up to and including ordr ξ. (d) Rpat part (c) for two idntical Frmi particls, of spin 1/, put in such an oscillator, on with spin up and th othr with spin down. () How dos th partition function from part (d) chang if th two frmions ar both in spin-up stats in th oscillator. roblm 5 h intrnal nrgy U and ntropy of an idal monatomic gas at tmpratur ar, rspctivly: U 3Nk / Nk [ ln( n / n) + 5 / ] Hr k is th oltzmann constant, N is th numbr of atoms, n N/ is th avrag concntration, and n ( Mk / πh ) 3 / o, with M th atomic mass. (a) how that th chmical potntial µ of th gas is rlatd to th prssur of th gas µ k ln n k. by ( ) o / wo such gass, X and Y, ar in quilibrium with surfac sits at which th gass bind to a mtal surfac. In th prsnc of X and Y simultanously thr ar just thr possibl configurations of ach surfac sit: (i) th surfac sit is mpty (dnotd by E blow) (ii) th surfac sit is occupid by on X atom, with nrgy ε rlativ to th mpty sit (iii) th surfac sit is occupid by on Y atom, with nrgy ε y rlativ to th mpty sit. Ecitd configurations at th sit ar not bound, and multipl occupancy is forbiddn. o E X Y (i) (ii) (iii) (b) h mtal surfac is at quilibrium with gass X and Y simultanously at tmpratur. In trms of th paramtrs dfind abov, and chmical potntials µ and µ y, writ down th grand partition function for th configurations of on sit. 4

5 5 (c) Calculat th probability that th sit is (i) mpty, (ii) occupid by X, (iii) occupid by Y. (d) At room tmpratur, and fid partial prssurs of y 0.5 atm, th sits ar occupid in th ratio n E :n X :n Y 1:1:. What is th nrgy diffrnc Y ε X ε ε? Ignor any diffrncs in th atomic masss M and M Y. Eprss your answr in units of k. () h partial prssurs ar now doubld to 1 atm ach. What is th nw valu of th ratio n E :n X :n Y? Equations and constants: k J/K N A R J/mol/K 1 atm N/m Hyprbolic functions sinh cosh + cosh sinh tanh Mawll's rlations