hermodynamcs and Knetcs of Solds 33 III. Statstcal hermodynamcs 5. Statstcal reatment of hermodynamcs 5.1. Statstcs and Phenomenologcal hermodynamcs. Calculaton of the energetc state of each atomc or molecular consttuent by makng use of mechancs/quantum mechancs. Because of the large number of speces: Consderaton of probabltes,.e. statstcs. Determnaton of the partton functon (e.g. elocty). In ew of the ery large number only one dstrbuton among all possble dstrbutons s most probable (Fg. 5.1: comparson of the relate probablty for throwng a certan number of spots wth 1,, 3 and many dces). Number of mcrostates for the realaton of a macrostate (total number of spots)..., e (r-states). - he number of partcles of the energy state e s N. he total number of partcles s N: N N (5.3) he total energy of the system s E (fxed alue) N e E (5.4) Of nterest s the most probable dstrbuton functon,.e. the dstrbuton for whch W N! (5.1) and under consderaton of the exchange of dces wthout new confguratons W has a maxmum. N! N!N 1!KN! (5.5) W N! N!N 1!N!K (5.) We consder lnw (whch has a maxmum at the same alue as W): Determnaton of the dstrbuton functon for the followng model: - N partcles - Partcles may be dstngushed. - Partcles are ndependent of each other (no mutual nfluence). - Each partcle has one of the energetc states e, e 1, e, lnw ln N!- ln N! (5.6) Consderng Strlng s formula (ln n! n ln n - n for large numbers of n) we hae lnw N lnn - N - N lnn + N (5.7) Consderng eq. (6.3) ths results n lnw N lnn - N ln N (5.8) A maxmum wth regard to N s obtaned from the deraton wth regard to N under consderaton of eqs. (5.3) and (5.4): Fg. 5.1. Comparson of the relate probabltes for throwng a certan number of spots when usng, 3 and many dces. dln W -d N lnn -dn - lnn dn (5.9) 1.8.97
34 hermodynamcs and Knetcs of Solds dn, sn ce dn (5.1) e dn, sn ce de (5.11) Applcaton of Lagrange s multpler method,.e. multplcaton of eqs. (5.1) and (5.11) by l and m (constant, but not fxed alues) and addng them to eq. (5.9): dn + or lnn dn + l dn + m e dn (5.1) (5.18) U N - e e -e / k e-e / k he denomnator s named partton functon Z Z (5.19) e - e / k (5.) he counter of eq. (5.19) s k dz d. Accordngly we hae dn ( 1+ lnn + l + me ) (5.13) U Nk 1 dz Z d dln Z Nk d (5.1) Snce the dn may be arbtrarly chosen, the quanttes n parenthess hae to dsappear: By knowlegde of Z the nner energy may be determned and from that all other thermodynamc functons. 1 + lnn + l + me (5.14) or N e - ( 1+l ) -me e (5.15) Accordng to eq. (5.3) we hae N N e - ( 1+l) e -me (5.16) and by makng use of eq. (5.15) follows N Ne -me e-me (5.17) m has to hae the nerse dmenson of energy. Under consderaton of the aerage oscllaton energy of a partcle, e k, we obtan N N e-e / k e / k e- (5.18) (more strct deraton of ths equaton by consderng quantum mechancs and gong to classcal mechancs). Equaton (5.18) represents the Boltmann s dstrbuton (Boltmann s e-relaton). Inner Energy from Boltmann s dstrbuton: Because of E U n eq. (5.4) we hae accordng eq. Entropy he entropy prodes nformaton about the drecton of rreersble processes and s a crterum for the presence of equlbrum. From a statstc pont of ew there s a transton to the most probable macrostate. he probablty plays accordngly the same role as the entropy. herefore, a functonal relatonshp s assumed: S S( W ) (5.) Deraton of ths relatonshp: ndependent systems of the same type of partcles (1 and ) are combned sothermally to the total system (1, ) In ths case, the entropes of the nddual systems (S 1, S 1 + S ) are added up, whle statstc weghts are beng multpled (W 1, W 1 W ): S 1, S( W 1, ) S( W 1 W ) S( W 1 ) + S( W ) (5.3) hs equaton may be only fulflled f S k * lnw (5.4) k * wll be later dentfed as Boltmann s constant k. 5.. he Varous Statstcs. Boltmann s statstcs: - partcles whch buld up the system are ndependent of each other and dstngushable - any number of partcles may occupy the same state. 1.8.97
hermodynamcs and Knetcs of Solds 35 - locaton (3-dm. space) - elocty or momentum (3-dm. momentum space) Both are combned to the 6-dmensonal phase space. Quantum mechancal treatment of a partcle n a cube wth the length a of each edge. Possble energes of the partcle: Fg. 5.. Possbltes of realng the total number 7 and 8 spots wth dstngushable dces (B) and undstngushable dces (BE) as well as prohbton of the same number of spots. Quantum statstcs: It s mpossble to determne smultaneously exactly locaton and momentum of a partcle. he possblty to dstngush of partcles s therefore questonable. Fg. 5..: Possbltes to throw dces wth a total number of 7 and 8 spots wth dstngushable dces (B), not dstngushable dces (BE) and prohbton of the same number of spots (FD). Partcles for whch the sum of numbers of electrons, protons and neutrons s een (H, D +, D, N, 4 He, photons) hae an nteger spn. Partcles for whch the sum of the numbers of electrons, protons and neutrons s odd (e -, H +, 3 He, NH 4 +, NO) hae a half-numbered spn. A system whch conssts of many partcles s descrbed n the case of an nteger-numbered spn by a symmetrc and n the case of a half-numbered spn by an antsymmetrc egenfuncton. e h 8ma n x + n ( y + n ) (5.5) By consderng the relatonshp between energy and momentum e 1 m p x + p ( y + p ) (5.6) eq. (5.5) results n the followng possble components of the momentum: p x p y h a n x h a ny p h a n n x, n y and n are the nteger quantum numbers. (5.7) By usng a cartesan coordnate system wth the unt h/a of the axs p x, p y and p, the states of the partcle n the cube are represented by the lattce ponts wth nteger numbered coordnate alues (Fg. 5.3.). In the frst case (nteger-numbered spn) any number of partcles may be present n the same energy state, whle n the latter case (half-numbered spn) each energy state may be only occuped by 1 partcle (Paul s law). he non-dstngushablty results n the followng quantum statstcs: - Bose-Ensten statstcs n the case of an ntegernumbered spn. - Ferm-Drac-Statstcs n the case of a half-numbered spn (Paul s law). 5.3. Momentum- and Phase space Classcal descrpton of the state of a partcle: Fg. 5.3. States of a partcle n a cube wth the length of a of each edge n the momentum space. 1.8.97
36 hermodynamcs and Knetcs of Solds Volume of each cell: h 3 /8a 3. Snce eq. (5.6) holds for poste and negate momenta, all 8 octants of the momentum space hae to be taken nto consderaton. Accordngly, a state of a speces corresponds to a cell of the olume h 3 /a 3 n the full 3-dmensonal momentum space. By consderng the phase space,.e. addng the physcal space to the momentum space, a state of the partcle corresponds n the 6-dmensonal phase space toa cell of the olume h 3, snce the speces wll occupy the olume a 3. Eq. (5.5) may be rewrtten n the followng way: a Ë h n x + n y a 8m e Ë h 8me + n a Ë h 8m e 1 (5.8) here exst as many dfferent quantum states that belong to the energy e as nteger solutons n x, n y, n are possble. All those nteger numbers n x, n y, n, that result n a alue < 1 for the left hand sde of eq. (5.8 belong to quantum states wth energy alues < e. Calculaton of the number of quantum states wth translatonal energes < e: Feedng eq. (5.7) nto eq. (5.8) results n: p x 1 ( 8me ) + p y 1 8me ( ) + p 1 ( 8me ) 1 (5.9) of translaton of a helum atom (m 6.7 1-7 kg) n a olume of 1 l at 3 K s under consderng the translaton of energy e 3 k 6. 1-1 J ( ): N( e) 1.3 1 8 whch s a contnuum n a frst approach. If there s not only 1 helum atom n the olume of 1 L, but 3 1 atoms under atmospherc pressure, the number of states s N( e) 3.3 1 5 Een under such condtons, the number of quantum states and accordngly the number of cells n the phase space s stll much larger than the number of speces. 5.4. DstrbutonFfunctons Snce the dscrete energy leels are ery close to each other, we do not consder the occupaton of the nddual leels but the occupaton of the total number of energy alues between e and e + de. he number of energy leels between e and e + de : A. hese are occuped by N speces. For the determnaton of the dstrbuton functon t has to be calculated by how many mcrostates a macrostate may be bult up. hat macrostate whch may be generated by the largest number of mcrostates s the most probable one and characterstc for the system. hs equaton s the surface of a bowl wth a radus 1 8m e and the olume 4 3 p 1 8 ( 8m e) 3. Æ he number of cells wth energes < e s: N( e) 4 3 p 1 8 ( 8m e) 3 / h 3 a 3 8 3 p V h 3 e 3 m3 (6.3) where (V a 3 ). he number of states wth energes between e and e + de s dn( e) D( e)de dn( e) de de (5.31) D(e) s the densty of states. Dfferentaton of eq. (5.3) results n dn( e) D( e)de 4 p V h 3 m 3 e 1 de (5.3) he order of magntude of the number of quantum states Bose-Ensten-Statstcs. he speces may not be dstngushed and any number of speces may occupy one state. he energy leels are: I, II,..., A. Fg. 5.4. Shows the dstrbuton of speces oer 3 cells. Dots are used n that fgure to ndcate that the speces may not be dstngushed. In general: Dstrbuton of N speces oer A cells: Number of the possbltes of dstrbutons: Fg. 5.4. Illustraton or the deraton of Bose-Enstens- Statstcs. wo not dstngushable speces (dots) are dstrbuted oer three cells (I, II, III) of a group of energy leels. 1.8.97
hermodynamcs and Knetcs of Solds 37 A ( A +1)L( A + N - 1) 1 LN Expanson of ths expresson by (A - 1)! results n the followng number of mcrostates ( N + A - 1)! N!( A - 1)! he same holds for all energy nterals. Snce each dstrbuton wthn one group may be combned wth any dstrbuton n another group, the number of dfferent mcrostates s N + A dln W dn + ln N + A ( N + A )dn or N N - dn - ln N dn (5.38) ln A +1 dn (5.39) Ë N wth the boundary condtons ( N W P + A -1)! N!( A - 1)! Condtons that hae to be fulflled: ) he total number of speces s constant (5.33) dn dn (5.4) and de e dn (5.41) Applcaton of Lagrange s multpler method: N N (5.34) ) he total energy of the system s constant E N e (5.35) hat macrostate s the most stable one for whch W or ln W takes up a maxmum under the boundary condtons eqs. (5.34) and (5.35). Equaton (5.33) results n ln W ln( ( N + A )! ) - ln( N!) - ln( A!) (5.36) Consderng Strlng s formula (ln n! n ln n - n for large numbers n): ln W ( N + A )ln( N + A ) -( N + A ) -N lnn + N - A lna + A (5.37) ( N + A ) ln( N + A )-N lnn - A lna (5.37a) dn ln A Í + 1 + a + be (5.4) Î Ë N hs results n or ln A + 1 + a + be (5.43) Ë N N 1 A e -a -be - 1 Determnaton of a and b: (5.44) Makng use of eq. (5.37a), the expresson S k ln W may be wrtten as: S k N ln N + A + A ln N + A Î Í (5.45) N and accordng to eq. (5.44) S k N ( lnb - be ) - A ln 1-1 B ebe Ë (5.46) A Maxmum: 1.8.97
38 hermodynamcs and Knetcs of Solds 1 wth B e -a. If B ebe <<1 (confrmaton later!), the followng approxmaton holds under consderng of ln (1-x) -x: S kí Î N ( lnb- be ) + A Be -be (5.47) Accordngly, the followng expresson holds for the entropy (5.49) S k NÍ lnb- b lnb Î b + 1 (5.55) For the nner energe U of the system of N partcles holds and because of eq. (5.44) for Be -be >> 1: Í S k lnb N 1 3 - b Í Ne + N Í N 14 4 3 E (5.48) U Ne N lnb b Because of determne b: U Ë S (5.56), eqs. (5.55) and (5.56) allow to Wth E Ne (e : aerage energy of each partcle) eq. (5.48) results n S kn[ lnb- be + 1] (5.49) e may be expressed by the dstrbuton functon (5.44): A E Ne N e e Be -be (5.5) - 1 Under consderng of N N Be -be -1 (5.51) A e becomes n the case Be -be >> 1: e Í b Î ln A e e be A e be b A e be A e be A e be (5.5) Accordng to eq. (5.51) results n the lmtng case Be -be >> 1 U Ë b S Ë b Æ U Ë S or b - 1 k N ln B b (5.57) -knb lnb b (5.58) - 1 kb (5.59) Accordngly, eq. (5.51) may be wrtten n the followng way A N Be e / k (5.6) -1 Changng from summaton to ntegraton: he number of states A has to be expressed as a functon of e: A,.e. the number of energy leels between e and e + de, s dentcal wth d N(e) n eq. (5.3): N 4 p V e 1 de h 3 m3 Ú (5.61) Be e k - 1 o Be e k >>1: ln N ln A e be - lnb (5.53) and eq. (5.5) may be rewrtten: e lnn + ln B b ( ) lnb (5.54) b B s a functon of b. or N 4 p V h 3 m 3 1 B e 1 Ú e -e k de (5.6) N 4 p V 1 h 3 m3 B ( k )3 Ú u e -u du (5.63) 1.8.97
hermodynamcs and Knetcs of Solds 39 he ntegral has the alue 1 4 hs results n p mk B ( )3 h 3 V N p Æ. (5.64) Applcaton of the same procedure as for the Bose- Ensten and Ferm-Drac-Statstcs. he cells may be occuped wthout lmtatons; all speces may be dstngushed from each other. Number of possbltes to dstrbute N speces oer A states: (B e -a ) A N Ferm-Drac-Statstcs Agan the speces may not be dstngushed, but n addton Paul s law holds,.e. each quantum state may only be occuped by one speces. Number of mcrostates: A ( A - 1) ( A - )L A - N +1 1 L N ( ) Expanson by (A - N )! results n the number of mcro states A! N!( A - N )! For all energy nterals holds A W! N!( A - N )! (5.65) Number of possbltes to dstrbute N speces n groups of N, N 1, N,... speces each wth the same properteson A, A 1, A, energy leels: N! N!N 1!LN!L Number of possbltes to reale the dstrbuton W N! N!N 1!LN!L A N N A 1 N 1 LA L N! (5.68) N! A N Wth the same boundary condtons and the same procedure as before, ths results n N 1 A e -a -be (5.69) Analogously results as aboe under the same boundary condtons For N 1 A e -a -be + 1 (5.66) Also for the Boltmann-Statstcs holds b - 1 k B ( p m k )3 h 3 V N (5.7) (5.71) b - 1 k (5.67) Comparson of the Statstcs and n the lmtng case B e -be >> 1 for B e -a the same alue holds as n the case of Bose-Ensten s statstcs. able 5.1. B-alues for the H -molecule and conductng electrons n sodum at dfferent temperatures and pressures. Boltmann-Statstcs 1.8.97
4 hermodynamcs and Knetcs of Solds Dfference n the dstrbuton functons: "1" n the denomnator. If e -a e e k >> 1, the quantum statstcs result n the Boltmann statstcs. When s that the case? For e -a e e k >>1 the rght hand sde of the dstrbuton functons becomes ery small,.e. N / A (occupaton probablty) becomes ery small. he number of quantum states s ery much larger than the number of speces (holds, e.g., for a gas under normal condtons). Snce e, e e k takes up alues between 1 and. ForB e e k >>1, B has to be suffcently large: large mass, hgh temperature, hgh dssoluton. B-alues for H and e - : able 1. 1. row: he same densty as at 73 K and p 1.13 bar s assumed at all temperatures.. + 3. row: p constant For H, the condton B>>1 s fulflled except for extremely low temperatures and hgh pressures. Electrons: small mass, large concentraton (n the case of metals) Æ Ferm-Drac-Statstcs. 5.5. Partton Functon and hermodynamc Potental Makng use of the dfferent statstcs, thermodynamc quanttes are dered. Accordng to Boltmann s statstcs, the rato of the number of speces N wth the energy e relate to the total number of speces N s N g e - e k g e -e k N (5.7) g : degree of degeneraton of the -th state (statstcal weght). : g e -e k (5.74) ("molecular partton functon") results from eq. 5.73, as may be easly shown by substtuton, e k / k ln (5.75) he aerage energy may be determned from the dfferentaton of the partton functon wth regard to the temperature. When we do not consder the occupaton probablty of an energy state e by a sngle speces and not the aerage energy of ths sngle speces, but a large number (n moles) of speces, then the energy ( nner energy) s analogously for the entre system U k lnz Ë Z: "system partton functon" (5.76) Eq. 5.76 allows to relate the statstcal treatment to phenomenologcal thermodynamcs by knowledge of the partton functon: ) Heat capacty C U Ë k lnz Ë [ ] lnz / 1/ - k ( ) Ë k lnz / ( 1/ ) Ë ( 1/ ) ) Entropy [ ] ln Z -k Ë ( 1 / ) [ ] - k lnz / 1/ ( ) Ë - k ln Z Ë ( 1/ ) (5.77) hs results n the aerage energy e of one speces: e N e N Wth the abbreaton e g e -e k g e - e k (5.73) ds C d (5.78) Integraton: C S - S Ú d (5.79) Makng use of eq. 5.77, ths results n 1.8.97
hermodynamcs and Knetcs of Solds 41 S - S Ú 1 Ë 1 Ú k lnz Í Ë k lnz d + k lnz Ë d k ln Z lnz Ú Ë d + Ú k d (5.8) Ë p V k lnz Ë lnv ) Enthalpy H U + pv k Ë H kí Î Ë ln Z lnz + ln Z Ë lnv + k lnz Ë lnv (5.89) (5.9) Partal ntegraton Æ S - S k ln Z Ë - k ln Z ln Z Ú d + k Ë Ú d Ë ) Gbbs Energy G F + p V -k lnz + k lnz Ë lnv k ln Z + k lnz Ë (5.81) G -k ln Z - ln Z Í Î Ë lnv (5.91) Makng use of eq. 5.76, ths results n S - S U + k lnz - ( k lnz ) (5.8) Æ S ( kln Z) (temperature ndependent) (5.83) Accordngly, we hae oder S U + k lnz (5.84) lnz S kí Î Ë ln + lnz (5.85) Relatonshp between the entropy S and the statstcal weght W because the entropy adds up whle the probablty s multpled when seeral systems are combned, the assumpton s made S ~ lnw or S k * lnw (5.9) It has been lnw N lnn - N ln N (5.93) Makng use of the partton functon (wthout degeneraton) ) Free Energy N N e- e k (5.94) F U - S (5.86) ths results n In ew of eq. (5.84) ths results n F U - U Ë + kln Z -k lnz (5.87) ) p V S k * N lnn - N e- e k ln N e- e k Í Ë k * Nln N - N e -e k Í ln N + N e-e k ln + N (5.95) e-e k e k p - F Ë V k lnz Ë V (5.88) k * Í Nln N - N ln N + N ln + N k e e -e k 1.8.97
4 hermodynamcs and Knetcs of Solds Makng use of eqs. (5.73) and (5.74) ths results n S k * N ln + Ne k k* N ln + U k (5.96) Comparson wth eq. (5.84) results n k * k (5.97) and Z N (5.98) Relatonshp between the molecular partton functon and the system partton functon. Accordng to eq. 5.98, the system partton functon (wthout degeneraton) may be wrtten as Z e - Ne k e - E k (5.99) E : energy egen alue of the -th quantum state of the macro system of N speces. Determnaton of Z from : ) Boltmann: he system conssts of N not nteractng dstngushable speces (ther exchange prodes a new state). Example: Crystal of N speces, whch may be dstngushed from each other because of the localaton at specfc lattce stes. Exchange of such speces prodes a new state E N e ; Z N (5.1) All speces of the system are equal to each other and hae the same energy egen alues. ) Bose-Ensten: he system conssts of N not nteractng and not dstngushable speces (ther exchange prodes no new state) Example: Ideal Gas of free molecules. he exchange of speces prodes not a new state. Under the assumpton that the number of states s much larger than the number of speces, t may be assumed that each quantum state s only occuped by 1 speces. 1.8.97