fiziks Institute for NET/JRF, GATE, IIT JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics Thermodynamics & Statistical Mechanics JEST-2012
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1 Q. monatomc dal gas at hrmodynamcs & Statstcal Mchancs JS- volum. h tmpratur aftr comprsson s ns. : (d) Soluton:. C (b) P costant, P R 7 C s adabatcally comprssd to /8 of ts orgnal 7 C (c).5 C (d) costant costant x 5 / mpratur n C Most approprat answr s opton (d) 887 C Q. Consdr a systm of partcls n thr dmnsons wth momntum p and nrgy c p, c bng a constant. h systm s mantand at nvrs tmpratur, volum and chmcal potntal. What s th grand partton functon of th systm? xp 8 / ch (c) xp 6 / ch ns. : Soluton: Canoncal partton functon, (b) 6 / ch 8 ch (d) / H N x y z z dp dp dp dxdydz h, pc z N h 4 p Grand canoncal partton functon, xp 8 hc pc 4 8 dp 4 p dp h h c hc z u xp z N xp 8 hc H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com
2 Q. Consdr a systm mantand at tmpratur, wth two avalabl nrgy stats and ach wth dgnracs g and g. If p and p ar probablts of occupancy of th two nrgy stats, what s th ntropy of th systm? ns. : Soluton: S k p lnp / g p lnp g / (b) S k p lnp g p p g ln g g (c) S k pln p pln p (d) S k / p lnp / g / p lnp g g p z ln p ln g ln z /, whr z s partton functon p F ln F ln z g p ln F g p g ln F U F U S p ln S, g S k p p p p ln k p ln kpln pln g g g g Q4. ffcncy of a prfctly rvrsbl (Carnot) hat ngn opratng btwn absolut tmpratur and zro s qual to (b).5 (c).75 (d) ns. : (d) Soluton: H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com
3 Q5. Consdr an dal gas of mass m at tmpratur whch s mxd sobarcally (.. at constant prssur) wth an qual mass of sam gas at tmpratur n a thrmally nsulatd contanr. What s th chang of ntropy of th unvrs? mc p ln (b) mc p ln (c) mc p ln (d) ns. : Soluton: Lt us consdr fnal tmpratur wll b mc mc S mcp mc p ln d Now, S S S S mcp mc S mcpln mcpln p S mcpln mcpln d S mc ln p Q6. collcton of N two-lvl systms wth nrgs and s n thrmal qulbrum at tmpratur. For, th spcfc hat approachs to, (b) Nk (c) Nk (d) ns.: Soluton: Z Z ln z ln U ln z ln H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com
4 Now, U C C C C Q7. thrmally nsulatd dal gas of volum and tmpratur xpands to anothr nclosur of volum through a porous plug. What s th chang n th tmpratur of th gas? (b) ln ns. : (c) Soluton: (c) ln (d) ln dh ds dp, for porous plug Joul homshon dh and ds snc t s thrmally nsulatd dal gas dp dp nrd d d d pd nrd d nrd d ln H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com 4
5 JS- Q8. Consdr a systm of two partcls and. ach partcl can occupy on of thr possbl quantum stats, and. h rato of th probablty that th two partcls ar n th sam stat to th probablty that th two partcls ar n dffrnt stats s calculatd for bosons and classcal (Maxwll-oltzmann) partcls. hy ar rspctvly, (b), (c), (d), ns.: (c) Soluton: For two partcl n sam stat: oson Classcal (Maxwll - oltzman) Probablty rato: / / For two partcl n dffrnt stats oson Probablty rato: / / Classcal (Maxwll-oltzmann) Q9. For a datomc dal gas nar room tmpratur, what fracton of th hat suppld s avalabl for xtrnal work f th gas s xpandd at constant prssur? 7 (b) 5 7 (c) 4 (d) 7 ns.: (d) Soluton: It s sobarc procss (constant prssur). hn Q ncp W nr H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com 5
6 In ths procss Q s hat xchang durng procss. Functon of hat suppld W nr R Q ncp R Cp R Cp C f f f [ f dgr of frdom, for datomc molcul f 5 ] Q. mtal bullt coms to rst aftr httng ts targt wth a vlocty of 8 m/ s. If 5% of th hat gnratd rmans n th bullt, what s th ncras n ts tmpratur? (h spcfc hat of th bullt 6 Joul / kg / C ) ns.: (c) 4 C (b).5 C (c) Soluton: Consrvaton of momntum 5% of C (d) C mv mc 88 C 4 6 Q. Consdr a partcl wth thr possbl spn stats: s and. hr s a magntc fld h prsnt and th nrgy for a spn stat s s hs. h systm s at a tmpratur. Whch of th followng statmnts s tru about th ntropy S? S ln at, and at hgh (b) ln at, and at hgh S (c) S at, and at hgh (d) at, and ln at hgh ns.: (d) Soluton: S k ln, whr numbr of mcrostats S S k ln, at hght and at, t s prfct ordrd.. S H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com 6
7 Q. Consdr thr stuatons of 4 partcls n on dmnsonal box of wdth L wth hard walls. In cas (), th partcls ar frmons, n cas () thy ar bosons, and n cas () thy ar classcal. If th total ground stat nrgy of th four partcls n ths thr cass ar F, and cl rspctvly, whch of th followng s tru? F cl (b) F cl (c) F cl (d) F cl ns.: (b) Soluton: For frmons, n -D box of wdth L, th ground stat nrgy for sngl partcl s wrttn as, ml For oson 4, For Maxwll 4 F cl H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com 7
8 JS-4 Q. monoatomc gas conssts of atoms wth two ntrnal nrgy lvls, ground stat and an xctd stat. h spcfc hat of th gas s gvn by (c) ns.: (c) k Soluton:, / (b) / k (d) / / k / / hn partton functon s z z ln z ln ln ln U z.. U C v C k k k If gas wll classcally allowd, thn C and quantum mchancally, C C k / / H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com 8
9 Q4. h tmpratur of a thn bulb flamnt (assumng that th rsstanc of th flamnt s narly constant) of radus r and lngth L s proportonal to ns.: / 4 / r L (b) L r (c) / 4 L r (d) r L Q5. Ic of dnsty mlts at prssur P and absolut tmpratur to form watr of dnsty. h latnt hat of mltng of gram of c s L. What s th chang n th ntrnal nrgy U rsultng from th mltng of gram of c? L P (b) L P (c) L P (d) L P ns.: (d) Soluton: du dq W dq Pd du ml Pd du L P d LP d d Q6. What s th contrbuton of th conducton lctrons n th molar ntropy of a mtal wth lctronc coffcnt of spcfc hat? ns.: (b) Soluton: C (c) (d) Q7. Consdr a systm of N non-ntractng spn / partcls ach fxd n poston and ns.: (b) carryng a magntc momnt. h systm s mmrsd n a unform magntc fld. h numbr of spn up partcls for whch th ntropy of th systm wll b maxmum s 4 (b) N (c) N (d) N / H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com 9
10 Soluton: Lt us consdr n numbr of spn out of N partcl hav spn up rmanng N n s down. N Cn forspn (up) Numbr of ways,, N CNn for spn (down) ntropy, N N S k ln S kln C kln C Nn n N! N ln ln! S k n! N n! n! N n! S k ln N! ln n! ln N n! S k N ln N N nln nn N n ln N n N n ln N! Nln N N! S k N ln N N nln nn N ln N n nln N n N n S knlnn nlnnnln N n nln N n Now for maxmum ntropy at qulbrum for spn up partcl, ds dn ds n N n k ln n ln N n dn n N n N n N n k ln n ln N n N n N n N n N n k lnn nlnn N n k ln n k N n ln n N n n N n n N H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com
11 Q8. For whch gas th rato of spcfc hats C p / C v wll b th largst? mono-atomc (b) d-atomc (c) tr-atomc (d) hxa-atomc ns.: CP Soluton:, whr f s dgr of frdom. C f For monoatomc: f, For datomc: f 6, For ratomc: f 9 For hxaatomc: f 8 H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com
12 JS-5 Q9. For a systm n thrmal qulbrum wth a hat bath at tmpratur, whch on of th followng qualts s corrct? (b) (c) (d) ns.: Soluton: Q. n dal gas s comprssd adabatcally from an ntal volum to a fnal volum and a work W s don on th systm n dong so. h fnal prssur of th gas wll b C C P W W (b) ns.: (c) Soluton: Work don n adabatc procss, P P W P P P P W W P P P P P W (c) W (d) H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com
13 Q. partcl n thrmal qulbrum has only possbl stats wth nrgs,,. If th systm s mantand at a tmpratur, partcl can b approxmatd to, k, thn th avrag nrgy of th (b) (c) ns.: (b) Soluton: (d) Q. h blackbody at a tmpratur of 6 K mts a radaton whos ntnsty spctrum paks at 6 nm. If th tmpratur s rducd to K, th spctrum wll pak at, m (b) m (c) mm (d)mm ns.: (b) Soluton: 66 nm m Q. h ntropy-tmpratur dagram of two Carnot ngns, and, ar shown n th fgur 4. h ffcncs of th ngns ar and rspctvly. Whch on of th ns.: (d) followng qualts s corrct? (b) (c) (d) S H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com
14 W Soluton:, whr W = ara undr th curv, Q = ara undr hgh tmpratur Q S S / /4 and 4 S 4 S 4 4 Q4. lctrons of mass m n a thn, long wr at a tmpratur follow a on-dmnsonal Maxwllan vlocty dstrbuton. h most probabl spd of ths lctrons s, ns.: (c) Soluton: m (b) m / mv m x x x; x f v dv v Most probabl spd vx 8 (c) (d) m f v x v x. H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com 4
15 JS-6 Q5. n dal gas wth adabatc xponnt undrgos a procss n whch ts prssur P s rlatd to ts volum by th rlaton P P, whr P and ar postv constants. h volum starts from bng vry clos to zro and ncrass monotoncally P to. t what valu of th volum durng th procss dos th gas hav maxmum ntropy? P P P P (b) (c) (d) ns.: (c) nrd Soluton: ds nc d Pd ds Pd For maxmum ntropy, ds For Idal gas, P nr Pd dp nrd Pd dp P dp ds Pd ds Pd nr Snc, P P dp d P d ds nrp nr ds Pd nr d P P ds For maxmum ntropy, P P d P CP Q6. n dal gas has a spcfc hat rato C. Startng at a tmpratur th gas undr gos an sothrmal comprsson to ncras ts dnsty by a factor of two. ftr ths an adabatc comprsson ncrass ts prssur by a factor of two. h tmpratur of th gas at th nd of th scond procss would b: (b) (c) (d) H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com 5
16 ns.: (b) Soluton: Durng th sothrmal procss, s constant Lt us assum, th adabatc procss startd at pont P and at pont th, coordnat s P,, t s gvn P P P P P P Q7. two dmnsonal box n a unform magntc fld contans N localsd spn- partcls wth magntc momnt, and N fr spnlss partcls whch do not ntract wth ach othr. h avrag nrgy of th systm at a tmpratur s: N Nsnh (b) N N tanh (c) N N tanh ns.: (c) (d) N N cosh N N N Soluton: For fr partcls n two dmnson, avrag nrgy s and for localzd spn- partcl, th avrag nrgy s Ntanh hn avrag nrgy of systm at tmpratur s N Ntanh. H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com 6
17 Q8. gas of N molculs of mass m s confnd n a cub of volum L at tmpratur. h box s n a unform gravtatonal fld gzˆ. ssum that th potntal nrgy of a molcul s U mgz whr z, L s th vrtcal coordnat nsd th box. h prssur Pz at hght z s: L mg z xp NmgL Pz (b) P z mgl snh N ns.: (c) Soluton: h partton functon of a systm s gvn by, (c) Pz (d) Pz Z N N mk k mgl xp h mgl k Hlmohtz fr nrgy s gvn by, N F ln Z N N L mg z xp NmgL mgl cosh N mgz Prssur s gvn by F P N, N H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com 7
18 JS 7 Part-: -Mark Qustons Q9. ftr th dtonaton of an atom bomb, th sphrcal ball of gas was found to b of 5 ns. : (c) mtr radus at a tmpratur of 5 K. Gvn th adabatc xpanson coffcnt 5/, what wll b th radus of th ball whn ts tmpratur rducs to 56m (b) 5m (c) 5m (d) m K? Soluton: / 5 R R 5 5 Q. If th man squar fluctuatons n nrgy of a systm n qulbrum at tmpratur s proportonal to, thn th nrgy of th systm s proportonal to (b) ns. : (c) (c) (d) Soluton: U C C U Q. Suppos that th numbr of mcrostats avalabl to a systm of N partcls dpnds on ns. : 5 N and th combnd varabl of th systm. h systm ntally has volum sntropc xpanson to volum Soluton: Hr, U N S Nkln U U, whr U s th ntrnal nrgy and s th volum m and nrgy J. It undrgos an 4m. What s th fnal prssur of th systm n SI unts? From law of thrmodynamcs, ds du Pd S U N U.() H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com 8
19 S P and P N U From quaton () and (),.() P U.() Now, from quaton (), U P U P atm.(4) s th gvn procss s sontropc, ds du Pd U S P and from quaton (), U P P P P P P P S On solvng abov quaton, w hav P K (constant) S P P f f P Pf 5atm f 4 Q. cylndr at tmpratur s sparatd nto two compartmnts and by a fr sldng pston. Compartmnts and ar flld by Frm gass mad of spn / and / partcls rspctvly. If partcls n both th compartmnts hav sam mass, th rato of qulbrum dnsty of th gas n compartmnt to that of gas n compartmnt s (b) /5 ns. : (c) Soluton: Follow Pathra Pag 98 quaton for nd quaton (8) at pags From quaton (8) at p 5 n f (c) /5 f (d) / H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com 9
20 / 6 n n 5 g m for qulbrum (usng quaton (4)) n / / n n n g g 5/ / n g n g g 5, g g n n /5 / n n 5 g 5 4 // H.No. 4 D, Ground Floor, Ja Sara, Nar II, Hauz Khas, Nw Dlh 6 Phon: / Wbst: mal:.physcs@gmal.com
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