Thermodynamics of solids 4. Statistical thermodynamics and the 3 rd law. Kwangheon Park Kyung Hee University Department of Nuclear Engineering

Transcription

1 Themodynamcs of solds 4. Statstcal themodynamcs and the 3 d law Kwangheon Pak Kyung Hee Unvesty Depatment of Nuclea Engneeng

2 4.1. Intoducton to statstcal themodynamcs Classcal themodynamcs Statstcal themodynamcs System U Q W In classcal themodynamcs, the atoms o molecules ae not consdeed ndvdually. du Q W In statstcal themodynamcs, the states (eneges) of ndvdual atoms o molecules ae consdeed, and the themodynamc popetes ae obtaned based on the nfomaton of the ndvdual atoms. ds T Q ev U SYS 1 S kln n Intoducton to statstcal TDs.

3 4.2. Mcostate, macostate, and entopy A smple example: Atoms at the poston a, b, c, d n a system. Possble enegy state of each atom: 1, 2 a b c d Numbe of possble state= 2 4 = 16 :numbe of mcostates Mcostate, macostate, entopy

4 Macostate Enegy of macostate I II III IV V ! 4! 1 4! 3!1! 4 4! 2!2! 6 4! 1!3! 4 Numbe of mcostates 4! 1 4! Enegy pe atom Homewok 4.1. Daw the gaph of the numbe of mcostates vs. enegy pe atom when the numbe of atoms s 10. Assume 1 =0, 2 = Mcostate, macostate, entopy

5 Numbe of mcostates n a macostate fo a system contanng lage numbe of atoms Macostate, J Enegy levels Numbe of atoms n 1 n 2 n 3 n 4 n N n n... n n Numbe of mcostates = N! N! n1! n2! n3! n4!... n! n! 1 J (macostate J) al numbe of mcostates n the system: N The pobablty of macostate J = J N Mcostate, macostate, entopy

6 The pobablty dstbuton fo macostates has an extemely shap peak fo systems wth lage numbes of patcles and enegy levels. The entopy of an solated system s a maxmum at equlbum, suggestng a connecton between entopy and the numbe of mcostates coespondng to a gven macostate. S kln whee k s the Boltzmann constant (=R/N avog ) Mcostate, macostate, entopy

7 4.3. Fundamentals fo statstcal themodynamc calculaton Lagange Multples A fomal pocedue fo detemnng a maxmum pont n a contnuous functon subject to one o moe constants. (,, ): contnuous functon. f x y z f f f df dx dy dz 0 at maxmum (o mnmum) xy z When x, y, z ae ndependent, changes dx, dy, dz ae also ndependent. Hence at maxmum, But x, y, z ae nteelated to each othe, f f f 0 xy z gxyz (,, ) 0 (*) Then eq.(*) s no longe vald! How can we get the condton fo maxmum (mnmum)? >Lagange multple method. g g g dx dy dz 0 x y z f g f g f g dx dy dz 0 x x y y z z Then, the condtons fo maxmum (o mnmum) f become, f g f g f g x x y y z z Lagange multple, should be chosen Fundamentals fo calculaton

8 Stlng s fomula ln N! Nln N N Impotant ntegals n ax x e 0 2 dx n=0, 1 n=1, 1 n=2, 1 n=3, 3 2 a 2a 4 a 1 2a 2 Homewok 4.2. Usng the method of Lagange multple method, fnd the maxmum (o mnmum) of the followng functons. (1) f(x,y)=x 2 y wth the constant of x+y=12. (2) f(x,y,z)=2x 2 +4y 2 +9z 2 wth the constant of x+y+z=31 (3) f(x,y,z)=xyz wth the constant of x2y3z 3 6 Homewok 4.3. Check Stlng s fomula by compang the elatve dffeence between the eal esult and the appoxmaton of the followng calculaton; 5!, 20!, 100! Fundamentals fo calculaton

9 4.4. Condtons fo equlbum n statstcal themodynamcs Enegy levels Evaluaton of entopy Numbe of atoms n 1 n 2 n 3 n 4 n N! S kln k ln N! ln n! k ln N! ln n! 1 1 n! 1 N n n... n n kn ln N N n ln n n kn ln N n ln n 1 1 usng N n N n k n ln N n ln n k n ln k n ln n 1 N 1 S n n kn ln 1 N N Condtons fo equlbum

10 ds=? ds kn d n n k d n n k d n n n N ln ln ln ln 1 N N 1 N kln n dn n dn ln N dn n dn 1 n N n k ln n ln N dn dn dn N n Snce dn 1 dn 1 1 N n ds kln dn 1 N Condtons fo equlbum

11 The constaned maxmum n the entopy functon Adabatc, close wall du 1 dn 0 dn 1 dn 0 Fnd the condton when the entopy of the system s the maxmum at gven constants. Maxmum: Constants: n ds k ln dn 0 1 N du 1 dn 0 dn 1 dn 0 Usng the Lagange multples method, ds dn du 0 n n k ln dn dn dn k ln dn 0 1 N N n k ln N 0 whee =1, 2, 3,..., Condtons fo equlbum

12 n k ln N 0 whee =1, 2, 3,..., n k k e e whee =1, 2, 3,..., N Evaluaton of the Lagange multples, and n N k k k k e e e e e k Z k e whee Z : patton functon Z k n e k n Then, e ds k ln dn k ln dn N Z 1 N 1 Z ds k ln Z dn dn k ln Z dn dn du 1 k Snce ds du T 1 T 1 e k Condtons fo equlbum

13 At equlbum (maxmum entopy condton), kt kt n e e Z kt N e 1 n N n N e kt Condtons fo equlbum

14 4.5. Calculaton of the macoscopc popetes fom the patton functon Patton functon, Z: Z 1 e kt kt n e S k n ln k n ln k n ln Z 1 N 1 Z 1 kt 1 1 n k ln Z n U kn ln Z T 1 1 T And, F U TS F N kt ln Z F And, S T V ln Z S NkT ln Z Nk ln Z Nk T T V V Calculaton fom patton functon

15 ln Z And, U F TS NkT ln Z T Nk ln Z NkT T 2 ln Z U NkT T V V 2 U ln Z 2 ln Z V 2 T V T V T V And, C 2N kt N kt F P V G F PV H U PV C P H T P T Calculaton fom patton functon

16 4.6. Applcaton of the algothm A model wth 2 enegy levels 4.6. Applcaton of the algothm 16

17 Ensten model of a cystal Assume that a cystal s composed of a system of atoms whch vbate as hamonc oscllatos all wth the same fequency,. And, each oscllato has thee degees of feedom wth egad to ts decton of vbaton. Thus a system of N 0 oscllatos n a thee dmensonal cystal coesponds to 3N 0 lnea oscllatos. 1 h 2 whee s an ntege. U 3 n Because 1 atom coesponds to 3 oscllatos n 3 d space. 1 h 2 hh2h3h kt kt kt kt kt kt 2 Z e e e 1 e e e... Snce 1 xx x... f x 1 1 x ln Z C N kt N kt h CV 3Nk kt Snce 1 atom acts as 3 oscllato. Z e h 2kT 1 e 2 2 V 2 2 T V T V h 2 kt E e 2 T h 2 E e O, CV 3 Nk whee 2 E kt E e 1 T T ln Z e 1 h kt h k 4.6. Applcaton of the algothm 17

18 Homewok 4.4. Based on the Ensten model, deve ntenal enegy fom the patton functon. Then obtan the specfc heat capacty by dffeentaton of the ntenal enegy. Also fnd the entopy. Consde 1 mole of sold. At T E Show that C V becomes 3R (Dulong Pett law) Applcaton of the algothm 18

19 Debye model of a cystal Atoms n a cystal ae not hamonc oscllatos wth the same fequency,. They ae oscllatos coupled wth othe atoms. The oscllatos can have many natual fequences; howeve, the total numbe of fequences should be 3 N. The total numbe of possble fequences of the oscllatos can have n a cystal, can be expessed as: 4 V 3 whee c s sound wave tavelng speed 3 3 c 4 V 3 3 N = the numbe of total oscllatos 3 3 c Max m 3N 3 9N 2 o, = 3 3 the numbe pe unt ange of fequency m m m xm h 0 x m kt D e 9N h T x dx U d N kt 1 e 1 h h m D h m whee x x m D : kt kt T k Debye tempeatue 4.6. Applcaton of the algothm 19

20 At hgh tempeatue egon (T> D ), T xm x dx T xm x 1 0 D e D U N kt N kt x dx N kt U CV 3Nk 3 R (heat capacty pe mole) T V At low tempeatue egon (T~0K), T x m x dx T 3 T U 9NkT 9N 0 x kt NkT D e 1 D 15 5 D C V 4 U 12 T R T V 5 D 3 (heat capacty pe mole) 4.6. Applcaton of the algothm 20

21 Monatomc gas model 1 2 mv 2 l z l x l y Z 4.6. Applcaton of the algothm 21

22 And, 4.6. Applcaton of the algothm 22

23 PV N kt nrt Applcaton of the algothm 23

24 Random mxng of atoms n sold ( N n) A nb Soluton[( N n) A, nb) 0 0 S S S S k ln ln ln m A, B A B A, B A B And, 1 A AB, B N0! ( N n)!n! N0! Sm kln A, B kln ( N0 n )!n! k N ln N N ( N n)ln( N n) ( N n) nlnnn k N ln N ( N n) ln( N n) nln n k N ln N nlnn nlnn ( N n)ln( N n) nln n n ( N0 n) n n ( N0 n) ( N0 n) k nln ( N0 n) ln kn0 ln ln N0 N0 N 0 N 0 N 0 N 0 S kn X ln X X ln X N n n 0 m 0 A A B B whee, X A N X B 0 N Applcaton of the algothm 24

25 Homewok 4.5. A system contang 500 patcles and 15 enegy levels s n the followng macostate: (14, 18, 27, 38, 51, 78, 67, 54, 32, 27, 23, 20, 19, 17, 15) Ths system expeences a pocess n whch the numbe of patcles n each enegy level changes by the followng amounts: (0, 0, 1, 1, 2, 0, +1, +1, +2, +2, +1, 0, 1, 1, 1) Homewok 4.6. Use the Ensten model to compute the change n ntenal enegy of cystal when t s heated evesbly at one atmosphee pessue fom 90 to 210K. Assume θ E =250K. Homewok 4.7. The low tempeatue mola specfc heat of damond vaes wth tempeatue as, C V 3 3 T J mol K D whee the Debye tempeatue D =1860K. What s the entopy change of 1g of damond when t s heated at constant volume fom 4K to 300K? 4.6. Applcaton of the algothm 25

26 4.7. The 3 d law of themodynamcs Nenst s statement: The entopy change n a pocess, between a pa of equlbum states, assocated wth a change n the extenal paametes tends to zeo as the tempeatue appoaches absolute zeo. (Hee, extenal paametes ae P, T, V, ) Planck s statement: The entopy of all pefect cystals s the same at the absolute zeo, and may be taken as zeo. Wong Coect G H TS 4.7. The 3 d law of themodynamcs 26

27 A mcoscopc vew ponts 4 3 N kt kt n e e Z kt 1 e At 0K, n N 0 1 kt f 0 At 0K At 0K, all atoms ae n the gound state. O, all atoms ae n the dentcal state. Hence, the numbe of mcostates s 1 (=1). S kln The 3 d law of themodynamcs 27

28 Some consequences of the 3 d law Themal expanson coeffcent: 1 V V S V T T P PT P 1 S V P Themal expanson coeffcent s zeo at 0K. T Heat capacty: C V S S T T V lnt V As T 0K, S 0, ln T - C V s zeo at 0K. Homewok 4.8. Show that C P becomes 0 at 0 K. 28

29 4.8 Empcal pedctons of entopy 298 o o o CP S298 S0 S298 dt 0 T An empcal elaton between S o 298 the mola weght (M W ) of vaous substances: o 3 J S298 Rln M W b 2 mol K whee b s a constant. The value of the tem b depends upon the natue of chemcal bondng wthn the phase Empcal pedcton of entopy

30 Empcal pedcton of entopy

31 Touton s ule, Rchads ule Touton s ule: An empcal elaton between the enegy of vapozaton of a cystal and the bolng tempeatue. The unt of L v = J/mol Empcal pedcton of entopy

32 Empcal pedcton of entopy

33 Homewok 4.9. Show that S o 3 fo CdSb, CuAl 2, and AuSn ae 5.7, 5.3, and 5.7 J/g atom K. 33