and Statistical Mechanics Material Properties

Transcription

1 Statstcal Mechancs and Materal Propertes By Kuno TAKAHASHI Tokyo Insttute of Technology, Tokyo , JAPA Phone/Fax Only for the students n TAKAHASHI s class. All rght reserved. Do not dstrbute to others. 013

2 Statstcal Mechancs and Thermodynamcs Thermodynamcs ( 0 th low : A.eq.B and B.eq.C -> A.eq.C ) 1 st low : Energy conservaton nd low : Free energy decreases 3 rd low : Entropy -> 0 at 0K Interface essental to utlze thermodynamcs Statstcal Mechancs Grave-stone of Boltzmann Ergodc theorem : tme average = space average, (n equlbrum)

3 You may have learnt Thermodynamcs Thermodynamcs ( 0 th low : A.eq.B and B.eq.C -> A.eq.C ) Everythng s 1 st low : Energy conservaton derved from these nd low : Free energy decreases prncples 3 rd low : Entropy -> 0 at 0K But! Defntons Temperature Free energy Entropy Please remember when you have heard them frst. How they were defned? Equlbrum Could you understand them as you could do n knetcs?

4 Defntons Temperature C : water Emprcal temp. F : salt water Temp. n scence ( physcs or knetcs ) s same as above? Free energy Helmholtz Gbbs F U TS G U PV TS Free energy s same as the energy n knetcs? Why they need to be defned as above? Entropy What s ths? Equlbrum What s the defnton?

5 You may have learnt Thermodynamcs Thermodynamcs ( 0 th low : A.eq.B and B.eq.C -> A.eq.C ) Everythng s 1 st low : Energy conservaton derved from these nd low : Free energy decreases prncples 3 rd low : Entropy -> 0 at 0K But! Defntons Temperature Free energy Entropy Equlbrum Understandng of them ( = statstcal mechancs ) lead you to utlze thermodynamcs Materal propertes Let s defne them step by step!

6 Temperature ( frst tral )

7 Example : Ideal gas (1) Assumpton umber of molecule : n volume : Velocty of molecule : c ( u, v, w) probablty : F c u f v f w Momentum change = Pressure on area A 0 mu f ( u) V f Audt du PAdt V c u u v w v w PV 0 mu f ( u) udu m u f ( u) du m u 1 u c 3 Comparson wth eq. of state : PV kt 1 mc 3 kt PV m 1 3 c 3 1 mc Law of equ-partton of energy 1 mu 1 mv 1 mw 1 kt

8 Example : Ideal gas () Probablty functon F c f u f v f w ln Fc ln f u ln f v ln f w takng log : c u v w dff. wth u : Const. ndependent of c,u,v,w : because t s probablty Average knetc energy (expectated value) : Maxwell dstrbuton : m m u kt f ( u) e kt d ln F du d ln f du usng d ln F dc 1 d ln F c dc eq. of state dc du d ln F dc 1 d ln f u du u c f ( u) d ln f du 1 d ln f v dv e 1 u 1 1 u mu 1 mu m f ( u) du 3 m mu F( u, v, w) e m kt 3 e du 1 m kt m u v w kt u therefor e 1 d ln f w dw 1 u du m kt

9 Example : Ideal gas (3) Maxwell dstrbuton F( u, v, w) m kt 3 e m u v w kt Probablty functon e knetcenergy kt f ( u) m kt e m u kt Only for Ideal gas? < Remember > Emprcal knowledge s used. Eq. of state for deal gas Relaton between Temp. and average Knetc energy Ths s the defnton of Temp.!? Let s consder more general case!

10 You may have learnt Thermodynamcs Thermodynamcs ( 0 th low : A.eq.B and B.eq.C -> A.eq.C ) Everythng s 1 st low : Energy conservaton derved from these nd low : Free energy decreases prncples. 3 rd low : Entropy -> 0 at 0K Defntons Temperature Free energy Entropy equlbrum etc Temperature s knetc energy? ( at equlbrum )

11 Temperature and equlbrum (general treatment)

12 Canoncal ensemble (1) Assumpton 1 ( a model of system ) um. of elements wth energy s. Total num. of elements (partcle, molecule, etc ) : Total energy : E n e e n n and E are gven., therefore n ( e ) s determned by the state. (densty of state) States wth same n ( e ) are ndstngushable. Assumpton ( most mportant!! ) The state s observed so as to maxmze the way of shufflng of the densty of state n ( e ) Why!?

13 Canoncal ensemble () the way of shufflng of the state To maxmze W usng the defnton of W usng the Strlng s formula,!!!! 3 1 n n n W max ln 1 E n e n W L E n e n n L 1!! ln x x x x! ln ln max ln ln 1 E n e n n n n L Lagrange multpler 0 L n L

14 Canoncal ensemble (3) L Here, and E are constant, therefore L n Usng n ln n 11 e 0 ln n 1 Comparng wth Maxwell dstrbuton n 1 exp e exp 1 e kt exp n ln n n n n e E max e exp e exp e kt 1 e e n exp 1 1 exp f ( u) m kt m u : Maxwell-Boltzmann dstrbuton e kt e s not lmted to be the knetc energy. knetc energy, potental energy, etc

15 Canoncal ensemble (3) L Here, and E are constant, therefore L n Usng n ln ln Comparng wth Maxwell dstrbuton n n ln n n n n e E max 1 n 11 1 e 0 n exp 1 e 1 exp e Temperature s energy! n ( used n physcs ) 1 n the exp most e probable exp e stuaton. exp e exp 1 e kt exp e kt f ( u) m kt m u : Maxwell-Boltzmann dstrbuton e kt e s not lmted to be the knetc energy. knetc energy, potental energy, etc

16 Canoncal ensemble (4) Probablty of the state of energy E : exp E kt Probablty of the state whose energy > E : exp E kt Assumpton The state s observed so as to maxmze the way of shufflng of the densty of state n ( e ) The most probable stuaton Equlbrum! Equlbrum s,,, the most probable state!!

17 You may have learnt Thermodynamcs Thermodynamcs ( 0 th low : A.eq.B and B.eq.C -> A.eq.C ) Everythng s 1 st low : Energy conservaton derved from these nd low : Free energy decreases prncples. 3 rd low : Entropy -> 0 at 0K Defntons Temperature Free energy Entropy equlbrum etc Temperature s energy! ( at equlbrum ) Equlbrum s the most probable state.

18 Defntons Temperature Emprcal temp. C : water defned one or two temp. F : salt water T. can be shfted and scaled, so as proportonal to energy! Free energy Helmholtz Gbbs F U TS G U PV TS Free energy s same as the energy n physcs? Why they need to be defned as above? Entropy What s ths? Equlbrum The most probable state! So, we can observe always

19 Free energy and entropy

20 Predcton of phenomena Physcsts ( knetcs ) as to decrease Potental Energy Chemsts ( chemstry ) as to decrease Free Energy m mgh Ideal gases A B h mxed A,B A,B After the phenomena Decreased Potental Energy ot changed ot changed Total energy ot changed? ( ot changed ) Free Energy Decreased Chemsts wanted to ntroduce somethng lke potental energy, to predct the phenomena.

21 Chemsts wanted to ntroduce somethng lke potental energy, to predct the phenomena. The somethng; related to the work or energy, used n thermodynamc technology, ( ex. steam engne, chemcal reacton, etc ) Defnton of free energy : Energy that can be converted nto a work n a process of constant temperature and volume (Helmholtz). constant temperature and pressure (Gbbs). ( Two of three T, P, &V determne ts state. ) F U TS G U PV TS Energy or work n knetcs What s ths!?? G U PV TS

22 Defnton of free energy : Energy that can be converted nto a work n a process G U PV TS What s ths!?? A B A,B A,B The TS term ncreases n the mxng process, even f knetc and potental energy never changes. The mxng process never goes back. Increase n TS term Decrease of G = Increase of the usable energy = Degrade of energy

23 Defnton of free energy : Energy that can be converted nto a work n a process G U PV TS What s ths!?? There exsts a part of energy, whch we can not take as a work. It s proportonal to the temperature. The degrade s proportonal to the temp Degrade of the energy exts S s defned as the entropy. Emprcal knowledge of chemsts What s the entropy!?? Ths s the defnton of entropy n thermodynamcs.

24 Boltzmann's entropy formula & Free energy Free energy of deal gas A work s taken out! What happened? P, V, T -> 1/P, V, T G U PV TS The degrade of energy exsts n lower process o work s taken out! The degrade s TS.

25 Boltzmann's entropy formula & Free energy Free energy of deal gas PV kt A work s taken out! V V PdV V V kt V dv kt ln P, V, T -> 1/P, V, T G U PV TS The degrade of energy exsts n lower process o work s taken out! Change n S s? S k ln

26 Statstcal Mechancs and Thermodynamcs Thermodynamcs ( 0 th low : A.eq.B and B.eq.C -> A.eq.C ) 1 st low : Energy conservaton nd low : Free energy decreases 3 rd low : Entropy -> 0 at 0K Boltzmann's entropy formula : S k lnw Statstcal Mechancs Grave-stone of Boltzmann Ergodc theorem : tme average = space average, (n equlbrum)

27 Boltzmann's entropy formula & Free energy W : umber of state S k lnw v : small control volume V/v : num. of the c. volume V/v : num. of the c. volume um. of the way to shuffle to dstrbute the atoms to the each c. volume W 0 ( V / v) W ( V / v) W0 P, V, T -> 1/P, V, T S0 k lnw 0 S S k ln W S 0 0 S 0 k ln o work s taken out! S k ln

28 Boltzmann's entropy formula & Free energy Free energy of deal gas PV kt A work s taken out! V V PdV V V kt V dv kt ln P, V, T -> 1/P, V, T S0 k lnw 0 S S k ln W S 0 0 S 0 k ln o work s taken out! S k ln

29 Boltzmann's entropy formula & Free energy Free energy of deal gas P, V, T -> 1/P, V, T Internal E. U 0 -> U 0 Free E. G 0 =U 0 +PV-TS 0 -> G 0 -TS Only entropy term must be changed as Entropy S 0 -> S 0 +S Entropy change Energy taken out : V V must be kt PdV dv kt ln S k ln V V V o change. Defnton of entropy : S k lnw If the entropy s defned as ths, t becomes consstent. S S 0 0 k lnw 0 S k ln W S k ln 0 0

30 Boltzmann's entropy formula & Free energy Free energy of deal gas PV kt A work s taken out! We can take out the nfnte work!???... although nternal energy must be fnte. P, V, T -> 1/P, V, T W depends on volume!???... Energy never depend on volume V V Check of understandng PdV V V kt V dv kt ln o work s taken out! S S 0 0 k lnw 0 S k ln W S k ln 0 0

31 Boltzmann's entropy formula & Free energy Free energy of deal gas A work s taken out! The energy s gven from outsde! so as to keep T const. P, V, T -> 1/P, V, T o work s taken out! Entropy s not energy, It s a ndex of randomness.

32 Introducton of Free energy & entropy & temperature Defnton of Free Energy : Energy that can be converted nto a work. ( There exst the energy part we can not take out. ) Internal energy = Knetc energy we can take out from outsde ( by heat transfer ) G U PV TS Defnton of Entropy Defnton of Temperature Related to the randomness S k lnw Degrade of the energy

33 Temperature Emprcal temp. C : water defned one or two temp. F : salt water T. can be shfted and scaled, so as proportonal to energy! Free energy Helmholtz Gbbs F U TS G U PV TS Free energy s energy that can be converted nto work. If defned as above, t become consstent. Entropy S k lnw G U PV TS correspondng to number of states,.e. randomness. Equlbrum Most probable state! So, we can observe always These are the nterface between Thermodynamcs and statstcal mechancs

34 Statstcal Mechancs and Thermodynamcs Thermodynamcs ( 0 th low : A.eq.B and B.eq.C -> A.eq.C ) 1 st low : Energy conservaton + nd low : Free energy decreases 3 rd low : Entropy -> 0 at 0K Boltzmann's entropy formula : S k lnw Statstcal Mechancs Grave-stone of Boltzmann Ergodc theorem : tme average = space average, (n equlbrum)

35 Based of statstcal mechancs, Thermodynamcs can be utlzed!

36 Actvaton energy and Arrhenus plots chemcal reacton and rate-determnng step dffuson coeffcent others (evaporaton, adsorpton, etc ) Arrhenus plots s to know the mechansm of rate-determnng step.

37 Specfc heat capacty (1: deal gasses ) Remember thermodynamcs! (How have you learnt?) C V 1 U T V C H 1 T why? p why? p Defnton Heat requred to ncrease unt temp. - under constant volume fxed Q G U C V T - under constant pressure Q G U P V T S U P V C P T Mayer s relaton C p C V k C T V PV C P T G U PV TS

38 Specfc heat capacty (: metals) Ensten s model Dulong-Pett s Low What s the U? Debye model - Ionc cores - Electrons = neglgble 1 1 U ek ep 3 kt 3 kt 3kT Q 3 k T Quantum statstcal mechancs

39 Phase dagram Stablty of phases Phase rule Phase dagram Calculaton of Phase Dagram (CALPHAD)