Thermodynamics of solids 6. Multicomponent homogeneous nonreacting systems: solutions

Transcription

1 Thermdynamis f slids 6. Multimpnent hmgeneus nnreating systems: slutins Kwanghen Par Kyung Hee University Department f Nulear Engineering

2 6.. Partial mlar prperties Multimpnent System Speies:,, 3, 4,.. n, n, n 3, n 4,.. Single phase Surrunding Open bundary Speies:,, 3, 4,.. n, n, n 3, n 4,.. Mass in Reatr V V( T, P, n, n, n3,.. n ) Mass ut V V V V V dv dt dp dn dn... dn T P n n n Pn, Tn, P,T, n, n,.. n P,T, n, n,.. n P,T, n, n,.. n 3 3 V V V dt dp dn T P n Thermal Expansin Ceffiient Pn, Tn, P,T, n ( ) Cmpressibility Ceffiient V V dv dt dp vdn T P Pn, Tn, where, v V n P,T, n ( ) :Partial mlar vlume 6.. Partial mlar prperties

3 Any analgus definitin may be devised fr any extensive prperties (= U, H, G, F, S) d dt dp bdn T P Pn, Tn, where, b n PT,, n ( ) :Partial mlar prperty Cnsider when the system is under the nditin f nstant T and nstant P, dv v dn TP, Fr the press under nsideratin, visualize the additin f all mpnents simultaneusly in the prprtins fund in the final mixture. (T, P, X : fixed) n n where, X n n ntt V v dn v dn v n n TP, 0 0 Hene, ttal value f the vlume fr the system is the weighted sum f the partial mlar vlumes. An analgus apprah is als pssible fr any extensive prperties (= U, H, G, F, S) d b dn T, P bn TP, Capital letters: extensive prperties. U, H, G, F, S Lwerase letters: intensive prperties. u, h, g, f, s (values per mle) 6.. Partial mlar prperties 3

4 Gibbs Duhem equatin d d b n b dn n db b dn ndb 0 : Gibbs Duhem equatin Mixing press: Let b : the value per mle f the prperty fr pure mpnent. Pure A n Pure C n 3 Pure n Pure D n 4 Pure E n 5 Hmgeneus mixture f A,,C,D,E n Tt = n +n + n 3 + n 4 + n 5 bn Sln bn After mixing at nstant T and nstant P, b n b n b b n b n mix Sln where, b b b 6.. Partial mlar prperties 4

5 d d b n b dn n db mix ndb 0 : Gibbs Duhem equatin Intrduing mle fratin, X whih is: X then, ntt n db bdx b bx Xdb 0 n n And, db b dx mix b b X mix Xdb Partial mlar prperties 5

6 6.. Evaluatin f partial mlar prperties Partial mlar prperties frm ttal prperties ( b, b ) When b mix is nwn as a funtin f X n, b an be btained. Fr mpnent system, db bdx b dx mix mix b b X b X And, X X r, dx dx db bdx b dx ( b b ) dx mix db mix b b dx db db bmix b ( X ) b X b ( X ) mix mix dx dx dbmix b bmix ( X) dx dbmix And, b bmix ( X) dx 6.. Evaluatin f partial mlar prperties 6

7 Example: hmix axx : find h, h dhmix h hmix ( X) dx mix ax ax a X X a X dh dx dx h ax X ( X ) a X a( X ) ax h ax X ( X ) a X a( X ) ax dx Che h h X h X ax X mix 6.. Evaluatin f partial mlar prperties 7

8 Partial mlar prperties frm nwn PMP f the ther: (nwn b b) Frm the Gibbs Duhem equatin, Xd b Xd b 0 X db db X X X db db dx b dx X0 X0 X dx X dx X X db Example: b ax db dx X X X a X ax b ( ) dx ax dx a X dx ax X0 X 0 dx Hmewr 6.. Given that the vlume hange n mixing f a slutin beys the relatin. Vmix.7 XX ml (a) Derive expressins fr the partial mlar vlumes f eah f the mpnents as funtins f mpsitin. (b) Demnstrate that yur result is rret by using it t mpute V mix demnstrating that the equatin abve is revered. () Use the partial mlar vlume, v mputed in (a) t demnstrate that the Gibbs Duhem equatin hlds by evaluating v. 6.. Evaluatin f partial mlar prperties 8

9 Hmewr 6.. In an imaginary system Pandemnium(Pn) Cndminium(Cn), the partial mlar heat f mixing f Pn may be fitted by the expressin, h,500 X X Pn Pn Cn J ml Calulate and plt the funtin that desribes the variatin f the heat f mixing with mpsitin fr this system. Hmewr 6.3. Titanium metal is apable f disslving up t 30 atmi perent xygen. Cnsider a slid slutin in the system Ti O ntaining an atm fratin, X O = 0.. The mlar vlume f this ally is 0.68 /ml. Calulate (a) The weight perent f O in the slutin. (b) The mlar nentratin (ml/) f O in the slutin () The mass nentratin (g/) f O in the slutin. 6.. Evaluatin f partial mlar prperties 9

10 6.3. Chemial ptential in multimpnent systems U U( S, V, n, n,..., n )... du TdS PdV dn dn dn TdS PdV dn U where : hemial ptential fr the mpnent, n STn,, ( ) Che: / du TdS PdV W TdS PdV dn W / dn H dh TdS VdP dn W dn / where and n SPn,, ( ) F df SdT PdV dn W dn / where and n TVn,, ( ) G dg SdT VdP dn W dn Chemial ptential, is defined by: / where and n T, P, n ( ) U H F G n n n n S, V, n ( ) S, P, n ( ) T, V, n ( ) T, P, n ( ) Chemial ptential in multimpnent systems

11 G n T, P, n ( ) g : nwn dg SdT VdP S G s n n T T, P, n ( ) Pn, T, P, n ( ) V G n n P T, P, n ( ) Tn, T, P, n ( ) v s v G G g n T T n T T Pn,,, ( ) Pn, Pn, T, P, n ( ) T P n Pn, G G g n P Pn P P Tn,,, ( ) Tn, Tn, T, P, n ( ) T P n Tn, h H H n n T, P, n ( ) S, P, n ( ) H h g Ts T n T T, P, n ( ) Pn, 6.3. Chemial ptential in multimpnent systems

12 U u h Pv T P n T T, P, n ( ) Pn, P Tn, F f u Ts P n P T, P, n ( ) Tn, dg d s dt v dp 6.3. Chemial ptential in multimpnent systems

13 6.4. Fugaities, ativities, and ativity effiients Ativity f mpnent, is defined in terms f the hemial ptential by the equatin, a is the ativity f in a slutin at a given T, P, and mpsitin. RT ln a : the hemial ptential (Gibbs free enrgy per mle) fr pure mpnent. Ativity effiient f mpnent, written, is defined by the equatin: a X And, u RTln X Fugaities, ativities, and ativity effiient

14 Prperties f ideal gas mixtures Pure A n Pure C n 3 Pure n Pure D n 4 Pure E n 5 Hmgeneus mixture f A,,C,D,E n Tt = n +n + n 3 + n 4 + n 5 Partial pressure f gas, : P XP Ttal pressure is the sum f the partial pressure f eah gas: P XPP Mixing f the ideal gases are dne under the nstant temperature (isthermal press, dt=0): d sdt vdp vdp V n v RT V RT P n P TPn,, ( ) P P RT P vdp dp RTln P P P P RTlnX g : Free energy differene f, befre and after mixing Fugaities, ativities, and ativity effiient

15 g s RlnX T T g v 0 P Tn, h Ts RTln X T( Rln X ) 0 u h Pv 000 f u Ts 0 T( Rln X ) RTln X g g X RT X ln X mix g s R X X mix mix ln T h h X 0 mix u u X 0 mix v v X 0 mix f fx RTln X X RT X ln X mix Fugaities, ativities, and ativity effiient

16 Mixtures f real gases: fugaity Fugaity, is defined by: RT ln P And, RT ln RT ln a a P P Partial mlar vlume f a real gas is different frm that f the ideal gas. The differene, is given, RT P RT P P v dp dp RTln RTln P P P P P P Pe P P dp ln a s Rln a RT T Pn, T Pn, ln a v RT P P Tn, Tn, ln a ln a h Ts RTln a T Rln a RT RT T T Pn, Pn, Fugaities, ativities, and ativity effiient

17 ln a ln a u h P v RT PRT T Pn, P Tn, ln a ln a ln a f u Ts RT PRT T R ln a RT ln a RT ln a PRT P T Pn, P Tn, T Pn, Tn, In a binary system, the ativity f mpnent is measured, then the ativity f mpnent an be btained by the Gibbs Duhem equatin. X d X dln a X X dx ln a dx X 0 X X 0 dx X dx Hmewr 6.4. Fr an ideal slutin it is nwn that, fr mpnent, g = RT lnx Use the Gibbs Duhem integratin t derive rrespnding relatin fr mpnent Fugaities, ativities, and ativity effiient

18 Use f the ativity effiient ( ) in desribing the real gases a X And, u RTln X g RTln X RTln g ln s Rln RT Rln X T Pn, T Pn, ln v RT P P Tn, Tn, ideal g exess ln ln h Ts RTln RTln X T Rln RT Rln X RT T Pn, T Pn, ln ln u h P v RT PRT T Pn, P Tn, exess g RTln ln ln ln f u Ts RT PRT T R ln RT R ln X ln RT ln PRT RT ln X P T Pn, P Tn, T Pn, Tn, Fugaities, ativities, and ativity effiient

19 In a binary system, the ativity effiient an be btained using the Gibbs Duhem equatin. Xd Xd 0 And, RT ln X RT ln dx d d( RTln X RTln ) RT RTdln X dx dx Xd X d XRT RTd ln X RT RTd ln X X RT ( dx dx ) X RTd ln X RTd ln X RTd ln X RTd ln XRTdln XRTdln 0 r, Xdln Xdln 0 ln X dln X dx X 0 X dx Fugaities, ativities, and ativity effiient

20 6.5. The behavir f dilute slutins Rault s law fr the slvent: Henry s law fr the slute: lim a X x lim a X where is the Henry's law nstant. x ehavir f dilute slutins 0

21 6.6. Slutin mdels Pure A n Pure C n 3 Pure n Pure D n 4 Pure E n 5 Hmgeneus mixture f A,,C,D,E n Tt = n +n + n 3 + n 4 + n 5 Slutin mdel Ideal slutin Regular slutin Real slutin s, h, g s R X ln X 0 mix ideal ln g RT X X Regular ideal exess g g g h mix exess g h mix Regular g hmix RT X ln X g g g Real ideal exess n exess g XiX i( Xi X ) i i v0 n Real g RTX ln X XiX i( Xi X ) i i v Slutin mdels

22 Regular slutin mdels The entrpy f mixing is the same as that fr an ideal slutin. smix R X ln X The enthalpy f the mixed slutin is nt zer (heat f mixing); but is sme funtin f mpsitin. h h ( X, X, X,... X ) g mix mix 3 4 exess Regular ideal exess g g g hmix RT X ln X Example: a binary slutin with the heat f mixing that is given by: h a X X g mix exess g a X X RT( X ln X X ln X ) mix 6.6. Slutin mdels

23 dhmix h hmix ( X) dx mix a X a X ax X a X dh dx dx dx h a X X ( X ) a X a ( X ) a X h a X 0 exess g RTln g RTln h a X r, e exess 0 g RTln h a X r, e exess 0 The ativity effiient fr the dilute slutin, r the Henry s law effiient fr mpnent, and : 0 ax RT 0 ax RT 0 x 0 x x 0 a0 ax RT lim lim lim e e RT 0 x 0 x x 0 a0 ax RT lim lim lim e e RT Slutin mdels 3

24 Real slutin mdels Subregular slutin mdel fr a binary slutin: h a X X ( X X ) mix 0 Nnregular slutin mdel fr a binary slutin: h a X X ( T) X ( T) X mix 0 Fr multimpnent slutins, the mst widely applied frm (the Redlih Kister equatin): n i i i i i v0 exess g X X ( X X ) n i v0 exess Fr a binary mixture, g X X ( X X ) When 0, regular slutin mdel When, subregular slutin mdel 6.6. Slutin mdels 4

25 Hmewr 6.5. The exess Gibbs free energy f mixing fr the system A is given: exess g a( bt) X X mix A where, a,500( J / ml), b0 K 4 (a) Draw the exess Gibbs free energy f mixing at 300, 500, and 700 K. (b) Draw the Gibbs free energy f mixing at 300, 500, and 700K. Hmewr 6.6. Given that Henry s law hlds fr the slute in a dilute real slutin, derive Rault s law fr the slvent. Hmewr 6.7. The system A frms a regular slutin with the heat f mixing given by: J h 3,500 X mix AX ml (a) Derive expressins fr the Henry s law nstant fr A as a slute in and as a slute in A (b) Plt bth Henry s law nstants as a funtin f temperature Slutin mdels 5

26 6.7. Atmisti mdels fr (slid) slutin behavir e A e e A A A e A e AA e AA A e A A e A e A e A e A e Types f bnding t the neighbr: A A,, A A e AA e A e e A A e A The energy f eah bnding type: e AA, e, e A The number f eah bnding type: P AA, P, P A Ttal number f bnds in the system: 0 Nz P P P P Ttal AA A where z : rdinatin number z=6 : simple ubi, z=8 : bdy entered ubi z= : fae entered ubi, and hp N N N And, 0 A Atmisti mdels fr slutin behavir

27 P AA P A zn A zn0 X A P P A zn zn0 X The internal energy f the slutin system: P zn X P P zn0x P AA 0 A A A U P e P e P e P e e e N z X e X e sln AA AA A A A A AA 0 A AA The hange in internal energy fr the mixing press: U U U U XU XU mix sln pure sln A A where, U N ze U N ze A 0 AA 0 U U U P e e e N z X e X e X A N0zeAA X N0ze Umix PA ea eaa e Hmix mix sln pure A A AA 0 A AA ( H U PV U mix mix mix mix V : is negligible in slid) Atmisti mdels fr slutin behavir

28 If A and are randmly mixed tgether, the prbabilities t have AA,, A bnding are given by, respetively: f X X, f X X, f X X X X X X AA A A A A A A P zn f zn X X A 0 A 0 A he: faa f fa ( X A X ) Hmix PA ea eaa e zn0 X AX ea eaa e zn0 ea eaa e X AX H X X In a regular slutin, the enthalpy f mixing is given by: then, * 0 zn0 e e e A AA mix 0 A If A and are nt randmly mixed (rdered, segregated), still the site balane must be met: P P zn X P f X f AA and f X f AA 0 A A AA A A zn0 h f e e e mix A A AA P A zn Atmisti mdels fr slutin behavir

29 Randm mixing f atms in slid N A N Slutin[ N A, N ) A A S S S S ln ln ln m A, A A, A And, ln A, A N0! N! N! A where, N0 NA N S ln ln N! N ln N N N ln N N N ln N N 0 m A, NA! N! N ln N N ln N N ln N 0 0 N ln N N ln N N ln N N ln N N ln N 0 0 A A A A A 0 0 ( N N ) ln N N ln N N ln N N ln N A A m 0 A A A A N N N N NAln N0 NAln NA Nln N0 Nln NN0 ln ln N N N N S N X ln X X ln X where, X A N N X N A 0 N0 A A Atmisti mdels fr slutin behavir

30 G H TS 0X X N0T X ln X X ln X * m mix mix A A A g 0 X X RT X ln X X ln X m A A A where 0 zn e e e AVO A AA Psitive mixing enthalpy Ideal slutin Negative mixing enthalpy Atmisti mdels fr slutin behavir

31 Hmewr 6.8. The A system exhibits a measured heat f mixing given by the relatinship: J hmix XAX 7,500XA 8,00 X ml The bnd energies (J/bnd) fr this system are estimated t be: e 6.50 e 5.30 e AA A Cmpute and plt the fratins f AA, A, and bnds in the system as a funtin f mpsitin. hint: h zn f e e e and f X f and f X f mix AVO A A AA AA A A A Atmisti mdels fr slutin behavir