fiziks Institute for NET/JRF, GATE, IIT JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics THERMODYNAMICS AND STATISTICAL PHYSICS

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1 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics HERMODYAMICS AD SAISICAL PHYSICS E/RF (UE-) Q. Considr th transition of liquid watr to stam as watr boils at a tmpratur of C undr a prssur of atmosphr. Which on of th following quantitis dos not chang discontinuously at th transition? h Gibbs fr nrgy h intrnal nrgy h ntropy h spcific volum Solution: In first ordr transition Gibbs fr nrgy is continuous. Q. A particl is confind to th rgion x by a potntial which incrass linarly as x u u x. h man position of th particl at tmpratur is k k / u u k u k u p mk k Solution: Partition function Z dp dx h x p ux m x dp dx p u x m dp dx ux and x xp xdxdpx x x x dx dx u t t dt t dt u u Q. A cavity contains blackbody radiation in quilibrium at tmpratur. h spcific hat pr unit volum of th photon gas in th cavity is of th form C, whr is a constant. h cavity is xpandd to twic its original volum and thn allowd to quilibrat at th sam tmpratur. h nw intrnal nrgy pr unit volum is Solution: du C v d du H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

2 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics Q. Considr a systm of non-intracting spins, ach of which has classical magntic momnt of magnitud. h Hamiltonian of this systm in an xtrnal magntic fild H is i. H i spin at tmpratur is, whr i is th magntic momnt of th i th spin. h magntization pr H H k coth k k H sinh H k H tanh k H cos cosxp sin d d Solution: For classical limit M H cos xp sin dd H M coth H Q5. Considr an idal os gas in thr dimnsions with th nrgy-momntum rlation s p with s. h rang of s for which this systm may undrgo a os-einstin condnsation at a non-zro tmpratur is s s s s E/RF (DEC-) bs Q6. h intrnal nrgy E of a systm is givn by E, whr b is a constant and othr symbols hav thir usual maning. h tmpratur of this systm is qual to bs bs bs S Solution: ds de Pd de ds Pd E S bs H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

3 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics Q7. Considr a Maxwllian distribution of th vlocity of th molculs of an idal gas. Lt mp and rms dnot th most probabl vlocity and th root man squar vlocity, rspctivly. h magnitud of th ratio / is / / / mp rms Solution: For Maxwllian distribution mp, rms m m Q8. If th numbr dnsity of a fr lctron gas in thr dimnsions is incrasd ight tims, its Frmi tmpratur will incras by a factor of dcras by a factor of incras by a factor of 8 dcras by a factor of 8 Solution: Frmi nrgy E F g, whr is numbr dnsity and g is dgnracy m E F K F F n F n n F n sinc 8. F n n Q9. A systm of non-intracting spin - particls is placd in an xtrnal magntic fild H. h bhavior of th ntropy of th systm as a function of nrgy is givn by S S mb rms H H E H E H S S H E H H E H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

4 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics S Solution: U U U U ln ln k, whr H. S is symmtrical about E. Q. A gas of non-intracting particls is in thrmal quilibrium at tmpratur. Each particl can b in any of th possibl non-dgnrat stats of nrgy, and. h avrag nrgy pr particl of th gas, whn, is / Solution: E, E, E, E whr. Z E o 6 Q. A on-dimnsional chain consists of a st of rods ach of lngth a. Whn strtchd by a load, ach rod can align ithr paralll or prpndicular to th lngth of th chain. h nrgy of a rod is whn prpndicular to it. Whn th chain is in thrmal quilibrium at tmpratur, its avrag lngth is a / a / k / k a / a Solution: Lt n no. of rods ar paralll and n no. of rods ar prpndicular. Enrgy of rod whn it is prpndicular Enrgy of rod whn it is paralll is. P Avrag lngth nap nap Sinc P P P and na na so n, n. a a H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

5 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics Q. h xcitations of a thr-dimnsional solid ar bosonic in natur with thir frquncy and wav-numbr k ar rlatd by k in th larg wavlngth limit. If th chmical potntial is zro, th bhavior of th spcific hat of th systm at low tmpratur is proportional to / s Solution: If disprsion rlation is k, At low tmpratur spcific hat /s / Q. Gas molculs of mass m ar confind in a cylindr of radius R and hight L (with R L ) kpt vrtically in th Earth s gravitational fild. h avrag nrgy of th gas at low tmpraturs (such that mgl k ) is givn by / / 5 / H Solution: Z dp xdpydpzdxdydz h px py pz L mgz m m m x y z Z dp dp dp dx dy dz mgz mk L Z R dz mgl k m Z R mg Z Z, E ln z 5, sinc mgl H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

6 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics E/RF (UE-) Q. Considr a systm of non-intracting particls in d dimnsional obying th disprsion s rlation Ak, whr is th nrgy, k is th wav vctor; s is an intgr and A is constant. h dnsity of stats, s d, is proportional to d s s d Solution: W can solv this problm with intuition for xampl Ak Dnsity of stat in -dimnsional (ε) Dnsity of stat in -dimnsional (ε) Dnsity of stat in -dimnsional (ε) s Dnsity of stat in d-dimnsional, whr Ak d Q5. h numbr of ways in which idntical bosons can b distributd in two nrgy lvls, is Solution: umbr of boson, umbr of nrgy lvl g d s s So numbr of ways to distribut boson into g lvl is, g W c sinc g. Q6. h fr nrgy of th gas of particls in a volum and at a tmpratur is 5 / F k ln a k /, whr a is a constant and k dnots th oltzmann constant. h intrnal nrgy of th gas is k 5 k k ln a k 5 / / k k ln a / k 5 / H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

7 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics Solution: F k ln a k 5 / /, F U S, U F S F df Sd Pd S or F k C 5 / ln whr ak C F S 5 / F U F F C 5 / ln 5 / F k C k k 5 / C 5 ln k C 5 / F 5 F k U F F 5 k. Q7. A systm has two normal mods of vibration, with frquncis and. What is th probability that at tmpratur, th systm has an nrgy lss than? [In th following x and Z is th partition function of th systm.] x / x x / Z x / x x / Z x / x / Z x / x x / Z Solution: hr is two normal mod so thr is two dgr of frdom. Enrgy of harmonic oscillator is E n n. E n n whr n...,,, and n,,,... 5 Ground stat nrgy E, first xcitd stat nrgy E. Scond xcitd stat 7 nrgy E which is doubly dgnrat stat so g, othr stat hav mor nrgy than. P 5 7 x x x E whr x Z Z Q8. os condnsation occurs in liquid tmpratur will os condnsation occur in. H kpt at ambint prssur at.7 K. At which H in gasous stat, th dnsity of which is tims smallr than that of liquid H? (Assum that it is a prfct os gas.).7 mk.7 mk.7 K.7 K H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

8 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics Solution: For bosons Q9. Considr black body radiation containd in a cavity whos walls ar at tmpratur. h radiation is in quilibrium with th walls of th cavity. If th tmpratur of th walls is incrasd to and th radiation is allowd to com to quilibrium at th nw tmpratur, th ntropy of th radiation incrass by a factor of Solution: For lack ody, nrgy is givn by F, 5 C S F 5 C 5 k. S, If tmprat incras from to thn ntropy will incas S to 8S. E/RF (DEC-) Q. h ntropy of a systm, S, is rlatd to th accssibl phas spac volum by S k ln E,, whr E, and ar th nrgy, numbr of particls and volum rspctivly. From this on can conclud that dos not chang during volution to quilibrium oscillats during volution to quilibrium is a maximum at quilibrium is a minimum at quilibrium Solution: Entropy is maximum at quilibrium. Q. Lt W b th work don in a quasistatic rvrsibl thrmodynamic procss. Which of th following statmnts about W is corrct? W is a prfct diffrntial if th procss is isothrmal W is a prfct diffrntial if th procss is adiabatic W is always a prfct diffrntial W cannot b a prfct diffrntial H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

9 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics Solution: Work don is prfct diffrntial in adiabatic procss. Q. h fr nrgy diffrnc btwn th suprconducting and th normal stats of a matrial is givn by F f S f whr is an ordr paramtr and and ar constants s.t. in ormal and in th supr conducting stat, whil always, minimum valu of F is Solution: F F Putting th valu, F F min Q. A givn quantity of gas is takn from th stat A C rvrsibly, by two paths, A C dirctly and A C as shown in th figur. During th procss A C th work don by th gas is and th hat 5 P A absorbd is 5. If during th procss A C th work don by th gas is, th hat absorbd is 8 8 Solution: During path AC, du dq dw 5 5 Sinc, intrnal nrgy is point function, so du will sam in all path In path AC, dq du dw 5 8. C Q. Considr a on-dimnsional Ising modl with spins, at vry low tmpraturs whn almost all spins ar alignd paralll to ach othr. hr will b a fw spin flips with ach flip costing an nrgy. In a configuration with r spin flips, th nrgy of th systm is E r and th numbr of configuration is ; r varis from to. h partition function is C r H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

10 fiziks Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics k / k Solution: Lt us considr only thr nrgy lvls, E. Q sinh k cosh k E r r i.. E, E and E E E C C C r Q cosh C r cosh Q cosh. E/RF (UE-) Q5. n grams of ic at C is addd to a bakr containing grams of watr at 5 C. What is th final tmpratur of th systm whn it coms to thrmal quilibrium? (h spcific hat of watr is C cal / gm / C and latnt hat of mlting of ic is 8 cal / gm ) 7.5 C.5 C Solution: h amount of hat rquird to mlt th ic of mass gm at C is.5 C QmL8 8Cal, whr L is th latnt hat of mlting of ic and m is th mass of th ic. h amount of hat availabl in watr of mass gm at QmC 5 75Cal v 5 C is Sinc th hat availabl is lss than th hat rquird to mlt th ic thrfor ic will not mlt as a rsult th tmpratur of th systm will b at C only. Q6. A vssl has two compartmnts of volum and, containing an idal gas at prssurs p and p, and tmpraturs and rspctivly. If th wall sparating th compartmnts is rmovd, th rsulting quilibrium tmpratur will b p p p p p / p / p p / H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

11 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics p p Solution:, n n n, U U U, nc v nc v nc v, n n n p p p p Q7. For tmpratur, th qualitativ tmpratur dpndnc of th probability distribution F v of th spd v of a molcul in thr dimnsions is corrctly rprsntd by th following figur: F(v) F(v) v v F(v) F(v) v Solution: Ara undr th F v is consrv and th man vlocity shift towards right for highr tmpratur. Q8. A systm of non-intracting spin-/ chargd particls ar placd in an xtrnal magntic fild. At low tmpratur, th lading bhavior of th xcss nrgy abov th ground stat nrgy, dpnds on as: ( c is a constant) v c H Solution: U tanh H c H Excss nrgy from th ground lvl H H H H c / c (is indpndnt of ) H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

12 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics H H H H H ( H) H H H H H H H H H At low tmpratur, th lowr valu, U C, whr C H. Q9. Considr a systm of two Ising spins S and S taking valus with intraction nrgy givn by S S, whn it is in thrmal quilibrium at tmpratur. For larg, th avrag nrgy of th systm varis asc / k, with C givn by Solution: h intraction nrgy is givn by E S S whr S and S taking valus. Possibl valus of th Enrgy of th systm ar E, E E, E Er Eg r r r U Er g r r U C (For larg, ) Q. Considr two diffrnt systms ach with thr idntical non-intracting particls. oth hav singl particl stats with nrgis, 5,. On systm is populatd by spin frmions and th othr by bosons. What is th valu of EF E whr E F and E ar th ground stat nrgis of th frmionic and bosonic systms rspctivly? and 6 Solution: Enrgy of Frmion = 5 Enrgy of boson = 5 EF E = H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

13 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics E/RF (DEC-) Q. hr idntical spin- frmions ar to b distributd in two non-dgnrat distinct nrgy lvls. h numbr of ways this can b don is 8 Solution: otal numbr of dgnracy g (umbr of nrgy stat (n))(umbr of dgnracy du to spin ( s )) n, s, g (. ) umbr of particl,. So numbr of ways, g c c Q. Considr th mlting transition of ic into watr at constant prssur. Which of th following thrmodynamic quantitis dos not xhibit a discontinuous chang across th phas transition? Intrnal nrgy Hlmholtz fr nrgy Gibbs fr nrgy ntropy Solution: Ic to watr: st ordr phas transition. So Gibbs fr nrgy is continuous, so it dosn t xhibit discontinuous chang. Q. wo diffrnt thrmodynamic systms ar dscribd by th following quations of stat: R 5R,,, and whr, and U ar rspctivly, th U U tmpraturs, th mol numbrs and th intrnal nrgis of th two systms, and R is th gas constant. Lt U tot dnot th total nrgy whn ths two systms ar put in contact and attain thrmal quilibrium. h ratio U U tot is Solution: R 5R and U U H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

14 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics () 5 ow Utot U U R R R U U tot 5R 5 At thrmal quilibrium, thus 5 Q. h spd v of th molculs of mass m of an idal gas obys Maxwll s vlocity distribution law at an quilibrium tmpratur. Lt vx, vy, v z dnot th componnts of th vlocity and and ar constants, is k th oltzmann constant. h avrag valu of v v k / m k / m k / m k / m Solution: Idal gas obys Maxwll vlocity distribution law at quilibrium tmpratur. hn avrag valu of v v x y x y, whr ow v v x v, v and x y y v v v x y v v v m x y z x v y hn v v x y v v v x y x v y v x vy m m m Q5. h ntropy S of a thrmodynamic systm as a function of nrgy S C E is givn by th following graph. h tmpraturs of th phass A, ndc dnotd by, and A A C rspctivly. E Satisfy th following inqualitis: C A A C C A A C H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

15 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics Solution: mpraturs of phas ar: A,, C Sinc, ds S C de ds A Hnc, will b slop, thn it will b zro for - phas de E So and in C and A phass, intrnal nrgy of C phas is mor, so C A ow C A Q6. A systm of classical non-intracting particls, ach of mass m, is at a tmpratur and is confind by th xtrnal potntial r Ar (whr A is a constant) in thr dimnsions. h intrnal nrgy of th systm is / A k k k ma k ln m m Solution: r Ar Ax y z it is harmonic oscillator. So its partition function will b z ln Z Intrnal nrgy, U Q7. A Carnot cycl oprats as a hat ngin btwn two bodis of qual hat capacity until thir tmpraturs bcom qual. If th initial tmpraturs of th bodis ar and, rspctivly and, thn thir common final tmpratur is / / Solution: For hat Carnot ngin th chang in ntropy for sourc and sink F d F F d F ds log and ds log F S ds ds log F log. Sinc, Carnot ngin is rvrsibl in natur, so F log F H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

16 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics E/RF (UE-) Q8. Which of th graphs blow givs th corrct qualitativ bhaviour of th nrgy dnsity r E of blackbody radiation of wavlngth at two tmpraturs and? E r E r E r E r Q9. A systm can hav thr nrgy lvls: E,. h lvl E is doubly dgnrat, whil th othrs ar non-dgnrat. h avrag nrgy at invrs tmpratur is tanh Solution: E,, E doubly dgnrat E i z g i ln z ln z tanh ow E ln z ln E tanh H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

17 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics Q. h fr nrgy F of a systm dpnds on a thrmodynamic variabl as F 6 a b with a, b. h valu of, whn th systm is in thrmodynamic quilibrium, is zro a / 6b / a / b / a / b / Solution: Frquncy F a b 6, a, b F is quilibrium i.. F, now F 5 a 6b F 5 a a a 6b b b Q. For a particular thrmodynamic systm th ntropy S is rlatd to th intrnal nrgy U Solution: and volum by S cu / / whr c is a constant. h Gibbs potntial G U S P for this systm is PU cu / S cu /, du ds Pd / zro US S U S c / / U U U c / / 5/ 5/ U S U S P P U c c S / 5/ U / / S / G U cu U U U U / c c Q. h prssur of a non-rlativistic fr Frmi gas in thr-dimnsions dpnds, at, on th dnsity of frmions n as 5 / n / / n / n / n H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

18 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics / Solution: Prssur P ne F, EF n, at / 5/ P nn n 5/ P n Q. h vandr Waals quation of stat for a gas is givn by whr P a b R P, and rprsnt th prssur, volum and tmpratur rspctivly, and a and b ar constant paramtrs. At th critical point, whr all th roots of th abov cubic quation ar dgnrat, th volum is givn by a 9b a 7b 8a 7bR b a P P P b R, for critical volum, a ab P Pb R P a ab P a 6ab P, a 6ab c b Solution: E/RF (DEC-) Q. h prssur P of a fluid is rlatd to its numbr dnsity by th quation of stat P a b whr a and b ar constants. If th initial volum of th fluid is, th work don on th systm whn it is comprssd, so as to incras th numbr dnsity from an initial valu of to is a a b a 7b a ln b Solution: n n P a b P a b n H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

19 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics d W Pd an bn d, whr, W naln b aln b Work don on th systm W ln, n a b Q5. An idal os gas is confind insid a containr that is connctd to a particl rsrvoir. Each particl can occupy a discrt st of singl-particl quantum stats. If th probability that a particular quantum stat is unoccupid is., thn th avrag numbr of bosons in that stat is 8 9 Q6. In low dnsity oxygn gas at low tmpratur, only th translational and rotational mods of th molculs ar xcitd. h spcific hat pr molcul of th gas is k k k Solution: otal D.O.F. = transition + rotation i.. f 5 5 U 5 U f C k Q7. Whn a gas xpands adiabatically from volum to by a quasi-static rvrsibl procss, it cools from tmpratur to. If now th sam procss is carrid out adiabatically and irrvrsibly, and is th tmpratur of th gas whn it has quilibratd, thn n n 5 k Q8. A random walkr taks a stp of unit lngth in th positiv dirction with probability / and a stp of unit lngth in th ngativ dirction with probability /. h man displacmnt of th walkr aftr n stps is n / n / 8 n / H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

20 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics Solution: P P For on stp, for n stp n Q9. A collction of non-intracting spins, i,,...,, S S is kpt in an i i xtrnal magntic fild at a tmpratur. h Hamiltonian of th systm is H is i. What should b th minimum valu of k S i? ln ln ln for which th man valu ln Solution: PS i, PS i S i Si tanh For particl S i tanh Si According to qustion, tanh ln E/RF (UE-5) Q5. A systm of non-intracting classical particls, ach of mass m is in a two dimnsional harmonic potntial of th form r x y whr is a positiv constant. h canonical partition function of th systm at tmpratur is : m m m m Ans. H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

21 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics Solution: r x y p x y p x y m m x y z dp dp dx dy h m m z h h m m z z h h Q5. A systm of distinguishabl particls, ach of which can b in on of th two nrgy lvls and, has a total nrgy n, whr n is an intgr. h ntropy of th systm is proportional to Ans. ln n nln Solution: o of ways for abov configuration is C n! ln n!! ln n! n! W n n Entropy= k ln n n Q5. h condition for th liquid and vapour phass of a fluid to b in quilibrium is givn by dp Q th approximat quation (Clausius-Claypron quation) whr v vap is th d v vap volum pr particl in th vapour phas, and Q is th latnt hat, which may b takn to b a constant. If th vapour obys idal gas law, which of th following plots is corrct? ln P ln P ln P O O O ln P O Ans. Solution: dp d Q v R dp QP dp Q d C ln P l l l, ap vap P d R P R H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

22 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics Q5. Considr thr Ising spins at th vrtics of a triangl which intract with ach othr with a frromagntic Ising intraction of strngth. h partition function of th systm at tmpratur is givn by : 6 6 Ans. 6 H S S S S S S Solution: S S S E z 6 cosh Q5. A larg numbr of rownian particls in on dimnsion start thir diffusiv motion from th origin at tim t. h diffusion cofficint is D. h numbr of particls crossing a point at a distanc L from th origin, pr unit tim, dpnds on L and tim t as Ans. Dt L Dt L Dt Dt L 6 Dt L Dt Dt L Solution: From Einstin Smoluchowski thory dx x pxdx xp Dt Dt L umbr of particl passing from point L at origin.xp Dt Dt H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

23 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics s Q55. An idal os gas in d -dimnsions obys th disprsion rlation k Ak Ans., whr A and s ar constants. For os-einstin condnsation to occur, th occupancy of xcitd stats ds s c d whr c is a constant, should rmain finit vn for. his can happn if d s d d s s d s ds s Solution: c d.e. condnsation is possibl in -D For matrlistic particl g For masslss particl d In both cass s d s d s s d s d g s s E/RF (DEC-5) Q56. h hat capacity of th intrior of a rfrigrator is. k / K. h minimum work that must b don to lowr th intrnal tmpratur from tmpratur is o 7 C will b o o 8 C to 7 C, whn th outsid. k.8 k. k. k Ans.: Q57. For a systm of indpndnt non intracting on-dimnsional oscillators, th valu of th Ans.: fr nrgy pr oscillator, in th limit, is H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

24 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics Solution: For th givn systm Z sinh F ln Z ln sinh ln ln ln ln F ln Q58. h partition function of a systm of Ising spins is Z whr and ar functions of tmpratur, but ar indpndnt of. If, th fr nrgy pr spin in th limit is ln Ans.: Solution: Z ln, F ln ln, it is givn F ln, F F ln ln ln ln Q59. h Hamiltonian of a systm of non intracting spin particls is z H Si, z whr S ar componnts of i th i spin along an xtrnal magntic fild. At a tmpratur such that. th spcific hat pr particl is 6 k 8 k ln ln 5 5 Ans.: Solution: For th givn systm E tanh 6 5 ln k k i C ksc h H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

25 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics k C ln 6 6 k k ln C k ln 5 k Q6. An nsmbl of non-intracting spin - particls is in contact with a hat bath at tmpratur and is subjctd to an xtrnal magntic fild. Each particl can b in on of th two quantum stats of nrgis. If th man nrgy pr particl is /, thn th fr nrgy pr particl is ln / ln / ln ln Ans.: Solution: For th givn systm, partition function, Z n cosh Man nrgy pr unit particl tanh put tanh ln ln ln F It is givn, ln cosh ln cosh b ac ln ln ln ln ln ln ln ln ln ln ln ln ln ln ln ln 6 ln 6 ln ln H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

26 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics F ln ln ln ln ln ln Q6. Which of th following graphs shows th qualitativ dpndnc of th fr nrgy f h, of a frromagnt in an xtrnal magntic fild h, and at a fixd tmpratur C, whr C is th critical tmpratur? f f f h h h Ans.: Solution: For supr conductor stat on will find two local minima f Option is corrct. h f h E/RF (UE-6) Q6. h spcific hat pr molcul of a gas of diatomic molculs at high tmpraturs is 8k.5k.5k k Solution: For high tmpratur all numbr ar xcitd so dgr of frdom for diatomic molcul is 7. f 7 Intrnal nrgy is, U U, C.5k Q6. Whn an idal monoatomic gas is xpandd adiabatically from an initial volum to, its tmpratur changs from to. hn th ratio is Solution: For adiabatic procss P k, k H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

27 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics 5 For monoatomic gas 5 v / Q6. A box of volum containing molculs of an idal gas, is dividd by a wall with a hol into two compartmnts. If th volum of th smallr compartmnt is, th varianc of th numbr of particls in it, is 9 Solution: Probability that on particl is in smallr compartmnt having volum, so p hr ar only two options ithr particl is in lft half or right half, so for on particl distribution is rnoulli for rnoulli s distribution p p. For particl distribution is p p, 9 Q65. A gas of non-rlativistic classical particls in on dimnsion is subjctd to a potntial x x (whr is a constant). h partition function is m m 8m m h h h h Solution: px x m x z dp dx h / m dx h / m h x z dx x H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

28 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics x x x dx dx dx z / m 8 m h h, put E of a systm at a fixd volum is found to dpnd on th Q66. h intrnal nrgy tmpratur as E a b. hn th ntropy S, as a function of tmpratur, is a b a b a Solution: From first law of thrmodynamics, ds de Pd, de ds Pd, it is givn d de ds ds de E a b de ad b d b a b ds ad b d b ad b d a Q67. Considr a gas of Cs atoms at a numbr dnsity of atoms/cc. whn th typical intrparticl distanc is qual to th thrmal d rogli wavlngth of th particls, th tmpratur of th gas is narst to (ak th mass of a Cs atom to b 6.7 kg ) 9 5 K 7 K K 8 K Solution: Whn d rogli wavlngth = thrmal wavlngth g/ z.6 m / / h n mk.6 h m.6 / h.6 / / 6.6 / / H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

29 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics / / 7.. / E/RF (DEC-6) Q68. h partition function of a two-lvl systm govrnd by th Hamiltonian H is sinh cosh cosh sinh cosh sinh Ans. : Solution: H h ign valu is givn by Z trac H E E and E E cosh Q69. Considr a gas of classical particls in a two-dimnsional squar box of sid L. If th total nrgy of th gas is E, th ntropy (apart from an additiv constant) is k ln LE k LE ln k L E ln Lk E ln Ans. : Solution: Z ml h E Z LE m Assum h H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

30 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics ln Z ln ln LE F ln Z E ln ln LE ln ln LE U F E F S k kln k kln L E LE L E L E k ln k ln k ln Q7. Considr a random walk on an infinit two-dimnsional triangular lattic, a part of which is shown in th figur blow. If th probabilitis of moving to any of th narst nighbour sits ar qual, what is th probability that th walkr rturns to th starting position at th nd of xactly thr stps? Ans. : Solution: For walkr to rturn to starting position it must mov along an quivalnt triangl in thr stps. For stps on any movmnt can rsult in quilatral triangl. For stp two, two out of six options will form quilatral triangl. For stp thr, only on out of six options will form quilatral triangl 6 otal probability Q7. An atom has a non-dgnrat ground-stat and a doubly-dgnrat xcitd stat. h nrgy diffrnc btwn th two stats is. h spcific hat at vry low tmpraturs Ans. : is givn by k k k k H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

31 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics Solution: Assum nrgy at ground stat is and nrgy at first xcitd stat is. h partition function is Z Enrgy Spcific hat, C U k k C k, Q7. h lctrons in graphn can b thought of as a two-dimnsional gas with a linar nrgy-momntum rlation E p v p p, p and v is a constant. If is th, whr x y numbr of lctrons pr unit ara, th nrgy pr unit ara is proportional to Ans. : / Solution: h numbr of k stat in rang k to k dk: L In D, it is givn by gkdk kdk Sinc, disprsion rlation is E Pv kv L g E de EdE L v v h numbr of lctron at K is EF EF / L gede EdE v L E. F v E v L F EF v L h avrag nrgy at K is H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

32 E fiziks Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics EF EF E g E de L LE F av E de v v L Eav v v v / L v / E av E v L L / E L / Q7. A thrmodynamic function E/RF (UE-7) GP,, US P is givn in trms of th intrnal nrgy U, tmpratur, ntropy S, prssur P, volum and th numbr of particls. Which of th following rlations is tru? (In th following is th chmical potntial.) Ans. : G S P, Solution: G U S P G S P, G P, G dg du ds sd Pd dp ds Pd ds Sd Pd dp dg Sd dp G, P S and G P, Q7. A box, sparatd by a movabl wall, has two compartmnts filld by a monoatomic gas CP of. Initially th volums of th two compartmnts ar qual, but th prssurs ar C P and P rspctivly. Whn th wall is allowd to mov, th final prssurs in th two compartmnts bcom qual. h final prssur is P, Ans. : P P Solution:,, / / P H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: / / P

33 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics P P, P P P P From (i) and (ii) / put th valu of in (i) / / / P / P P P / P / / / Q75. A gas of photons insid a cavity of volum is in quilibrium at tmpratur. If th Ans. : tmpratur of th cavity is changd to, th radiation prssur will chang by a factor of 6 8 Solution: For dimnsional systm P P P P P p 6 6P Q76. In a thrmodynamic systm in quilibrium, ach molcul can xist in thr possibl stats with probabilitis /, / and /6 rspctivly. h ntropy pr molcul is k ln k ln kln k ln kln k ln kln 6 Ans. : Solution: S k Pi ln Pi i P, P / and P /6 S k ln/ / ln/ / 6ln/ 6 k b 6 ln ln ln ln ln ln 6 H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

34 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics k ln ln ln ln 6 6 k ln ln ln ln 6 6 ln ln ln ln ln ln S k 6 6 k k ln ln 6 6 Q77. h singl particl nrgy lvls of a non-intracting thr-dimnsional isotropic systm, Ans. : lablld by momntum k, ar proportional to P to th nrgy dnsity at a fixd tmpratur, is k. h ratio P / of th avrag prssur / / s Solution: E p, whr p is momntum se P, whr P is prssur P s. E In problm, E k, so, s E E prssur P P at fixd. Q78. h Hamiltonian for thr Ising spins S, S and S, taking valus, is Ans. : H S S S If th systm is in quilibrium at tmpratur, th avrag nrgy of th systm, in trms of, is cosh cosh sinh / Solution: H S S S SS SS sinh cosh S S S S S S E H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /

35 Institut for E/RF, GAE, II AM, M.Sc. Entranc, ES, IFR and GRE in Physics E E g g E g Ei i i Ei g i Eg U U sinh U cosh 8sinh cosh H.o. D, Ground Floor, ia Sarai, ar II, Hauz Khas, w Dlhi 6 Phon: /