PH605. Thermal and. Statistical Physics. M.J.D.Mallett. P.Blümler

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1 PH605 : hmal and Statistical Physics 2 Rcommndd txt books: PH605 hmal and Statistical Physics M.J.D.Malltt P.lüml Finn C..P. : hmal Physics Adkins C.J. : Equilibium hmodynamics Mandl F: Statistical Physics HERMODYNAMICS...4 Rviw of Zoth, Fist, Scond and hid Laws...4 hmodynamics...4 h zoth law of thmodynamics,...4 mpatu,...4 Hat, Q...4 Wok, W...4 Intnal ngy, U...5 h fist law of thmodynamics,...5 Isothmal and Adiabatic Expansion...6 Hat Capacity...6 Hat capacity at constant volum, C V...7 Hat capacity at constant pssu, C P...7 Rlationship btwn C V and C P...8 h scond law of thmodynamics,...8 Hat Engins...9 Efficincy of a hat ngin...10 h Canot Cycl...11 h Otto Cycl...13 Concpt of Entopy : lation to disod...15 h dfinition of Entopy...16 Entopy latd to hat capacity...16 h ntopy of a ubb band...17 h thid law of thmodynamics,...18 h cntal quation of thmodynamics...18 h ntopy of an idal gas...18 hmodynamic Potntials : intnal ngy, nthalpy, Hlmholtz and Gibbs functions, chmical potntial...19 Intnal ngy...20 Enthalpy...20 Hlmholtz f ngy...20 Gibbs f ngy...21 Usful wok...21 Chmical Potntial...22 h stat functions in tms of ach oth...22 Diffntial lationships : th Maxwll lations...23 Maxwll lation fom U...23 Maxwll lation fom H...24 Maxwll lation fom F...24 Maxwll lation fom G...25 Us of th Maxwll Rlations...26 Applications to simpl systms...26 h thmodynamic divation of Stfan s Law...27 Equilibium conditions : phas changs...28 Phas changs...28 P- Diagams...29 PV Sufac...29 Fist-Od phas chang...30 Scond-Od phas chang...31

2 PH605 : hmal and Statistical Physics 3 PH605 : hmal and Statistical Physics 4 Phas chang causd by ic skats...31 h Clausius-Claypon Equation fo 1 st od phas changs h Ehnfst quation fo 2 nd od phas changs...33 ASIC SAISICAL CONCEPS...35 Isolatd systms and th micocanonical nsmbl : th oltzmann-planck Entopy fomula...35 Why do w nd statistical physics?...35 Macostats and Micostats...35 Classical vs Quantum...36 h thmodynamic pobability, Ω...36 How many micostats?...36 What is an nsmbl?...37 Stiling s Appoximation...39 Entopy and pobability...39 h oltzmann-planck ntopy fomula...40 Entopy latd to pobability...40 h Schottky dfct...41 Spin half systms and paamagntism in solids...43 Systms in thmal quilibium and th canonical nsmbl : th oltzmann distibution...45 h oltzmann distibution...45 Singl paticl patition function, Z, and Z N fo localisd paticls : lation to Hlmholtz function and oth thmodynamic paamts...47 h singl paticl patition function, Z...47 h patition function fo localisd paticls...47 h N-paticl patition function fo distinguishabl paticls...47 h N-paticl patition function fo indistinguishabl paticls...48 Hlmholtz function...49 Adiabatic cooling...50 hmodynamic paamts in tms of Z...53 hmodynamics Rviw of Zoth, Fist, Scond and hid Laws hmodynamics Why study thmal and statistical physics? What us is it? h zoth law of thmodynamics, If ach of two systms is in thmal quilibium with a thid, thn thy a also in thmal quilibium with ach oth. his implis th xistnc of a popty calld tmpatu. wo systms that a in thmal quilibium with ach oth must hav th sam tmpatu. mpatu, h 0 th law of thmodynamics implis th xistnc of a popty of a systm which w shall call tmpatu,. Hat, Q In gnal tms this is an amount of ngy that is supplid to o movd fom a systm. Whn a systm absobs o jcts hat th stat of th systm must chang to accommodat it. his will lad to a chang in on o mo of th thmodynamic paamts of th systm.g. th tmpatu,, th volum, V, th pssu, P, tc. Wok, W Whn a systm has wok don on it, o if it dos wok itslf, thn th is a flow of ngy ith into o out of th systm. his will also lad to a chang in on o mo of th thmodynamics paamts of th systm in th sam way that gaining o losing hat, Q, will caus a chang in th stat of th systm, so too will a chang in th wok, W, don on o by th systm. Whn daling with gass, th wok don is usually latd to a chang in th volum, dv, of th gas. his is paticulaly appant in a machin such as a cas ngin.

3 PH605 : hmal and Statistical Physics 5 PH605 : hmal and Statistical Physics 6 If a thmally isolatd systm is bought fom on quilibium stat to anoth, th wok ncssay to achiv this chang is indpndnt of th pocss usd. W can wit this as, du đ W Adiabatic Not : whn w consid wok don w hav to dcid on a sign convntion. y convntion, wok don on a systm (ngy gain by th systm) is positiv and wok don by th systm (loss of ngy by th systm) is ngativ. Intnal ngy, U h intnal ngy of a systm is a masu of th total ngy of th systm. If it w possibl w could masu th position and vlocity of vy paticl of th systm and calculat th total ngy by summing up th individual kintic and potntial ngis. N N n 1 n 1 U KE+ PE Howv, this is not possibl, so w a nv abl to masu th intnal ngy of a systm. What w can do is to masu a chang in th intnal ngy by coding th amount of ngy ith nting o laving a systm. In gnal, whn studying thmodynamics, w a intstd in changs of stat of a systm..g. đ W + PdV : compssion of gas in a pump ( of gas incass). đ W PdV : xpansion of gas in an ngin ( of gas dcass). Isothmal and Adiabatic Expansion Whn w consid a gas xpanding, th a two ways in which this can occu, isothmally o adiabatically. Isothmal xpansion : as it s nam implis this is whn a gas xpands o contacts at a constant tmpatu ( iso -sam, thm - tmpatu). his can only occu if hat is absobd o jctd by th gas, spctivly. h final and initial stats of th systm will b at th sam tmpatu. which w usually wit, U Q+ W du đq + đ W h ba though th diffntial, đ, mans that th diffntial is inxact, this mans that th diffntial is path dpndnt i.. th actual valu dpnds on th out takn, not just th stat and finish points. Adiabatic xpansion : this is what happns whn no hat is allowd to nt o lav th systm as it xpands o contacts. h final and initial stats of th systm will b at diffnt tmpatus. Hat Capacity h fist law of thmodynamics, As a systm absobs hat it changs its stat (.g. P,V,) but diffnt systms bhav individually as thy absob th sam hat so th must b a paamt govning th hat absoption, this is known as th hat capacity, C.

4 PH605 : hmal and Statistical Physics 7 PH605 : hmal and Statistical Physics 8 h hat capacity of a matial is dfind as th limiting ation of th hat, Q, absobd, to th is in tmpatu,, of th matial. It is a masu of th amount of hat quid to incas th tmpatu of a systm by a givn amount. Q C limit 0 Whn a systm absobs hat its stat changs to accommodat th incas of intnal ngy, thfo w hav to consid how th hat capacity of a systm is govnd whn th a stictions placd upon how th systm can chang. In gnal w consid systms kpt at constant volum and constant tmpatu and invstigat th hat capacitis fo ths two cass. Hat capacity at constant volum, C V If th volum of th systm is kpt fixd thn no wok is don and th hat capacity can b wittn as, C V đq U V d Hat capacity at constant pssu, C P h hat capacity at constant pssu is thfo analogously, đqp C P d W now us a nw stat function known as nthalpy, H, (which w discuss lat). H U + PV dh du + PdV + VdP dh đq + VdP Using this dfinition w can wit, V Rlationship btwn C V and C P h intnal ngy of a systm can b wittn as, du đq + đw đq du - PdV Assuming th chang of intnal ngy is a function of volum and tmpatu, U U( V, ), i.. w hav a constant pssu pocss, this can b wittn as, U U đ Q dv + d + PdV V which lads to, đqp U V U V C P P + + d V U V C C + P P V + V P V P V P his is th gnal lationship btwn C V and C P. In th cas of an idal gas th intnal ngy is indpndnt of th volum (th is zo intaction btwn gas paticls), so th fomula simplifis to, V C C P + P V P C C R h scond law of thmodynamics, h Klvin statmnt of th 2 nd law can b wittn as, P V It is impossibl to constuct a dvic that, opating in a cycl, will poduc no ffct oth than th xtaction of hat fom a singl body at a unifom tmpatu and th pfomanc of an quivalnt amount of wok. C P đq H P d P

5 PH605 : hmal and Statistical Physics 9 PH605 : hmal and Statistical Physics 10 It would b usful to convt all th hat, Q H, xtactd into usful wok but this is disallowd by th 2 nd law of thmodynamics. A mo concis fom of this statmnt is, A pocss whos only ffct is th complt convsion of hat into wok is impossibl. Anoth fom of th 2 nd law is known as th Clausius statmnt, If this pocss w possibl it would b possibl to join two hat ngins togth, whos sol ffct was th tanspot of hat fom a cold svoi to a hot svoi. It is impossibl to constuct a dvic that, opating in a cycl, will poduc no ffct oth than th tansf of hat fom a cold to a hott body. Efficincy of a hat ngin Hat Engins Hat ngins convt intnal ngy to mchanical ngy. W can consid taking hat Q H fom a hot svoi at tmpatu H and using it to do usful wok W, whilst discading hat Q C to a cold svoi C. W can dfin th fficincy of a hat ngin as th atio of th wok don to th hat xtactd fom th hot svoi. W Q Q Q η 1 Q Q Q H C C H H H Fom th dfinition of th absolut tmpatu scal 1, w hav th lationship, Q C C Q H H 1 Fo a poof of this s Finn CP, hmal Physics,

6 PH605 : hmal and Statistical Physics 11 PH605 : hmal and Statistical Physics 12 On way of dmonstating this sult is th following. Consid two hat ngins which sha a common hat svoi. Engin 1 opats btwn 1 and 2 and ngin 2 opats btwn 2 and 3. W can say that th must b a lationship btwn th atio of th hat xtactd/absobd to th tmpatu diffnc btwn th two svois, i.. Q Q Q f f f Q Q Q ' '' ( θ, θ ), ( θ, θ ), ( θ, θ ) hfo th ovall hat ngin can b considd as a combination of th two individual ngins. ' ( θ, θ ) ( θ, θ ) ( θ, θ ) '' f f f Howv this can only b tu if th functions factoiz as, f ( θ, θ x y) ( θ ) x ( θ y ) h Canot cycl is a closd cycl which xtacts hat Q H fom a hot svoi and discads hat Q C into a cold svoi whil doing usful wok, W. h cycl opats aound th cycl A C D A Wh (θ) psnts a function of absolut, o thmodynamic tmpatu. hfo w hav th lationship, Q Q 1 ( θ1) ( θ ) 2 2 hfo w can also wit th fficincy lation as, η 1 C H h fficincy of a vsibl hat ngin dpnds upon th tmpatus btwn which it opats. h fficincy is always <1. h most fficint hat ngin is typifid by th Canot cycl. W can consid this cycl in tms of th xpansion/contaction of an idal gas. h Canot Cycl

7 PH605 : hmal and Statistical Physics 13 PH605 : hmal and Statistical Physics 14 A hat ngin can also opat in vs, xtacting hat, Q C fom a cold svoi and discading hat, Q H, into a hot svoi by having wok don on it, W, th total hat discadd into th hot svoi is thn, h 4-stok ptol ngin follows th Otto cycl ath than th Canot cycl. h actual cycl diffs slightly fom th idalisd cycl to accommodat th intoduction of fsh ptol/ai mixtu and th vacuation of xhaust gass. Q Q + W H C his is th pincipl of th figato. h Otto Cycl h Canot cycl psnts th most fficint hat ngin that w can contiv. In ality it is unachivabl. wo of th most common hat ngins a found in vhicls, th 4-stok ptol ngin and th 4-stok disl ngin. h 4-stok cycl can b considd as: 1. Induction : Ptol/Ai mixtu dawn into th ngin cylind. 2. Compssion : Ptol/Ai mixtu compssd to a small volum by th ising piston. 3. Pow : Ignition of ptol/ai mixtu causs apid xpansion pushing th piston down th cylind 4. Exhaust : Exhaust gass vacuatd fom th cylind by th ising piston. h Otto cycl and th Disl cycl can b appoximatd by PV diagams. Otto cycl

8 PH605 : hmal and Statistical Physics 15 PH605 : hmal and Statistical Physics 16 đq R 0 h inquality of an ivsibl pocss is a masu of th chang of ntopy of th pocss. Disl cycl final S Sfinal Sinitial initial so fo an infinitsimal pat of th pocss w hav, đ Q ds đq h dfinition of Entopy An ntopy chang in a systm is dfind as, Concpt of Entopy : lation to disod W shall dal with th concpt of ntopy fom both th thmodynamic and th statistical mchanical aspcts. Suppos w hav a vsibl hat ngin that absobs hat Q 1 fom a hot svoi at a tmpatu 1 and discads hat Q 2 into a cold svoi at a tmpatu 2, thn fom th fficincy lation w hav, Q Q but fom th 2 nd law w know that w cannot hav a tu vsibl cycl, th is always a hat loss, thfo w should wit this lationship as, Q Q < h hat absobd in on complt cycl of th hat ngin is thfo, đq his is known as th Clausius inquality. 0 If w had a tuly vsibl hat ngin thn this would b, ds đq h ntopy of a thmally isolatd systm incass in any ivsibl pocss and is unaltd in a vsibl pocss. his is th pincipl incasing ntopy. h ntopy of a systm can b thought of as th invitabl loss of pcision, o od, going fom on stat to anoth. his has implications about th diction of tim. h fowad diction of tim is that in which ntopy incass so w can always dduc whth tim is volving backwads o fowads. Although ntopy in th Univs as a whol is incasing, on a local scal it can b dcasd that is w can poduc systms that a mo pcis o mo odd than thos that poducd thm. An xampl of this is cating a cystallin solid fom amophous componnts. h cystal is mo odd and so has low ntopy than it s pcusos. On a lag scal lif itslf is an xampl of th duction of ntopy. Living oganisms a mo complx and mo odd than thi constitunt atoms. Entopy latd to hat capacity Suppos th hat capacity of a solid is C P JK -1. What would b th ntopy chang if th solid is hatd fom 273 K to 373 K?

9 PH605 : hmal and Statistical Physics 17 PH605 : hmal and Statistical Physics 18 Knowing th hat capacity of th solid and th is in tmpatu w can asily calculat th hat input and thfo th ntopy chang. ds đq W intgat ov th tmpatu ang to dtmin th total ntopy chang. final dq S Sfinal Sinitial initial final CPd initial final 1 CP ln 39.2 JK initial h ntopy of a ubb band A ubb band is a collction of long chain polym molculs. In its laxd stat th polyms a high disodd and ntangld. h amount of disod is high and so th ntopy of th systm must b high. Sttching foc h thid law of thmodynamics, h ntopy chang in a pocss, btwn a pai of quilibium stats, associatd with a chang in th xtnal paamts tnds to zo as th tmpatu appoachs absolut zo. O mo succinctly, h ntopy of a closd systm always incass. An altnativ fom of th 3 d law givn by Planck is, h ntopy of all pfct cystals is th sam at absolut zo and may b takn as zo. In ssnc this is saying that at absolut zo th is only on possibl stat fo th systm to xist in so th is no ambiguity about th possibility of it xisting in on of sval diffnt stats. his concpt bcoms mo vidnt whn w consid th statistical concpt of ntopy. h cntal quation of thmodynamics h diffntial fom of th fist law of thmodynamics is, du đq + đ W Using ou dfinition fo ntopy and assuming w a daling with a compssibl fluid w can wit this as, du ds - PdV If th ubb band is sttchd thn th polyms bcom lss ntangld and align with th sttching foc. hy fom a quasi-cystallin stat. his is a mo odd stat and must thfo hav a low ntopy. h total ntopy in th sttchd stat is mad up of spatial and thmal tms. S S + S otal Spatial hmal If th tnsion in th band is apidly ducd thn w a pfoming an adiabatic (no hat flow) chang on th systm. h total ntopy must main unchangd sinc th is no hat flow, but th spatial ntopy has incasd so th thmal ntopy must dcas this mans th tmpatu of th ubb band dops. his is mo usually wittn as, ds du + PdV his assums that all th wok don is du to changs of pssu, ath than changs of magntisation tc. h ntopy of an idal gas h spcific hat capacity at constant volum fo a gas is, C V U du d V

10 PH605 : hmal and Statistical Physics 19 PH605 : hmal and Statistical Physics 20 Substituting this into th cntal quation givs, ds C d + PdV V If w consid on mol of an idal gas and us low cas ltts to f to mola quantitis thn w can wit this as, R ds c d + dv V v d dv ds c + R v v Intgating both sids givs us, s c ln + Rln v+ s So th ntopy of an idal gas has th main tms, v 1. A tmpatu tm latd to th motion, and thfo kintic ngy of th gas 2. A volum tm latd to th positions of th gas paticls 3. A constant tm th intinsic disod tm which is un-masuabl. As an xampl of this can b usd, consid gas insid a cylind of volum, V 0. Suppos th volum of th cylind is suddnly doubld. What is th incas in ntopy of th gas? Assuming this chang occus at constant tmpatu, w can wit, s s s Rln 2V RlnV 2V0 V0 0 0 R 2V 0 ln Rln 2 V0 If w w daling with mo than on mol of gas w could wit this as, s nrln 2 Nk ln 2 Wh n is th numb of mols and N is th numb of molculs. W will tun to this sult whn w look at th statistical dfinition of ntopy. 0 h quilibium conditions of a systm a govnd by th thmodynamic potntial functions. hs potntial functions tll us how th stat of th systm will vay, givn spcific constaints. h diffntial foms of th potntials a xact bcaus w a now daling with th stat of th systm. Intnal ngy his is th total intnal ngy of a systm and can b considd to b th sum of th kintic and potntial ngis of all th constitunt pats of th systm. n 1 n 1 U KE+ PE his quantity is pooly dfind sinc w a unabl to masu th individual contibutions of all th constitunt pats of th systm. Using this dfinition of intnal ngy and th 2 nd law of thmodynamics w a abl to combin th two togth to giv us on of th cntal quations of thmodynamics, ds du + PdV his nabls us to calculat changs to th intnal ngy of a systm whn it undgos a chang of stat. Enthalpy du ds PdV his is somtims onously calld th hat contnt of a systm. his is a stat function and is dfind as, H U + PV W a mo intstd in th chang of nthalpy, dh, which is a masu of th hat of action whn a systm changs stat. In a mchanical systm this could b whn w hav a chang in pssu o volum. In a pdominantly chmical systm this could b du to th hat of action of a chang in th chmisty of th systm. dh du + PdV + VdP hmodynamic Potntials : intnal ngy, nthalpy, Hlmholtz and Gibbs functions, chmical potntial Hlmholtz f ngy

11 PH605 : hmal and Statistical Physics 21 PH605 : hmal and Statistical Physics 22 h Hlmholtz f ngy of a systm is th maximum amount of wok obtainabl in which th is no chang in tmpatu. his is a stat function and is dfind as, F U S h chang of Hlmholtz f ngy is givn by, Gibbs f ngy df du ds Sd PdV Sd h Gibbs f ngy of a systm is th maximum amount of wok obtainabl in which th is no chang in volum. his is a stat function and is dfind as, G H S h chang of Gibbs f ngy is givn by, dg dh ds Sd VdP Sd It is obvious that th Hlmholtz and Gibbs f ngis a latd, G F + ( PV) and th coct on to us has to b asctaind fo th systm in hand. Fo xampl, a mtal undgos vy small volum changs so w could us th Gibbs function whas a gas usually has lag volum changs associatd with it and w hav to chos th function dpnding upon th situation. Usful wok Suppos w hav a systm that dos wok and that pat of that wok involvs a volum chang. If th systm tuns to its initial stat of pssu and tmpatu at th nd of it doing som wok thn th is no tmpatu chang, i.. Initial tmpatu and pssu 0 and P 0 Final tmpatu and pssu 0 and P 0 hn bcaus th is no ovall tmpatu chang, th maximum amount of wok don by th systm is givn by th dcas in th Hlmholtz f ngy, F, of th systm. Wotal F Howv, som of th wok don is uslss wok, suppos th actual gas xpansion was a by poduct of som chmical action, thn w would want to know how much usful wok has actually bn don. WUsful Wotal WUslss W P V otal 0 F P V 0 ( F + PV 0 ) W G Usful hfo, th dcas in th Gibbs f ngy tlls us how much usful wok was don by this pocss. Chmical Potntial his is impotant whn th quantity of matt is not fixd (.g. w a daling with a changing numb of atoms within a systm). Whn this happns w hav to modify ou thmodynamic lations to tak account of this. du ds PdV + µ dn df PdV Sd + µ dn dg VdP Sd + µ dn his mans that th a sval ways of witing th chmical potntial, µ. U F G µ N N N SV, V,, P W can also show that th chmical potntial can b wittn, G µ N h chmical potntial, µ, is th Gibbs f ngy p paticl, povidd only on typ of paticl is psnt. h stat functions in tms of ach oth W can wit infinitsimal stat functions fo th intnal ngy, U, th nthalpy, H, th Hlmholtz f ngy, F and th Gibbs f ngy, G. du ds PdV dh ds + VdP df Sd PdV

12 PH605 : hmal and Statistical Physics 23 PH605 : hmal and Statistical Physics 24 dg Sd + VdP y inspction of ths quations it would appa that th a natual vaiabls which govn ach of th stat functions. Fo instanc, fom th fomula fo th Hlmholtz f ngy w can assum its natual vaiabls a tmpatu and volum and thfo w can wit, F S V and F P V and fom th fomula fo th Gibbs f ngy, assuming its natual vaiabls a tmpatu and pssu, w hav, G S P and G V P his mans that if w know on of th thmodynamic potntials in tms of its natual vaiabls thn w can calculat th oth stat functions fom it. Suppos w know th Gibbs f ngy, G, in tms of its natual vaiabls and P, thn w can wit, G G U G PV + S GP P G H G+ S G P G F G PV GP P P Diffntial lationships : th Maxwll lations h Maxwll lations a a sis of quations which w can div fom th quations of stat fo U, H, F and G. Maxwll lation fom U W alady hav an quation of stat fo du, du ds PdV his suggsts that U is a function of S and V, thfo w could wit this as, U U du ds + dv S V V S which would thn man that w can wit, moov w can thn wit, his is th fist Maxwll lation. Maxwll lation fom H U U and P S V V S P S V V S W alady hav an quation of stat fo dh, dh ds + VdP his suggsts that H is a function of S and P, thfo w could wit this as, H H dh ds + dp S P P S which would thn man that w can wit, moov w can thn wit, his is th scond Maxwll lation. Maxwll lation fom F H H and V S P P S V S P P S W alady hav an quation of stat fo df,

13 PH605 : hmal and Statistical Physics 25 PH605 : hmal and Statistical Physics 26 df Sd PdV his suggsts that F is a function of and V, thfo w could wit this as, F F df d + dv V V which would thn man that w can wit, F F S and P V V Us of th Maxwll Rlations Consid applying pssu to a solid (vy small volum chang), vsibly and isothmally. h pssu applid changs fom P 1 to P 2 at a tmpatu,. h pocss is vsibl so w can wit, ds đqr Sinc th only vaiabls w hav a pssu, P and tmpatu,, w can wit th ntopy chang of th systm as a function of ths two vaiabls. moov w can thn wit, his is th thid Maxwll lation. P S V V so thfo, S S ds dp + d P P S dq ds dp R P Maxwll lation fom G W alady hav an quation of stat fo dg, dg VdP Sd his suggsts that G is a function of P and, thfo w could wit this as, G G dg dp + d P P which would thn man that w can wit, moov w can thn wit, G G V and S P P V S P P h scond tm is zo sinc th pocss is isothmal, d0. Using th Maxwll lation divd fom th Gibbs f ngy, V S P P w can thn wit, V dq dp V β dp R P Wh β is th cofficint of xpansion. Intgating this givs, P2 Q Vβ dp P1 ( ) V β P P h appoximation sign assums β to b constant. Applications to simpl systms 2 1 his is th fouth Maxwll lation.

14 PH605 : hmal and Statistical Physics 27 PH605 : hmal and Statistical Physics 28 h thmodynamic divation of Stfan s Law h ngy dnsity of thmal adiation (black-body adiation) is dpndnt only on th tmpatu of th black body. h ngy dnsity at a paticula wavlngth is thn, u λ u λ (λ,) and th total ngy dnsity is, uu() h Planck fomula fo th spctal ngy dnsity is givn by, wh β is a constant. u λ ( λ, ) 5 β 1 hc λ λk 1 h pfct gas thoy tlls us that th pssu of a gas can b xpssd in tms of th man vlocity, c. wh A is a constant. 1 du 1 u u 3 d 3 4 du u 3 3 d d du 4 u 4 u A P 1 ρc 3 2 but whn w a daling with thmal adiation w can us th Einstin massngy lation, thfo, E mc u ρc 2 2 u P 3 h ngy quation fo a PV systm is, substituting into this w hav, U P P V V Equilibium conditions : phas changs Phas changs A chang of phas of a systm occus whn th systm changs fom on distinct stat into anoth. his chang of phas can b causd by many diffnt factos.g. tmpatu changs can caus a phas chang btwn a solid and a liquid, applid magntic filds can caus a phas chang btwn a supconducto and a nomal conducto.

15 PH605 : hmal and Statistical Physics 29 PH605 : hmal and Statistical Physics 30 P- Diagams In a simpl systm compising a singl substanc w can constuct a Pssu-mpatu diagam (P- diagam) showing how changs in pssu o tmpatu can affct th phas of th systm. oth th P and PV diagams can b obtaind fom th PV sufac by pojction. PV Sufac h P diagam is a spcialisd cas of th mo gnal PV sufac. his givs us all th thmodynamic infomation w qui whn dtmining th availabl phas changs. Fist-Od phas chang A fist-od phas chang of a substanc is chaactisd by a chang in th spcific volum btwn th two phas, accompanid by a latnt hat.

16 PH605 : hmal and Statistical Physics 31 PH605 : hmal and Statistical Physics 32 Som typical xampls of fist-od phas changs a: h tansition of a solid mlting into a liquid. h tansition of a liquid boiling into a gas. h chang fom supconducto to nomal conducto, povidd th chang occus in an applid magntic fild. Scond-Od phas chang Scond-Od phas chang of a substanc is chaactisd by no chang in th spcific volum btwn th two phass and no accompanying latnt hat. Som typical xampls of a scond-od phas chang a: h tansition fom fo-magnt to paa-magnt at th Cui tmpatu. h tansition fom supconducto to nomal conducto, povidd th is no applid magntic fild. h chang fom nomal liquid 4 Hlium to supfluid liquid 4 Hlium blow 2.2K Phas chang causd by ic skats Wat has a paticulaly intsting phas diagam. It is on th fw matials that xpands on fzing (o contacts on mlting), this is a consqunc of th ffcts of Hydogn-bonding within th matial. at constant tmpatu and th phas of th ic cosss th mlting lin on th P diagam. h skat is now standing on a thin lay of wat which acts as a lubicant btwn th skat and th bulk of th ic. If th tmpatu is too low thn th incasd pssu of th skat on th ic sufac is insufficint to caus th ic to coss th mlting lin and so ic skating is not possibl. Skiing is not a pssu-mlting ffct sinc th sufac aa of th sky is fa to lag to caus mlting of snow, instad th ffct is causd by fictional hating and wax lubicant applid to th ski. h Clausius-Claypon Equation fo 1 st od phas changs. Whn a phas chang occus w a mostly intstd in how it affcts th Gibbs f ngy. At thmodynamic quilibium th Gibbs function is at a minimum, so at th tansition lin on th P diagam th spcific Gibbs ngy is th sam fo both phass. g ( P, ) g ( P, ) 1 2 Futh along th tansition lin w must also hav th sam condition. g ( P + dp, + d ) g ( P + dp, + d ) 1 2 Expanding this to fist od using a aylo appoximation, g1 g1 g2 g2 g1( P, ) + d + dp g2( P, ) + d + dp P P P P So thfo, g g g g d dp P P P P Fom ou pvious discussion about th natual vaiabls fo th Gibbs function, w can wit, G G V and S P P and thfo considing th spcific quantitis, w hav his has implications which can b usd to ou advantag to allow ic skating. h hollow-gound dg of an ic skat causs an nomous pssu on th ic sufac of 100 atmosphs o mo. his pssu incass occus

17 PH605 : hmal and Statistical Physics 33 PH605 : hmal and Statistical Physics 34 dp s s S S d v v V V h latnt hat associatd with this phas chang is latd to th ntopy chang, ( ) L Q S S 2 1 Which givs us th Clausius-Claypon quation which is th gadint of th phas tansition lin fo a fist od phas chang. his is th fist Ehnfst quation fo a 2 nd od phas chang, th scond Ehnfst quation can b divd by considing th continuity of th volum of th two phass, which givs, dp d β κ wh κ is th bulk modulus of th phas. β κ dp L d V V ( ) 2 1 h Ehnfst quation fo 2 nd od phas changs In a scond od phas chang th is no latnt hat, and so no ntopy chang, and no volum chang. So using a simila agumnt as th on fo th Clausius-Claypon quation, s ( P, ) s ( P, ) 1 2 Futh along th tansition lin w must also hav th sam condition. s ( P + dp, + d ) s ( P + dp, + d ) 1 2 Expanding this to fist od using a aylo appoximation, s1 s1 s2 s2 s1( P, ) + d + dp s2( P, ) + d + dp P P P P Now substituting in xpssions fo th spcific hat capacity and th volum xpansion, w can wit, s 1 v c and β v P dp C C d V β β P1 P2 ( ) 1 2 P

18 PH605 : hmal and Statistical Physics 35 PH605 : hmal and Statistical Physics 36 asic statistical concpts Isolatd systms and th micocanonical nsmbl : th oltzmann-planck Entopy fomula Why do w nd statistical physics? h a two ways that w can look at th bhaviou of matt 1. Mchanical viwpoint : masuing th positions and vlocitis of atoms at th micoscopic lvl. 2. hmodynamic viwpoint : masuing th bulk poptis of matt at th macoscopic lvl. Howv, whn w ty and concil th two viwpoints w un into a poblm. hotically w should gt th sam answs fom both viwpoints. Howv th is a poblm with th diction of tim, Micoscopic tim : vsibl.g. th laws of motion look idntical whichv way tim gos. Macoscopic tim : non-vsibl, th is a pfd diction of tim dfind by th incas of ntopy of th Univs. h two viwpoints can b concild by looking at systms fom a statistical o pobabilistic viwpoint. Macostats and Micostats Whn w a daling with a systm,.g. a gas insid a contain, w can dtmin vaious poptis of th systm. In ou xampl of a gas w can dtmin, i.. masu, th volum, V, th pssu, P, th tmpatu,, th molcula dnsity, ρ, plus oths. h stat of th systm can thfo b dfind by quantitis such as V,P, & ρ. his is calld a macostat. In thmodynamic tms w usually only consid macostats. Howv, this givs us no knowldg of th poptis of th individual gas paticls, fo instanc thi positions o vlocitis. h a sval, ffctivly infinitly many, stats of a systm which all hav th sam macostat. Each of ths individual stats a known as micostats. hy hav diffnt positions, vlocitis tc. fo th individual paticls but all hav th sam thmodynamic poptis such as volum, pssu and tmpatu. Classical vs Quantum h classical viw of th univs allows us an infinit choic of position and vlocity. his implis that fo any givn macostat th must b an infinit numb of micostats that cospond to it. h quantum viwpoint howv tlls us that at th atomic lvl w a abl to assign quantum numbs to th poptis of paticls. So fo a closd systm th a only a finit numb of stats that th systm can occupy. In this scnaio th most pobabl macostat fo a systm is th on that has th most micostats simply bcaus w hav th gatst chanc of finding th systm in that micostat. h thmodynamic pobability, Ω h numb of possibl micostats fo a givn macostat is calld th thmodynamic pobability, Ω, o somtims, W. his is not a pobability in th nomal sns sinc it has a valu gat than o qual to on. Consid a systm of two paticls A and that can both xist in on of two ngy lvls, E 1 and E 2. h macostat of this systm can b dfind by th total ngy of th systm. Macostat (1) E 1 + E 1 (2) E 1 + E 2 (3) E 2 + E 2 Micostat A(E 1 ),(E 1 ) A(E 1 ),(E 2 ) A(E 2 ),(E 1 ) A(E 2 ),(E 2 ) Ω hfo if both ngy lvls, E 1 and E 2 a qually likly th systm has a 50% chanc of bing in macostat (2) and a 25% chanc ach of bing in macostats (1) and (3). Howv, in gnal not vy ngy lvl is qually likly so th most likly macostat is also govnd by th pobability of ngy lvl occupation. his lads on to th concpt of th patition function, Z, fo a systm, which w will cov lat. How many micostats? Suppos w a daling with a paamagntic solid. his mans w hav atoms aangd in a gula cystallin lattic and ach atom acts as a magntic dipol bcaus it has an unpaid lcton.

19 PH605 : hmal and Statistical Physics 37 PH605 : hmal and Statistical Physics 38 In a magntic fild th magntic dipol can ith point along th fild o against th fild. hfo th a two possibl stats fo ach dipol, spin-up o spin-down. hfo, fo N atoms (N~10 23 ) th must b 2 N possibl micostats fo th magntic dipols. How long would it tak fo th systm to sampl all th possibl micostats? Assuming a dipol changs ointation vy sconds th would b a nw micostat gnatd vy sconds, so vy micostat would b sampld aft ~ ( ) 2 33 sconds h liftim of th Univs is only ~10 17 sconds! Instad of considing on systm and watching it chang w us th concpt of an nsmbl. What is an nsmbl? An nsmbl is a vy lag numb of plica systms with idntical spcifications.g. volum, tmpatu, chmisty, numb of paticls tc. h nsmbl psnts th macostat of th systm whil ach individual plica psnts on of th possibl micostats. Systm Ensmbl i n h a sval diffnt nsmbls that w might ncount. h typ of nsmbl is govnd by th masuabl paamts. 1. Mico-canonical nsmbl : isolatd systms, th total intnal ngy, U, and numb of paticls, N, is wll dfind. 2. Canonical nsmbl : systms in thmal quilibium, th tmpatu,, and numb of paticls, N, is wll dfind. 3. Gand canonical nsmbl : systms in thmal and chmical contact, th tmpatu,, and chmical potntial, µ, is wll dfind.

20 PH605 : hmal and Statistical Physics 39 PH605 : hmal and Statistical Physics 40 Stiling s Appoximation W a now daling with vy lag numbs! Almost any macoscopic systm will hav on th od of paticls. h mathmatics of such lag numbs can bcom vy involvd but fotunatly th a som simpl appoximations that wok wll whn th N is vy lag. h a two paticulaly usful appoximations known as Stiling s two-tm and th-tm appoximations; thy a, Stiling s 2-tm appoximation ( ) ln N! Nln N N Exampl ln( 100! ) 100ln ( Stilings appoximation) ( Actual valu) Stiling s 3-tm appoximation ( ) ln N! Nln N N + ln 2π N Exampl ln( 100! ) 100ln ln 200π ( Stilings appoximation) ( Actual valu) hs appoximations a paticulaly usful whn daling with thmodynamic pobabilitis. Entopy and pobability In statistical mchanics ach paticl is sn as having its own dynamic stat, a position in spac, and a spatial vlocity o momntum. In th-dimnsional spac this givs th paticl 3 dgs of fdom. h position coodinats and th momntum coodinats plac ach paticl somwh in th six-dimnsional phas spac. If th is mo than on paticl you can consid th systm of thos paticls as having 6 tims N coodinats and hnc a singl position in a 6N-dimnsional phas spac. Altnativly you can stick with th 6-dimnsional phas spac and hav N points in it. In this latt pictu you might cut up th phas spac into clls and count how many paticls a in ach cll. Fom this pictu of phas spac cll occupation numbs it is a small stp to th thmodynamic pobability. If th total numb of paticls is N and th numb of paticls in ach cll is N j, thn th thmodynamic pobability is givn by, N! Ω xp Nln N N ln N j j N! j j j h quantity, Ω, is known as th thmodynamic pobability o thmodynamic wight of th stat of an nsmbl (Not: Ω is somtims wittn as W). h stats with th lagst valus of Ω a thos that a most likly to occu. Howv th stats with th lagst valu of Ω a also thos with th most disod simply bcaus th a so many configuations of th micostats to giv on macostat. his mans that th must b lationship btwn th thmodynamic wight of a systm and th ntopy of a systm. hfo if thn, Ω tnds to a maximum, S tnds to a maximum. h oltzmann-planck ntopy fomula h oltzmann-planck quation fo ntopy is wittn, S k ln Ω wh Ω is th thmodynamic pobability, o th numb of aangmnts, of th stat of th systm. h stats with th lagst valu of Ω will b th ons most likly to occu. his quation is cavd on oltzmann s tombston in Vinna. Entopy latd to pobability o div this w fist consid an nsmbl of a lag numb, v, of plica systms. Each of ths individual systms can xist in on of,, micostats, and fo ach micostat th is an associatd pobability, p, of th systm bing in that micostat.

21 PH605 : hmal and Statistical Physics 41 PH605 : hmal and Statistical Physics 42 hfo, within th nsmbl, th numb of systms in th micostat,, is simply, ν ν p h thmodynamic pobability of th nsmbl is thn givn by, ν! Ω ν ν! ν! ν!... ν! Applying this to th oltzmann-planck quation givs us, Sν k lnω ν ν! k ln ν1! ν2! ν3!... ν!... k ν lnν ν lnν which has bn simplifid using Stiling s 2-tm appoximation. h ntopy of th nsmbl is thus latd to th ntopy of a singl systm. h Schottky dfct 1 S Sν k ln ln ν ν ν ν ν k lnν ν p ln p k p ln p (fo lagν ) At absolut zo th atoms in a cystallin solid a pfctly odd. hy inhabit gula lattic positions within th cystal. As th tmpatu incass th atoms gain ngy and a abl to thmally vibat. What a known as dfcts can occu within th cystal, wh an atom has movd away fom its oiginal odd position. hs a known as point dfcts. On paticula typ of point dfct is th Schottky dfct. In this cas a displacd atom will migat to th sufac of th cystal, laving a vacancy bhind it. Suppos w hav a cystal composd of N atoms. Insid this cystal th a n dfcts. Each dfct has an associatd ngy ε, sinc it taks ngy to mov th atom fom th intio to th sufac. h ngy associatd with all th dfcts is, E nε What is th thmodynamic pobability (o statistical wight) of this systm? N! Ω ( n) n! N n! ( ) h ntopy sulting fom th Schottky dfcts will thn b, N! Sn ( ) k ln Ω ( n) k ln n! ( N n)! h tmpatu of th cystal can b xpssd in tms of th changs in ntopy and ngy of th cystal, 1 S S Q E ds( n) dn dn de 1 ds( n) ε dn Using Stiling s two-tm appoximation w can wit, [ ] Sn ( ) k Nln Nnln n( Nn)ln( Nn) ds( n) k[ ln n+ ln( N n) ] dn So th tmpatu thn bcoms, Raanging this givs us, 1 k N n ln ε n

22 PH605 : hmal and Statistical Physics 43 PH605 : hmal and Statistical Physics 44 n N 1 ε k n N + 1 ε k his mans that instad of xpssing th tmpatu as a function of th numb of Schottky dfcts w can now wok out how many dfcts w can xpct to find at a givn tmpatu. h dfct ngy is usually of th od of ε~1v. mpatu (K) n/n 0 (absolut zo) (oom tmpatu) Spin half systms and paamagntism in solids Ω N!!! ( N ) ( N ) ut w can wit th spin-up and spin-down stats in tms of th diffnc btwn thm i.., n N N N N + N hfo, th thmodynamic wight, Ω, is, which looks somthing lik this. N! Ω ( n) N + n N n!! 2 2 Ω(n) Suppos w hav a systm of N molculs in th fom of a cystallin solid. If on of th atoms of th molcul has an unpaid lcton thn w can consid th ffct of an xtnal magntic fild applid to th solid. An lcton has an angula momntum s1/2. In an applid magntic fild th lcton spin will align with th magntic fild ith paalll o anti-paalll to th fild. Intgal 2 N All spins paalll (nn) W thfo hav a systm of N spins, ach of which can xist in ith a spin-up o spin-down stat. h ngis associatd with th spin-up and spin-down stats spctivly a, so th total ngy of th systm is, U µ N U µ N U µ N + µ N ( N N ) µ h will b N spins in th spin-up stat and N in th spin-down stat. So fo this two stat systm th thmodynamic wight will b, All spins anti-paalll (n -N) h stat quation fo th intnal ngy of a systm givs us, du ds PdV but whn w a daling with a solid th volum chang, dv, is ngligibl and th numb of paticls, N, is constant. du ds du ds 1 ds du n

23 PH605 : hmal and Statistical Physics 45 PH605 : hmal and Statistical Physics 46 Using th oltzmann-planck ntopy quation, W know that, thfo, Raanging this w gt, which can b wittn, wh, Using th hypbolic idntity, ( ln Ω( n) ) 1 ds d k du du d( ln Ω( n) ) dn k dn du k N n dn ln 2 N + ndu dn du 1 µ 1 k 1 N n ln 2 N + nµ µ 1 N n ln k 2 N+ n N n N + n µ x k 2 x x x n tanh( x) x x + N S k ln Ω w can wit, w finally hav an xpssion fo th total magntisation of th solid. µ M nµ Nµ tanh k Systms in thmal quilibium and th canonical nsmbl : th oltzmann distibution Consid th thmal quilibium btwn a systm (1) and a lag thmal svoi (2) (hat bath). At a paticula point in tim th systm (1) will b in a ctain micostat,, with an associatd ngy, E. h total ngy of th systm is E 0 so th ngy of th thmal svoi (2) is, E E E 2 0 h pobability of finding th systm (1) in th micostat,, is latd to th thmodynamic wight. P () Ω ( E E ) ( E E ) 2 0 Ω 2 0 Using th aylo xpansion of ln( ( E E 2 0 )) Ω w gt, ln( Ω ( E 2 0 E )) ln 2( 0) ln ( 2( 0 ) Ω E Ω E E ) E E +... Howv w hav th dfinition that, So w can wit, 1 ln Ω k E ( ( E 2 0 )) ( ) ( ) Ω E E Ω E 0 0 Substituting this into th quation fo th pobability of finding systm (1) in a micostat,, givs, P () Ω Ω which givs th oltzmann distibution, ( E ) 0 ( E ) 0 E k E k E k h oltzmann distibution

24 PH605 : hmal and Statistical Physics 47 PH605 : hmal and Statistical Physics 48 P () Singl paticl patition function, Z, and Z N fo localisd paticls : lation to Hlmholtz function and oth thmodynamic paamts h singl paticl patition function, Z h patition function, Z, is a wightd count of th numb of possibl micostats, n, that a paticl can achiv, that taks account of th difficulty of aching thm. E k E k Z 3 all possibilitis ( Ea + Eb+ Ec) k Ea Eb Ec k k k h paticls all hav th sam st of ngy lvls so this bcoms, 3 E a k Z Z So fo N distinguishabl paticls w hav, ZN ( Z ) 1 N Z h patition function fo localisd paticls n Whn w a daling with a lag numb of paticls th patition function has to tak into account all th possibl combinations of th micostats fo ach paticl. Howv, th valu of th patition function is dpndnt upon whth o not w can distinguish th paticls w a daling with. 1. Distinguishabl paticls : paamagntic ions in a solid, distinguishabl by thi lattic positions. 2. Indistinguishabl paticls : gas paticls. h N-paticl patition function fo distinguishabl paticls Suppos w hav th localisd paticls (a,b,c) with a distinct st of possibl ngy lvls fo ach paticl (E a,e b,e c ). h patition function fo ths th paticls is givn by, En k h N-paticl patition function fo indistinguishabl paticls If th paticls a indistinguishabl thn th is no maning to assigning spaat ngis to thm sinc w cannot idntify thm. h choic of ngis E a,e b,e c is thn psntd by on tm in th sum ov micostats, so th patition function fo N indistinguishabl paticls is small by a facto N! than fo N distinguishabl paticls. Distinguishabl paticls, Z diffnt ngis 2 diffnt ngis 1 diffnt ngis ( Ea + Eb+ Ec) k ( E + E + E ) a b c k ( E + E + E ) a b c k Indistinguishabl paticls,

25 PH605 : hmal and Statistical Physics 49 PH605 : hmal and Statistical Physics 50 1! Z3 3! 2! + 3! 3! + 3! 3 diffnt ngis 2 diffnt ngis 1 diffnt ngis ( Ea + Eb+ Ec) k ( E + E + E ) a b c k ( E + E + E ) a b c k hfo fo lag N, wh w can assum that no two paticls sha th sam micostat, w hav, Z N Z ( indistinguishabl) N ( distinguishabl) N! So fo indistinguishabl paticls, th N-paticl patition function is givn by, Z N ( Z ) 1 N N! his is valid in th smi-classical appoximation. Hlmholtz function W alady hav and xpssion fo th Hlmholtz function, F U S and an xpssion fo th ntopy of a systm in tms of th pobability P, wh, so th ntopy can b wittn, S k ln P P E k P Z S k E E k k ln Z Z E k k E k ln Z + k Z k Z Simplifying this givs us, E S + k ln Z so, kln Z E S U S F kln Z h Hlmholtz f ngy, F, is a minimum fo a systm in thmodynamic quilibium at constant,v,n. Adiabatic cooling h most common fom of cooling is by figation using pocss basd on th Lind Liquifi. Hat is xchangd by compssing and xpanding a woking fluid. At tmpatus vy clos to absolut zo this opation bcoms impossibl sinc almost all systms will xist in a solid stat. Adiabatic dmagntisation is an impotant tchniqu fo aching xtmly low tmpatus. h disod and hnc ntopy of a paamagntic salt is tmpatu dpndnt and magntic fild dpndnt. S hmal is popotional to tmpatu S Magntic is invsly popotional to magntic fild. E S S + S otal Magntic hmal So if w duc th magntic fild adiabatically, and so S Magntic incass, th total ntopy must main constant and th thmal ntopy must dcas with a cosponding tmpatu dop. h sampl is nclosd in a liquid Hlium hat bath fom which it can b in contact o thmally isolatd and placd within th pol pics of an lctomagnt. h magntic fild can b apidly switchd btwn a high valu ( 1 ) and a low valu ( 0 ).

26 PH605 : hmal and Statistical Physics 51 PH605 : hmal and Statistical Physics 52 U F S U + k ln N Z h singl paticl patition function is asy to calculat sinc ach paticl can only b in on of two micostats, spin-up o spin-down. 1 1 µ µ k k Z + Simplifying this by making th substitution, µ x k x x Z1 + 2cosh x h intnal ngy of th systm can also b xpssd in simila tms. U µ N ( [ µ ] [ µ ]) p + p + N + x Z Z 1 1 x x µ N x x + µ N tanh x So th ntopy of th systm is givn by, x ( ) µ N S Nk ln 2cosh x xtanh x Fo a systm in thmal quilibium th Hlmholtz function is a minimum, F U S W can xpss th ntopy of th systm in tms of th patition function, So w s th ntopy of th systm is ssntially a function of x, so fo th adiabatic chang, hfo if w duc th magntic fild adiabatically (no chang in ntopy bcaus no hat flow) th tmpatu of th systm must duc. his

27 PH605 : hmal and Statistical Physics 53 nabls us to duc th tmpatu of th paamagntic solid blow that of th suounding Hlium bath. hmodynamic paamts in tms of Z h Hlmholtz function tlls us, F U S and fom this w can div th stat quation, df PdV Sd Md F P V F S, V, F M V,