Chap.5 Statistical Thermodynamics

Transcription

1 Chap5 Statstcal Thermodynamcs () Free expanson & adabatc from macroscopc: Δ S dq T R adabatc Q, free expanson W rrev rrev ΔU, deal gas ΔT If reversble & sothermal QR + WR ΔU 因 Uf(T) RT QR WR ( Pd ) Pd d ΔS dq T R T from mcroscopc: statstcal thermodynamc: d( RT ln ) R ln R ln > RT ln *defnton: Boltzmann & Gbbs thermodynamc probablty, Ω: n! Ω The no of dfferent ways that a (n!)(n!)(n!) unque macro-confguraton can be acheved!, where n: total no of partcles n : partcle no at state Edted by Prof Yung-Jung Hsu

2 S KlnΩ From mcroscopc (Boltzmann hypothess), where K: Boltzmann constant R / o Ω Δ S S - S Kln Ω!! Ω ntal, Ω fnal!*! ( /)!*( /)! Ωfanl ()! ΔS Kln Kln K[ln(!) - ln(!)] Ω ntal (!) sterlng s formula: lnx! XlnX - X (by 泰勒展開式 ) ΔS K[ln - - ( ln - )] K ln Kln Rln > ( /) () Entropy of mxng From macroscopc vewpont: S M -R X ln X From mcroscopc vewpont: At ntal state: pure substance A & pure substance B! For A: ΩA SA K ln Ω A!! For B: ΩB SB K lnω B! for ntal state S S + S A B Edted by Prof Yung-Jung Hsu

3 At fnal state: deal soluton A-B! Ω A -B S K lnω (!)(!) the entropy change of mxng f A-B Δ S ΔS K ln - ΔS! (!)( f Kln ΔS ΔSM K[ln!-!-ln!] K[( + ) ln - ln - ln] R[X lnx - X lnx ] A A B B!)! (!)(!) (3) The thrd law for a perfectly ordered crystal of the compound AB at K: *hgh-ordered arrangement of atoms no randomness at all ( 無亂度 ) the locaton of atoms on the lattce ponts s fxed only a sngle mcroscopc state exsts to represent the mcrostate Ω( 在特定巨觀狀態下, 所能存在之微觀狀態之數目 ) By Boltzmann hypothess: S K ln Ω (TK & perfect crystallnty) the concept of 3 rd Law for crystals wth mperfectons at K: Crystal defect: vacances, ntersttals, dslocatons Compostonal defect: non-stochometry( 非當量 ), sotopes *random arrangement of atoms 有亂度 (S>) some randomness exsts on the placement of atoms a few mcrostates may exst Ω>, S> 3 Edted by Prof Yung-Jung Hsu

4 (4) Thermodynamc probablty and probablty Defnton: P The probablty that the system wll be n mcrostate 在特定巨觀狀態下, 微觀狀態 所出現之機率 Ω The total number of mcrostate exsted n a specfc macroconfguraton 在特定巨觀狀態下, 所能存在之微觀狀態數目 Correlaton: Assume two black & two whte balls are arranged n a lne, how many confguratons wll exst?! 4! Ω 6( 有 6 種排法 )!!!! b w Assumpton: for a unque confguraton, each arrangement s equally possble to exst In the current example, the probablty of each arrangement s /6 As a result, we realze P Ω & P 3 Gbbs Hypothess: For a closed system wth constant T and energy (canoncal ensemble), the energy could be expressed as: S -K P ln P, where P s the probablty that the system wll be n mcrostate 4 at equlbrum state: Macroscopc states may possbly consst of many mcrostates, then whch one wll be observed when the system s at equlbrum? From macroscopc vewpont: energetc consderaton From mcroscopc vewpont: possblty consderaton The macroscopc state we observed at equlbrum s the one wth the most possble mcrostates! 4 Edted by Prof Yung-Jung Hsu

5 5 Boltzmann dstrbuton (entropy calculaton for energy level) a knetc molecular theory by Maxwell: ( 分子動力學 ) - mv f(v) α v *exp( ) The probablty of a molecule havng energy E s proportonal -E to e b defnton: Consder partcles are put nto m boxes, havng dfferent energy levels, and there are no lmt on the number of partcles exstng at a specfc energy level Combned wth Maxwell s hypothess The probablty for a partcular partcle to be found n the energy E s gven by: E exp( ) P, E exp( ) The probablty for a partcle beng n a partcular state at energy E s proportonal to -E e Boltzmann dstrbuton c Ideal gas molecular dstrbuton: Consder a column wth specfc volume contans gas molecules: By Boltzmann dstrbuton: P P P E + E exp P P ote that mgh exp mgh exp exp( E ), P E exp( ) 5 Edted by Prof Yung-Jung Hsu

6 If the gas s deal, the pressure at the top & bottom wll be proportonal to & o, respectvely Pressure, mgh Mgh exp exp Barometrc equaton Pressure, RT d Degeneracy 簡倂態 Consder an energy E level; there may be a number of dstngushable states g, each havng the same energy We need to modfy Boltzmann dstrbuton The probablty that a partcle wll be n a specfc energy level possessng g dstngushable states should be modfed accordngly: P E g * exp( ) E, g * exp( ) e Thermodynamc propertes: (no degeneracy) Consder a system wth energy levels n equlbrum: From mcroscopc vewpont: to maxmze the entropy S S -K P ln P P, : number of partcles found at th energy level P, : total number of partcles E E, E: total energy of system E < E > P E, <E>: average energy for a partcle n system 6 Edted by Prof Yung-Jung Hsu

7 By maxmzng the entropy term assocated wth other relatons, we can obtan the expresson of S for a system contanng one partcle: < E > S Kln +, <E>: average energy nternal energy T E exp( < ) E E > P E E E exp( ) U U ln S Kln + ln F (A) U-TS, 將上式帶入 F -ln f Dstngush ablty of partcle 可識別性 Consder four partcles are put n two boxes, there are fve macro confguratons For mcrostate (): If the four partcles are total dfferent (dstngushable), there wll be four way of arrangements Eg atom n sold state If the four partcles are dentcal (ndstngushable), there wll be one way to reach mcrostate () Eg atom n gas or lqud state On the bass of ths consderaton, the Dstngush ablty s needed to be checked when a system contans many partcles ow, consder a system contans two partcles; the grand partton functon s gven by Φ 7 Edted by Prof Yung-Jung Hsu

8 Take partcles nto account, we can obtan Φ, f these partcles are dentcal and ndstngushable, the grand partton functon should be modfed to ensure we do not over-count the number of mcrostates Φ (partcles are dentcal, for gases)! For one partcle exstng n system: ln S Kln + F -ln ln U For partcles exstng n system: Sold Gas ln ln S Kln + S K[ln( ) + ] + F -ln Φ ln U F -(ln( ) + ) Φ! ln U 6 Other approach * Before 9, Boltzmann dstrbuton: P E g *exp( ) E g *exp( ) o lmtaton on the number of partcles exstng at E Wthout takng care of the over-count ssue (usually assumng dstngushable partcles) * After the development of quantum mechancs at 9 8 Edted by Prof Yung-Jung Hsu

9 Bose-Ensten dstrbuton: g E - μ exp( ) - especally vald for 輻射能量之分佈 o lmtaton on the number of partcles exstng at E Consder the partcles are ndstngushable * Ferm-Drac dstrbuton: g E - μ exp( ) + especally vald for 金屬與半導體中電子之分佈 Subject to Paul Excluson Prncple Consder the partcles are ndstngushable * At hgh temp & low chemcal potental (low densty of partcles), both B-E & F-D become M-B model E - μ E - μ E (exp( ) >>, exp( ) exp( )) 9 Edted by Prof Yung-Jung Hsu